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Sliding-Mode Control: Advanced Design Techniques

by

DUY-KY NGUYEN, M.E.

A Thesis submitted in Partial Fulfilment of the Requirements for the Degree of

DOCTOR OF PHILOSOPHY

1998

In Memory of my absolutely beloved Parents Nguyn Tß¶ng Din

Nguyn Th¸ Ngát who lived and even . . . died for their poor children

Nguyn Th¸ Liu Nguyn Th¸ Nhiu Nguyn Duy Hy Nguyn Duy KÖ

Nguyn Th¸ Kim Cß½ng Nguyn Th¸ Tß¶ng Vân

NDK

In Memory of all my Teachers, all my Sisters and Brothers,

who had Hard Time for my Good Time. NDK

Acknowledgments

From the bottom of my heart, I am greatly indebted to my Supervisors, Professor Hung T. Nguyen and Professor Barry S. Thornton, for their sincere discussions and warm support. I would like to thank the Staff of the Control Laboratory, Instrumentation & Control Post-Graduate Research Laboratory at the School of Electrical Engineering, University of Technology, Sydney. I am also grateful to authors mentioned in this thesis whose works have opened my mind for research and development of this work. Duy-Ky Nguyen

Table of Contents

Acknowledgments Table of Contents List of Symbols Abstract

Advanced Sliding-Mode Controller Design: Introduction

1. Motivation....................................................................................................................................................1

2. Methodology

3. Results

3.1. Chapter 1: Sliding Mode Control Background......................................................................................2

3.2. Chapter 2: Sliding Hyperplane Design

3.3. Chapter 3: A New Variable-Structure System Controller Design .........................................................3

3.4. Chapter 4: A New Robust Sliding Mode Controller-Observer Design

3.5. Chapter 5: A New Robust Discrete-Time Sliding-Mode Controller-Observer Design...........................4

3.6. Chapter 6: General Sliding-Mode Controller Design for MIMO Uncertain Nonlinear Systems.............5

3.7. Chapter 7: Advanced Sliding-Mode Controller Design: Experimental Results

Chapter 1 Sliding Mode Control Background

1.1. Introduction ..........................................................................................................................................1.1

1.2. SMC Background

1.2.1. Hyperplane Equation

1.2.2. Sliding Condition............................................................................................................................1.2

Remark 1.1: Negativeness of Sliding Condition ...............................................................................1.3

Remark 1.2: Stability of Sliding Mode

1.2.3. Sliding Dynamics

1.2.4. Equivalent Control

Sliding-Mode Control: Advanced Design Techniques 2

1.3. Numerical Examples ..........................................................................................................................1.4

1.3.1. Example 1.1: Variation of System Structure

Remark 1.3: Example 1.1...........................................................................................................1.5

Remark 1.4: Dimension of Sliding Hyperplane

1.3.2. Example 1.2: Uncertain System..........................................................................................................1.6

Remark 1.5: Example 1.2...........................................................................................................1.7

1.3.3. Example 1.3: Nonlinear System..........................................................................................................1.7

Remark 1.6: Example 1.3...........................................................................................................1.8

1.4. Conclusions .........................................................................................................................................1.9

Chapter 2 Sliding Hyperplane Design

2.1. Introduction..........................................................................................................................................2.1

2.2. Sliding Dynamics ................................................................................................................................2.2

2.2.1. Sliding Dynamics Equation

Theorem 2.1: Sliding Dynamics Equation

2.2.2. Sliding Eigenvalue ......................................................................................................................2.3

Theorem 2.2: Sliding Eigenvalues

Remark 2.1: Relation between Sliding Eigenvalues and Hyperplane Eigenvalues............................2.4

Remark 2.2: Hyperplane is of Reduced-Order

2.3. Eigenvalue Allocation........................................................................................................................2.5

2.3.1. Full-State Technique

2.3.1.1. Partition Transformation Method

Corollary 2.1: Hyperplane Design by Partition Transformation Method

2.3.1.2. Direct Allocation Method...............................................................................................2.6

Corollary 2.2: Hyperplane Design by Direct Allocation Method

Remark 2.3: Application of Direct Allocation Method.................................................................2.7

Remark 2.4: Extensibility of Hyperplane Design by Direct Allocation Method to Nonlinear Systems

2.3.1.3. Direct Calculation Method

Corollary 2.3: Hyperplane Design by Direct Calculation Method

Remark 2.5: Hyperplane Normalization in Direct Calculation Method............................................2.8

Remark 2.6: Dimension of Sliding Variable and Extensibility to MIMO Systems

2.3.2. Canonical Transformation

Theorem 2.3: Computation of Canonical Transformation Matrix

2.3.3. I/O-State Technique....................................................................................................................2.9

Theorem 2.4: Hyperplane Design by I/O-State Technique

Remark 2.7: Extensibility of I/O-State Technique to Nonlinear Systems ......................................2.10

Table of Contents 3

Remark 2.8: Decomposition of I/O-State Technique

2.3.4. Proposition 2.1: Application of I/O-State Technique ...........................................................................2.11

2.4. Optimal Sliding Mode

2.4.1. Error-Optimal Sliding Mode (Optimal Hyperplane 1)

Lemma 2.1: Performance Index of Error-Optimal Sliding Mode

Theorem 2.5: Optimal State of Error-Optimal Sliding Mode ......................................................2.12

2.4.2. Solution to the Optimal Hyperplane 1. ......................................................................................2.13

Corollary 2.4: Hyperplane Design by Error-Optimal Sliding Mode Method

2.4.3. Error/Energy-Optimal Sliding Mode (Optimal Hyperplane 2).....................................................2.14

Theorem 2.6: Hyperplane Design by Error/Energy-Optimal Sliding Mode Method

2.4.4. Solution to the Optimal Hyperplane 2 .......................................................................................2.16

Theorem 2.7: Gradient of Performance Index in Error/Energy-Optimal Sliding Mode

2.4.5. Numerical Algorithm for Solution to Optimal Hyperplane 2 .......................................................2.17

2.4.6. Proposition 2.2: Selection of Weighting Matrices Q and R ...................................................................2.18

2.5. Normalization of a Hyperplane Equation....................................................................................2.20

2.6. Numerical Examples.

Remark 2.9: Summary of Hyperplane Designs

2.6.1. Example 2.1: Direct Allocation.......................................................................................................2.21

2.6.2. Example 2.2: Partition Transformation

2.6.3. Example 2.3: Direct Calculation

2.6.4. Example 2.4: I/O-State Technique for Linear Systems ........................................................................2.22

2.6.5. Example 2.5: I/O-State Technique for Nonlinear Systems

2.6.6. Example 2.6: Error-Optimal Sliding Mode.........................................................................................2.23

2.6.7. Example 2.7: Error/Energy-Optimal Sliding Mode

2.7. Conclusions .......................................................................................................................................2.24

Chapter 3 A New Variable-Structure System Controller Design

3.1. Introduction ..........................................................................................................................................3.1

3.2. SMC Stability Criterion

Theorem 3.1: SMC Stability Criterion ...................................................................................3.2

Remark 3.1: Stability Test

3.3. A New VSS Controller Design

Lemma 3.1: Compact Control Function .................................................................................3.3

Remark 3.2: Simplification of Control Function By Hyperplane Normalization

Theorem 3.2: Switching VSS Controller Design

Corollary 3.1: Conventional Reaching Controller Design ...........................................................3.4


Corollary 3.2: New Reaching Controller Design ......................................................................3.5

Remark 3.3: Sliding Mode as Fast Mode for Invariance

3.4. Proposition 3.1: SMC Design Rule

3.5. Solution to Chattering Problem ......................................................................................................3.6

3.5.1. Switching Function......................................................................................................................3.7

3.5.2. Saturate Function

3.5.3. Unitvector Function

3.5.4. Hyperbolic Tangent Function (Tanh)

Remark 3.4: Pseudo-SMC with Boundary Layer......................................................................3.8

3.5.5. Performance of Sliding Functions (Saturate, Unitvector and Tanh Functions)

Proposition 3.2: Application of Sliding Function

Proposition 3.3: Solution to Problem of Chattering and Steady-State Error....................................3.9

3.6. Integral VSS Controller Design

Theorem 3.3: Integral VSS Controller Design by Integral Sliding Condition

Remark 3.5: Applicability of SMC Design Rule to Integral VSS .................................................3.10

Theorem 3.4: Integral VSS Controller Design by Augmented System

3.7. Numerical Examples ........................................................................................................................3.11

Remark 3.6: Summary of VSS Designs...............................................................................3.12

3.7.1. Example 3.1: Canonical System ....................................................................................................3.13

3.7.1.1. Original Design

3.7.1.2. New Designs

(a) Switching VSS ........................................................................................................3.14

(b) Saturate VSS

(c) Unitvector VSS

(d) Tanh VSS ...............................................................................................................3.15

(e) Switching VSS by I/O-state of Relative Degree r ....................................................3.16

(f) Error-Optimal Sliding Mode

3.7.1.3. Performances .............................................................................................................3.17

3.7.2. Example 3.2: Non-canonical System


3.7.2.2. New Designs ..............................................................................................................3.18

(a) Switching VSS

(b) Saturate VSS

(c) Unitvector VSS........................................................................................................3.19

(d) Tanh VSS

(e) Switching VSS by I/O-state of Relative Degree r

(f) Error-Optimal Sliding Mode......................................................................................3.20

3.7.2.3. Performances compared with the Normal Case .........................................................3.21

Table of Contents 5

3.7.3. Example 3.3: Equal Excursion Sliding Mode


3.7.3.2. New Designs ..............................................................................................................3.22

(a) Switching VSS

(b) Saturate VSS

(c) Unitvector VSS........................................................................................................3.23

(d) Tanh VSS

(e) Switching VSS by I/O-state of Relative Degree r ....................................................3.24


3.7.3.3. Performances .............................................................................................................3.25

Remark 3.7: Comparison between Proposed Design and Equal Excursion Sliding-Mode Design

3.7.4. Example 3.4: Non-linear Hyperplane


3.7.4.2. New Designs ..............................................................................................................3.26

(a) Switching VSS

(b) Saturate VSS ..........................................................................................................3.27

(c) Unitvector VSS

(d) Tanh VSS ...............................................................................................................3.28


(f) Error-Optimal Sliding Mode......................................................................................3.29

3.7.4.3. Performances compared with the Normal case

Remark 3.8: Comparison between Proposed Design and Nonlinear Hyperplane Design

3.7.5. Example 3.5: Efficiency of New Design

3.7.5.1. Conventional Design...................................................................................................3.30

3.7.5.2. New Design ................................................................................................................3.31

Remark 3.9: Performances of New Design...........................................................................3.32

3.7.6. Example 3.6: Error/Energy-Optimal VSS

3.7.7. Example 3.7: Applications of I/O-State Method (Section 2.3.4) ..............................................................3.33

3.7.7.1. Relative Degree less than System Order

3.7.7.2. Relative Degree equal to System Order .....................................................................3.34

3.7.8. Example 3.8: A Simple Test of SMC Design Rule (Section 3.4)

3.7.8.1. Violation 1 in SMC Design Rule: Sliding Time much smaller than Reaching Time

3.7.8.2. Violation 2 in SMC Design Rule: Reaching Time much smaller than Sliding Time...................3.35

3.7.9. Example 3.9: Steady-State Error when Reducing Sliding Gain ..............................................................3.36

3.7.9.1. Saturate VSS

3.7.9.2. Unitvector VSS ...........................................................................................................3.37

3.7.9.3. Tanh VSS

3.7.9.4. Performances:

3.7.10. Example 3.10: Integral VSS Control using Integral of Sliding Variable...................................................3.38



3.7.10.2. New Design

3.7.10.3. Performances ...........................................................................................................3.39

3.7.11. Example 3.11: Integral VSS Control using Integral of Output Error


3.7.11.2. New Design ..............................................................................................................3.41

3.7.11.3. Performances ...........................................................................................................3.42

3.8. Conclusions

Chapter 4 A New Robust Sliding Mode Controller-Observer Design

4.1. Introduction ..........................................................................................................................................4.1

4.2. Robustness in SMC............................................................................................................................4.2

4.2.1. Invariance Condition to Uncertainties

Theorem 4.1: Invariance Condition to Uncertainty

Remark 4.1: Form of Matching Condition .......................................................................................4.3

Corollary 4.1: Alternative Form of Matching Condition .......................................................................4.4

Corollary 4.2: Canonical System with Matching Condition

4.2.2. Invariance Condition to Disturbances. ...........................................................................................4.5

Theorem 4.2: Sufficient Invariance Condition to Disturbance

Remark 4.2: Invariance Property of Sliding Eigenvalues under Matching Condition

4.3. A New Robust VSS Controller Design (Discontinuous SMC)

Assumption 4.1: System Constraint on Parametric Variation..............................................................4.6

Theorem 4.3: Robust VSS Controller Design under Uncertainty and Disturbance

Remark 4.3: Robust VSS Controller Design under No Perturbations.....................................................4.7

Remark 4.4: Equivalent Control in Robust VSS Controller

Corollary 4.3: Robust VSS Controller under Reduced Parametric Variation ............................................4.8

4.4. A New Robust Linear SMC Design

4.4.1. Linear SMC....................................................................................................................................4.9

Theorem 4.4: Linear SMC Design

Theorem 4.5: Closed-Loop Eigenvalues composed of Hyperplane Eigenvalues and Sliding Margin

Remark 4.5: Reduced Order of Sliding Mode ................................................................................4.10

4.4.2. Robust Linear SMC

4.4.2.1. Robust Linear SMC under External Disturbances ............................................................4.11

Theorem 4.6: Robust Linear SMC Design under External Disturbance

Remark 4.6: Disturbance Condition in Sliding Mode........................................................................4.12

Remark 4.7: Validity of Theorem 4.6 under No Disturbance

Table of Contents 7

4.4.2.2. Robust Linear SMC under Parametric Uncertainties

Lemma 4.1: Computation of Special Eigenvalues

Theorem 4.7: Robust Linear SMC Design under Parametric Uncertainty .............................................4.13

Remark 4.8: Robust Linear SMC Design under No Uncertainty .........................................................4.14

Corollary 4.4: Robust Linear SMC Design under Reduced Parametric Variation ....................................4.15

Remark 4.9: Equivalent Control in Robust SMC Design

4.4.2.3. Robust Linear SMC under Matched Parametric Uncertainties

Theorem 4.8: Robust Linear SMC Design under Matched Uncertainty

Proposition 4.1: Robust Linear SMC Design Rule .........................................................................4.16

Remark 4.10: Advantage of Robust Design Rule

4.4.2.4. Robust Linear SMC under Un-matched Uncertainty and Disturbances

Corollary 4.5: Robust Linear SMC Design under Un-matched Uncertainty and Disturbance

4.4.2.5. Alternative Formula for Robust Linear SMC.....................................................................4.17

Lemma 4.2: Alternative Computation of Special Eigenvalues

Theorem 4.9: Alternative Robust Linear SMC Design .....................................................................4.18

Corollary 4.6: Alternative Robust Linear SMC Design under Reduced Parametric Variation .....................4.19

4.5. Robust Integral SMC Design..........................................................................................................4.20

Theorem 4.10: Robust Integral Linear SMC Design under Uncertainty

4.6. A New Robust Linear Sliding-Mode Observer Design ............................................................4.21

Theorem 4.11: Robust Linear Sliding-Mode Observer Design under Uncertainty

4.7. Analysis of Matched and Unmatched Uncertainties ...............................................................4.22

4.7.1. Transfer Function Model

Proposition 4.2: Uncertainty in Transfer Function Model .................................................................4.23

4.7.2. State-Space Model ......................................................................................................................4.24

4.7.2.1. Analysis of Uncertain State-Space Model

4.7.2.2. Conversion of Uncertain State-Space Model ...................................................................4.25

Proposition 4.3: Conversion of Unmatched to Matched Uncertain System ...........................................4.26

4.8. Numerical Examples

Remark 4.11: Summary of Robust SMC Designs

4.8.1. Example 4.1: No Perturbation ..........................................................................................................4.27


4.8.1.2. New Design

4.8.2. Example 4.2: Matched Uncertainties..................................................................................................4.29

Remark 4.12: Efficiency of Robust Linear SMC Design Rule (Proposition 4.1) ......................................4.31

4.8.3. Example 4.3: Un-matched Uncertainty

4.8.4. Example 4.4: Un-matched Uncertainty and External Disturbance..............................................................4.33

4.8.5. Example 4.5: Linear SMC for Nonlinear Systems..................................................................................4.35

4.8.6. Example 4.6: Robust Integral SMC under Uncertainty and Disturbance......................................................4.36

4.8.6.1. Original Design.................................................................................................................4.37


4.8.6.2. New Robust Integral SMC Design....................................................................................4.38

4.8.7. Example 4.7: Robust Sliding-Mode Observer under Un-Matched Uncertainties ..........................................4.39

Remark 4.13: Performance of Pseudo-SMC

4.8.8. Example 4.8: Robust Sliding-Mode Observer under Converted Matched Uncertainties ................................4.41

Remark 4.14: Comparison between Unmatched and Converted Matched Uncertainty. ...........................4.42

4.9. Conclusions .......................................................................................................................................4.43

Chapter 5 A New Robust Discrete-Time Sliding Mode Controller-Observer Design

5.1. Introduction ..........................................................................................................................................5.1

5.2. Digital Control Systems ....................................................................................................................5.2

5.2.1. Some Terminologies

5.2.2. ADC as Physical Sampler..............................................................................................................5.3

5.2.3. Starred Transform as Ideal Sampler..............................................................................................5.4

5.2.4. Sampling Characteristics and Signal Reconstruction ....................................................................5.5

Theorem 5.1: Laplace Transform of Sampled Signal

5.2.4.1. Signal Reconstruction ........................................................................................................5.6

Proposition 5.1: Choice of Sampling Time

5.2.4.2. Aliasing or Folding..............................................................................................................5.7

5.2.5. Z-Transform...................................................................................................................................5.8

5.2.6. Some Typical S-Z Transformations

5.2.7. Discretization of Continuous-Time State-Space Equations .........................................................5.10

Theorem 5.2: Discretization of State-Space Model

5.3. Discrete-Time Sliding Hyperplane Design .................................................................................5.12

Proposition 5.2: Discrete-Time Sliding-Mode Controller Design

5.3.1. Hyperplane Equation ...................................................................................................................5.13

5.3.2. Equivalent Control

5.3.3. Sliding-Eigenvalues

Theorem 5.3: Discrete-Time Sliding Eigenvalues

5.4. A Robust Discrete-Time Discontinuous SMC (VSS) Design .................................................5.15

Theorem 5.4: Robust Discrete-Time Discontinuous SMC Design under Matched Uncertainty

Remark 5.1: Validity of Convergence Condition in Discrete-Time SMC................................................5.16

5.5. A New Robust Discrete-Time Linear Sliding Mode Controller Design ...............................5.17

5.5.1. Robust Discrete-Time Linear SMC under External Disturbance

Theorem 5.5: Robust Discrete-Time Linear SMC Design under External Disturbance

5.5.2. Robust Discrete-Time Linear SMC under Parametric Uncertainties ............................................5.18

5.5.2.1. Matched and Unmatched Parametric Uncertainties

Table of Contents 9

Assumption 5.1: System Constraint on Parametric Variation

Theorem 5.6: Robust Discrete-Time Linear SMC Design under Parametric Uncertainty ..........................5.19

Remark 5.2: Sampling Condition in Discrete-Time SMC ..................................................................5.22

5.5.2.2. No Uncertainty

Corollary 5.1: Robust Discrete-Time Linear SMC Design under No Perturbation

Theorem 5.7: Closed-Loop Eigenvalues composed of Hyperplane Eigenvalues and Sliding Margin ...........5.23

Remark 5.3: Sampling Condition equivalent to Stability Criterion with Unit Circle

5.5.2.3. Matched Uncertainty ........................................................................................................5.24

Theorem 5.8: Robust Discrete-Time Linear SMC Design under Matched Uncertainty

Proposition 5.3: Robust Discrete-Time Linear SMC Design Rule ......................................................5.25

5.5.3. Robust Discrete-Time Linear SMC under Uncertainties and Disturbances

Corollary 5.2: Robust Discrete-Time Linear SMC Design under Uncertainty and Disturbance

Proposition 5.4: Robust Discrete-Time VSS Controller Design under Uncertainty and Disturbance ...........5.26

5.6. A Robust Integral Discrete-Time Linear SMC Design .............................................................5.27

Theorem 5.9: Robust Discrete-Time Linear SMC Design under Uncertainty

5.7. A New Robust Discrete-Time Linear Sliding-Mode Observer Design.................................5.28

Theorem 5.10: Robust Discrete-Time Linear Sliding-Mode Observer Design under Uncertainty

5.8. Numerical Examples ........................................................................................................................5.29

5.8.1 Example 5.1: No Uncertainty ............................................................................................................5.30


5.8.1.2. New Design......................................................................................................................5.31

5.8.1.3. Performances...................................................................................................................5.32

5.8.2. Example 5.2: Matched Uncertainty


5.8.2.2. New Design......................................................................................................................5.33

5.8.2.3. Performances...................................................................................................................5.34

5.8.3. Example 5.3: Integral Discrete-Time SMC


5.8.3.2. New Design......................................................................................................................5.35

5.8.3.3. Performances

5.8.4. Example 5.4: Deterministic System

Remark 5.5: Comparison of Continuous and Discrete Hyperplane......................................................5.36

5.8.5. Example 5.5: Matched Uncertain System............................................................................................5.38

5.8.6. Example 5.6: Un-Matched Uncertain System Under Disturbance..............................................................5.40

5.8.7. Example 5.7: Robust Sliding-Mode Observer under Un-Matched Uncertainty and Disturbance ........................5.42

5.8.8. Example 5.8: Robust Sliding-Mode Observer under Converted Matched Uncertainty and Disturbance ..............5.44

5.9. Conclusions .......................................................................................................................................5.45


Chapter 6

General Sliding-Mode Controller Design for MIMO Uncertain Nonlinear Systems

6.1. Introduction ..........................................................................................................................................6.1

6.2. Hyperplane and Stability

6.2.1. Hyperplane Design .....................................................................................................................6.2

Theorem 6.1: Hyperplane Design for Canonical Nonlinear Systems

Remark 6.1: Hyperplane Design for Canonical Linear and Nonlinear Systems ................................6.3

6.2.2. SMC Stability Criterion for Nonlinear Systems

Corollary 6.1: Stability Criterion

6.3. Robust Discontinuous SMC Design for SISO Uncertain Nonlinear Systems

6.3.1. Discontinuous SMC Design for SISO Nonlinear Systems

Definition 6.1: Nonlinear System States Vector

Theorem 6.2: Discontinuous SMC Design for Nonlinear Systems................................................6.4

Remark 6.2: Reaching Control

6.3.2. Robust Discontinuous SMC Design for SISO Uncertain Nonlinear Systems


Theorem 6.3: Robust Discontinuous SMC Design for Uncertain Nonlinear Systems ........................6.5

Remark 6.3: Nonlinearity as Uncertainty

Remark 6.4: Alleviation of Chattering....................................................................................6.6


6.4. Robust Continuous SMC Design for SISO Uncertain Nonlinear Systems

6.4.1. Continuous SMC Design for SISO Nonlinear Systems

Theorem 6.4: Continuous SMC Design for Nonlinear Systems

6.4.2. Robust Continuous Pseudo-SMC Design for SISO Uncertain Nonlinear Systems .....................6.7

Corollary 6.2: Robust Continuous Pseudo-SMC Design for Uncertain Nonlinear Systems

6.5. General SMC Design for SISO Nonlinear Systems

Theorem 6.5: General Discontinuous SMC Design for Nonlinear Systems

Theorem 6.6: General Continuous SMC Design for Nonlinear Systems ........................................6.9

6.6. General SMC Design for MIMO Uncertain Nonlinear Systems .............................................6.10

Theorem 6.7: General Discontinuous SMC Design for MIMO Nonlinear Systems

Theorem 6.8: General Continuous SMC Design for MIMO Nonlinear Systems .............................6.11

Remark 6.6: Singular System Jacobian

Remark 6.7: A Robust SMC Design for General MIMO Uncertain Nonlinear Systems

6.7. Numerical Examples. .......................................................................................................................6.12

Remark 6.8: Summary of SMC for Uncertain MIMO Systems

6.7.1. Example 6.1: SISO Nonlinear System

6.7.1.1. Discontinuous SMC ....................................................................................................6.13

6.7.1.2. Continuous Pseudo-SMC

Table of Contents 11

6.7.1.3. Continuous SMC ........................................................................................................6.14

6.7.1.4. Performance

6.7.2. Example 6.2: SISO Uncertain Nonlinear System

6.7.2.1. Robust Discontinuous SMC........................................................................................6.15

6.7.2.2. Robust Continuous Pseudo-SMC...............................................................................6.16

6.7.3. Example 6.3: Tracking SISO Nonlinear System



6.7.3.3. Continuous SMC ........................................................................................................6.18


6.7.4. Example 6.4: MIMO Nonlinear System


6.7.4.2. Continuous Pseudo-SMC ...........................................................................................6.20

6.7.4.3. Continuous SMC ........................................................................................................6.21

6.7.4.4. Performance...............................................................................................................6.22

6.7.5. Example 6.5: MIMO Nonlinear Robot Manipulators


6.7.5.2. Continuous Pseudo-SMC ...........................................................................................6.24

6.7.5.3. Continuous SMC ........................................................................................................6.25

6.7.5.4. Performance...............................................................................................................6.26

6.7.6. Example 6.6: MIMO Uncertain Nonlinear Robot Arms

6.7.6.1. Robust Discontinuous SMC........................................................................................6.28

6.7.6.2. Robust Continuous Pseudo-SMC...............................................................................6.29

Remark 6.8: Steady-State Error due to Low Sliding Gain and Efficiency of Design Scheme (Proposition 3.3)6.31


6.8. Conclusions

Chapter 7

Advanced Sliding-Mode Controller Design: Experimental Results

7.1. Introduction ..........................................................................................................................................7.1

7.2. System Description ............................................................................................................................7.2

7.3. Instrumentation ...................................................................................................................................7.3

7.4. System Identification .........................................................................................................................7.4

7.4.1. Theoretical System Identification

7.4.1.1. Ball-Hoop Transfer Function ..............................................................................................7.6

7.4.1.2. DC Servo Motor Transfer Function ....................................................................................7.7

7.4.2. Model Verification ..........................................................................................................................7.8


7.4.2.1. Separated System

7.4.2.2. Combined System

7.5. Digital Sliding-Mode Controller-Observer Designs .................................................................7.10

7.5.1. Robust Sliding-Mode Observer Design

7.5.2. Robust Sliding-Mode Controller Designs .....................................................................................7.11

7.5.2.1. Robust Switching SMC

Remark 7.1: Excitation of Unmodelled High-Frequency due to Switching Function .................................7.12

7.5.2.2. Robust Saturate SMC ......................................................................................................7.13

Remark 7.2: Slow-Down System Response due to Low Sliding Gain ..................................................7.14

7.5.2.3. Robust Unitvector SMC....................................................................................................7.15

7.5.2.4. Robust Tanh SMC............................................................................................................7.17

Remark 7.3: Performance of Saturate and Tanh SMC .....................................................................7.18

7.5.2.4. Robust Linear SMC..........................................................................................................7.19

Remark 7.4: Experimental Performance of Pseudo-SMC (Unitvector & Tanh SMC) and Linear SMC..........7.20

7.6. Conclusions .......................................................................................................................................7.21

Advanced Sliding-Mode Controller Design: Conclusions

Appendix

A.1. Eigenvalues, and Similarity Transformation .............................................................................. A.1

A.1.1. Eigenvalues of a square matrix

A.1.2. Similarity Transformation

A.2. Pole-Placement Method

A.2.1. Problem Statement

A.2.2. Ackermann's Formula................................................................................................................ A.2

A.3. Matrix Operations.............................................................................................................................. A.3

A.3.1. Differentiation and Derivative of a Matrix

A.3.1.1. Differentiation

A.3.1.2. Derivative

A.3.1.3. Jacobian

A.3.2. Trace of an n×n Matrix .............................................................................................................. A.4

A.3.3. The Caley-Hamilton theorem

A.3.4. Inversion of Matrices

A.3.4.1. Nonsingular Matrix and Singular Matrix

A.3.4.2. Matrix Inversion Lemma.............................................................................................. A.5

A.3.4.3. Block Matrix Inversion

Table of Contents 13

A.3.5. Gradient Matrices and Frechet Derivative Operator

A.3.5.1. Frechet Derivative Operator.

A.3.5.2. Rule for Differentiating Functions Involving the Trace Operator.................................. A.6

A.3.6 Some Gradient Matrices

A.4. Some Properties of a Determinant ............................................................................................... A.8

A.5. Lie Algebra.......................................................................................................................................... A.9

A.5.1. Covector Field Differential

A.5.2. First Type of Differential Operation

A.5.3. Second Type Differential Operation

A.5.4. Third Type of Differential Operation ........................................................................................ A.10

A.5.5. Lie algebra

A.5.6. Properties.

A.5.7. Example .................................................................................................................................. A.11

Bibliography

List of Symbols

( ) { }( ) { }0

011 <ℜ∈=ℜ

<ℜ∈=ℜ××

−

−

inn x

xx

x

∀ : for any

∃ : for some x∈ℜ ×n 1 : system state variable

A A A A, , ~, ~∆ ∆ ∈ℜ ×n n : first system matrix

B B B B, , ~, ~∆ ∆ ∈ℜ ×n m : second system matrix

u ∈ℜ ∈ℜ×m u1; 1

1

×m n

: control function

y p∈ℜ ∈ℜ ×1; y : output state

H H∈ℜ ∈ℜ×1 n or : hyperplane s = Hx or s Hx=

where ( ) (x Ax Bu A A x B B u= + = + + + )~ ~ ~ . ~ .∆ ∆

~ ~ A A A A A B B B B− ≤ ≤ + − ≤ ≤ +∆ ∆ ∆, ~ ~

B∆

∆ ∆ ∆ ∆,A A B≤ ≤ B

)

( )jiijm ,M= : an element at row i-th and column j-th of a matrix M

M i* : row i-th of a matrix M

M* j : column j-th of a matrix M

( kj::,M : a matrix extracted from column j-th to column k-th of a matrix

: absolute of a matrix

( )M M M= ∈ℜ ∈ℜ ×det , n n : determinant of a square matrix

M = ∈ℜ ×mijn m

P Q R= ⊗ : elementwise matrix multiplication, ie. p q rij ij ij= × .

( ) [ ]{ }nnn

×ℜ∈=−λλλ= MMIM ,0,,eig 1 L=

λH : hyperplane-eigenvalues

λ S : sliding-eigenvalues

λC : closed-loop system eigenvalues

x = xi : absolute of a vector

x = ∑ xii

2 : Euclidean norm of a vector

( ) [ ] ( ) ( )[ ]sgn sgn sgn sgnv = =v v v vn n1 1, , , ,L L ;

( )sgn, if, if, if

ξ

ξ

ξ

ξ

=

>

=

− <

1 00 0

1 0

Abstract

In this thesis, advanced design techniques in sliding-mode control (SMC) are presented. All the proposed

designs are developed in a unified manner for robust analog/digital sliding-mode controllers/observers,

robust integral SMC, continuous and discontinuous SMC designs, and for multi-input multi-output (MIMO)

uncertain nonlinear systems.

A hyperplane design is fully presented for both eigenvalue allocation and optimal sliding-mode

approaches for linear and nonlinear systems. Two eigenvalue types are identified: sliding-eigenvalues and

hyperplane-eigenvalues which are desired sliding-eigenvalues. The concept of sliding-eigenvalues is

employed to establish a stability criterion and a sufficient invariance condition.

The proposed analysis of the chattering problem is more comprehensive and precise than other works in

the current literature. Moreover my approach is in a unified manner for the saturate function and unitvector

function. In addition, the introduction of the hyperbolic tangent to replace the saturate and unitvector

functions is more convenient and efficient.

In the proposed continuous SMC design, the control function is not only continuous but also linear. The

continuous nature of this control function helps to eliminate the chattering problem found in discontinuous

SMC which is known as variable-structure system (VSS) control. The advantage of the control function

being linear is that some fundamental concepts of SMC design can be explained from linear control theory

frame-work. This leads me to propose some novel SMC designs such as a new discontinuous SMC which

alleviates the chattering problem, discrete-time SMC with both continuous and discontinuous control

functions, continuous-time and discrete-time sliding-mode observers, etc.

A new robust discrete-time sliding-mode controller-observer design is presented for both discontinuous

SMC (VSS) and continuous SMC control functions. A high sampling rate is proved to be a necessary

condition for a discrete-time SMC. Under no uncertainty this condition reduces to the stability criterion of

unit circle in the Z-domain.

All theoretical results are consistent with experimental results determined on a Ball-Hoop system. For

example, an infinite gain of the switching function may cause excitation of unmodelled high-frequency

dynamics and slow-down system responses due to low gain of sliding function. In addition, the improved

performances of continuous SMC and continuous pseudo-SMC (Tanh SMC) are as expected.

Advanced Design Techniques in Sliding-Mode Control: Introduction

1. MOTIVATION In real dynamical systems it is impossible to avoid uncertainties due to imperfect modelling and the

environment, such as temperature and pressure, and other external disturbances. The crucial demand is

therefore a solution to the robust control problem for uncertain systems. One solution to this problem is H-

Infinity Control, however it can deal with uncertain linear systems only (Francis 1988, Kwakernaak 1993,

Zhou 1995). A more general solution is Sliding-Mode Control (SMC) since it is simpler than H-Infinity

Control and it can deal with both uncertain linear and nonlinear systems (Bartolini et al 1995, DeCarlo et al.

1988, Fernandez et al. 1987).

2. METHODOLOGY My approaches have been kept as simple and practically precise as possible compared to complicated and

precise approaches (Ryan et al 1987, Spurgeon et al. 1993, Edwards et al. 1996). Proposed designs have

been put in theorems for clarity, results can be quickly found in theorems and can be checked in their proofs

and verified in simulations of extensive numerical examples. All experimental results have been consistent

with the theoretical results.

3. RESULTS New robust analog/discrete discontinuous/continuous sliding mode controller-observer designs have

been presented. All the proposed designs have been developed in a unified manner for robust analog-

discrete-time sliding-mode controllers, robust analog-discrete sliding-mode observers, continuous and

discontinuous designs, and for MIMO uncertain nonlinear systems.

In the discontinuous design, the chattering problem has been analyzed comprehensively as opposed to the

analysis in the current literature. As an alternative to the current literature (Slotine et al 1983 for saturate

function, Ambrosino et al. 1984 and Spurgeon 1991 for unit-vector function), my proposed approach has

been in a unified manner for all sliding functions (saturate, unitvector and hyperbolic tangent) where the

hyperbolic tangent function has been proposed due to its availability as a standard mathematical function.

Based on the sliding-mode mechanism, a design rule has been proposed to guarantee the existence of a

sliding mode since the invariance property exists with this mode. A new reaching control design has been

proposed to include system dynamics for efficient SMC designs, which has not been reported elsewhere to

the best of my knowledge.

In the continuous design, the control function is not only continuous but also linear. The continuous

nature of this control function helps to eliminate the chattering problem found in the discontinuous sliding-

mode control (SMC) which is known as the variable-structure system (VSS) control. The advantage of the


control function being linear is that some fundamental concepts of SMC design can be explained from the

linear control theory frame-work.

3.1. Chapter 1: Sliding Mode Control Background In the sliding-mode control theory, control dynamics have 2 sequential modes (Utkin 1977, Utkin 1992).

The first is the reaching mode and the second is the sliding mode. The Lyapunov sliding condition forces

system states to reach a hyperplane and keeps them sliding on this hyperplane, so a SMC design is

composed of 2 phases, hyperplane design and controller design. A hyperplane is first designed via the pole-

placement approach as in state-space control, a controller design is then based on the sliding condition

(Utkin et al. 1979). Stability is guaranteed by the sliding condition (Lyapunov Stability Criterion Theorem)

and by a stable hyperplane (stable designer-chosen pole-placement). In the reaching mode, the control

dynamics depend on system parameters, but in the sliding mode they depend on the hyperplane; this is the

invariance property of the sliding mode.

In this chapter, I has revised some SMC terminologies such as: hyperplane, sliding condition, sliding

margin, equivalent control, the reaching and sliding modes. Some numerical examples have been used to

illustrate how SMC can handle uncertain systems and non-linear systems.

3.2. Chapter 2: Sliding Hyperplane Design In this chapter, a hyperplane design has been fully presented for both eigenvalue allocation and optimal

sliding-mode approaches in a unified manner for linear and nonlinear systems. I have identified 2 types of

eigenvalues: sliding-eigenvalue and hyperplane-eigenvalue. Sliding-eigenvalues determines the dynamics of

the system states in the sliding mode. Hyperplane-eigenvalues are the desired sliding-eigenvalues that

represent the expected dynamics of the system states in the sliding mode. In addition, the concept of sliding-

eigenvalue is conveniently applied in the stability problem and the invariance property (Chapter 3, 4). On the

basis of the partition transformation method for linear systems in Utkin et al. 1979, I have proposed a direct

allocation method that may be the simplest way to see what the invariance property really means (Chapter 3)

and that may be extendable into nonlinear systems (Chapter 6). Moreover, I have proposed a direct

calculation method to reduce the computational effort compared to the partition transformation method in a

special case where a system matrix A is non-singular. A hyperplane normalization has been proposed to

simplify computation of a control function.

Another approach to designing a hyperplane has been based on the optimal sliding dynamics: error and

error/energy optimizations. To complement the work in the Utkin paper on the error/energy optimization

(Utkin et al. 1979), proofs of the theorems and a solution to the optimal problem have been presented in this

chapter.

Advanced Design Techniques in Sliding-Mode Control: Introduction 3

3.3. Chapter 3: A New Variable-Structure System Controller Design In this chapter, a Variable-Structure System (VSS) controller design has been fully developed in a

unified manner that is extendable into robust control (Chapter 4) and MIMO nonlinear system control

(Chapter 6). A VSS control approach is a discontinuous sliding mode control approach (SMC). Based on the

concept of sliding-eigenvalues introduced in Chapter 2, I have proposed a stability criterion (Section 3.2)

which is much simpler than the current one (Utkin 1992). By the sliding condition, a control function is first

computed in an if-form as in Utkin 1977. I have then proposed a technique to convert this form into a

compact form to facilitate controller implementations (Lemma 3.1). I have also proposed a sufficient

condition as a design rule to ensure the reaching mode will terminate in a finite time (Proposition 3.1), hence

the sliding mode can exist, because the invariance property exists with the sliding mode only. A new

reaching control design has been proposed to include system dynamics for an efficient SMC design

(Corollary 3.2), which has not been reported elsewhere to the best of my knowledge.

As an alternative to the chattering problem analysis in Slotine et al 1983, I have proposed a

comprehensive analysis (Section 3.5) that allows a unified approach for both the saturate (Slotine et al.

1983) and unit-vector functions (Ryan et al. 1984, Spurgeon 1992). In addition, the introduction of the

hyperbolic tangent (Tanh) to replace the saturate and unit-vector functions is more convenient as Tanh is a

standard mathematical function. The performance resulting from using the sliding functions (saturate,

unitvector and hyperbolic tangent functions) in place of the switching function has been analyzed in terms of

steady-state error and response speed (Section 3.5.5). A design scheme has been proposed to cope with

limitations arising from these replacements (Example 6.6).

A VSS control approach is based on a state-space model and an integral action may be required to

eliminate a steady-state error. In Chern et al. 1991, an integral of error was used, however it is a pseudo-

SMC because the larger gain of a controller, the closer the sliding mode. In Chang 1991, an integral sliding

condition was proposed using the same system order, however this approach had some limitations.

Alternatively, I have proposed an integral VSS control approach by first augmenting the order of a system

model. The controller has been then designed in a unified manner as in the normal case without integral

action. It can solve the problem of pseudo-SMC in Chern et al. 1991 and cope with the limitations in Chang

1991 (Example 3.9).

3.4. Chapter 4: A New Robust Sliding Mode Controller-Observer Design In this chapter, a new robust sliding mode controller (SMC) design has been fully presented in a unified

manner for both discontinuous SMC (VSS) and continuous SMC control functions under parametric

uncertainty and external disturbance. The control function has been partitioned into 3 components:

equivalent control, reaching control, and perturbation control.

For the robust discontinuous SMC, the design proposed in Chapter 3 has been extended to deal with

uncertain systems. This approach can be directly applied for uncertain MIMO nonlinear systems and it may

be the simplest in the current literature (Fu 1992, Chapter 6).


In the proposed linear SMC design, the control function is not only continuous but also linear. The

continuous nature of this control function helps to eliminate the chattering problem in the standard VSS

design. The advantage of the control function being linear is that some fundamental concepts of SMC design

can be explained from the linear control theory frame-work. For example, the sliding margin in the literature

can be shown to function as the reaching eigenvalue, the closed-loop system eigenvalues are composed of

the sliding margin (reaching eigenvalues) and hyperplane eigenvalues.

As a SMC is based on a state-space model, an integral action may be required to eliminate steady-state

errors. I have proved that the integral VSS control (discontinuous SMC) in Theorem 3.4 can be still applied

to both robust discontinuous and linear continuous SMC controls in this chapter.

Since the principal operating mode of a VSS control is the sliding mode, the VSS control can be seen as a

discontinuous subset of the SMC. In this chapter, the state-space control can be seen as a deterministic subset

of the SMC.

As SMC is a state-space control approach, it requires an observer to estimate unavailable states. In

Bondarev et al. 1985, a linear Luenberger observer was used as an observer of a VSS controller for

deterministic linear systems (no uncertainties). In Walcott et al. 1987 and Yaz et al. 1993, a Lyapunov

sliding condition was used to design an observer for a class of systems under matched uncertainty restricted

to a certain system structure. In Slotine et al. 1987, a sliding patch condition was used to have a region of

direct attraction where uncertainty is not fully tackled. In fact, in Walcott et al. 1987, Slotine et al. 1987 and

Yaz et al. 1993, to cope with uncertainty, a Lyapunov sliding condition was employed to include a switching

component into a linear Luenberger observer where a linearized model was used for a nonlinear system. In

Edwards et al. 1995, an observer was implemented using output feedback technique.

In this chapter, I have proposed a novel robust sliding mode observer design for a wide class of systems

under both matched and unmatched uncertainty. This design has been a development of the proposed robust

linear sliding mode controller design.

To complete the work in Drazenovic 1969, I have presented necessary and sufficient invariance

conditions. They have been valid for both discontinuous and continuous SMC.

3.5. Chapter 5: A New Robust Discrete-Time Sliding-Mode Controller-Observer Design

In this chapter, a new robust discrete-time sliding mode controller (SMC) and observer design have been

fully presented for both discontinuous SMC (VSS) and continuous SMC control functions. It can be seen as

a discrete-time extension of the previous chapter. A high sampling rate has been proved to be a necessary

condition for a discrete-time SMC. For deterministic systems, this condition has reduced to the stability

criterion of unit circle in the Z-domain.

Advanced Design Techniques in Sliding-Mode Control: Introduction 5

Based on the work in Pieper et al 1992 for matched uncertain dynamical systems, an alternative design

has been presented without the constraint stated therein. On the basis of the new robust sliding mode

controller design in the previous chapter, a new robust discrete-time sliding mode controller design has been

developed to deal with uncertain systems. This design has thus inheritted all features of the previous

continuous-time design (reaching, sliding, hyperplane and closed-loop eigenvalues; robust SMC and integral

SMC, and a robust sliding-mode observer.)

3.6. Chapter 6: General Sliding-Mode Controller Design for MIMO Uncertain Nonlinear Systems

In this chapter, I have presented a SMC design for SISO nonlinear systems, and also a design for MIMO

systems. So far I have only considered the case where the output is the first system state. In this chapter I

have considered the case where the output is a nonlinear function of all system states.

For a MIMO SMC, the hierarchical control technique was used in Utkin 1977 for linear systems.

Alternatively, I have used a decoupling technique which is applicable for MIMO nonlinear systems. This

technique allows a MIMO system to be considered as a collection of SISO subsystems. As a consequence,

all SISO results developed so far can be applicable, including the SISO nonlinear SMC in this chapter.

For the general case of multivariable nonlinear SMC, in the SMC literature (Fernandez et al. 1987, Sira-

Ramirez 1989, Chen et al. 1992, Spurgeon et al. 1993, Edwrads et al. 1995), the hyperplane design was

based on the Input-Output Linearization technique (Hunt et al. 1983, Kravaris et al. 1986, Isidori 1989) to

transform a nonlinear system into a canonical nonlinear system. By the nature of a hyperplane that it is of

reduced-order (Remark 1.4, Remark 2.2, Remark 4.5), I have designed the hyperplane via the direct

allocation approach in Chapter 2. The controller has been designed in a unified manner as in other cases. The

robust design developed here may be the simplest approach in the literature (Fu 1992, Sira-Ramirez 1996).

In addition, the design scheme in Proposition 3.3 has been proved to be efficient in solving the chattering

problem and steady-state error (Example 6.6).

3.7. Chapter 7: Advanced Sliding-Mode Controller Design: Experimental Results In this chapter, results from experiments conducted on a Ball-Hoop system are presented to validate my

anticipations from the theory (Section 3.5.5) such as infinite gain of the switching function may cause

excitation of unmodelled high-frequency dynamics and may slow-down system responses due to a low gain

of saturate function; performances of continuous SMC, and continuous pseudo-SMC (Tanh SMC).


Advanced Sliding-Mode Controller Design: Introduction

1. Motivation ......................................................................................................................................................1

2. Methodology ..................................................................................................................................................1

3. Results ............................................................................................................................................................1

3.1. Chapter 1: Sliding Mode Control Background ...................................................................................2

3.2. Chapter 2: Sliding Hyperplane Design ...............................................................................................2

3.3. Chapter 3: A New Variable-Structure System Controller Design ......................................................3

3.4. Chapter 4: A New Robust Sliding Mode Controller-Observer Design...............................................3

3.5. Chapter 5: A New Robust Discrete-Time Sliding-Mode Controller-Observer Design ......................4

3.6. Chapter 6: General Sliding-Mode Controller Design for MIMO Uncertain Nonlinear Systems........5

3.7. Chapter 7: Advanced Sliding-Mode Controller Design: Experimental Results..................................5

Chapter 1

Sliding Mode Control Background

1.1. INTRODUCTION In real dynamical systems, it is impossible to avoid uncertainties due to imperfect modelling and due to

the environment factors such as temperature, pressure and other external disturbances. So the crucial

demand is a solution to the robust control problem for uncertain systems. H-Infinity Control (Doyle et al.

1992) and Adaptive Control (Astrom et al. 1989) can be used to deal with this problem, however a more

general solution is Sliding-Mode Control (SMC) since it can deal with both uncertain linear and nonlinear

systems efficiently in a unified frame-work (Utkin 1977, Utkin 1992).

In the sliding-mode control theory, control dynamics have 2 sequential modes, the first is the reaching

mode and the second is the sliding mode (Utkin 1977, Utkin 1992). In particular, the Lyapunov sliding

condition forces system states to reach a hyperplane and keeps them sliding on this hyperplane. Essentially,

a SMC design is composed of 2 phases: hyperplane design and controller design. A hyperplane is first

designed via the pole-placement approach as in the state-space control (Utkin et al. 1979), a controller

design is then based on the sliding condition. The stability is guaranteed by the sliding condition (Lyapunov

Stability Criterion Theorem) and by a stable hyperplane (stable designer-chosen pole-placement). In the

reaching mode, the control dynamics depend on system parameters; but in the sliding mode they depend on

the hyperplane, this is the invariance property of the sliding mode (Drazenovic 1969).

In this chapter, I revise some SMC terminologies such as: hyperplane, sliding condition, sliding margin,

equivalent control, the reaching and sliding modes (Utkin 1977, Utkin et al. 1979, Zhou et al. 1992). Some

numerical examples are used to illustrate how SMC can handle uncertain systems and non-linear systems.

The most well-known application of sliding mode control is the variable-structure system (VSS) control

whose principal operating mode is the sliding mode in which the dynamics are independent from the system

parameters (Utkin 1992). A VSS control is a discontinuous subset of the SMC, the concept of VSS is

illustrated using system eigenvalues in Example 1.1.

1.2. SMC BACKGROUND First I revise the sliding condition that guarantees the sliding mode to occur on a hyperplane. In this

mode the control function reduces to the equivalent control.

1.2.1. Hyperplane Equation Consider a SISO linear system & . . , .x A x B C x= + =u y (1.1)

and a hyperplane H is defined by (Utkin 1977, DeCarlo et al. 1988, Utkin 1992)

s = H x. (1.2)

Sliding-Mode Control: Advanced Design Techniques 1.2

where H = >h, ,1 h 0

and

x B A H, , , , , ,∈ℜ ∈ℜ ∈ℜ ∈ℜ× × ×2 1 2 2 1 2 u h sfor a 2-order system.

1.2.2. Sliding Condition Choose a positive definite function for a Lyapunov candidate (Utkin 1977, DeCarlo et al. 1988, Utkin 1992) V s V s= ⇒ =1

22 & . &s (1.3)

Define a sliding condition as

s s. & < 0 (1.4)

then

000

0 →⇒ →⇒

<

>∞→∞→ tt sV

V

V&

⇒ V must reduce to zero ⇒ s must reduce toward zero

For s → in a finite time to achieve the sliding mode 0 ( )0=s , the sliding condition should be strictly

where . That is the larger δ is, the faster the sliding mode is attained. Strictly speaking, the

sliding mode exists only asymptotically.

s s. & < −δ δ > 0

Hence the sliding condition is analogous to the Lyapunov's direct stability criterion.

s

y

sy

tR tStT

t tT R=t tS R<<

Fig.1.1: Reaching mode and sliding mode

where

s is sliding variable from Eq.(1.2)

y is system output t t tT R= + S with

tT is total time

tR is reaching time ( ) 0. <ss &

tS is sliding time ( )0=s

In the figure above, there is a sliding mode in the first case, but practically not in the second.

1. SMC Background 1.3

When s ≠ 0, the system is in the ( )0. <ss mode reaching & , and the sliding condition then guarantees the

( )0= sliding s mode .

Remark 1.1: Negativeness of Sliding Condition

The sliding condition must be negative enough to guarantee that the reaching mode terminates in a

finite time for the sliding mode to exist (Section 3.4).

Remark 1.2: Stability of the Sliding Mode

A stable sliding mode implies a stable system only if the sliding condition is satisfied. The sliding mode

stability is based on 6 theorems in Utkin 1977 where the proofs were referred to other works in the Russian

literature. In the latest book by Utkin 1992, these 6 theorems were omitted and the sliding-mode stability was

based instead on 2 complex theorems. Alternatively, I will propose a simple stability criterion in Section 3.2.

1.2.3. Sliding Dynamics From Eq.(1.1), if the output y and the state vector x are defined as

x =

=

xx

yy

1

2 &

then I have the following sliding dynamics equation s h x x y h y= ⇒ + = ⇒ + =0 01 2. & . 0 (1.5)

( ) ( ) theyty ..0 −=⇒ (1.6)

Therefore, once s = 0, the output state y decays as described by Eq.(1.6). Note that Eq.(1.5) is a characteristic equation whose root is −h , so the hyperplane-eigenvalue is defined as λ . This

hyperplane-eigenvalue determines the sliding dynamics. The sliding dynamics depend only on this

hyperplane-eigenvalue and not on the system parameters or on how the sliding mode is reached: that is the

property of invariance in the sliding mode (Drazenovic 1969).

H h= −

1.2.4. Equivalent Control Consider a hyperplane ( ) ( )[ ]HAxHBHBBAxHxHHx 1. −+=+==⇒= uuss &&

where

H x∈ℜ ∈ℜ× ×1 n , 1n

).

Assuming ( is invertible, the equivalent control is defined as (Utkin 1977) HB

( ) HAxHB 1−−=equ (1.7)

thus ( )equus −= HB& (1.8)

It is important to note that in the sliding mode s s u ueq= ⇒ = ⇒ =0 0& .


1.3. NUMERICAL EXAMPLES I will use the standard if-form of control functions as in the VSS literature. Alternatively, the compact

form will be developed in Lemma 3.1 (Chapter 3).

1.3.1. Example 1.1: Variation of System Structure Consider an example in Hong et al. 1989

& .x A x B.= + u

where

. A =− −

=

0 11 2

01

, B

If a hyperplane-eigenvalue is chosen as λH = −3

then the corresponding hyperplane is characterized by H = 3 1,

The equivalent control u can be found from eq

( ) ( )[ ]HAxHBHBHBHAxBAxHxHHx 1. −+=+=+==⇒= uuuss &&

&s x x u u x xeq= − + + ⇒ = −1 2 1 2

A typical VSS control function can be chosen to be similar ("equivalent") to the equivalent control u x x= +ψ ψ1 1 2 2

then ( ) ( ) 2211 11 xxs +ψ+−ψ=& .

To satisfy the sliding condition s s. & < 0, I have

ψ 1

if if

< >>

1 01

1

1

,,

,sxsx < 0

ψ 2

if if

< − >> − <

1 01 0

2

2

,,

sxsx

or

ψ

δδ

δ1

if if

, =− >+ <

>1 01 0

01

1

,,

,sxsx

ψδδ

δ2

if if

>0=− − >− + <

1 01 0

2

2

,,

,sxsx

In general, the value of δ termed the sliding margin (Zhou et al. 1992) will affect the control effort u.

A value of δ = is arbitrarily chosen. 4

Therefore u x x= +ψ ψ1 1 2 2

where

ψ ψ

<+>−

=0 if ,50 if ,3

1

11 sx

sx

<+>−

=0 if ,30 if ,5

2

22 sx

sx


• If sx sx1 0> and 2 0> , then u x x= − −3 51 2 , hence the closed-loop system eigenvalues are

−0 5616 3 5616. , . : positive feedback

• If sx sx1 0< and 2 0> , then u x x= −5 51 2 , hence the closed-loop system eigenvalues are

1 5000 1 9365. .± j : positive feedback

• If sx sx1 0> 2 0< and , then u x x= − +3 31 2 , hence the closed-loop system eigenvalues are

0 3723 5 3723. , .− : positive feedback

• If sx sx1 0 0< and 2 < , then u x x= +5 31 2 , hence the closed-loop system eigenvalues are −2 3, :

negative feedback

0 1 2 3 40

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 1 2 3 4-10

-5

0

5

10

15

Time [s]

Con

trol

0 1 2 3 4-1

0

1

2

3

Time [s]

Slid

ing

Trajectory Sliding Line

0 0.5 1-3

-2

-1

0

Output

Cha

nge

of O

utpu

t

VSS Control for 2-nd Order System

Fig.1.2: VSS Control for Example 1.1.

Remark 1.3: Example 1.1

• The chattering problem exists in the control effort;

• As shown in the trajectory diagram, the output starts from the position of 1 unit and reaches to and slides

on the sliding line.

Remark 1.4: Dimension of Sliding Hyperplane

In the sliding mode, the system effectively becomes another system of reduced order. Therefore, a 2-nd

order system has a sliding line, a 3-rd order system has a sliding plane, a higher order system has a sliding

hyperplane.


1.3.2. Example 1.2: Uncertain System Note that uncertainties and nonlinearities may be taken into account in satisfying the sliding condition,

so the sliding mode is guaranteed even in such cases.

Consider the above example under uncertainties

& . .x A x B= + u

where

A B=

=

= − ± =

−−

= − ± =−−

0 1 01

10 515

2 113a b

a b, ,..

, 0.5




( ) ( )[ ]HAxHBHBBAxHxHHx 1. −+=+==⇒= uuss &&

( ) uxbaxs +++= 21 3&

( ) 21221121 20

5.05.1

3 xxxxxbaxueq −+=φ+φ=+−−=

A typical VSS control function can be chosen to be similar ("equivalent") to the equivalent control u x x= +ψ ψ1 1 2 2

then ( ) ( ) 2211 3 xbxas ++ψ++ψ=& .

To satisfy the sliding condition s s. & < 0, I have

( )( )

<φ>>φ<

0 if ,sup0 if ,inf

11

111 sx

sxψ

( )( )

<φ>>φ<

0 if ,sup0 if ,inf

22

222 sx

sxψ

ψ ψ

<>><

0 if ,5.10 if ,5.0

1

11 sx

sx

<>>−<

0 if ,00 if ,2

2

22 sx

sx

ψ ψ

<δ+>δ−

=0 if ,5.10 if ,5.0

1

11 sx

sx0

0 if ,0 if ,2

2

22 >δ

<δ>δ−−

=sxsx

In general, the value of ,δ termed the sliding margin, will affect the control effort u. For a reasonable

control effort, a value of δ = is chosen. Therefore 4 u x x= +ψ ψ1 1 2 2

where

ψ1

if if

=− >

<

3 5 05 5 0

1

1

. ,. ,

sxsx

ψ2

if if

=− >

<

6 04 0

2

2

,,

sxsx


0 1 2 3 40

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 1 2 3 4-20

-10

0

10

20

Time [s]

Con

trol

0 1 2 3 4-1

0

1

2

3

Time [s]

Slid

ing

Lower Bound Upper Bound

0 0.5 1-2.5

-2

-1.5

-1

-0.5

0

Output

Cha

nge

of O

utpu

t

VSS Control for 2-nd Order System under Uncertainty

Fig.1.3: VSS Control under Uncertainty for Example 1.2, 2 output curves coincide.


• Existence of chattering problem

• Robustness: there is not much differences between the 2 boudaries as shown in all diagrams in Fig.1.3

1.3.3. Example 1.3: Nonlinear System Consider a nonlinear system in Zhou et al. 1992 & .x f g= + u

where

f =+ +

= +

xx x x tx x

2

1 2 12

1 120

1sin( ), g .




s = Hx ( ) ( )[ ]HfHgHggfHxH 1.. −+=+== uus &&

( )1

2122111

1

)(2)sin(

x

xxxxtxueq

+

+++=−= − HfHg


where

sin( ) ,txx x1

1 11

11

11

10+

=− +

=

thus

u x x x x x x x xeq = + + + =−

+ + +φ φ φ φ0 1 2 2 1 2 3 12

2 1 221

120

20

10 1

x

A typical VSS control function can be chosen to be similar ("equivalent") to the equivalent control u x x x= + + +ψ ψ ψ ψ0 1 2 2 1 2 3

21

then

( )( )

ψφ

φ0

0

0

0

0

<

> <

inf ,

sup ,

if

if

s

s

> ( )

( )ψ

φ

φ1

1 2

1 2

0

0

< >

> <

inf ,

sup ,

if

if

sx

sx

( )( )

ψφ

φ2

2 1 2

2 1 2

0

0

< >

> <

inf ,

sup ,

if

if

sx x

sx x ( )

( )ψ

φ

φ3

3

312

0

00

< >

> <

>

inf ,

sup ,

if

if

s

sx

or

ψ0

1 01 0

< − >> <

,,

if if

ss

ψ12

2

0 02 0

< >

> <

,, if if

sxsx

ψ21 2

1 2

0 02 0

< >

> <

,, if if

sx xsx x

ψ3

0 01 0

< >> <

,,

if if

ss

thus

ψδ

δ0

11 0

=− − >

+ <

,,

if if

ss

0ψ

δ

δ12

2

02 0

=− >

+ <

,,

if if

sxsx

ψδ

δ21 2

1 2

02 0

=− >

+ <

,,

if if

sx xsx x

ψδ

δ3

01 0

=− >

+ <

,,

if if

ss

In general, the value of, δ termed the sliding margin, will affect the control effort u. A value of δ = 0 5.

is arbitrarily chosen. Therefore u x x x x= + + +ψ ψ ψ ψ0 1 2 2 1 2 3

21

where

ψ0

1 5 01 5 0

=− >

<

. ,. ,

if if

ss

ψ12

2

0 5 02 5 0

=− >

<

. ,. ,

if if

sxsx

ψ21 2

1 2

0 5 02 5 0

=− >

<

. ,. ,

if if

sx xsx x

ψ3

0 5 015 0

=− >

<

. ,. ,

if if

ss


• Existence of chattering problem;

• Nonlinearity is considered as uncertainty in a linear system. However, this is not true in general (Chapter

6).


0 1 2 3 40

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 1 2 3 4-10

-5

0

5

Time [s]

Con

trol

0 1 2 3 4-0.5

0

0.5

1

1.5

2

Time [s]

Slid

ing

0 0.5 1-2

-1.5

-1

-0.5

0

0.5

Output

Cha

nge

of O

utpu

t

VSS Control for 2-nd Order Nonlinear System

Fig.1.4: Nonlinear VSS Control for Example 1.3.

1.4. CONCLUSIONS 1s

1s

ψ1

ψ 2

s

x1rx2

Fig.1.5: VSS Control (Discontinuous SMC)

where

s: sliding variable from Eq.(1.2)

2,1,0 if ,

0 if ,=

<−

>+=ψ

−

+

isxk

sxk

ii

iii

To satisfy the sliding condition , all its terms are set negatively. This approach may be considered

conservative. For each state variable, the feedback gain is switching between positive and negative values,

i.e. positive and negative feedback, hence the name of variable-structure system. So the VSS control is a

discontinuous subset of the SMC. A SMC easily deals with uncertain dynamical systems (Example 1.2),

especially with nonlinear systems (Example 1.3).

ss& < 0


Chapter 1 Sliding Mode Control Background

1.1. Introduction .................................................................................................................................................1

1.2. SMC Background........................................................................................................................................1

1.2.1. Hyperplane Equation .......................................................................................................................1

1.2.2. Sliding Condition.............................................................................................................................2

Remark 1.1: Negativeness of Sliding Condition.......................................................................3

Remark 1.2: Stability of the Sliding Mode ...............................................................................3

1.2.3. Sliding Dynamics.............................................................................................................................3

1.2.4. Equivalent Control ...........................................................................................................................3

1.3. Numerical Examples ...................................................................................................................................4

1.3.1. Example 1.1: Variation of System Structure....................................................................................4

Remark 1.3: Example 1.1..........................................................................................................5

Remark 1.4: Dimension of Sliding Hyperplane ........................................................................5

1.3.2. Example 1.2: Uncertain System.......................................................................................................6

Remark 1.5: Example 1.2...........................................................................................................................7

1.3.3. Example 1.3: Nonlinear System.......................................................................................................7

Remark 1.6: Example 1.3...........................................................................................................................8

1.4. Conclusions .................................................................................................................................................9

Chapter 2

Sliding Hyperplane Design

2.1. INTRODUCTION. There are 2 stages in the sliding mode control (SMC) design: the hyperplane design and the controller

design. In the hyperplane design stage, a sliding hyperplane is computed on the basis of the nominal model, .

In the controller design associated with this hyperplane, a control function is achieved by satisfying the

sliding condition which is in the form of the Lyapunov's direct stability criterion.

In this chapter, a hyperplane design is fully presented for both eigenvalue allocation and optimal sliding-

mode approaches in a unified manner for linear and nonlinear systems. On the basis of the partition

transformation method for linear systems in Utkin et al. 1979, I propose a direct allocation method that may

be the simplest way to see what the invariance property really means (Corollary 4.2) and is extendable into

nonlinear systems (Section 6.2). Alternatively, I propose a direct calculation method to reduce the

computational effort as opposed to the partition transformation method above. A hyperplane normalization is

proposed to greatly simplify a control function in the controller design stage. There are 2 types of

eigenvalues to be proposed: sliding-eigenvalue and hyperplane-eigenvalue. Sliding-eigenvalues determine

the dynamics of the system states in the sliding mode. Hyperplane-eigenvalues are the desired sliding-

eigenvalues that represent the expected dynamics of the system states in the sliding mode. In addition, the

concept of sliding-eigenvalue is conveniently applied in the stability problem (Section 3.2) and the

invariance property (Section 4.2).

In Utkin et al. 1979, there are 2 optimal sliding-mode methods to design a hyperplane: error and

error/energy optimizations. The first optimal problem is a linear quadratic control problem with a Riccati

equation. The second optimal problem is a constant output feedback problem. Computational algorithms to

solve these optimal problems can be classified into 3 groups: Levine-Athans algorithm (Levine et al. 1970,

1971; Toivonen 1985), Davidon-Fletcher-Powell numerical method (Choi et al. 1974, Horisberger et al.

1974) and sequential algorithm with the Newton-Raphson method (Bingulac et al. Calovic et al. 1973,

Bingulac et al. 1973, 1975). In these optimal problems, 2 Lyapunov equations must be solved. There has

been no mechanism in these algorithms to guarantee solutions due to the singularity problem in the

Lyapunov equations.

To complement the work in the Utkin paper on the second optimal problem, proofs of the theorems and a

solution to the optimal problem are proposed in this thesis. To prove the theorems, I will use the concise

Lagrangian derivations (Kwakernaak et al. 1972, Mendel 1974), the gradient matrix operations (Athans

1968) and the Frechet derivative operators (Geering 1976). For a solution to the second optimal problem, I

use the line-search algorithm in Kuhn et al. 1987 with the well-known BFGS (Broyden-Fletcher-Goldfarb-

Shannon) numerical method (Fletcher 1987). In general, the BFGS algorithm requires a line-search

procedure to circumvent the calculation of a Hessian (second-order derivative matrix), but the general line-


search algorithm cannot be used in my problem due to the problem of singularity in Lyapunov equations.

The line-search in Kuhn et al. 1987 can be used instead since it guarantees the solution of singular

Lyapunov equations (Section 2.4.5).

2.2. SLIDING DYNAMICS This section concerns the sliding dynamics where the sliding condition guarantees the existence of the

sliding mode defined by the sliding dynamics equation. This equation determines the sliding-eigenvalue via

the first sliding-eigenvalue formula based on the partition transformation. Alternatively I propose the second

sliding-eigenvalue formula without such determination. First I start with the sliding condition to guarantee

the existence of the sliding mode derived from the condition s = 0.

2.2.1. Sliding Dynamics Equation A control function is designed to satisfy the sliding condition Eq.(1.4) in order to guarantee the sliding

mode. From Utkin et al. 1979, I have the following theorem on the sliding dynamics equation

Theorem 2.1: Sliding Dynamics Equation

Consider a system

&x Ax Bu= + (2.1)

and a hyperplane

s H x= .

where ( ) 111 ,rank , , , , ××××× ℜ∈=ℜ∈ℜ∈ℜ∈ℜ∈ mmnnnmn m sBBAux .

Define a similarity transformation as

( )

( ) ( ) ( )

~~~

~, ~ , ~

~ ~ ~~ ~ , ~

~ , ,

~ , ~ ~ ~

x Mxxx

x M x x x M M

A MAM A AA A

B MB0

BB B

A A A A

= =

⇔ = ∈ℜ ∈ℜ ∈ℜ ≠

= =

= =

∈ℜ ≠

∈ℜ ∈ℜ ∈ℜ ∈ℜ

− − × × ×

− ×

− × − − × × − ×

1

2

11

12

1

1 11 12

21 22 22 2

0

0

, ,

, , 11 12( )

21 22

n m m n n

m m

n m n m n m m m n m m m

(2.2)

then a sliding dynamics equation can be determined by ( )~& ~ ~ ~ ~ .~x A A H H x1 11 12 2

11 1= − − (2.3)

where ( )~ , ~ , ~H H1 2 0∈ℜ ∈ℜ ≠× − ×m n m m m H2 (2.4)

Proof

By the coordinate transformation Eq.(2.2), a system equation can be written as

~& ~~ ~ ~ ~~ ~

~~ ~x Ax Bu A A

A Axx

0B

u= + =

+

11 12

21 22

1

2 2

2. Sliding Hyperplane Design 2.3

thus ~& ~ ~ ~ ~x A x A x1 11 1 12 2= + (2.1.a)

~ ~ ~ ~& ~ ~x A x A x B2 21 1 22 2 2= + + u (2.4.b)

then

( ) ( ) [ ]s Hx H M x HM x Hx H Hxx

H x H x1= = = = =

= +− −~ ~ ~~ ~ ~ .

~~

~ ~ ~ ~11 2

1

21 1 2 2 (2.5)

where [ ]HM H H H− = =1

1 2~ ~ ~ (2.6)

In the sliding mode, from Eq.(2.5) I obtain s 0 x H H x= ⇒ = − −~ ~ ~ ~

2 21

1 1 (2.7)

Q.E.D.

2.2.2. Sliding Eigenvalue From Eq.(2.3), the sliding-eigenvalues can be computed from ( )λ S = − −eig ~ ~ ~ ~A A H H11 12 2

11 (2.8)

However, the transformation matrix M in Eq.(2.2) must be determined. Alternatively, I propose the

following theorem to determine the sliding-eigenvalues without the determination of such transformation

matrix M. The following theorem is for SISO systems and it will be extended into MIMO systems in

Chapter 5.

Theorem 2.2: Sliding Eigenvalues

Consider a system

& .x A x B.= + u

and a hyperplane

s = H x. then the sliding-eigenvalues λ can be determined from S

( )[ ] { }eig A B HB HA− =−1 0, λ S (2.9)

Proof: For an n-order SISO system, by Eq.(2.8) there are only ( )1−n sliding-eigenvalues λ , so I need to

prove that the n-th eigenvalue of Eq.(2.9) is zero, ie. to prove that S

( )[ ] 0det =− HAHBBA 1− . Since

eigenvalues are invariant under a similarity transformation, I use the similarity transformation defined in

Eq.(2.2).

Consider a system after the similarity transformation Eq.(2.2)

& .x A x B.= + u

and a hyperplane

s = H x.


where

=ℜ∈ℜ∈ℜ∈ ××

22

11 = , ,, , ,,B0

Bxx

xABx sunnn

. ( ) ( ) ( ) ( ) ℜ∈ℜ∈ℜ∈ℜ∈

= −××−−×−

22211

12111

1211

112221

1211 , ,, , ,, xAxAAAAAAA

A nnnn

I have

( ) [ ] [ ]B HB HA0

BH H

0B

H HA AA A

H0 0

H A H A H A H A−

−

−=

= + +

1

21 2

2

1

1 211 12

21 222

1

1 11 2 21 1 12 2 22,

thus

( )[ ]A B HB HAA AA A

H0 0

H A H A H A H AA A

H H A H H A− =

− + +

= − −

− −− −

1 11 12

21 222

1

1 11 2 21 1 12 2 22

11 12

21

1 11 21

1 12, ,

By the following property of a determinant

DAD0BA

DCBDAD

ABCADA

DCBA

. 0 if ,.

0 if ,.1

1

=⇒

≠−

≠−=

−

−

then

(2.10) ( )[ ]det detA B HB HAA A

H H A H H A− =

− −

=

−− −

1 11 12

21

1 11 21

1 120

,

Q.E.D.

Remark 2.1: Relationship between Sliding Eigenvalues and Hyperplane Eigenvalues

In the sliding mode, all states are sliding on the hyperplane, hence the full name of "sliding hyperplane".

Hyperplane-eigenvalues are the desired sliding-eigenvalues, and by convention it is usually chosen to be the

sliding-eigenvalue of the nominal mode under perturbation in the system dynamics:

( )[ ] [ ]Heig ,0 1 =− − HAHBBA (2.11)

where A, B are nominal values of uncertain dynamical system matrices A respectively. % , %B

The hyperplane-eigenvalues are exactly the same as the sliding- eigenvalues if there is no perturbation at

all to the system dynamics.

Remark 2.2: Hyperplane is of Reduced-Order

As presented in Remark 1.4, in Theorem 2.2 and its proof and as mentioned in Remark 2.1, an n-th order system has its hyperplane of eigenvalues. A hyperplane is therefore stated to have reduced-order in

sliding-mode control literature.

( 1−n )


2.3. EIGENVALUE ALLOCATION The hyperplane-eigenvalues are the desired sliding-eigenvalues for the nominal model. Based on these

sliding-eigenvalues, this section relates to a hyperplane design by an eigenvalue allocation approach. From

the first sliding-eigenvalue formula Eq.(2.8), I have the partition transformation method where the direct

allocation method is its special case for canonical systems. From the second sliding-eigenvalue formula

Eq.(2.9), I have the direct calculation method that involves no partition transformation. By the direct

allocation method, I directly obtain the I/O technique in Section 2.3.3 to design a hyperplane for the SMC

general case. This technique is much simpler and hence more comprehensive than that in the current SMC

and VSS literature (Fernandez et al. 1987, Sira-Ramirez 1989, Chen et al. 1992, Spurgeon et al. 1993,

Edwrads et al. 1995) where the hyperplane design is based on the input-output linearization technique

(Hunt et al. 1983, Kravaris et al. 1986, Isidori 1989).

In this approach of eigenvalue aloocation, I can identify 2 techniques: full-state and I/O-state. In the full-

state technique, the general method is the partition transformation where the direct allocation method is the

special case for a canonical system. I can also use this direct allocation method for a non-canonical system

via a canonical transformation. If a system matrix A is non-singular, I propose the direct calculation method

to circumvent this partition transformation. Based on the direct allocation method, I will develop the I/O-

state technique to be used in the general case of SMC (MIMO nonlinear systems) in Chapter 6, rather than

based on the input-output linearization technique as in the current SMC and VSS literature (Isidori 1989).

2.3.1. Full-State Technique In this technique, there are 3 methods: the first one is general, the second is a special case where the

system matrix A is canonical. It is necessary to determine the partition transformation for the first method.

The third method is for a system of non-singular matrix A where such determination will not be necessary.

2.3.1.1. Partition Transformation Method By Theorem 2.1, I have the following corollary to design a hyperplane where %H is arbitrarily

chosen, hence it is possible to choose H such that HB

I

I2 =

%2 ≠ I= as in Section 2.5 on the normalization of a

hyperplane.

Corollary 2.1: Hyperplane Design by Partition Transformation Method

Consider a linear system

& . .x A x B u= +

let ~ (~ , ~ ,H A A1 11 12= place λ H ) (2.12)

then a hyperplane can be determined by [ ]H H I M= ×

~1 (2.13)


where

place: pole-placement procedure (Appendix) : n−1 desired Hurwitz hyperplane-eigenvalues ( )

( 11 −×−ℜ∈ n

H)

and M is a transformation matrix such that

MB0

BA A A

A AMAM=

=

= −~ , ~ ~ ~

~ ~2

11 12

21 22

1 .

Proof Let H in Eq.(2.3), I have %

2 = I

( )~& ~ ~ ~ .~x A A H x1 11 12 1= − 1 (2.14)

It is shown in Utkin et al. 1979, Utkin 1992 that if the pair ( )~ ~A B, is controllable, then so is the pair

( )~ , ~A A11 12 . Thus I can use the pole-placement procedure place in Appendix to determine %H1 as in Eq.(2.12).

Because the desired poles λ [H = −eig ]~ ~ ~A A H11 12 1 are Hurwitz, ~x1 is stabilized by Eq.(2.14), hence so is ~x2

by Eq.(2.7). By the similarity transformation by Eq.(2.2), both x1 and x2 are stabilized.

Q.E.D.

2.3.1.2. Direct Allocation Method

By Theorem 2.1, I propose the following corollary to design a hyperplane for a canonical system

Corollary 2.2: Hyperplane Design by Direct Allocation Method

If the system is canonical, with desired hyperplane-eigenvalues [ ] ( )

( )1111 ,, −×

−− ℜ∈λλ= nnH L

then the hyperplane is determined by H = − ∈ℜ ×a an

n0 2

11, , ,L (2.15)

where (2.16) ( ) ( ) 01

22

111 aaa n

nn

n +λ++λ+λ=λ−λλ−λ −−

−− LL

Proof

Consider a canonical system

. & .x =

− − − −

+

0 1 0 00 0 1 0

0 0 0 1

00

011 2 3

L

L

M M M O M

L

L

M

a a a a

u

n

.x

]

so the determination of the similarity transformation Eq.(2.2) is not necessary, then I have

[~ , ~ , ~ , ,A A A A H H11 11 12 12 1 1 1 1

0 1 0 00 0 1 0

0 0 0 10 0 0 0

00

01

= =

= =

= = −

L

L

M M M O M

L

L

M Lh hn


thus

[ ]~ ~ ~ , ,A A H11 12 1 1 1

1 2 3 1

0 1 0 00 0 1 0

0 0 0 10 0 0 0

00

01

0 1 0 00 0 1 0

0 0 0 1− =

−

=

− − − −

−

−

L

L

M M M O M

L

L

M L

L

L

M M M O M

L

L

h h

h h h h

n

n

hence the eigenvalues are determined by ( )λ λ λ λI A A H− − = + + + + =−

−−~ ~ ~

11 12 11

12

2 1 0nn

nh h hL .

I have the following characteristic equation associated with the hyperplane-eigenvalues . ( ) ( ) 001

22

111 =+λ++λ+λ=λ−λλ−λ −

−−

− aaa nn

nn LL

Equating the coefficients in these 2 equations yields h a i ni i= = −−1 1, , ,L 1.

Since H H H H I H= = =1 2 1 1 1

hence Eq.(2.15).

Q.E.D.

Remark 2.3: Application of Direct Allocation Method

• Notice that the hyperplane for a canonical system is independent of the system parameters. This is

because a canonical system satisfies the matching invariance condition (Drazenovic 1969);

• I can apply this method for a non-canonical system by using the canonical transformation.

Remark 2.4: Extensibility of Hyperplane Design by Direct Allocation Method to Nonlinear Systems

It will be shown in the following chapters that the hyperplane design is independent of the controller

implementation, for both linear and nonlinear systems. The above proof of Corollary 2.2 has been derived

from Theorem 2.1 for linear systems where hi are the coefficients of the desired characteristic equation.

Alternatively, I will obtain exactly the same result by proposing a proof in Chapter 6 that is valid for both

linear and nonlinear canonical systems.

2.3.1.3. Direct Calculation Method

In the above partition transformation method, I have to determine the partition transformation. To

circumvent such determination, based on Eq.(2.11), I propose the following method to determine a

hyperplane for a system with a non-singular system matrix A.

Corollary 2.3: Hyperplane Design by Direct Calculation Method

If the system matrix A is non-singular, then the hyperplane can be determined by [ ]( )HAA ,0,,place,. 1 BAHAHH == − (2.17)


with: place is a pole-placement procedure defined in the appendix, A : nominal values of uncertain dynamical system matrices A respectively, B, % , %B

λ : desired hyperplane-eigenvalues. H

Proof

It is a direct development from Eq.(2.11).

Q.E.D.

Remark 2.5: Hyperplane Normalization in Direct Calculation Method

The normalization is implied in the direct calculation method.

Remark 2.6: Dimension of Sliding Variable and Extensibility to MIMO Systems

For all 3 methods above:

• It is quite possible for s to include less than n states, but in that case it is well-known that one is not free

to obtain the desired eigenvalues allocation in the whole LHP for the sake of control quality. Moreover it

can be shown that, in such a case, there are some constraints on the control function;

• The above approach is applicable to SISO systems. For MIMO systems, a decoupling technique may be

used first to transform a MIMO system into a multiple SISO subsystems (Sections 6.8.4 to 6.8.6).

2.3.2. Canonical Transformation I can use the direct allocation method for a noncanonical system using the following canonical

transformation (Ogata 1987).

Theorem 2.3: Computation of Canonical Transformation Matrix

Consider a system

& . .

. .x A x B

C x D= += +

UVWu

y u (2.18)

If the above system is controllable, it can be transformed into a canonical system by means of the

following transformation T M R= . (2.19)

where M B AB A B A B= −2 L n 1 : controllability matrix (2.20)

(2.21) R =

−

−

a a aa a

a

n

n

1 2 1

2 3

1

11 0

1 01 0 0 0

L

L

M M N M M

L

L

0

with a are the coefficients of the characteristic equation i

λ λ λ λI A− = + + + +−−n

nna a1

11L a0


Proof

Following the proof in Ogata 1987, from Eqs.(2.19), (2.20) and (2.21), I have

T AT MR A MR R M AMR− − − −

−

= = =

− − − −

L

N

MMMMMM

O

Q

PPPPPP

1 1 1 1

0 1 2

0 1 0 00 0 1 0

0 0 0 1a f a f

L

L

M M M O M

L

La a a an 1

(2.22)

T B− =1 0 0 0 1, , , ,LT (2.23)

Q.E.D.

2.3.3. I/O-State Technique As I have seen in Section 2.3.1, in a canonical system, the state variables are the sequential derivatives of

the output until the first appearance of the control term u. Similarly, now based on the definition of the

relative degree (Isidori 1989), I develop a hyperplane design relating to the output state for a linear system.

This approach will be directly extended to a case where an output state is a nonlinear function of the states

in a nonlinear system (Chapter 6).

I propose the following theorem to design a hyperplane using I/O technique

Theorem 2.4: Hyperplane Design by I/O-State Technique

Consider a system

& . .

,x A x B= +=

RSTu

y x r1 with the relative degree of (2.24)

then a hyperplane can be determined by H H M H x= =

~. , . s (2.25)

where

(2.26) MA

A A

A=

−

1 0 0

11

12

1

L

M*

*

*, :

r

row

and %H is computed by the direct allocation method for the order of (r−1).

Proof

Let

( ) ( )

( ) ( )

& &

&& &&

.

*

*

*

* *

y xy x

y x

y x

r r r

r r r r

= == =

= =

= = +

− − −

− −

1 1

1 1

11

11

2

1 11

12

A xA Ax

A A x

A A x A A B

L

u

BA B

A A B

A A B

1

1

13

12

00

0

0

*

*

*

*

==

=

≠

−

−

Lr

r


or, using the notation of the Lie derivative (Isidori 1989) I have

( )

( )

&

.

*

*

*

* *

y L x

y L x

y L x

y L x L L x u

r r r

r r r r r

= =

= =

= =

= + = +

⇒− − −

− − −

f

f

f

f g f

A x

A Ax

A A x

A A x A A B

relative degree

1 12

1 1

1 11 1

2

11

1 11

12

L r : of the system

hence the new coordinate is

( )

( ) ( )

y y xy y

y y

y yy y

y y

y ur

r r r r

r rr r

1 1

2 1

11

2

1 2

2 3

1

11

12

= == =

= =

⇒

==

=

= + = +

− − −

− −

&

&

&

&

& . .

*

** *

A x

A A xx x A A x A A B

LL

α β u

thus the coordinate transformation is

. y Mx MA

A A

= =

−

; *

*

where

1 0 0

1

12

L

Mr

then the system in the new coordinate can be written as

( )

&~ ~ , ~ , ~y Ay B A B

x

= + =

=

u

r r r rr

where

0 1 0 00 0 1 0

0 0 0 1

00

0

1 2 3

L

L

M M M O M

L

L

M

α α α α β

Since the matrix ~A is canonical, the hyperplane can be designed as in the Section 2.3.1 by the direct

allocation method, say ~H , ( ) ( )s = = =

~ ~ ~Hy H Mx HM x . (2.27)

Q.E.D.

Remark 2.7: Extensibility of I/O-State Technique to Nonlinear Systems

Despite the complexity, the I/O-state technique is the most convenient one because it can replace the

well-known partition method in Section 2.3.1.1. Moreover, it can handle the general case of multivariable

nonlinear systems where the output is a nonlinear function of all the system states. Note that the proof above

is based on linear systems, in Chapter 6 the proof will be modified so that this technique is applicable to both

linear and nonlinear systems.

Remark 2.8: Decomposition of I/O-State Technique

By means of the I/O-state technique with r < n, the system clearly appears decomposed into a linear

system of dimension r, which is solely responsible for the input-output behaviour, and a possibly nonlinear

system of dimension n−r, whose behaviour however does not affect the output (Isidori 1989).


2.3.4. Proposition 2.1: Application of I/O-State Technique By the I/O technique, I can allocate r −1a hyperplane-eigenvalues only, whereas it is necessary to

allocate a hyperplane-eigenvalues. If r < n, it is required to check the stability of sliding mode via its

sliding-eigenvalues, because if there is a non-Hurwitz sliding-eigenvalue, then the system is unstable

(stability criterion by Theorem 3.1).

ffn −1

2.4. OPTIMAL SLIDING MODE In Utkin et al. 1979, there are 2 optimal sliding-mode methods to design a hyperplane: error and

error/energy optimizations. A solution to the first optimal problem is obtained via a Riccati equation in the

optimal linear quadratic control. A solution to the second optimal problem is achieved via the conjugate

gradient BFGS (Broyden-Fletcher-Goldfarb-Shannon) numerical method where the required gradient is

determined by several Lyapunov equations. To complement the work in the Utkin paper on the second

optimization, proofs of the theorems and a solution to the optimal problem will be developed in this chapter.

To prove the theorems, I will use the concise Lagrangian derivations (Kwakernaak et al. 1972, Mendel

1974), the gradient matrix operations (Athans 1968) and the Frechet derivative operators (Geering 1976).

For a solution to the optimal problem, I will use the line-search algorithm in Kuhn et al. 1987 with the well-

known BFGS numerical method. In general, the BFGS algorithm requires a line-search procedure to

eliminate the calculation of a Hessian (second-order derivative matrix), but the general line-search algorithm

cannot be used in my problem due to the problem of singularity. The line-search in Kuhn et al. 1987 may be

used instead since it guarantees the solubility of singular Lyapunov equations.

2.4.1. Error-Optimal Sliding Mode (Optimal Hyperplane 1) Based on the proof of Theorem 2.1, I propose the following lemma,

Lemma 2.1: Performance Index of Error-Optimal Sliding Mode

Consider a problem of minimizing a quadratic performance index

J dtT

t

n

s

111

2= ∈ℜ

∞× n n∈ℜ ×z x Q x x Q. . , , (2.28)

where Q , i.e. Q is symmetric positive-definite, tQ≥ 0, T Q= s is the time at which the sliding mode

begins. This problem can be interpreted as a linear optimal state regulator problem in minimizing the

functional J1 which can be converted into the following form

( )[ ]J T T T

ts

1 1 11 12 221

12 1 2212

= − +−∞

∫ ~ ~ ~ ~ ~.~

~x Q Q Q Q x v Q v dt (2.29)

where v x Q Q x x v Q Q x= + ⇔ = −− −~ ~ ~ ~ ~ ~ ~ ~

2 221

12 1 221

12 1T

2T (2.30)

and

( )~ ~ ~~ ~ , ~ ~ ~ ~ ~ ~Q M QM Q QQ Q

x Q x x Q x= =

=− −1 1 11 12

21 222 21 1 1 12 2

T T T (2.31)


Proof

From the proof of Theorem 2.1 by x M x= −1~ , I have ( ) ( )x Qx M x Q M x x Q x x Q x x Q x x Q xT T T T T T= = = + +− −1 1

1 11 1 1 12 2 2 22 22~ ~ ~ ~.~ ~ ~ ~ ~ ~ ~ ~ ~ ~ .

Let

( )v x Q Q x v x x Q Q= + ⇒ = +−~ −~ ~ ~ ~ ~ ~ ~2 22

112 1 2 1 12 22

1T T T TT

then ( )~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~x Q Q Q Q x v Q v x Q x x Q x x Q x1 11 12 22

112 1 22 1 11 1 1 12 2 2 22 22T T T T T− + = + +− T ;

thus

(J T T T

ts

1 1 11 1 1 12 2 2 22 212

2= + +∞

∫ ~ ~ ~ ~ ~ ~ ~ ~ ~x Q x x Q x x Q x ) dt

and by Eq.(2.30), I obtain Eq.(2.29).

Q.E.D.

Next, I will prove the following theorem given by Utkin et al. 1979. First I need a definition of

"detectable" mentioned in this theorem.

Definition (Utkin 1977, Utkin 1978, Utkin 1992)

All the asymptotically stable systems and observable systems are detectable.

Theorem 2.5: Optimal State of Error-Optimal Sliding Mode If

~Q (positive definite), the pair (A, B) is stabilizable, and the pair 22 0> ( )~ ~ ,A A Q Q D11 12 22

112− − T is

detectable, where D D Q Q Q QT = − − T~ ~

11 12 221

12 (2.32)

then the optimal vector ~x2 can be found from

( )~ ~ ~ .~x Q A P Q2 221

12 12 1= − +− T T x (2.33)

where P is the unique non-negative solution of the Riccati equation

( ) ( )P A A Q Q A A Q Q P P A Q A P D D 0. ~ ~ ~ ~ . ~ ~11 12 22

112 11 12 22

112 12 22

112− + − − +− − −T T

TT T = (2.34)

Proof

Consider a linear quadratic regulator with its main results as follows

(

&x Ax Bu

xQx uRu

= +

= +

∞

∫Jt

12

0

) dt

i

then the optimal solut on can be found from

Optimal control function :

Steady - state Riccati equation :

u K x K R B P

PA A P PBR B P Q 0

* * *,= − =

+ − + =

RS|T|−

−

1

1

T

T T

From Eqs.(2.4.a) and (2.30), the system equation can be written as ( )~& ~ ~ ~ ~ ~ ~ ~ ~ ~ ~x A x A x A A Q Q x A v1 11 1 12 2 11 12 22

112 1 12= + = − +− T


By Eq.(2.29), the performance index is

( )[J T T T

ts

1 1 11 12 221

12 1 2212

= − +−∞

∫ ~ ~ ~ ~ ~ ~ ~x Q Q Q Q x v Q v] dt

hence based on the results in the linear quadratic regulator above, the optimal value of ~ *K s is

v K x

K Q A

* *

* P

~ ~

~ ~ ~= −

=

−

s

sT

1

221

12

(2.35)

then, from Eq.(2.30), I obtain Eq.(2.33).

Q.E.D.

2.4.2. Solution to the Optimal Hyperplane 1. The optimal hyperplane 1 can be determined by the following corollary

Corollary 2.4: Hyperplane Design by Error-Optimal Sliding Mode Method

The optimal hyperplane 1 can be found from

( )[ ]H Q A P Q I M= +− ×~ ~ ~

221

12 12T T (2.36)

where P is a solution of the following Riccati equation PA A P PB R B P Q 0~ ~ ~ ~ ~ ~

s sT

s s sT

s+ − + =−1

with ~ ~ ~ ~ ~

, ~ ~ ~ ~ ~ ~ ~, ~ ~

A A A Q Q B A Q Q Q Q Q R QsT

s sT

s= − = = − =− −11 12 22

112 12 11 12 22

112 22 ,

and

( )

& ,

~~ ~ ~

~ ~~ ~ ~

~ ~

x Ax Bu x Qx

MB0

BA MAM A A

A AQ M QM Q Q

Q Q

= + =

=

= =

= =

∞

− − −

∫

, ,

J dtT

t

T

s

1

2

1 11 12

21 22

1 1 11 12

21 22

12

.

Proof

Consider a hyperplane x M x= −1~

s = Hx

where H defined by Eq.(2.36), then in the sliding mode I have ( )[ ]{ } ( )[ ]s T T T T= ⇒ + × = + =− −0 022

112 12 22

112 12

~ ~ ~ ~ ~ ~ .Q A P Q I M x Q A P Q I Mx .

By ~x Mx= in Eq.(2.2), I have

( )[ ] ( )[ ]~ ~ ~.~

~ ~ ~.

~~Q A P Q I x Q A P Q Ixx22

112 12 22

112 12

1

2

0− −+ = +

=

T T T T

hence ( )~ ~ ~

.~ ~Q A P Q x x221

12 12 1 2 0− + +T T =

then the optimal vector ~x 2 in Eq.(2.33) is satisfied.

Q.E.D.


2.4.3. Error/Energy-Optimal Sliding Mode (Optimal Hyperplane 2) I will prove the following theorem in Utkin et al. 1979

Theorem 2.6: Hyperplane Design by Error/Energy-Optimal Sliding Mode Method

Consider a linear system

&x Ax Bu= +

and a performance index

J TeqT

eqts

212

= + dt∞z x Q x u R u. .d .i (2.37)

then the hyperplane can be determined by H H I M= %

1 × (2.38)

where ~H is determined by the following set of matrix equations 1

( ) [ ]

( ) ( )( )

( ) ( ) ( )

~& ~ ~ ~ ~ .~ , ~ , ~ .~ .

~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~ ~ ~ :

x A A H H x H I x W x P

H A A H A A H

W Q Q H H Q H S S B RB

A A H P P A A H W 0

1 11 12 21

1 1 2 2 1 1

1 11 21 1 12 22 1

11 12 1 1 22 1 21

21

11 12 1 11 12 1

12

12

2

= − = = =

= + − +

= − + + =

− + − + =

⇒

−∞

− −

∫

Lyapunov equation

J dt trT

t

T TT

T

s

Γ

Γ Γ,

~H1

with

( )MB0

BA MAM A A

A AQ M QM Q Q

Q Q=

= =

= =

− − −~~ ~ ~

~ ~~ ~ ~

~ ~2

1 11 12

21 22

1 1 11 12

21 22

, , T

Proof (a) First I prove the equality W Q Q H H Q H S= − + +

~ ~ ~ ~ ~ ~11 12 1 1 22 12 T TΓ Γ

I have ( ) ( )& &

~ ~& ~ ~& ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~s Hx H x H x H A H A x H A H A x H B u= = + = + + + +1 1 2 2 1 11 2 21 1 1 12 2 22 2 2 2

so the equivalent control is

( ) ( ) ( )[ ]u H B H A H A x H A H A xeq = − + + +−~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

2 2

1

1 11 2 21 1 1 12 2 22 2 (2.39) ~ From Eq.(2.39), as in the partition transformation, it is arbitrary to choose H I2 =

( ) ( )[ ]u B H A A x H A A xeq = − + + +−~ ~ ~ ~ ~ ~ ~ ~ ~2

11 11 21 1 1 12 22 2

with

( )S B RB= − −~ ~2

12

1T

(2.40)

then

( ) ( ) ( ) ( )( ) ( )

x Qx u Ru x Q H A A S H A A x x Q H A A S H A A x

x Q H A A S H A A x

TeqT

eqT

TT

T

TT

+ = + + +

+ + + +

+ + +

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~ ~ ~ ~ ~ ~

1 11 1 11 21 1 11 21 1 1 12 1 11 21 1 12 22 2

2 22 1 12 22 1 12 22 2

2 L

+

in the sliding mode I have s 0 x H x x x H= ⇒ = − ⇒ = −~ ~ ~ ~ ~ ~

2 1 1 2 1T T

1T


hence x Qx u Ru x WxT

eqT

eqT+ = ~ ~1 1 (2.41)

where

( ) ( )

( ) ( ) ( ) ( )W Q Q H H Q H H A A S H A A

H A A S H A A H H H A A S H A A H

= − + + + +

− + + + + +

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~11 12 1 1 22 1 1 11 21 1 11 21

1 11 21 1 12 22 1 1 1 12 22 1 12 22 1

2

2

T T

T T T

L

Let ( ) ( )Γ = + − +

~ ~ ~ ~ ~ ~ ~H A A H A A H1 11 21 1 12 22 1 (2.42)

so

( ) ( )Γ TT

TT

= + − +~ ~ ~ ~ ~ ~ ~H A A H H A A1 11 21 1 1 12 22

then

( ) ( ) ( ) ( )

( ) ( )Γ ΓT

T T

TT

S H A A S H A A H A A S H A A H

H H A A S H A A H

= + + − + +

+ + +

~ ~ ~ . ~ ~ ~ ~ ~ ~ . ~ ~ ~ ~

~ ~ ~ ~ . ~ ~ ~ ~1 11 21 1 11 21 1 11 21 1 12 22 1

1 1 12 22 1 12 22 1

2 L

where

( ) ( ) ( ) ( )~ ~ ~ ~ . ~ ~ ~ ~ ~ ~ . ~ ~ ~ ~H H A A S H A A H A A S H A A H1 1 12 22 1 11 21 1 11 21 1 12 22 1T

T T+ + = + +

therefore W Q Q H H Q H S= − + +

~ ~ ~ ~ ~ ~11 12 1 1 22 12 T Γ ΓT (2.43)

(b) Now I prove the Lyapunov equation ( ) ( )~ ~ ~ ~ ~ ~A A H P P A A H W11 12 1 11 12 1− + − +T

0=

The solution to this minimization problem for the performance index J2, if it exists, is found by applying

gradient matrix operations (Athans 1968) to the following Lagrangian (Mendel 1974)

[ ] ( ) ( )L = + − + − +

12

12 11 12 1 11 12 1tr tr ~ ~ ~ ~ ~ ~P A A H P P A A H W

TTL (2.44)

where L is an n matrix of Lagrange multipliers, then the necessary conditions n× ∂ ∂L ~H1 = 0 (2.45.a)

∂ ∂L L 0= (2.45.b)

∂ ∂L P 0= (2.45.c) By ∂ (Appendix), from Eqs.(2.44) and (2.45.b), I obtain the following Lyapunov equation ∂tr TAX X A=

( ) ( )~ ~ ~ ~ ~ ~A A H P P A A H W11 12 1 11 12 1− + − +T

0= (2.46)

Q.E.D.


2.4.4. Solution to the Optimal Hyperplane 2 Based on Eqs.(2.45.a) and (2.45.b), I will prove the following theorem on the gradient of the

performance index, because this gradient will be used to solve the problem for an optimal hyperplane by the

BFGS numerical method.

Theorem 2.7: Gradient of Performance Index in Error/Energy-Optimal Sliding Mode

The gradient of the performance index is determined by

( ) ( ) ( )∂

∂

J T TT T

2

112 12 22 1 11 12 1 1 12 22~

~ ~ ~ ~ . . . . ~ ~ ~ ~ ~ ~ . . .H

A P Q Q H L S L A A H H A A S L= + − − − + +Γ Γ (2.47)

Proof

Rearrange Eq.(2.44), I have

[ ] ( ) ( )[ ] [L = + −

+ − +1

212 11 12 1

12 11 12 1

12tr tr ~ ~ ~ tr ~ ~ ~ trP A A H PL P A A H L WL

TT T ]T

By the properties of the trace of a matrix (Appendix), I obtain

[ ] ( ) ( )[ ] [L = + −

+ − +1

212 11 12 1

12 11 12 1

12tr tr ~ ~ ~ tr ~ ~ ~ trP L A A H P A A H L P WLT

TT ]T

by ∂ ∂tr X X = I and ∂ ∂tr AX X A= T (Appendix), from Eq.(2.45.c), I obtain the following Lyapunov

equation ( ) ( )~ ~ ~ . . ~ ~ ~A A H L L A A H I11 12 1 11 12 1− + − + 0= (2.48)

First, by the properties of the trace of a matrix (Appendix) and by P P L LT T= =; , I rearrange

Eq.(2.44) to facilitate the operations [ ] [ ] [ ] [ ]L = + + − +J T T T T T T T

212 11

12 11

12 12 1

12tr ~ tr ~ tr ~ ~ trA PL PA L A PL H WL (2.49)

thus by Eq.(2.45.a), and by ∂ (Appendix), I obtain ∂tr TAX X A=

[∂

∂∂

∂

JtrT2

112

1

12~

~ . ~ .H

A P LH

W L= − ]T

by Eq.(2.43), I have

( )[ ]∂

∂∂

∂

JtrT T2

112

111 12 1 1 22 1

12

2~~ . ~

~ ~ ~ ~ ~ ~ .H

A P LH

Q Q H H Q H S L= − − + + Γ ΓT T

or

[ ] [ ] [ ] [ ]∂

∂∂

∂∂

∂∂

∂∂

∂

Jtr tr tr trT T T2

112

111

112 1

122 1 1

1

12

12

12~

~ . ~~

~~ ~

~~ ~ ~

~HA P L

HQ L

HLQ H

HQ H LH

HS L= − + − − Γ ΓT T (2.50)

By ∂ and ∂tr TAX X A= ∂ ∂ (Appendix), I obtain tr T T TAXBX X A XB AXB= +

( ) [ ] ( ) [ ]∂

∂

∂

∂

∂

∂

Jtr trT T T T T T T T T T2

112 12 22 1 22 1

112 12 22 1

1

12

12

12~

~ .~ ~ ~ ~ ~

~~ ~ ~ ~

~HA P L Q L Q H L Q H L

HS L A P Q Q H L

HS L= + − + − = + − −Γ Γ Γ Γ

By Eq.(2.42), to facilitate the operations, let X H=

~1 , I obtain

~ ~ ( ) ( )Γ = + − +XA A XA A X~ ~11 21 12 22 (2.51)

thus I have to compute ∂∂X

S Ltr TΓ Γ. . T , where Γ by Eq.(2.51).

Let

( ) ( ) ( ) ( )[ ] ( ) ( )[ ]f T TT

TX S L XA A XA A X S XA A XA A X L= = + − + + − +

tr trΓ Γ. . ~ ~ ~ ~ . . ~ ~ ~ ~ .11 21 12 22 11 21 12 22 (2.52)

and by the differentiation technique in Geering 1976 (Appendix), I have


( ) ( ) ( )[ ] ( ) ( )[ ]f d d dT

T T TX X XA A XA A X S L S XA A XA A X L, ~ ~ ~ ~ . . . . . ~ ~ ~ ~ .= + − + + + − +

tr 11 21 12 22 11 21 12 22Γ Γ

where ( ) ( )[ ] ( ) ( )d dXA A XA A X X A A X XA A Xd~ ~ ~ ~ . ~ ~ ~ ~ .11 21 12 22 11 12 12 22+ − + = − − +

thus

( ) ( ) ( )

( ) ( )[ ]f d d d

d d

TT T T

TT

T T T

X X A A X X S L X XA A S L

S X A A X L S XA A X L

, ~ ~ . . . . ~ ~ . . .

. . . ~ ~ . . . ~ ~ . .

= − − +

+ − − +

tr

tr

11 12 12 22

11 12 12 22

Γ Γ

Γ Γ

L

T (2.53)

By the Frechet derivative operator in Geering, 1976 (Appendix), I have

( ) ( ) ( )

( ) ( )

f d d d

d d

T TT

TT

T

T TT

TT

T

X X X S L A A X X XA A S L

X S L A A X X XA A S L

, . . . ~ ~ ~ ~ . . .

. ~ ~ . ~ ~ . .

= −

− +

+ −

− +

tr tr

tr tr

Γ Γ

Γ. . Γ.

11 12 12 22

11 12 12 22

L

(2.54)

thus

[ ] ( ) ( )∂∂X

S L S L A A X XA A S Ltr T TT

Γ Γ Γ Γ. . . . . ~ ~ . ~ ~ . . .= − − +2 211 12 12 22

T

L

(2.55)

where S S LT T= =, Therefore, by X H=

~1 , I achieve

( ) ( ) ( )∂

∂

J T TT T

2

112 12 22 1 11 12 1 1 12 22~

~ ~ ~ ~ . . . ~ ~ ~ ~ ~ ~ . . .H

A P Q Q H L S L A A H H A A S L= + − − − + +Γ Γ (2.56)

where L is determined by Eq.(2.48).

Q.E.D.

2.4.5. Numerical Algorithm for Solution to Optimal Hyperplane 2 I now re-arrange the relevant results in Theorems 2.6 and 2.7, and in their proofs, into the sequence to

compute the gradient of the performance index to be used in the next BFGS numerical algorithm.

&x Ax Bu= +

J TeqT

eqts

212

= + dt∞z x Qx u Rud i (2.37)

( )S B RB= − −~ ~2

12

1T

(2.40)

where

( )MB0

BA MAM A A

A AQ M QM Q Q

Q Q=

= =

= =

− − −~~ ~ ~

~ ~~ ~ ~

~ ~2

1 11 12

21 22

1 1 11 12

21 22

, , T

~with ~H1 known, then ∂ ∂J2 H is calculated as follows 1

( ) ( )Γ = + − +~ ~ ~ ~ ~ ~ ~H A A H A A H1 11 21 1 12 22 1 (2.42)

~ ~ ~ ~ ~ ~ W Q Q H H Q H S= − + +11 12 1 1 22 12 T Γ ΓT (2.43) ~ ~ ~ ( ) ( ) ( )~ ~ ~ . .A A H L L A A H I 0 L11 12 1 11 12 1− + − + = ⇒ Lyapunov equation (2.48)

( ) ( ) ( )~ ~ ~ ~ ~ ~A A H P P A A H W 0 P11 12 1 11 12 1− + − + = ⇒T

Lyapunov equation (2.46)

( ) ( ) ( ) ( )∇ = = + − − − + +JJ T T

T T

2 12

112 12 22 1 11 12 1 1 12 22

~~

~ ~ ~ ~ . . . ~ ~ ~ ~ ~ ~ . . .HH

A P Q Q H L S L A A H H A A S L∂

∂Γ Γ (2.47)


Having obtained the gradient of the performance index, I can use the following algorithm for a solution

to design an optimal sliding hyperplane 2. In general, the BFGS algorithm requires a line-search procedure

to eliminate the calculation of a Hessian (second-order derivative matrix), but the general line-search

algorithm cannot be used in my problem due to the problem of singularity. The line-search in Kuhn et al.

1987 may be used instead, since it guarantees the solubility of singular Lyapunov equations.

The Algorithm BFGS (Fletcher 1980) with Line-Search Algorithm (Kuhn et al. 1987) Step 1 : Initialize

Hs I B

H A A P P A

1

11

11 12 11 12

= ×

=−n m

h h

n m

place

,

, , , ,

Initial Hessian matrix, :

where : any Hurwitz Eigenvalues to stabilize b g b A g Step k :

( )Sd H Hs= −∇ ×J k2 1

k~ ~ ( ) ( )J Jk

2 1 0 2 1k~ min*H Sd H+ × = + ×

>α

αSdα : Line Search,

~ ~ , *H H S12

1k k= + ×α d ~ ~ ∆H H H Sd= − = ×, *

12

1k k α

~ ~ ( ) ( )∆G H H= ∇ − ∇J Jk2 1

22 1

, k

Hs Hs Hs G H H G HsH G

G Hs GH G

H HH G

k kk T T k

T

k T

T

T

T+ = −

++ +

1 1. . . .

.. .

..

.∆ ∆ ∆ ∆

∆ ∆∆ ∆∆ ∆

∆ ∆∆ ∆

Terminate H H H 0 ∆ = − ≈~ ~,

12

1k k (Consecutive values)

2.4.6. Proposition 2.2: Selection of Weighting Matrices Q and R For a weighting matrix Q n , I propose the following form ∈ℜ ×n

Q =

qq

q

1

2

2

0 00 0

0 0

L

L

M M O M

L

because I are concerned with the output state x1 as an error, regardless the other states. The more the error is

minimized, the faster the system is stabilized.

For a weighting matrix R, in the case of SISO

R = r1

I propose 3 cases: (1) q q r1 2 11 0= = =, 1. : the error is minimized only, regardless of the control effort;

(2) q q r1 2 10 1 1= = =. , : the control effort is minimized only, regardless of the error;

(3) The intermediate case to compromise between the error and the control effort.


Start : Line Search Algorithm

α αi i, ,3 2= α αi i, ,1 2=

αi ,0 0=

α α αi i, , ,2 1 1 3=

α αi ih,1 = α αi i,*

2 =

α αi i,*

0 =

α αi i, ,,1 3 : found

αi ,2 : found

αi* found :

α αi i, ,0 010=

α αi i, ,3 010= ?

α αi i, ,1 0=α αi i, ,2 0=

α αi i, ,.1 00 1= ?

α αi i, ,.0 00 1=

α αi i, ,3 0=

α α αi i i, ,;0 1 12 0= =−

( ) ( )[ ] ( ) ( )

( ) ( )

( ) ( )( ) ( )2,1,1

2,211,2,2,

*

2,1,212

2,1,1,2,

2,1,1

2

.

3

iiiii

iiiiiiii

iiiiii

iiiiii

iiiii

ffhfhh

ffhh

ffff

h

α∇+α∇+

α∇+−α−α−α=α

α∇α∇−=

α∇+α∇+α−α

α−α=

αα

α

αα

αα

Stop

Yes No

Yes No

Yes No

YesNo

Yes No

Yes No

Yes

No

No Yes

( ) 02, >α∇ α iif

( ) ( ) ( ) 0*11

* ≈α∇+α∇ −α−α iiii ff

( ) ?0* >α∇ α iif

( ) ? unstable"" 0,iif α

( ) 00, >α∇ α iif

( ) ? unstable"" 2,iif α


2.5. NORMALIZATION OF A HYPERPLANE EQUATION Throughout all the hyperplane design methods, for both eigenvalue allocation and optimal approaches,

one coefficient of a hyperplane may be freely chosen. This section concerns a normalization technique that

greatly simplifies the sliding mode control function.

So far, except in the direct calculation method of a hyperplane, without loss of generality, for SISO

system it has arbitrarily assumed that H H H H2 1 2 11= = ⇒ 1 1= = −h hn , , ,L hn , . (2.57)

If HB = ≠α 0

then, from Eq.(2.57), a normalization of a hyperplane can be found from H H= = −α α αh hn1 1 1, , ,L α

so HB H B HB= =αb g a f. =α 1 (2.58)

and the hyperplane equation is unchanged since

s h x h x x h x h xn n n n n n= = ⇒ + + = ⇒ + + =− − − −H x. 0 0 1 01 1 1 1 1 1 1 1L L xα

b g .

It will be shown in the following chapters that a sliding mode control function always includes the following

terms HB HBa f− = ⇒ =1 1 s s (2.59)

Similarly, I have the normalization of a hyperplane for a MIMO system as HB I sHB sa f− = ⇒ =1 (2.60)

thus, it is convenient to normalize a hyperplane as above.

2.6. NUMERICAL EXAMPLES This section relates to numerical examples that illustrate all of the above issues. The hyperplane design is

the first stage of the SMC design while the controller design is the second. In this chapter there is no control

function, hence there is no simulation for assessment of the different approaches to designing a hyperplane.

This assessment is postponed to the next chapter onwards. I have built a SMC toolbox in the MATLAB

language and all the following hyperplanes are determined by the corresponding program in that toolbox.

Remark 2.9: Summary of Hyperplane Designs

In the numerical examples below, and in this work generally, hyperplane eigenvalues will be chosen at

the same value for simplicity except in case of the optimal sliding mode approach. This matter will be

discussed in the following chapters on designing sliding-mode controllers.

Recall that 2 hyperplanes which are different by a scale factor produce the same sliding dynamics.

• Corollary 2.1 is used to compute a hyperplane for general cases;

• Corollary 2.2 is used to compute a hyperplane for canonical systems;

• Corollary 2.3 is used to compute a hyperplane when the system matrix A has its determinant different

from zero.

• Theorem 2.4 and Proposition 2.1 are employed to compute a hyperplane using I/O technique.


2.6.1. Example 2.1: Direct Allocation Consider a canonical system in Lee et al. 1991

. & .x x=−

+

0 1 00 0 10 1 0

00

10u.

Choose hyperplane-eigenvalues as λH = − −3 3,

so the characteristic equation is . p p p+ = + +3 62 2a f 9

Thus by Corollary 2.2, the hyperplane is defined as s = =H x H. , , , 9 6 1 ;

or, by the normalization technique, the above hyperplane becomes H = 0 9 0 6 0 1. , . , . .

2.6.2. Example 2.2: Partition Transformation Consider a system in Spurgeon 1991

. &

. . .. . . .x =

− − −− −

−

+

0 277 32 98 5 4320 365 0 319 9 49

0 0 5

00

10u.x

Choose hyperplane-eigenvalue as λH = − −3 3,

by the partition transformation method. First I have to determine a transformation matrix> Since the matrix B

is in the required form, I have the following transformation matrix

. M I= =

3

1 0 00 1 00 0 1

Then by Corollary 2.1, the hyperplane is computed as H = − −0 0018 0 0559 0 1. , . , . .

Notice that A is non-singular, so that I can apply the direct calculation method by Corollary 2.3 without

the partition transformation above.

2.6.3. Example 2.3: Direct Calculation Consider a system in Sivaramakrishnan et al. 1984

& ..

.x Ax B x= + =

−−

− − −

+

u u

0.05 6 0 00 3.333 3.333 0

5.208 0 12.5 12.50.6 0 0 0

00

12 50

If I use the partition transformation method, I have to determine the transformation matrix M since the

matrix B is not in the required form. Because the matrix A is non-singular, I can apply the direct calculation.


Choose hyperplane-eigenvalues as λH = − − −2 2 2, , ,

then by Corollary 2.3, the hyperplane is computed as H = 0 0468 0 0628 0 08 0 0533. , . , . , . .

2.6.4. Example 2.4: I/O-State Technique for Linear Systems Consider the modified system for the same system above in Spurgeon 1991 so that the relative degree is

equal to the system order of 3.

&

. .. . . .x Ax B

0x= + =

− −− −

−

+

u u0 277 32 98

0 365 0 319 9 490 0 5

00

10. where Ar1 0 277 32 98 0= − −. , . ,

then n r= =3; and ⇒3 choose a hyperplane-eigenvalue as usual by Proposition 2.1, it is not necessary to

check the stability of the sliding-eigenvalue [ ] [ ]λ H = − − ⇒ =3 3 9 6 1, ~ , ,H .

Notice that %H is computed via the direct allocation method in Corollary 2.2 due to the canonical form. Then

by the I/O-state technique using Theorem 2.4, a hyperplane is H = −0 0015 0 0569 0 1. , . , . .

2.6.5. Example 2.5: I/O-State Technique for Nonlinear Systems Consider a nonlinear system in Zhou et al. 1992

& . ,

sin( ),

x f g

f g

= + =

=+ +

= +

u y x

xx x x tx x

1

2

1 2 12

1 120

1.

I have y x= 1

& &y x x= =1 2

&& & sin( ) .y x x x x tx x u r= = + + + + ⇒ =2 1 2 12

1 12 1 2

g

: relative degree

since n=2, the relative degree r is equal to the system order n, I can apply the I/O-state method to design a hyperplane in the coordinates b g . Choose the hyperplane-eigenvalue as y y x x, & ,b= 1 2

λH = −2

then, by Corollary 2.2, the hyperplane is determined as [ ]H = 2 1, .


2.6.6. Example 2.6: Error-Optimal Sliding Mode Consider a system in Sivaramakrishnan et al. 1984

& ..

.x Ax B x= + =

−−

− − −

+

u u

0.05 6 0 00 3.333 3.333 0

5.208 0 12.5 12.50.6 0 0 0

00

12 50

first I must find a partition transformation matrix as

M =

1 0 0 00 1 0 00 0 0 10 0 1 0

for the weighting matrices, choose

Q =

1 0 0 00 0 1 0 00 0 0 1 00 0 0 0 1

..

.

thus by Section 2.4.2, the optimal hyperplane 1 is H = 0 2597 0 2160 0 08 0 08. , . , . , . .

2.6.7. Example 2.7: Error/Energy-Optimal Sliding Mode Consider a system in Sivarakrishnan et al. 1984

& ..

.x Ax B x= + =

−−

− − −

+

u u

0.05 6 0 00 3.333 3.333 0

5.208 0 12.5 12.50.6 0 0 0

00

12 50

first I must determine a transformation matrix, then choose the weighting matrices

M =

1 0 0 00 1 0 00 0 0 10 0 1 0

, , Q =

10 0 0 00 10 0 00 0 0 1 00 0 0 0 1

..

R = 1

thus by Section 2.4.5, the optimal hyperplane 2 given by the BFGS numerical method H = 0 04 0 0649 0 08 0 0543. , . . , . .


2.7. CONCLUSIONS The controller has been not mentioned in the hyperplane design, so the hyperplane design procedure is

independent of the controller implementation. It will be shown in the following chapters that there are

numerous different types of controller implementation, such as VSS (discontinuous control function),

pseudo-VSS (continuous control function), continuous SMC. Particularly, the I/O-state method is applicable

to design a hyperplane for both linear and nonlinear systems.

The partition method is general but it is necessary to determine a partition transformation. To simplify the

design task, a direct allocation method has been developed for canonical systems, and a direct calculation

method for systems of non-singular matrix A. It will not be necessary to determine a partition

transformation in these 2 proposed methods. Most control problems may be classified in 2 cases: either the

system matrix A is singular and canonical or it is nonsingular and noncanonical. In fact, all my numerical

examples in this work have used these 2 methods. Moreover, using the direct allocation method, I can see the

hyperplane is independent of the system parameters, because a canonical system satisfies the matching

invariance condition (Drazenovic 1969).

Note that the optimal hyperplane design is optimal in sliding dynamics, not optimal in system dynamics.

By the nature of SMC design that the invariance property is in the sliding mode, therefore the more dominant

is the sliding mode, the more robust is the controller.

Based on an arbitrary choice of H2 for all hyperplane designs in the eigenvalue allocation and in the

optimal sliding mode approaches, I have proposed a normalization of a hyperplane that will greatly simplify

the sliding mode control function, cf. Eq.(2.59).

I have identified 2 types of eigenvalues: hyperplane-eigenvalues and sliding-eigenvalues to clarify the

mechanism of the sliding-mode in SMC, especially for stability test and for invariance condition. The

sliding-eigenvalues represent the sliding dynamics of the system state variables. The hyperplane-eigenvalues

are the desired sliding-eigenvalues.

( )[ ] [ ]

( ) [ ]

eig

eig

H

S

A B HB HA

A B HB HA

− =

−

=

−

−

1

1

0

0

,

~ ~ ~ ~ ,

λ

λ

where A, B are nominal values of uncertain dynamical system matrices A . % , %B

Note that for the sake of connectivity, the direct allocation method has been proved to be a special case of

the partition transformation method for canonical linear systems. This direct allocation method is still

applicable for canonical nonlinear systems by the modified proof in Chapter 6. The proof in Chapter 6 is

valid for both canonical linear and nonlinear systems.


For the SMC general case where the system may be MIMO nonlinear and the outputs may be some

nonlinear functions of the system state variables, a hyperplane design has been developed on the basis of the

input/output linearization technique (Hunt et al. 1983, Kravaris et al. 1986, Isidori 2ed. 1989) as in the

current SMC and VSS literature (Fernandez et al. 1987, Sira-Ramirez 1989, Chen et al. 1992, Spurgeon et

al. 1993, Edwrads et al. 1995). Based on the direct allocation method, I have designed a hyperplane for a

SMC general case via the I/O-state technique applicable for both linear and nonlinear systems.


Chapter 2 Sliding Hyperplane Design

2.1. Introduction. ................................................................................................................................................1

2.2. Sliding Dynamics ........................................................................................................................................2

2.2.1. Sliding Dynamics Equation .............................................................................................................2

Theorem 2.1: Sliding Dynamics Equation ................................................................................2

2.2.2. Sliding Eigenvalue...........................................................................................................................3

Theorem 2.2: Sliding Eigenvalues ............................................................................................3

Remark 2.1: Relationship between Sliding Eigenvalues and Hyperplane Eigenvalues............4

Remark 2.2: Hyperplane is of Reduced-Order..........................................................................4

2.3. Eigenvalue Allocation .................................................................................................................................5

2.3.1. Full-State Technique........................................................................................................................5

2.3.1.1. Partition Transformation Method .........................................................................................5

Corollary 2.1: Hyperplane Design by Partition Transformation Method .................................5

2.3.1.2. Direct Allocation Method.....................................................................................................6

Corollary 2.2: Hyperplane Design by Direct Allocation Method .............................................6

Remark 2.3: Application of Direct Allocation Method.............................................................7

Remark 2.4: Extensibility of Hyperplane Design by Direct Allocation Method to Nonlinear

Systems .....................................................................................................................................7

2.3.1.3. Direct Calculation Method ...................................................................................................7

Corollary 2.3: Hyperplane Design by Direct Calculation Method............................................7

Remark 2.5: Hyperplane Normalization in Direct Calculation Method....................................8

Remark 2.6: Dimension of Sliding Variable and Extensibility to MIMO Systems ..................8

2.3.2. Canonical Transformation ...............................................................................................................8

Theorem 2.3: Computation of Canonical Transformation Matrix ............................................8

2.3.3. I/O-State Technique.........................................................................................................................9

Theorem 2.4: Hyperplane Design by I/O-State Technique.......................................................9

Remark 2.7: Extensibility of I/O-State Technique to Nonlinear Systems...............................10

Remark 2.8: Decomposition of I/O-State Technique..............................................................10

2.3.4. Proposition 2.1: Application of I/O-State Technique ....................................................................11

2.4. Optimal Sliding Mode ...............................................................................................................................11

2.4.1. Error-Optimal Sliding Mode (Optimal Hyperplane 1)...................................................................11

Lemma 2.1: Performance Index of Error-Optimal Sliding Mode ...........................................11

Theorem 2.5: Optimal State of Error-Optimal Sliding Mode .................................................12

2.4.2. Solution to the Optimal Hyperplane 1. ..........................................................................................13

Corollary 2.4: Hyperplane Design by Error-Optimal Sliding Mode Method .........................13

2.4.3. Error/Energy-Optimal Sliding Mode (Optimal Hyperplane 2) ......................................................14

Theorem 2.6: Hyperplane Design by Error/Energy-Optimal Sliding Mode Method..............14

2.4.4. Solution to the Optimal Hyperplane 2 ...........................................................................................16


Theorem 2.7: Gradient of Performance Index in Error/Energy-Optimal Sliding Mode .........16

2.4.5. Numerical Algorithm for Solution to Optimal Hyperplane 2 ........................................................17

2.4.6. Proposition 2.2: Selection of Weighting Matrices Q and R...........................................................18

2.5. Normalization of a Hyperplane Equation..................................................................................................20

2.6. Numerical Examples .................................................................................................................................20

Remark 2.9: Summary of Hyperplane Designs.......................................................................20

2.6.1. Example 2.1: Direct Allocation .....................................................................................................21

2.6.2. Example 2.2: Partition Transformation..........................................................................................21

2.6.3. Example 2.3: Direct Calculation....................................................................................................21

2.6.4. Example 2.4: I/O-State Technique for Linear Systems .................................................................22

2.6.5. Example 2.5: I/O-State Technique for Nonlinear Systems............................................................22

2.6.6. Example 2.6: Error-Optimal Sliding Mode....................................................................................23

2.6.7. Example 2.7: Error/Energy-Optimal Sliding Mode .......................................................................23

2.7. Conclusions ...............................................................................................................................................24

Chapter 3

A New Variable-Structure System Controller Designs

3.1. INTRODUCTION

In this chapter, a VSS controller design will be fully developed in a unified manner that is extendable into

robust control (Chapter 4) and MIMO nonlinear systems (Chapter 6) where a VSS control is a discontinuous

sliding mode control (SMC). Based on the concept of sliding-eigenvalues introduced in Chapter 2, we

propose a stability criterion which is much simpler than the current approach. By the sliding condition

Eq.(1.4), a control function is first computed in a standard form as in Utkin 1977. I then propose a technique to

convert this form into a compact form to facilitate controller implementations. The sliding condition is a

necessary condition to guarantee the existence of the sliding mode. I propose a sufficient condition as a SMC

design rule that the reaching mode must terminate in a finite time so that the sliding mode can exist, because

the invariance property is with the sliding mode only. A new reaching control design is proposed to include

system dynamics for efficient SMC designs; this is not the case in the current literature.

Alternative to the chattering problem analysis in Slotine et al 1983, I propose an analysis that allows a

unified approach for both the saturate (Slotine et al. 1983) and unitvector functions (Ryan et al. 1984,

Spurgeon 1991). In addition, the hyperbolic tangent function is proposed to cope with limitations of saturate

and unitvector functions. Since using the saturate, unitvector and hyperbolic tangent functions may cause a

steady-state error, a design scheme (Proposition 3.3) is proposed to solve this problem (Example 7.6).

A VSS control is based on a state-space model, so an I-action may be required to eliminate a steady-state

error. In Chern et al. 1991, an integral of error has been used, however it is a pseudo-SMC because the larger

controller gain the closer the sliding mode. In Chang 1991, an integral sliding condition has been proposed

using the same system order, however this approach has some limitation with steady-state error.

Alternatively, I propose an integral VSS control by first augme nting the order of a system model. The

controller is then designed in a unified manner as in the normal case without integral action. This can solve

the problem of pseudo-SMC in Chern et al. 1991 and cope with limitation in Chang 1991 (Example 3.11).

3.2. SMC STABILITY CRITERION

To the best of my knowledge, there has been a little attention paid to the stability problem in the VSS

literature. There are only 2 works in Itkis 1976 and Utkin 1977 where a stable sliding mode implies a stable

closed-loop system. These two works are the same approach to deal with this problem. They are based on

numerous complicated theorems and hence are not convenient to apply for a practical stability test. Also in

these 2 works above, a controller design has been based on (n−1) system states for a n-ordered system, so

there is always one constraint while the hyperplane still does include all the n-states, hence there is is no

point to be constrained.


Alternatively, I analyze this stability problem based on the mechanism of the SMC, then I propose a

much simpler stability criterion as a practical test for SMC.

Theorem 3.1: SMC Stability Criterion

Consider a SISO linear system

& . .x A x B= + u

where

x B A, , ,∈ℜ ∈ℜ ∈ℜ× ×n n n u1

if a control function u is designed to satisfy the sliding condition

s s. & < 0 (3.1)

and if there exists a stable sliding mode

( )[ ] { } ( )( )λ Sn= − − ⊂ ℜ−

−× −eig A B HB HA1 1 10 (3.2)

then the system is stable.

Proof:

I have

s sddt

s s. & < ⇒ < ⇒ →0 0 02c h

since s 2 must monotonically reduce to zero as s 2 0> .

The sliding mode is characterized by the sliding-eigenvalues determined by Eq.(2.9). If these eigenvalues are Hurwitz-stable, then ~x1 converges to zero by Eq.(2.14), as does ~x2 by Eq.(2.7). By the similarity

transformation in Eq.(2.2), x x x= 1 2T converges to zero: the system is stable.

Q.E.D.

Remark 3.1: Stability Test

• All that a SMC has to do is to satisfy the sliding condition s s. & < 0. How negative this value may be is

based on the SMC design rule in Proposition 3.1 in Section 3.4.

• This stability test is particularly useful and frequently employed in robust SMC in the next chapter.

3.3. A NEW VSS CONTROLLER DESIGN

In the VSS literature, a control function is either in the standard if-form (Slotine et al. 1983, White et al

1984, Sivaramakrishnan et al 1984, Panicker et al 1985, Baley et al. 1987, DeCarlo et al. 1988, Hikita 1988,

Hong et al 1989) or in the compact form (Young 1978, Ryan 1983, Ambrosino et al. 1984, Fernandez et al.

1987, DeCarlo et al. 1988, Lee et al 1991). The derivation of the compact form has been either too

complicated to be applied (Ryan 1983) or not clear how to derive (Young 1978, Ambrosino et al. 1984, Lee et

al 1991). In Fernandez et al. 1987, the derivation has been based on the input/output linearization technique

for nonlinear systems without perturbation. In DeCarlo et al. 1988, the derivation is valid only for linear

system without perturbation.

3. VSS Controller Design 3.3

The standard if-form in Utkin 1977 is a convenient form to write a control function that satisfies the

sliding condition. I first propose the following lemma to simplify the standard if-form of a VSS control

function.

Lemma 3.1: Compact Control Function

For a switching control function of the following standard if-form

u xx

xi i

i

n

ii i

i i

= =>

<

RS|T|=

+

−∑Ψ Ψ1

0

0,

, .

, .

ψ σ

ψ σ

if

if (3.3)

where σ can be any scalar function, then the if-form control function above becomes the compact form as

follows

u x xi ii

i ii

i

n

=+

+−L

NMOQP

+ − + −

=∑ ψ ψ ψ ψ

σ2 21

.sgn a f . (3.1)

Proof

Let

Ψi ii i

i ii i i

x

xx= +

+ >

− <RST

UVW = +ψψ σ

ψ σψ ψ σ0

1

10 1

0

0

,

,.

if

if sgnb g

then

ψ ψ ψ

ψ ψ ψ

ψψ ψ

ψψ ψ

i0

i0

+ =

− =

UV|W| ⇒=

+

=−

RS||T||

+

−

+ −

+ −

i i

i i

ii i

ii i

1

1

0

1

2

2

thus

u x x x x x x xi i i ii

n

i i i i ii

n

i i i ii

n

= + = + = += = =

∑ ∑ ∑ψ ψ σ ψ ψ σ ψ ψ σ0 11

0 11

0 11

. . . . .sgn sgn sgnb g b g a f .

Q.E.D.

Remark 3.2: Simplification of Control Function By Hyperplane Normalization

It is usual that σ = sHB , and σ = s if the hyperplane is normalized (Section 2.5).

For a switching VSS controller design, I propose the following theorem

Theorem 3.2: Switching VSS Controller Design

For a linear system

& . .x A x B= + u

with a hyperplane

s = Hx

then a VSS control function can be combined as u u ue r= + (3.4)

where

• equivalent control

ue e e= − = −K x K HB HA, a f 1 (3.4.a)


• reaching control u sr r r n= − =−HB K x Ka f a f1

1. . ,sgn δ δL (3.4.b)

with x B A, , , , ,∈ℜ ∈ℜ ∈ℜ ∈ℜ× ×

+n n n

iu s1 δ a f: sliding margin.

Proof

I will check the sliding condition, from Eq.(1.8) I have & .s u ueq= −HB d i

where

u xeq i ii

n

= − =−

=∑HB HAxa f 1

1

α , where αi are determined by the system parameters

and

u xi ii

n

==∑ψ

1

, where ψi are to be determined

thus

ss s u u s xeq i i ii

n

& = − = −=∑HB HBd i b gψ α

1

To satisfy the sliding condition, I obtain

sss x

s x

s x

s xi

i i

i i

i

i i i

i i i

i& ,< ⇒< >

> <

RS|T|⇒ =

− >

<

RS|T|

>00

0

0

00ψ

α

αψ

α δ

α δδ

, if

, if

, if

+ , if

HB

HB

HB

HB

a fa f

a fa f

by Lemma 3.1, I have ( )ψ α δi i i i i ix x x s= − . .sgn HB

then

( ){ } ( ) ( ) ( ) ( )u x x s x s x si i i ii

n

i ii

n

i ii

n

= − = − −

= − +

=

−

=

−

=∑ ∑ ∑α δ δ δ. .sgn .sgnsgn HB HB HAx HB HB HAx

1

1

1

1

1

where δ δi i= >. HB 0 for the sake of simplicity. Q.E.D.

To design a VSS controller using Theorem 3.1, the reaching control constant vector Kr needs to be

determined. For an n-order system, there are n design parameters δ i to be determined. The following two

corollaries propose designs using only 1 design parameter δ.

Corollary 3.1: Conventional Reaching Controller Design

The reaching control in Theorem 3.1 can be determined with

K r = δ δL (3.4.c)

where δ > 0 is a s liding margin.

Proof

As in the literature, I can choose δ δi = since the sufficient condition for the existence of a sliding mode

is δ i > 0 as in the proof of Theorem 3.1.

Q.E.D.


In the previous approach, the system dynamics are not included in the reaching mode whose dynamics

totally depend on the system dynamics as the same sliding margin is used for all system states in all systems

(slow and fast systems). The following corollary proposes a more efficient design where the system dynamics

are included in the reaching control (Example 3.5)

Corollary 3.2: New Reaching Controller Design

The reaching control in Theorem 3.1 can be determined with

K Hr = δ. (3.4.d)

where δ > 0 is a sliding margin.

Proof

All elements of H are associated with all system states, and H is designed based on the system matrices A and B. To included the system dynamics, choosing δ δi ih= . satisfies the sufficient condition of δ i > 0 as

in the proof of Theorem 3.1 since hi are coefficient of the system differential equation.

Q.E.D.

Remark 3.3: Sliding Mode as Fast Mode for Invariance

The sliding margin δ determines the reaching mode. It may be high for the required faster reaching mode

as opposed to the sliding mode. So in the literature, the reaching mode is also called the fast mode, and the

sliding mode is called the slow mode. By the nature of VSS design the invariance property is in the sliding

mode, therefore a longer sliding time (Section 3.4) implies more robust controller.

3.4. PROPOSITION 3.1: SMC DESIGN RULE

From Remark 3.3, for a proper reconciliation between the reaching mode and the sliding mode, I propose

the following design rule applicable to all systems (linear or nonlinear, SISO or MIMO) and all approaches

(continuous or discontinuous SMC or continuous pseudo-SMC, continuous-time or discrete-time SMC,

and conventional or integral SMC)

s

y

s

y

tR tStT

t tT R=t tS R<<

Fig. 3.1: SMC Design Rule

where

s: sliding variable from Eq.(1.2)

y: system output

t t tT R S= + , tT : total time, tR : reaching time s s. & < 0a f, tS : sliding time s = 0a f In the figure above, there is a sliding mode in the first case, but practically not in the second.


Response Time = Reaching Time(higher δ → lower tR ) + Sliding Time(higher p1 → lower tS ).

where p1 is the magnitude of the dominant hyperlane eigenvalue.

• If t t tS S R≈ <<0 :b g , then increasing p1 does not make the response faster but does increase the control

effort only; • Otherwise, if t t tR R S≈ <<0 :b g, then increasing δ does not make the response faster but does increase

the control effort only .

So I propose the following design rule that will be effective in both simulation and experiment.

SMC Design Rule: Ensure tS<tR, but not t t tS S R≈ <<0 :b g and not t t tR R S≈ <<0 :b g .

3.5. SOLUTION TO CHATTERING PROBLEM

In the VSS literature, there are 2 approaches to eliminate the chattering by using the saturate function

(Slotine et al 1983) and the unitvector function (Ambrosino et al. 1984, Spurgeon 1991). The analysis of

chattering in Slotine et al. 1983 raises 2 questions. Firstly why this chattering still exists even in a simulation

where the switching controller is practically free from imperfection. Secondly, based on this analysis, it is not

clear how to choose the width of the boundary layer. The elimination of chattering is implied by using the

unitvector function in Ambrosino et al. 1984 and Spurgeon 1991 because this function is continuous. The

derivation of results that follow from the choice of this function is either unavailable (Ambrosino et al. 1984)

or too mathematically complicated to be applied ( Ryan et al. 1984, Spurgeon 1991), and it is not clear how to

choose δ which is the width of the boundary layer in the sense in Slotine et al. 1983.

Because of the infinite gain of a relay represented by the sign function of a typical VSS control function

as in Eq.(3.5), normally there is a problem of chattering . My approach answers the questions raised above in

Slotine et al. 1983, Amb rosino et al. 1984, Spurgeon 1991. To eliminate this problem, I may use either a

saturation method (this section) or continuous VSS (Chapter 4).

I analyze this chattering problem from another viewpoint: an analysis that allows a unified design for both

saturate and unitvector functions. In addition, I propose the hyperbolic tangent function since it is a

standard mathematical function and it may outperform the saturate and unitvector functions.

s

Saturating: sat(s, ks) UnitVector: smt(s, ks)Hyperbolic Tangent: tanh(ks*s)

s

ε

Switching: sgn(s)

k s = ∞

ks < ∞

s

ε

ks+a +a+a

−a −a −a

0 0 0

Fig. 3.2: SMC Sliding Functions


3.5.1. Switching Function

A switching function is defined as

sgn if if if

sa s

sa s

a f =+ >

=− <

RS|T|

,,,

00 0

0 (3.5)

3.5.2. Saturate Function

The sliding gain k s is reduced from ks=∞ as it is in a switching function, so a saturate function can be

defined as

sat

if

sgn if s k

k s sk

s sk

s

ss

s

,

. ,

,b g

a f=

≤

>

RS||

T||

1

1 (3.6)

3.5.3. Unitvector Function

Alternatively, a switching function in Eq.(3.5) can be defined as:

sgn( )sss

=

Hence, to avoid an infinite gain as above, a unitvector can defined as follows

smt s ks

ks

s

s

,b g =+1 smt for smooth since its appearance (3.7)

since

ss

sr s

→+

where

s rs

r sss

sgn s

s r sr s r

s

>>+

≈ =

<<+

≈

RS||

T||

:

:

( )

1

with

kss

s

ss : the gain reduced from because ∞ =

∞+

, 1

3.5.4. Hyperbolic Tangent Function (tanh)

To avoid an infinite gain, a hyperbolic tangent function can be used instead:

tanh k ssb g (3.8)

and

d k s

dsks

s

s

tanh b g=

=0


Remark 3.4: Pseudo-SMC with Boundary Layer

To satisfy the sliding condition, a VSS control function takes the form of a switching function that is defined by Eq.(3.5), so there are only 3 valid values: − +a a, ,0 .

For a sliding function, outside the band of ε are the valid values − +a a, , so the sliding condition is

satisfied. Except the origin, inside the band of ε are invalid values, so the sliding condition is violated in

this band. There is no sliding mode in this band, therefore a VSS control using a sliding function is called a

pseudo-SMC.

3.5.5. Performance of Sliding Functions (Saturate, Unitvector and Tanh Functions) (1) The switching function has infinite gain while the sliding functions have finite gains. The less gain is, the

less chattering is. However, the more possible it is that steady-state error and a slow-down response may

result.

(2) As it will be shown in the experimental result of the final chapter, the real danger of the chattering

problem is that it excites high-frequency unmodelled plant dynamics. So there is always an upper bound for

the sliding margin, this bound is the lowest for a switching function and the highest for a unitvector function.

So, by Proposition 3.1 in Section 3.4 above, for a switching function, it is possible that the upper bound is

low enough to slow down the system response even with the fast eigenvalue. (3) For a saturate function, the gain is constant (ks of the saturate function in the figure above) within the

boundary layer. For unitvector and hyperbolic functions, the gain decreases when the value of the unitvector function increases (k ks s

* < of the unitvector function in the figure above), thus reducing the possibility for an

oscillation (chattering) to occur

• This is a disadvantage of the unitvector and hyperbolic tangent over the saturate function, because it is

more possible for a steady-state error to occur.

• On the other hand, this is an advantage of the unitvector and hyperbolic tangent over the saturate

function. The upper bound of the saturate function is lower than that of a unitvector function. For the

saturate function, this bound may be low enough to slow down the system response even with the faster

eigenvalue.

Since the gains of all sliding functions above are lower than that of the sign function (infinite gain), a

steady-state error may occur. In addition, the chattering is excited by a large error. The following proposition

is used to solve this problem

Proposition 3.2: Application of Sliding Function To eliminate the chattering problem, the design procedure is exactly the same, but I simply replace the

switching function by a saturate, a unitvector or a tanh function. The gain may be chosen just low enough to

eliminate the chattering. The lower is the gain, the wider is the boundary layer in which a pseudo-VSS

control exists (Remark 3.4).


The following proposition eliminates the steady-state error arising from in using the sliding function to

eliminate the chattering (Example 7.6).

Proposition 3.3: Solution to Problem of Chattering and Steady-State Error

if error > err_tol

u is based on the saturate, unitvector or hyperbolic function

else

u is based on the sign function

where err_tol is chosen small enough to eliminate the chattering and error.

3.6. INTEGRAL VSS CONTROLLER DESIGN

A VSS control is based on a state-space model, so that an I-action may be required to eliminate a steady-

state error. In Chern et al. 1991, an integral of error has been used, however it is a pseudo-SMC because the

larger controller gain the closer the sliding mode. In Chang 1991, an integral sliding condition has been

proposed using the same system order, however this approach can be applied to some systems but not all

(Example 3.11). Alternatively, I propose an integral VSS control by augmenting the order of a system model.

The controller is then designed in a unified manner as in the case without integral action. It can solve the

problem of pseudo-SMC in Chern et al. 1991 and cope with limitation in Chang 1991.

Based on the integral sliding condition in Chang 1991, I have the following theorem to design an integral

SMC.

Theorem 3.3: Integral VSS Controller Design by Integral Sliding Condition

Consider the following system

x Ax B= + u (3.9)

and a hyperplane

s = Hx . If ~u is a SMC of the system in Eq.(3.9), then there exists a constant δ i to determine an integral SMC as

( )u u s dti

t

i= − >− ∫~ . ,HB 1

0

0δ δ (3.10)

where δ i is a positive constant as an integral sliding margin and can be determined by the SMC design rule in

Proposition 3.1.

Proof

A Lyapunov function can be defined as

V s s dti

t

= +

∫1

22 1

20

2

δ (3.11)

so

& &V s s s dti

t

= +

∫δ

0


then from Eq.(3.9), I have

& .V s u s dti

t

= + +

∫HAx HB δ

0

by Eq.(3.10), I have ( )& . ~V s u= +HAx HB

thus

&V < 0 (3.12)

since ~u is a SMC of Eq.(3.9).

Q.E.D.

Remark 3.5: Applicability of SMC Design Rule to Integral VSS

In Theorem 3.3, the control ~u is designed normally as a SMC and an integral of sliding variable s is

included with an integral sliding margin δ i determined by the SMC design rule in Proposition 3.1.

Alternatively, I have the following theorem to design an integral SMC where the system model is

augmented and an integral of error is used instead of the sliding variable to improve an I-action. The

advantage of the design in Theorem 3.3 is simple since the system model is not used in the I-action, however

the following design can cope with its limitation (Example 3.10 and 3.11) because all system states are

integrated in Theorem 3.3 via the sliding variable but only the error of output in the following theorem.

Theorem 3.4: Integral VSS Controller Design by Augmented System

Consider a system

&x Ax B

Cx= +=

uy

(3.13)

and its augmented-order system with a reference input of r

&x A x Bi i i i iu= + (3.14)

where

( )xx

AC

0 AB

Bi

t

in

iy r dt= −

=

=

∫

×0

1

0 0, , . (3.14.a)

Then an integral VSS control with a hyperplane H i can be determined by

u u ui= + ~ (3.15)

where

( ) [ ]~ ,, , ,u h r h hi i i i i i n= =−

+H B H1

1 1 1L (3.15.a)

and ui is a VSS control for Eq.(3.14) and can be determined by Theorem 3.2 or 3.3.

with x B A C, , , , , ~, , ,∈ ℜ ∈ ℜ ∈ℜ ∈ ℜ× × ×n n n n

iu u u y r1 1


Proof

Let

( )x y r dt x y rt

00

0= − ⇒ = −∫ &

from Eq.(3.13), I have

&&&

xx Ax B

CxAx B 0

C0 A x B 0i

n n n

x y ru u

r xu r=

=

−+

=

+

−

=

+

+

−

× × ×

0

1 1

0

1

0 0 0 1

or, by Eq.(3.14a)

&x A x B0i i i i

nu r= + +

−

×

1

1

(3.16)

thus

& & . . ,s u r u h ri i i i i i i i in

i i i i i i= = + +−

= + −

×

H x H A x H B H0

H A x H B1

11 (3.17)

by Eq.(3.15), I have ( )& ~s u u ui i i i i i i i i i i i= + − == +H A x H B H A x H B (3.18)

since ui is a VSS control for Eq.(3.14), the sliding condition s si i& ≤ 0 is satisfied using Eq.(3.18)

Q.E.D.

3.7. NUMERICAL EXAMPLES

For the purpose of comparison only, my design keeps the control effort in the same order as in the

original design, although in some examples it is rather high. Based on the SMC design rule in Proposition 3.1

(Section 3.4), my proposed control functions are designed in a unique approach, while the original control

functions have been designed by the following approaches: equal excursion sliding-mode, non-linear

hyperplane.

Hyperplane designs are omitted here as they can be found in the previous chapter on hyperplane design.

In fact, I have built a VSS toolbox in MATLAB language and all the following hyperplanes are determined by

the corresponding function in that toolbox. It is necessary to emphasize that H is normalized such that HB=I.

By this choice, I not only simplify the design calculation, but also obtain a unique equation for a hyperplane

rather than a different equation by a scaling factor.

All of the hyperplane design methods in the previous chapter are used: the eigenvalues allocation

method, the optimal sliding mode method, and the direct calculation method. Since the original control

function is in a switching form, the corresponding proposed control functions are also in the switching form

for the purpose of comparison. To see that the hyperplane design is independent of the controller design,

there are different VSS control functions for each hyperplane design method.

A saturate VSS or unitvector VSS design exactly the same as its switching counterpart with the extra

introduction of Ks whose value may be chosen just low enough to eliminate the chattering.


The first 4 examples are designed in all methods for an assessment Each starts with the original design

from the VSS literature, next follows with new designs for switching VSS then saturate, unitvector and Tanh

VSS to solve the chattering problem. Each concludes with I/O-State and optimal VSS designs in switching

control functions to compare with the original design which is also in switching control function. Other

designs (saturate, unitvector and Tanh VSS) are not presented to avoid a lengthy presentation.

The next 3 examples are used to illustrate some issues such as the efficiency of the new design, the

validity of the application of the I/O-state method (Section 2.3.4), the validity of the SMC design rule (Section

3.4) and the problem of a steady-state error when reducing the gain in the saturation approach (Section 3.5).

The following example is used to show the efficiency of the proposed integral VSS control. As mentioned in

the previous chapter on hyperplane design, assessment of all the hyperplane design methods must be made

in this chapter where the VSS control function is available. So in this chapter, there are comparisons not only

for the hyperplane design methods (eigenvalue allocation method, I/O-state method, optimal method), but

also for the controller design methods (switching, saturate, unitvector and tanh functions).

Remark 3.6: Summary of VSS Designs

In the numerical examples below, and in this work generally, hyperplane eigenvalues will be chosen at the

same unique value for simplicity except in case of the optimal sliding mode approach. Different multiple values

may be attempted to compromise between the response speed and overshoot.

• Hyperplane designs are referred to the previous chapter;

• Sliding margin δ is chosen on the basis of Proposition 3.1;

• New Switching VSS control functions are computed by Theorem 3.2 and Corollary 3.2 where the

switching function is given by Eq.(3.5);

• Saturate, Unitvector and Tanh VSS control functions are given by Eqs.(3.6) to (3.8), respectively, based

on Proposition 3.2;

• Integral VSS control functions can be determined by Theorem 3.3 (Integral VSS) or Theorem 3.4

(Augmented System)


3.7.1. Example 3.1: Canonical System


Consider a system in White et al. 1984

&x x=− − −

+

0 1 00 0 16 11 6

001

u

with the hyperplane

s = =Hx H, , , 38 9 1

and the original control function is given as

usx

sx

i

i

= − + = − =+ + + >

− − − <

RS|T|K K x K K∆ ∆a f , , , ,

, ,

, ,

, if

, if 120 0 0

150 30 5 0

150 30 5 0

0 1 2 3-0.5

0

0.5

1

Time [s]

Out

put

0 1 2 3-600

-400

-200

0

200

Time [s]

Con

trol

0 1 2 3-10

0

10

20

30

40

Time [s]

Slid

ing

-0.5 0 0.5 1-3

-2

-1

0

1

Output

Cha

nge

of O

utpu

t

Original VSS Control for 3-rd Order System

Fig. 3.3: Original VSS Control for Example 3.1.


Choose [ ]λ H = − −7 7, , and I have

[ ]H = 49 14 1, ,

thus [ ]K e = −6 38 8, ,

Choose δ = 2 5. , hence Theorem 3.2 and Corollary 3.2 yield [ ]Kr = 98 28 2


(a) Switching VSS

0 1 2 30

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 1 2 3-100

0

100

200

300

Time [s]

Con

trol

0 1 2 3-20

0

20

40

60

Time [s]

Slid

ing

0 0.5 1-2.5

-2

-1.5

-1

-0.5

0

OutputC

hang

e of

Out

put

Switching VSS Control for 3-rd Order System

Fig. 3.4: Switching VSS Control for Example 3.1.

(b) Saturate VSS

Choose ks = 0 75. , then

0 1 2 30

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 1 2 3-100

-50

0

50

100

Time [s]

Con

trol

0 1 2 3-20

0

20

40

60

Time [s]

Slid

ing

0 0.5 1-2.5

-2

-1.5

-1

-0.5

0

Output

Cha

nge

of O

utpu

t

Saturate VSS Control for 3-rd Order System

Fig. 3.5: Saturate VSS Control for Example 3.1.


(c) Unitvector VSS

Choose ks = 1 75. , then

0 1 2 30

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 1 2 3-100

-50

0

50

100

Time [s]

Con

trol

0 1 2 3-20

0

20

40

60

Time [s]

Slid

ing

0 0.5 1-2

-1.5

-1

-0.5

0

Output

Cha

nge

of O

utpu

t

Unitvector VSS Control for 3-rd Order System

Fig. 3.6: Unitvector VSS Control for Example 3.1.

(d) Tanh VSS

Choose ks = 125. , then

0 1 2 30

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 1 2 3-100

-50

0

50

100

150

Time [s]

Con

trol

0 1 2 3-20

0

20

40

60

Time [s]

Slid

ing

0 0.5 1-2.5

-2

-1.5

-1

-0.5

0

Output

Cha

nge

of O

utpu

t

TanH VSS Control for 3-rd Order System

Fig. 3.7: Tanh VSS Control for Example 3.1.



I have

n r= =3 3,

so the resulting hyperplane by the I/O state method is similar to the above hyperplane due to the canonical

system

0 1 2 30

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 1 2 3-100

0

100

200

300

Time [s]

Con

trol

0 1 2 3-20

0

20

40

60

Time [s]

Slid

ing

0 0.5 1-2.5

-2

-1.5

-1

-0.5

0

Output

Cha

nge

of O

utpu

t

I/O-State VSS Control for 3-rd Order System

Fig. 3.8: I/O State Switching VSS Control for Example 3.1.


0 2 4 6 8-0.5

0

0.5

1

Time [s]

Out

put

0 2 4 6 8-100

-50

0

50

100

Time [s]

Con

trol

0 2 4 6 8-5

0

5

10

Time [s]

Slid

ing

-0.5 0 0.5 1-1.5

-1

-0.5

0

0.5

Output

Cha

nge

of O

utpu

t

Error-Optimal VSS Control for 3-rd Order System

Fig. 3.9: Error Optimal VSS Control for Example 3.1.


Choose Q =

10 0 00 0 1 00 0 0 1

..

, then

[ ]H = 10 4 5826 1, . ,

thus [ ]Ke = − − −6 1 14174, , .

choose δ = 3 , hence Theorem 3.2 and Corollary 3.2 yield

[ ]Kr = 30 0 13 7477 3 0. . .


• I/O-State method: the same since the system is canonical.

• Error-optimization: comparable to the normal approach since the control effort reduces by half but the

response time doubles.

3.7.2. Example 3.2: Non-canonical System


Consider a system in Sivaramakrishnan et al. 1984

&

.

. .. . .

.

..x x=

−−

− − −

+

0 05 6 0 0

0 3 333 3 333 05 208 0 12 5 12 5

0 6 0 0 0

00

12 5

0

u

with the hyperplane

s = =Hx H, . , . , 5 155 4 385 1 16

and the original control function

us x

s xi

i

= − =+ + + >

− − − <

RS|T|K x K. ,

, , , , .

, , , , .

if

if

6 6 2 0 0

6 6 2 0 0

0 2 4 6 8-0.5

0

0.5

1

Time [s]

Out

put

0 2 4 6 8-20

-10

0

10

20

Time [s]

Con

trol

0 2 4 6 8-2

0

2

4

6

Time [s]

Slid

ing

-0.5 0 0.5 1-1

-0.5

0

0.5

Output

Cha

nge

of O

utpu

t

Original VSS Control for 4-th Order System




Choose [ ]λ H = − − −6 6 6, , , then

[ ]H = 0 4285 03508 0 0800 14401. , . , . , .

thus [ ]K e = −0 4260 1 4014 0 1694 1. , . , . ,


[ ]Kr = 4 2845 3 5084 0 80 14 4014. . . .

(a) Switching VSS

0 1 2 3-0.5

0

0.5

1

Time [s]

Out

put

0 1 2 3-10

-5

0

5

10

15

Time [s]

Con

trol

0 1 2 3-0.2

0

0.2

0.4

0.6

Time [s]

Slid

ing

-0.5 0 0.5 1-1

-0.5

0

0.5

Output

Cha

nge

of O

utpu

t

Switching VSS Control for 4-th Order System


(b) Saturate VSS

Choose ks = 12 , then

0 1 2 3-0.5

0

0.5

1

Time [s]

Out

put

0 1 2 3-8

-6

-4

-2

0

2

Time [s]

Con

trol

0 1 2 3-0.2

0

0.2

0.4

0.6

Time [s]

Slid

ing

-0.5 0 0.5 1-1

-0.5

0

0.5

Output

Cha

nge

of O

utpu

t

Saturate VSS Control for 4-th Order System



(c) Unitvector VSS


0 1 2 3-0.5

0

0.5

1

Time [s]

Out

put

0 1 2 3-6

-4

-2

0

2

Time [s]

Con

trol

0 1 2 3-0.2

0

0.2

0.4

0.6

Time [s]

Slid

ing

-0.5 0 0.5 1-1

-0.5

0

0.5

Output

Cha

nge

of O

utpu

t

Unitvector VSS Control for 4-th Order System


(d) Tanh VSS


0 1 2 3-0.5

0

0.5

1

Time [s]

Out

put

0 1 2 3-10

-5

0

5

Time [s]

Con

trol

0 1 2 3-0.2

0

0.2

0.4

0.6

Time [s]

Slid

ing

-0.5 0 0.5 1-1

-0.5

0

0.5

Output

Cha

nge

of O

utpu

t



I have

n r S= = ⇒ = − −4 3 6 6 0, , , λ : stable, so I continue to design a controller


Choose [ ]λ H IO, ,= − −6 6 , then

[ ]H = 0 1416 0 2068 0 08 0. , . , .

thus [ ]Ke = − − −0 4237 01604 0 3106 1. , . , . ,


[ ]Kr = 0 2832 0 4137 01600 0. . .

0 1 2 30

0.2

0.4

0.6

0.8

Time [s]

Con

trol

0 1 2 3-0.05

0

0.05

0.1

0.15

Time [s]

Slid

ing

0 0.5 1-0.4

-0.3

-0.2

-0.1

0

Output

Cha

nge

of O

utpu

t

I/O-State VSS Control for 4-th Order System

0 1 2 30

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

Fig. 3.15: I/O-State VSS Control for Example 3.2.


0 1 2 3-0.5

0

0.5

1

Time [s]

Out

put

0 1 2 3-5

0

5

10

Time [s]

Con

trol

0 1 2 3-0.5

0

0.5

1

Time [s]

Slid

ing

-0.5 0 0.5 1-1

-0.5

0

0.5

Output

Cha

nge

of O

utpu

t

Error-Optimal Control for 4-th Order System

Fig. 3.16: Error-Optimal VSS Control for Example 3.2.


Choose Q =

10 0 0 00 0 1 0 0

0 0 0 1 0

0 0 0 0 1

.

.

.

, then

[ ]H = 08008 0 4134 0 08 0 08. , . , . , .

thus [ ]K e = − −0 4087 3 4270 0 3779 1. , . , . ,


[ ]Kr = 4 0041 2 0671 0 4000 0 4000. . . .

3.7.2.3. Performances compared with the Normal Case

• I/O-State method: same speed, no overshoot, much higher efficiency.

• Error-optimization: faster and smaller overshoot.

3.7.3. Example 3.3: Equal Excursion Sliding Mode


Consider a system in Hong et al. 1989

& . .x x=− −LNM

OQP +

−LNM

OQP

0 11 2

03

u

with the hyperplane

s = =Hx H, . , 0 7 1


u k x x ksx

sx= − =

>

− <RST1 1 2 1

1

1

0 430 6 0

0. ,

. ,

if

1.26, if

0 5 100

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 5 10-1

-0.5

0

0.5

1

Time [s]

Con

trol

0 5 10-0.2

0

0.2

0.4

0.6

0.8

Time [s]

Slid

ing

0 0.5 1-0.8

-0.6

-0.4

-0.2

0

Output

Cha

nge

of O

utpu

t

Original VSS Control for 2-nd Order System




Choose [ ]λ H = −1 , then

[ ]H = − −03333 03333. , .

thus [ ]K e = 0 3333 0 3333. , .

choose δ =1 5. , hence Theorem 3.2 and Corollary 3.2 yield

[ ]Kr = 0 5 0 5. , .

(a) Switching VSS

0 5 100

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 5 10-1

-0.5

0

0.5

1

Time [s]

Con

trol

0 5 10-0.4

-0.3

-0.2

-0.1

0

0.1

Time [s]

Slid

ing

0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

Output

Cha

nge

of O

utpu

t

Switching VSS Control for 2-nd Order System


(b) Saturate VSS


0 5 100

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 5 10-0.2

0

0.2

0.4

0.6

0.8

Time [s]

Con

trol

0 5 10-0.4

-0.3

-0.2

-0.1

0

0.1

Time [s]

Slid

ing

0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

Output

Cha

nge

of O

utpu

t

Saturate VSS Control for 2-nd Order System



(c) Unitvector VSS


0 5 100

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 5 10-0.2

0

0.2

0.4

0.6

0.8

Time [s]

Con

trol

0 5 10-0.4

-0.3

-0.2

-0.1

0

0.1

Time [s]

Slid

ing

0 0.5 1-0.8

-0.6

-0.4

-0.2

0

Output

Cha

nge

of O

utpu

t

Unitvector VSS Control for 2-nd Order System


(d) Tanh VSS


0 5 100

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 5 10-0.2

0

0.2

0.4

0.6

0.8

Time [s]

Con

trol

0 5 10-0.4

-0.3

-0.2

-0.1

0

0.1

Time [s]

Slid

ing

0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

Output

Cha

nge

of O

utpu

t

TanH VSS Control for 2-nd Order System




I have n r= =2 2,

similar to the conventional case due to the canonical system.

0 5 100

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 5 10-1

-0.5

0

0.5

1

Time [s]

Con

trol

0 5 10-0.4

-0.3

-0.2

-0.1

0

0.1

Time [s]

Slid

ing

0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

Output

Cha

nge

of O

utpu

t

I/O-State VSS Control for 2-nd Order System



0 5 100

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 5 10-0.6

-0.4

-0.2

0

0.2

Time [s]

Slid

ing

0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

Output

Cha

nge

of O

utpu

t

Error-Optimal VSS Control for 2-nd Order System

0 5 10-1

-0.5

0

0.5

1

Time [s]

Con

trol



Choose Q =

0 2 00 0 1.

., I have then

[ ]H = − −0 4714 0 3333. , .

thus [ ]Ke = 0 333 01953. , .

choose δ = 1 25. , hence Theorem 3.2 and Corollary 3.2 yield

[ ]Kr = 0 5893 0 4167. .


• I/O-State method: the same since the canonical system

• Error-optimization: comparable to the normal approach.

Remark 3.7: Comparison between Proposed Design and Equal Excursion Sliding-Mode Design

In the original design, a technique of an equal excursion sliding mode was proposed and illustrated by

the example. I use my normal design approach and get the same result of equal excursion.

3.7.4. Example 3.4: Non-linear Hyperplane


Consider a system in Lee et al. 1991

&x x=−

+

0 1 00 0 10 1 0

0010

u

0 2 4 60

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 2 4 6-10

-5

0

5

10

Time [s]

Con

trol

0 2 4 6-5

0

5

10

Time [s]

Slid

ing

0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

Output

Cha

nge

of O

utpu

t

Original Switching VSS Controller for 3-rd Order System



with the non-linear hyperplane

s x x x x x x x x x= + + + + − +9 6 0 33 1 79 0 42 0 481 2 3 13

12

2 1 22

23. . . .


[ ] [ ]

u x x x x x x x x x x x x x si ii

= − + + + + + +

= =

=∑φ ψ ψ ψ ψ ψ ψ

φ ψ1

3

1 12

2 2 12

3 3 1 22

4 1 2 3 5 23

6 22

3

0 1 1 1 0 2 0 2 0 5 0 1 01 0 2

sgn( )

. , , , . , . , . , . , . , .

3.7.4.2. New Designs Choose [ ]λ H = − −3 3, , then

[ ]H = 0 9 0 6 01. , . , .

thus [ ]K e = 0 0 8 0 6, . , .

choose δ =1 5. , hence Theorem 3.2 and Corollary 3.2 yield [ ]Kr = 13500 0 9000 0 1500. . .

(a) Switching VSS

0 2 4 60

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 2 4 6-2

0

2

4

6

Time [s]

Con

trol

0 2 4 6-0.5

0

0.5

1

Time [s]

Slid

ing

0 0.5 1-1.5

-1

-0.5

0

Output

Cha

nge

of O

utpu

t

Switching VSS Controller for 3-rd Order System



(b) Saturate VSS


0 2 4 6-2

-1

0

1

2

3

Time [s]

Con

trol

0 2 4 60

0.2

0.4

0.6

0.8

1

Time [s]

Slid

ing

0 0.5 1-1.5

-1

-0.5

0

Output

Cha

nge

of O

utpu

t

Saturate VSS Controller for 3-rd Order System

0 2 4 60

0.2

0.4

0.6

0.8

1

Time [s]

Out

put


(c) Unitvector VSS


0 2 4 60

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 2 4 6-2

-1

0

1

2

3

Time [s]

Con

trol

0 2 4 60

0.2

0.4

0.6

0.8

1

Time [s]

Slid

ing

0 0.5 1-1.5

-1

-0.5

0

Output

Cha

nge

of O

utpu

t

Unitvector VSS Controller for 3-rd Order System



(d) Tanh VSS


0 2 4 60

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 2 4 6-2

-1

0

1

2

3

Time [s]

Con

trol

0 2 4 60

0.2

0.4

0.6

0.8

1

Time [s]

Slid

ing

0 0.5 1-1.5

-1

-0.5

0

Output

Cha

nge

of O

utpu

t

TanH VSS Controller for 3-rd Order System



I have

n r= =3 3,

similar to the conventional case due to the canonical system.

0 2 4 60

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 2 4 6-2

0

2

4

6

Time [s]

Con

trol

0 2 4 6-0.5

0

0.5

1

Time [s]

Slid

ing

0 0.5 1-1.5

-1

-0.5

0

Output

Cha

nge

of O

utpu

t

I/O-State Switching VSS Controller for 3rd-Order System




Choose Q =

10 0 00 0 1 00 0 0 1

..

, then

[ ]H = 1 0 4583 01, . , .

thus [ ]Ke = 0 0 9 0 4583, . , .

choose δ =1 5. , hence Theorem 3.2 and Corollary 3.2 yield [ ]Kr = 15000 0 6874 01500. . .

0 2 4 6-0.5

0

0.5

1

Time [s]

Out

put

0 2 4 6-2

0

2

4

6

Time [s]C

ontr

ol

0 2 4 6-0.5

0

0.5

1

Time [s]

Slid

ing

-0.5 0 0.5 1-1.5

-1

-0.5

0

0.5

Output

Cha

nge

of O

utpu

t

Error-Optimal VSS Controller for 3rd-Order System


3.7.4.3. Performances compared with the Normal case

• I/O-State method: the same since the canonical system.

• Error-optimization: the normal approach is even better because it has no overshoot!

Remark 3.8: Comparison between Proposed Design and Nonlinear Hyperplane Design

In the original design, a nonlinear hyperplane was prposed to make the system responses faster. I use my

normal design and get the same result, perhaps ours might be better because it is as fast as the original but

perhaps with lower control effort.

3.7.5. Example 3.5: Efficiency of New Design

Consider the following DC servo system

G ss s

a f a f=+

600

20


with its state-space model

& . .x x=−

+

0 10 20

0600

u .

Choose λ H = −60 then

[ ]H = 0 1 0 0017. .

thus [ ]Ke = 0 0 0667.

for all designs below.

3.7.5.1. Conventional Design

By the SMC design rule, choose δ = 0 4. , hence Theorem 3.2 and Corollary 3.1 yield [ ]K r = 0 4 0 4. .

0 0.1 0.2 0.3 0.40

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 0.1 0.2 0.3 0.4-30

-20

-10

0

10

20

Time [s]

Con

trol

0 0.1 0.2 0.3 0.4-0.1

-0.05

0

0.05

Time [s]

Slid

ing

0 0.5 10

20

40

60

Output

Cha

nge

of O

utpu

t

Conventional VSS Control for 2-nd Order System

Fig. 3.31: Conventional VSS Control for Example 3.5 with δ = 0 4.

Choose ks = 50 , I have

0 0.1 0.2 0.3 0.40

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 0.1 0.2 0.3 0.4-10

-5

0

5

10

Time [s]

Con

trol

0 0.1 0.2 0.3 0.4-0.1

-0.05

0

0.05

Time [s]

Slid

ing

0 0.5 10

10

20

30

40

50

Output

Cha

nge

of O

utpu

t

Conventional TanH VSS Control for 2-nd Order System

Fig. 3.32: Conventional Tanh VSS Control for Example 3.5


3.7.5.2. New Design

By the SMC design rule, choose δ = 40 , hence Theorem 3.2 and Corollary 3.2 yield [ ]K r = 4 0000 0 0667. .

0 0.1 0.2 0.3 0.40

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 0.1 0.2 0.3 0.4-10

-5

0

5

Time [s]

Con

trol

0 0.1 0.2 0.3 0.4-0.1

-0.05

0

0.05

Time [s]

Slid

ing

0 0.5 10

10

20

30

40

Output

Cha

nge

of O

utpu

t

New VSS Control for 2-nd Order System

Fig. 3.33: New VSS Control for Example 3.5 with δ = 40

Choose ks = 200 , I have

0 0.1 0.2 0.3 0.40

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 0.1 0.2 0.3 0.4-6

-4

-2

0

2

4

Time [s]

Con

trol

0 0.1 0.2 0.3 0.4-0.1

-0.05

0

0.05

Time [s]

Slid

ing

0 0.5 10

10

20

30

40

Output

Cha

nge

of O

utpu

t

New TanH VSS Control for 2-nd Order System

Fig. 3.34: New Tanh VSS Control for Example 3.5


Remark 3.9: Performances of New Design

The new design has less chattering and requires lower control effort than the conventional approach. In

addition, the new Tanh VSS design has much higher sliding gain so there is less possibility of steady-state

error. The reason is that the system dynamics are very fast. These dynamics are included in H where

H H1 2 60a f a f = , so that the new design includes this fact in the reaching control whereas the conventional

design ignores them and uses the same sliding margin for all system states.

3.7.6. Example 3.6: Error/Energy-Optimal VSS

Consider a system in Sivaramakrishnan et al. 1984

&

.

. .. . .

.

..x x=

−−

− − −

+

0 05 6 0 0

0 3 333 3 333 05 208 0 12 5 12 5

0 6 0 0 0

00

12 5

0

u

choose

Q =

10 0 0 00 10 0 0

0 0 0 1 0

0 0 0 0 1

.

.

, and R = 1

then by the error/energy optimal sliding mode method, a hyperplane can determined as H = 0 04 0 0649 0 08 0 0543. , . . , .

thus [ ]Ke = − − −0 3861 0 0259 0 7835 1. , . , . ,


[ ]Kr = 0 2828 0 4543 05600 0 3801. . . .

0 2 4 6-0.5

0

0.5

1

Time [s]

Out

put

0 2 4 6-1

-0.5

0

0.5

1

Time [s]

Con

trol

0 2 4 6-0.02

0

0.02

0.04

0.06

Time [s]

Slid

ing

-0.5 0 0.5 1-0.3

-0.2

-0.1

0

0.1

Output

Cha

nge

of O

utpu

t

Error/Energy-Optimal VSS Control for 4-th Order System

Fig. 3.35: Error/Energy-Optimal VSS Control for Example 3.6.

The control effort is very low as expected, however the system response is slow and there is a large

overshoot.


3.7.7. Example 3.7: Applications of I/O-State Method (Section 2.3.4)

As mentioned in the previous chapter on hyperplane design, the following examples are included to

illustrate the validity of applying the I/O-State technique in the design of a hyperplane (Section 2.3.4). When

the relative degree is less than the system order, the stability test in Section 3.2 must first be used. Note that

this method is applicable for both linear and nonlinear systems. The following illustrations are for linear

systems to investigate into the I/O-state method, Chapter 6 will be for nonlinear systems.

3.7.7.1. Relative Degree less than System Order

By the stability criterion, I can have the following 2 cases

(a) Instability found by Stability Criterion: I/O-State Method Inapplicable

Consider a system in Spurgeon 1991

&. . .. . . . .x Ax B x= + =

− − −− −

−

+

u u0 277 32 98 5 432

0 365 0 319 9 490 0 5

00

10.

By the normal method, choose λH = − −3 3, , and by the direct calculation technique, I have

[ ]H = − −0 0018 0 0559 01. . .

thus [ ]Ke = −0 0199 0 0783 0 0404. . . .

Choose δ = 2 , hence Theorem 3.2 and Corollary 3.2 yield [ ]Kr = 0 0037 01118 0 2000. . . .

0 1 2 30

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 1 2 3-0.01

0

0.01

0.02

0.03

Time [s]

Con

trol

0 1 2 3-20

-15

-10

-5

0

5x 10-4

Time [s]

Slid

ing

0 0.5 10

0.01

0.02

0.03

0.04

Output

Cha

nge

of O

utpu

t

Switching VSS Controller for 3-rd Order System


By the I/O-state method with a relative degree r, choose the same λH above, I have

[ ]H = −0 0501 0 6071 0 1000. . .

thus [ ]Ke = −0 2355 14596 59895. . .


so

n r S= = ⇒ = −3 2 3 57 2989, , . λ : unstable!


[ ]Kr = 0 1003 1 2143 0 2000. . .

0 1 2 3-4

-2

0

2

4x 1053

Time [s]

Out

put

0 1 2 3-8

-6

-4

-2

0x 1057

Time [s]

Con

trol

0 1 2 3-2

-1

0

1

2x 1054

Time [s]

Slid

ing

-4 -2 0 2 4x 1053

0

0.5

1

1.5

2x 1056

Output

Cha

nge

of O

utpu

t

I/O-State VSS Controller for 3-rd Order System: n=3; r=2; UnStable Sliding mode

Fig. 3.37: I/O-State VSS Control for Example 3.7: n=3, r=2; Unstable Sliding Mode

This method fails in this case.

(b) Stability found by Stability Criterion: I/O-State Method Applicable

In Example 3.2, I had n r= =4 3, , the relative degree is less than the system order, so the sliding

eigenvalues need be checked [ ]λ S = − −6 6

since they are stable, the I/O method is applicable.

3.7.7.2. Relative Degree equal to System Order

In Examples 3.1, 3.3 and 3.4, I had n r= , so the I/O method is always applicable as the system stability is

guaranteed.

3.7.8. Example 3.8: A Simple Test of SMC Design Rule (Section 3.4)

Consider a system in White et al. 1984 as in Example 3.1.

3.7.8.1. Violation 1 in SMC Design Rule: Sliding Time much smaller than Reaching Time

The sliding margin is kept the same, δ = 2 5. , instead of [ ]λ H = − −7 7 , the hyperplane eigenvalues are

increased to violate the design rule

[ ]λ H = − −35 35

thus [ ]H = 1225 70 1


and [ ]Ke = −6 1214 64

with δ = 2 5. , hence Theorem 3.2 and Corollary 3.2 yield

[ ]Kr = 3062 5 175 2 5. .

0 1 2 30

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 1 2 3-4000

-3000

-2000

-1000

0

1000

Time [s]

Con

trol

0 1 2 30

500

1000

1500

Time [s]

Slid

ing

0 0.5 1-3

-2

-1

0

Output

Cha

nge

of O

utpu

t

Faster Eigenvalues violating Design Guideline for 3-rd Order System

Fig. 3.38: Cf. Fig. 3.4, Switching VSS Control violating SMC Design Rule for Example 3.8

RESULTS: the eigenvalues were chosen to be about 5 times faster, the time response is the same (about 1.2

secs), but the maximum control effort is much higher (3000 compared to 220).

3.7.8.2. Violation 2 in SMC Design Rule: Reaching Time much Smaller than Sliding Time

0 1 2 30

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 1 2 3-4000

-2000

0

2000

4000

Time [s]

Con

trol

0 1 2 3-40

-20

0

20

40

60

Time [s]

Slid

ing

0 0.5 1-2.5

-2

-1.5

-1

-0.5

0

Output

Cha

nge

of O

utpu

t

Higher Sliding Margin violating Design Guideline for 3-rd Order System

Fig. 3.39: Cf. Fig.3.4, Higher Sliding Margin violating SMC Design Rule for Example 3.7.


The hyperplane eigenvalues are kept the same, [ ]λ H = − −7 7 , thus the hyperplane and the equivalent

control are unchanged [ ]H = 49 14 1

thus [ ]Ke = −6 38 8 .

Instead of δ = 2 5. , the sliding margin is increased to δ = 25 to violate the design rule, hence Theorem 3.2 and

Corollary 3.2 yield [ ]Kr = 1225 350 25

RESULTS: the time response is a bit faster (1 sec compared to 1.2 secs), however the maximum control effort is

much higher (3800 compared to 220)

3.7.9. Example 3.9: Steady-State Error when Reducing Sliding Gain Consider a system in Lee et al. 1991 as in Example 3.3 where the sliding gain ks of saturate, unitvector

and tanh are reduced to unnecessarily low to see steady-state errors.

3.7.9.1. Saturate VSS

Choose ks = 1 , instead of ks = 50

0 2 4 60

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 2 4 6-1.5

-1

-0.5

0

0.5

Time [s]

Con

trol

0 2 4 60

0.2

0.4

0.6

0.8

1

Time [s]

Slid

ing

0 0.5 1-0.8

-0.6

-0.4

-0.2

0

Output

Cha

nge

of O

utpu

t

Saturate VSS Controller with Very Low Sliding Gain for 3-rd Order System

Fig. 3.40: Saturate VSS Control with Very Low Sliding Gain for Example 3.9.


3.7.9.2. Unitvector VSS


0 2 4 60

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 2 4 6-0.8

-0.6

-0.4

-0.2

0

0.2

Time [s]

Con

trol

0 2 4 60

0.2

0.4

0.6

0.8

1

Time [s]

Slid

ing

0 0.5 1-0.6

-0.4

-0.2

0

Output

Cha

nge

of O

utpu

t

Unitvector VSS Controller with Very Low Sliding Gain for 3-rd Order System

Fig. 3.41: Unitvector VSS Control with Very Low Sliding Gain for Example 3.9.

3.7.9.3. Tanh VSS


0 2 4 60

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 2 4 6-1

-0.5

0

0.5

Time [s]

Con

trol

0 2 4 60

0.2

0.4

0.6

0.8

1

Time [s]

Slid

ing

0 0.5 1-0.8

-0.6

-0.4

-0.2

0

Output

Cha

nge

of O

utpu

t

TanH VSS Controller with Very Low Sliding Gain for 3-rd Order System

Fig. 3.42: Tanh VSS Control with Very Low Sliding Gain for Example 3.9.

3.7.9.4. Performances:

There are steady-state errors when the sliding gains are reduced much more than necessary.


3.7.10. Example 3.10: Integral VSS Control using Integral of Sliding Variable

In Example 3.3, the reference input is equal to 0, if a unit reference input is used then there is a noticeable

steady-state error. I will use an integral VSS in Theorem 3.3 to eliminate this error.


Consider a system in Hong et al. 1989 as in Example 3.3

& . .x x=− −

+

−

0 11 2

03

u

with the hyperplane

s = =Hx H, . , 0 7 1


u k x x ksx

sx= − =

>

− <RST1 1 2 1

1

1

0 430 6 0

0. ,

. ,

if

1.26, if

0 5 100

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 5 10-1

-0.5

0

0.5

1

Time [s]

Con

trol

0 5 10-0.8

-0.6

-0.4

-0.2

0

0.2

Time [s]

Slid

ing

0 0.2 0.4 0.6 0.8-0.2

0

0.2

0.4

0.6

0.8

Output

Cha

nge

of O

utpu

t

Original Design with Unit Reference Input for 2-nd Order System

Fig. 3.43: Original Design with Unit Reference Input in Example 3.10: Steady-State Error


Choose the same hyperplane-eigenvalue as in Example 3.3, based on the SMC design rule, now the

sliding margin is changed to δ = 1 and includes an integral sliding margin of δ i = 0 15. in Theorem 3.3


0 5 100

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 5 10-2

-1.5

-1

-0.5

0

Time [s]

Con

trol

0 5 10-0.2

0

0.2

0.4

Time [s]

Slid

ing

0 0.5 10

0.2

0.4

0.6

0.8

1

Output

Cha

nge

of O

utpu

t

Integral VSS Control using Sliding Condition for 2-nd Order System

Fig. 3.44: Integral VSS Control using Integral Sliding Condition in Example 3.10: No Steady-State Error

Recall that the integral sliding condition in Eq.(3.11) has been used instead of the conventional sliding

condition s s. & < 0 .


Steady-state error is eliminated.

3.7.11. Example 3.11: Integral VSS Control using Integral of Output Error

Consider the following system from Lin et al. 1992 where perturbation is ignored for the time -being and it

will be considered in the next chapter

&&xx

xx

u1

2

1

2

25 313 84

4858

=

− −−

+

−−


Using the linear quadratic optimal control, an original control has been determined as

u x x= − +1 078 4 8191 2. .


0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-1.5

-1

-0.5

0

0.5

Time [s]

Con

trol

Original Design for 2-nd Order System

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

0.5

1

1.5

Time [s]

Out

put

Fig. 3.45: Original Design in Example 3.11.

For a unit step response, I have

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-1

-0.5

0

0.5

1

1.5

Time [s]

Con

trol

Original Design with Unit Step Input for 2-nd Order System

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-0.2

0

0.2

0.4

0.6

0.8

Time [s]

Out

put

Fig. 3.46: Original Design with Unit Step Input in Example 3.11: Steady-State Error.

If the integral sliding condition is used, choose

λH = − ⇒ = −50 H 0.0073 0. 0233

then Theorem 3.3 yields

( )u s dt si

t

e= − − −∫δ δ0

1K x x sgn

where

[ ]Ke i= − = =0 12 1 98 50 106. , . , ,δ δ

Note that δ i in Theorem 3.3 is chosen as large as possible to reduce the steady-state error as much as

possible.


0 0.05 0.1 0.15 0.2-1

-0.5

0

0.5

1

Time [s]O

utpu

t

0 0.05 0.1 0.15 0.2-10

-5

0

5

10

Time [s]

Con

trol

0 0.05 0.1 0.15 0.2-0.01

-0.005

0

0.005

0.01

Time [s]

Slid

ing

-1 -0.5 0 0.5 1-0.8

-0.6

-0.4

-0.2

0

Output

Cha

nge

of O

utpu

t

Integral VSS Control using Sliding Condition for 2-nd Order System

Fig. 3.47: Integral VSS Control using Integral Sliding Condition in Example 3.11: Steady-State Error

Note that the integral sliding condition in Eq.(3.11) has been used instead of the conventional sliding

condition.


Choose the same hyperplane-eigenvalue as above, recall that the system order is augmented [ ] [ ]λ H i= − − ⇒ = −50 50 0 5944 0 0234 0 0366, , ,H . . .

for a unit step input, then Theorem 3.4 yields

( ) ( )u r s x r dt x xe i i i i

t T

= − − = −

∫0 5944 1

01 2. sgn , , ,K x x xδ

where Ke = − =0 0 4856 3 1433 0 1, , .. . δ

0 0.05 0.1 0.15 0.2-0.5

0

0.5

1

1.5

Time [s]

Out

put

0 0.05 0.1 0.15 0.2-1

-0.5

0

0.5

1

Time [s]

Con

trol

0 0.05 0.1 0.15 0.2-1

-0.5

0

0.5

1x 10 -4

Time [s]

Slid

ing

-0.5 0 0.5 1 1.5-0.4

-0.3

-0.2

-0.1

0

Output

Cha

nge

of O

utpu

t

New Integral VSS Control using Unit Step Input for 2-nd Order System

Fig. 3.48: New Integral VSS Control with Unit Step Input for Example 3.11: No Steady-State Error


Note that the sliding variable of an augmented system in Eq.(3.14) has been used instead of the sliding

variable of the original system. The magnitude of s seems to increase and the sliding condition is thus not

satisfied, but it is not the case due to the chattering and its maximum magnitude is less than 5 10 4× − .


The control effort in the new integral VSS control is comparable to the case of regulator in Fig.3.47, but

there is no steady-s tate error, cf. Fig.3.48.

3.8. CONCLUSIONS

In the VSS literature so far, designs have been obtained using a variety of techniques. All the VSS control

functions have to satisfy the sliding condition s s. & < 0. SISO discontinuous VSS designs are White et al.

1984 where a control function is designed to satisfy the sliding condition with a given hyperplane (Example

3.1), Sivaramakrishnan et al. 1984 to apply the hyperplane design by partition of Utkin et al. 1978 (Example

3.2), Hong et al. 1989 with a proposed equal excursion sliding mode design to obtain an ideal sliding mode

(Example 3.3), Lee et al. 1991 to propose a nonlinear hyperplane for a fast system response (Example 3.4).

Continuous pseudo-VSS designs are Slotine et al. 1983 where a saturate function is proposed, Ambrosino

et al. 1984 and Spurgeon 1991 with a unitvector function.

In this chapter, a unified VSS controller design methodilogy has been developed. It is a simple and

unified technique with a SMC design rule applicable to all systems (linear or nonlinear, SISO or MIMO) and

all approaches (continuous or discontinuous VSS or continuous pseudo-VSS, continuous-time or discrete-

time VSS, and conventional or integral VSS) . I have proposed some new features below

• Different from the current stability criterion, I have analyzed the stability problem based on the mechanism

of the SMC, then I have proposed a simpler stability criterion for a practical test applicable to SMC.

• I propose a SMC design rule to choose a sliding margin to guarantee the existence of the sliding mode and

thereby achieve the invariance property. This rule is very effective both in simulations and in experiments

(Chapter 7). There is a simple illustration in Example 3.7.

• The standard if-form in Utkin 1977 is a convenient form to write a control function that satisfies the sliding

condition. My derivation of a compact form is simple, valid and unified for linear or nonlinear systems with

or without perturbation (to be used in Chapter 4, 6). First, based on the sliding condition I obtain the

standard if-form, then by Lemma 3.1 I achieve the compact form;


• The new design may include the system dynamics into the reaching control. The conventional design

may fail in fast systems since it uses the same sliding margin for all system states regardless the system

dynamics. The faster are the system dynamics, the better are the new designs in terms of lower control

effort and higher sliding gain, that is less potential steady-state error (Example 3.1 to 3.5)

• There are 2 components in a conventional VSS control: reaching and sliding controls. The proposed

integral VSS control is composed of 3 components where the first two are exactly the same as those in a

conventional VSS control and the third is an additional integral control. The proposed integral VSS

control with augmented system order has been shown to be efficient. It can cope with the limitation in

using the integral sliding condition in Chang 1991 (Example 3.11) and solve the problem of pseudo-SMC

using integral of error in Chern et al. 1991.

• The proposed analysis of the chattering problem is more comprehensive and precise than the works in

Slotine et al. 1983, Ambrosino et al. 1984, Spurgeon 1991. Moreover my approach is in a unified manner

for the saturate function and unitvector function. In addition, the introduction of the hyperbolic tangent

to replace the saturate and unitvector functions for more convenient since Tanh is a standard

mathematical function. The performance of the sliding functions is also analyzed and verified by

simulations in Example 3.9 and by experimental results in Chapter 7. Note that this approach is a pseudo-

VSS (Remark 3.4).

• For the I/O-state method, I have seen that it is crucial to do a stability test (Section 3.2) when the relative

degree is less than the system order.

• In Section of numerical examples, all original designs in different approaches have been represented to

compare with my proposed design in a unified approach. The following is a table to summarize the results

on the performance in all the numerical examples above. The I/O-state method and the optimal method

can be implemented in all methods (switching, saturate, unitvector and Tanh VSS). For simplicity, the I/O-

state method and the optimal method are implemented in the switching VSS control, so they may be

compared to the equivalent switching control the normal case

The following table sumarizes results using I/O-state and optimal I methods

I/O-state Optimal I

Ex 3.1 same as Switching same as Switching

Ex 3.2 better than Switching better than Switching




Chapter 3

Variable-Structure System Controller Design

3.1. Introduction...................................................................................................................................................................1

3.2. SMC Stability Criterion................................................................................................................................................1

Theorem 3.1: SMC Stability Criterion ...................................................................................................2

Remark 3.1: Stability Test.......................................................................................................................2

3.3. A New VSS Controller Design....................................................................................................................................2

Lemma 3.1: Compact Control Function................................................................................................3

Remark 3.2: Simplification of Control Function By Hyperplane Normalization..............................3

Theorem 3.2: Switching VSS Controller Design..................................................................................3

Corollary 3.1: Conventional Reaching Controller Design .................................................................4

Corollary 3.2: New Reaching Controller Design .................................................................................5

Remark 3.3: Sliding Mode as Fast Mode for Invariance....................................................................5

3.4. Proposition 3.1: SMC Design Rule.............................................................................................................................5

3.5. Solution to Chattering Problem ..................................................................................................................................6

3.5.1. Switching Function..........................................................................................................................................7

3.5.2. Saturate Function .............................................................................................................................................7

3.5.3. Unitvector Function.........................................................................................................................................7

3.5.4. Hyperbolic Tangent Function (tanh)............................................................................................................7

Remark 3.4: Pseudo-SMC with Boundary Layer.................................................................................8

3.5.5. Performance of Sliding Functions (Saturate, Unitvector and Tanh Functions).....................................8

Proposition 3.2: Application of Sliding Function...............................................................................8

Proposition 3.3: Solution to Problem of Chattering and Steady-State Error...................................9

3.6. Integral VSS Controller Design...................................................................................................................................9

Theorem 3.3: Integral VSS Controller Design by Integral Sliding Condition .................................9

Remark 3.5: Applicability of SMC Design Rule to Integral VSS..................................................... 10

Theorem 3.4: Integral VSS Controller Design by Augmented System.......................................... 10

3.7. Numerical Examples .................................................................................................................................................... 11

Remark 3.6: Summary of VSS Designs................................................................................................ 12

3.7.1. Example 3.1: Canonical System..................................................................................................................... 13

3.7.1.1. Original Design.................................................................................................................................. 13

3.7.1.2. New Designs ...................................................................................................................................... 13

(a) Switching VSS.................................................................................................................................. 14

(b) Saturate VSS..................................................................................................................................... 14

(c) Unitvector VSS................................................................................................................................. 15

(d) Tanh VSS .......................................................................................................................................... 15

(e) Switching VSS by I/O-state of Relative Degree r........................................................................ 16

(f) Error-Optimal Sliding Mode ............................................................................................................ 16


3.7.1.3. Performances ...................................................................................................................................... 17

3.7.2. Example 3.2: Non-canonical System............................................................................................................ 17

3.7.2.1. Original Design.................................................................................................................................. 17

3.7.2.2. New Designs ...................................................................................................................................... 18

(a) Switching VSS.................................................................................................................................. 18

(b) Saturate VSS..................................................................................................................................... 18

(c) Unitvector VSS................................................................................................................................. 19

(d) Tanh VSS .......................................................................................................................................... 19



3.7.2.3. Performances compared with the Normal Case............................................................................. 21

3.7.3. Example 3.3: Equal Excursion Sliding Mode............................................................................................... 21

3.7.3.1. Original Design.................................................................................................................................. 21

3.7.3.2. New Designs ...................................................................................................................................... 22

(a) Switching VSS.................................................................................................................................. 22

(b) Saturate VSS..................................................................................................................................... 22

(c) Unitvector VSS................................................................................................................................. 23

(d) Tanh VSS .......................................................................................................................................... 23



3.7.3.3. Performances ...................................................................................................................................... 25

Remark 3.7: Comparison between Proposed Design and Equal Excursion Sliding-Mode Design25

3.7.4. Example 3.4: Non-linear Hyperplane ............................................................................................................ 25

3.7.4.1. Original Design.................................................................................................................................. 25

3.7.4.2. New Designs ...................................................................................................................................... 26

(a) Switching VSS.................................................................................................................................. 26

(b) Saturate VSS..................................................................................................................................... 27

(c) Unitvector VSS................................................................................................................................. 27

(d) Tanh VSS .......................................................................................................................................... 28


(f) Error-Optimal Sliding Mode ..................................................................................................................... 29

3.7.4.3. Performances compared with the Normal case.............................................................................. 29

Remark 3.8: Comparison between Proposed Design and Nonlinear Hyperplane Design .......... 29

3.7.5. Example 3.5: Efficiency of New Design....................................................................................................... 29

3.7.5.1. Conventional Design........................................................................................................................ 30

3.7.5.2. New Design........................................................................................................................................ 31

Remark 3.9: Performances of New Design......................................................................................... 32

3.7.6. Example 3.6: Error/Energy-Optimal VSS ...................................................................................................... 32

3.7.7. Example 3.7: Applications of I/O-State Method (Section 2.3.4).............................................................. 33


3.7.7.1. Relative Degree less than System Order........................................................................................ 33

3.7.7.2. Relative Degree equal to System Order.......................................................................................... 34

3.7.8. Example 3.8: A Simple Test of SMC Design Rule (Section 3.4)............................................................... 34

3.7.8.1. Violation 1 in SMC Design Rule: Sliding Time much smaller than Reaching Time.................. 34

3.7.8.2. Violation 2 in SMC Design Rule: Reaching Time much Smaller than Sliding Time ................. 35

3.7.9. Example 3.9: Steady-State Error when Reducing Sliding Gain ................................................................. 36

3.7.9.1. Saturate VSS....................................................................................................................................... 36

3.7.9.2. Unitvector VSS................................................................................................................................... 37

3.7.9.3. Tanh VSS............................................................................................................................................ 37

3.7.9.4. Performances:..................................................................................................................................... 37

3.7.10. Example 3.10: Integral VSS Control using Integral of Sliding Variable ................................................. 38

3.7.10.1. Original Design ................................................................................................................................ 38

3.7.10.2. New Design ...................................................................................................................................... 38

3.7.10.3. Performances .................................................................................................................................... 39

3.7.11. Example 3.11: Integral VSS Control using Integral of Output Error ...................................................... 39

3.7.11.1. Original Design ................................................................................................................................ 39

3.7.11.2. New Design ...................................................................................................................................... 41

3.7.11.3. Performances .................................................................................................................................... 42

3.8. Conclusions................................................................................................................................................................. 42

Chapter 4

A New Robust Sliding Mode Controller-Observer Design

4.1. INTRODUCTION

In the current sliding mode control (SMC) literature, robust discontinuous SMC's have been

developed for matched uncertain dynamical decouplable systems in Iyer et al. 1988 and for matched uncertain

dynamical canonical systems in Emali et al. 1992. In Zhou et al. 1992 and Spurgeon et al. 1993, robust

continuous pseudo-SMC with nonlinear control functions was developed for matched and unmatched

uncertain systems. The former used high order power functions and the latter used a unitvector function in

place of a sign function. For these controllers, the sliding condition was satisfied within a boundary layer.

In this chapter, a new robust SMC design is fully presented in a unified manner for both discontinuous

SMC (VSS) and continuous SMC control functions under both matched and unmatched uncertainties. The

VSS design proposed in Chapter 3 is extended to deal with uncertain systems for VSS control functions. The

control function is partitioned into 3 components: equivalent control, reaching control, and perturbation

control. In the proposed linear SMC design, the control function is not only continuous but also linear. The

continuous nature of this control function helps to eliminate the chattering problem in the standard VSS

design. The advantage of the control function being linear is that some fundamental concepts of SMC design

can be explained from the linear control theory framework. For example, the sliding margin can be shown to

relate to the system eigenvalues.

As a SMC is based on a state-space model, an I-action may be required to eliminate a steady-state error. I

will prove that the integral VSS control (discontinuous SMC) in Theorem 3.4 & 3.5 can be still applied to both

robust discontinuous and linear SMC controls in this chapter.

Since the principal operating mode of a VSS control is the sliding mode, the VSS control can be seen as a

subset of the SMC using discontinuous control functions.

Similar to state-space control design, SMC usually requires an observer for estimation of system states.

In Bondarev et al. 1985, a linear Luenberger observer has been used as an observer of a VSS controller for

deterministic linear systems (no uncertainties). In Walcott et al. 1987 and Yaz et al. 1993, a Lyapunov sliding

condition has been used to design an observer for a class of systems under matched uncertainty restricted to

a certain system structure. In Slotine et al. 1987, a sliding patch condition has been used to have a region of

direct attraction where uncertainty is not fully tackled. In fact, in Walcott et al. 1987, Slotine et al. 1987 and

Yaz et al. 1993, to cope with uncertainty, a Lyapunov sliding condition has been employed to include a

switching component into a linear Luenberger observer where a linearized model is used for a nonlinear

system. In Edwards et al. 1996, an observer has been implemented using an output feedback technique.


In this chapter, I propose a novel robust sliding mode observer design for a wide class of systems under

both matched and unmatched uncertainty. This design is a dual extension of the new robust sliding mode

controller design mentioned above.

Based on the work on invariance property in Drazenovic 1969, I prove that the matching condition is

necessary and sufficient condition. This condition is valid for both discontinuous and continuous SMC .

4.2. ROBUSTNESS IN SMC

In real dynamical systems, it is impossible to avoid uncertainties (due to imperfect modelling, due to the

environment such as temperature, pressure...) and external disturbances. So the crucial demand is a solution

to the robust control problem for uncertain dynamical systems. In the SMC literature, it is generally known

that the main feature of SMC is its invariance to perturbations. This section shows what this "invariance"

really means.

4.2.1. Invariance Condition to Uncertainties

Theorem 4.1: Invariance Condition to Uncertainty

Consider an uncertain dynamical system

( )&~

. .~

. .x A x B A A x B= + = + +u u∆ (4.1)

where

x B A A, , ,~

,∈ℜ ∈ℜ ∈ℜ× ×n n n u1 ∆ .

The sliding mode is invariant to parameter uncertainties if and only if the variation ∆~A satisfies the

matching condition ∆

~. ,A B z z= ∀ ∈ℜ × 1 n (4.2)

Proof

Consider a system

( )& ~ ~x Ax B A A x B= + = + +u u∆

and a hyperplane

s = Hx .

From Theorem 2.2, the sliding dynamics can be determined by ( )[ ]s = = ⇒ = − −Hx x I B HB H Ax0 1& .

~

or

( )[ ] ( )& .~

x I B HB H A A x= − +−1 ∆ (4.3)

4. Robust Sliding-Mode Controller-Observer Design 4.3

(a) Necessary Condition (If Clause)

By Eq.(4.3), the sliding dynamics are invariant if ( )[ ]I B HB H A x− =−1 0.

~.∆

for any x, so I obtain 2 cases as follows

( )[ ] ( ) ( )

( ) ( )I B HB H A x

B HB H I

B HB H A x A x− = ⇒

=

=

−−

−

11

10. ~.

. ~. ~.∆

∆ ∆

:

:

α

β

• Case αa f a f: B HB H I− =1

[ ] ( )B H HB=

= =+ + +

= ≠−

bb

b

h h hb h b h b h

c

n

nn n

1

21 2

1

1 1 2 2

1 0M

LL

, ,

thus

( ) [ ]B HB H BH− = =

=

1

1

21 2

1 1 1 2 1

2 1 2 2 2

1 2

c c

bb

b

h h h c

b h b h b hb h b h b h

b h b h b hn

n

n

n

n n n n

. . .M

L

L

L

M M O M

L

hence

c

b h b h b hb h b h b h

b h b h b h

n

n

n n n n

1 1 1 2 1

2 1 2 2 2

1 2

1 0 00 1 0

0 0 1

L

L

M M O M

L

L

L

M M O M

L

=

;

by c ≠ 0,then

[ ]

[ ]cb h b h i ncb h b h i j n

i i i i

i j i j i j

= ⇒ ≠ ≠ ∀ ∈

= ⇒ = = ∀ ∀ ∈

≠

1 0 0 10 0 0 1

AND OR

, ,, , , : contradiction!

• Case βb g : ( )B HB H A x A x− =1 . ~. ~.∆ ∆ , then there exists at least one solution

∆~ . ,A B z z= ∈ℜ × 1 n

(b) Sufficient Condition (Only-If Clause) Eq.(4.2) gives ( )[ ] ( )[ ] [ ]I B HB H A I B HB H Bz B B 0− = − = − =− −1 1. ~ . .∆ z

thus the sliding mode is invariant by Eq.(4.3).

Q.E.D.

Remark 4.1: Form of Matching Condition Note that z takes an arbitrary value, so the matching condition requires the form rather the actual value.


Corollary 4.1: Alternative Form of Matching Condition

The system

( ) ( )&~ ~ ~ ~

x Ax B A A x B B= + = + + +u u∆ ∆

satisfies a matching condition if

∆

~. . ,A x B= ∀ ∈ℜv v1 1

(4.4.a)

and ∆

~. ,B B= ∀ ∈ ℜv v2 2 (4.4.b)

then

( )&x Ax B= + +u v (4.4.c)

Proof:

Post-multiplying x into the matching condition in Eq.(4.2), I have ∆

~.

~. .

~.A x B z x B= = v1 .

Since x and z are arbitrary, so is v1 . I also have

∆~

. . . .B B Bu v u v= =2 3

thus Eq.(4.4.c) is achieved.

Q.E.D.

Corollary 4.2: Canonical System with Matching Condition

Any canonical system satisfies the matching uncertainty condition.

Proof

Consider an uncertain dynamical canonical system

( ) ( )&~ ~ ~ ~

,x Ax B A A x B B= + = + + +u u∆ ∆

where

A A B B=

=

=

=

0 1 0 00 0 1 0

0 0 0 1

0 0 0 00 0 0 0

0 0 0 0

00

0

00

0

1 2 3 1 2 3

LL

M M M O MLL

LL

M M M O MLL

M M

a a a a a a a a b bn n n n

,~

, ,~

∆

∆ ∆ ∆ ∆

∆

∆

thus the matchin g condition is satisfied, since

∆ ∆~

.~

. .A x B B+ =u v

Q.E.D.


4.2.2. Invariance Condition to Disturbances.

Theorem 4.2: Sufficient Invariance Condition to Disturbance

Consider a linear system under disturbances

& . . .x A x B W= + +u v .

The s liding mode is invariant to disturbances if and only if the matching condition is satisfied W B= ∀ ∈ℜ. ,z z (4.5)

where

x B W A, , , , ,∈ℜ ∈ℜ ∈ℜ× ×n n n u v1 .

Proof

In the sliding mode

Hx Hx H Ax B W HAx HB HW= ⇒ = ⇒ + + = ⇒ + + =0 0 0 0& . . . .u v u va f

so the equivalent control

u veq = − +−HB H Ax Wa f a f1 .

then the sliding equation

& . .x I B HB H Ax W

Hx

= − +

=

RS|T|−a f a f1

0

v

thus, similarly to the case of uncertainties above, the sliding mode is invariant to disturbances if and only if I B HB H W 0 W B− = ⇔ = ∀ ∈ℜ−a f 1 . . . ,v z z

Q.E.D.

Remark 4.2: Invariance Property of Sliding Eigenvalues under Matching Condition

By Theorem 3.1 on a stability test, I have

• In case of the system without any perturbation, the sliding-eigenvalues are equal to the hyperplane-

eigenvalues, hence the Hurwitz hyperplane-eigenvalues do guarantee the stability of the system.

• Under perturbations satisfying the matching condition, the sliding-eigenvalues are absolutely unchanged

and equal to the hyperplane-eigenvalues, so the Hurwitz hyperplane-eigenvalues also do guarantee the

stability of the system.

4.3. A NEW ROBUST VSS CONTROLLER DESIGN (D ISCONTINUOUS SMC)

A discontinuous control is also called a switching control . As mentioned in Chapter 3, there are 2 VSS

control types, but only the first type can be extended into a robust VSS control. A robust VSS control

function consists of 3 components, namely equivalent, reaching and perturbation controls, where the first

two are similar to the normal VSS and the third additional one is to deal with perturbations. Different from the

VSS literature so far (White et al. 1984, Sivaramakrishnan et al. 1984, Hong et al. 1989, Lee et al. 1991), my

proposed design is extended from Chapter 3 where all designs are done in a unified manner.


First I propose an assumption which is practically satisfied in the real world

Assumption 4.1: System Constraint on Parametric Variation A system matrix B takes any variation such that the polarity of HB%e j is unchanged, i.e.

( ) ( )sgn sgnHB HB B B~ ~= ⇒ <∆ ∆ (4.6)

where

~ ~B B B= + ∆

Under Assumption 4.1, I propose the following theorem to design a robust VSS controller

Theorem 4.3: Robust VSS Controller Design under Uncertainty and Disturbance

For a linear system under perturbations (uncertainties and disturbances) ( ) ( )&

~.

~. .

~.

~. .x A x B W A A x B B W= + + = + + + +u v u v∆ ∆

and a hyperplane

s = H x.

with ∆ ∆ ∆ ∆

~,

~,A A B B≤ = = v v

then, under Assumption 4.1, there exists a constant δ > 0 for a VSS control function to be determined as u u u ue r p= + + (4.7)

where

• equivalent control ( )ue e e= − = −K x K HB HA. , 1 (4.7.a)

• reaching control ( ) ( )u sr r r= − =−HB K x K H1 . . , .sgn δ (4.7.b)

• perturbation control

( ) ( )( ) ( )

[ ]u K s Kv

xp p p p p i= − + = = =0 0K xH W

HB HBK

H A

HB HBx. . ,

.

inf ~ .sgn,

.

inf ~ .sgn,sgn

∆

(4.7.c)

with x B B W A A H, , , , , , , , , ,∆ ∆∈ℜ ∈ℜ ∈ℜ ∈ℜ ∈ℜ× × ×

+n n n n u s v1 1 δ a f

Proof I have

& &~ ~s u v= = + +Hx HAx HB HW

substituting Eq.(4.7) gives

( ) ( ) ( )&~ ~ ~

. . .

~

inf ~ . . . .s v s v s= + − − − +HAx HW HBHB

HAx HBHB

H xHB

HBH W H A xδ sgn sgn∆

( ) ( ) ( )&~

. . .

~

inf ~ . . . ~.~ ~

inf ~ . . . .s s v v s s= − + −

+ − −

HBHB

H x HWHB

HBH W H A x HB

HBHAx

HB

HBH A xδ sgn sgn sgn∆ ∆


or

ss s s s v s v s s& .~

. .~

.

~

inf~ . .

~. .

~

inf~ . . .= − − + −

+ −

HBHB

H xHBHB

HAx HWHB

HBH W H A x

HB

HBH A xδ ∆ ∆

then the Assumption 4.1 guarantees the existence of a sliding margin δ > 0 such that ss& < 0 for a bounded-

input bounded-output system.

Q.E.D.

Remark 4.3: Robust VSS Controller Design under No Perturbations

If there is no perturbation, I have

~,

~, , ,A A B B A 0 B 0= = = = =∆ ∆ v 0

then by Eq.(4.7.c), the perturbation control vanishes up = 0

and the control function in Eq.(4.7) becomes

u u ue r= +

it is exactly the same as in Eq.(3.4) for the normal case without perturbation.

Remark 4.4: Equivalent Control in Robust VSS Controller

Consider a system under no perturbations

&x Ax B= + u

and a hyperplane

s = Hx .

Then, by Eq.(3.4), a discontinuous VSS control function can be determined by

u u ue r= +

with

( )ue = −HB HAx1 : equivalent control

( ) ( )ur = −HB H x1δ. . .sgn s : reaching control.

Once in the sliding mode, s s= ⇒ =0 0sgnb g , by definition, so

u ue=

Now, under perturbation, A and B are unknown, so the equivalent control above, ( )ue =−

HB HAx~ ~1

, is

undefined. It is impossible to design a control function by the above formula. By Eq.(4.7), it is possible to

determine a control function since ( )ue = − −HB HA x1 . is defined. This component is termed "equivalent

control" since the control function reduces to this component once in the sliding mode, by definition it must

be the equivalent control.


Corollary 4.3: Robust VSS Controller under Reduced Parametric Variation

If

∆B 0=

then the VSS control function in Theorem 4.3 becomes u u u ue r p= + + (4.8)

where

• equivalent control ( )ue e e= − = −K x K HB HA. , 1 (4.8.a)

• reaching control ( ) ( )u sr r r= − =−HB K x K H1 . . , .sgn δ (4.8.b)

• perturbations control ( ) ( ) ( ) ( ) [ ]u K s K v xp p p p p i= − + = = =− −

0 01 1K x HB H W K HB H A x. . , . , . ,sgn ∆

(4.8.c)

with x B W A A H, , , , , , , , ,∈ℜ ∈ℜ ∈ℜ ∈ℜ ∈ℜ× × ×

+n n n n u s v1 1 ∆ δ a f

4.4. A NEW ROBUST LINEAR SMC DESIGN

First I start with a basic linear SMC without any perturbation where the control function is composed of

2 components: an equivalent control and a reaching control. Next I present a robust linear SMC under an un-

matched perturbation where the control function is composed of 3 components: an equivalent control, a

reaching control and a perturbation control. Then the matched perturbation is a special case of the un-

matched one. To the best of my knowledge, there is only one work on the robust continuous SMC in Zhou et

al. 1992.

Current Design Proposed Design

• pseudo-SMC: derivation unavailable • SMC: precise derivation

• non-linear continuous control function • linear continuous control function

• both sliding margin, and boundary layer are

used in design

• only sliding margin is used in design

• N. A. • sliding margin and hyperplane-eigenvalues

are system-eigenvalues

• N. A. • design separation: equivalent control,

reaching control, and perturbation control


4.4.1. Linear SMC Different from the SMC literature so far (DeCarlo et al. 1988, Zhou et al. 1992), it is shown that a

sliding margin and hyperplane-eigenvalues are the closed-loop system-eigenvalues. This is a criterion which

is used to choose a sliding margin in designing a linear SMC control function.

Theorem 4.4: Linear SMC Design

For a linear system

&x Ax B= + u

with a hyperplane

s = Hx

then a linear SMC function can be determined by u e r= − = − +Kx K K xb g. (4.9)

where

• equivalent control K HB HAe =

−a f 1 (4.9.a)

• reaching control K HB Hr =

−a f 1 .δ (4.9.b)

with x B A H, , , , , ,∈ℜ ∈ℜ ∈ℜ ∈ℜ ∈ℜ× × ×

+n n n n u s1 1 δ a f

Proof From a hyperplane equation & . & . . .s u u= = + = +−H x H A x B HB HB HAxa f a f 1

by the above control function u

s= − = − −− −Kx HB HAx HB Hxa f a f 11 1 . ;δ

hence & . . . . . & .s s s s s s= − = − ⇒ = − ≤−δ δ δHB HBa f 1 2 0

thus the sliding condition is satisfied.

Q.E.D.

To design a SMC, I must choose δ. To do so, I propose the following theorem.

Theorem 4.5: Closed-Loop Eigenvalues composed of Hyperplane Eigenvalues and Sliding Margin

If a system has a sliding margin δ and a hyperplane-eigenvalue λH λH

n∈ℜ −× −

a fa f1 1

then the system closed-loop eigenvalues are determined by { }λ λC H

n= − ∈ℜ ×δ, 1 (4.10)

Proof: By definition of eigenvalues: eig M I Ma f n s= − =λ λi idet 0 , from (4.9.a)

A BK A B HB HA H A B HB H A I− = − + = − +− −a f b g a f b g1 1. . .δ δ


To prove − δ is one of the eigenvalues, I have to prove

det det− − − = ⇒ − + =−δ δb g a f a f b g{ }. . .I A BK HB BH I A I0 01 (4.11)

By the properties of a determinant

P Q P Q Q P P Q. . . , ,= = ∈ℜ × n n

and I P Q I Q P P Qn m

n m m n+ = + ∈ℜ ∈ℜ× ×. . , ,

then Eq.(4.11) is satisfied since HB BH I HB HB Ia f a f− −− = − =1 1 0 .

Thus I can write

λ δ λ δ λI A B HB HA HB H− + + = + ×− − −a f a f b g c h1 1 1. P* n . (4.12)

Eq.(4.12) holds for all δ , so it must hold for δ = 0, then Eq.(4.12) becomes

λ λ λI A B HB HA− + = ×− −a f d i1 1P * n (4.13)

From Eq.(2.10), I obtain

λ λ λ λ λI A B HB HA− − = × = =−a f d i d i{ }1 1 1 0P PnH i

n- - : λ (4.14)

by Eq.(4.13), I achieve

P P* λ λn n− −≡1 1d i d i

this proves that λ H are the remaining eigenvalues.

Q.E.D.

Remark 4.5: Reduced Order of Sliding Mode

In the sliding mode, the system states are sliding on the hyperplane. Eq.(4.10) shows that the sliding

mode is reduced order.

4.4.2. Robust Linear SMC

In Section 4.3, both disturbances and uncertainties are simultaneously taken into account in designing a

robust VSS control function (discontinuous SMC). To design a robust linear SMC function, because of the

form of the mathematical derivations, disturbances and perturbations are considered separately.

For uncertain dynamical systems in Zhou et al. 1992 and Spurgeon et al. 1993, the control function is not

only complicated, but also leads to a pseudo-sliding mode since the sliding condition is not satisfied within

the boundary layer (Remark 3.4). Both the sliding margin and boundary layer are used in the design and it is

not clear how to determine them. In fact, there is no proof for the proposed formulas! Moreover, the control

function is nonlinear even for a linear system; and for both cases of disturbances and uncertainties, it is a

pseudo-SMC only!


I propose an alternative approach to designing a linear SMC function for the case of uncertainties. It is

simple for linear systems: only the sliding margin is used with the design rule (Section 3.4) and is not pseudo-

SMC. Moreover, the control function is linear for a linear system and even for a certain class of non-linear

systems. For the case of disturbances, I can obtain the pseudo-SMC only.

4.4.2.1. Robust Linear SMC under External Disturbances

Theorem 4.6: Robust Linear SMC Design under External Disturbance

Consider a linear system under external disturbances

& . . . ,x A x B W= + + ≤u v v v (4.15)

with a hyperplane equation

s = Hx

where

x B W A H, , ; ; ; , ,∈ℜ ∈ℜ ∈ℜ ∈ℜ× × ×n n n n u v s1 1 .

Then a robust linear SMC function can be found from u e r p= − = + +K x K K K K. , (4.16)

where

• equivalent control K HB HAe = −a f 1 (4.16.a)

• reachin g control

K HB Hr = −a f 1 .δ (4.16.b)

• perturbation control ( ) ( )K HB H HWp p v= −1 δ . sup . (4.16.c)

if the following disturbance condition is satisfied

( )

sv

vp

>+

sup .sup .HW

HWδ δ (4.17)

where δ p is a sliding margin for disturbance.

Proof

From Eq.(4.16), I have

( ) ( ) ( ) ( ) ( ){ }s s s v s vp. & . sup .= − − + + +− − − −HB HB HAx HB HW HB HAx HB HW1 1 1 1δ δ

or

( ){ } ( )s s s v s v v s v sp p. & . sup . sup . .= − + + = − + +δ δ δ δHW HW HW HW2

so

( ){ } ( )s s s v s v v s v sp p. & . sup . sup . .= − + + = − + +δ δ δ δHW HW HW HW2

since sup sup .HW HWv v=

From ( )δ δ+ >p vsup .HW 0


I consider the following 2 cases

• if HWHW

HWv

vvp

> ⇒+

>0 0δ δ sup .

s −∞ − 0 + HW

HWv

vpδ δ+ sup . + +∞

s . s& −∞ − 0 + 0 − −∞

• if HWHW

HWv

vvp

< ⇒+

<0 0δ δ sup .

s −∞ − HW

HWv

vpδ δ+ sup . − 0 + +∞

s . s& −∞ − 0 + 0 − −∞

Therefore, in the most conservative case, the sliding condition is satisfied when

sv

vp

>+

HWHW.

sup .δ δ

Q.E.D.

Remark 4.6: Disturbance Condition in Sliding Mode

The disturbance condition in Eq.(4.17) means that there is no sliding mode in the boundary layer of width sup .

sup .HW

HWv

vpδ δ+, so strictly speaking, it is a pseudo-sliding mode.

Remark 4.7: Validity of Theorem 4.6 under No Disturbance

If there is no disturbance, I have

sup .HW v = 0

then by Eq.(4.17.c), the perturbation control vanishes

K 0p =

and the control function in Eq.(4.17) becomes exactly the same as in Eq.(4.8) for the case without perturbation.

4.4.2.2. Robust Linear SMC under Parametric Uncertainties

Lemma 4.1: Computation of Special Eigenvalues

For any H ∈ℜ ×1 n and any M ∈ℜ ×n n , I have

Eig H HM H HM 0T T T+LNM OQP = ±d i α β , (4.18)

where α β= =HMH HH HM HMT T T, a fa f (4.18.a)

Proof: I will prove by induction. By the definition of eigenvalues Eig P I Pa f n s= − =λ λi i 0 , I can prove that

Eq.(4.18) is true for n = 2 . By induction, Eq.(4.18) is true up to n.

Q.E.D.

I propose the following theorem for both matched and un-matched uncertain dynamical systems.


Theorem 4.7: Robust Linear SMC Design under Parametric Uncertainty

Consider a linear system under both matched and unmatched parametric uncertainty

( ) ( )&~

.~

.~

.~

.x A x B A A x B B= + = + + +u u∆ ∆

and a hyperplane

s = H x.

where

∆ ∆ ∆ ∆~

,~

A A B B≤ ≤

with

x A A B B H∈ℜ ∈ℜ ∈ℜ ∈ℜ ∈ℜ× × × ×n n n n n u s1 1 1, , , , , , , ∆ ∆ .

then under Assumption 4.1, there exists a sliding margin δ > 0 for a robust SMC to be determined as u e r p= − = + +K x K K K K. , (4.19)

where


( )K HB HAe = −1 (4.19.a)

• reaching control

( )K HB Hr = −1 .δ (4.19.b)


( )

( )( )K

H A B K K

HB HBHp

e r=

+ +

⊗. .

inf~

.sgnsgn

∆ ∆ (4.19.c)

with

K K K Me r pn

ijm, , , ,∈ℜ ∈ℜ =×+

1 δ a f , [ ] [ ] [ ]p q p qij ij ij ij⊗ = .

Proof

I have

( ) ( )& ~ ~.

~. .

~.s u u u= + + + = + + +H A A x H B B HAx H A x HB H B∆ ∆ ∆ ∆

by Eq.(4.19), I obtain ( )& .

~. .

~. .

~s s e r p= − − + − + −HAx HAx H A x H B K K x HBK xδ ∆ ∆

so

( )[ ]ss s s sp e rT T& .

~ ~.

~.= − − + + − = − −δ δ2 2H BK B K K A x x H HM x∆ ∆ (4.20)

where ( )M BK B K K A= + + −

~ ~.

~p e r∆ ∆ . (4.21)

The quadratic form requires a symmetric matrix, Eq.(4.20) can be read as

ss s T T T T& = − − +LNM OQPδ 2 1

2 x H HM H HM xd i (4.22)

The sliding mode ss& < 0 is satisfied if the quadratic term is non-negative, that is the matrix is positive-semi-

definite, so all eigenvalues must be non-negative. By Lemma 4.1, this can be achieved if both sum and product

of 2 eigenvalues are non-negative, that is

2 0α ≥ (4.23.a)

α β2 0− ≥ (4.23.b)


I will use the first condition in Eq.(4.23.a) to compute the perturbation control and the second condition will be

used to check.

(a)

Eq.(4.18.a) can be written as

α = H M H. . T (4.24)

In view of Eqs.(4.19.c) and (4.21), I get

( )

( )( ) ( )[ ]α = =

+ +

⊗

− − +HMH HB

H A B K K

HB HBH H H A B K K HT e r T

e rT~ . .

inf~

.signsgn . .

~ ~. .

∆ ∆∆ ∆

or

( ) ( )[ ]α = + + − − +HB

HBH A B K K H H A B K K H

inf~ . . . .

~ ~. .∆ ∆ ∆ ∆e r

Te r

T

since HB

HBinf %>1 by Assumption 4.1, so

( ) ( )[ ]α > + + − − + >H A B K K H H A B K K H. . . .~ ~

. .∆ ∆ ∆ ∆e rT

e rT 0

(b)

By Eq.(4.18.a), Eq.(4.23.b) can be written as

α β2 2 2 2= =LH H LTd i , (4.25)

where

L HM= (4.25.a)

By Schwartz's inequality LH L HT ≤ ⇒ ≤. α β (4.26)

so there is a negative eigenvalue. However, both sides of Eq.(4.26) are the same order, so the negative

eigenvalue can be expected too small compared to the positive one, ie. the quadratic term in Eq.(4.20) will

contribute negative value to the sliding condition. In addition, the first term in Eq.(4.20) will even make more

negative.

Q.E.D.

Remark 4.8: Robust Linear SMC Design under No Uncertainty

If there is no uncertainty, I have

~

,~

,~

,~

A A B B A 0 B 0= = = =∆ ∆

then by Eq.(4.26.c), the perturbation control vanishes, and the control function in Eq.(4.26) is exactly the same

as in Eq.(4.8) for the case without perturbation.

Corollary 4.4: Robust Linear SMC Design under Reduced Parametric Variation

If

∆B 0=

then the linear SMC in Theorem 4.7 becomes


u e r p= − = − + +K x K K K x. d i (4.27)

where

• equivalent control ( )K HB HAe = −1 (4.27.a)

• reaching control ( )K HB Hr = −1 .δ (4.27.b)

• perturbation control ( ) ( ) ( )K HB H A Hp = ⊗−1 . sgn∆ (4.27.c)

Remark 4.9: Equivalent Control in Robust SMC Design

The first component in Eq.(4.27) is termed "equivalent control" due to the same the argument in Remark

4.4, that is the control function reduces to the equivalent control once in the sliding mode.

4.4.2.3. Robust Linear SMC under Matched Parametric U ncertainties

Under uncertainties, all that can be guaranteed is that if the sliding-eigenvalues are Hurwitz, then so are

the system-eigenvalues, ie. the closed-loop system is stable (Theorem 3.1). Under the matching condition, I

have very interesting characteristics by the following theorem.

Theorem 4.8: Robust Linear SMC Design under Matched Uncertainty

If a matched uncertain system has all elements of H of the same polarity, then the system-eigenvalues at

the upper boundary are { }λ λC H, ,2 = −δ (4.28)

Proof

From Eq.(4.3), a matched uncertain system takes the form

( )& ~. .x A A x B= + +∆ u

where

∆~

.A B v=

From Eq.(4.19.c) with ∆ ~B 0= and with hi of the same polarity, without loss of generality, assume positive

polarity ( ) ( )K HB H A HB H B v vp = = ± = ±− −1 1. . .∆ (4.29)

Thus the system-eigenvalues at the upper boundary, A A A= + ∆ , are ( )[ ] ( )[ ] [ ]λ C e r p er er er e r2 = + − + + = ± − + = − = +eig eig eigA A B K K K A B v B K v A B K K K K∆ . . . . ,

Recall that, under no perturbation I have ( ) { }eig .A B K− = −er Hδ, λ

so [ ] { }λ λC er H2 = − = −eig A B K. ,δ

Q.E.D.


Theorem 4.8 has shown that system eigenvalues of an uncertain dynamical system at the upper boundary ( )A A+ ∆ are equal to those of a deterministic dynamical system. I propose the following design rule to move

these eigenvalues inside the bounds by shifting down the nominal values of the system matrix.

Proposition 4.1: Robust Linear SMC Design Rule

Theorem 4.8 is valid for matched uncertain systems and H has all elements of the same polarity. In

general, choose the nominal system matrices as

A A A B B B→ + → +12

12∆ ∆,

in designing a hyperplane may produce a more robust controller.

Remark 4.10: Advantage of Robust Design Rule

An advantage from the proposition above is that the system-eigenvalues of the nominal model are { }−δ, λ H for matched uncertain systems and H has all elements of the same polarity

4.4.2.4. Robust Linear SMC under Un-matched Uncertainty and Disturbances

From Theorem 4.6 and 4.7, I propose the following corollary to design a robust linear SMC for systems

under unmatched uncertainty and disturbances

Corollary 4.5: Robust Linear SMC Design under Un-matched Uncertainty and Disturbance

Consider a linear system under unmatched parametric uncertainty and external disturbances

( ) ( )&~

.~

. .~

.~

. .x A x B W A A x B B W= + + = + + + +u v u v∆ ∆ (4.30)

and a hyperplane

s = H x.

where

∆ ∆ ∆ ∆~

,~

,A A B B≤ ≤ ≤ v v

with

x A A B B H∈ ℜ ∈ ℜ ∈ ℜ ∈ ℜ ∈ ℜ× × × ×n n n n n u s1 1 1, , , , , , , ∆ ∆ .

Then, under Assumption 4.1, a robust linear SMC function is determined by u e r p= − = + +K x K K K K. , (4.31)

where


( )K HB H Ae = −1 . (4.31.a)


( )K HB Hr = −1 .δ (4.31.b)

• perturbation control K K Kp u d= + (4.31.c)

* uncertainty control


( )

( )( )K

H A B K K

HB HBHu

e r=

+ +

⊗. .

inf~

.sgn. sgn

∆ ∆ (4.31.d)

* disturbance control ( ) ( )K HB H HWd p v= −1δ . sup . (4.31.e)

if the following disturbance condition is satisfied

sv

vp

>+

sup .sup .HW

HWδ δ (4.31.f)

with

( ) [ ]K K K K K Me r p u dn

p ijm, , , , , , ,∈ℜ ∈ℜ =×+

1 δ δ

Proof

Eqs.(4.31.a), (4.31.b) and (4.31.d) can be determined by Theorem 4.7. Since the uncertainty control tackles

uncertainty, Eq.(4.30) can be read as

& . . .x A x B W= + +u v (4.32)

so Eq.(4.31.e) can be determined from Eq.(4.32) using Theorem 4.6.

Q.E.D.

4.4.2.5. Alternative Formula for Robust Linear SMC

Alternative to Theorem 4.7, under Assumption 4.1, I propose a theorem for a robust linear SMC. I first

present the following lemma

Lemma 4.2: Alternative Computation of Special Eigenvalues

For any V ∈ℜ ×1 n and any M ∈ℜ ×n n , I have [ ] [ ]Eig . , , ,V V M VMVT T= 0 0L (4.33)

Proof:

I will prove by induction. By the definition of eigenvalues ( ) { }Eig P I P= − =λ λi i 0 , I can prove that

Eq.(4.33) is true for n = 3 .

Let

P V VM3

1 11

3

1 21

3

1 31

3

2 11

3

2 21

3

2 31

3

3 11

3

3 21

3

3 31

3

= =

= = =

= = =

= = =

∑ ∑ ∑

∑ ∑ ∑

∑ ∑ ∑

T

i ii

i ii

i ii

i ii

i ii

i ii

i ii

i ii

i ii

v v m v v m v v m

v v m v v m v v m

v v m v v m v v m

. (4.34)


so

λ λ λI V VM− = −FHG

IKJ==

∑∑Ti j ij

ji

v v m2

1

3

1

3

(4.35)

Assume that Eq.(4.33) is true up to n −1 . Note that the non-zero eigenvalue in Eq.(4.35) is equal to the

sum of diagonal elements in Eq.(4.34). So I can prove that Eq.(4.33) is also true for n by using some properties

of the trace of a matrix as follows

tr Pn iii

n

pb g ==∑

1

tr , EigP Pn ii

n

n ib g b g l q= ==

∑λ λ1

tr tr trP P P Pa b a b+ = +b g b g b g

where

[ ]P V VMnT

n

n

n

n

n n nn

n

n

n n n n

n

n

n n nn

vv

v

v v v

m m mm m m

m m m

v v v v v vv v v v v v

v v v v v v

m m mm m m

m m m

= =

=

1

21 2

11 12 1

21 22 2

1 2

1 1 1 2 1

2 1 2 2 2

1 2

11 12 1

21 22 2

1 2

ML

LL

M M O ML

LL

M M O ML

LL

M M O ML

or

Pn

i ii

n

i ii

n

i ini

n

i ii

n

i ii

n

i ini

n

n i ii

n

n i ii

n

n i ini

n

v v m v v m v v m

v v m v v m v v m

v v m v v m v v m

=

= = =

= = =

= = =

∑ ∑ ∑

∑ ∑ ∑

∑ ∑ ∑

1 11

1 21

11

2 11

2 21

21

11

21 1

L

L

M M O M

L

and

P P P P P Pn a b a n b

i ini

n

i ini

n

n i ii

n

n i ii

n

n i ini

n

v v m

v v m

v v m v v m v v m

= + = =

−

=

=

= = =

∑

∑

∑ ∑ ∑

, ,1

11

21

11

21 1

0 0

0 0

L

M O M

L M

L

Q.E.D.

I then have the following theorem

Theorem 4.9: Alternative Robust Linear SMC Design


( ) ( )& ~ ~ ~ ~,

~,

~x Ax B A A x B B A A B B= + = + + + ≤ ≤u u∆ ∆ ∆ ∆ ∆ ∆ (4.36)

and

s = Hx .

Then under Assumption 4.1, a robust SMC can be determined by u e r p= − = + +Kx K K K K, (4.37)

where



( )K HB HAe = −1 (4.37.a)


( )K HB Hr = −1 δ (4.37.b)


( )( )( )

K HB HH A B K K H

HB H B HHp

e rT

T= =

+ +

−−

−

1

11γ γ,

.

.

∆ ∆

∆ (4.37.c)

Proof:

From Eq.(4.37), I have

& ~ ~.

~. .

~.s u u u= + = + + +HAx HB HAx H A x HB H B∆ ∆

by Eq.(4.37), I obtain ( )& .

~. .

~. .

~s s e r p= − − + − + −HAx HAx H A x H B K K x HBK xδ ∆ ∆

so

( )[ ] ( )[ ]ss s s sp e rT T

p e r& .~ ~

.~

.~ ~

.~

= − − + + − = − − + + −δ δ2 2H BK B K K A x x H H BK B K K A x∆ ∆ ∆ ∆

thus ss& < 0 only if ( )[ ]H H BK B K K ATp e r.

~ ~.

~+ + −∆ ∆ is semi-positive definite. If so required, Lemma 4.2

yields

( )[ ]λ = + + − >H BK B K K A H.~ ~

.~

p e rT∆ ∆ 0 (4.38)

In view of Eq.(4.19.c), Eq.(4.38) can be read as

( )[ ] ( )[ ]λ = + + − = − − +H BK B K K A H HBK H H A B K K H.~ ~

.~ ~ ~ ~

.p e rT

pT

e rT∆ ∆ ∆ ∆

( )

( ) ( )[ ]λ =+ +

−− − +

−

H A B K K H

HB H BH A B K K H

∆ ∆

∆∆ ∆

.

.

~ ~.

e rT

e rT

1 1

or ( ) ( )[ ]λ > + + − − +H A B K K H H A B K K H∆ ∆ ∆ ∆.

~ ~.e r

Te r

T

since Eq.(4.6) in Assumption 4.1 gives

H B HB H B HB HB H B HB H B.~

.∆ ∆ ∆ ∆< ⇒ < < ⇒ < < ⇒ < − <− −0 0 1 0 1 11 1

so Eq.(4.38) can be achieved.

Q.E.D.

If ∆~B 0= , I have the following corollary

Corollary 4.6: Alternative Robust Linear SMC Design under Reduced Parametric Variation


( )& ~ ~,

~x Ax B A A x B A A= + = + + ≤u u∆ ∆ ∆

and

s = Hx

then a robust SMC can be determined by u e r p= − = + +Kx K K K K,

where



( )K HB HAe = −1


( )K HB Hr = −1 δ


( )K HB HH A H

HHp

T

T= =−1 γ γ,. .∆

4.5. ROBUST INTEGRAL SMC DESIGN

A SMC is based on a state-space model, so an I-action may be required to eliminate a steady-state error. I

have the following corollary to design a robust integral SMC using Theorem 4.3 or 4.9.

Theorem 4.10: Robust Integral Linear SMC Design under Uncertainty

Consider an uncertain dynamical linear system system

( ) ( )& ~ ~ ~ ~,

~,

~x Ax B A A x B B A A B B

Cx

= + = + + + ≤ ≤

=

u u

y

∆ ∆ ∆ ∆ ∆ ∆

and its augmented-order system with a reference input of r

& ~.

~.x A x Bi i i i u= + (4.39)

where

~

~ ,~

~AC

0 AB

Bin

i=

=

×

0 0

1. (4.39.a)

Then a robust integral SMC with a hyperplane Hi can be determined by u u u i= +*

~ (4.40)

where ( ) [ ]u h r h hi i i i i i n* , , ,,= =−+H B H1

1 1 1L (4.40.a)

and

~u i is a robust SMC for Eq.(4.39) and can be determined by Theorem 4.3 or 4.7.

with

x B A C, , , , , , ~ , ,*∈ℜ ∈ℜ ∈ℜ ∈ℜ× × ×n n n niu u u y r1 1 : sliding margin.

Proof

Let

( )x y r dt x y rt

00

0= − ⇒ = −∫ &

then

&& ~ ~ ~ ~ . ~ ~ . .

x y ru u

rx

u rn n n

0

1 1

0

1

0 1 0 0 1

x Ax BCxAx B 0

C

0 A x B 0

=

−+

=

+

+

−

=

+

+

−

× × ×

from Eq.(4.39.a), I have

& . . .x A x B0i i i i

n

u r= + +−

×

1

1 (4.41)


since the uncertain component of u can tackle any uncertainty ∆ %Bi , Eq.(4.41) is still hold for ∆ %B 0i = so it can

be read as

&~

.~~

. .~

. ~ . .x A x B0

A x B0i i i

ni i

n

u r u r= + +−

= + +

−

× ×

1 1

1 1 (4.41.a)

thus

& &~ ~ . .

~ ~,s u r u h ri i i i i i i i i

ni i i i i i= = + +

−

= + −

×

H x H A x H B H0

H A x H B1

11

by Eq.(4.40.a), I have

&~ ~ ~ ~ ~

*s u u ui i i i i i i i i i i i i i= + − = +H A x H B H B H A x H B (4.42)

since ~u i is a robust SMC for Eq.(4.39), the sliding condition ~~&s s ≤ 0 is satisfied using Eq.(4.42)

Q.E.D.

4.6. A NEW ROBUST LINEAR SLIDING-MODE OBSERVER DESIGN

As in the state-space observer design where an the response of an observer is 3 to 10 faster than that of

a controller, to design a robust linear Sliding-Mode Observer (SMO), I propose the following theorem

Theorem 4.11: Robust Linear Sliding-Mode Observer Design under Uncertainty

Consider the following uncertain dynamical system

( ) ( )

( )& ~ ~ ~ ~

~ ~

x Ax B Ax A x B B

Cx C C x

= + = + + +

= = +

u u

y

∆ ∆

∆ (4.43)

where

∆ ∆ ∆ ∆~

,~

A A B B≤ ≤ .

If the system is observable, then a robust linear SMO can be found from

( )$& $ $ $ ~ $x Ax B L y y , y Cx= + + − =u (4.44)

where L L L L= + +e r p (4.44.a)

with

( )L AH CHe o o=−/ / 1

(4.44.b)

( ) ( )L H CHr o o o o= = ×−

δ δ δ/ / , ..1

3 10 (4.44.c)

( ) ( )( )

L HA L L C H

H C H Cp o

e r o

o o

= ⊗+ +

sgn. .

inf~

.sgn

/ /

/ /

∆ ∆ (4.44.d)

or the alternative

( ) [ ]( )

L H CHH A C L L H

H C H C H Hp o o o o

o e r o

o o o o

= =+ +

−

−

−γ γ/ /

/ / /

/ / /,

. . .

. . .

1

11

∆ ∆

∆ (4.44.e)

and

Ho is determined by the eigenvalue allocation with the observer hyperplane-eigenvalues

( ) ( )λ λHo Hc o Ho= × ⇒ =3 10.. hyper , ,/ /H A C λ

$ , , , , , , , , , ,x y A A B B C C H∈ℜ ∈ℜ ∈ℜ ∈ℜ ∈ℜ ∈ℜ ∈ℜ× × × × × ×n p n n n p n n u s1 1 1 1 ∆ ∆ ∆


Proof

Since L will be determined under uncertainty, Eq.(4.44) can be perturbed to read as

( ) ( ) ( ) ( )$& ~ $ ~. $ ~ $ ~

. $x A A x B B L y y Ax B L y y= + + + + − = + + −∆ ∆ u u

then an error equation e x x= −$ can be determined by

( ) ( )& $& & ~ $ ~. $ ~

.~

.~ ~

.e x x Ax B L y y A x B A LC e= − = + + − − − = −u u (4.45)

To determine L, I will find a function v to nullify the error by the following equation

& ~ ~/ /e A e C= + v (4.46)

By Theorem 4.7, I can find

v = −L e/ (4.47)

where L/ is determined by Eqs.(4.44), so Eq.(4.46) can be read

( )& ~ ~ ~ ~ ~/ / //

e A e C L e A LC e= − = − (4.48)

since eig eigM Ma f c h= / , I have the following mapping to complete the proof

~ ~

,~ ~

, ,/ / /A A B C H H K L→ → → →o (4.49)

so Eqs.(4.44) have been proved.

Q.E.D.

4.7. ANALYSIS OF MATCHED AND UNMATCHED UNCERTAINTIES

I will analyze matched and unmatched uncertainties in transfer function and state-space models.

4.7.1. Transfer Function Model

Consider a system

G sb b s

s a s aa f b g

=+

+ +0 1

21 0

1 (4.50)

then the minimal realization is

&x Ax B

Cx D= += +

uy u

(4.51)

where

A B C D=− −LNM

OQP =

LNM

OQP = =

0 1 01 0

0 1 01a a b

b, , , , (4.51a)

I want to find a transformation T

x Tx= % (4.52)

such that CT = 1 0, (4.53)

thus

T T=−

LNM

OQP ⇔ =

−LNM

OQP

−10 1

10 1

1 1 1b b (4.54)

then y x= = = = =Cx C Tx CT x C x% . % % . % %b g a f 1 (4.55)


and Eq.(4.51) becomes

%& % % %

% %

x Ax B

Cx

= +

=

RS|T|u

y (4.56)

where

% , % , % ,A T AT B T B C CT= =− − + −

− +

LNM

OQP = =

−LNM

OQP = =− −1 1 0 1 1 1

20

0 1 1 0

1 1 0

0

11 0

b a b a b a

a a b a

b b

b (4.57)

(a) If b1 0= , then Eqs.(4.50) & (4.57) gives

G sb

s a s aa f =

+ +0

21 0

(4.58)

and

% , % , % ,A B C=−−

LNM

OQP =

−LNM

OQP =

0 1 01 0

0 1 0a a b (4.59)

Thus, if all parameters are uncertain, I have matched uncertain system.

(b) If b1 0≠ , assume that

u k kxx

k x k x= − = −LNM

OQP = − −Kx% .

%%

% %1 21

21 1 2 2 (4.60)

then Eqs.(4.56) & (4.57) give

%& $ % $

% %

x Ax B

Cx

= +

=

RS|T|u

y (4.61)

where

$ , $A B=− − − + − −

− +

LNM

OQP =

−LNM

OQP

b a k b b b a b a k b b

a a b a b1 0 1 1 0 1 1 1

20 2 1 0

0 1 1 0 0

1 0 (4.62)

thus it is unmatched uncertain system.

I can generalize the above analysis to have the following proposition

Proposition 4.1: Uncertainty in Transfer Function Model

In general,

• A matched uncertain system has the model

G sb

s a s a s ann

na f =+ + + +−

−

%

% % %0

11

1 0L (4.63)

where the minimal realization can be used.

• An unmatched uncertain system has the model in Eq.(4.63), but its minimal realization cannot be applied

due to inaccessibility of system states, or it has the model

G sb s b s b

s a s a s an

n

nn

na f =+ + +

+ + + +−

−

−−

% % %

% % %1

11 0

11

1 0

LL

(4.64)

where % , %a bi i are uncertain.


4.7.2. State -Space Model

I will analyze an uncertain state-space model and consider whether I can convert an unmatched uncertain

system into a matched uncertain system.

4.7.2.1. Analysis of Uncertain State-Space Model

Consider an uncertain system

& % % % . % .x A A x B B Ax A x B B= + + + = + + +∆ ∆ ∆ ∆e j e ju u u (4.65)

Let

∆ %A Wz= (4.66)

and

∆ % . .B Bu v= 1 (4.67)

where

∆ % ,A W∈ℜ ∈ℜ× ×n n n 1

and ∀ ∈ℜ ∀ ∈ℜ×z 11

n v, are uncertainties . Then Eq.(4.63) becomes

& .x Ax Wz x B B= + + +u v1 (4.68)

Since z is an uncertainty, let

zx = v2 (4.69)

where v2 is also an uncertainty; then Eq.(4.68) can be written as

&x Ax B W= + + +u v v1 2b g (4.70)

Therefore,

• for a matched uncertain system, I have

W B= (4.71)

so Eq.(4.70) becomes

&x Ax B Ax B= + + + = + +u v v u v1 2b g a f (4.72)

where v v v= +1 2 , thus I have a disturbance to u.

• for an unmatched uncertain system, I have both disturbances to u and to &x . However, to determine the

bound of v2 , I need to know the bound of x, ie. the ranges of x i ni , , , ,= 1 2 L . Eq.(4.69) reveals that the

bound of v2 depends on the bound of xi which are proportional to the reference, for example, while it is

not the case for a normal disturbance. For example, consider an unmatched uncertain system

A B A B=L

NMMM

O

QPPP

=L

NMMM

O

QPPP

=L

NMMM

O

QPPP

=L

NMMM

O

QPPP

0 1 0

0 1

0

0

0 0 0

0 0

0 0

0

01

2 3 4 1

1

2 1

a

a a a b

a

a b

, , % %

%, %

%∆ ∆

∆∆

∆ (4.73)

then choosing

W z=L

NMMM

O

QPPP

=0

1

1

0 1 2, % %∆ ∆a a (4.74)


gives, by Eq.(4.69)

v a a

x

x

x

x a x a2 1 2

1

2

3

2 1 3 20= =L

NMMM

O

QPPP

= +zx ∆ ∆ ∆ ∆% % . . % . % (4.75)

so

v x a x a2 2 1 3 2= +. .∆ ∆ (4.76)

where v v x x x x a a a a2 2 1 1 2 2 1 1 2 2≤ ≤ ≤ ≤ ≤, , , % , %∆ ∆ ∆ ∆

4.7.2.2. Conversion of Uncertain State -Space Model

Consider the state-space model

&x Ax B X AX B I A X B= + ⇒ = + ⇒ − =u s U s Ua f (4.77)

then

y Y s U G sYU

s= ⇒ = = − ⇒ = = −− −Cx CX C I A B C I A Ba f a f a f1 1 (4.78)

where

A B=L

NMMM

O

QPPP

=L

NMMM

O

QPPP

0 1 0

0 1

0

01

2 3 4 1

%

%,

%a

a a a b (4.79)

thus

G sb

s a s a a s a a aa f b g=

− − + + −

%

% % %1

34

23 1 1 4 2

(4.80)

so

%

% % %, %

%, %A B C=

− +

L

NMMM

O

QPPP

=L

NMMM

O

QPPP

=0 1 00 0 1

00 1 0 0

2 1 4 3 1 4 1a a a a a a bb g b g (4.81)

I will find a transformation T such that

%x Tx= (4.82)

then

%& % % %x Ax B= + u (4.83)

where

% , %A TAT B TB= =−1 (4.84)

thus solving the equation

%

%

AT TA

B TB

=

=

RS|T| (4.85)

to obtain

T =L

NMMM

O

QPPP

1 0 0

0 1 0

0 11%a

(4.86)


Due to uncertainty, %a1 is unknown, and so is the transformation T. It is therefore impossible to transform the

unmatched uncertain model in Eq.(4.79) by Eq.(4.82) into the matched uncertain system in Eq.(4.81).

Therefore, the above analysis yields the following proposition

Proposition 4.2: Conversion of Unmatched to Matched Uncertain System

In general, it is impossible to find a (constant) transformation from an unmatched to matched uncertain

system. However, if all system states are inaccessible except the output and an observer is required, then

there may exist a conversion via transfer function to transform an unmatched to matched uncertain system

(Example 4.8).

4.8. NUMERICAL EXAMPLES In order to see the elimination of chattering, the discontinuous SMC (VSS) will be included. The Tanh

VSS control will be included to compare with the linear SMC since both are able to eliminate chattering, but

the former is a pseudo-sliding mode while the latter is a true sliding mode. So I sumarize the results as

follows

Remark 4.11: Summary of Robust SMC Designs


the same unique value for simplicity. Different multiple values may be attempted to compromise between the

response speed and overshoot.

• Nominal model can be determined using proposition 4.1 in designing a hyperplane (Chapter 2)

• Sliding margin δ is chosen on the basis of Proposition 3.1;

• New Robust Discontinuous SMC functions are computed by Theorem 4.3 where the switching function

is given by Eq.(3.5). Saturate, Unitvector and Tanh VSS control functions are given by Eqs.(3.6) to

(3.8), respectively, based on Proposition 3.2;

• New Robust Linear (Continuous) SMC functions are computed by Corollary 4.5 deduced from

Theorems 4.6 and 4.7 for a general case under uncertainty and disturbance where a special case without

perturbation (uncertainty and/or disturbance) will make the corresponding control component(s) vanish;

• Robust Integral SMC functions are computed by Theorem 4.10;

• New robust sliding-mode observers are computed by Theorem 4.11


4.8.1. Example 4.1: No Perturbation

Consider a linear system (Sivaramakrishnan et al. 1984)

& . . ,

.

. .. . .

.

.x A x B A B= + =

−−

− − −

=

u

0 05 6 0 0

0 3 333 3 333 05 208 0 12 5 12 5

0 6 0 0 0

00

12 5

0

,


From Sivaramakrishnan et al. 1984, I have the original control function as

u x x x x= + + +ψ ψ ψ ψ1 1 2 2 3 3 4 4

where

ψ

ψ

11

1

33

3

6 06 0

2 02 0

=− >+ <

RST=

− >+ <RST

, ., .

, ., .

if if

if if

s xs x

s xs x

ψ

ψ

22

2

43

3

6 06 0

0 00 0

=− >+ <

RST=

><

RST

, ., .

, ., .

if if

if if

s xs x

s xs x

and s = =H x H. , . , . , 5 155 4 385 1 16

0 1 2 3-0.5

0

0.5

1

Time [s]

Out

put

0 1 2 3-20

-10

0

10

20

Time [s]

Con

trol

0 1 2 3-2

0

2

4

6

Time [s]

Slid

ing

-0.5 0 0.5 1-1

-0.5

0

0.5

Output

Cha

nge

of O

utpu

t

Original Switching SMC Control for 4-th Order System

Fig. 4.1: Original Switching SMC Control for Linear System in Example 4.1.

4.8.1.2. New Design To compare to the original design, choose [ ]λ H = − − −6 6 6, , , then

[ ]H = 0 4285 0 3508 0 0800 14401. , . , . , .

thus [ ]K e = −0 4260 1 4014 0 1694 1. . .

and the sliding-eigenvalues λ λS H= = − − −6 6 6, ,


• Tanh SMC (Continuous Pseudo-SMC)

Choose ks = 20 and δ = 10 , then Theorem 4.3 yields

[ ]K r = 4 2845 3 5084 0 8 14 4014. . . .

0 1 2 3-0.5

0

0.5

1

Time [s]

Out

put

0 1 2 3-10

-5

0

5

Time [s]

Con

trol

0 1 2 3-0.2

0

0.2

0.4

0.6

Time [s]

Slid

ing

-0.5 0 0.5 1-1

-0.5

0

0.5

Output

Cha

nge

of O

utpu

t

TanH SMC Control for 4-th Order System

Fig. 4.2: Tanh SMC Control for Linear System without Perturbation in Example 4.1.

• Linear SMC

Choose δ = 18 , then Corollary 4.5 yields

[ ]K r = 7 7122 6 3152 1 4400 25 9226. , . , . , .

thus [ ]K = 81382 77165 16094 24 9226. , . , . , .

and the system-eigenvalues

λC = − − − −18 6 6 6, , ,

0 1 2 3-0.5

0

0.5

1

Time [s]

Out

put

0 1 2 3-10

-5

0

5

Time [s]

Con

trol

0 1 2 3-0.2

0

0.2

0.4

0.6

Time [s]

Slid

ing

-0.5 0 0.5 1-1

-0.5

0

0.5

Output

Cha

nge

of O

utpu

t

Linear SMC Control for 4-th Order System

Fig. 4.3: Linear SMC for Linear System without Perturbation in Example 4.1.


• Performance

The performance of the Tanh SMC control is comparable to that of the linear SMC. Note that the former is

a pseudo-sliding mode, whereas the latter is a true sliding mode. Moreover, the latter is a linear control so the

linear control theory can be applied to obtain more insight such as the closed-loop system eigenvalues.

The following are modified models to include perturbations where their hyperplane equations, and

equivalent controls are exactly the same as in the Example 4.1 above, while reaching control is modified for

proper reaching dynamics due to the design rule (Proposition 3.1)

4.8.2. Example 4.2: Matched Uncertainties

Consider a matched uncertain linear system in Coleman et al. 1994

( ) ( )&~

.~

.x A A x B B= + + +∆ ∆ u

where

A A B B=− −

=

=

=−

0 1 00 0 10 1 2 5

0 0 00 0 00 1 0 5

0010

005.

,.

, , ∆ ∆

Choose [ ]λ H = − −2 2 , then

[ ]H = 0 5254 0 5293 0 1333. . .

thus [ ]Ke = 0 0 4587 0 2293. .

due to the matching condition, the sliding-eigenvalues are absolutely unchanged under matched uncertainties

and equal to the hyperplane-eigenvalues: [ ]λ λ λS S H1 2 2 2= = = − −

I have [ ]K p = 0 0 2 0 1. .

choose δ = 3 , then Theorem 4.3 yields

[ ]K r = 15761 1 5880 0 4. . .

0 2 4 60

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 2 4 6-6

-4

-2

0

2

4

Time [s]

Con

trol

0 2 4 6-0.6

-0.4

-0.2

0

0.2

Time [s]

Slid

ing


0 0.5 10

0.2

0.4

0.6

0.8

Output

Cha

nge

of O

utpu

t

Robust Switching SMC Control for Matched Uncertain 3-rd Order System

Fig. 4.4: Robust SMC Control for Matched Uncertain System in Example 4.2.


• Tanh SMC (Continuous Pseudo-SMC)

Choose ks = 25

0 2 4 60

0.5

1

1.5

Time [s]

Out

put

0 2 4 6-2

-1

0

1

2

3

Time [s]

Con

trol

0 2 4 6-0.6

-0.4

-0.2

0

0.2

Time [s]

Slid

ing


0 0.5 1 1.50

0.2

0.4

0.6

0.8

OutputC

hang

e of

Out

put

Robust TanH SMC Control for Matched Uncertain 3-rd Order System

Fig. 4.5: Robust Tanh SMC Control for Matched Uncertain System in Example 4.2.

• Linear SMC

I have [ ]K p = 2 6268 33054 0 9960. . .


[ ]K r = 26268 2 6467 0 6667. . .

thus [ ]K = 5 2536 6 4107 1 8920. . .

but the system-eigenvalues always vary with the uncertainties [ ] [ ]λ λC C1 21 41 2 27 98 1 69 2 7 77= − − − = − − −. . , . .

0 2 4 60

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 2 4 6-2

0

2

4

6

Time [s]

Con

trol

0 2 4 6-0.6

-0.4

-0.2

0

Time [s]

Slid

ing


0 0.5 10

0.2

0.4

0.6

0.8

Output

Cha

nge

of O

utpu

t

Robust Linear SMC for Matched Uncertain 3-rd Order System

Fig. 4.6: Robust Linear SMC for Matched Uncertain Dynamical System in Example 4.2.


• Performance

The performance of the Tanh SMC control is comparable to that of the linear SMC. Note that the forme r is

a pseudo-sliding mode, whereas the latter is a true sliding mode. Moreover, the latter is a linear control so


Remark 4.12: Efficiency of Robust Linear SMC Design Rule (Proposition 4.1)

Without using Proposition 4.1, I have the following system eigenvalues [ ] [ ]λ λC C1 21 44 2 20 47 2 2 5= − − − = − − −. . ,

which is less robust than the above eigenvalues calculated using the proposition. Also note that λ C2 verifies

Theorem 4.6.

4.8.3. Example 4.3: Un-matched Uncertainty

Consider an un-matched uncertain linear system in Chen et al. 1989 where disturbance is deferred to the

next example

( ) ( )&~

.~

.x A A x B B= + + +∆ ∆ u

where

A B A B=− −

=

=

=

0 1 04 0 18 7 8

00

1 3

0 0 02 0 05 0 0

00

0 3,

., ,

. ∆ ∆


[ ]H = 20 6897 6 8966 0 6897. . .

thus [ ]Ke = 417241 158621 13793. . .

due to un-matched uncertainties, the sliding-eigenvalues are changed under un-matched uncertainties, but

they are still Hurwitz: [ ] [ ]λ λS Sj1 273 4 6= − ± = − −5 1 . ,

• Robust Switching SMC (Discontinuous SMC)

I have [ ]K p = 25 0 0

Choose δ = 2 , then Theorem 4.3 yields

[ ]K r = 413793 137931 13793. . .


0 1 2 3 4-0.5

0

0.5

1

1.5

Time [s]

Out

put

0 1 2 3 4-200

-100

0

100

200

Time [s]

Con

trol

0 1 2 3 4-10

0

10

20

30

Time [s]

Slid

ing


-0.5 0 0.5 1 1.5-2

-1.5

-1

-0.5

0

0.5

Output

Cha

nge

of O

utpu

t

Robust Switching SMC Control for Un-Matched Uncertain 3-rd Order System

Fig. 4.7: Robust SMC Control for Un-Matched Uncertain System in Example 4.3.

• Robust Tanh SMC (Continuous Pseudo-SMC) Choose k s = 2

0 1 2 3 4-0.5

0

0.5

1

1.5

Time [s]

Out

put

0 1 2 3 4-150

-100

-50

0

50

Time [s]

Con

trol

0 1 2 3 4-10

0

10

20

30

Time [s]

Slid

ing


-0.5 0 0.5 1 1.5-2

-1.5

-1

-0.5

0

0.5

Output

Cha

nge

of O

utpu

t

Robust TanH VSS Control for Un-Matched Uncertain 3-rd Order System

Fig. 4.8: Robust Tanh SMC Control for Un-Matched Uncertain Dynamical System in Example 4.3.

• Robust Linear SMC

I have [ ]K p = 995862 25 4483 2 4828. . .


[ ]K r = 2068966 68 9655 6 8966. . .


thus [ ]K = 348 2069 110 2759 10 7586. . .

but the system-eigenvalues always vary with the uncertainties [ ] [ ]λ λC Cj j1 210 64 406 353 15 29 4 96 103= − − ± = − − ±. . . , . . .

0 1 2 3 4-0.5

0

0.5

1

1.5

Time [s]

Out

put

0 1 2 3 4-400

-300

-200

-100

0

100

Time [s]

Con

trol

0 1 2 3 4-10

0

10

20

30

Time [s]

Slid

ing


-0.5 0 0.5 1 1.5-3

-2

-1

0

1

Output

Cha

nge

of O

utpu

t

Robust Linear SMC for Un-Matched Uncertain 3-rd Order System

Fig. 4.9: Robust Linear SMC for Un-Matched Uncertain System in Example 4.3.

• Performance

The performance of the Tanh SMC control is comparable to that of the linear SMC. Note that the former is

a pseudo-sliding mode, whereas the latter is a true sliding mode. Moreover, the latter is a linear control so the


4.8.4. Example 4.4: Un-matched Uncertainty and External Disturbance

Consider an un-matched uncertain linear system under disturbance in Chen et al. 1989

( ) ( )& ~ ~,x A A x B B W= + + + +∆ ∆ u v v ≤ 1

where

A B A B W=− −

=

=

=

=

0 1 04 0 18 7 8

00

1 3

0 0 02 0 05 0 0

00

0 3

100

,.

, ,.

, ∆ ∆

with the disturbance is chosen as follows for simulation

( )v t= sin 50

All design parameters are the same as in Example 4.3 with an additional disturbance control.


• Robust Switching SMC

Eq.(4.7.c) yields Kp0 30=

0 1 2 3 4-0.5

0

0.5

1

1.5

Time [s]

Out

put

0 1 2 3 4-400

-200

0

200

400

Time [s]

Con

trol

0 1 2 3 4-10

0

10

20

30

Time [s]

Slid

ing


-0.5 0 0.5 1 1.5-3

-2

-1

0

1

Output

Cha

nge

of O

utpu

t

Robust Switching SMC Control for Un-Matched Uncertain 3-rd Order System under Disturbance

Fig. 4.10: Robust SMC Control under Unmatched Uncerta inties and Disturbances in Example 4.3.

• Robust Tanh SMC

Choose k s = 2

0 1 2 3 4-0.5

0

0.5

1

1.5

Time [s]

Out

put

0 1 2 3 4-150

-100

-50

0

50

100

Time [s]

Con

trol

0 1 2 3 4-10

0

10

20

30

Time [s]

Slid

ing


-0.5 0 0.5 1 1.5-3

-2

-1

0

1

Output

Cha

nge

of O

utpu

t

Robust TanH SMC Control for Un-Matched Uncertain 3-rd Order System under Disturbance

Fig. 4.11: Robust Tanh SMC Control under Unmatched Uncertainties and Disturbances in Example 4.3.



Choose δ p = 0 2. , then Corollary 4.5 yields

[ ]Kd = 124 1379 413793 41379. . .

thus [ ]K = 472 3448 151 6552 14 8966. . .

but the system-eigenvalues always vary with the uncertainties [ ] [ ]λ λC Cj j1 213 74 4 58 314 21 83 5 0 51= − − ± = − − ±. . . , . .

0 1 2 3 4-0.5

0

0.5

1

1.5

Time [s]

Out

put

0 1 2 3 4-600

-400

-200

0

200

Time [s]

Con

trol

0 1 2 3 4-10

0

10

20

30

Time [s]

Slid

ing


-0.5 0 0.5 1 1.5-3

-2

-1

0

1

Output

Cha

nge

of O

utpu

t

Robust Linear SMC for Un-Matched Uncertain 3-rd Order System under Disturbance

Fig. 4.12: Robust Linear SMC under Unmatched Uncertainties and Disturbances in Example 4.3.

• Performance

The performance of the Tanh SMC control is comparable to that of the linear SMC one. Note that the

former is a pseudo-sliding mode, whereas the latter is a true sliding mode. Moreover, the latter is a linear

control so the linear control theory can be applied to obtain more insight such as the closed-loop system

eigenvalues.

4.8.5. Example 4.5: Linear SMC for Nonlinear Systems

Consider a nonlinear from Zhou et al. 1992

& .x f g= + u

where

f g=+ +

= +

xx x x tx x

2

1 2 12

1 12

01sin( )

,

since any physical system is bounded-input and bounded-output, I can have the following assumption x1 1≤


the above system can be rewritten as

&

& .

x x

x ax bx c u1 2

2 1 22

=

= + +RST

or

& . . ,,

,x A x B A B= + =

=

u

a b c0 1

20

where

− ≤ ≤ − ≤ ≤ ≤ ≤1 2 1 1 1 2a b c, ,

then

( ) ( )& . . . . ,,.

,.

,,

.,

.x A x B A A x B B A B A B= + = + + + =

=

=

=

u u∆ ∆ ∆ ∆

0 10 5 0

01 5

0 01 5 2

00 5


H = 2 1,

thus [ ]Ke = −0 2 08. .

choose δ = 4 , hence Corollary 4.5 yields [ ]K rp = 11 7 2.

so a linear SMC is [ ]u = − =K x K. , ,10 8

0 2 4 60

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 2 4 6-15

-10

-5

0

5

Con

trol

0 2 4 6-0.5

0

0.5

1

1.5

2

Slid

ing

0 0.5 1-2

-1.5

-1

-0.5

0

Output

Cha

nge

of O

utpu

t

Linear SMC for Nonlinear System

Fig. 4.13: Linear SMC for Nonlinear System in Example 4.5.

4.8.6. Example 4.6: Robust Integral SMC under Uncertainty and Disturbance

Consider the following system under uncertainty from Lin et al. 1992 where a disturbance is included to

illustrate the design

( ) ( )& ~, sinx A A x B W= + + + =∆ u v v t10 500


where

A B A W=− −−

=

−−

=

=

≤

25 313 84

4858

48 058 0

10

10, , , ,∆ v


Using linear quadratic optimal control to minimize the perturbation effect, an original control was

u x x= − +1 078 4 8191 2. .

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-0.5

0

0.5

1

1.5

Time [s]

Out

put


0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-2

-1

0

1

2

Time [s]

Con

trol

Original Design for Robust Control of 2-nd Order System

Fig. 4.14: Original Control under Uncertainty and Disturbance in Example 4.6.

If a unit step input is used

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-0.5

0

0.5

1

1.5

2

Time [s]

Out

put


0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2-3

-2

-1

0

1

2

Time [s]

Con

trol

Original Design for Robust Control of 2-nd Order System with Unit Step Input

Fig. 4.15: Original Control under Uncertainty and Disturbance with Unit Step Input in Example 4.6: Steady-State Error


4.8.6.2. New Robust Integral SMC Design Choose [ ]λ H = −50 , then

[ ]~, ,H = −0 5944 0 0234 0 0366. . .

thus Ke = −0 0 9856 3 1433, . , .

• Robust Integral SMC (Discontinuous SMC)

Choose δ = 3 , , then Theorem 4.10 yields

[ ]K K Krp r p= + = 89105 13505 0 5487. . .

and k0 5 9439= .

0 0.05 0.1 0.15 0.2-0.5

0

0.5

1

1.5

Time [s]

Out

put

0 0.05 0.1 0.15 0.2-10

-5

0

5

10

Time [s]

Con

trol

0 0.05 0.1 0.15 0.2-0.01

-0.005

0

0.005

0.01

Time [s]

Slid

ing


-0.5 0 0.5 1 1.5-1

-0.5

0

0.5

Output

Cha

nge

of O

utpu

t

Robust Integral SMC Control under Uncertainty and Disturbance for 2-nd Order System

Fig. 4.16: Robust Integral SMC Control under Uncertainty and Disturbance with Unit Step Input in

Example 4.6: No Steady-State Error

• Robust Integral Tanh SMC (Continuous Pseudo-SMC), ks = 250

0 0.05 0.1 0.15 0.2-0.5

0

0.5

1

1.5

Time [s]

Out

put

0 0.05 0.1 0.15 0.2-2

-1

0

1

2

Time [s]

Con

trol

0 0.05 0.1 0.15 0.2-2

-1.5

-1

-0.5

0x 10 -3

Time [s]

Slid

ing

Upper Bound Lower Bound

-0.5 0 0.5 1 1.5-0.4

-0.3

-0.2

-0.1

0

0.1

Output

Cha

nge

of O

utpu

t

Robust Integral TanH SMC Control under Uncertainty and Disturbance for 2-nd Order System

Fig. 4.17: Robust Integral Tanh SMC Control under Uncertainty and Disturbance with Unit Step Input

in Example 4.6: No Steady-State Error


• Robust Integral Linear SMC


[ ]K = −237 7556. 9.8353 17.7775, ,

0 0.05 0.1 0.15 0.2-1

-0.5

0

0.5

1

1.5

Time [s]

Out

put

0 0.05 0.1 0.15 0.2-5

0

5

10

Time [s]

Con

trol

0 0.05 0.1 0.15 0.2-0.03

-0.02

-0.01

0

0.01

Time [s]

Slid

ing


-1 0 1 2-1

-0.8

-0.6

-0.4

-0.2

0

Output

Cha

nge

of O

utpu

t

Robust Integral SMC under Uncertainty and Disturbance for 2-nd Order System

Fig. 4.18: Robust Integral Linear SMC under Uncertainty and Disturbance with Unit Step Input in

Example 4.6: No Steady-State Error

• Performance

There is a serious problem on steady-state error and robustness in the original design. The proposed

design has solved this problem.

4.8.7. Example 4.7: Robust Sliding-Mode Observer under Un-Matched

Uncertainties

Consider an un-matched uncertain linear system under external disturbance in Example 4.4 above, using

observer dynamics 2 times faster than that of the robust linear SMC, then Theorem 4.11 yields

L A A LC=−

= − =−

−− −

71 2090241 3402110 0076

71 2090 1 02373402 0 1

118 0076 7 8

...

..

., e

This observer can be used for both robust discontinuous and continuous sliding-mode controllers in Example

4.4. To compare with Linear SMC, the Tanh SMC Control will be used rather than the Switching SMC Control.


In the linear SMC which is a true SMC, the sliding variable s asymptotically goes to zero, but it is never

equal to zero. Strictly speaking, the linear SMC is also a pseudo-SMC, that is why the performance of linear

SMC is similar to that of pseudo-SMC such as Tanh-SMC in all simulations above.


0 1 2 3 4-0.5

0

0.5

1

1.5

Time [s]

Out

put

0 1 2 3 4-600

-400

-200

0

200

Time [s]

Con

trol

0 1 2 3 4-10

0

10

20

30

Time [s]

Slid

ing


-0.5 0 0.5 1 1.5-3

-2

-1

0

1

Output

Cha

nge

of O

utpu

t

Robust Sliding-Mode Observer for Linear SMC under Un-Matched Uncertainty and Disturbance

Fig. 4.19: Robust Sliding-Mode Observer for Linear Sliding-Mode Controller in Example 4.7.

0 1 2 3 4-0.5

0

0.5

1

1.5

Time [s]

Out

put

0 1 2 3 4-150

-100

-50

0

50

100

Time [s]

Con

trol

0 1 2 3 4-10

0

10

20

30

Time [s]

Slid

ing


-0.5 0 0.5 1 1.5-2

-1.5

-1

-0.5

0

0.5

Output

Cha

nge

of O

utpu

t

Robust Sliding-Mode Observer for TanH SMC under Un-Matched Uncertainty and Disturbance

Fig. 4.20: Robust Sliding-Mode Observer for Tanh SMC Controller in Example 4.7.

Performance

Peformances are almost comparable between the case using this sliding-mode observer and the ideal case

where all states are available (Example 4.4)


4.8.8. Example 4.8: Robust Sliding-Mode Observer under Converted Matched

Uncertainties

Consider an un-matched uncertain linear system under external disturbance in Example 4.4 above, Section

4.7.2.2 converts the unmatched uncertain system into a matched system as

A B A B=− −

L

NMMM

O

QPPP

=L

NMMM

O

QPPP

=L

NMMM

O

QPPP

=L

NMMM

O

QPPP

0 1 00 0 140 3 8

00

1 3

0 0 00 0 021 2 0

00

0 3,

., ,

.∆ ∆

For the same maximum control effort, choose hyperplane eigenvalues as [ ]λH = − −6 6 , then

[ ]H = 24 8276 8 2759 0 6897. . .

thus [ ]Ke = 348276 23 4483 2 7586. . .


For the same maximum control effort as in Example 4.4, choose δ = 6, then Corollary 4.5 yields [ ]Kr = 148 9655 49 6552 41379. . .

and choose δ p = 0 2. to have

[ ]Kp = 761379 239310 2 0690. . .

• Robust Tanh SMC

Theorem 4.3 yields [ ]Kp pK= =21 2 0 360,

and choose δ = 2 , then [ ]Kr = 49 6552 16 5517 13793. . .

Choose observer dynamics 2 times faster than that of the robust linear SMC, then Theorem 4.11 yields

L A A LC=...

...

3314542840 4938240 6432

331 4542 1 0840 4938 0 1200 6432 3 8

= − =−−− − −

, e

This observer can be used for both robust discontinuous and continuous sliding-mode controllers in the

sequence.


0 1 2 3 4-0.5

0

0.5

1

1.5

Time [s]

Out

put

0 1 2 3 4-600

-400

-200

0

200

Time [s]

Con

trol

0 1 2 3 4-10

0

10

20

30

Time [s]

Slid

ing


-0.5 0 0.5 1 1.5-3

-2

-1

0

1

Output

Cha

nge

of O

utpu

t

Robust Sliding-Mode Observer for Linear SMC under Converted Matched Uncertainty and Disturbance

Fig. 4.21: Cf. Fig. 4.19, Robust Sliding-Mode Observer for Linear SMC Controller in Example 4.8.

0 1 2 3 4-0.5

0

0.5

1

1.5

Time [s]

Out

put

0 1 2 3 4-200

-100

0

100

200

Time [s]

Con

trol

0 1 2 3 4-10

0

10

20

30

Time [s]

Slid

ing


-0.5 0 0.5 1 1.5-3

-2

-1

0

1

Output

Cha

nge

of O

utpu

t

Robust Sliding-Mode Observer for TanH SMC under Converted Matched Uncertainty and Disturbance

Fig. 4.22: Cf. Fig. 4.20, Robust Sliding-Mode Observer for Tanh SMC Controller in Example 4.8.

Remark 4.14: Comparison between Unmatched and Converted Matched Uncertainty.

Performances of SMC's for converted matched uncertain system are similar to the original unmatched

ones (Cf. Figs. 4.19 & 4.21, 4.20 & 4.22).


4.9. CONCLUSIONS

It has been shown that the sliding eigenvalues are unchanged while the system eigenvalues are always

changed under the perturbations. Even though the system eigenvalues vary with the perturbations, under

matching conditions, the sliding eigenvalues are absolutely unchanged and equal to the hyperplane-

eigenvalues chosen to be Hurwitz previously. Due to the stability criterion (Theorem 3.1), the system is stable

regardless how large the perturbations are. Without the matching conditions, if the sliding-eigenvalues are

still Hurwitz then the system is stable; but they may migrate into the right-half plane (RHP) and then the non-

Hurwitz sliding-eigenvalues make the system unstable even though the sliding condition is still satisfied and

the hyperplane is stable. In this case, I must do a stability test (Theorem 3.1).

I have identified 3 types of eigenvalues to clarify the mechanism of the sliding mode in a SMC with a

sliding margin of δ : hyperplane-eigenvalues, sliding-eigenvalues and system-eigenvalues. The sliding-

eigenvalues represent the sliding dynamics of the system state variables in the sliding mode. T he hyperplane-

eigenvalues are the desired sliding-eigenvalues. The first 2 types are applicable for both linear and nonlinear

systems, the third is only for linear systems.

( )[ ] { }

( ) { }

[ ] [ ] [ ] { }

λ

λ

λ λ λ

H

S

C H H

= − −

= −

−

= − = − ⇒ = − − −

−

−

eig

eig

eig eig

A B HB HA

A B HB HA

A BK A BK

1

1

0

0~ ~ ~ ~

,δ δ

By the stability criterion (Theorem 3.1) with the design rule (Proposition 3.1), it is simple enough to do a

stability test. If the sliding-eigenvalues are Hurwitz but its simulation is unstable, then the sliding margin

should be reduced.

The unified design in Chapter 3 is extended to deal with perturbations in this chapter. Moreover, there is

a unified approach to deal with discontinuous or continuous SMC, conventional or integral SMC. From a

discontinuous control function of the discontinuous SMC, I can get a continuous pseudo-SMC by using

either a saturated function or a Tanh function in exactly the same manner. For a linear SMC, it is only pseudo-

SMC for the case of disturbances, but it is not for uncertainties. Also by the simulations, I have seen that the

performance of the Tanh SMC control is comparable to that of the linear SMC. Note that the former is a

pseudo-sliding mode, the latter is not. Moreover, the latter is a linear control so the linear control theory can

be applied for a more insight such as the closed-loop system eigenvalues.

I have also seen that it is possible for a linear control law for a certain class of nonlinear systems. For a

more general class of nonlinear systems, a nonlinear system is linearized around an operating point, then the

proposed robust linear SMC is applicable to that linearized system where the nonlinearity is considered as a

special type of an uncertainty within the bound in that operating range.


By my new design of linear control function for linear systems under uncertainties, I can use the well-

established linear control theory for assessments and for comparisons in order to have a closer look at the

true nature of SMC as mentioned so far. In fact, the proposed robust linear SMC can be considered as a

superset of the linear state-space control. Some other features are the following:

(i) In systems under no uncertainties, sliding-eigenvalues are equal to hyperplane-eigenvalues.

Moreover, system-eigenvalues comprise hyperplane-eigenvalues and the negative of the sliding margin,

− δ .

(ii) The invaria nce condition really means the sliding-eigenvalues are invariant to matched uncertainties

and are equal to the hyperplane-eigenvalues, while the system-eigenvalues vary with the uncertainties. The

Hurwitz sliding-eigenvalues do guarantee the system stability because they are equal to the Hurwitz

hyperplane-eigenvalues. Moreover, by Theorem 4.5, the negative of the sliding margin, −δ , and the

hyperplane-eigenvalues are the system-eigenvalues at one boundary of matching uncertainties.


Chapter 4

Robust Sliding Mode Controller-Observer Design

4.1. Introduction...................................................................................................................................................................1

4.2. Robustness in SMC.....................................................................................................................................................2

4.2.1. Invariance Condition to Uncertainties..........................................................................................................2

Theorem 4.1: Invariance Condition to Uncertainty............................................................................2

Remark 4.1: Form of Matching Condition ............................................................................................3

Corollary 4.1: Alternative Form of Matching Condition....................................................................4

Corollary 4.2: Canonical System with Matching Condition..............................................................4

4.2.2. Invariance Condition to Disturbances..........................................................................................................5

Theorem 4.2: Sufficient Invariance Condition to Disturbance.........................................................5

Remark 4.2: Invariance Property of Sliding Eigenvalues under Matching Condition...................5

4.3. A New Robust VSS Controller Design (Discontinuous SMC) ..............................................................................5

Assumption 4.1: System Constraint on Parametric Variation...........................................................6

Theorem 4.3: Robust VSS Controller Design under Uncertainty and Disturbance.......................6

Remark 4.3: Robust VSS Controller Design under No Perturbations..............................................7

Remark 4.4: Equivalent Control in Robust VSS Controller................................................................7

Corollary 4.3: Robust VSS Controller under Reduced Parametric Variation ...................................8

4.4. A New Robust Linear SMC Design...........................................................................................................................8

4.4.1. Linear SMC........................................................................................................................................................9

Theorem 4.4: Linear SMC Design.........................................................................................................9

Theorem 4.5: Closed-Loop Eigenvalues composed of Hyperplane Eigenvalues and Sliding

Margin.......................................................................................................................................................9

Remark 4.5: Reduced Order of Sliding Mode .................................................................................... 10

4.4.2. Robust Linear SMC........................................................................................................................................ 10

4.4.2.1. Robust Linear SMC under External Disturbances........................................................................ 11

Theorem 4.6: Ro bust Linear SMC Design under External Disturbance........................................ 11

Remark 4.6: Disturbance Condition in Sliding Mode....................................................................... 12

Remark 4.7: Validity of Theorem 4.6 under No Disturbance ........................................................... 12

4.4.2.2. Robust Linear SMC under Parametric Uncertainties.................................................................... 12

Lemma 4.1: Computation of Special Eigenvalues ............................................................................. 12

Theorem 4.7: Robust Linear SMC Design under Parametric Uncertainty..................................... 13

Remark 4.8: Robust Linear SMC Design under No Uncertainty .................................................... 14

Corollary 4.4: Robust Linear SMC Design under Reduced Parametric Variation........................ 15

Remark 4.9: Equivalent Control in Robust SMC Design................................................................. 15

4.4.2.3. Robust Linear SMC under Matched Parametric Uncertainties................................................... 15

Theorem 4.8: Robust Linear SMC Design under Matched Uncertainty....................................... 15


Proposition 4.1: Robust Linear SMC Design Rule ........................................................................... 16

Remark 4.10: Advantage of Robust Design Ru le ............................................................................. 16

4.4.2.4. Robust Linear SMC under Un -matched Uncertainty and Disturbances ................................ 16

Corollary 4.5: Robust Linear SMC Design under Un-matched Uncertainty and Disturbance.. 16

4.4.2.5. Alternative Formula for Robust Linear SMC................................................................................. 17

Lemma 4.2: Alternative Computation of Special Eigenvalues ........................................................ 17

Theorem 4.9: Alternative Robust Linear SMC Design .................................................................... 18

Corollary 4.6: Alternative Robust Linear SMC Design under Reduced Parametric Variation... 19

4.5. Robust Integral SMC Design.................................................................................................................................... 20

Theorem 4.10: Robust Integral Linear SMC Design under Uncertainty....................................... 20

4.6. A New Robust Linear Sliding-Mode Observer Design........................................................................................ 21

Theorem 4.11: Robust Linear Sliding-Mode Observer Design under Uncertainty ..................... 21

4.7. Analysis of Matched and Unmatched Uncertainties ............................................................................................ 22

4.7.1. Transfer Function Model.............................................................................................................................. 22

Proposition 4.2: Uncertainty in Transfer Function Model.............................................................. 23

4.7.2. State-Space Model......................................................................................................................................... 24

4.7.2.1. Analysis of Uncertain State-Space Model .................................................................................... 24

4.7.2.2. Conversion of Uncertain State-Space Model................................................................................ 25

Proposition 4.3: Conversion of Unmatched to Matched Uncertain System ................................ 26


Remark 4.11: Summary of Robust SMC Designs.............................................................................. 26

4.8.1. Example 4.1: No Perturbation........................................................................................................................ 27

4.8.1.1. Original Design.................................................................................................................................. 27

4.8.1.2. New Design........................................................................................................................................ 27

4.8.2. Example 4.2: Matched Uncertainties ............................................................................................................ 29

Remark 4.12: Efficiency of Robust Linear SMC Design Rule (Proposition 4.1) ........................... 31

4.8.3. Example 4.3: Un-matched Uncertainty......................................................................................................... 31

4.8.4. Example 4.4: Un-matched Uncertainty and External Disturbance ........................................................... 33

4.8.5. Example 4.5: Linear SMC for Nonlinear Systems ....................................................................................... 35

4.8.6. Example 4.6: Robust Integral SMC under Uncertain ty and Disturbance............................................... 36

4.8.6.1. Original Design.................................................................................................................................. 37

4.8.6.2. New Robust Integral SMC Design.................................................................................................. 38

4.8.7. Example 4.7: Robust Sliding-Mode Observer under Un -Matched Uncertainties ............................... 39

Remark 4.13: Performance of Pseudo-SMC....................................................................................... 39

4.8.8. Example 4.8: Robust Sliding-Mode Observer under Converted Matched Uncertainties .................. 41

Remark 4.14: Comparison between Unmatched and Converted Matched Uncertainty.............. 42

4.9. Conclusions................................................................................................................................................................. 43

Chapter 5

A New Robust Discrete-Time Sliding Mode Controller-Observer Design

5.1. INTRODUCTION In the current discrete-time sliding-mode control (SMC), a convergence condition has been proposed in

addition to the sliding condition in Sarpturk et al 1987 (Remark 5.1). There have been some comments on

applicability of this approach in Kotta 1989, and these conditions have been shown to be sufficient but not

necessary in Spurgeon 1990. In Utkin et al 1989, Furuta 1990, the proposed controls have been pseudo

sliding mode with a boundary layer using discontinuous switch control function. In the current continuous-

time SMC, a SMC can be obtained using a discontinuous switch control function and a pseudo-SMC is used

to smooth out the discontinuity of the switch function in solution of the chattering problem. In Su et al. 1993,

pre-filtering and post-filtering have been implemented to reduce chattering in sliding-mode controllers. In

Spurgeon 1991 and Pieper et al 1992, robust discrete-time discontinuous pseudo SMC designs have been

proposed for matched uncertain systems. In Pan et al. 1993, a discrete-time model has been transformed into

a canonical one then a robust discrete-time SMC has been designed for this transformed canonical model

under matched uncertainty. In Wang et al. 1994, a robust discrete-time SMC has been designed for a system

where its transformed model has matched uncertainty.

In this chapter, a new robust discrete-time sliding mode controller (SMC) and observer design is

presented for both discontinuous SMC (VSS) and continuous SMC control functions, it can be seen as a

discrete-time version of the previous chapter. A high sampling rate is proved to be a necessary condition for

a discrete-time SMC.

Based on the work in Pieper et al 1992 for matched uncertain systems, an alternative design will be

presented without the constraint stated therein. On the basis of the new robust sliding mode controller design

in the previous chapter, a new robust discrete-time sliding mode controller design will be proposed to deal

with both matched and unmatched uncertainty without chattering.

As a SMC is based on a state-space model, an I-action may be required to eliminate a steady-state error.

We will also prove in this chapter that the integral VSS control (discontinuous SMC) in Theorem 3.4 can be

still applied to both robust discontinuous and linear continuous SMC controls. Also as a state-space

approach, SMC requires an observer to estimate unavailable states, we present a new robust discrete-time

sliding-mode observer design.


As a state vector is necessary for SMC, an observer may be required for estimation of unavailable states.

In Bondarev et al. 1985, a linear Luenberger observer has been used as an observer of a VSS controller for

deterministic linear systems (no uncertainties). In Walcott et al. 1987 and Yaz et al. 1993, a Lyapunov

sliding condition has been used to design an observer for a class of systems under matched uncertainty

restricted to a certain system structure. In Slotine et al. 1987, a sliding patch condition has been used to have

a region of direct attraction where uncertainty is not fully tackled. In fact, in Walcott et al. 1987, Slotine et

al. 1987 and Yaz et al. 1993, to cope with uncertainty, a Lyapunov sliding condition has been employed to

include a switching component into a linear Luenberger observer where a linearized model is used for a

nonlinear system.

5.2. DIGITAL CONTROL SYSTEMS Nowadays, most controllers are implemented as digital controllers rather than analog ones since digital

controllers are much more flexible and they can be used to implement highly complicated control functions.

Simple control functions can be implemented with analog controllers, however they are less reliable due to

the drifting of op-amps, the leakage of capacitors and the aging of analog components such as resistors,

capacitors etc.

The z-transform (Section 5.2.5) and a discretization of a state-space model (Section 5.2.7) will be used in the

latter sections on discrete-time SMC. Although these results are well-known in the digital control literature,

this section is included to review digital control systems in a compact, concise and logical manner. We first

start with a sampling process to convert an analog signal into digital signal in deriving the starred transform

which provides a basis to develop the Z-transform. The Z-transform will then define a mapping between the

continuous-time s-plane and the discrete-time z-plane as a mathematical tool to design a digital controller, as

the Laplace transform is used to design an analog controller.

5.2.1. Some Terminology In this subsection, we are concerned with the difference between discrete-time systems and digital

systems whereas they both are discretized from an analog system. In the real world, all signals are analog

signals whose amplitudes take infinite values within some ranges. Their values can be any real numbers with

fraction and the difference between successive values is infinitesimal.

Analog signals are continuous-time signals. Discrete-time signals, or discrete signals for short, take finite

values of any real numbers, hence the difference of successive values is finite and is a real number.

A digital signal is a subset of discrete-time signals. A values of a digital signal must be an integer, thus the

difference of successive values is finite and is an integer. These integers are within the range determined by the length of registers in a ADC, the range is 0 2~ n or [ ]− −− −2 2 11 1n n~ for a n-bit ADC.

5. Discrete-Time Robust Sliding Mode Controller-Observer Design 5.3

Discrete-time control theory is used to design a discrete-time controller which is employed to implement a

digital controller for a digital control system.

A continuous-time system is discretized using discrete-time control theory, there is no digitization of a

system since it is a special case of discretization as a real number is rounded off to an integer. An analog

signal is digitized, also known as quantized, by an ADC. There is no discretization of an analog signal since

there is no such converter available nowadays.

DigitalController Processyd

ed udyue

−ADC DAC

Controller

Fig. 5.1: Digital Control System

where y y e ud, , , and are analog signals while e ud dand are digital signals. ADC and DAC are analog-to-

digital and digital-to-analog converters, respectively. Digital controller can be either a micro-controller or a

digital signal processor (DSP) where ADC and DAC are on-chip (internal) or a general purpose computer

with external ADC/DAC. For 8-bit processors such as 8051 Intel family or 68HC11 Motorola family, ADC

and DAC are also 8-bit. However, higher resolution ADC’s are expensive, so that 10-bit or 12-bit ADC are

usually used for 16-bit processors such as 80196 Intel family, 68000 Motorola family, 32-bit 486-PC, 64-bit

Pentium-PC.

5.2.2. ADC as Physical Sampler Practically, ADC is a zero-order-hold sampler of successive approximation type

Sampler andData Hold

( )e t

( )E s

( )e t

( )e t

( )E s

( )e t

0 T 2T 3T 4T 5T 6T 7T 8T 9T

Amplitude

Time Fig. 5.2: Sample and Hold Signal

The sampled and zero hold signal ( )e t can be mathematically expressed as

( ) ( ) ( ) ( )[ ]{ }e t e kT t kT t k Tk

= − − − +=−∞

∞

∑ U U 1 (5.1)


where ( )U t is the unit step function, that is

( )U ttt

=≥<

1 00 0,,

(5.2)

Using the shifting theorem of Laplace transform gives

( ) ( )( )

( )[ ]E s e kT es

es

es

e kT ekTs k Ts

k

TskTs

k

= −

=−

− − +

=−∞

∞ −−

=−∞

∞

∑ ∑1 1 (5.3)

5.2.3. Starred Transform as Ideal Sampler On the basis of Eq.(5.3), the starred transform is defined as

( ) ( )E s e kT e kTs

k

* = −

=−∞

∞

∑ (5.4)

then Eq.(5.3) can be written as

( ) ( )E s es

E sTs

=−

−1 * (5.5)

( )E s( )E s*

T

( )E s1− −e

s

Ts

Eq.(5.4) gives the inverse Laplace transform of ( )E s* as

( ) ( ){ } ( ) ( )[ ]∑∞

−∞=

− −δ==k

kTtkTesEte *1* L (5.6)

where ( )δ t is a Dirac delta function.

0 T 2T 3T 4T 5T 6T 7T

e t* a f e ta f

e 0a f e 1a f e 2a f e 3a f e 4a f e 5a f e 6a f e 7a f

Fig. 5.3: A Representation of ( )e t*


5.2.4. Sampling Characteristics and Signal Reconstruction This subsection founds a basis for choosing sampling rate and necessity of an analog low-pass filter as

an anti-aliasing filter before ADC. Sampling characteristics and signal reconstruction is determined by the

following theorem

Theorem 5.1: Laplace Transform of Sampled Signal

An alternative expression for the starred transform is given by

( ) ( )E sT

E s jk sk

* = +=−∞

∞

∑1ω (5.7)

thus the Laplace transform of a sampled signal is equal to an infinite sum of Laplace transform of the

Laplace transform of its original signal. The s-plane may be divided into a primary strip and complementary

strips as shown in Fig. 5.4.

Proof

Following the proof in Lockhart et al. 1989, let

( ) ( )S t t kTk

= −=−∞

∞

∑δ (5.8)

then Eq.(5.6) can be written as ( ) ( ) ( )e t e t t* .= S (5.9)

j s0 5. ω

j s1 5. ω

− j s1 5. ω

− j s0 5. ω

σ

jω

Primary Strip

Complementary Strip

Complementary Strip

Complementary Strip

Complementary Strip

− +σ ω1 1j

( )− + +σ ω ω1 1j s

( )− + − +σ ω ω1 1j s

( )− − +σ ω ω1 1j s

( )− − − +σ ω ω1 1j s

( )− + +σ ω ω1 1 2j s

( )− + − +σ ω ω1 1 2j s

( )− − +σ ω ω1 1 2j s

( )− − − +σ ω ω1 1 2j s

− −σ ω1 1j

x

x

x

x

x

x

x

x

x

x

Fig. 5.4: Primary and Complementary Strips


Since ( )S t is a periodic function ( )δ t whose period is T, its Fourrier series is

( ) ( )S tT

t e d eT

ej k TT

j kt T

k

jk t

k

s=

=−

=−∞

∞

=−∞

∞

∫∑ ∑1 12

0

2δ τπ τ π ω (5.10)

where

ωπ

s T=

2 (5.11)

thus Eq.(5.9) can be read as

( ) ( )e tT

e t e jk t

k

s* ==−∞

∞

∑1 ω (5.12)

and its Laplace transform is

( ) ( ){ } ( ) ( ) ( ) ( ) ( )E s e tT

e t e dtT

e t e dtT

E s jks jk t

k

s jk t

ks

k

s s* *= = = = +− −

=−∞

∞∞− −

∞

=−∞

∞

=−∞

∞

∑∫ ∫∑ ∑L 1 1 1

0 0

ω ω ω

Q.E.D.

5.2.4.1. Signal Reconstruction To see the significance of Theorem 5.1, consider the case of frequency response, when s j= ω Eq.(5.7)

gives

( ) ( )[ ]E jT

E j k sk

* ω ω ω= +=−∞

∞

∑1 (5.13)

thus the spectrum of sampled signal consists of an infinite number of replicas of the analog signal spectrum scaled by the factor 1 T and the frequency shifted by multiples of ω s .

If the sampling time T is low enough such that ω ωmax < s 2 , then the sampled spectrum retains the shape

of the analog one and the analog signal can be reconstructed from its sampled spectrum within the band [ ]−ω ωs s2 2, .

For signal reconstruction, the sampling rate must satisfy the sampling theorem, also known as Shannon's Theorem which states that the sampling frequency ω s must be at least twice the value of the highest

significant frequency in the signal. Since an ideal low-pass reconstruction filter cannot be implemented, one

rule of thumb is to choose T as one-tenth of the smallest process time constant or the desired closed-loop

time constant. Another convenient rule suggests sampling at the rate of 6 to 10 times per cycle.

Proposition 5.1: Choice of Sampling Time In this work, the sampling time is chosen such that ω ωs ≥ 10 max . With this choice, ignoring the sampling

effect will introduce a maximum tolerance of gain about 0.12% and of phase about 30 .

Proof: We have

( )

( )e e

e

sT sT

sT sT

sT

sTsT

sT

sT−

−

= =− + −

+ + +≈

−

+

2

2

2

2

2

2

12 2 2

12 2 2

12

12

!

!

L

L


then Eq.(5.5) gives

( )G s es s

sT

sTT

T sh

sT

=−

≈ −−

+

=+

−1 1 11

2

12 2

1

Since the overall DC gain will be determined at the final design stage, the sampling time will be discarded to

have

( )G s T sh ≈

+

1

21

thus ignoring the sampling effect will introduce a maximum tolerance of

( )G jj jh ωmax . .

. .≈+

=+

= ∠ −1

1 0 12

11 0 05

0 9988 2 860

Q.E.D

( )E jω

ωmax

ωs

2

ωs

2

ωs

ω ωmax < s 2

ω ωmax > s 2

ωs

−ωs

2

−ωs

2

−ωs

−ωs

−ωmax

( )E j* ω

A

A T

A T

0

0

0

( )E j* ω

−32

ω s

−32

ω s

32

ω s

32

ω s

Fig. 5.5: Spectrum of Analog and Sampled Signal

5.2.4.2. Aliasing or Folding Eq.(5.7) reveals that if ( )E s has a pole at p0 , then ( )E s* will have poles at ( )p jn s0 + ω where

n = ± ±0 1 2, , ,L For example, suppose that ( )E s has one pair of poles located at − ±a j s0 6. ω , outside the

primary strip. The sampling will then “fold” this pair back into the primary strip, to the location − ±a j s0 4. ω . So ( )e t* contains components inside the primary strip that do not occur in ( )e t . This property

applies for poles only, not for zeros, since poles of a transfer function component in Eq.(5.7) will be poles of

the whole transfer function; however, zeros of a component are not necessarily the zeros of the whole.


An anti-aliasing analog filter is used to solve this problem by filtering out all noise. It is a low-pass filter

with the cut-off frequency is based on the highest frequency of the analog signal.

5.2.5. Z-Transform On the basis of Eq.(5.4), the Z transform is defined as

( ) ( ){ } ( )∑∞

−∞=

−==k

kzkekezE Z (5.14)

where the mapping between the s-plane and the z-plane is defined as

( )z e sT

zTs= ⇔ =1 ln (5.15)

since comparing Eqs.(5.4) and (5.14) yields ( ) ( ) Tsez

zEsE=

=* (5.16)

The Laplace transform is used in obtaining a control transfer function for implementing a analog

controller. We will see that the Z-transform will be employed in obtaining a control difference equation for

implementing a digital controller.

Two of the most important properties of the Z-transform are the linearity and the real are given below ( ) ( ){ } ( ){ } ( ){ }kebkeakbekae 2121 ZZZ +=+ (5.17)

and ( ){ } ( ){ }keznke n ZZ −=− (5.18)

where n is a positive integer.

Since by the definition of the Z-transform, we have

( ) ( ){ } ( ) ( )[ ] ( ) ( ) ( ){ } ( ){ }tebteazkebzkeazkbekaekbekaek

k

k

k

k

k21212121 ZZZ +=+=+=+ ∑∑∑

∞

−∞=

−∞

−∞=

−∞

−∞=

−

and

( ){ } ( )[ ] ( ) ( ) ( ) ( ){ }kezzmezzmeznkenke n

k

mn

m

mn

k

k ZZ −∞

−∞=

−−∞

−∞=

+−∞

−∞=

− ===−=− ∑∑∑

where m k n= − .

5.2.6. Some Typical S-Z Transformations For completeness, this subsection presents a brief derivation of 2 well-known transformations: backward

difference method and bilinear method. The Z-transform defines the mapping in Eq.(5.15) as the exact

transformation between the s-plane and the z-plane. This transform should be used to convert a continuous-

time system into a discrete-time one. A direct substituting Eq.(5.15) into a transfer function in s-plane will

produce an infinite difference equation due to the Taylor expansion of the natural logarithmic function. This

necessitates an approximate transformation to obtain finite difference equations if the substitution method is used in place of the z-transform. From a single term s, we can have s and 1 s corresponding to a

differentiator and an integrator, respectively.


( )y t ( )y t

t tt0 0 tt T− t T−

Approximation of Differnetiator as a Chord Approximation of Integrator as an Area

Fig. 5.6: Approximate Transformations for Differentiator and Integrator

The backward difference transformation approximates a differentiator as the slope of a chord

( ) ( ) ( )ddt

y ty t y t T

T≈

− −

taking Laplace transform gives

( ) ( ) ( )sY sY s e Y s

T

Ts

≈− −

so

s eT

zT

zTs

Ts

≈−

=−

⇔ ≈−

− −1 1 11

1

(5.19)

σσ

jωjω

σ

jωs-plane

z-planeExact Transformation

Bilinear Transformationz-plane

Backward Transformation

Fig. 5.7: Stability Regions

The bilinear transformation approximates an integrator as the area of a trapezoidal

( ) ( ) ( ) ( )[ ]y d y d T y t T y tt t T

τ τ τ τ. .0 0 2∫ ∫− ≈ − +

−

taking Laplace transform gives

( ) ( ) ( ) ( )[ ]1 12s

Y ss

e Y s T e Y s Y sTs Ts− ≈ +− −


thus

( )( )

12

11 2

11

22

1

1sT e

eT z

zz

T sT s

Ts

Ts≈+−

=+−

⇔ ≈+−

−

−

−

− (5.20)

The stability region in the s-plane is the RHP, to find its mapping in the z-plane, substituting the frequency contour s j= ω into Eqs.(5.15), (5.20) and (5.21) produces

z e j T= ω (5.21)

( )zj T

j Tj T

e Tj=−

= ++−

= + = −11

12

12

11

12

12

2 21

ωωω

θ ωθ , tan (5.22)

( )( ) ( )zT j TT j T

e j≈+−

= = −22

2 21ωω

θ ωθ , tan (5.23)

Thus a stable controller could be unstable under the backward difference transformation. This is a limitation

of this transformation.

5.2.7. Discretization of Continuous-Time State-Space Equations The z-transform is used to discretize a continuous-time transfer-function model for designing a discrete-

time controller in the z-domain. Alternatively, a transfer-function model can be modified by taking into

account the sampling process, and a continuous-time controller can be designed in the s-plane then

discretized to obtain a discrete-time controller. To discretize a continuous-time state-space model, from the

digital control literature, we have the following theorem:

Theorem 5.2: Discretization of State-Space Model Consider the continuous-time state equation and output equation ( ) ( ) ( )&x A x B ut t tc c= + (5.24)

( ) ( ) ( )y Cx Dut t t= + (5.25)

under the assumption of zero-order hold that ( ) ( ) ( )u ut kT kT t k T= ≤ ≤ +, 1 . (5.26)

Then a discrete-time representation of Eq.(5.24) will take the form ( )[ ] ( ) ( ) ( ) ( )x A x B uk T T kT T kTd d+ = +1 . (5.27)

( ) ( ) ( )y Cx DukT kT kT= + (5.28)

where ( )A A

dTT e c= (5.29)

and

( )B BAd

tT

cT e dtc=

∫

0

(5.30)

with T is a constant of sampling time, ( ) ( )A Bd dT and T are thus constant matrices.


Proof

Following the proof in Ogata 1987, the Laplace transform of Eq.(5.24) give ( ) ( ) ( ) ( )s s s sc cX x A X B U− = +0

( ) ( ) ( ) ( )s s sc cI A X x B U− = +0

or ( ) ( ) ( ) ( ) ( )X I A x I A B Us s s sc c c= − + −

− −1 10 (5.31)

Note that

( )ssc

cn

nn

cI AA

A I− = =−

+=

∞

∑11

0

0, (5.32)

hence its inverse Laplace transform is

( ){ } ( )L − −

=

∞

− = = =∑1 1

00 1s

tn

ecc

n

n

tcI AA A

!, ! (5.33)

thus Eq.(5.32) becomes ( ) ( ) ( )X x B UA As e e sc ct t

c= +0 (5.34)

then its inverse Laplace transform is

( ) ( ) ( )x x B uA A At e e e dc c ct tc

t

= + −∫00

τ τ τ (5.35)

This equation gives

( )[ ] ( ) ( ) ( ) ( )( )

x x B uA A Ak T e e e dc c ck T k Tc

k T

+ = ++ + −+

∫1 01 1

0

1τ τ τ (5.36)

and

( ) ( ) ( )x x B uA A AkT e e e dc c ckT kTc

kT

= + −∫00

τ τ τ (5.37)

Multiplying Eq.(5.37) by e cTA and subtracting from Eqs.(5.26) and (5.36) yields

( )[ ] ( ) ( ) ( )( )

x x B uA A Ak T e kT e e kT dc c cT k Tc

kT

k T

+ = + + −+

∫1 11

τ τ

then

( )[ ] ( ) ( ) ( )x x B uA Ak T e kT e dt kT t k Tc cT tc

T

+ == +

= + −∫1 1

0

, τ (5.38)

Substituting t kT= in Eq.(5.25) gives Eq.(5.28).

Q.E.D.


5.3. DISCRETE-TIME SLIDING HYPERPLANE DESIGN Sliding-mode control has 2 phases of dynamics: the reaching and the sliding modes. The sliding

condition forces the system states to reach the hyperplane and keeps them sliding on it. In this section, the

hyperplane design and the sliding dynamics will be presented.

Proposition 5.2: Discrete-Time Sliding-Mode Controller Design

If design parameters such as hyperplane, control function (continuous and discontinuous SMC) are

available in continuous-time domain, then they can be used for the discretized system to keep the discrete-

time controller parameters close to the continuous-time counterparts. If they are unavailable in the case

where a continuous-time model is unavailable, then the rest of this chapter will be used to design a discrete-

time sliding-mode controller.

A discrete-time sliding-mode observer must be designed since it will be used in a discrete-time observer

equation to estimate system states.

Proof: By Theorem 5.2, we have

( ) ( ) ( )( ) ( )

x A x B uy Cx

t t t

t tc c= +

=

then

( ) ( ) ( )( ) ( )

x Ax Buy Cx

k k k

k k

+ = +

=

1

where A, B are discretizations of A Bc c, , respectively.

In this sense, consider

( ) ( ) ( )( ) ( )

( )( )

( ) ( ) ( )[ ]

x A x B us Hx

uKx

K x K x s

t t t

t t

tt

t t t

c c

e r

= +

=

=−

− −

.sign

then we have

( ) ( ) ( )( ) ( )

( )( )

( ) ( ) ( )[ ]

x Ax Bus Hx

uKx

K x K x s

k k k

k k

kk

k k ke r

+ = +

=

=−

− −

1

.sign

Q.E.D.


5.3.1. Hyperplane Equation The exact transformation in Eq.(5.15) converts a desired continuous-time hyperplane-eigenvalues Pc into

a discrete-time counterpart P , then Chapter 2 will be used with the discrete-time model { }A B, to design a

hyperplane for a discrete-time sliding-mode controller.

Continuous-Time Hyperplane Design Discrete-Time Hyperplane Design ( )H A B Pc c c chyper= , , ( )H A B P= hyper , ,

where

P P= eT c ;

hyper is exactly the same hyperplane design procedure presented in Chapter 2 (Remark 2.9).

5.3.2. Equivalent Control Consider a hyperplane ( ) ( )kks Hx= .

Then ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( )kukkkksksks ..1.1 HBxIAHxxH +−=−+=−+=∆

where

H x∈ℜ ∈ℜ× ×1 1n n, Assume ( )HB is invertible, then the equivalent control is defined as

( ) ( ) ( ) ( )kkueq xIAHHB .1 −−= − (5.39)

thus ( ) ( ) ( ) ( ) ( )[ ]kukuksksks eq−=−+= .1 HB∆ (5.40)

It is important to note that in the sliding mode ( ) ( ) ( ) ( )kukuksks eq=⇒=⇒= 00 ∆ .

5.3.3. Sliding-Eigenvalues We have the following corollary as a discretization of sliding-eigenvalues in Theorem 2.2

Theorem 5.3: Discrete-Time Sliding Eigenvalues Consider a SISO discrete-time state equation ( ) ( ) ( )kukk BxAx +=+ 1

and a hyperplane ( ) ( )kks xH=

In the sliding mode ( ) ( ) ( ) ( ) ( )IAHHBKxK −=−== −1,. eee kkuku

then the sliding-eigenvalue λ S can be determined by

[ ] ( ) ( )[ ] { }eig eig A BK A B HB H A I− = − − =−e S

1 1, λ (5.41)


Proof: For a reduced order of an n-ordered system, there are only ( )1−n sliding-eigenvalues λ S , so we need to

prove that the remaining n-th eigenvalue of Eq.(5.41) is one. By definition of eigenvalues: ( ) [ ]{ }0deteig =−λλ= MIM ii , thus we have to prove that ( ) ( )[ ]{ } 0det 1 =−−− − IAHHBBAI or

equivalently ( )[ ]det ~ ~A B HB HA− =−1 0 .

where

~A A I= −

Since eigenvalues are invariant under a similarity transformation, we will use the similarity

transformation as in Utkin et al. 1979.

Consider a system after the similarity transformation ( ) ( ) ( )x A x Bk k u k+ = +1 ~. .

and a hyperplane ( ) ( )kks xH.=

where

x B A B0

B, , ~ , , ,∈ℜ ∈ℜ ∈ℜ

× ×n n n u s1

2 =

and

( ) ( ) ( ) ( )xxx

AA AA A

A A A x A x=

=

∈ℜ ∈ℜ ∈ℜ ∈ℜ− × − − × × −1

2

11 12

21 22

1 1 1 121 1

1 12, ~

~ ~~ ~ , ~ ~ ~ , , ~ , , , 11 12 22

n n n n

We have

( ) [ ] [ ]B HB HA0

BH H

0B

H HA AA A

H0 0

H A H A H A H A−

−

−=

=

+ +

1

21 2

2

1

1 211 12

21 222

1

1 11 2 21 1 12 2 22

~~ ~~ ~ ~ ~ , ~ ~

thus

( )[ ]~ ~~ ~~ ~ ~ ~ , ~ ~

~ ~~ , ~A B HB HA

A AA A

H0 0

H A H A H A H AA A

H H A H H A− =

−

+ +

=

− −

− −− −

1 11 12

21 222

1

1 11 2 21 1 12 2 22

11 12

21

1 11 21

1 12

By the property of a determinant

A BC D

A D CA B A

D A BD C D

A B0 D

A D=− ≠

− ≠

⇒ =

−

−

. ,

. ,.

1

1

0

0

if

if

then

( )[ ] 0~,~~~

det~~det121

12111

12

12111 =

−−=−

−−

−

AHHAHHAAAHHBBA

Therefore ( ) ( )[ ] ( ) ( ) ( ){ }01 1-1-1 =λλ=⇒λ×−λ=−−−λ − n

iHn PPIAHHBBAI (5.42)

Q.E.D.


5.4. A ROBUST DISCRETE-TIME DISCONTINUOUS SMC (VSS) DESIGN Based on the work in Pieper et al 1992 for matched uncertain systems, we propose the following

theorem without the constraint stated therein

Theorem 5.4: Robust Discrete-Time Discontinuous SMC Design under Matched Uncertainty

Consider a discrete-time linear system under matched parametric uncertainty (Corollary 4.1) ( ) ( ) ( ) ( ){ }kvkukk ++=+ BxAx 1 (5.43)

where the disturbance ( )kv must satisfy the following bound max

iiv v≤ (5.44)

and a hyperplane equation ( ) ( )kks xH=

where x B A H, , , , ,∈ℜ ∈ℜ ∈ℜ ∈ℜ× × ×n n n n u s1 1

To satisfy the discrete-time sliding condition ( ) ( ) ( ) ( )ksksksks 22 11 <+⇒<+ (5.45)

a control can be determined by ( ) ( ) ( )kukuku rpe += (5.46)

where

• equivalent control ( ) ( ) ( ) ( )IAHHBKxK −=−= −1, eee kku (5.46.a)

• reaching and perturbation control

( ) ( )( ){ } ( )

u k k ks k x s k v x

rp rp rpii i

i

n

= − => = +

=∑K x

HB HB,

sgn , if ,

,

δ β β δ12

1

0 otherwise

(5.46.b)

The design parameter δ > 0 can be considered as a sliding margin and determined due to a reasonable

control effort as in the continuous domain.

Proof:

Let ( ) ( ) ( ) ( )[ ]kvkuukkss rpq +=== HBxx ,,

then from Eqs.(5.40) and (5.46), we have ( ) ( ) ( ) ( )[ ] qrp ukvkuksks =+=−+ HB1

so ( ) ( ) suksks q 21 +=++

thus ( ) ( ) ( )suuksks qq 21 22 +=−+

To satisfy the sliding condition Eq.(5.45), we may choose ( ) vussvusu rprpq +>⇒<+⇒< HBHB 2

122


By Eq.(5.46.b), we obtain

s v xii

n

> +=∑1

21

HB δ

If

s v xii

n

≤ +=∑1

21

HB δ

then

HB HB HBv v v x sii

n

≤ ≤ + ==∑δ

1

2

or HB v s≤ 2

by Eq.(5.46.b), we have K 0r =

so ( ) ( ) ( )svvksks 21 22 +=−+ HBHB

hence ( ) ( )ksks 22 1 ≤+

Q.E.D.

Remark 5.1: Validity of Convergence Condition in Discrete-Time SMC

In Sarpturk et al 1987, both sliding and convergence conditions have been proposed as

( ) ( ) ( ){ }( ) ( ) ( ){ }

≥++<−+

condition econvergenc :01condition sliding :01

ksksksksksks

(5.47)

The sliding condition in Eq.(5.45) can be read ( ) ( )[ ] ( ) ( )[ ] ( ) ( ) ( )[ ] ( ) ( )[ ]s k s k s k s k s k s k s k s k s k+ − + + < ⇒ + − + + <1 1 0 1 1 02

or ( ) ( ) ( )[ ]{ } ( ) ( ) ( )[ ]{ } 011 <++−+ ksksksksksks

so

( ) ( ) ( )[ ]( ) ( ) ( )[ ]

( ) ( ) ( )[ ]( ) ( ) ( )[ ]

<++>−+

>++<−+

0101

0101

ksksksksksks

ksksksksksks

or (5.48)

Thus Eq.(5.47) is only part of the discrete-time sliding condition Eq.(5.45) so it may be too conservative

to be used.


5.5. A NEW ROBUST DISCRETE-TIME LINEAR SLIDING MODE CONTROLLER DESIGN In this section, a robust SMC design will be proposed for uncertain linear systems under parametric

uncertainty and external disturbance.

5.5.1. Robust Discrete-Time Linear SMC under External Disturbance To design a SMC under external disturbance, we have the following theorem

Theorem 5.5: Robust Discrete-Time Linear SMC Design under External Disturbance

Consider a linear system under disturbances ( ) ( ) ( ) ( ) vvkvkukk ≤++=+ ,...1 WBxAx (5.49)

with a hyperplane equation ( ) ( )kks xH.=

where

x B W A H, , ; ; ; , ,∈ℜ ∈ℜ ∈ℜ ∈ℜ× × ×n n n n u v s1 1

then a robust linear SMC function can be found from u e r p= − = + +K x K K K K. , (5.50)

where

• equivalent control ( ) ( )IAHHBK −= −1

e (5.50.a)

• reaching control ( )K HB Hr = −1ρ. (5.50.b)

• perturbation control ( )K HB HW Hp p v= −1ρ sup . . (5.50.c)

if the following sampling condition is satisfied (Remark 5.2) 0 1< + <ρ ρ p vsup .HW (5.50.d)

and if the following disturbance condition is satisfied

( ) [ ]s ks v

vv

vp

p

p

p

>+ − +

sup

sup .sup .

,sup .

sup .

ρ

ρ ρ

ρ

ρ ρ

HWHW

HW

HW1 (5.51)

where ρ ρ, p are discrete-time sliding margins equivalent to the continuous-time sliding margins δ δ, p .

Proof From Eqs.(5.49) and (5.50), we have ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )kvkukkkksks ....1.1 HWHBxIAHxxH ++−=−+=−+

or

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

ξ+α−=++ξ+α−=−+

kskskskksksks

.11.1

so ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )kkskksksks 2222 ..21.11 ξ+ξα−+α−α−=−+ (5.52)

where ( ) ( )α ρ ρ ξ= + =p v k v ksup . , .HW HW (5.53)


By the sampling condition in Eq.(5.50.d), we have ( ) 0110 >α−α⇒<α<

then we consider the following 2 cases

• If ( ) ( )α

ξ<

α−ξ− kk

1, then

( )ks ( )α−

ξ−1

k ( )α

ξ k

( ) ( )ksks 22 1 −+ −∞ − 0 + 0 − −∞

• If ( ) ( )α−

ξ−<

αξ

1kk , then

( )ks ( )α

ξ k ( )α−

ξ−1

k

( ) ( )ksks 22 1 −+ −∞ − 0 + 0 − −∞

So, in the worst case, the discrete-time sliding condition is satisfied if

( )

α−

ξ

α

ξ>

1,supks

By Eq.(5.53), we obtain Eq.(5.51).

Q.E.D.

5.5.2. Robust Discrete-Time Linear SMC under Parametric Uncertainties In this subsection, a general case of parametric uncertainty is considered first, then the cases under no

uncertainty and under matched uncertainty are considered as the special cases.

5.5.2.1. Matched and Unmatched Parametric Uncertainties We propose the following theorem for both non-matched and matched uncertain dynamical systems.

Assumption 5.1: System Constraint on Parametric Variation A system matrix ~B takes any variation such that the polarity of ( )HB~ is unchanged, ie.

( ) ( )sgn sgnHB HB B B~ ~= ⇒ <∆ ∆ (5.54)

where

~ ~B B B= + ∆ .


Under Assumption 5.1, we propose the following theorem to design a robust linear discrete-time sliding

mode controller for both matched and unmatched uncertainties.

Theorem 5.6: Robust Discrete-Time Linear SMC Design under Parametric Uncertainty

Consider an uncertain dynamical linear discrete-time system ( ) ( ) ( ) ( ) ( ) ( ) ( )x A x B A A x B Bk k u k k u k+ = + = + + +1 ~. ~. ~ . ~ .∆ ∆ (5.55)

with a hyperplane ( ) ( )kks xH.=

where ∆ ∆ ∆ ∆

~,

~A A B B≤ ≤

with x A A B B H∈ℜ ∈ℜ ∈ℜ ∈ℜ ∈ℜ× × × ×n n n n n u s1 1 1, , , , , , , ∆ ∆ .

Then, under Assumption 5.1, a robust discrete-time linear sliding mode control function can be determined

by ( ) ( ) prekku KKKKxK ++=−= ,. (5.56)

where

• equivalent control ( ) ( )K HB H A Ie = −−1 (5.56.a)

• reaching control ( )K HB Hr =

−1ρ. (5.56.b)


( )

( )KH A B K K

HBHBp

e r=

+ +

⊗∆ ∆

inf ~ sgn (5.56.c)

if the following sampling condition is satisfied (Remark 5.2)

( )1 12

− ≥ +

ρ

supinf

max.

HBHB i

i

ih∆ (5.57)

where ( )( )∆ ∆ ∆i i col ei riK K= + +H A B. . (5.57.a)

K K K Me r pn

ijm, , ,∈ℜ =×1 , [ ] [ ] [ ]p q p qij ij ij ij⊗ = .

Proof To keep equivalent and reaching controls similar to the case without uncertainty, we will prove a

perturbation control in Eq.(5.57.c) based on Eqs.(5.57.a) and (5.57.b).

Let K K Ker e r= +

by Eq.(5.57), we have ( ) ( ) ( ) ( )[ ] ( ) ( ) ( )[ ] ( ) ( ) ( )s k s k k k k u k k u k+ − = + − = − + = − +1 1H x x H A I x B H A I x HB. . ~ . ~. ~ . ~.

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )s k s k k k k k ker p er p+ − = − − + = − − −1 H A I x HB K K x H A I x HB K x HB K x. . .~

.~

. .~

. .

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )s k s k k k k k ker er p+ − = − − + − −1 H A I x HB K x H A x H B K x HB K x. . . . ~. . ~. . ~. .∆ ∆


( ) ( ) ( ) ( ) ( ) ( ) ( )s k s k k k ke e r p er+ − = − + − + −1 HBK x HB K K x HB K H B K H A x. . .~

. .~

. .~

.∆ ∆

( ) ( ) ( ) ( ) ( )s k s k k kr p er+ − = − − + −1 HB K x HB K H B K H A x. . ~. . ~. . ~ .∆ ∆

or ( ) ( ) ( ) ( ) ( )s k s k s k kp er+ − = − − + −1 ρ. . . . . .HB K H B K H A x∆ ∆

so ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )s k s k s k k s k s k s k k+ − = − − + + = − −1 1 2ρ ρR x R x,

where ( ) ( )R HB K H B K H A= + −

~.

~.

~p er∆ ∆ (5.58)

thus ( ) ( ) ( ) ( ){ } ( ) ( ) ( ){ }s k s k s k k s k k2 21 2+ − = − − − −ρ ρR x R x

( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( )[ ] ( )[ ]s k s k s k s k k k k2 2 21 2 2+ − = − − + − − + +ρ ρ ρ ρ R x R x R x.

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )[ ]s k s k s k s k k k k2 2 21 2 2 1+ − = − − − − +ρ ρ ρ R x R x R x.

( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )s k s k s k s k k k2 2 21 2 2 1+ − = − − − − −ρ ρ ρ R x R x

( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )s k s k s k k k k2 2 21 2 2 1+ − = − − − − −ρ ρ ρ H x R x R x

or ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )s k s k s k k kT T T2 2 21 2 2 1+ − = − − − − −ρ ρ ρx H R R x

To satisfy the sliding condition ( ) ( ) ( ) ( ) ( ) ( ) 0111 2222 <−+⇒<+⇒<+ ksksksksksks

we choose ( ) ( )[ ] ( )x H R R R xT T Tk k2 1 0− − ≥ρ

For a symmetric matrix, it can be shown that ( )[ ] ( )[ ]{ } [ ]Eig H R R R H R R R 01

2122 1 2 1− − + − − = ±ρ ρ α βT T T T T

, (5.59)

where

( )[ ] ( )TTn

ii

TT RHRRHR −γ=λ⇒−γ=α ∑=1

21 (5.60)

and ( )γ ρ= −2 1 (5.61)

so a necessary condition for a positive-definite matrix is

( ) 01

≥−γ∑=

n

iiii rrh (5.62)

we may choose ( ) nirrh iii ,,1,0 L=≥−γ (5.63)

We will prove that ( )γ ρ= − >2 1 0 (5.64)


(a) If hi > 0, then from Eq.(5.63) we have

(5.65) ( ) ( ) ( )0 < + − <HB H B H A~ ~ ~

K K hpi eri i col i∆ ∆ γ

( )( ) ( ) ( ) ( ) ( )H A H B HB H A H B∆ ∆ ∆ ∆~ ~ ~ ~ ~

i col eri pi i i col eriK K h K− < < + −γ

• If HB~ > 0 , then ( )( ) ( ) ( ) ( )H A H B HB H A H B∆ ∆ ∆ ∆

~ ~ ~ ~ ~i col eri pi i i col eriK K h K− < < + −γ

by Eq.(5.57.a), we have

∆ ∆i

pii iK

h

inf ~ sup ~HB HB≤ ≤

−γ (5.66)

Assume that

γ hi i i−

≥∆ ∆

sup ~ inf ~HB HB (5.67)

γ γ γh hhi i i i i

i

i

− ≥ ⇒ ≥ +

⇒ ≥ +

∆ ∆ ∆

∆sup~

inf~

sup~

inf~

sup~

inf~

HB

HB

HB

HB

HB

HB1 1

γ ≥ +

max

sup ~

inf ~i

i

ih∆

1HB

HB

thus we obtain the condition in Eq.(5.57). By Eq.(5.57) and hence Eq.(5.67), we can choose

( ) ( )K hpii i

i= =∆ ∆

inf ~ inf ~ sgn sgnHB HB

HB (5.68.a)

under Assumption 5.1.

• If HB~ < 0 , then ( )( ) ( ) ( ) ( )H A H B HB H A H B∆ ∆ ∆ ∆

~ ~ ~ ~ ~i col eri pi i i col eriK K h K− < − < + −γ

then under Assumption 5.1, we have

( ) ( )∆ ∆ ∆ ∆

inf ~ sup ~ inf ~ inf ~ sgn sgnHB HB HB HB

HB< − <−

⇒ = − =Kh

K hpii

pi i

γ (5.68.b)

since Eq.(5.57) can be read as

γ hi i i−

≥∆ ∆

sup~

inf~HB HB

(b) If hi < 0, then − < <γ h ri i 0

From Eq.(5.58), we have ( ) ( ) ( ) ( ) ( )− + − < < + −γ h K K Ki i col eri pi i col eriH A H B HB H A H B∆ ∆ ∆ ∆

~ ~ ~ ~ ~


• If HB~ > 0 , then we have ( ) ( ) ( ) ( )− + − < < + −γ h K K Ki i col eri pi i col eriH A H B HB H A H B∆ ∆ ∆ ∆

~ ~ ~ ~ ~

by the condition Eq.(5.57), under Assumption 5.1, we have

( )

( ) ( )− −

≤ ≤−

⇒ =−

=γ h

K K hipi pi i

∆ ∆ ∆ ∆

sup ~ inf ~ inf ~ inf ~ sgn sgnHB HB HB HB

HB (5.68.c)

• If HB~ < 0 , then ( ) ( ) ( ) ( )− + − < − < + −γ h K K Ki i col eri pi i col eriH A H B HB H A H B∆ ∆ ∆ ∆

~ ~ ~ ~ ~

by the condition Eq.(5.57), under Assumption 5.1, we have

( )

( ) ( )− −

< − <−

⇒ = =γ h

K K hipi pi i

∆ ∆ ∆ ∆

sup ~ inf ~ inf ~ inf ~ sgn sgnHB HB HB HB

HB (5.68.d)

Q.E.D.

Remark 5.2: Sampling Condition in Discrete-Time SMC

• The conditions in Eqs.(5.50.d) and (5.57) are sampling conditions because they depend on H, W and B

which in turn depend on the sampling rate.

• The sampling conditions may be too complicated to check, in particular Eq.(5.57). The implication is

that a faster sampling rate may be chosen for a lager uncertainty.

5.5.2.2. No Uncertainty Under no uncertainties, we have the following corollary to design a linear discrete-time sliding mode

controller

Corollary 5.1: Robust Discrete-Time Linear SMC Design under No Perturbation

A linear discrete-time sliding mode controller for a deterministic dynamical system can be determined by ( ) ( ) ( ) ( )kkku re xKKxK .. +−=−= (5.69)

where


• reaching control ( )K HB Hr = −1ρ. (5.69.b)

and the sampling condition in Eq.(5.57) reduces to ( )1 0− ≥ρ (5.70)

with x B A H, , , , ,∈ℜ ∈ℜ ∈ℜ ∈ℜ× × ×n n n n u s1 1

Proof If there is no uncertainty, we have ~ , ~ , ,A A B B A 0 B 0= = = =∆ ∆

then by Eq.(5.56.c), the perturbation control vanishes.

Q.E.D.


To design a linear discrete-time sliding mode controller, we must choose ρ. To do so, we propose the

following theorem.

Theorem 5.7: Closed-Loop Eigenvalues Composed of Hyperplane Eigenvalues and Sliding Margin

If a system has a discrete-time sliding margin ρ and a hyperplane-eigenvalue ( )[ ]HAHBBA 1eig −−⊂S , then the eigenvalues of the closed loop can be determined by

{ }λ λC Hn= − ∈ℜ ×1 1ρ, (5.71)

Proof: By definition of eigenvalues : ( ) [ ]{ }0deteig =−λλ= MIM ii , from Eq.(5.69)

( ) ( )[ ] ( ) ( )[ ]A BK A B HB H A I H A B HB H A I− = − − + = − − −− −1 1 1ρ ρ. .

To prove ( )1 − ρ is one of the eigenvalues, we have to prove

( ) ( )[ ] ( )[ ] ( )( ){ }det det1 0 1 01− − − = ⇒ − − − =−ρ ρ. . .I A BK HB BH I A I (5.72)

By the properties of a determinant P Q P Q Q P P Q. . . , ,= = ∈ℜ × n n

and I P Q I Q P P Qn m

n m m n+ = + ∈ℜ ∈ℜ× ×. . , ,

then Eq.(5.72) is satisfied since ( ) ( ) 011 =−=− −− IHBHBIBHHB

Thus we can write ( ) ( ) ( ) ( ) ( )λ ρ λ ρ λI A B HB H A I HB H− + − + = − + ×− − −1 1 11 P* n (5.73)

Eq.(5.73) holds for all ρ , so it must hold for ρ = 0 , then Eq.(5.73) becomes

( ) ( ) ( ) ( )11 1 −− λ×−λ=−+−λ n*PIAHHBBAI (5.74)

compared to Eq.(5.42), we have

( ) ( )11 −− λ≡λ nn PP* (5.75)

this proves that λ H are the remaining eigenvalues. Q.E.D

Remark 5.3: Sampling Condition Equivalent to Stability Criterion with Unit Circle Since ( )1 − ρ is a closed loop eigenvalue, to stabilize the system we must place it within the unit circle

1 1− <ρ (5.76)

hence the sampling condition in Eq.(5.70) is consistent with this condition. So the condition Eq.(5.70) is also

the condition of stability. In terms of a continuous-time sliding margin δ, we have the following relationship

ρ δ= − −1 e T (5.77)

The sampling condition Eq.(5.57) and hence Eq.(5.70) can be relaxed with a fast sampling rate, i.e. the

faster sampling rate is the easier this sampling condition satisfies.


5.5.2.3. Matched Uncertainty To determine the performance of a robust control design, we propose the following theorem

Theorem 5.8: Robust Discrete-Time Linear SMC Design under Matched Uncertainty

Consider an uncertain dynamical linear discrete-time system ( ) ( ) ( ) ( ) ( ) ( )x A x B A A x B A Ak k u k k u k+ = + = + + ≤1 ~. ~. ~ . . , ~

∆ ∆ ∆


where ∆ ∆ ∆ ∆

~,

~A A B B≤ ≤

If ∆A satisfies the matching uncertainties condition then the system-eigenvalues at the upper boundary { }λ λC B H, ,= −1 ρ (5.78)

where ( )

( )x B A A H. , , , , , , ,( )∈ℜ ∈ℜ ∈ℜ ∈ℜ ∈ℜ ∈ℜ× × × × −n n n nH

n u s1 11

1 11 ∆ λ δ

with ( )

( )111

)1( , ℜℜ −× n : unit circle

Proof From Eq.(5.56.c) with ∆B 0=

( )

( ) ( ) ( )KH A B K

HBHB H A

HBHB HB H Ap

er=

+= = −∆ ∆ ∆

∆.

inf ~ sgn . sgn .1 (5.79)

Since all the elements of ∆A are positive constant, from the matching uncertainties condition ∆A B v v= ∈ℜ ×. , 1 n

so Eq.(5.79) becomes ( )K HB H B v vp = =−1 . .

Thus the system-eigenvalues at the upper boundary, ~A A A= + ∆ , are ( ) ( )[ ] ( )[ ] ( )[ ]λ C e r p e r e r2 = − = + − + + = + − + + = − +eig eig eig eig2A B K A A B K K K A B v B K K v A B K K. . . . .∆

Recall that, under no perturbation we have ( ) { }eig .A B K− = −er H1 ρ, λ

so ( )[ ] { }λ λC e r H2 1= − + = −eig A B K K. ,ρ

Q.E.D.

Theorem 5.8 has shown that system eigenvalues of an uncertain dynamical system at the upper boundary ( )A A+ ∆ are equal to those of a deterministic dynamical system. We propose the following design rule to

move these eigenvalues inside the bounds by shifting down the nominal values of the system matrix.


Proposition 5.3: Robust Discrete-Time Linear SMC Design Rule Theorem 5.8 is valid for matched uncertain systems and H has all element the same polarity. In general,

choosing the nominal system matrices as A A A B B B→ + → +1

212∆ ∆,

in designing a hyperplane may produce a more robust controller.

5.5.3. Robust Discrete-Time Linear SMC under Uncertainties and Disturbances From Theorem 5.4 and 5.5, we have the following corollary to design a robust SMC for systems under

uncertainty and disturbance.

Corollary 5.2: Robust Discrete-Time Linear SMC Design under Uncertainty and Disturbance

Consider a discrete-time system under uncertainty and disturbances ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )x A x B W A A x B B Wk k u k v k k u k v k v k v+ = + + = + + + + ≤1

~.

~. .

~ ~. . ,∆ ∆ (5.80)


where ∆ ∆ ∆ ∆

~,

~A A B B≤ ≤

with x A A B B H∈ℜ ∈ℜ ∈ℜ ∈ℜ ∈ℜ× × × ×n n n n n u s1 1 1, , , , , , , ∆ ∆ .

Then, under Assumption 5.1, a robust discrete-time linear sliding mode control function can be determined

by ( ) ( ) prekku KKKKxK ++=−= ,. (5.81)

where


• reaching control ( )K HB Hr

Te= = −− −1 1ρ ρ δ. , (5.81.b)

• perturbation control K K Kp u d= + (5.81.c)

* uncertainty control

( )

( )KH A B K K

HBHBu

e r=

+ +

⊗∆ ∆

inf ~ sgn (5.81.d)

* disturbance control ( )K HB HW Hd v=

−1sup . . (5.81.e)

if the following sampling conditions is satisfied

( ) ( )( )1 12

− ≥ +

= +δ

supinf

max.

, .HBHB

H A Bi

i

ii i col erih

K∆∆ ∆ ∆ (5.81.f)


and 0 1< + <ρ ρ p vsup .HW (5.81.g)

and if the following disturbance condition is satisfied

( ) ( )s k

vv

v

vp

p

p

p

>+ − +

sup

sup .sup .

,sup .

sup .

ρ

ρ ρ

ρ

ρ ρ

HWHW

HW

HW1 (5.81.h)

Proof

Eqs(5.81.a), (5.81.b) and (5.81.d) can be determined using Theorem 5.5. Since the uncertainty control

can tackle uncertainty, Eq.(5.80) can be read as ( ) ( ) ( ) ( )x A x B Wk k u k v k+ = + +1 . . . (5.82)

the Eq.(5.81.e) can be determined by Eq.(5.82) using Theorem 5.4.

Q.E.D.

Based on the results on the robust VSS controller design Eq.(4.7) in Chapter and on the results on the

robust discrete-time linear SMC Eq.(5.80) above, we have the following proposition for the robust discrete-

time VSS controller design

Proposition 5.4: Robust Discrete-Time VSS Controller Design under Uncertainty and Disturbance For a linear system under perturbations (uncertainties and disturbances) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )x A x B W A A x B B Wk k u k v k k u k v k v k v+ = + + = + + + + ≤1

~.

~. .

~ ~. . ,∆ ∆

and a hyperplane ( ) ( )kks xH.=

with ∆ ∆ ∆ ∆

~,

~,A A B B≤ = = v v

then, under Assumption 4.1, there exists a constant δ > 0 for a VSS control function to be determined as u u u ue r p= + + (5.83)

where


• reaching control ( ) ( )u s er r r

T= − = = −− −HB K x K H1 1. . , . ,sgn ρ ρ δ (5.83.b)


( ) ( )( ) ( )

[ ]u K s Kv

xp p p p p i= − + = = =0 0K xH W

HB HBK

H A

HB HBx. . ,

.

inf ~ .sgn,

.

inf ~ .sgn,sgn

∆

(5.83.c)

with ( )+

××× ℜ∈δℜ∈ℜ∈ℜ∈ℜ∈ ,,, , ,, ,,,, 11 vsunnnn HAAWBBx ∆∆


5.6. A ROBUST INTEGRAL DISCRETE-TIME LINEAR SMC DESIGN A SMC is based on a state-space model, an I-action may be required to eliminate a steady-state error.

We have the following corollary to design a robust integral discrete-time SMC using Theorem 5.5.

Theorem 5.9: Robust Discrete-Time Linear SMC Design under Uncertainty

Consider an uncertain dynamical linear system system

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

x A x B A A x B B A A B B

C x

k k u k k u k

y k k

+ = + = + + + ≤ ≤

=

1 ~. ~. ~ . ~ . , ~ , ~

.

∆ ∆ ∆ ∆ ∆ ∆

and its augmented-order system with a reference input of r ( ) ( ) ( )x A x Bi i i i ik k u k+ = +1 ~ . ~ . (5.84)

where

~ ~~ ,

~ ~~A CA

0 AB CB

Bin

i=

=

×

1

1

(5.84.a)

then a robust integral discrete-time SMC with a hyperplane ~H can be determined by ( ) ( ) ( )u k u k u ki= + ~ (5.85)

where

( ) ( ) ( ) [ ]u k h r k h hi i i i i i i n= + =−

+H B H1

1 1 11, , ,, L (5.85.a)

and ( )~u k is a robust SMC for Eq.(5.84) and can be determined by Theorem 5.5.

with x B A C, , , , , ~, , ,∈ℜ ∈ℜ ∈ℜ ∈ℜ× × ×n n n n

iu u u y r1 1 .

Proof

With an input reference of r, define an integral as

( ) ( )x t y r dtt

00

= −∫

then its discretization is

( ) ( ) ( ) ( ) ( ) ( ) ( )zrzyzxzz

zrzyzx −=−⇒−

−= −

− 01

10 .11

or ( ) ( ) ( ) ( )111 00 +−+=−+ krkykxkx

thus ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( )x k x k k r k k u k r k k u k r k0 01 1 1 1 1+ − = + − + = + − + = + − +C x C A x B CA x CB. .

~.

~.

~.

~.

or ( ) ( ) ( ) ( ) ( )x k x k k u k r k0 01 1+ = + + − +CA x CB~. ~.

so we obtain the following augmented system

( )( )

( )( )

( )( ) ( )x

xCA

0 A xCBB 0i

n

kx k

kx k

ku k r k+ =

++

=

+

+−

+

×

111

1 110

1

0~

~ .~

~ . .


from Eq.(5.84.a), we have

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )x A x B 0 A A x B B 0i i i in

i i i i in

k k u k r k k u r k+ = + +−

+ = + + + +

−

+

× ×1

11

11

1 1

~ . ~ . . ~ . ~ . .∆ ∆ (5.86)

as the uncertainty component of u can tackle any uncertainty ∆~Bi , Eq.(5.85) is holds for ∆

~B 0i = so it can

be read as

( ) ( ) ( ) ( ) ( ) ( ) ( )x A A x B 0 A x B 0i i i i in

i i in

k k u k r k u k r k+ = + + + +−

= + +

−

+

× ×1 1

1 11

1 1∆~ . . . ~ . . . (5.86.a)

thus

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )s k k k u k r k k u k h r ki i i i i i i i in

i i i i i i+ = + = + +−

+ = + − +

×1 1

11 1

11H x H A x H B H 0 H A x H B. ~ . . . . . ~,

by Eq.(5.85.a), we have ( ) ( ) ( ) ( ) ( ) ( )s k k u k u k k u ki i i i i i i i i i i i i i+ = + − = +1 H A x H B H B H A x H B~ . . . ~ . .~

or ( ) ( ) ( )[ ]s k k u ki i i i i+ = +1 H A x B~ . .~ (5.87)

since ~u is a robust SMC for Eq.(5.84), the sliding condition ( ) ( )s k s ki i2 21 0+ − ≤ is satisfied using Eq.(5.87)

Q.E.D.

5.7. A NEW ROBUST DISCRETE-TIME LINEAR SLIDING-MODE OBSERVER DESIGN As in the state-space observer design where the estimator response is 3 to 10 faster than that of the

controller response, to design a robust observer, we propose the following theorem

Theorem 5.10: Robust Discrete-Time Linear Sliding-Mode Observer Design under Uncertainty

Consider a system state equation

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

x A x B A A x B B

y C x C C x

k k u k k u k

k k k

+ = + = + + +

= = +

1 ~. ~. ~ . ~ .~ ~

:

∆ ∆

∆ available output (5.88)

where ∆ ∆ ∆ ∆

~,

~A A B B≤ ≤

with x y A A B B C C H∈ℜ ∈ℜ ∈ℜ ∈ℜ ∈ℜ ∈ℜ ∈ℜ× × × × × ×n p n n n p n n u s1 1 1 1, , , , , , , , , , ∆ ∆ ∆

if the system is observable, then a robust SMO can be found from ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )$ $ . $ $

~$x Ax B L y y , y C. xk k u k k k k k+ = + + − =1 (5.89)

where L L L L= + +e r p (5.90)

with ( ) ( )L A I H CHe o o= −

−/ / 1 (5.91.a)

( ) ( )L H CHr o o o oT

oe o= = − = ×− −ρ ρ δ δδ/ / , , ..

11 3 10 (5.91.b)

( ) ( )( )

L HA L L C H

H C H Cp o

e r o

o o

= ⊗+ +

sgn. .

inf~

.sgn

/ /

/ /

∆ ∆ (5.91.c)


Ho is determined by the eigenvalue allocation method (Section 2.3) with the observer hyperplane-

eigenvalues ( ) ( )λ λHo Hc o Ho

= × ⇒ =3 10.. hyper , ,/ /H A C λ

Proof Since L will be determined under uncertainty, Eq.(5.89) can be perturbed to read as ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ]$

~$

~. $x A A x B B L y yk k u k k k+ = + + + + −1 ∆ ∆

then an error equation ( ) ( ) ( )11ˆ1 +−+=+ kkk xxe can be determined by

( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( ) ( )[ ] ( ) ( )e x x A x B LC x x A x B A LC ek k k k u k k k k u k k+ = + − + = + + − − + = −1 1 1$~

. $~

.~

$~

.~

.~ ~ ~

.

(5.92)

To determine L, we will find a function v to nullify the error by the following equation ( ) ( ) ( )e A e Ck k v k+ = +1 ~ ~/ / (5.93)

By Theorem 5.5, we can find ( ) ( )kkv eL/−= (5.94)

where L/ is determined by Eqs.(5.93), so Eq.(5.92) can be read

( ) ( ) ( ) ( ) ( )e A e C L e A LC ek k k k+ = − = −1 ~ ~ ~ ~ ~/ / //

(5.95)

we have the following mapping to complete the proof ~ ~

,~ ~

, ,/ / /A A B C H H K L→ → → →o (5.96)

so Eqs.(5.90) have been proved.

Q.E.D.

5.8. NUMERICAL EXAMPLES Some original designs from the literature will be used to compare with the proposed design. Sliding

parameters are the same as in the previous chapter, if available. Reaching parameters may be changed, if

necessary, to conform with the sliding-mode design rule.

Remark 5.4: Summary of Robust Discrete SMC Designs


the same unique value for simplicity. Different multiple values may be investigated to find a compromise

between the response speed and overshoot.

• If continuous-time sliding-mode controllers (continuous and discontinuous SMC) are available, then

Proposition 5.2 can be used to obtain discrete-time sliding-mode controllers. However, discrete-time

sliding-mode observers must be designed in the discrete-time domain;

• Nominal model can be determined using proposition 5.3 in designing a hyperplane (Chapter 2)

• A sampling rate can be chosen on the basis of Proposition 5.1 and Remark 5.2; • Sliding margin δ is chosen on the basis of Proposition 3.1, and the mapping is ρ δ= − −1 e T ;

• New Robust Discrete-Time Discontinuous SMC functions are computed by Theorem 5.4 or Proposition

5.4 where the latter is a discretization of Theorem 4.3;

• New Robust Discrete-Time Linear (Continuous) SMC functions are computed by Corollary 5.2 deduced

from Theorems 5.5 and 5.6 for a general case under uncertainty and disturbance where a special case


without perturbation (uncertainty and/or disturbance) will make the corresponding control component(s)

vanish;

• Robust Discrete-Time Integral SMC functions are computed by Theorem 5.9;

• New discrete-time robust sliding-mode observers are computed by Theorem 5.10

5.8.1 Example 5.1: No Uncertainty

Consider a SISO discrete-time state equation in Furuta 1990

A B=−

=

=

0.9953, 0.09050.0905, 0.8144

0.00470.0905

, , . secTs 0 1


The original control function is proposed in Furuta1990 as

( ) ( ) ( ) [ ]1,1, =+−= ere kku KxKK

where

( ){ } ( )

δ=ββ>δ

=⇒=δ ∑=

else,0

,if,sgn5 1

21

n

iii

ri

xksxksk

HBHB

0 5 100

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 5 100

0.02

0.04

0.06

0.08

0.1

Time [s]

Con

trol

0 5 100

0.2

0.4

0.6

0.8

1

Time [s]

Slid

ing

0 0.5 1-0.4

-0.3

-0.2

-0.1

0

Output

Cha

nge

of O

utpu

t

Original Discrete SMC Design for 2-nd Order System

Fig. 5.8: Original Discrete SMC Design in Example 5.1


5.8.1.2. New Design

Choose hyperplane-eigenvalues as p1 0 8 8 4899 10 6127= − ⇒ =. ,H . .

• Linear SMC

Choose the reaching dynamics 1.5 times faster than the sliding dynamics, then Corollary 5.2 yields

( ) ( ) [ ]1161.0,0772.0,. =−= KxK kku

0 5 100

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 5 100

0.01

0.02

0.03

0.04

Time [s]

Con

trol

0 5 100

2

4

6

8

10

Time [s]

Slid

ing

0 0.5 1-0.4

-0.3

-0.2

-0.1

0

Output

Cha

nge

of O

utpu

t

New Discrete Linear SMC Design for 2-nd Order System

Fig. 5.9: New Discrete Linear SMC Design in Example 5.1

• Discontinuous SMC

Choose δ = 0 4. , then Theorem 5.4 yields

[ ]K r = −0 4 0 4. .

0 5 100

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 5 10-0.5

0

0.5

1

Time [s]

Con

trol

0 5 10-5

0

5

10

Time [s]

Slid

ing

0 0.5 1-0.4

-0.3

-0.2

-0.1

0

Output

Cha

nge

of O

utpu

t

Discrete Discontinuous SMC Design for 2-nd Order System

Fig. 5.10: Discrete Discontinuous SMC in Example 5.1



The performances of proposed designs are comparable to those of the original design. Compared to the

original control effort, the Linear SMC has less control effort and Discontinuous SMC has higher value.

5.8.2. Example 5.2: Matched Uncertainty

Consider the following system from Sarpturk et al. 1987 ( ) ( ) ( ) ( )[ ]kpkukk ++=+ ..1 BxAx

where

( ) ( )A B=−

=

=

1 0.050.055 1.2

,.

, sin .0

0 0550 1p k kπ

Note that this form of perturbation has also appeared in Spurgeon 1992.


The proposed control function from the original design is

( ) ( ) ( )kxkxku 2211 ψ−ψ−=

with

( ) ( )( ) ( )

( ) ( )( ) ( )

<+>+

=ψ

<−>+

=ψ0.,14

0.,36,

0.,140

0.,138

2

22

1

11 kxksif

kxksif

kxksif

kxksif

and the hyperplane is

( ) ( ) [ ]1,20,. == HxH kks

0 1 2 3 4-0.5

0

0.5

1

Time [s]

Out

put

0 1 2 3 4-400

-200

0

200

400

Time [s]

Con

trol

0 1 2 3 4-10

0

10

20

Time [s]

Slid

ing

-0.5 0 0.5 1-15

-10

-5

0

5

10

Output

Cha

nge

of O

utpu

t

Original Robust Discrete Switching SMC Design for 2-nd Order System

Fig. 5.11: Original Robust Discrete Switching SMC Design under Matched Uncertainty in Example 5.2.


5.8.2.2. New Design


[ ]H = 50.65 18.18

thus [ ]K e = −1 4 4885.


Choose δ δ= =9 50and p , then Corollary 5.2 yields

[ ]K = 10 6703 16 9391. .

0 1 2 3 4-0.5

0

0.5

1

Time [s]

Out

put

0 1 2 3 4-15

-10

-5

0

5

Time [s]

Con

trol

0 1 2 3 4-5

0

5

10

15

20

Time [s]

Slid

ing

-0.5 0 0.5 1-0.8

-0.6

-0.4

-0.2

0

0.2

Output

Cha

nge

of O

utpu

t

Robust Discrete Linear SMC for 2-nd Order Matched Uncertaint System

Fig. 5.12: Robust Linear SMC under Matched Uncertainty in Example 5.2.

• Robust Discontinuous SMC


[ ]K r = 3 3

0 1 2 3 4-0.5

0

0.5

1

Time [s]

Out

put

0 1 2 3 4-2

0

2

4

6

8

Time [s]

Con

trol

0 1 2 3 4-5

0

5

10

15

20

Time [s]

Slid

ing

-0.5 0 0.5 1-1

-0.5

0

0.5

Output

Cha

nge

of O

utpu

t

Robust Discrete Discontinuous SMC for 2-nd Order Matched Uncertaint System

Fig. 5.13: Robust Discrete Discontinuous SMC under Matched Uncertainty in Example 5.2



The proprosed design has solved the chattering problem in the original design and it has much less

control effort than the original design.

5.8.3. Example 5.3: Integral Discrete -Time SMC

Consider a SISO discrete-time state equation in Chan 1991

A B=−

=

=

. , .. , .

..

0 99532116 0 09048374180 0904827418 0 8142533676

0 004678840160 090904837418

01, , . secTs


The original control function is proposed in Chan 1991 as

( ) ( ) ( ) ( ) [ ]0477.0,9958.0,8, 00 −−==ψ+−ψ−= ere kksku KxKK

where

( ){ } ( )δ

ρ

δρ

β β δ==

⇒ = −> =

=∑3

0 41

0

12

1.sgn , if ,

,k

s k x s k xri

i ii

n

HB HB

otherwise

0 1 2 3 40

2

4

6

Time [s]

Out

put

0 1 2 3 40

10

20

30

40

Time [s]

Con

trol

0 1 2 3 4-5

-4

-3

-2

-1

0

Time [s]

Slid

ing

0 2 4 60

2

4

6

8

Output

Cha

nge

of O

utpu

tOriginal Discrete Integral SMC Design for 2-nd Order System

Fig. 5.14: Original Discrete Integral SMC Design for Servo-System in Example 5.3.


5.8.3.2. New Design

Choose [ ]λH = − −35 35. . for the augmented system defined by Eq.(5.84), then

[ ]Hi = 9 8592 50 4187 7 9119. . .

and if the reaching dynamics are chosen to be 5 times faster than the sliding dynamics, then Theorem 5.9

yields

( ) [ ]( )[ ]

u k h rx k r

i i i i i= − − = =−

∑, , ,1

17 9229 50 2498 10 4972K x K x

x. . .

0 1 2 3 4-2

0

2

4

6

Time [s]

Out

put

0 1 2 3 4-20

0

20

40

60

Time [s]C

ontr

ol

0 1 2 3 4-300

-200

-100

0

100

Time [s]

Slid

ing

-2 0 2 4 6-2

0

2

4

6

8

Output

Cha

nge

of O

utpu

t

Discrete Integral SMC for 2-ndOrder System

Fig. 5.15: Discrete Integral SMC for Servo-System in Example 5.2.


The performances of proposed design are comparable to those of the original design.

5.8.4. Example 5.4: Deterministic System

Consider a linear 4-th order system (Sivaramakrishnan et al. 1984) as in Example 4.1

& ,x A x B A B= + =

−−

− − −

=

c c c cu

0.0500, 6.0000, 0, 0 0, 3.3330, 3.3330, 0

5.2080, 0, 12.5000, 12.5000

0.6000, 0, 0, 0

,

0 0

12.5000

0

Choose [ ]λ H = − − −6 6 6, , , then

[ ]H = . . . .0 4285 0 3508 0 0800 14401

thus [ ]K e = −0 4260 1 4014 0 1694 1 0000. . . .

and the sliding-eigenvalues are unchanged [ ]λ S , , = − − −6 6 6 .

Choose the sampling time as Ts = 0 01. sec .


Since this is about 17 times faster than the desired dynamics λ H , the corresponding discrete-time system is

( ) ( ) ( )x A x B A Bk k u k+ = + =− −− − −

=

1

0 9995 0 0590 0 0009 0

0 0008 0 9672 0 0308 0 00200 0493 0 0015 0 8825 0 1175

0 0060 0 0002 0 1

00 0020

0 1175

0

, ,.

.

. . .

. . . .

. . . .

. .

Remark 5.5: Comparison of Continuous and Discrete Hyperplane

Using the above discrete model to design a hyperplane, we obtain [ ]~

H = 42 3857 34 7105 7 9120 142 4628. . . . .

This discrete hyperplane differs by the above continuous one only by a scale factor of 100, so they are

essentially the same (Section 2.5).

• Discrete Linear SMC

Based on the robust practical design rule, choose δ = 18 , then Corollary 5.2 yields [ ]K r = 7 7122 6 3152 1 4400 25 9226. , . , . , .

thus [ ]K = 81382 77165 16094 24 9226. , . , . , .

and the closed-loop system-eigenvalues are [ ]λ C = − − − −6 6 6 18, , ,

0 1 2 3 4-0.5

0

0.5

1

Time [s]

Out

put

0 1 2 3 4-10

-5

0

5

Time [s]

Con

trol

0 1 2 3 4-0.2

0

0.2

0.4

0.6

Time [s]

Slid

ing

-0.5 0 0.5 1-0.8

-0.6

-0.4

-0.2

0

0.2

Output

Cha

nge

of O

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t

Discrete Linear SMC for 4-th Order System

Fig. 5.16: Discrete Linear SMC for 4-th Order System in Example 5.4.

• Discrete Switching SMC

Choose δ = 10 , then Proposition 5.4 yields [ ]K r = 4 2845 3 5084 0 8 14 4014. . . .


0 1 2 3 4-0.5

0

0.5

1

Time [s]

Out

put

0 1 2 3 4-10

-5

0

5

10

15

Time [s]

Con

trol

0 1 2 3 4-0.2

0

0.2

0.4

0.6

Time [s]

Slid

ing

-0.5 0 0.5 1-1

-0.5

0

0.5

Output

Cha

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of O

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t

Discrete Switching SMC Control for 4-th Order System

Fig. 5.17: Discrete Switching SMC for 4-th Order System in Example 5.4

• Discrete Tanh SMC

Choose ks = 20

0 1 2 3 4-0.5

0

0.5

1

Time [s]

Out

put

0 1 2 3 4-10

-5

0

5

Time [s]

Con

trol

0 1 2 3 4-0.2

0

0.2

0.4

0.6

Time [s]

Slid

ing

-0.5 0 0.5 1-1

-0.5

0

0.5

Output

Cha

nge

of O

utpu

t

Discrete TanH SMC Control for 4-th Order System

Fig. 5.18: Discrete Tanh SMC for 4-th Order System in Example 5.4

• Performances

The performances of the discrete-time SMC are comparable to those of the continuous-time SMC in

Example 4.1 for a deterministic 4-th order system.


5.8.5. Example 5.5: Matched Uncertain System

Consider a matched uncertain linear system in Coleman et al. 1994 as in Example 4.2 ( ) ( )& ~

.~

.x A A x B B= + + +c c c c u∆ ∆

where

A A B Bc c c c=− −

=

=

=−

0 1 00 0 10 1 2 5

0 0 00 0 00 1 0 5

0010

005.

,.

, , ∆ ∆


[ ]H = 0 5254 0 5293 01333. . .

thus [ ]K e = 0 0 4587 0 2293. .

Choose the sampling time as Ts = 0 01. sec

since it is about 50 times faster than the desired dynamics ( )λ H , then

( ) ( ) ( ) ( ) ( )& ~.

~.x A A x B Bk k u k+ = + + +1 ∆ ∆

where

A A B B=−

=

=

= −−

1 0 0100 00 1 0 00990 0 0099 0 9753

0 0 00 0 00 0 0099 0 0049

00 00050 0986

00 00020 0491

..

. .

. . .

. .

., , ,∆ ∆

• Robust Discrete Linear SMC

We have [ ]K p = 2 6268 33054 0 9960. . .


[ ]K r = 26268 2 6467 0 6667. . .

thus [ ]K = 5 2536 6 4107 1 8920. . .

0 2 4 60

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 2 4 6-2

0

2

4

6

Time [s]

Con

trol

0 2 4 6-0.6

-0.4

-0.2

0

Time [s]

Slid

ing


0 0.5 10

0.2

0.4

0.6

0.8

Output

Cha

nge

of O

utpu

t

Robust Discrete Linear SMC for 3-rd Order Matched Uncertain System

Fig. 5.19: Robust Discrete Linear SMC under Matched Uncertain 3-rd Order System in Example 5.5.

•


Robust Discrete Switching SMC

We have [ ]K p = 0 0 2 0 1. .

choose δ = 5 , then Proposition 5.4 yields

[ ]K r = 2 6667 2 6667 0 6667. . .

0 2 4 60

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 2 4 6-10

-5

0

5

Time [s]

Con

trol

0 2 4 6-0.6

-0.4

-0.2

0

0.2

Time [s]

Slid

ing


0 0.5 10

0.2

0.4

0.6

0.8

Output

Cha

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of O

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t

Robust Discrete Switching SMC Control for Matched Uncertain 3-rd Order System

Fig. 5.20: Robust Discrete Switching SMC under Matched Uncertain 3-rd Order System in Example 5.5.

• Robust Discrete Tanh SMC

Choose ks = 40

0 2 4 60

0.5

1

1.5

Time [s]

Out

put

0 2 4 6-2

-1

0

1

2

3

Time [s]

Con

trol

0 2 4 6-0.4

-0.3

-0.2

-0.1

0

0.1

Time [s]

Slid

ing


0 0.5 1 1.5-0.2

0

0.2

0.4

0.6

0.8

Output

Cha

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of O

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t

Robust Discrete TanH SMC Control for Matched Uncertain 3-rd Order System

Fig. 5.21: Robust Discrete Tanh SMC under Matched Uncertain 3-rd Order System in Example 5.5.

•


Performances


Example 4.2 for a matched uncertain 4-th order deterministic system.

5.8.6. Example 5.6: Un-Matched Uncertain System Under Disturbance

Consider an un-matched uncertain linear system under disturbance in Chen et al. 1989 as in Example 4.4

( ) ( )& ~ ~,x A A x B B W= + + + +∆ ∆ u v v ≤ 1

where

A B A B W=− −

=

=

=

=

0 1 04 0 18 7 8

00

1 3

0 0 02 0 05 0 0

00

0 3

100

,.

, ,.

, ∆ ∆

with the disturbance is chosen as follows for simulation

( )v t= sin 50


[ ]H = 20 6897 68966 0 6897. . .

thus [ ]K e = 41 7241 15 8621 1 3793. . .

Choose the sampling time as Ts = 0 01. sec

since it is about 50 times faster than the desired dynamics ( )λ H , then

A B A B W=

−

=

=

=

=

1 0 01 00 0404 1 0 0096

0 0755 0 0669 0 9228

00 0001

0 0125

0 0001 0 00 0202 0 0001 0

0 0474 0 0002 0

00

0 0029

0 010 0002

0 0004

., , , ,

.. .

. . .

.

.

.

. .

. .

.

.

.

∆ ∆

• Robust Discrete Linear SMC

We have [ ]K p = 995862 25 4483 2 4828. . .

Choose δ δ= =10 0 2and .p , then Corollary 5.2 yields

[ ]K r = 2068966 68 9655 6 8966. . .

and [ ]K d = 124 1379 41 3793 4 1379. . .

thus [ ]K = 472 3448 1516552 148966. . .


0 1 2 3 4-0.5

0

0.5

1

1.5

Time [s]

Out

put

0 1 2 3 4-600

-400

-200

0

200

Time [s]

Con

trol

0 1 2 3 4-10

0

10

20

30

Time [s]

Slid

ing


-0.5 0 0.5 1 1.5-3

-2

-1

0

1

Output

Cha

nge

of O

utpu

t

Robust Discrete Linear SMC for 3-rd Order under Un-Matched Uncertainty and Disturbance

Fig. 5.22: Robust Discrete Linear SMC under Un-Matched Uncertainty and Disturbance in Example 5.6.

• Robust Discrete Switching SMC

We have [ ]K p = 25 0307 0 1247 0 0004. . .

and k o = .29 9396

choose δ = 2 , then Proposition 5.4 yields

[ ]K r = 45 2387 15 0796 15079. . .

0 1 2 3 4-0.5

0

0.5

1

1.5

Time [s]

Out

put

0 1 2 3 4-200

-100

0

100

200

Time [s]

Con

trol

0 1 2 3 4-10

0

10

20

30

Time [s]

Slid

ing


-0.5 0 0.5 1 1.5-3

-2

-1

0

1

Output

Cha

nge

of O

utpu

t

Robust Discrete Switching SMC for 3-rd Order under Un-Matched Uncertainty and Disturbance

Fig. 5.23: Robust Discrete Switching SMC under Un-Matched Uncertainty and Disturbance in Example 5.6.


• Robust Discrete Tanh SMC

Choose k s = 2 , then

0 1 2 3 4-0.5

0

0.5

1

1.5

Time [s]

Out

put

0 1 2 3 4-150

-100

-50

0

50

100

Time [s]

Con

trol

0 1 2 3 4-10

0

10

20

30

Time [s]

Slid

ing


-0.5 0 0.5 1 1.5-2

-1.5

-1

-0.5

0

0.5

OutputC

hang

e of

Out

put

Robust Discrete TanH SMC for 3-rd Order under Un-Matched Uncertainty and Disturbance

Fig. 5.24: Robust Discrete Tanh SMC under Un-Matched Uncertainty and Disturbance in Example 5.6.

• Performances


Example 4.4 for an unmatched uncertain 4-th order deterministic system under disturbance.

5.8.7. Example 5.7: Robust Sliding-Mode Observer under Un-Matched Uncertainty

and Disturbance

The controller parameters as in Example 5.6 above, choose the discrete-time sliding-mode observer

dynamics 2 times faster than the linear SMC dynamics, then Theorem 5.10 yields

L A A LC=−

⇒ = − =−−

−

1 686514 1978

4 0141

0 6863 0 01 014 1574 0 9999 0 00964 0896 0 0669 0 9228

...

.. . .

. . .o

.


0 1 2 3 4-0.5

0

0.5

1

1.5

Time [s]

Out

put

0 1 2 3 4-600

-400

-200

0

200

Time [s]

Con

trol

0 1 2 3 4-10

0

10

20

30

Time [s]

Slid

ing


-0.5 0 0.5 1 1.5-3

-2

-1

0

1

Output

Cha

nge

of O

utpu

t

Robust Discrete Sliding-Mode Observer for Linear SMC under Un-Matched Uncertainty and Disturbance

Fig. 5.25: Robust Discrete Sliding-Mode Observer for Linear SMC under Un-Matched Uncertainty and

Disturbance in Example 5.7.

0 1 2 3 4-0.5

0

0.5

1

1.5

Time [s]

Out

put

0 1 2 3 4-200

-100

0

100

200

Time [s]

Con

trol

0 1 2 3 4-10

0

10

20

30

Time [s]

Slid

ing


-0.5 0 0.5 1 1.5-2

-1.5

-1

-0.5

0

0.5

Output

Cha

nge

of O

utpu

t

Robust Discrete Sliding-Mode Observer for TanH SMC under Un-Matched Uncertainty and Disturbance

Fig. 5.26: Robust Discrete Sliding-Mode Observer for Tanh SMC under Un-Matched Uncertainty and


• Performances

The performances of the case using sliding-mode observer are comparable to those of the ideal case

where all states are assumed available in the pervious example.


5.8.8. Example 5.8: Robust Sliding-Mode Observer under Converted Matched

Uncertainty and Disturbance

Example 4.8 is discretized with Ts = 0 01. sec to have

=

=

=

−=

0029.000

,0001.00202.02018.000001.0001.0000

,0125.00001.00

,9230.00269.03844.00096.010019.0001.01

BABA ∆∆

By Proposition 5.2, we can use the analog SMC in Example 4.8 as a discrete SMC for this example.

However, we have to design a discrete observer by choosing sliding-mode observer dynamics 2 times faster

than linear SMC, then Theorem 5.10 yields

−−−−

=−=

=

9230.00269.02146.30096.015221.14001.09330.0

,5990.35241.149330.1

LCAAL e

0 1 2 3 4-0.5

0

0.5

1

1.5

Time [s]

Out

put

0 1 2 3 4-600

-400

-200

0

200

Time [s]

Con

trol

0 1 2 3 4-10

0

10

20

30

Time [s]

Slid

ing


-0.5 0 0.5 1 1.5-3

-2

-1

0

1

Output

Cha

nge

of O

utpu

t

Robust Discrete Sliding-Mode Observer for Linear SMC under Converted Matched Uncertainty and Disturbance

Fig. 5.27: Robust Discrete Sliding-Mode Observer for Linear SMC under Converted Matched Uncertainty

and Disturbance in Example 5.8.


0 1 2 3 4-0.5

0

0.5

1

1.5

Time [s]

Out

put

0 1 2 3 4-200

-100

0

100

200

Time [s]

Con

trol

0 1 2 3 4-10

0

10

20

30

Time [s]

Slid

ing


-0.5 0 0.5 1 1.5-3

-2

-1

0

1

Output

Cha

nge

of O

utpu

t

Robust Discrete Sliding-Mode Observer for TanH SMC under Converted Matched Uncertainty and Disturbance

Fig. 5.28: Robust Discrete Sliding-Mode Obs erver for Tanh SMC under Converted Matched Uncertainty and


• Performances

An unmatched uncertain system can be conveted into a matched one using Proposition 4.3. The

performances of these 2 systems are comparable using the proposed sliding-mode observer (Cf. Figs. 5.27 &

5.28 with 5.25 & 5.26)

5.9. CONCLUSIONS In both designs from Utkin and Furuta, there is a boundary layer corresponding to the term sk < β , so

there is no chattering, but strictly speaking it is pseudo-sliding mode control, while this is not the case for the

new linear discrete sliding mode control. In the proposed design in Section 2, there is no condition on β as in

Furuta's approach (Furuta 1990, Pieper et al 1992). The main advantages of the proposed discrete linear SMC

are that the control function is simple and the performance is predictable (system-eigenvalues). Furthermore, it

is applicable to both matched and unmatched uncertainty. There is a condition on the sampling rate for

uncertain dynamical systems but not for deterministic systems. However, this condition can be relaxed by

increasing the sampling rate as shown in Example 2 above.

A convergence condition has been proposed in Sarpturk et al. 1987 to guarantee system stability while

the hyperplane has not been based on hyperplane-eigenvalues. However, although my simulation violates

this convergence condition, stable responses are still produced. In Spurgeon 1992, the convergence

condition was also presented, however there was no proof to support this condition.


In the current SMC literature, a discrete-time SMC has been presented in Sarpturk et al. 1987, Sira-Ramirez

1991 and Spurgeon 1992, but a robust discrete-time SMC has not been fully developed until this work. A new

robust discrete-time linear SMC design has been fully developed. Its analysis has been used to explore the

true nature of SMC including the invariance property and the stability problem. In the New Robust Discrete-

Time Linear SMC Design , the control function is partitioned into 3 components: equivalent control, reaching

control, and perturbation control. The control function is not only continuous but also linear. The continuous

nature of this control function helps to eliminate the chattering problem in the standard SMC design. The

advantage of the control function being linear is that some fundamental concepts of SMC design can be

explained from the linear control theory frame-work.


Chapter 5

A New Robust Discrete -Time Sliding Mode Controller-Observer Design

5.1. Introduction...................................................................................................................................................................1

5.2. Digital Control Systems ...............................................................................................................................................2

5.2.1. Some Terminology...........................................................................................................................................2

5.2.2. ADC as Physical Sampler................................................................................................................................3

5.2.3. Starred Transform as Ideal Sampler...............................................................................................................4

5.2.4. Sampling Characteristics and Signal Reconstruction .................................................................................5

Theorem 5.1: Laplace Transform of Sampled Signal..........................................................................5

5.2.4.1. Signal Reconstruction.........................................................................................................................6

Proposition 5.1: Choice of Sampling Time...........................................................................................6

5.2.4.2. Aliasing or Folding..............................................................................................................................7

5.2.5. Z-Transform ......................................................................................................................................................8

5.2.6. Some Typical S-Z Transformations...............................................................................................................8

5.2.7. Discretization of Continuous-Time State-Space Equations..................................................................... 10

Theorem 5.2: Discretization of State-Space Model .......................................................................... 10

5.3. Discrete-Time Sliding Hyperplane Design.............................................................................................................. 12

Proposition 5.2: Discrete-Time Sliding-Mode Controller Design................................................... 12

5.3.1. Hyperplane Equation ..................................................................................................................................... 13

5.3.2. Equivalent Control......................................................................................................................................... 13

5.3.3. Sliding-Eigenvalues ....................................................................................................................................... 13

Theorem 5.3: Discrete-Time Sliding Eigenvalues ............................................................................. 13

5.4. A Robust Discrete-Time Discontinuous SMC (VSS) Design.............................................................................. 15

Theorem 5.4: Robust Discrete-Time Discontinuous SMC Design under Matched Uncertainty15

Remark 5.1: Validity of Convergence Condition in Discrete-Time SMC....................................... 16

5.5. A New Robust Discrete-Time Linear Sliding Mode Controller Design.............................................................. 17

5.5.1. Robust Discrete-Time Linear SMC under External Disturbance............................................................. 17

Theorem 5.5: Robust Discrete-Time Linear SMC Design under External Disturbance............... 17

5.5.2. Robust Discrete-Time Linear SMC under Parametric Uncertainties ....................................................... 18

5.5.2.1. Matched and Unmatched Parametric Uncertainties ..................................................................... 18

Assumption 5.1: System Constraint on Parametric Variation......................................................... 18

Theorem 5.6: Robust Discrete-Time Linear SMC Design under Parametric Uncertainty ........... 19

Remark 5.2: Sampling Condition in Discrete-Time SMC.................................................................. 22

5.5.2.2. No Uncertainty................................................................................................................................... 22

Corollary 5.1: Robust Discrete-Time Linear SMC Design under No Perturbation ...................... 22

Theorem 5.7: Closed-Loop Eigenvalues Composed of Hyperplane Eigenvalues and Sliding

Margin..................................................................................................................................................... 23

Remark 5.3: Sampling Condition Equivalent to Stability Criterion with Unit Circle..................... 23


5.5.2.3. Matched Uncertainty........................................................................................................................ 24

Theorem 5.8: Robust Discrete-Time Linear SMC Design under Matched Uncertainty ............. 24

Proposition 5.3: Robust Discrete-Time Linear SMC Design Rule.................................................. 25

5.5.3. Robust Discrete-Time Linear SMC under Uncertainties and Disturbances.......................................... 25

Corollary 5.2: Robust Discrete-Time Linear SMC Design under Uncertainty and Disturbance25

Proposition 5.4: Robust Discrete-Time VSS Controller Design under Uncertainty and

Disturbance ............................................................................................................................................ 26

5.6. A Robust Integral Discrete-Time Linear SMC Design.......................................................................................... 27

Theorem 5.9: Robust Discrete-Time Linear SMC Design under Uncertainty.............................. 27

5.7. A New Robust Discrete-Time Linear Sliding-Mode Observer Design............................................................... 28

Theorem 5.10: Robust Discrete-Time Linear Sliding-Mode Observer Design under Uncertainty28


5.8.1 Example 5.1: No Uncertainty.......................................................................................................................... 30

5.8.1.1. Original Design.................................................................................................................................. 30

5.8.1.2. New Design........................................................................................................................................ 31

5.8.1.3. Performances ...................................................................................................................................... 32

5.8.2. Example 5.2: Matched Uncertainty.............................................................................................................. 32

5.8.2.1. Original Design.................................................................................................................................. 32

5.8.2.2. New Design........................................................................................................................................ 33

5.8.2.3. Performances ...................................................................................................................................... 34

5.8.3. Example 5.3: Integral Discrete-Time SMC................................................................................................... 34

5.8.3.1. Original Design.................................................................................................................................. 34

5.8.3.2. New Design........................................................................................................................................ 35

5.8.3.3. Performances ...................................................................................................................................... 35

5.8.4. Example 5.4: Deterministic System............................................................................................................... 35

Remark 5.5: Comparison of Continuous and Discrete Hyperplane................................................ 36

5.8.5. Example 5.5: Matched Uncertain System.................................................................................................... 38

5.8.6. Example 5.6: Un-Matched Uncertain System Under Disturbance........................................................... 40

5.8.7. Example 5.7: Robust Sliding-Mode Observer under Un -Matched Uncertainty and Disturbance ..... 42

5.8.8. Example 5.8: Robust Sliding-Mode Observer under Converted Matched Uncertainty and

Disturbance............................................................................................................................................................... 44

5.9. Conclusions................................................................................................................................................................. 45

Chapter 6

General Sliding-Mode Controller Design for

MIMO Uncertain Nonlinear Systems

6.1. INTRODUCTION

In this chapter, I present a sliding-mode control (SMC) design for SISO and MIMO nonlinear systems. So

far I have only considered the case where the output is the first system state. In this chapter I will consider

the general case where the output is a nonlinear function of all system states.

For a MIMO SMC, the hierarchical control technique has been used in Utkin 1977 for linear systems.

Alternatively, I will use a decoupling technique which is applicable for MIMO nonlinear systems. This

technique allows a MIMO system to be considered as a set of SISO subsystems. As consequence, all SISO

results developed so far can be applied, including this chapter.

For the general SMC design for MIMO nonlinear systems, in the SMC literature (Fernandez et al. 1987;

Chen et al. 1992), the hyperplane design has been based on the Input-Output Linearization technique (Hunt

et al. 1983, Kravaris et al. 1986, Isidori 1989, Edwards et al. 1995) to transform a nonlinear system into a

canonical nonlinear system. By the nature of a hyperplane that it is of reduced-order, I will design the

hyperplane via the direct allocation approach in Chapter 2. The controller will be designed in a unified manner

as in other cases. The proposed robust design may be the simplest approach in the literature (Fu 1992, Sira-

Ramirez 1996). In addition, the design scheme in Proposition 3.3 is proved to be efficient in solving the

chattering and steady-state error (Example 6.6).

Part A : SMC Design for SISO Uncertain Nonlinear Systems

I start with the hyperplane design and a nonlinear system stability test. Next I present a discontinuous

SISO nonlinear SMC and a continuous SISO nonlinear SMC. I then present a general SMC design for

nonlinear systems where the output does not have to be the first state, y x≠ 1, but can be a nonlinear function

of system states, y = η xa f . I then present a robust discontinuous SMC design and robust continuous

pseudo-SMC design for SISO uncertain nonlinear systems.

6.2. HYPERPLANE AND STABILITY

In Chapter 2 on the hyperplane design, the proof of the direct allocation method is based on linear

systems. In Chapter 3, the proof of the stability test is based on linear systems. Now both proofs will be

modified so that they are also applicable to nonlinear systems. As consequence, the I/O-state technique can

be applied for both linear and nonlinear systems.


6.2.1. Hyperplane Design

Theorem 6.1: Hyperplane Design for Canonical Nonlinear Systems

For a n-ordered canonical linear or nonlinear system, if the hyperplane-eigenvalues are λ H n= −λ λ λ1 2 1, , ,L (6.1)

then a hyperplane can be found by s h h hn= = −H x H. , 1 2 1 1L (6.2)

where hi are the coefficients of the following polynomial

λ λ λ λ λ λ λ λ λ λ− − − = + + + +−−

−−

1 2 11

12

2 1c hc h c hL Lnn

nnh p (6.3)

Proof

Consider the following n-ordered canonical linear or nonlinear system with output y x= 1

& &

&& &

&

& ,

y x x

y x x

y x x

y x

nn n

nn

= =

= =

= =

= =

R

S

||||

T

||||−

−

1 2

2 3

11

L

a f

a f a f a fφ φx x x : linear or nonlinear function of the system state variable

and a hyperplane

s h h

x

x

x

h x h x xnn

n

n n n= =

L

N

MMMMM

O

Q

PPPPP= + + +−

−− −Hx 1 1

1

11 1 1 11, , , .L

ML

In the sliding mode, I have the following linear differential equation s h x h x x h y h y yn n n n

n n= ⇒ + + + = ⇒ + + + =− − −− −0 0 01 1 1 1 1 1

2 1L L a f a f (6.4)

then the corresponding characteristic equation is

λ λnn

nh h−−

−+ + + =11

21 0L (6.5)

thus if the roots of this equation Eq.(6.5) are Hurwitz, then the output decays according to the above linear

differential equation Eq.(6.4).

Therefore, to design a hyperplane for a canonical linear or nonlinear system, first choose the n −1b g desired hyperplane-eigenvalues to be Hurwitz λ H n= ⊂ ℜ− −

λ λ1 1, ,L a f

then the hyperplane is determined by the coefficients of the characteristic equation as follows λ λ λ λ λ λ λ− − = + + + +−

−−

−1 1

11

22 1c h c hL Ln

nn

nh h h

Q.E.D.

6. General Sliding-Mode Controller Design for MIMO Uncertain Nonlinear Systems 6.3

Remark 6.1: Hyperplane Design for Canonical Linear and Nonlinear Systems

• Note that this proof is more general than that in Section 2.3.1 because it is for both linear and nonlinear

systems.

• Note that φ xa f is not taken into account in the hyperplane design, but it will be in the controller design.

This is consistent with the invariance condition because a canonical form satisfies the matching condition.

6.2.2. SMC Stability Criterion for Nonlinear Systems

Based on Theorem 6.1 above, I have the following corollary on a stability test for both linear and

nonlinear systems.

Corollary 6.1: Stability Criterion

For an n-ordered canonical linear or nonlinear system, if there exists a control function to satisfy the

sliding condition s s. & ≤ 0 on the hyperplane determined by Theorem 6.1, then the system is stable.

Proof

The control function satisfying the sliding condition will drive the system into the sliding mode. Then, by

Theorem 6.1, the system is stable.

Q.E.D.

6.3. ROBUST DISCONTINUOUS SMC DESIGN FOR SISO UNCERTAIN NONLINEAR

SYSTEMS

In this section, I will consider a robust SMC design for SISO uncertain nonlinear systems.

6.3.1. Discontinuous SMC Design for SISO Nonlinear Systems

I first define a nonlinear system states vector to be used in a SMC for a nonlinear system

Definition 6.1: Nonlinear System States Vector

Consider a SISO nonlinear system

& , , .x t f x g xa f a f a f= +t t u (6.6)

then the nonlinear system states vector is composed of every term in f x,ta f. For example in Example 6.1, I have

& . ,&

&,

sin( ),x f g x f g= + =

LNMM

OQPP =

+ +

LNMM

OQPP =

+

LNMM

OQPP

ux

x

x

x x x tx x1

2

2

1 2 12

1 12

0

1

then w = x x x x tx2 1 2 1

21, , , sin( )


I then have the following theorem to design a discontinuous SMC for a nonlinear system

Theorem 6.2: Discontinuous SMC Design for Nonlinear Systems

Consider a SISO nonlinear system in Eq.(6.6). Then a discontinuous SMC control function can be found

by u u ue r= + (6.7)

where

• equivalent control u we e i= − = − = =−Hg Hf K w w w x xa f a f a f1 . , (6.7.a)

• reaching control u sr r r= − =−Hg K w Ka f a f1 . . ,sgn δ (6.7.b)

with x f g H, , ; ; , ;∈ℜ ∈ℜ ∈ℜ ∈ℜ× ×

+n u s1 1 n δ a f : sliding margin.

Proof

Consider a hyperplane s s u u u ueq= ⇒ = = + = + = −−Hx Hx H f g Hg Hg Hf Hg& & . . .a f a f d i1

then from the definition of w above, I have − = = ×− ∑u k weq ei iHg Hf xa f a f1

As usual, let

u k wi i= ×∑ xa f

then, to satisfy the sliding condition, I obtain

ss kk s w

k s wk

k s w

k s wi

e i i

e i ii

e i i

e i i

&,

,

*,

*,< ⇒

< − >

> − <

RS|T| ⇒ =− − >

− + <

RS|T|0

0

0

0

0

if

if

if

if

Hg

Hg

Hg

Hg

δ

δ

thus by Lemma 3.1, a discontinuous SMC control function is u k w w s k w w sei i i ei i i= − × − × = − × − ×∑ ∑ ∑ ∑−x x x Hg xa f a f a f a f a f a f a fδ δ* . .sgn sgn1

where

δ δ= Hg . *

Q.E.D.

Remark 6.2: Reaching Control

I have chosen equal sliding margin for the reaching control since I have no knowledge of the dynamics of

w. This remark applies for all designs in the sequel.

6.3.2. Robust Discontinuous SMC Design for SISO Uncertain Nonlinear Systems

I have the following assumption


A system matrix g takes any variation such that the polarity of Hg%b g is unchanged, i.e.

sgn sgnHg Hg%b g a f= (6.8)

where g is the nominal value of %g .


Under Assumption 6.1, I have the following theorem

Theorem 6.3: Robust Discontinuous SMC Design for Uncertain Nonlinear Systems

Consider a canonical uncertain nonlinear SISO system

& % , % , .x f x g xt t t ua f a f a f= + (6.9)

then, under Assumption 6.1, a robust dis continuous SMC control function is determined by u u u ue r p= + + (6.10)

where


ue e ei i= − =+

K w K. ,sup inf

2 (6.10.a)


u sr r r= − =K w Hg K. . ,sgna f δ (6.10.b)


u sp p pi i= − =−

K w Hg K. . ,sup inf

sgna f2

(6.10.c)

with w ,sup , infi i defined by

− = = = = =− ∑u w weq

i

ii

i

i

iiHg Hf

SupInf

wSupInf

w w x x% % .sup

inf. ,

sup

inf,b g a f a f1

=

where x f g H, % , % , , , ,∈ℜ ∈ℜ ∈ℜ ∈ℜ× ×

+n u s1 1 n δ a f : sliding margin.

Proof

Consider a hyperplane

s s u u u ueq= ⇒ = = + = + = −−

Hx Hx H f g Hg Hg Hf Hg& & % % . %. % % %.e j b g d i1

then

− = = ×− ∑u weq

i

iiHg Hf x% % sup

infb g a f1

Let u k wi i= ×∑ xa f

then, to satisfy the sliding condition

ss ks w

s wk

s w

s wii i

i ii

i i

i i

&sup , %

inf , %

sup , %

inf , %< ⇒

< − >

> − <

RST| ⇒ =− − >

− + <

RST|00

0

0

0

if

if

if

if

Hg

Hg

Hg

Hg

δ

δ

thus, by Lemma 3.1, under Assumption 6.1, the discontinuous SMC control function is

u w w si ii

i ii= −

+FH IK × − +−FH IK ×∑ ∑sup inf sup inf

.2 2

x x Hga f a f a fδ sgn

Q.E.D.

Remark 6.3: Nonlinearity as Uncertainty

I can apply this robust control formula for the nonlinear case where the nonlinearity is considered as an

uncertainty within an operating range.


Remark 6.4: Alleviation of Chattering

To eliminate the chattering problem, for simplicity, the discontinuous SMC control functions in Eqs.(6.7)

and (6.8) are smoothed out by the following substitution of a hyperbolic tangent function (Section 3.5.4) sgn s k ssa f c h→ tanh (6.11)

where ks is chosen low enough to eliminate the chattering problem. This is a continuous pseudo-SMC

(Remark 3.4)


As stated in Remark 4.13, there is negligible difference between a true-SMC and a pseudo-SMC.

6.4. ROBUST CONTINUOUS SMC D ESIGN FOR SISO UNCERTAIN NONLINEAR SYSTEMS

In this section, I will consider a SMC design for SISO nonlinear systems, then a robust pseudo-SMC

design for SISO uncertain nonlinear systems.

6.4.1. Continuous SMC Design for SISO Nonlinear Systems

Under no perturbation, I have the following theorem.

Theorem 6.4: Continuous SMC Design for Nonlinear Systems

Consider a canonical nonlinear SISO system

& , , .x t f x g xa f a f a f= +t t u

then a continuous SMC control function is determined by u u ue r= + (6.12)

where


ue = − −Hg Hfa f 1 , (6.12.a)


u sr = − −Hga f 1 . .δ (6.12.b)

where x f g H, , , , , ,∈ℜ ∈ℜ ∈ℜ ∈ℜ× ×

+n n u s1 1 δ a f : sliding margin.

Proof

Since the system is in the canonical form, direct eigenvalue allocation is applied for a hyperplane s s u u= ⇒ = = + = + −Hx Hx H f g Hg Hg Hf& & . .a f a f 1

.

By the above continuous SMC control function, I have

ss s& .= − <δ 2 0: the sliding condition is satisfied.

Q.E.D.


6.4.2. Robust Continuous Pseudo-SMC Design for SISO Uncertain Nonlinear Systems

For an uncertain nonlinear system, I have only a pseudo-SMC which is determined by the following

corollary

Corollary 6.2: Robust Continuous Pseudo-SMC Design for Uncertain Nonlinear Systems

Consider a canonical uncertain dynamical nonlinear SISO system & % , % , .x f x g xt t t ua f a f a f= +

then, under Assumption 6.1, a discontinuous SMC control function is determined by u u u ue r p= + + (6.13)

where


ue e ei i= − =+

K w K. ,sup inf

2 (6.13.a)

• reaching control u k sr r s r= − =K w Hg K. . tanh ,b g δ (6.13.b)


u k sp p s pi i= − =−K w Hg K. . tanh ,

sup infb g2

(6.13.c)

with w , sup , infi i defined by

− = = = = =− ∑u w weq

i

ii

i

i

iiHg Hf

SupInf

wSupInf

w w x x% % .supinf

. ,supinf

,b g a f a f1 =

where x f g H, % , % , , , ,∈ℜ ∈ℜ ∈ℜ ∈ℜ× ×

+n n u s1 1 δ a f : sliding margin.

Proof By Section 3.5.4, the sign function is replaced by the hyperbolic tangent function to eliminate the

chattering problem.

Q.E.D.

6.5. GENERAL SMC DESIGN FOR SISO NONLINEAR SYSTEMS For the general case, the output does not have to be the first state, y x≠ 1, but can be a possibly nonlinear

function of system states, y = η xa f . First I have a discontinuous SMC, then a continuous SMC. For a

discontinuous SMC, I have the following theorem

Theorem 6.5: General Discontinuous SMC Design for Nonlinear Systems

Consider a general nonlinear SISO system

& , , .x f x g x

x

= +

=

RS|T|

t t u

y

a f a fa fη

(6.14)

with the relative degree of n, then a discontinuous control function can be found by u u ue r= + (6.15)


where


u L L Len n

n n= − +− −

−g f f H z1 1

1 1 2η ηd i e ja f a f: : , (6.15.a)


u L L s srn= − =− −

g f H z1 1η δd i a f. . , .sgn (6.15.b)

with

x f g z x

x x x

H z

, , , , ,

, , , , , , , , ,: :

∈ℜ ∈ℜ ∈ℜLNMM

OQPP

= =

×+

−

−

−

−

− − − −

nn

n

n

n

T

n n n n n n

T

u yd

dt

d

dt

d

dt

h h h h z z z z

12

2

1

1

1 1 1 2 2 1 2 2 3 1

, =

δ ηη η η

a f

a f a f

a f a f a f a fL

L L

;

Proof

By the I/O state method, keep differentiating y until u first appears, since the relative degree r, under the

Lie derivative notation:

&

&&

. ,

y L

y L

y L

y L L L u L L

n n

n n n n

=

=

=

= + ≠

R

S

||||

T

||||− −

− −

f

f

f

f g f g f

η

η

η

η η η

2

1 1

1 1 0

L

a f

a f with

thus

z z=

L

N

MMMMMMM

O

Q

PPPPPPP

=

L

N

MMMMMMM

O

Q

PPPPPPP

⇒ =

L

N

MMMMMMM

O

Q

PPPPPPP

=

L

N

MMMMMMM

O

Q

PPPPPPP−−

−

−−

z

z

z

z

y

y

y

y

z

z

z

z

y

y

y

y

n

n

n

n

n

n

n

n

1

2

12

1

1

2

11

M M M M

&

&

&

&

&

&

&

&&

a fa f

a fa f

then

&

.

.z

f g f f g f

=

L

N

MMMMMMM

O

Q

PPPPPPP

=

+

L

N

MMMMMMM

O

Q

PPPPPPP

=

L

N

MMMMMMM

O

Q

PPPPPPP

+

L

N

MMMMMMM

O

Q

PPPPPPP− −

z

z

z

v

z

z

z

L L L u

z

z

z

L

z

z

z

L L

u

r r

n nr

n

r

n

2

3

2

3

1

2

3

2

3

1

M M M M

η η η η

where

v L L L u u L L v Ln n n n= + ⇔ = −− − −

f g f g f fη η η η1 1 1. d i d i

then by the direct allocation method, I have a hyperplane s h h s v v vn n n eq= = ⇒ = + = −− −

Hz z H z1 1 1 1 21, , , . &

: :L a f a f

with veq n n

= −−

H z1 1 2: :a f a f

To satisfy the sliding condition, s s s v veq. & < ⇒ − <0 0d i


then the control function is v v s s s seq= − ⇒ = − <δ δ. . & .sgn a f 0

thus

u L L v s L L L s Lneq

n nn n

n= − − = − + +− − − −

−g f f g f fH z1 1 1 1

1 1 2η δ η η δ ηd i a fd i d i a fe ja f a f. .

: :sgn sgn

Q.E.D.

Now for a continuous SMC, I have the following theorem

Theorem 6.6: General Continuous SMC Design for Nonlinear Systems

Consider a general nonlinear SISO system

& , , . ;x f x g x x= + =t t u ya f a f a f η

with the relative degree of n, then a continuous control function can be found by u u ue r= + (6.16)

where


u L L Len n

n n= − +− −

−g f f H z1 1

1 1 2η ηd i e ja f a f: : (6.16.a)


u L L s srn= − =− −

g f H z1 1η δd i . . , . (6.16.b)

where

x f g z xx x x

, , , , ,∈ℜ ∈ℜ ∈ℜLNMM

OQPP

×

+

−

−

−

−

nn

n

n

n

T

u yd

dt

d

dt

d

dt

12

2

1

1 =δ η

η η ηa f a f a f a f a f

L

H z1 1 1 2 2 1 2 2 3 1: :, , , , , , , , ,n n n n n n

Th h h h z z z z− − − −= =a f a fL L .

Proof

Similar to the above proof, but now for a continuous SMC, to satisfy the sliding condition, the control

function is v v s s s seq= − ⇒ = − <δ δ. . & . 2 0

thus

u L L v s L L L s Lneq

n nn n

n= − − = − + +− − − −

−g f f g f fH z1 1 1 1

1 1 2η δ η η δ ηd i d i d i e ja f a f. .: : .

Q.E.D.


Part B : General SMC Design for MIMO Uncertain Nonlinear Systems

In this part, I will use a simple decoupling method to convert a MIMO system into multiple SISO

subsystem. Then all SISO results developed so far can be applied, including the robust results and even the

recent nonlinear result.

6.6. GENERAL SMC DESIGN FOR MIMO UNCERTAIN NONLINEAR SYSTEMS

This section will generalize the SISO technique above for MIMO nonlinear systems

Theorem 6.7: General Discontinuous SMC Design for MIMO Nonlinear Systems

Consider a general nonlinear MIMO system

& . ;

;,

;, ;

, ,x f x G x u

y xx R

u y RG x g x g x u y

= +

=

UVW∈∈

UVW = =L

NMMM

O

QPPP

=L

NMMM

O

QPPP

a f a fa f a f a f a f

η ,

n

m m

m m

u

u

y

y1

1 1

L M M (6.17)

If the system Jacobian L LG fr x−1ηa f is non-singular, then a discontinuous control function can be found from

u u u= +e r (6.18)

where


u H zG fr 1

fr

1:r 1e nL L L= − +− −

−η ηd i e ja f a f1

2: , (6.18.a)


u s s H zG fr 1

r L L= − =− −η δd i a f1

. . , .sgn (6.18.b)

Proof

Similarly to the case of SISO

&

&&

.

&

&&

.

y L

y L

y L

y L L L u

y L

y L

y L

y L L L u

r r

r r r

m m

m m

m

r rm

m

r r rm

m m

m m

m

m

1 1

12

1

1

1 11

1 11

1

2

1 1

11

1 1

1 1

1

1

=

=

=

= +

R

S

|||||

T

|||||

=

=

=

= +

R

S

|||||

T

|||||

− −

−

− −

−

f

f

f

f g f

f

f

f

f g f

η

η

η

η η

η

η

η

η η

L L L

c h

c h

c h

c h

where r = r rm1 L : relative degree of the system

then

y v v ur

fr

G fra f

= = + −, L L Lη η1 .

with

L

L

L

L

L L

L L h

L L h

L L h

L L h L L h

L L h L L h

L L

r

r

rm

r

r

r

r r

r r

rm m

m

m

m

fr

f

f

f

G fr

G f

G f

G f

g f g f

g f g f

g f

x

x

x

x

x x

x xη η=

L

N

MMMMMM

O

Q

PPPPPP=

L

N

MMMMMM

O

Q

PPPPPP=−

−

−

−

− −

− −

−

1

2

1

2

1

1 1

1

2 2

1

1

2 1

11

11

11

11

11

11

11

1

η

η

η

M M

L

L

M N M, a f

a fa f

a f

a f a fa f a f

h L L hm

mr1

11x xg fa f a fL −

L

N

MMMMMM

O

Q

PPPPPP


since the system Jacobian L LG fr x−1ηa f is non-singular, so

u vG fr

fr= −− −

L L L1 1η ηd i d i

thus the control function, elementwisely

u L L L s

s

i

r r

r

in

i ii

ii i

i i

i= − + +LNM OQP=

RS|T|

− −

−G f f H z

H z

1 1

1 2η ηe j c hc ha f

a fa f a f

a f a f

: .

.

δ sgn

Q.E.D.

For a continuous SMC control function, I have

Theorem 6.8: General Continuous SMC Design for MIMO Nonlinear Systems

Consider a general nonlinear MIM O system

&

,,

, ,x f x G x u

y x

x R

u y RG x g x g x u y

= +

=

UV|W|∈

∈

UV|W| = =

L

N

MMMM

O

Q

PPPP=

L

N

MMMM

O

Q

PPPPa f a fa f a f a f a f

η ,

n

m m

m m

u

u

y

y1

1 1

L M M

If the system Jacobian L LG fr x−1ηa f is non-singular, then the continuous control function is determined

by u u u= +e r (6.19)

where


u H zG fr 1

fr

1:r 1e nL L L= − +− −

−η ηd i e ja f a f

1

2: , (6.19.a)


u s s H zG fr 1

r L L= − =− −η δd i 1

. . , . (6.19.b)

Proof.

Similar to the above proof of Theorem 6.7.

Q.E.D.

Remark 6.6: Singular System Jacobian

If the system Jacobian L LG fr x−1ηa f is singular, then the Structure Algorithm (Silverman 1969) and the

error dynamic equation (Chen et al. 1992) must be used.

Remark 6.7: A Robust SMC Design for General MIMO Uncertain Nonlinear Systems

For a class of MIMO systems whose system Jacobians are non-singular, I can use a simple decoupling

technique to get a multiple SISO subsystem. All of the robust SISO results so far are then directly applicable

for a robust general SMC design for MIMO uncertain nonlinear systems. It is a very direct extension; Example

6.6 illustrates this issue.


6.7. NUMERICAL EXAMPLES .

Because it is quite feasible for a SMC to deal with nonlinear systems, so there is no differences, and

hence no difficulties in the design approach between the regulating control and the tracking control . Thus

it is illustrated by an example without any special development.

This section is crucial in the fact that it clarifies something left open previously for the sake of simplicity,

such as the design of a tracking control, the definition of w in an equivalent control (Section 6.3), the

decoupling technique to convert a MIMO system into a multiple SISO subsystem, a robust MIMO general

SMC controller design (Remark 6.7).

There is no original design available for all the following examples, the proposed designs are based on

the design guideline in Proposition 3.1.

For MIMO examples, Example 6.4 to 6.6, each has 2 simulations to check interactions after decoupling, only

the first output is under control in the first simulation, then both are under control in the second simulation.

Remark 6.8: Summary of SMC for Uncertain MIMO Systems

I will consider only canonical nonlinear system

• Hyperplane for a SISO canonical nonlinear system can be computed using Theorem 6.1;

• Continuous SMC for SISO nonlinear systems can be designed using Theorem 6.4. There is no such

design for uncertain systems;

• Discontinuous SMC for SISO uncertain nonlinear systems can be designed using Theorem 6.3 where

deterministic systems are special cases (or Theorem 6.2 can be attempted);

• For general SISO nonlinear systems where the output is a function of system states, discontinuous and

continuous SMC can be designed using Theorem 6.5 and 6.6, respectively; while Theorem 6.7 and 6.8 for

MIMO systems.

6.7.1. Example 6.1: SISO Nonlinear System

Consider a system in Zhou et al. 1992:

& .

,&&

,sin ( )

,x f g

x f g= +=

RST =LNM

OQP =

+ +LNM

OQP =

+LNM

OQP

uy x

x

x

x

x x x tx x1

1

2

2

1 2 12

1 12

0

1

choose a hyperplane-eigenvalue λ H = − ⇒ =2 2 1H

By Eq.(6.7), the equivalent control is

− = = +FH IK + + +−−

u x x x x x txeq Hg Hfa f 11

1

2 1 2 12

11 2 2 sin ( )

thus K we x x x x x tx= + × =−

1 2 2 1 11

1

2 1 2 12

1e j , , , , , , , sin( )


6.7.1.1. Discontinuous SMC

Theorem 6.3 yields a discontinuous SMC as

u se r= − −K w K w. . .sgna f.

Choose δ = 2 to have

Kr x= + ×−

1 2 2 2 21

1e j , , ,

0 1 2 3 40

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 1 2 3 4-2

0

2

4

6

Time [s]

Con

trol

0 1 2 3 4-0.5

0

0.5

1

1.5

2

Time [s]

Slid

ing

0 0.5 1-1.5

-1

-0.5

0

Output

Cha

nge

of O

utpu

t

Regulating Discontinuous SMC for Nonlinear System

Fig. 6.1: Regulating Discontinuous Nonlinear SMC for Example 6.1.

6.7.1.2. Continuous Pseudo-SMC In the discontinuous control function above, let ( ) ( )sgn tanhs k ss→ and choose ks = 15 . Then I have

0 1 2 3 40

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 1 2 3 4-2

-1

0

1

2

3

Time [s]

Con

trol

0 1 2 3 40

0.5

1

1.5

2

Time [s]

Slid

ing

0 0.5 1-1.5

-1

-0.5

0

Output

Cha

nge

of O

utpu

t

Regulating TanH-Continuous SMC for Nonlinear System

Fig. 6.2: Regulating Tanh-Continuous Nonlinear SMC for Example 6.1.


6.7.1.3. Continuous SMC

Theorem 6.4 yields a continuous control function as

u s xx

x x x txs= − + = − +FH IK + +

LNMM

OQPP+

RS|T|UV|W|

−−

Hg Hfa f 1

1

1 2

1 2 12

1

1 2 12

δ δ. , .sin( )

.

so

u x x x x x tx s= − + + + + +−

1 2 21

1

2 1 2 12

1e j m rsin( ) .δ .

Choose δ = 4 , then I have

0 1 2 3 40

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 1 2 3 4-6

-4

-2

0

2

Time [s]C

ontr

ol

0 1 2 3 40

0.5

1

1.5

2

Time [s]

Slid

ing

0 0.5 1-1.5

-1

-0.5

0

Output

Cha

nge

of O

utpu

t

Regulating Continuous SMC for Nonlinear System

Fig. 6.3: Regulating Continuous Nonlinear SMC of Form 3 for Example 6.1.


The performance of the continuous SMC is comparable to that of the continuous pseudo-SMC.

6.7.2. Example 6.2: SISO Uncertain Nonlinear System

Consider a system in Zhou et al. 1992 which is modified with uncertainties

&

& sin( ), ,

.

.

x

x

x

ax x bx b tx x ua b

1

2

2

1 2 12

1 11

3

1

1 5

0 5

LNMM

OQPP

=+ + + +FH IK

LNMM

OQPP = = .

Choose a hyperplane-eigenvalue λ H = − ⇒ =2 2 1H

then the hyperplane

s = H x.

and

& . & sin( )s x ax x bx b tx x u= = + + + + +FH IKH x 2 12 1 2 12

1 1


or

& sin( )s x u x x ax x bx b tx= +FH IK + +FH IK + + +RST

UVW−

1 1 21 1

1

2 1 2 12

1 .

So

− = +FH IK + + + = +FH IK + + +−

LNMM

OQPP

− −

u x x ax x bx b tx x x x x xeq 1 2 12

2

3

1

1 5

0 5

1 5

1 51

1

2 1 2 12

1 1

1

2 1 2 12sin( )

.

.

.

.

thus w = 1 2 1 2 1

2, , ,x x x x

6.7.2.1. Robust Discontinuous SMC

Theorem 6.3 yields a control function as u se rp= − −K w K w. . .sgna f

where

K e x= +−

1 0 2 2 11

1e j , , , .

Choose δ = 0 2. , I have

K K Krp r p x= + = +−

1 1 7 0 2 1 2 0 71

1e j . , . , . , .

0 1 2 3 40

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 1 2 3 4-5

0

5

10

Time [s]

Con

trol

0 1 2 3 4-0.5

0

0.5

1

1.5

2

Time [s]

Slid

ing


0 0.5 1-1.5

-1

-0.5

0

0.5

Output

Cha

nge

of O

utpu

t

Robust Discontinuous SMC for Uncertain Nonlinear System

Fig. 6.4: Robust Discontinuous Nonlinear SMC for Example 6.2, 2 curves coincide.


6.7.2.2. Robust Continuous Pseudo-SMC

In the discontinuous control function above, let ( ) ( )sgn tanhs k ss→ and choose ks = 30 , then I have

0 1 2 3 40

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 1 2 3 4-4

-2

0

2

Time [s]

Con

trol

0 1 2 3 40

0.5

1

1.5

2

Time [s]

Slid

ing


0 0.5 1-1.5

-1

-0.5

0

OutputC

hang

e of

Out

put

Robust TanH-Continuous SMC for Uncertain Nonlinear System

Fig. 6.5: Robust Tanh-Continuous Nonlinear SMC for Example 6.2, 2 curves coincide.

6.7.3. Example 6.3: Tracking SISO Nonlinear System

Consider a system in Zhou et al. 1992 which is modified with a reference output:

&

& sin( )

ln ln

x

x

x

x x x tx x u

y t z y y x tref ref

1

2

2

1 2 12

1 1

1

2 1

1 1

LNMM

OQPP =

+ + + +FH IKLNMM

OQPP

= + ⇒ = − = − +

RS||

T|| a f a f

then

z x t

z xt

z x x x txt

x u

= − +

= −+

= + + ++

+ +FH IK

R

S

||||

T

||||

1

2

1 2 12

1 2 1

1

1

1

21

11

ln

&

&& sin ( )

a f

a f

.

Choose a hyperplane-eigenvalue λ H = − ⇒ =2 2 1H

then the hyperplane

s = H z.

so

& & && sin( )s z z xt

x x x txt

x u= + = −+

FHG

IKJ + + + +

++ +FH IK2 2

1

12

1

112 1 2 1

21 2 1a f

or

& sin( )s x u x xt

x x x txt

= +FH IK + +FH IK −+

FHG

IKJ + + + +

+

LNMM

OQPP

RS|T|

UV|W|

−

1 1 21

12

1

11 1

1

2 1 2 12

1 2a f


hence

u x xt

x x x txt

eq = − +FH IK −+

FHG

IKJ + + + +

+

LNMM

OQPP

−

1 21

12

1

11

1

2 1 2 12

1 2sin( ) a f


By Theorem 6.3, a discontinuous SMC is determined by

u x xt

x x x txt

s= − +FH IK −+

FHG

IKJ + + + +

++

RS|T|UV|W|

−

1 21

12

1

11

1

2 1 2 12

1 2sin( ) .a f a fδ sgn .

Choose δ = 2 , I have

0 1 2 3 40

0.5

1

1.5

2

Time [s]

Out

put

0 1 2 3 4-3

-2

-1

0

1

2

Time [s]C

ontr

ol

0 1 2 3 4-0.5

0

0.5

1

1.5

Time [s]

Slid

ing

Response Reference

0.8 1 1.2 1.4 1.6-0.4

-0.2

0

0.2

0.4

Output

Cha

nge

of O

utpu

t

Tracking Discontinuous SMC for Nonlinear System

Fig. 6.6: Tracking Discontinuous Nonlinear SMC for Example 6.3.

6.7.3.2. Continuous Pseudo-SMC In the discontinuous control function above, let ( ) ( )sgn tanhs k ss→ and choose ks = 60 , then

0 1 2 3 40

0.5

1

1.5

2

Time [s]

Out

put

0 1 2 3 4-1.5

-1

-0.5

0

0.5

1

Time [s]

Con

trol

0 1 2 3 4-0.5

0

0.5

1

1.5

Time [s]

Slid

ing

Response Reference

0.8 1 1.2 1.4 1.6-0.4

-0.2

0

0.2

0.4

Output

Cha

nge

of O

utpu

t

Tracking TanH-Continuous SMC for Nonlinear System

Fig. 6.7: Tracking Tanh-Continuous Nonlinear SMC for Example 6.3.



By Theorem 6.4, a continuous SMC is determined by

u x xt

x x x txt

s= − +FH IK −+

FHG

IKJ + + + +

++

RS|T|UV|W|

−

1 21

12

1

11

1

2 1 2 12

1 2sin( ) .a f δ .

Choose δ = 4 , I have

0 1 2 3 40

0.5

1

1.5

2

Time [s]

Out

put

0 1 2 3 4-3

-2

-1

0

1

Time [s]

Con

trol

0 1 2 3 4-0.5

0

0.5

1

1.5

Time [s]

Slid

ing

Response Reference

0.5 1 1.5 2-0.6

-0.4

-0.2

0

0.2

0.4

Output

Cha

nge

of O

utpu

t

Tracking Continuous SMC for Nonlinear System

Fig. 6.8: Tracking Continuous Nonlinear SMC for Example 6.3.


The performance of the continuous SMC is comparable to that of the continuous pseudo-SMC.

6.7.4. Example 6.4: MIMO Nonlinear System

Consider a system in Chen et al. 1992

&

&

&

&

&

,

x x x x u u

x x u u

x x x x

x x x x x x x x u

x x u

y x x

y x

1 2 2 3 1 2

2 3 1 2

3 4 32

52

4 5 3 4 32

52

5 5 2

5 5 2

1 1 22

2 3

2

2 2 2

2

= + + +

= + +

= − −

= + − − + − −

= − −

R

S

||||

T

||||

= −

=

RS|T|

c h

d i c h

then

&&

&&&

,

y x u u

y x u

L L

Lx

x1 3 1 2

2 5 2

1

3

52

1 1

0 1

2

2 3

= + +

= − −

RS|T| ⇒

= =−

LNM

OQP

= =−

LNMM

OQPP

=

R

S

|||||

T

|||||

−M x

M x

r

g G fr

f fr

η

η

a f

a f


thus u M v Mg f= −−1 c h .

Hence

y x x

y x

y y

y v

y x

y x x x

y x

y y

y y

y v

11 1 22

12 2

11 12

12 1

21 3

22 4 32

52

23 5

21 22

22 23

23 2

= −

=

RS|T|

⇒=

=

RS|T|

=

= − −

=

RS||T||

⇒

=

=

=

RS||T||

&

&,

&

&

&

where

&

&

y

y

v

v

12

23

1 1

2

LNMM

OQPP = + = + − = =

LNMM

OQPP

−M M u M M M v M vf g f g g fc h

thus

s h y y

s h y y h y v

s h y h y y

s h y h y y h y h y v

1 11 21 12

1 11 11 12 11 12 1

2 21 21 22 22 23

2 21 21 22 22 23 21 22 22 23 2

= +

= + = +

RS|T|= + +

= + + = + +

RS|T|& & &,

& & & &.

Choose the hyperplane-eigenvalues as

λ λH H1 1 2 22 2 1 2 2 4 4 1= − ⇒ = = − − ⇒ =H H,

then the control function is u M M Mg f s= − +−1c h

with

M Mg f=−

LNM

OQP =

−

LNMM

OQPP

1 1

0 1 2

3

5

, x

x


Theorem 6.7 yields

MH

H Hs

y s

y y s=

+

+ +

LNMM

OQPP = =1 12 1 1

2 22 2 23 2 21 2

1

1 24 0 2

a f c ha f a f c h

.

. ., , .

δ

δδ δ

.sgn

.sgn

0 1 2 3 4-0.5

0

0.5

1

Time [s]

Out

put

0 1 2 3 4-4

-2

0

2

4

Time [s]

Con

trol

0 1 2 3 4-0.5

0

0.5

1

1.5

2

Time [s]

Slid

ing

Subsystem 1 Subsystem 2

-0.5 0 0.5 1-1

-0.5

0

0.5

Output

Cha

nge

of O

utpu

t

Discontinuous SMC for MIMO Nonlinear System: Y1(0) = 1, Y2(0) = 0

Fig. 6.9: MIMO Discontinuous Nonlinear SMC for Example 6.4: Y1(0) = 1, Y2(0) = 0


0 1 2 3 4-0.5

0

0.5

1

Time [s]

Out

put

0 1 2 3 4-5

0

5

Time [s]

Con

trol

0 1 2 3 4-0.5

0

0.5

1

1.5

2

Time [s]

Slid

ing


-0.5 0 0.5 1-1.5

-1

-0.5

0

0.5

Output

Cha

nge

of O

utpu

t

Discontinuous SMC for MIMO Nonlinear System: Y1(0) = Y2(0) = 1

Fig. 6.10: MIMO Discontinuous Nonlinear SMC for Example 6.4: Y1(0) = Y2(0) = 1


In the above discontinuous functions, let sgn tanhs k ssa f b g→ and choose k ks s1 250 1000= =, to have

0 1 2 3 40

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 1 2 3 4-2

-1

0

1

2

Time [s]

Con

trol

0 1 2 3 4-0.5

0

0.5

1

1.5

2

Time [s]

Slid

ing


0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

Output

Cha

nge

of O

utpu

t

TanH-Continuous SMC for MIMO Nonlinear System: Y1(0) = 1, Y2(0) = 0

Fig. 6.11: MIMO Tanh-Continuous Nonlinear SMC for Example 6.4: Y1(0) = 1, Y2(0) = 0


0 1 2 3 40

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 1 2 3 4-6

-4

-2

0

2

4

Time [s]

Con

trol

0 1 2 3 4-0.5

0

0.5

1

1.5

2

Time [s]

Slid

ing


0 0.5 1-1.5

-1

-0.5

0

Output

Cha

nge

of O

utpu

t

TanH-Continuous SMC for MIMO Nonlinear System: Y1(0) = Y2(0) = 1

Fig. 6.12: MIMO Tanh-Continuous Nonlinear SMC for Example 6.4: Y1(0) = Y2(0) = 1


Theorem 6.8 yields

MH

H Hs

y s

y y s=

+

+ +

LNMM

OQPP

= =1 12 1 1

2 22 2 23 2 11 2

1

1 24

a fa f a f

.

. .,

δ

δδ δ

.

.

0 1 2 3 40

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 1 2 3 4-8

-6

-4

-2

0

2

Time [s]

Con

trol

0 1 2 3 4-0.5

0

0.5

1

1.5

2

Time [s]

Slid

ing


0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

Output

Cha

nge

of O

utpu

t

Continuous SMC for MIMO Nonlinear System: Y1(0) = 1, Y2(0) = 0

Fig. 6.13: MIMO Continuous Nonlinear SMC for Example 6.4: Y1(0) = 1, Y2(0) = 0


0 1 2 3 40

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 1 2 3 4-6

-4

-2

0

2

Time [s]

Con

trol

0 1 2 3 4-0.5

0

0.5

1

1.5

2

Time [s]

Slid

ing


0 0.5 1-1.5

-1

-0.5

0

Output

Cha

nge

of O

utpu

t

Continuous SMC for MIMO Nonlinear System: Y1(0) = Y2(0) = 1

Fig. 6.14: MIMO Continuous Nonlinear SMC for Example 6.4: Y1(0) = Y2(0) = 1


• The decoupling is effective as the second output is almost unaffected. The second output is essentially

equal to zero if it is initially at zero and under no control. It goes to zero from an initial value if it is under

control as required.

• The performance of the continuous SMC is comparable to that of the continuous pseudo-SMC since for

the latter, to eliminate the chattering completely, it is necessary to reduce the gain ks further, but the

steady-state error will be noticeable. This is consistent with Remark 3.5.

6.7.5. Example 6.5: MIMO Nonlinear Robot Manipulators

Consider a robot system in Young 1978

& ;

&

sin . . cos cos

cos sin . cos

. cos cos;

x x

x

x x x x x x x u

x x x x x u

x x

1 2

2

3 4 2 4 1 1 3 1

3 3 22

1 3 2

32

3

9 5 2 66 15 49

4 5 5 49

125 75 50 25

=

=

+ − − + +

− + − − + +

RS|T|

UV|W|

+ −

c h c hc h c h

L

and

& ;

&

cos sin . . cos cos

. cos sin . cos

. cos cos;

x x

x

x x x x x x x x u

x x x x x u

x x

3 4

4

3 3 4 2 4 1 1 3 1

3 3 22

1 3 2

32

3

4 5 5 2 66 15 49

15 75 10 5 49

125 75 50 25

=

=

− + + − − + +

+ + − − + +

RS|T|

UV|W|

+ −

c h c h c hc h c h

L

then the control function u M M M= − +−

g f s1d i;


where

M g x xx

x x= + −

− +

− + +

LNMM

OQPP

−125 75 50 25

9 4 5

4 5 15 75 1032

3

1 3

3 3

. cos coscos

cos . cosd i c h

c h ;

and

M Mf g

x x x x x x x

x x x x=

+ − − +

− − +

LNMM

OQPP

5 2 66 15 49

5 493 4 2 4 1 1 3

3 22

1 3

sin . . cos cos

sin . cos

c h c hc h

Choose

λ

λ

H

H

T

T

s

s

x x

x x

1 1

2 2

1

2

1 1 2

2 3 4

1 1 1

1 1 1

= − ⇒ =

= − ⇒ =

UV|W|

⇒ =LNMM

OQPP =

LNMM

OQPP

H

Hs

H

H

θ

φr2

r1

g


Theorem 6.7 yields

MH

Hs

x s

x s=

+

+

LNMM

OQPP = =1 2 1 1

2 4 2 21 2

1

10 5

a f c ha f c h

.

.; . ;

δ

δδ δ

.sgn

.sgn

0 2 4 6 8-0.5

0

0.5

1

Time [s]

Out

put

0 2 4 6 80

50

100

150

Time [s]

Con

trol

0 2 4 6 8-0.5

0

0.5

1

Time [s]

Slid

ing


-0.5 0 0.5 1-0.6

-0.4

-0.2

0

0.2

Output

Cha

nge

of O

utpu

t

Discontinuous SMC for MIMO Robotic Nonlinear System: Y1(0) = 1, Y2(0) = 0

Fig. 6.15: MIMO Discontinuous Nonlinear SMC for Example 6.5: Y1(0) = 1, Y2(0) = 0


0 2 4 6 80

0.5

1

1.5

Time [s]

Out

put

0 2 4 6 8-50

0

50

100

150

Time [s]

Con

trol

0 2 4 6 8-0.5

0

0.5

1

1.5

Time [s]

Slid

ing


0 0.5 1 1.5-0.6

-0.4

-0.2

0

0.2

Output

Cha

nge

of O

utpu

t

Discontinuous SMC for MIMO Robotic Nonlinear System: Y1(0) = 1, Y2(0) = 1.5

Fig. 6.16: MIMO Discontinuous Nonlinear SMC for Example 6.5: Y1(0) = 1, Y2(0) = 1.5


In the above discontinuous functions, let sgn tanhs k ssa f b g→ and choose k ks s1 2 250= = to have

0 2 4 6 80

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 2 4 6 820

40

60

80

100

120

Time [s]

Con

trol

0 2 4 6 8-0.5

0

0.5

1

Time [s]

Slid

ing


0 0.5 1-0.6

-0.4

-0.2

0

0.2

Output

Cha

nge

of O

utpu

t

TanH-Continuous SMC for MIMO Robotic Nonlinear System: Y1(0) = 1, Y2(0) = 0

Fig. 6.17: MIMO Tanh-Continuous Nonlinear SMC for Example 6.5: Y1(0) = 1, Y2(0) = 0


0 2 4 6 80

0.5

1

1.5

Time [s]

Out

put

0 2 4 6 8-50

0

50

100

150

Time [s]

Con

trol

0 2 4 6 8-0.5

0

0.5

1

1.5

Time [s]

Slid

ing


0 0.5 1 1.5-0.5

-0.4

-0.3

-0.2

-0.1

0

Output

Cha

nge

of O

utpu

t

TanH-Continuous SMC for MIMO Robotic Nonlinear System: Y1(0) = 1, Y2(0) = 1.5

Fig. 6.18: MIMO Tanh-Continuous Nonlinear SMC for Example 6.5: Y1(0) = 1, Y2(0) = 1.5


Theorem 6.8 yields

MH

Hs

x s

x s=

+

+

LNMM

OQPP

= =1 2 1 1

2 4 2 21 2

1

12

a fa f

.

.,

δ

δδ δ

.

.

0 2 4 6 80

0.2

0.4

0.6

0.8

1

Time [s]

Out

put

0 2 4 6 820

40

60

80

100

120

Time [s]

Con

trol

0 2 4 6 80

0.2

0.4

0.6

0.8

1

Time [s]

Slid

ing


0 0.5 1-0.6

-0.4

-0.2

0

0.2

Output

Cha

nge

of O

utpu

t

Continuous SMC for MIMO Robotic Nonlinear System: Y1(0) = 1, Y2(0) = 0

Fig. 6.19: MIMO Continuous Nonlinear SMC for Example 6.5: Y1(0) = 1, Y2(0) = 0


0 2 4 6 80

0.5

1

1.5

Time [s]

Out

put

0 2 4 6 8-50

0

50

100

150

Time [s]

Con

trol

0 2 4 6 80

0.5

1

1.5

Time [s]

Slid

ing


0 0.5 1 1.5-0.8

-0.6

-0.4

-0.2

0

Output

Cha

nge

of O

utpu

t

Continuous SMC for MIMO Robotic Nonlinear System: Y1(0) = 1, Y2(0) = 1.5

Fig. 6.20: MIMO Continuous Nonlinear SMC for Example 6.5: Y1(0) = 1, Y2(0) = 1.5


• The decoupling is effective as the second output is almost unaffected. The second output is essentially

equal to zero if it is initially at zero and under no control. It goes to zero from an initial value if it is under

control as required.

• The performance of the continuous SMC is comparable to that of the continuous pseudo-SMC.

6.7.6. Example 6.6: MIMO Uncertain Nonlinear Robot Arms

Consider a robot system in Fu et al. 1990

&& . & ,

&& . . & & ,

. , .

y M M R M y M M u

y I M R M y y I u

R M M y M y a I M J J M y M y a

c c

c c

1

1

22

0 1

21

1 2 0 2

1 1 1 2 12

1

2

2

= + + +

= − +

= + + = + + + +

RS||

T||

−

−

c h a f c ha f a fc h

a f c h a f c h

Let

x y

x y

x y

x y

11 1

12 1

21 2

22 2

=

=

RS|T|=

=

RS|T|&,

&

then with 0 100kg M kg≤ ≤

&

&. ,

&

&.

x x

x x xM x u

M

x x

x

x M x x x

x x u

x M x

11 12

12 11 222 22

21

21 22

22

11 11 11 12

112

11

2

2

112

11

2

200

100

2 100 1

200 100 100 1

200 100 1

=

= ++

+

RS|T|

=

=

− + +

+ + + +

RS|T|

UV|W|

+ + +

R

S|||

T|||

c hc hc h

L


• For the subsystem 1:

&

&. .

x x

x x xM x u

M

11 12

12 11 222 22

21200

100

=

= ++

+

RS|T|

so

& & &. .

s h x x h x x xM x u

M Mu ueq1 1 11 12 1 12 11 22

2 222

11 1

200

100

200

100= + = + +

+

+=

+−

thus

− =+

+ + = + +uM

h x x xM

x x h x xeq1 1 12 11 222

222

11 1 12 222100

200 200

2

1

0 5

0d i c h .

then

λ H

e p

x x x1 1 1 11 12 222

1 1 1 1

2 2 1

1 5 1 5 1 0 25 0 5 0 5 1 0 25

= − ⇒ = =

= × = ×

RS|T|

H w

K H K H

, , ,

. , . . , . , . .

,

a f a f

• For the subsystem 2:

&

&.

x x

xx M x x x x x u

x M x

21 22

22

11 11 11 12 112

11

2

2

112

11

2

2 100 1 200 100 100 1

200 100 1

=

=− + + + + + +

+ + +

RS||

T||

c h c hc h

so

& & &.

s h x x h xx M x x x x x u

x M x2 2 21 22 2 22

11 11 11 12 112

11

2

2

112

11

2

2 100 1 200 100 100 1

200 100 1= + = +

− + + + + + +

+ + +

c h c hc h

thus

&sx x

x M xu ueq2

112

11

2

112

11

2 2 2

200 100 100 1

200 100 1=

+ + +

+ + +−

c hc h d i

hence

− =+ + + +

+ + +−

+ +

+ + +u

M Mx M x

x xh x

M x M x

x xxeq 2

11 112

112

11

2 2 2211 11

112

11

2 12

200 2 100

200 100 100 1

2 100

200 100 100 1

a f a fc h

a fc h

since

200 100 1

200 100 100 1

1

0 4112

11

2

112

11

2

+ + +

+ + +=

x M x

x x

c hc h .

or

200 100 100 1 2501

200 100 100 1

1 250

0112

11

2

112

11

2+ + + > ⇒+ + +

=x xx x

c h c h

so

− = +−

+−

= +−

+−

uh

x x x x xh

hx x x x xeq 2

2

22 11 12 12 112 2

222 11 12 12 11

20 8

0

0

0 8

0

1 6

0 4

0 4

0 4

0 4

0 8

0 8

.

. .

.

.

.

.

.

.

then

λ H

e p

x x x x x2 2 2 22 11 12 12 112

2 2 2 2

2 2 1

0 4 1 0 0 0 4 1 0 4 0 8

= − ⇒ = =

= × = ×

RS|T|

H w

K H K H

, , ,

. , , . . , .

,

a f a f


6.7.6.1. Robust Discontinuous SMC

Theorem 6.7 yields

u se rp

e

rp r p p

1 1 1 1 1 1

1

1 1 1 1 1 1

1 5 3 0 25

0 5 1 1 5 0 75

= − −

=

= ⇒ = + = + =

RS||

T||

K w K w

K

K K K K

. .

. , , .

. , . , .

sgn c h

δ δ

and

u se rp

e

rp r p p

2 2 2 2 2 2

2

2 2 2 2 2 2

0 8 0 0

1 1 8 1 4 1 8

= − −

=

= ⇒ = + = + =

RS||T||

K w K w

K

K K K K

. .

. , ,

. , . , .

sgn b g

δ δ

0 1 2 3 40

0.5

1

1.5

Time [s]

Out

put

0 1 2 3 4-6

-4

-2

0

2

4

Time [s]

Con

trol

0 1 2 3 4-2

-1.5

-1

-0.5

0

0.5

Time [s]

Slid

ing

Lower Bound 1

Lower Bound 2 Upper Bound 2

Upper Bound 1

-0.5 0 0.5 1-0.5

0

0.5

1

1.5

Output

Cha

nge

of O

utpu

t

Robust Discontinuous SMC for MIMO Uncertain Nonlinear Robotic System: Ref_1 = 1, Ref_2 = 0

Fig. 6.21: Robust Discontinuous SMC for MIMO Uncertain Nonlinear System in Example 6.6:

Ref_1 = 1.0, Ref_2 = 0

0 1 2 3 40

0.5

1

1.5

Time [s]

Out

put

0 1 2 3 4-10

-5

0

5

Time [s]

Con

trol

0 1 2 3 4-3

-2

-1

0

1

Time [s]

Slid

ing

Lower Bound 1


Upper Bound 1

0 0.5 1 1.50

0.5

1

1.5

2

2.5

Output

Cha

nge

of O

utpu

t

Robust Discontinuous SMC for MIMO Uncertain Nonlinear Robotic System: Ref_1 = 1, Ref_2 = 1.5

Fig. 6.22: Robust Discontinuous SMC for MIMO Uncertain Nonlinear System in Example 6.6:

Ref_1 = 1, Ref_2 = 1.5


6.7.6.2. Robust Continuous Pseudo-SMC

In the above discontinuous functions, let sgn tanhs k ssa f b g→ and choose k ks s1 24 5= =, to have

0 1 2 3 40

0.5

1

1.5

Time [s]

Out

put

0 1 2 3 4-2

-1

0

1

2

3

Time [s]

Con

trol

0 1 2 3 4-2

-1.5

-1

-0.5

0

Time [s]

Slid

ing

Lower Bound 1 Upper Bound 1 Lower Bound 2 Upper Bound 2

-0.5 0 0.5 1-0.5

0

0.5

1

1.5

Output

Cha

nge

of O

utpu

t

Robust TanH-Continuous SMC for MIMO Uncertain Nonlinear Robotic System: Ref_1 = 1, Ref_2 = 0

Fig. 6.23: Robust Tanh-Continuous SMC for MIMO Uncertain Nonlinear System in Example 6.6:

Ref_1 = 1.0, Ref_2 = 0: Steady-State Error

0 1 2 3 40

0.5

1

1.5

Time [s]

Out

put

0 1 2 3 4-6

-4

-2

0

2

4

Time [s]

Con

trol

0 1 2 3 4-3

-2

-1

0

1

Time [s]

Slid

ing


0 0.5 1 1.5 2-1

0

1

2

3

Output

Cha

nge

of O

utpu

t

Robust TanH-Continuous SMC for MIMO Uncertain Nonlinear Robotic System: Ref_1 = 1, Ref_2 = 1.5

Fig. 6.24: Robust Tanh-Continuous SMC for MIMO Uncertain Nonlinear System in Example 6.6;

Ref_1 = 1, Ref_2 = 1.5: Steady-State Error


The design scheme in Proposition 3.3 with Err_Tol = 0.05 eliminates the steady-state error as follows

0 1 2 3 40

0.5

1

1.5

Time [s]

Out

put

0 1 2 3 4-2

-1

0

1

2

3

Time [s]

Con

trol

0 1 2 3 4-2

-1.5

-1

-0.5

0

0.5

Time [s]

Slid

ing


Upper Bound 2 Lower Bound 2

-0.5 0 0.5 1-0.5

0

0.5

1

1.5

Output

Cha

nge

of O

utpu

t

Robust TanH-Continuous SMC for MIMO Uncertain Nonlinear Robotic System: Ref_1 = 1, Ref_2 = 0


Ref_1 = 1.0, Ref_2 = 0: using Proposition 3.3 with Err_Tol = 0.05 to eliminate Steady-State Error

0 1 2 3 40

0.5

1

1.5

Time [s]

Out

put

0 1 2 3 4-6

-4

-2

0

2

4

Time [s]

Con

trol

0 1 2 3 4-3

-2

-1

0

1

Time [s]

Slid

ing


0 0.5 1 1.5 2-1

0

1

2

3

Output

Cha

nge

of O

utpu

t

Robust TanH-Continuous SMC for MIMO Uncertain Nonlinear Robotic System: Ref_1 = 1, Ref_2 = 1.5


Ref_1 = 1, Ref_2 = 1.5: using Proposition 3.3 with Err_Tol = 0.05 to eliminate Steady-State Error


Remark 6.9: Steady-State Error due to Low Sliding Gain and Efficiency of Design Scheme (Proposition 3.3)

• As expected in Section 3.5.5, low sliding gains cause steady-state errors using a sliding function to replace

the switching function in eliminating the chattering;

• The design scheme in Proposition 3.3 is efficient in eliminating these steady-state errors


The decoupling is effective as the second output is almost unaffected. The second output is essentially

equal to zero if it is initially at zero and under no control. It goes to the desired position if it is under control as

required.

6.8. CONCLUSIONS

For a nonlinear system without perturbation, I can get a continuous SMC. But for an uncertain dynamical

nonlinear system, I can get a continuous pseudo-SMC only. Note that the continuous pseudo-SMC is also

applicable to a nonlinear system without any perturbation, so I have got continuous pseudo-SMC controllers

for all numerical examples above to compare with available continuous SMC controllers, and I have found out

that the difference is negligible.

For the general case, if the system Jacobian is non-singular, I can use the decoupling technique to deal

with a MIMO system in the unified manner as with a SISO system. Otherwise, I have to use the complicated

Structure-Algorithm (Silverman 1969) and the error dynamic equation (Chen et al. 1992). I have dealt with the

system in Chen et al. 1992 by using the simple decoupling technique where any perturbation can be taken

into account. Note that this system has been considered by using the structure-algorithm in the above work

where it is not clear how to take perturbations into account.

All the numerical examples above have the relative degree equal to the system order, so they are in the

canonical form, thus the matching condition is satisfied, so the system stability is guaranteed by the stable

hyperplane. If the relative degree is less than the system order, a stability test can be made for a linear system,

but not for a nonlinear system. If this is the case, that nonlinear system should be linearized as usual with the

nonlinearity is considered as uncertainties and a stability test can be done.

A SMC design involves 2 design stages: hyperplane design and controller design. The hyperplane

design is independent of the controller design because it is regardless of the implementation of the controller

in terms of its form (continuous SMC or discontinuous SMC or continuous pseudo-SMC). Based on the

nominal model, a hyperplane is designed in such a way that it is stable, thus it is chosen to be linear with its

Hurwitz eigenvalues, hence a stable hyperplane even for a nonlinear system. Based on the actual model

where any potential perturbation is taken into account, a control function is designed in such a way that it

satisfies the sliding condition to make the system states slide on the hyperplane even under perturbations.

Only then the system is stable if the sliding mode is stable.


For a linear or nonlinear canonical system, the hyperplane is independent of the system parameters, so a

stable hyperplane does guarantee the system stability. For a linear system, if the matching condition is

satisfied, then a stable hyperplane also guarantees the system stability; otherwise it is necessary to check if

the sliding-eigenvalues are Hurwitz. For a nonlinear system, it should be linearized as usual with the

nonlinearity is considered as uncertainties in order to calculate the sliding-eigenvalues, then it is necessary to

check if the sliding-eigenvalues of the linearized system are Hurwitz in the operating range. Note that the

invariance conditions in Section 4.2 are valid for linear systems, not for nonlinear systems, because they are

based on linear systems. It is crucial to note that the invariance property is valid for both canonical linear

and nonlinear systems, and for non-canonical linear systems satisfying the matching condition. This property

means that the sliding dynamic is invariant to perturbations. For a linear system, the sliding-eigenvalues are of

the actual model, while the hyperplane-eigenvalues are the desired sliding-eigenvalues of the nominal model.

In other words, every actual model has its own sliding plane and if the invariance condition is satisfied then

all these sliding planes coincide with the hyperplane, hence the invariance property. For canonical linear or

nonlinear systems, there is only one hyperplane possible, so the invariance property must be implied.


Chapter 6

General Sliding-Mode Controller Design for MIMO Uncertain Nonlinear Systems

6.1. Introduction....................................................................................................................................................................1

6.2. Hyperplane and Stability ..............................................................................................................................................1

6.2.1. Hyperplane Design...........................................................................................................................................2

Theorem 6.1: Hyperplane Design for Canonical Nonlinear Systems................................................2

Remark 6.1: Hyperplane Design for Canonical Linear and Nonlinear Systems ...............................3

6.2.2. SMC Stability Criterion for Nonlinear Systems ............................................................................................3

Corollary 6.1: Stability Criterion .............................................................................................................3

6.3. Robust Discontinuous SMC Design for SISO Uncertain Nonlinear Systems .....................................................3

6.3.1. Discontinuous SMC Design for SISO Nonlinear Systems .........................................................................3

Definition 6.1: Nonlinear System States Vector...................................................................................3

Theorem 6.2: Discontinuous SMC Design for Nonlinear Systems ...................................................4

Remark 6.2: Reaching Control.................................................................................................................4

6.3.2. Robust Discontinuous SMC Design for SISO Uncertain Nonlinear Systems .........................................4

Assumption 6.1: System Constraint on Parametric Variation............................................................4

Theorem 6.3: Robust Discontinuous SMC Design for Uncertain Nonlinear Systems ..................5

Remark 6.3: Nonlinearity as Uncertainty...............................................................................................5

Remark 6.4: Alleviation of Chattering....................................................................................................6

Remark 6.5: Performance of Pseudo-SMC............................................................................................6

6.4. Robust Continuous SMC Design for SISO Uncertain Nonlinear Systems...........................................................6

6.4.1. Continuous SMC Design for SISO Nonlinear Systems ...............................................................................6

Theorem 6.4: Continuous SMC Design for Nonlinear Systems ........................................................6

6.4.2. Robust Continuous Pseudo-SMC Design for SISO Uncertain Nonlinear Systems................................7

Corollary 6.2: Robust Continuous Pseudo-SMC Design for Uncertain Nonlinear Systems ........7

6.5. General SMC Design for SISO Nonlinear Systems ...................................................................................................7

Theorem 6.5: General Discontinuous SMC Design for Nonlinear Systems ....................................7

Theorem 6.6: General Continuous SMC Design for Nonlinear Systems ..........................................9

6.6. General SMC Design for MIMO Uncertain Nonlinear Systems ...........................................................................10

Theorem 6.7: General Discontinuous SMC Design for MIMO Nonlinear Systems .....................10

Theorem 6.8: General Continuous SMC Design for MIMO Nonlinear Systems ..........................11

Remark 6.6: Singular System Jacobian ................................................................................................11

Remark 6.7: A Robust SMC Design for General MIMO Uncertain Nonlinear Systems ..............11

6.7. Numerical Examples.....................................................................................................................................................12

Remark 6.8: Summary of SMC for Uncertain MIMO Systems .........................................................12

6.7.1. Example 6.1: SISO Nonlinear System............................................................................................................12

6.7.1.1. Discontinuous SMC...........................................................................................................................13


6.7.1.2. Continuous Pseudo-SMC .................................................................................................................13

6.7.1.3. Continuous SMC................................................................................................................................14

6.7.1.4. Performance.........................................................................................................................................14

6.7.2. Example 6.2: SISO Uncertain Nonlinear System..........................................................................................14

6.7.2.1. Robust Discontinuous SMC.............................................................................................................15

6.7.2.2. Robust Continuous Pseudo-SMC...................................................................................................16

6.7.3. Example 6.3: Tracking SISO Nonlinear System ...........................................................................................16



6.7.3.3. Continuous SMC................................................................................................................................18

6.7.3.4. Performance.........................................................................................................................................18

6.7.4. Example 6.4: MIMO Nonlinear System.........................................................................................................18



6.7.4.3. Continuous SMC................................................................................................................................21

6.7.4.4. Performance.........................................................................................................................................22

6.7.5. Example 6.5: MIMO Nonlinear Robot Manipulators ..................................................................................22



6.7.5.3. Continuous SMC................................................................................................................................25

6.7.5.4. Performance.........................................................................................................................................26

6.7.6. Example 6.6: MIMO Uncertain Nonlinear Robot Arms..............................................................................26

6.7.6.1. Robust Discontinuous SMC.............................................................................................................28

6.7.6.2. Robust Continuous Pseudo-SMC...................................................................................................29

Remark 6.8: Steady-State Error due to Low Sliding Gain and Efficiency of Design Scheme

(Proposition 3.3) .....................................................................................................................................31

6.7.6.3. Performance.........................................................................................................................................31

6.8. Conclusions..................................................................................................................................................................31

Chapter 7

Advanced Sliding-Mode Controller Design:

Experimental Results

7.1. INTRODUCTION

In this chapter, results from experiments conducted with a Ball-Hoop system are presented to validate my

thoeretical predictions such as infinite gain of the switching function may cause excitation of unmodelled

high-frequency dynamics; performances of the saturate, unitvector and hyperbolic tangent functions;

performances of continuous SMC, and continuous pseudo-SMC (Tanh SMC).

One of the consequences of the trend towards the high speed transportation of bulk liquids is that the

influence of the cargo upon the vehicle must be fully accounted for. This has always been true in the static

sense, but in recent years the dynamical interaction between the material being transported and its container

has grown in importance. In fact, the behaviour of a vehicle during a manoeuvre is very often a joint function

of the liquid cargo and vehicle dynamics. This problem, referred to as " liquid slop", is known to be a problem

in:

(i) High speed road and rail transport of bulk liquids

(ii) Maritime transportation, especially in oil tankers

(iii) Liquid fuelled missiles

Actually, "liquid slop" is not confined to vehicular transport, any arrangement which involves the rapid

movement of large quantities of fluid can exhibit the characteristic oscillations which are associated with

" liquid slop". The pumping of concrete in the civil engineering and building industry is a fair example.

An essential aim of the Ball-Hoop system is to illustrate the dynamical behaviour and control problems

associated with liquid slop. In particular, if a cylindrical vessel is considered, then to a first approximation, the

essential dynamical character of liquid movement in the cylinder (Fig. 7.1.a) is captured by the motion of a

body rolling inside a hoop (Fig. 7.1.b).


Liquid Ball

a) Cross-section through a Cylindrical Vessel b) Ball rolling inside a Hoop

Cylindrical Vessel Hoop

Fig. 7.1: Ball-Hoop motion

7.2. SYSTEM DESCRIPTION

In the system the vehicle motion is introduced by allowing the hoop to rotate under the action of a direct

drive servo-motor, while the liquid motion is modelled by the oscillations of a ball rolling in the inner periphery

of the hoop.

The Ball-Hoop system consists of a D.C. servo-motor which is equipped with an integral tachometer and a

hoop angular position transducer mounted in a vertical frame. A large annular hoop is directly fixed to the

motor shaft and constitutes an inertial load on the motor. A metal ball is placed inside the hoop so that it rolls

in the groove on the inner periphery of the hoop. The ball angular position is measured using a potentiometer

as the ball position transducer mounted on the front of, and coaxial with, the hoop. A wiper assembly is

connected to the potentiometer shaft and registers on the ball via a yoke. This provides a measurement of the

angular deviation of the ball from vertical, which is referred to as the "slop angle".

Motor Hoop Ball

Tacho-Generator

Hoop AngleTransducer

Slop AngleTransducer

Vin

VH&VH VB

BH&H

Fig. 7.2: Block diagram of the Ball-Hoop system

where

B : Angular deviation of the ball from the vertical (slop angle)

H : Angular position of the hoop

&H : Angular velocity of the hoop

7. Advanced Sliding-Mode Controller Design: Experimental Results 7.3

7.3. INSTRUMENTATION

The instrumentation box has

1 input : Servo-motor drive voltage

3 outputs: (i) Hoop velocity signal, derived from tachometer

(ii) Hoop angular position signal, derived from the rotary position transducer

attached to the rear motor shaft

(iii) Ball angular position signal, referred to as the "slop angle", derived from the

rotary potentiometer mounted in front of the hoop.

Slop AngleTransducer Motor Tacho-

GeneratorHoop AngleTransducer

Motor DriveInput

MotorInput Preset

Slop AnglePreset Slop Angle

Output

Hoop VelocityOutput

Hoop AngleOutput

HoopAnglePreset

Hoop

Ball

Fig. 7.3: Instrumentation box

The instrumentation associated with the Ball-Hoop system is typical of that found in a D.C. servo position

control system. Namely,

(i) A D.C. servo-motor

(ii) Tacho-generator

(iii) Angular position transducer

In addition, the system is equipped with a rotary potentiometer for measuring the angular deflection of

the ball.


7.4. SYSTEM IDENTIFICATION

Theoretical and practical system identifications will be presented in this section. In the theoretical

method, based on only one typical actual response from some experiments, a model verification will be used to

identify all unknown model parameters to obtain a system model.

7.4.1. Theoretical System Identification

R

r

rb

φB

H

Fig. 7.4: Description of Ball-Hoop Dynamics

where

Ia: Moment of Inertia of the Hoop

Ib: Moment of Inertia of the Ball

Mb: Mass of the Ball

Bb: Coefficient of Rolling Friction of the Ball

Bm: Coefficient of Rotational Friction of the Motor Assembly

Th: Torque

R: Radius of the Hoop

r: Radius of the Ball

B: Ball angle against the vertical axis

H: Hoop angle against vertical axis


I will use Lagrange approach to derive theoretically a mathematical model of the system using the

following extended Lagrangian equations

ddt

Lq

Lq

Jq

F L T Ui i i

i∂∂

∂∂

∂∂& &

,

− + = = − (7.1)

where

T : total co-kinetic energy

U : total potential energy

J : total co-content energy qi : generalized coordinate

Equation for the transformation of coordinates:

( )φ = −Rr

H B (7.2)

Translational velocity of the ball

v R r B= −( ) & (7.3)

The hoop rotates only, but the ball rotates and translates as well. The total co-kinetic energy of the

system

T I H I M va b b= + +12

12

12

2 2 2& &φ (7.4)

Eqs.(7.2) and (7.3) give

( )T I IRr

H IRr

HB IRr

M R r Ba b b b b= +

−

+

+ −

12

12

22

2 22 2& & & & (7.5)

The total potential energy of the system: ( ) ( )U M g R r Bb= − −. . cos1 (7.6)

For the ball and hoop, although the ball translates, assuming that there is no translational friction for the

ball. The total co-content energy of the system

J B H Bm b= +12

12

2 2& &φ (7.7)

Eq.(7.2) yields

J B BRr

H BRr

HB BRr

Bm b b b= +

−

+

12

12

22

2 22& & & & (7.8)

Hence the system Lagrangian equations are

ddt

Lq

Jq

Lq

Fn n n

n∂∂

∂∂

∂∂& &

+ − = (7.9)

where

( ) ( ) ( )L T U I IRr

H IRr

HB IRr

M R r B M g R r Ba b b b b b= − = +

−

+

+ −

− − −

12

12

12

22 2

2 2& & & & . . cos

(7.10)

then

∂∂

∂∂

LH

I IRr

H IRr

Bddt

LH

I IRr

H IRr

Ba b b a b b&& &

&&& &&= +

−

⇒

= +

−

2 2 2 2


and

( ) ( )∂∂

∂∂

LB

IRr

H IRr

M R r Bddt

LB

IRr

H IRr

M R r Bb b b b b b&& &

&&& &&= −

+

+ −

⇒

= −

+

+ −

2 22

2 22

and

∂∂

LH

= 0

( )∂∂

LB

M g R r Bb= − −. sin

Eq.(7.8) gives

∂∂

JH

B BRr

H BRr

Bm b b&& &= +

−

2 2

∂∂

JB

BRr

H BRr

Bb b&& &= −

+

2 2

Thus

(1) q H1 =

( )I IRr

H B BRr

HRr

I B B B Ta b m b b b+

+ +

−

+ =

2 2 2&& & && & (7.11)

(2) q B2 =

( ) ( ) ( )Rr

I H B H IRr

M R r B BRr

B M g R r Bb b b b b b

+ −

+ −

−

− − =2 2

22

0&& & && & sin (7.12)

7.4.1.1. Ball-Hoop Transfer Function

Assume that

sin

:

B Br r

I M r

b

b b b

≈≈

≈

2

52 ball is a solid sphere

then from Eq.(7.12)

&& & && &HBI

H BBI

BR

gBb

b

b

b

+

− − − =

72

52

0

or

B sH s

s s BM r

sB

M rs

gr

b

b

b

b

( )( )

=

+

+

+

27

52

57

57

2

22

(7.13)

where g = 9.81 m/s2


By a simple measurement of Hoop radius: R = 8.75 cm

In fact, r ≈ 0.70 cm, so R >> r as the above assumption

rb ≈ 0.75 cm, so rb ≈ r as the above assumption

From Eq.(7.13):

B sH s

s sA

sA

s

( )( )

=+

+ +

27

5

7802

(7.14)

where :

AB

M rb

b

=5

2 (7.15)

Hence ωn rad s= =80 8 944. /

Let A n7 2 17 888 18= = ≈δω δ δ. (7.16)

where δ is a damping factor

So ( )( )A A5 7 5 7 25= ≈ δ (7.17)

and Eqs.(7.14) & (7.15) give

( )δ = =5

126 2

BM r

f Bb

bb (7.18)

Eq.(7.13) yields

( )B s

H ss s

s s( )( )

.=+

+ +0 286

2518 802

δδ

(7.19)

Therefore, with only 1 simple measurement, I have got the transfer function with only 1 unknown δ :

Damping Factor.

7.4.1.2. DC Servo Motor Transfer Function

A servo motor has a transfer function as

H sV s

K

s sin

m

m

( )( )

=+

1τ

(7.20)


7.4.2. Model Verification

I first consider each motor and ball subsystems separately, then combine them to see the effect of

interactions.

7.4.2.1. Separated System

Using a step of 2, I obtain the following model for the motor

( )G ss sm =

+×

1 051 5

85..

(7.21)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35-5

0

5

10

Time [s]

Hoo

p

Actual Model

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.5

1

1.5

2

Time [s]

Hoo

p_D

ot

Model of Hoop

Fig. 7.5: Modelling Motor Subsystem

and for ball-hoop subsystem

( ) ( )G ss

s sb* .

.=

++ +0 3 1

0 72 812 (7.22)

it is very close to Eq.(7.18) where δ = 0 04. .

7.4.2.2. Combined System

The ball-hoop subsystem is driven by the motor. Due to interaction between the ball and the hoop, the

ball-hoop subsystem is modified to be

( )G ss sm =

+×

0 9871 4

76..

(7.23)

and

( )G ss

s sb* .

.=

−+ +

2 550 6 100

2

2 (7.24)

Note the different numerators in Eqs. (7.22) & (7.24).


Actual Model

Actual Model

0 0.5 1 1.5 2 2.5 3-10

-5

0

5

10

Time [s]

With

out I

nter

actio

n

0 0.5 1 1.5 2 2.5 3-10

-5

0

5

Time [s]

Und

er In

tera

ctio

n

Model of Ball Subsystem

Fig. 7.6: Modelling Individual Ball-Hoop Subsystem

I thus have the model of the ball-hoop system as

0 987

1 4

.

.s +

76

s

x VH1 = x VB4 = &

x VB2 =

x VT3 =

u −

+ +

2 55

0 6 100

2

2

.

.

s

s s

Fig. 7.7: Model of Ball-Hoop System


7.5. DIGITAL SLIDING-MODE CONTROLLER-OBSERVER DESIGNS

All sliding-mode controllers (SMC) described in previous chapters will be designed and implemented:

switching discontinuous SMC, saturate and Tanh pseudo-continuous SMC, linear continuous SMC, and

sliding-mode fuzzy controller. The Ball-Hoop system is an 4-th order system, however there are only 3 states

available: hoop position , hoop velocity and ball position. An observer is thus required to estimate the

missing ball velocity and this observer will be used for all types of SMC. A PC and an ADC card will be used

to implement the controller-observer, so discrete-time controller-observers will be designed.

7.5.1. Robust Sliding-Mode Observer Design

For the model whose system states defined in Fig.7.7, a state-space model is

( ) ( )&~ ~

x A A x B B= + + +∆ ∆ u

where

A A B B=−

− −

=

−

=

−

=−

0 0 77 5 00 0 0 10 0 1375 00 121 265 625 0 66

0 0 2 5 00 0 0 00 0 0 125 00 11 15 625 0 03

0

01 0050192 2500

0

00 39570 25

.

..

.

.. .

..

...

, , ,∆ ∆ .

Choose [ ]λ H = − − −8 8 8, , then

[ ]H = −0 0692 0 1176 2 0204 0 0040. . . .

thus [ ]Ke = − −0 0 4642 38329 01202. . .

choose T ss = 0 005. , since it is 25 faster than the desired dynamics ( )λ H , then

A Bd d=

−

=−

−

1 0 0 3862 00 0 9985 0 0033 0 0050

0 0 0 9931 0

0 0 6037 13207 0 9952

0 00100 0024

0 0050

0 9557

.. . .

.

. . .

..

.

.

,

and

∆ ∆A Bd d=−

−

=

−

−

0 0 0 0126 00 0 0001 0 0002 0

0 0 0 0006 0

0 0 0548 0 0771 0 0003

0 00040 0009

0 0020

0 3489

.. .

.

. . .

..

.

.

,

and choose the dynamics of a robust discrete-time sliding-mode observer 3 times faster than those of sliding-mode controller, thus [ ] [ ]λ λHO HO= − − − ⇒ =24 24 24 0 8869 0 8869 08869* . . . , then

L A A LC=

⇒ = − =− −−− −

0 3989 00 0165 13607

0 3282 0

1 2531 4 7037

0 6011 0 0 3862 00 0165 0 3622 0 0033 0 0050

0 3282 0 0 9931 0

12531 5 3074 1 3207 0 9952

.

. .

.

. .

. .. . . .

. .

. . . .

~

where

C =

1 0 0 00 1 0 0

and the observer equation is

( ) ( ) ( ) ( )~ ~.~ . .x A x B Lk k u k y kd+ = + +1

This observer equation will be used to estimate the ball velocity for all the following SMC's.


7.5.2. Robust Sliding-Mode Controller Designs

All controller types will use the same hyperplane above.

7.5.2.1. Robust Switching SMC

I have [ ]K p = 0 0 0596 0 6580 0 0002. . .

choose δ = 6 , then

[ ]K r = 0 4151 0 7057 12 1223 0 0241. . . .

then the simulation results are

0 1 2 30

1

2

3

4

5

Time [s]

Ho

op

0 1 2 3-10

-5

0

5

Time [s]

Con

trol

0 1 2 3-0.3

-0.2

-0.1

0

0.1

Time [s]

Slid

ing


0 1 2 3-1.5

-1

-0.5

0

0.5

Time [s]

Bal

l

Robust Switching Sliding-Mode Controller-Observer for Ball-Hoop System

Fig. 7.8: Robust Switching Sliding-Mode Controller-Observer Design for Ball-Hoop System


and the experimental results are

0 1 2 30

2

4

6

Time [s]

Hoo

p

0 0.5 1 1.5 2 2.5-5

0

5

10

Time [s]

Con

trol

0 0.5 1 1.5 2 2.5-0.4

-0.2

0

0.2

0.4

Time [s]

Slid

ing

0 0.5 1 1.5 2 2.5-4

-2

0

2

4

Time [s]

Ba

ll

Experimental Results of Robust Switching Sliding-Mode Controller-Observer for Ball-Hoop System

Fig. 7.9: Experimental Results of Robust Switching Sliding-Mode Controller-Observer for Ball-Hoop

System

Remark 7.1: Excitation of Unmodelled High-Frequency due to Switching Function


Due to infinite gain of switching function, the unmodelled high-frequency dynamics are excited to cause

oscillations. These high-frequency dynamics may come from the interactions between the ball and hoop in a

ball-hoop system.


7.5.2.2. Robust Saturate SMC

Controller parameters are the same as the switching SMC above where ( ) ( )sign sat , ,s s k ks s→ = 30


0 1 2 30

1

2

3

4

5

Time [s]

Hoo

p

0 1 2 3-3

-2

-1

0

1

2

Time [s]

Con

trol

0 1 2 3-0.3

-0.2

-0.1

0

0.1

Time [s]

Slid

ing


0 1 2 3-1.5

-1

-0.5

0

0.5

Time [s]

Bal

l

Robust Saturate Sliding-Mode Controller-Observer for Ball-Hoop System

Fig. 7.10: Robust Saturate Sliding-Mode Controller-Observer Design for Ball-Hoop System



0 1 2 30

2

4

6

Time [s]

Hoo

p

0 0.5 1 1.5 2 2.5-2

-1

0

1

2

3

Time [s]

Con

trol

0 0.5 1 1.5 2 2.5-0.8

-0.6

-0.4

-0.2

0

0.2

Time [s]

Slid

ing

0 0.5 1 1.5 2 2.5-0.8

-0.6

-0.4

-0.2

0

0.2

Time [s]

Bal

l

Experimental Results of Robust Saturate Sliding-Mode Controller-Observer for Ball-Hoop System

Fig. 7.11: Experimental Results of Robust Saturate Sliding-Mode Controller-Observer for Ball-Hoop

System

Remark 7.2: Slow-Down System Response due to Low Sliding Gain

In experiments, the sliding gain had to be reduced to reduce to 0.3 to alleviate chattering and the system

response is slowed down due to this lower gain as expected in theory (Section 3.5.5)


7.5.2.3. Robust Unitvector SMC

Controller parameters are the same as the switching SMC above where ( ) ( )sign smt , ,s k s ks s→ = 120


0 1 2 30

1

2

3

4

5

Time [s]

Hoo

p

0 1 2 3-4

-2

0

2

Time [s]

Con

trol

0 1 2 3-0.3

-0.2

-0.1

0

0.1

Time [s]

Slid

ing


0 1 2 3-1.5

-1

-0.5

0

0.5

Time [s]

Bal

l

Robust Unitvector Sliding-Mode Controller-Observer for Ball-Hoop System

Fig. 7.12: Robust Unitvector Sl iding-Mode Controller-Observer Design for Ball -Hoop System



0 1 2 30

2

4

6

Time [s]

Hoo

p

0 0.5 1 1.5 2 2.5-1

0

1

2

Time [s]

Con

trol

0 0.5 1 1.5 2 2.5-0.8

-0.6

-0.4

-0.2

0

0.2

Time [s]

Slid

ing

0 0.5 1 1.5 2 2.5-1.5

-1

-0.5

0

0.5

Time [s]

Bal

l

Experimental Results of Robust Unitvector Sliding-Mode Controller-Observer for Ball-Hoop System

Fig. 7.13: Experimental Results of Robust Unitvector Sliding-Mode Controller-Observer for Ball-Hoop

System


7.5.2.4. Robust Tanh SMC

Controller parameters are the same as the switching SMC above where ( ) ( )sign tanh ,s k s ks s→ = 50


0 1 2 30

1

2

3

4

5

Time [s]

Ho

op

0 1 2 3-4

-2

0

2

Time [s]

Con

trol

0 1 2 3-0.3

-0.2

-0.1

0

0.1

Time [s]

Slid

ing


0 1 2 3-1.5

-1

-0.5

0

0.5

Time [s]

Bal

l

Robust TanH Sliding-Mode Controller-Observer for Ball-Hoop System

Fig. 7.14: Robust Tanh Sliding-Mode Controller-Observer Design for Ball-Hoop System



0 1 2 30

1

2

3

4

5

Time [s]

Hoo

p

0 1 2 3-1

0

1

2

3

Time [s]

Con

trol

0 1 2 3-0.6

-0.4

-0.2

0

0.2

Time [s]

Slid

ing

0 1 2 3-1.5

-1

-0.5

0

0.5

Time [s]

Ba

ll

Experimental Results of Robust TanH Sliding-Mode Controller-Observer for Ball-Hoop System

Fig. 7.15: Experimental Results of Robust Tanh Sliding-Mode Controller-Observer for Ball-Hoop System

Remark 7.3: Performance of Saturate and Tanh SMC

As mentioned in Section 3.5.5, the sliding gain of a Tanh SMC can be larger than that of a Saturate SMC.

In this experiment, this gain has been chosen 50 for Tanh SMC and 30 for Saturate SMC, thus the system

response of Tanh SMC is faster than Saturate SMC.


7.5.2.4. Robust Linear SMC

I have [ ]K p = −14101 31323 47 4146 0 0932. . . .

choose δ = 14 , then

[ ]K r = −0 9686 16466 282853 0 0563. . . .

thus [ ]K = −2 3787 52431 79 5329 0 0293. . . .


0 1 2 30

1

2

3

4

5

Time [s]

Hoo

p

0 1 2 3-5

0

5

10

Time [s]

Con

trol

0 1 2 3-0.3

-0.2

-0.1

0

0.1

Time [s]

Slid

ing


0 1 2 3-1.5

-1

-0.5

0

0.5

Time [s]

Bal

l

Robust Linear Sliding-Mode Controller-Observer for Ball-Hoop System

Fig. 7.16: Robust Linear Sliding-Mode Controller-Observer Design for Ball-Hoop System



0 1 2 30

1

2

3

4

5

Time [s]

Hoo

p

0 1 2 3-5

0

5

10

Time [s]

Con

trol

0 1 2 3-0.8

-0.6

-0.4

-0.2

0

0.2

Time [s]

Slid

ing

0 1 2 3-3

-2

-1

0

1

Time [s]

Bal

l

Experimental Results of Robust Linear Sliding-Mode Controller-Observer for Ball-Hoop System

Fig. 7.17: Experimental Results of Robust Linear Sliding-Mode Controller-Observer for Ball-Hoop System

Remark 7.4: Experimental Performance of Pseudo-SMC (Unitvector & Tanh SMC) and Linear SMC

The system responses are comparable, however the control effort of Pseudo-SMC is smaller.


7.6. CONCLUSIONS

By this experiment I can see the real danger of the switching SMC where it has excited the unmodelled

high-frequency dynamics. These high-frequency dynamics may come from the interactions between the ball

and hoop in a ball-hoop system. Note that there are some oscillations in the switching SMC controller while it

is not the case for other SMC controllers. The limitation of Saturate SMC is its low gain to slow down the

response due to the longer reaching mode, as expected in theory. I have attempted with both the saturate and

Tanh SMC to reduce the gain from infinite gain of the switching SMC. As expected, the performance of the

Tanh SMC is better than that of the saturate SMC because for the Tanh SMC, the higher sliding margin is, the

lower gain is; so the oscillation is harder to exist and there is no steady-state error. In other words, the

saturate SMC has the upper bound for the sliding margin low enough to slow down the response even with

the same fast eigenvalues as in the Tanh and linear SMC.

All the experimental results are consistent with the proposed theoretical results, such as the excitation of

unmodelled high-frequency dynamics by a discontinuous SMC control function, the performances of a

saturate and Tanh SMC control functions (steady-state error, slow-down response), the performances of a

saturate SMC, a Tanh SMC and a linear continuous SMC in eliminating the chattering problem.


Chapter 7:

Sliding-Mode Controller: Experimental Results

7.1. Introduction...................................................................................................................................................................1

7.2. System Description ......................................................................................................................................................2

7.3. Instrumentation.............................................................................................................................................................3

7.4. System Identification...................................................................................................................................................4

7.4.1. Theoretical System Identification ..................................................................................................................4

7.4.1.1. Ball-Hoop Transfer Function .............................................................................................................6

7.4.1.2. DC Servo Motor Transfer Function..................................................................................................7

7.4.2. Model Verification............................................................................................................................................8

7.4.2.1. Separated System................................................................................................................................8

7.4.2.2. Combined System................................................................................................................................8

7.5. Digital Sliding-Mode Controller-Observer Designs.............................................................................................. 10

7.5.1. Robust Sliding-Mode Observer Design ..................................................................................................... 10

7.5.2. Robust Sliding-Mode Controller Designs.................................................................................................. 11

7.5.2.1. Robust Switching SMC .................................................................................................................... 11

Remark 7.1: Excitation of Unmodelled High-Frequency due to Switching Function.................. 12

7.5.2.2. Robust Saturate SMC....................................................................................................................... 13

Remark 7.2: Slow-Down System Response due to Low Sliding Gain............................................ 14

7.5.2.3. Robust Unitvector SMC................................................................................................................... 15

7.5.2.4. Robust Tanh SMC............................................................................................................................. 17

Remark 7.3: Performance of Saturate and Tanh SMC...................................................................... 18

7.5.2.4. Robust Linear SMC........................................................................................................................... 19

Remark 7.4: Experimental Performance of Pseudo-SMC (Unitvector & Tanh SMC) and Linear

SMC......................................................................................................................................................... 20

7.6. Conclusions................................................................................................................................................................. 21

Advanced Sliding-Mode Controller Designs: Conclusions

In the sliding-mode control theory, control dynamics have 2 sequential modes. The first is the

reaching mode and the second is the sliding mode. The Lyapunov sliding condition forces system states

to reach a hyperplane and keeps them sliding on this hyperplane. A SMC design is composed of 2

phases, hyperplane design and controller design. First, a hyperplane is designed via the pole-placement

approach as in the state-space control, then a controller design is based on the sliding condition. Stability

is guaranteed by the sliding condition (Lyapunov Stability Criterion Theorem) and by a stable hyperplane

(stable designer-chosen pole-placement). In the reaching mode, the control dynamics depend on system

parameters; but in the sliding mode they depend on the hyperplane, this is the invariance property of the

sliding mode. On this basis, a design rule (Section 3.4) has been proposed to guarantee that the reaching

mode terminates in a finite time for the existence of the sliding mode. In addition, the modified design

(Corollary 3.2) has been presented to include system dynamics into the reaching control to reinforce this

guarantee.

We have identified the sliding-eigenvalues and the hyperplane-eigenvalues. These 2 eigenvalue types

are of reduced-order for both linear and nonlinear systems. The sliding-eigenvalues determine the system

dynamics in the sliding mode. The hyperplane-eigenvalues are the desired sliding-eigenvalues for a

sliding hyperplane of a certain model which is usually chosen to be the nominal one. In other words, the

hyperplane-eigenvalues represent the desired sliding plane on which the state variables are expected to

slide. A direct calculation method has been proposed to ease designing a hyperplane where the system

matrix A is nonsingular. A normalization technique has been presented to simplify SMC functions.

The stability criterion proposed in Section 3.2 is much simpler for practical tests than the works in

Utkin 1992, Itkis 1983. This stability approach has been extended to nonlinear systems in Corollary 7.1.

To complete the work in Drazenov 1969, a sufficient invariance condition has been presented in Theorem

4.1 and 4.2.

For the discontinuous SMC well-known as VSS, a new reaching controller design in Corollary 3.2

includes the system dynamics to reinforce the guarantee for the existence of the sliding mode to enhance

the invariance property. The proposed robust VSS design approach in Section 4.3 is much simpler than

the current approach (Edwards et al. 1996) and applicable for MIMO uncertain nonlinear systems in

Section 7.3.2 (Fu 1992).

The main limitation of VSS is the chattering problem. The analysis of the chattering problem

presented in Section 3.5 is more precise than the work in Slotine 1983, its development to use the saturate

and unitvector functions is in a unified manner (Saturate function in Slotine 1983, Unitvector function in


Ryan et al. 1987, Spurgeon 1993) and much simpler than the work in Ryan et al. 1987 and Spurgeon

1993. The hyperbolic tangent function has been proposed to solve the chattering problem more

convenient since it is a standard mathematical function. Its performance is better than that of saturate and

unitvector function. The performances of sliding functions (saturate, unitvector and TanH functions) have

been analyzed in detail (Section 3.5.5) and are consistent with experimental results in Chapter 8. A

steady-state error may occur when using a sliding function to eliminate chattering. Proposition 3.3

completely solves the chattering problem and its consequence (steady-state error).

The proposed analysis of continuous SMC has been used to explore the true nature of SMC including

the invariance property and the stability problem. The control function is not only continuous but also

linear. The continuous nature of this control function helps to eliminate the chattering problem in the

discontinuous SMC. The advantage of the control function being linear is that some fundamental

concepts of SMC design can be explained from the linear control theory framework. In the current SMC

literature, a continuous SMC has been presented in DeCarlo et al. 1987 and Zhou et al. 1992, but the

robust continuous SMC has been considered only in Zhou et al. 1992. The following are the main new

features of this work:

• the negative of a sliding margin and the hyperplane-eigenvalues are the closed-loop system

eigenvalues for linear systems under no perturbation. This concludes the fact that a sliding margin

plays a key role in the reaching mode. If this absolute value is larger, the sliding mode is more

dominant;

• under matching perturbations, the negative of a sliding margin and the hyperplane-eigenvalues are

the closed-loop system eigenvalues for linear systems at one boundary. This may be used to estimate

the control performance;

• the control function is linear continuous for linear systems with or without perturbations, and for a

certain class of nonlinear systems. In the SMC literature, the control function is linear continuous for

linear system without perturbation, but it is non-linear for linear systems under perturbations;

For a general class of uncertain nonlinear systems, a continuous pseudo-SMC may be used to

eliminate the chattering problem. In fact, this continuous pseudo-SMC is a discontinuous SMC design

where a switching function is replaced by a sliding function. Strictly speaking, a pseudo-SMC is not in

the sliding mode within a boundary laye. The fact that the sliding mode exists asymptotically means that

the difference between a SMC and a pseudo-SMC is not noticeable.

A general case in SMC has been developed where an output is a nonlinear function of state variables

for multi-input multi-output (MIMO) systems. A hyperplane has been derived from the direct allocation

method rather than from the input/output linearization technique (Hunt et al. 1983, Isidori 1985,

Kravaris et al. 1986) as in the SMC literature (Fernandez et al. 1987; Chen et al. 1992). Since the SMC

can deal with nonlinear systems conveniently, tracking control problems can be solved without

Advanced Sliding-Mode Controller Designs: Conclusion 3

difficulties. For a MIMO SMC, the hierarchical control technique has been used in Utkin 1977 for linear

systems. Here an alternative approach used a decoupling technique which is applicable for MIMO

nonlinear systems. This technique allows a MIMO can be considered as a collection of SISO subsystems,

therefore all my proposed SISO results are applicable. This technique is also applicable for a certain class

of uncertain MIMO systems (section 7.7.6).

All the experimental results are consistent with the proposed theoretical results. These are as

follows:

• the excitation of unmodelled high-frequency by a discontinuous SMC control function;

• the performances of a saturate, unitvector, Tanh and linear SMC in eliminating the chattering

problem in terms of steady-state error, slow-down response. A linear continuous SMC is free from

these 2 disadvantages.

Appendix

A.1. EIGENVALUES, AND SIMILARITY TRANSFORMATION

A.1.1. Eigenvalues of a square matrix eig A I A Ab g m r= − = ∈ℜ ×λ λi

n n0 , (1.1)

A.1.2. Similarity Transformation The matrices A B, ∈ℜ ×n n are said to be similar if a non-singular matrix P exists such that

P A P B A P BP− −= ⇔ =1 . 1. (1.2)

The eigenvalues are invariant under a similarity transformation, suppose that

P A P A− =1 .

then B P A P P A P A P P A P P A I A= = = = = =− − − −1 1 1 1. . . . . . .

and λ λ λ λ λ λI B I P A P P I P P A P P I A P P I A P I A− = − = − = − = − = −− − − − −1 1 1 1 1. . . . . .b g b g

A.2. POLE-PLACEMENT METHOD

A.2.1. Problem Statement Consider a linear system

& .x Ax B= + u (2.1)

under a control function

u = −Kx (2.2)

then & .x A BK x= −b g (2.3)

where desired closed-loop poles are eig A BK P− = = p p pn1 2, , ,L (2.4)

The problem is to find K satisfying Eq.(2.4). We define a procedure place as follows

& .

, ,x A BK x

A BK PK A B

= −

− =

UV|W|⇒ =

b g beig

place P g (2.5)

where this procedure can be determined by a couple of techniques in the literature. Among them is the

Ackermann' formula which is the simplest one and applicable to SISO systems only. To illustrate the

procedure place above, the following is a derivation of Ackermann's formula.

Sliding-Mode Control: Advanced Design Techniques A.2

A.2.2. Ackermann's Formula (Ogata 1987) Let A A BKf = − (2.6)

Since the Caley-Hamilton Theorem states that Af satisfies its own characteristic equation, we have A A A A I Af

nfn

fn

n f n f+ + + + + =− −−α α α α1

12

21L 0=P d i (2.7)

hence

I I

A A BK

A A BK A ABK BKA

A A BK A A BK ABKA BKA

=

= −

= − = − −

= − = − − −

f

f f

f f

2 2 2

3 3 3 2

a fa f f

2

f f

f f4

f

1

A A BK A A BK A BKA ABKA BKA

A A BK A A BK A BKA A BKA ABKA BKA

A A BK A A BK BKA

f f

f f f

fn n n n

fn

4 4 4 3 2 2 3

5 5 5 4 3 2 2 3

1 1

= − = − − − −

= − = − − − − −

= − = − − −− −

a fa f

a f. . .

L

Multiply by α α , respectively: αn n, , , ,−1 1 1L

α α α α α α α α α α

α α αn n f n f f

nn n n n

nn n n

n n f n fn

fn

I A A A I A A A A BK ABK BKA

A BK ABKA BKA A BK BKA

+ + + + = + + + + + − − −

− − − − − − −− − − − − − − −

− − −− −

1 22

1 22

33

1 2 2

32

3 32 1 1

L L

L L

or equivalently P PA A B K AB K B KA A B K AB KA B KA A B K B KAf n n n f n n f n f

nfnd i a f a f a f a f d i a f a f d i a f= − − − − − − − − − −− − − − − −

− −α α α α α α1 2 2 32

3 32 1L L

P PA A B AB A B

K KA KA

K KA KA

K

fn

n n f fn

n n fn

d i a f= −

+ + +

+ + + f

L

N

MMMMMM

O

Q

PPPPPP

−

− −−

− −−

L

L

L

L

1

1 21

2 32

.

α α

α α

with P A fd i = 0 (2.8)

and M B AB A Bc

n= −L 1 : Controllability Matrix (2.9)

then

P ( )A M

K KA KA

K KA KA

K

=

+ + +

+ + +

L

N

MMMMMM

O

Q

PPPPPP

− −−

− −−

c

n n f fn

n n fn

α α

α α1 2

1

2 32

L

L

L

f

A

(2.10)

if Mc is invertible (controllable), then

(2.11)

α α

α αn n f f

n

n n f fn

c

− −−

− −−

−

+ + +

+ + +

L

N

MMMMMM

O

Q

PPPPPP

=

1 21

2 32

1

K KA KA

K KA KA

K

M

L

L

L.P ( )

Sliding-Mode Controller Designs: Appendix A.3

Premultiplying both side by 0 0 0 1, , , ,L

0 0 1 0 0 1

1 21

2 32

1L

L

L

LL. .

α α

α αn n f f

n

n n f fn

c

− −−

− −−

−

+ + +

+ + +

L

N

MMMMMM

O

Q

PPPPPP

=

K KA KA

K KA KA

K

M .P ( )A (2.12)

Therefore, the procedure place in Eq.(2.5) calculates K by the Ackermann's formula as follows K M= −0 0 0 1L c

1 . (A)P (2.13)

A.3. MATRIX OPERATIONS

A.3.1. Differentiation and Derivative of a Matrix

A.3.1.1. Differentiation d dA B A Bd+ = +a f (3.1)

d dAB A B A Ba f d= +. . (3.2)

Eq.(3.2) gives 0 I A A A A A A= = = +− −d d d d. . .1 1d i d −1 i

A−thus 1 (3.3) d dA A A− −= −1 1d i . .

A.3.1.2. Derivative

ddt

V t V ddt

T

xx

xb gc h = FHG

IKJ

∂∂

(3.4)

GRADIENT

V V

Vx

Vxn

∇ = =

L

N

MMMMM

O

Q

PPPPPx

xb g ∂

∂

∂∂

∂∂

1M

HESSIAN

V V

Vx x

Vx x

Vx x

Vx x

Vx x

Vx x

n

n n n

∇ = =

× × ×

× × ×

L

nN

MMMMM

O

Q

PPPPP2

2

2

2

1 1

2

1 2

2

1

2

1

2

2

2x

xb g ∂

∂

∂∂ ∂

∂∂ ∂

∂∂ ∂

∂∂ ∂

∂∂ ∂

∂∂ ∂

L

M

L

A.3.1.3. Jacobian

∇ = =

L

N

MMMMM

O

Q

PPPPPf f

x∂∂

∂∂

∂∂

∂∂

∂∂

fx

fx

fx

fx

m

n

m

n

1

1

1

L

M N M

L

1 (3.5)

∂∂

∂∂

∂∂

∂∂x

Ax Ax

x Ax Ax A xx

x Ay Ayy

x Ay A x= = + =T T T T T, , , = T


A.3.2. Trace of an n×n Matrix (3.6) tr Aa f =

=∑aiii

n

1

(i) (3.7) tr tr tr trA A AT Td i a f a fl q a f= , A=

(ii) tr A B A B+ = +a f a f atr tr f

f

(3.8) (iii) tr AB BAa f a= tr (3.9) (iv) (3.10) tr trM AM A− =1d i a f

T i

i

(v) (3.11) x Qx xx Q QxxT T= =tr trd i d (vi) (3.12) tr tr tr trAX XA A X X Aa f a f d i d= = =T T T T

Trace operator is unchanged under : swap, transpose (each individually).

A.3.3. The Caley-Hamilton theorem Let A be square matrix, and let s s sn n

nI A− = + + +−α11L α

0=

(3.13)

then A A In n

n+ + +−α α11L (3.14)

A.3.4. Inversion of Matrices

A.3.4.1. Nonsingular Matrix and Singular Matrix

A square matrix A is called a nonsingular matrix if a matrix B exists such that

BA AB I= =If such a matrix B exists, then it is denoted by A −1 . A −1 is called the inverse of A. The inverse matrix A −1

exists if A is nonzero. If A does not exist, A is said to be singular. −1

If A and B are nonsingular matrices, then the product AB is a nonsingular matrix and AB B Aa f− − −=1 1 1 (3.15)

Also A AT Td i d i− −=

1 1 (3.16)

For a 2 matrix A, where 2×

A A=LNM

OQP

= − ≠a b

c dad bc, 0

the inverse matrix is given by

AA

A− −=−

−

LNM

OQP=

−

−

LNM

OQP

1 11 d b

c a

d b

c a


A.3.4.2. Matrix Inversion Lemma

If A, B, C and D are, respectively, an n n× , an n m× , an m m× and an m n× matrix, then A BCD A A B C DA B DA+ = − +− − − − − − −a f d i1 1 1 1 1 1 1 (3.17)

provided the indicated inverses exist.

A.3.4.3. Block Matrix Inversion

If A, B, C and D are, respectively, an n n× , an n m× , an m n× and an m m× matrix, then

(3.18) A B

C D

A A B D CA B CA A B D CA B

D CA B CA D CA B

LNM

OQP

=+ − − −

− − −

L

NMM

O

QPP

− − − − − − − − −

− − − − −

1 1 1 1 1 1 1 1 1

1 1 1 1

d i d id i d i

,

,−

1

provided A ≠ 0 and D CA B− ≠1 0

L, or

(3.19) A B

C D

A BD C A BD C BD

D C A BD C D C A BD C BD D

LNM

OQP

=− − −

− − − +NMM

O

QPP

− − − − − −

− − − − − − − −

1 1 1 1 1 1

1 1 1 1 1 1 1 1

d i d id i d i

,

,−1provided D ≠ 0 and A BD C− ≠ 0

In particular, if C or B , then Eqs.(3.18) and (3.19) can be simplified as follows 0= 0=

A B

0 D

A A BD

0 D

LNM

OQP

=−L

NMM

OQPP

− − −

−

1 1 1

1

,

,

−1

(3.20)

or

A 0

C D

A 0

D CA D

LNM

OQP

=−

LNMM

OQPP

− −

− − −

1 1

1 1 1

,

, (3.21)

A.3.5. Gradient Matrices and Frechet Derivative Operator. (H. P. Geering, AC-21/8, 1976)

A.3.5.1. Frechet Derivative Operator. Consider a scalar-valued function f Rn mXb g: R× →

f TX Aa f d i= tr X

T

then the First Frechet derivative at X with increment dX is df d d dTX X A X X A,b g d i d i= =tr tr

and the First Frechet derivative at X with resp ct to X is e

∂∂

∂∂

ff

fxij

XX

XX

Ab g b g b g

= =LNMM

OQPP =

'


A.3.5.2. Rule for Differentiating Functions Involving the Trace Operator Consider f R of the form Rn m : × → f gXb g b g= tr X , where g R Rn m r r : × ×→ , g is a matrix polynomial

in X. The First Frechet derivative at X of is obtained as followed f Xb g(1) Differentiate separately each term of the polynomial

(2) For each term, form the partial Frechet differential for each factor X separately by placing it by dX. To

each partial differential, apply Id.(vi) until d appears at the extreme left. Hence, each partial differential is TXof the form tr d TX Md i for some M ∈ ×Rn m . The partial derivative corresponding to this partial differential is

M. (3) The derivative is obtained by summing over all of the partial derivatives of all of the terms of the f ' Xb gpolynomial.

A.3.6 Some Gradient Matrices. (M. Athans, Inf. Ctl. 11, 1968) The followings are derivations of some gradient matrices in Athans 1968 by using the above Frechet

derivative operator

(1) ∂

∂XX I X Xtr tr= ⇒ =f a f

df d d d fTX X X X I X I,a f a f= = ⇒tr tr ' =

(2) ∂

∂XAX A X AXtr tr= ⇒ =T f a f

df d d d fT T TX X A X X A X A, . 'a f a f= = ⇒ =tr tr

(3) ∂

∂XAX A X AXtr trT f= ⇒ =a f T

df d d d fT TX X A X X A X A, . 'a f a f= = ⇒tr tr =

(4) ∂

∂XAXB A Btr = T T

f df d d d fT T T T TX AXB X X A X B X A B X A Ba f a f a f a f= ⇒ = = ⇒ =tr tr tr, . . . ' .

(5) ∂

∂XAX B BA X AX Btr trT f= ⇒ =a f T

df d d d d d fT T T TX X A X B B A X B A X X B A X B A, . . . . . . . 'a f d i d i a f a f a f= = = = ⇒tr tr tr tr .=

(6) ∂

∂XAX A

Ttr =

∂

∂

∂

∂XAX A

XAX Atr tr= ⇒ =T

T

(7) ∂

∂XAX A

TTtr = T

∂

∂

∂

∂XAX A

XAX Atr trT

TT= ⇒ = T


(8) ∂

∂XAXB BA

Ttr =

∂

∂

∂

∂XAXB A B

XAXB BAtr tr= ⇒ =T T

T

(9) ∂

∂XAX B A B

TT Ttr = T

∂

∂

∂

∂XAX B BA

XAX B A Btr trT

TT T= ⇒ = T

(10) ∂

∂XXX X X XXtr tr= ⇒ =2 T f a f

df d d d d dT T T TX X X X X X X X X X, . . .a f = + = +tr tr tr tr .T

f T T' X X X Xa f = + = 2

(11) ∂

∂XXX X X XXtr trT f= ⇒ =2 a f T

df d d d d dT T TX X X X X X X X X X, . . .a f = + = +tr tr tr tr T .

X

f ' X X Xa f = + = 2

(12) ∂

∂XX Xtr n n T

n= −1d i

(13) ∂

∂XA X X A Xtr . . .n i n i

i

nT

=FHG

IKJ

− −

=

−

∑ 1

0

1

(14) ∂

∂XAXBX A X B B X A X AXBXtr tr= + ⇒ =T T T T T T f a f

df d d d d d

f

T T T T T T T T

T T T T T T

X X A X BX AXB X X A X B X B X A

X A X B B X A

, . . .

'

a f a fa f

= + = +

= +

tr tr tr tr

(15) ∂

∂XAXBX A XB AXB X AXBXtr trT T T f= + ⇒ =a f T

df d d d d d

f

T T T T T

T T

X X A X BX AXB X X A XB X AXB

X A XB AXB

, . . .

'

a fa f

= + = +

= +

tr tr tr tr T

(16) ∂

∂XX Xtr e e

T

=

f ek

kX I X X XXb g = = + + + + +LNM

OQPtr tr 1

212

! !L L

df d d d dk

d d e dk k T T

X X 0 X X X X X X X X X X XX X,!

. .!

.b g b g d i= + + + + + + + +LNM

OQP = =− −tr tr tr1

21 1 1L L L d e

(17) ∂

∂XX X X Xtr − − −= − = −1 1 1d i dT −2 iT

d d d d d d

f df d d d d T T

X X I X X X X X X X X

X X X X X X X X X X X

− − − − − −

− − − − −

= = = + ⇒ = −

= ⇒ = = − = − L T−

NMOQP

1 1 1 1 1 1

1 1 1 1

0d i a f d i d i

a f a f d i d i d i

.

, .tr tr tr tr 1 1


(18) ∂

∂XAX B X BAX X AX Btr tr− − −= − ⇒ =1 1 1d i a fT

f −1

df d d d d d

f

T T T T T

T

X X A X B AX X X B X AX X B X X BAX

X X BAX

, . .

'

a f d i d i d i d i

a f d i

= = − = − LNM

OQP = − L

NMOQP

=

− − − − − −

− −

tr tr tr tr1 1 1 1 1 1

1 1

−1

A.4. SOME PROPERTIES OF A DETERMINANT The determinant of an n matrix has the following properties: n×1. If 2 rows (or 2 columns) of the determinant are interchanged, only the sign of the determinant is changed.

2. The determinant is invariant under the addition of a scalar multiple of a row (or column) to another row

(or column).

3. If an n matrix has 2 identical rows (or columns), then the determinant is zero. n×

4. For an n matrix A, n× AT = A

5. The determinant of a product of 2 square matrices A and B is the product of their determinants: A B A B B A. . .= =

6. If a row (or column) is multiplied by k, then the determinant is multiplied by k.

7. If all elements of an n matrix are multiplied by n× k n ; that is k k n. A A= 8. If the eigenvalues of A are λ , then i i, ∈ 1, n

A = λ λ λ1 2K n

Hence A implies ≠ 0 λ i i n≠ ∀ ∈0 1, , 9. If matrices A B C D∈ℜ ∈ℜ ∈ℜ ∈ℜ× × × ×n n n m m n m m, , , and , then

A BC D

A D CA B A

D A BD C C=

− ≠

− ≠

RS|T|

−

−

. ,

. ,

1

1

0

0

if

if ⇒

A B0 D

A 0C D

A D= = .

10. For an A B∈ℜ ∈ℜ× ×n m m n,

I AB I BAn m+ = + ⇒ I AB BAn m+ = + =1 , for 1


A.5. LIE ALGEBRA. (A. Isidori 2ed. 1989).

A.5.1. Covector Field Differential The CoVector field differential, or gradient dλ : λ(x) : Real-Valued function

dx x xn

λ∂λ∂

∂λ∂

∂λ∂

∂λ∂

xx

b g = =LNM

OQP1 2

L (5.1)

to be mathematically precise

d d dx

dxi

iλ λ λ λ∂λ∂

x x x x xb g b g b g b g→∇ ⇒ = ∇ = ∑.

thus

∇ =LNM

OQP

λ∂λ∂

∂λ∂

∂λ∂

xb gx x xn1 2

L

and

∇ =

L

N

MMMMMMM

O

Q

PPPPPP

f xb g P

∂∂

∂∂

∂∂

∂∂

∂ ∂

∂∂

∂∂

∂∂

fx

fx

fx

fx

f f

fx

fx

fx

n

n

m m m

n

1

1

1

2

1

2

1

2

2

2

1 2

L

M M N M

L

∂ ∂x xL

A.5.2. First Type of Differential Operation Lie Differentiation : Derivative of a real-valued function λ along the vector fields f

Lx

fi

ii

n

f x x f x x f xx

f x xλ λ λ∂λ∂

∂λ∂

b g b g b g b g b g b g b g= ∇ = ∇ = ==∑, .

1 (5.2)

For a repeated differentiating (5.3) L L L L Lg f g f fx xλ λ λb g b gc h b g b= = ∇ .g xg

xλor in a recursion L L L Lk k k

f f f fx f xλ λb g d i b g b g= ∇ =−1 . −1 (5.4)

where

L L L

L L

L L

f f f

f f

f f

x x x f x f

x f x f x f x

x f x f x f x f x

0 1 0

2 1

3 2

λ λ λ λ λ

λ λ λ

λ λ λ

b g b g b g d i b g b gb g d i b g b g b gb g d i b g b g b g b g

= = ∇ =

= ∇ = ∇ ∇

= ∇ = ∇ ∇ ∇

R

S|||

T|||

, .

. . .

. . . .

L

x∇ .

A.5.3. Second Type Differential Operation Derivative of a CoVector field along a Vector field, compare to (2) for a Scalar instead of a Vector. L T T T

f x f x xω ω ωb g b g b g= ∇ +. f∇. (5.5)


A.5.4. Third Type of Differential Operation Lie bracket (Lie product) of both Vector fields f,g f g x g f x f g x, . .b g b g b g= ∇ −∇ (5.6)

for a repeated bracketing f f f g, , ... , , , to avoid by the following definition:

ad ad kk kf fg x f g xb g b g= −, ,1 1 ≥ (5.7)

ad

ad ad

ad ad

ad ad

ad adk k

f

f f

f f

f f

f f

g x g x

g x f g f g

g x f g f f g

g x f g f f f g

g x f g f f f g

0

1 0

2 1

3 2

1

b g b gb gb gb g

b g

=

= =

= =

= =

= =

U

V

|||||

W

|||||−

;

, ,

, , ,

, , , ,

, , , ... , ,

L

(5.8)

A.5.5. Lie algebra (1) Bilinear : r r n

1 2 1 2 1 2, , , , ,∈ℜ ∈ℜ real - valued functions vector fieldsb g bf f g g g

r r r r

r r r r1 1 2 2 1 1 1 1 2 2 1

1 1 1 2 2 1 1 1 2 2 1

f f g f g f g

f g g f g f g

+ = +

+ = +

RS|T|

, ,

, ,

,

, (5.9)

(2) Skew Commutative f g g f, ,= − (5.10) (3) Jacobian Identity : f g h, , ∈ℜn

f g h g, h, f h, f , g 0, , + + =

(5.11)

A.5.6. Properties.

L

L T T T

f

f

x x f x

x f x x

f g x g f x f g x

λ λb g b g b gb g b g b gb g b g b g

= ∇

= ∇ +

= ∇ −∇

RS||

T||

,

.

, . .

ω ω ω f∇

L L L

ad ad k

ad

k k

k k

f f f

f f

f

x x

g x f g x

g x g x

λ λb g b gb g b gb g b g

=

=

=

R||

T||

−

−

1

1

0

1, , ≥

x

S

(1) if λ ∈ ℜ ∈ℜ, f n

L Lf fx∇ = ∇λ λb g b g (5.12) (2) if α λ, , ,∈ℜ ∈ℜ f g n

L Lα λ λ αf fxb g b g b gx= . (5.13) (3) if λ ∈ℜ ∈ℜ, , f g n

L L L L L

L L

f g f g g f

f g

x

f g g f,

, , , ,

λ λ λ

λ λ

b g = −

∇ = ∇ − ∇

RS|T| λ

(5.14)


(4) if f, g : vector fields; ω ; co-vector field ω, ω , ω ,f g g gf f, = −L L (5.15) (5) if α β, , ,∈ℜ ∈ℜ f g n α β α β β α α β. . . . , . . . .f g f g gf g, = + −L Lb g fd i (5.16) (6) if α β, ,∈ℜ ∈ℜ f n ; ω : co-vector field

L L Lα β α β β α β αf f fx fω ω ω, ωb g b g b g= + ∇ +. . . . . . (5.17)

A.5.7. Example.

Lα λ∂ λ∂

α∂ λ∂

α λf xx

fx

fb g b g b g= = L αfFHG

IKJ =. . . .

L TT T

TT

TT

TT T

T TT

α β α∂ β

∂β

∂ α∂

α β∂∂

∂ β∂

β α∂∂

∂α∂

α β∂∂

β α∂∂

α∂ β∂

β∂α∂

f fx

fx

fx x

fx x

f

fx

fx

fx x

f

ω =ω

ωω

ω ω

ωω. ω ω

b g b g b gLNMM

OQPP + = +

LNM

OQP+ +

FHG

IKJ

=LNM

OQP+

RS|T|

UV|W|+

LNM

OQP+

=LNM

OQP+

RS|T|

UV|W|+

LNM

OQP+

= +FHG

IKJ + = +

αβ∂∂

∂∂

α∂ β∂

β∂α∂

α β α∂ β∂

β∂α∂

α β α β β α

fx

fx

fx x

f

fx x

f ff f

TT T

T TT

TT

L L L

ωω. ω ω

ω ω ω ω ω + ω. . . . . .f d ,


Appendix

A.1. Eigenvalues, and Similarity Transformation ..............................................................................................1

A.1.1. Eigenvalues of a square matrix .......................................................................................................1

A.1.2. Similarity Transformation...............................................................................................................1

A.2. Pole-Placement Method..............................................................................................................................1

A.2.1. Problem Statement ..........................................................................................................................1

A.2.2. Ackermann's Formula (Ogata 1987) ...............................................................................................2

A.3. Matrix Operations.......................................................................................................................................3

A.3.1. Differentiation and Derivative of a Matrix .....................................................................................3

A.3.1.1. Differentiation .....................................................................................................................3

A.3.1.2. Derivative ............................................................................................................................3

A.3.1.3. Jacobian ...............................................................................................................................3

A.3.2. Trace of an n×n Matrix ...................................................................................................................4

A.3.3. The Caley-Hamilton theorem..........................................................................................................4

A.3.4. Inversion of Matrices ......................................................................................................................4

A.3.4.1. Nonsingular Matrix and Singular Matrix ............................................................................4

A.3.4.2. Matrix Inversion Lemma .....................................................................................................5

A.3.4.3. Block Matrix Inversion........................................................................................................5

A.3.5. Gradient Matrices and Frechet Derivative Operator. (H. P. Geering, AC-21/8, 1976) ..................5

A.3.5.1. Frechet Derivative Operator. ...............................................................................................5

A.3.5.2. Rule for Differentiating Functions Involving the Trace Operator.......................................6

A.3.6 Some Gradient Matrices. (M. Athans, Inf. Ctl. 11, 1968)................................................................6

A.4. Some Properties of a Determinant..............................................................................................................8

A.5. Lie Algebra. (A. Isidori 2ed. 1989). ...........................................................................................................9

A.5.1. Covector Field Differential .............................................................................................................9

A.5.2. First Type of Differential Operation ...............................................................................................9

A.5.3. Second Type Differential Operation ...............................................................................................9

A.5.4. Third Type of Differential Operation............................................................................................10

A.5.5. Lie algebra ....................................................................................................................................10

A.5.6. Properties. .....................................................................................................................................10

A.5.7. Example. .......................................................................................................................................11

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Bibliography B.1. Sliding-Mode Control.................................................................................................................................1

B.2. Optimization & Numerical Methods and Matrix Theory ...........................................................................7

B.3. Fuzzy Identification and Control ................................................................................................................9

B.4. Robust Control and Optimal H∞ Control .................................................................................................10

B.5. Adaptive Control ......................................................................................................................................12

B.6. Nonlinear Control .....................................................................................................................................12

B.7. Digital Signal Processing..........................................................................................................................13

B.8. Neural Network ........................................................................................................................................14

B.9. General......................................................................................................................................................14

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