A Thesis submitted in partial fullfilment of the requirements for the degree of

Master of Science in Electronic Engineering

Department of Electronic Engineering

Faculty of Engineering and Applied Sciences

Muhammad Ali Jinnah University, Islamabad

Qadeer Ahmed

June 2009

ii

In the Name of ALLAH, the Most Gracious,

the Most Merciful

iii

To my Family

iv

Declaration

I, Qadeer Ahmed, honestly declare that I have worked out my Master of Science thesis

individually and all resources that I have used are mention in the references.

_____________________

Qadeer Ahmed

MT081011

v

Abstract

This thesis deals with robust control strategies from linear and nonlinear

techniques for helicopter system. This system is prone to highly disturbing

interstate cross-couplings and perturbations in center of gravity that affects

smooth flights. The presented controllers offer solutions for smooth

tracking in presence of strong cross-couplings and disturbing torques

caused by perturbation in center of gravity. The H∞ controller employed

from linear robust control theory involves traditional and Hadamard

weights in controller synthesis process. However the designed controller

works for linear range only and robustness is achieved at the cost of

performance or vice versa. The second attempt involves the sliding mode

controllers from nonlinear theory. This technique delivers solution against

cross-couplings and disturbing torques caused by variation in center of

gravity is solved by using 2-sliding mode controller. These nonlinear

controllers control the system in nonlinear range. Meanwhile an attempt to

design sliding surface from Linear Matrix inequalities algorithms delivers

the solution which is not suitable for practical implementation. The

designed controllers are validated by implementing on helicopter model,

after successful numerical simulations.

vi

Acknowledgments

First and foremost I would like to thank Allah Subhana Wataallahu, who gave me

the courage, guidance and atmosphere to carry on my postgraduate studies in Pakistan.

The perseverance and determination granted by Almighty Allah helped me to bear the

hard times to produce this thesis.

I acknowledge the efforts of my parents who kept me motivated, guided and

focused throughout my MS. Their help in various regards contributed in keeping my

moral high. Apart from this, I would admire my spouse for being cooperative and

supportive during my master’s tenure. Her responsible nature made me work free of

deviation and stress.

I consider myself blessed that I found a supervisor like Dr. Aamer Iqbal Bhatti

and mentor like Mr. Sohail Iqbal. The way they developed my skills in control systems

has really contributed in my advance theoretical and practical skills. Their wealth of

ideas, clarity of thoughts, enthusiasm and energy have made my working with them an

exceptional experience. I cannot overstate my gratitude and appreciation for their

encouragement, support and cooperation.

My special thanks to Mr. Nadeem Javaid, who granted me the access to the

helicopter model under his supervision. There I was able to apply my theoretical ideas

and verify my results. That later on contributed in various publications in international

conferences and journals.

I am also grateful to Control and Signal Processing research group members: Ijaz

Kazmi, Mudassar Rizvi, Khubaib Ahmed, Armaghan Mohsin, Muhammad Iqbal, Qudrat

Khan and many others, whose constructive comments and suggestions contributed in

clarifying various concepts.

vii

List of Publications

International Conferences:

Q. Ahmed, A. I. Bhatti, M. A. Rizvi, LMI Based Sliding Mode Control Design for Twin

Rotor System to be presented in SIAM Conference on Control and Its Applications 2009,

Colorado, USA.

Q. Ahmed, A. I. Bhatti, S. Iqbal Nonlinear Robust control design for Decoupling of Twin

Rotor System to be presented in Asian Control Conference (ASCC'09), Hong Kong.

Q. Ahmed , A. I. Bhatti, S. Iqbal, Robust Decoupling Control Design for Twin Rotor

System using Hadamard Weights, to be presented in CCA, MSC 2009, St. Petersburg,

Russia.

Qadeer Ahmed, Aamer Iqbal Bhatti, Sohail Iqbal, Syed Ijaz Kazmi 2-Sliding Mode Based

Robust Control for 2-DOF Helicopter, International workshop on Variable Structure

System VSS 2010, Mexico

viii

Table Of Contents

Abstract ................................................................................................................ v

Acknowledgments ............................................................................................................. vi

List of Publications ........................................................................................................... vii

Table Of Contents ............................................................................................................ viii

List of Figures ............................................................................................................... xi

List of Tables ............................................................................................................. xiv

Chapter 1 Introduction ......................................................................................... - 2 -

Chapter 2 Helicopter Modeling & Analysis ........................................................ - 6 -

2.1 Helicopter Dynamics ...................................................................................... - 7 -

2.1.1 Elevation Dynamics: ............................................................................... - 8 -

2.1.1.1 Gravitational and Centrifugal Torque ................................................. - 8 -

2.1.1.2 Main rotor Torque ............................................................................... - 8 -

2.1.1.3 Gyroscopic Torque............................................................................ - 10 -

2.1.1.4 Frictional Torque .............................................................................. - 11 -

2.1.2 Azimuth Dynamics ............................................................................... - 12 -

2.1.3 Motor and Rotor Dynamics .................................................................. - 12 -

2.1.4 Sensor Dynamics .................................................................................. - 13 -

2.2 Mathematical model ...................................................................................... - 13 -

2.2.1 Non-Linear Model ................................................................................ - 14 -

2.2.2 Linear Model ......................................................................................... - 15 -

2.2.3 Re-Formulation of Mathematical Model .............................................. - 17 -

2.2.4 CE150 Helicopter Model ...................................................................... - 18 -

2.2.5 Physical system description .................................................................. - 19 -

2.2.6 Model Validation .................................................................................. - 21 -

2.3 Model Analysis ............................................................................................. - 21 -

2.3.1 Root locus ............................................................................................. - 22 -

2.3.2 Bode plots ............................................................................................. - 23 -

2.3.3 Singular Values plot .............................................................................. - 24 -

ix

2.3.4 Phase portrait ........................................................................................ - 25 -

2.3.5 Controllability ....................................................................................... - 26 -

2.3.6 Observability ......................................................................................... - 26 -

2.4 Control Challenges ........................................................................................ - 27 -

Chapter 3 Control Algorithms & Decoupling Techniques ................................ - 28 -

3.1 Existing Controllers ...................................................................................... - 29 -

3.1.1 Classical Controllers ............................................................................. - 29 -

3.1.1.1 PID Controller ................................................................................... - 29 -

3.1.1.2 State feedback control ....................................................................... - 30 -

3.1.2 Non-Linear Predictive Control ............................................................. - 30 -

3.1.3 Feedback linearization .......................................................................... - 31 -

3.1.4 Time Optimal and Robust control ......................................................... - 31 -

3.1.5 Sliding Mode Control ........................................................................... - 32 -

3.1.6 Higher Order sliding mode (HOSM) control ........................................ - 33 -

3.2 Decoupling Techniques ................................................................................ - 34 -

3.2.1 Multi Variable Decouple control .......................................................... - 34 -

3.2.1.1 Boksenbom & Hood Decoupling Technique .................................... - 35 -

3.2.1.2 Zalkin & Lyben Decoupling Technique ........................................... - 36 -

3.2.2 State Space Approach for Decoupling .................................................. - 38 -

3.2.2.1 Static Decoupling .............................................................................. - 38 -

3.2.2.2 Dynamic Decoupling ........................................................................ - 39 -

3.2.3 Near Decoupling Techniques ................................................................ - 40 -

3.2.3.1 Near-Decoupling: State Feedback .................................................... - 41 -

3.2.4 Hadamard Weights in LSDP for Robust Decoupling ........................... - 42 -

3.3 Conclusion .................................................................................................... - 43 -

Chapter 4 H∞ Controller Design ....................................................................... - 44 -

4.1 General Control Problem Formulation for H∞ Control ................................. - 45 -

4.2 Mixed sensitivity procedure .......................................................................... - 46 -

4.2.1 Choice of Weights and controller design .............................................. - 48 -

4.2.2 Simulation Results ................................................................................ - 50 -

4.3 Loop shaping design procedure (LSDP) ....................................................... - 51 -

x

4.3.1 Choice of Weights and Controller Design ............................................ - 52 -

4.3.2 Hadamard weight .................................................................................. - 54 -

4.3.3 Simulations Results ............................................................................... - 55 -

4.4 Experimental test Results .............................................................................. - 57 -

4.5 Performance Evaluation ................................................................................ - 62 -

4.6 Conclusion .................................................................................................... - 64 -

Chapter 5 Nonlinear Control Algorithms ......................................................... - 65 -

5.1 Lyapunov Theory .......................................................................................... - 66 -

5.2 Sliding mode control ..................................................................................... - 68 -

5.2.1 Sliding Surface Design 1 ...................................................................... - 69 -

5.2.2 Sliding Surface Design 2 ...................................................................... - 70 -

5.2.2.1 Simulation Results ............................................................................ - 73 -

5.2.2.2 Experimental Test Results ................................................................ - 76 -

5.2.2.3 Performance evaluation .................................................................... - 78 -

5.2.3 Sliding Surface Design via LMIs .......................................................... - 79 -

5.2.3.1 Simulation Results ............................................................................ - 83 -

5.3 Higher order sliding mode control ................................................................ - 84 -

5.3.1 Super Twisting Algorithm .................................................................... - 85 -

5.3.1.1 Simulation Results ............................................................................ - 88 -

5.3.1.2 Experimental Results ........................................................................ - 89 -

Chapter 6 Conclusion & Future Work ............................................................. - 94 -

Chapter 7 Appendix ................................................... - 98 -

7.1 MATLAB code of modeling of Helicopter Model ....................................... - 99 -

7.2 MATLAB code for H∞ with Traditional weights ....................................... - 100 -

7.3 MATLAB code for H∞ with Hadamard weights ........................................ - 102 -

7.4 SIMULINK Diagram for H∞ Implementation on Helicopter Model .......... - 104 -

7.5 SIMULINK Diagram for H∞ Implementation on Helicopter Model .......... - 105 -

7.6 SIMULINK Block Diagram for 2-SMC ..................................................... - 108 -

xi

List of Figures

Figure 2-1: Symbolic representation of helicopter with 2-DOF ..................................... - 7 -

Figure 2-2: Gravitational and centrifugal forces acting of helicopter in vertical plane .. - 9 -

Figure 2-3: Main rotor torque caused by ω1 in vertical plane ....................................... - 10 -

Figure 2-4: Gyroscopic torque caused due to rate of change of azimuth in vertical plane .. -

10 -

Figure 2-5: Net torques acting on the helicopter model in vertical plane ..................... - 11 -

Figure 2-6: Mechanical Torques produced in horizontal plane ................................... - 12 -

Figure 2-7: Block diagram of nonlinear model of twin rotor system ........................... - 14 -

Figure 2-8: CE150 HUMUSOFT Helicopter Model .................................................... - 19 -

Figure 2-9: Schematic diagram of helicopter model ..................................................... - 19 -

Figure 2-10: Linear model validation in vertical plane against an Impulse input ........ - 21 -

Figure 2-11: Root locus of SISO systems given in G(s) ............................................... - 23 -

Figure 2-12: Bode plots of SISO systems given in G(s) ............................................... - 24 -

Figure 2-13: Singular values of MIMO system ............................................................ - 25 -

Figure 2-14: Phase portrait of Elevation ....................................................................... - 25 -

Figure 2-15: Phase portrait of Azimuth ........................................................................ - 26 -

Figure 3-1: General Structure to decouple the system .................................................. - 35 -

Figure 3-2: Decoupling control system (Boksenbom and Hood) ................................. - 35 -

Figure 3-3: Non-interacting decoupling control structure (Zalkind &Luyben) ............ - 38 -

Figure 4-1 General control configuration for H∞ control ............................................. - 45 -

Figure 4-2: One degree of freedom configuration ........................................................ - 47 -

Figure 4-3: S/KS mixed sensitivity Plant configuration for tracking control .............. - 48 -

Figure 4-4 W1, Weighting function for S ...................................................................... - 49 -

Figure 4-5 W2, Weighting function for KS ................................................................... - 49 -

Figure 4-6 Step response (Mixed sensitivity) ............................................................... - 50 -

Figure 4-7 Control Signal (Mixed sensitivity) .............................................................. - 51 -

Figure 4-8: H∞ Robust Stabilization Problem ............................................................... - 51 -

Figure 4-9 LSDP implementation ................................................................................. - 53 -

Figure 4-10: Modified Singular Values using Traditional Weights in LSDP .............. - 54 -

Figure 4-11: Modified Singular Values with Hadamard Weights in LSDP ................. - 55 -

xii

Figure 4-12: Step response of the system with Traditional Weighted H∞ controller ... - 56 -

Figure 4-13: Control Effort of Traditional Weighted H∞ controller ............................. - 56 -

Figure 4-14: Step response of the system with Hadamard Weighted H∞ controller..... - 57 -

Figure 4-15: Control Effort of Hadamard Weighted H∞ controller .............................. - 57 -

Figure 4-16: Actual System response with Traditional Weighted H∞ controller, when

exposed to coupling at 32 sec. ...................................................................................... - 58 -

Figure 4-17: Traditional Weighted H∞ controller effort to over come coupling effects

introduced at 32 sec in azimuth plane. .......................................................................... - 59 -

Figure 4-18: Actual System response with Traditional weighted H∞ controller to attain

equilibrium position when initialized in nonlinear range ............................................. - 59 -

Figure 4-19: Traditional Weighted H∞ controller effort to acquire equilibrium position

when actual system was initialized in nonlinear range. ................................................ - 60 -

Figure 4-20: Traditional Weighted H∞ controller response to coupling when robustness

was compromised with performance ............................................................................ - 60 -

Figure 4-21: Actual System response with Hadamard Weighted H∞ controller, when

exposed to coupling at 32 sec ....................................................................................... - 61 -

Figure 4-22: Hadamard Weighted H∞ controller effort to over come coupling effects

introduced at 32 sec in azimuth plane ........................................................................... - 62 -

Figure 4-23: Undershoots in the multi step response (Minimum phase behavior) of

helicopter system with 2-DOF ...................................................................................... - 62 -

Figure 5-1: Regulation control of Helicopter outputs ................................................... - 74 -

Figure 5-2: Phase Portrait for Elevation dynamics ....................................................... - 74 -

Figure 5-3: Phase portrait for Azimuth Dynamics ........................................................ - 75 -

Figure 5-4: Sliding mode controller effort for regulation ............................................. - 75 -

Figure 5-5: Sliding manifolds convergence which ensures states convergence ........... - 76 -

Figure 5-6: Response of Helicopter system with sliding mode controller when exposed to

coupling at 22 seconds .................................................................................................. - 77 -

Figure 5-7: Sliding mode Controller effort to decouple when exposed to coupling at 22

seconds. ......................................................................................................................... - 77 -

Figure 5-8: Response of Actual system with sliding mode control when initialized in

nonlinear range.............................................................................................................. - 78 -

xiii

Figure 5-9: Sliding mode Controller effort to reach equilibrium position when released in

nonlinear range.............................................................................................................. - 78 -

Figure 5-10: Output States response for LMI based Sliding mode control .................. - 83 -

Figure 5-11: LMI based sliding mode controller effort ................................................ - 84 -

Figure 5-12: LMI based sliding surface convergence ................................................... - 84 -

Figure 5-13: Regulation response of output states of dynamical model of helicopter . - 88 -

Figure 5-14: Phase portraits of elevation and azimuth dynamics ................................. - 88 -

Figure 5-15: Response of helicopter model in Elevation when exposed uncertainty in

center of gravity ............................................................................................................ - 89 -

Figure 5-16: Variations in center of gravity serving as parametric uncertainty ........... - 90 -

Figure 5-17: 2-SMCController effort to acquire and maintain equilibrium position in

elevation plane .............................................................................................................. - 90 -

Figure 5-18: Elevation dynamics sliding manifold convergence ................................. - 91 -

Figure 5-19 Response of helicopter model with 2-SMC controller in horizontal plane . - 92

-

Figure 5-20: 2-SMC Controller effort to maintain equilibrium position in azimuth .... - 92 -

Figure 5-21: Azimuth dynamics sliding surface convergence ...................................... - 93 -

Figure 7-1 SIMULINK Diagram for H∞ Implementation on Helicopter Model ........ - 104 -

Figure 7-2 SIMULINK Diagram for H∞ Implementation on Helicopter Model ........ - 105 -

Figure 7-3 SMC Controller Block .............................................................................. - 106 -

Figure 7-4 Elevation Controller Block ....................................................................... - 106 -

Figure 7-5 Azimuth Controller Block ......................................................................... - 107 -

Figure 7-6 SIMULINK Block Diagram for 2-SMC ................................................... - 108 -

Figure 7-7 2-SMC controller Block ............................................................................ - 109 -

xiv

List of Tables

Table 2-1 System specifications (HUMUSOFT CE 150 Manual) ............................... - 21 -

Table 2-2: Eigenvalues of helicopter model ................................................................. - 22 -

Table 4-1: Performance Indices .................................................................................... - 63 -

Table 5-1: Performance indices of Elevation Dynamics .............................................. - 79 -

Table 5-2: Performance indices of Azimuth Dynamics ................................................ - 79 -

Table 6-1: Comparison between proposed controllers in this thesis ............................ - 96 -

___________________________________- 2 -_______________________________

Chapter 1

Introduction

Chapter Objectives:

Background

Motivation

___________________________________- 3 -_______________________________

Helicopter is an aircraft that is lifted and propelled by one or more horizontal rotors,

each rotor consisting of two or more rotor blades. Helicopters are classified as rotorcraft

or rotary-wing aircraft to distinguish them from fixed-wing aircraft because the helicopter

achieves lift with the rotor blades which rotate around a mast. The primary advantage of a

helicopter is due the rotor which provides lift without the aircraft needing to move

forward, allowing the helicopter take off and land vertically without a runway. For this

reason, helicopters are often used in congested or isolated areas where fixed-wing aircraft

cannot take off or land. The lift from the rotor also allows the helicopter to hover in one

area, more efficiently than other forms of vertical takeoff and landing (VTOL) aircraft,

allowing it to accomplish tasks that fixed-wing aircraft cannot perform [1]

Helicopters are under-actuated an mechanical system that means we have to

perform maneuvers in all six degrees of freedom and available actuators to perform these

tasks are limited to just two in number. This task inherently induces cross-couplings in

the system’s dynamics i.e. each actuator must have some of its affect in all of the six

degrees of freedom. However, these cross-couplings should be under control of the

operator so that desired maneuvers can be performed at ease, else uncontrolled cross-

coupling can lead to fatal accidents causing human life losses.

Moreover, the disturbance torque caused by perturbations in center of gravity

(CG) also affects helicopter flight adversely, which certainly adds responsibilities for the

on board pilot. This unwanted torque must be compensated for the sake of smooth and

comfortable flight. These disturbing moments can occur as a consequence of weight

variations loaded on board during flight like turbulence in the fuel tank, coolant tanks and

hydraulic fluids tanks. Other causing agents may include the movements of passengers

during flight, wind gusts etc. Similarly relief luggage not loaded about the center of

gravity will continuously cause torque on the helicopter, forcing the nose tip to either

bend down or tilt up thus disturbing the normal helicopter flight. The ideal condition is to

have the helicopter in such perfect balance that the fuselage will remain horizontal in

hovering flight. The fuselage acts as a pendulum suspended from the rotor. Any change

in the center of gravity changes the angle at which it hangs from this point of support and

introduces additional torques that disturbs the flight [2], [3]. This demands a skillful pilot

and adds more responsibilities for the on board pilot.

___________________________________- 4 -_______________________________

The thesis considers solutions for above mentioned problems. The smooth flight

in the presence of cross-coupling and CG perturbations is at stake. However, solutions

from robust control theory may solve the problems and deliver best out of the available

mechanical helicopter structure. The first step involved for above mentioned problems is

modeling of the helicopter system. This modeling will later on contribute in indentifying

the core factors causing disturbance in helicopter flight. The cross-couplings in helicopter

dynamics can be modeled or treated as disturbances; therefore we can attain the solution

to this factor. However, perturbations in CG can be dealt under robust control theory by

declaring it as parametric uncertainty.

The robust controllers employed to handle the above mentioned problems are

from linear and nonlinear theory. H∞ controller [4] from linear control theory has proved

it robustness over the past few decades. This controller offers robustness against the

cross-coupling and parametric uncertainty at the cost of performance. The more the

cross-coupling and parametric uncertainty is catered the more the system looses it

performance. To overcome this problem, Hadamard weighting technique [5] has been

employed that slightly improves the performance and offers robustness at the same time.

Therefore, the two H∞ designing techniques with Traditional and Hadamard weights are

considered for the solution. The traditional weighted H∞ controller offers the control even

in nonlinear domain along with robustness but performance is slightly reduced, however

Hadamard weights do not operate in nonlinear domain but caters cross-coupling along

with desired performance.

Sliding mode control theory [6] has been exploited from the nonlinear control theory.

This technique has also proved its robust nature in the control history. The nonlinear

model of the system is utilized for the development of control algorithm. The control law

evolved as a result is usually not suitable for implementation due to chattering in it. This

problem is resolved by utilizing concepts from Higher Order Sliding Mode. The resultant

controller offers better performance and robustness as compared to its linear counterparts.

The thesis keeps the scope of the problem limited to cross-coupling affecting

horizontal and vertical plane dynamics only, along with uncertainty in CG. H∞ and

___________________________________- 5 -_______________________________

sliding mode controllers have been employed to deliver robust solutions. The verification

of the designed algorithms has been carried out first in simulation and later on Humusoft

Twin rotor system is used to authenticate the control laws. The laboratory helicopter

presents higher coupling between dynamics of the rigid body and dynamics of the rotors

and yields a highly nonlinear, coupled dynamics. Additionally, it can be proved that

characteristic dynamics of the system is non minimum phase, exhibiting unstable zero

dynamics. This system has been extensively investigated yielding a number of control

applications that range from linear robust control techniques to more recent nonlinear

approaches a [7] ~ [17]. Therefore after analyzing the system in detail we will apply few

of the following discussed robust control algorithms to extract the desired performance

from the laboratory helicopter model.

This chapter delivered a brief idea about the overview and motivation for the thesis work.

The rest of the thesis includes: the detailed helicopter modeling and its in-depth analysis

is discussed in Chapter 2, strategies to handle couplings in the dynamical system will be

discussed in Chapter 3 along with literature review of the control algorithms that have

already been implemented to deliver the solutions for the fore-discussed problems.

Finally, proposed robust control algorithms from linear and nonlinear theory will cover

Chapter 4 and Chapter 5. These chapters will include controllers’ derivation and their

validation based on simulation and implementations carried on the helicopter model

___________________________________- 6 -_______________________________

Chapter 2

Helicopter Modeling

&

Analysis

Chapter Objectives:

Dynamical modeling of helicopter

Model Validation

Analysis of mathematical model

___________________________________- 7 -_______________________________

This chapter deals with mathematical model formulation and its analysis in detail.

The basic physical concepts of moment generation have been utilized to develop

differential equations for helicopter dynamics both in vertical and horizontal planes.

These differential equations are then utilized for dynamical analysis of the system after

developing its state space model and transfer function. The detailed analysis as a result

will formulate the basic objectives for the controller synthesis.

2.1 Helicopter Dynamics

The helicopter dynamics can be reduced to vertical and horizontal plane dynamics

which can be approximated as twin rotor dynamics as shown in Figure 2-1. An attempt to

model the system dynamics in detail leads to extremely complicated, not readable and not

useful model. In our case the model will be used for investigating the system dynamics

with respect to control tasks. The system will operate in some working conditions only

and not all of the dynamical properties will be invoked. This leads to the assumptions

which will simplify the derivation of the model. We propose two ways model the system.

The first one is a systematic modeling method based on variational approach, i.e.

Lagrange's equations. The second approach is a direct derivation of the model by

computing the force balances. Both methods lead to the model in the form of nonlinear

differential equations [7]. System identification techniques are also used to model the

system, but the resultant model has linear dynamics only thus prevent us to understand

actual working of the system in detail. Several other authors have discussed the helicopter

dynamics in detail in new approach, the details of which can be found in [8].

Figure 2-1: Symbolic representation of helicopter with 2-DOF

___________________________________- 8 -_______________________________

2.1.1 Elevation Dynamics:

The modeling in elevation of helicopter is carried out using standard physics laws of

angular momentum. Considering the free body diagram of helicopter model, different

torques produced by different forces are balanced about the pivot point. The different

torques produced are

a) Gravitational Torque

b) Frictional Torque

c) Centrifugal Torque

d) Main rotor Torque

e) Gyroscopic Torque

2.1.1.1 Gravitational and Centrifugal Torque

Consider the free body diagram shown in Figure 2-2, the weight of the helicopter

and centrifugal force produce respective torques about the pivot point. Eq. (2.1) describes

the gravitational torque produced by the model weight.

sinw l w (2.1)

Where 2

( . )

( )

( . / sec )

( )

w Gravitational torque N m

Elevation Angle rad

w Weight of helicopter kg m

l Moment Arm m

Eq. (2.2) describes the centrifugal torque produced by centrifugal force during rotation in

horizontal plane.

c c= l F cos (2.2)

Where 2F ml sinc

(2.3)

( / sec)

( )c

AngularVelocity in horizontal plane rad

F Centrifugal Force N

2.1.1.2 Main rotor Torque

The main rotor force produced is the consequence of its angular speed as shown

in Figure 2-3. The more the angular speed of rotor the more force will be induced on the

___________________________________- 9 -_______________________________

Figure 2-2: Gravitational and centrifugal forces acting of helicopter in vertical plane

helicopter body, which will produce angular torque about the pivot point. Therefore we

can say that

1 1 1( )F (2.4)

Where 1

1

1

( )

( / sec)

( . )

F Main rotor Force N

Main rotor angular velocity rad

Main rotor torque N m

We can calculate the force caused by the angular velocity as,

1

2

2

rF m a

va

rv r

a r

21 rF m r (2.5)

Where ( )

( )r

r= radius of main rotor m

m = mass of main rotor kg

Finally we have the main rotor torque as

21 1k (2.6)

Where 1k m r l

Pivot Point

___________________________________- 10 -_______________________________

Figure 2-3: Main rotor torque caused by ω1 in vertical plane

2.1.1.3 Gyroscopic Torque

Gyroscopic torque occurs as a result of Coriolis forces acting on helicopter

elevation dynamics. This torque results when moving main rotor changes its position in

azimuth. Thus resultant gyroscopic torque caused by the main rotor and azimuth rotation

can be calculated from the Figure 2-4 as

G 2 1= k cos (2.7)

Where

( )

( / sec)

( . / sec)2

= Azimuth Angle rad

= Angular velocity in Azimuth rad

k = Contant of proportionality N m

Figure 2-4: Gyroscopic torque caused due to rate of change of azimuth in vertical plane

___________________________________- 11 -_______________________________

The Eq. (2.7) is based on the fact that main rotor speed is very high as compared to rate

of change of azimuth i.e. 1 .

2.1.1.4 Frictional Torque

The frictional torque can be estimated from the following equation

1f = B (2.8)

Where 21 ( . / sec)B = Damping Constant kg m

Considering all the torques produced on the helicopter body as discussed above, the net

torque produced as shown in Figure 2-5 is

1 1 wc G fI (2.9)

Where 21 ( . )I moment of inertia of the helicopter body around horizontal axis kg m

Figure 2-5: Net torques acting on the helicopter model in vertical plane

In calculating the elevation dynamics some influences are neglected, e.g. stabilizing

motor reaction torque and varying air resistance depending on the turnings of the main

propeller. While the influence of the side motor on the elevation angle is almost

negligible, varying damping of body oscillation in elevation is noticeable. The influence

of the speed of the main propeller on friction torque in elevation is hardly to be modeled

analytically and must be evaluated by an experiment and, if significant, nonlinear

coupling must be introduced [7].

___________________________________- 12 -_______________________________

2.1.2 Azimuth Dynamics

The net torques produced in horizontal plane as seen in Figure 2-6 are

2 2 2r fI (2.10)

Where

2

2

22

( . )

( . )

( . )

( . )

f

r

Side rotor torque N m

Frictional troque N m

Main motor reaction torque N m

I Moment of Inertia inverticle plane Kg m

Figure 2-6: Mechanical Torques produced in horizontal plane

The fictional torque and side rotor torque are calculated similarly to elevation dynamics

as they are proportional to rate of change of angular position and rotor speed respectively.

The main rotor reaction torque acting on azimuth can be estimated by first order transfer

function shown in Eq. (2.11).

orr

pr

T s+1T(s)= K

T s+1 (2.11)

2.1.3 Motor and Rotor Dynamics

The details of motors modeling are available in [7]., in our case DC motor has

been estimated against a simple first order transfer function whose time constant has been

identified by motor behavior. Eq. (2.12) shows the transfer function of motor

___________________________________- 13 -_______________________________

11

1

1M

T s

(2.12)

Where 1T Main motor timeconstant

And the nonlinearity caused by the rotor can be estimated as second order polynomial

whose constants have been indentified as explained in [7].. Finally the torque induced in

helicopter body via motor can be given by the following equation.

21 1 1a u bu (2.13)

Where 1u Output of the motor

2.1.4 Sensor Dynamics

Incremental sensor is installed in the model to measure the angular position in

elevation and azimuth. These parts have no dynamics and are considered to be linear in

the whole extent of measured angles. The 10-bit encoder gives 1024 pulse against 1

degree and the encoder can be modeled with the following equations

o

y k y (2.14)

y k (2.15)

Where o

y Elevation output angle

y Initial Elevation output angle

y Azimuth output angle

k Elevation constant

k Azimuth constant

2.2 Mathematical model

The above discussed dynamics of different components of helicopter model help us

in formulating the mathematical model which can be seen in Figure 2-7. The different

states of the model can be figured out on the basis of their function, like each motor has

one state that will give the information of angular speed of the motor, similarly the

angular position and rate of change of angular positions give rise to few other states.

Finally our system’s states and outputs come out to be

___________________________________- 14 -_______________________________

2

3

4

5

6

7 7

1

2

1 Mainmotor speedxxxxxx

x Angular Moment caused by ux

Elevation AngleAngular speed in Elevation

Sidemotor speedXAzimuth Angle

Angular speed in Azimuth

1on Azimuth

(2.16)

2

5

x Elevation AngleY

x Azimuth Angle

(2.17)

Based on these states and above discussed dynamical equations we can now proceed for

the dynamical model for helicopter model.

Figure 2-7: Block diagram of nonlinear model of twin rotor system

2.2.1 Non-Linear Model

The non-linear model based on above states can be developed from the basic

equation derived from physical laws is,

1

1( )1 1 1x = x u

T (2.18)

___________________________________- 15 -_______________________________

2 3

23 1 1 1 1 1 3 g 2 gyro 1 6 2

1

x = x

1x = ((a x ) +b x - B x -T sinx - K u x cosx )

I

(2.19)

2

1( )4 4 2x = -x +u

T (2.20)

5 6

26 2 4 2 4 2 6 pr 7 r or 1

2

x = x

1x = ((a x ) +b x - B x +T x - K T u )

I

(2.21)

7 pr 7 r or 1x = -T x K T u (2.22)

Eq. (2.18) represents the main motor dynamics estimated by 1st order transfer function,

Eq. (2.19) explains the elevation dynamics derived from physical laws, same

phenomenons have been utilized for equation of side motor and azimuth dynamics. Eq.

(2.22) describes the angular momentum caused by first input in horizontal plane. The

constants values can be found in [7]..

2.2.2 Linear Model

The non-linear model can be linearized around a set point to develop linear model

that would be functional in linear range. Conventional linearization technique by taking

the Jacobean of non-linear model and replacing the set points values has been utilized to

develop the linear model. Eq. (2.23) explains the linearization procedure mathematically

( )

0f x

xx

(2.23)

For example we can linearize Eq. (2.19) as

62

3 1 1 1 1 1 3 g 2 gyro 1 21

1x = (a x +b x - B x -T sinx - K u x cosx )

I

3 1 11

1[ 0 0 0 0]gx b T B X

I (2.24)

Where X StatesVector

Finally the linearized model of helicopter is formulated as below

___________________________________- 16 -_______________________________

X AX BU

Y CX DU

(2.25)

Where

7 1

2 1

2 1

:

:

:

X R StatesVector

Y R OutputVector

U R InputVector

1

1 1

1 1 1

7 7

2

2 2

2 2 2

1

1

0

0

g

pr

pr

- 0 0 0 0 0 0T

0 0 1 0 0 0 0

Tb B0 0 0 0

I I I

A R0 0 0 0 - 0 0

T

0 0 0 0 0 0 1

Tb B0 0 0

I I I

0 0 0 0 0 T

(2.26)

1

7 2

2

1

1

r or

r or

0T

0 0

0 0

B R0

T

0 0

-K T 0

K T 0

(2.27)

2 7 0 1 0 0 0 0 0C R

0 0 0 0 1 0 0

(2.28)

2 2 0 0D R

0 0

(2.29)

Based on state space model and the parameter values in Table 2-1, the 2 by 2 transfer

function of the helicopter model can be written as follows.

___________________________________- 17 -_______________________________

11 12

21 22

( )g g

G sg g

(2.30)

Where

4 3 2

11 7 6 5 4 3 2

7.141 s + 53.07 s + 118 s + 79.94 s( )

s + 11.19 s + 54.6 s + 178 s + 427.2 s + 596.6 s + 327 sg s

6 3 2

12 7 6 5 4 3 2

-1.776e-15 s + 5.684e-14 s + 1.137e-13 s( )

s + 11.19 s + 54.6 s + 178 s + 427.2 s + 596.6 s + 327 sg s

6 5 4 3 2

21 7 6 5 4 3 2

3.553e-15 s - 1.401 s - 11.37 s - 39.24 s - 110.8 s - 199.4 s - 59.68( )

s + 11.19 s + 54.6 s + 178 s + 427.2 s + 596.6 s + 327 sg s

6 4 3 2

22 7 6 5 4 3 2

-1.776e-15 s + 28.41 s + 144.5 s + 431 s + 1215 s + 1106( )

s + 11.19 s + 54.6 s + 178 s + 427.2 s + 596.6 s + 327 sg s

2.2.3 Re-Formulation of Mathematical Model

The differential model obtained in Eq.(2.18) – Eq. (2.22) can be reformulated in

standard mechanical systems dynamical equation as

Mq Cq G (2.31)

Where

2 2

2 2

2 1

2 1

2 1

:

:

:

:

:

M R Inertial Matrix

C R Coriolis Matrix

G R Gravitational Matrix

q R State Vector

R Input TorqueVector

The matrices values for helicopter model are

1

2

0

0

IM

I

(2.32)

1

2

0

0

BC

B

(2.33)

sin

0gT

G

(2.34)

___________________________________- 18 -_______________________________

q

(2.35)

11 12

22 21

(2.36)

11 and 22 are the torques generated by main and side motors and their affect on the

elevation and azimuth respectively. 12 and 21 are the cross coupled torques generated

by side motor on elevation and main motor on azimuth respectively. The torques

equations can be computed from the Eq.(2.18) – Eq.(2.22).

2.2.4 CE150 Helicopter Model

The CE150 Helicopter Model shown in Figure 2-8 is designed for the theoretical

study and practical investigation of basic and advanced control engineering principles.

This includes system dynamics modeling, identification, analysis and various controllers

design by classical and modern methods [7]. The twin rotor system resembles a

simplified behavior of a real helicopter with fewer degrees of freedom. In real helicopters

the control is generally achieved by tilting appropriately the blades of the rotors with the

collective and cyclic actuators, while keeping constant rotor speed. In order to simplify

the mechanical design of the system, the laboratory setup employed, is designed slightly

differently. In this case, the blades of the rotors have a fixed angle of attack, and control

is achieved by controlling the speeds of the rotors. As a first consequence of this, the

laboratory helicopter presents higher coupling between dynamics of the rigid body and

dynamics of the rotors than a conventional helicopter, and yields a highly nonlinear,

coupled dynamics. Additionally, it can be proved that characteristic dynamics of the

system is non minimum phase, exhibiting unstable zero dynamics. This system has been

extensively investigated yielding a number of control applications that range from linear

robust control techniques to more recent nonlinear approaches a [7] ~ [17]. In order to

elaborate the coupling affect in dynamics, while implementation this thesis involves the

practice of first restricting twin rotor system to 1-DOF (Elevation) and after some time 2-

DOF (Azimuth) is introduced to exaggerate coupling effects.

___________________________________- 19 -_______________________________

Figure 2-8: CE150 HUMUSOFT Helicopter Model

Figure 2-9: Schematic diagram of helicopter model

2.2.5 Physical system description

The laboratory helicopter is commonly known as twin rotor multi input multi

output (MIMO) system (TRMS). This system is hinged as the based, thus restricting the

six degrees of motion to just two degrees of freedom. This educational model consists of

two DC motors which drive upper and side propeller by generating torques perpendicular

to their rotation and a servo mechanism used to manipulate the center of gravity. The

system has two degrees of freedom i.e. Elevation ( ) in vertical plane and azimuth ( ) in

horizontal plane, which are measured precisely by incremental encoders installed inside

___________________________________- 20 -_______________________________

the helicopter body. The model is interfaced with desktop computer via Humusoft

MF624 data acquisition PCI card which is accessible in MATLAB Simulink environment

through Real-time Toolbox and Real Time Windows Target Toolbox. These toolboxes

provide us the liberty to access the encoder values and issue commands to DC motors and

servo system. The schematic diagram shown in Figure 2-9 gives a brief idea about the

helicopter model interfacing. The system is controlled by changing the angular velocities

of the rotors. This kind of action involves the generation of resultant torque on the body

of double rotor system that makes it to rotate in perpendicular direction of the rotor.

Some of the specifications are shown in Table 2-1 , more details can be found in [7].

System Outputs 50o in elevation

±40o in Azimuth

Main Motor ‘1’ DC motor with permanent magnet

Max Voltage 12V

Max Speed 9000 RPM

Side Motor ‘2’ DC motor with permanent magnet

Max Voltage 6V

Max Speed 12000 RPM

System Parameters T1 = 0.3 s

a1 = 0.105 N.m/MU

b1 = 0.00936 N.m/MU2

I1 = 4.37e-3 Kg.m2

B1 = 1.84e-3 Kg.m2/s

Tg = 3.83e-2 N.m

T2 = 0.25 s

a2 = 0.033 N.m/MU

b2 = 0.0294 N.m/MU2;

Tor = 2.7 s

Tpr = 0.75 s

Kr = 0.00162 N.m/MU

I2 = 4.14e-3 Kg.m2

B2 = 8.69e-3 Kg.m2/s

___________________________________- 21 -_______________________________

Kgyro = 0.015 Kg.m/s

Table 2-1 System specifications (HUMUSOFT CE 150 Manual)

2.2.6 Model Validation

The linear model extracted from the non-linear model is validated against impulse

signal. The elevation dynamics in vertical plane are asymptotically stable in open loop

and can be verified in open loop against any validation signal. However, azimuth

dynamics in horizontal plane are not asymptotically stable and can be verified by

trapping it in feedback to make it asymptotically stable and applying any validating

signal with proportional gain in the forward path. Figure 2-10 shows the validation of

linear model against impulse signal given for 2 seconds at input of the elevation in open

loop. The validation response clearly shows the non-linearities impact which were

ignored during linearization e.g. the gravitational torque caused by weight of the body is

non-linear terms, its difference is clearly visible in impulse response when helicopter

declines after the impulse input ends.

20 25 30 35 40 45 50 55 60 65-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12Impulse response

Time

Ele

vatio

n (r

ad)

Actual Response

Linear model ResponseError

Figure 2-10: Linear model validation in vertical plane against an Impulse input

2.3 Model Analysis

The linear model and non-linear model are ready for analysis. Table 2-2 shows the

Eigenvalues of the linear model and their properties

___________________________________- 22 -_______________________________

Eigenvalues Damping Freq. (rad/s)

0.00 -1 0.00

-1.33 1 1.33

-2.10 1 2.10

-.211 + 2.95i 0.0711 2.96

-.211- 2.95i 0.0711 2.96

-3.33 1 3.33

-4.00 1 4.00

Table 2-2: Eigenvalues of helicopter model

It can be seen that one pole of the system is on origin which will affect the stability of

our system and conjugate poles are having less damping ratio which will affect the

performance of the model. Few other techniques have been employed to have through

analysis of the system, so that we have detailed insight of the system behavior.

2.3.1 Root locus

The root locus [18] of each component of G(s) is shown in Figure 2-11. The

elevation root locus gives the picture that small increase in gain will make the system to

go unstable by shifting the poles on right half plane. The second and fourth root locus

point out the right half plane zeros, which make s our system non-minimum phase

system. After going through root locus we clearly get the idea that we cannot give high

gain to our system and we have to be careful about right plane zeros in future designing

procedures.

___________________________________- 23 -_______________________________

-10 -5 0 5-10

-5

0

5

10Root Locus

Real Axis

Imag

inar

y A

xis

-5 0 5 10 15-10

-5

0

5

10Root Locus

Real Axis

Imag

inar

y A

xis

-10 -5 0 5-4

-3

-2

-1

0

1

2

3

4Root Locus

Real Axis

Imag

inar

y A

xis

-2 -1 0 1 2 3 4

x 108

-1

-0.5

0

0.5

1x 10

8 Root Locus

Real Axis

Imag

inar

y A

xis

Figure 2-11: Root locus of each SISO systems given in G(s)

2.3.2 Bode plots

The bode plots [18] shown in Figure 2-12 give us the idea of each component of

G(s). The first bode show the behavior of 1st input to 1st output, second bode plot show us

the 1st output and 2nd input transfer function behavior and so on. The helicopter dynamics

in elevation have poor gain and phase margins however these margins are fine in azimuth

dynamics. The effect of 1st input on azimuth is significant at lower frequencies but the

elevation has very less impact of 2nd input. The bode plots give us the idea of coupling in

helicopter dynamics which will be discussed further in detail.

___________________________________- 24 -_______________________________

-200

-100

0

100

200

Mag

nitu

de (

dB)

10-1

100

101

102

-360

-180

0

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

-350

-300

-250

Mag

nitu

de (

dB)

10-1

100

101

102

0

360

720

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

-100

-50

0

50

100

Mag

nitu

de (

dB)

10-2

10-1

100

101

102

0

90

180

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

-100

-50

0

50

100

Mag

nitu

de (

dB)

10-2

10-1

100

101

102

0

180

360

Pha

se (

deg)

Bode Diagram

Frequency (rad/sec)

Figure 2-12: Bode plots of SISO systems given in G(s)

2.3.3 Singular Values plot

The singular values [19] shown in Figure 2-13 exhibits the poor tracking of the

system and the slope of the singular values at zero crossing is around 2 which founds the

base for poor tracking performance. The slight peak at zero crossing is also indicating the

oscillatory behavior of the helicopter. However, the upper and lower singular values at

higher frequencies are desirable.

___________________________________- 25 -_______________________________

10-2

10-1

100

101

102

-120

-100

-80

-60

-40

-20

0

20

40

60Singular Values

Frequency (rad/sec)

Sin

gul

ar V

alue

s (

dB)

Figure 2-13: Singular values of Twin Rotor MIMO System

2.3.4 Phase portrait

Phase portrait [6] give us the clear picture that helicopter dynamics in vertical

plane are inherently asymptotically stable and elevation eventually reaches to zero with in

finite time whatever the initial conditions are given. The elevation phase portrait can be

seen in Figure 2-14.

-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Elevation

Rat

e of

cha

nge

of e

leva

tion

Phase portrait of Elevation

Figure 2-14: Phase portrait of Elevation

The azimuth dynamics are stable. It can be observed in Figure 2-15 that the azimuth

reaches to certain value instead of converging to zero. Rate of change of azimuth when

___________________________________- 26 -_______________________________

approaches to zero, determines the current azimuth value i.e. -0.58 radians, as shown in

Figure 2-15.

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Azimuth

Rat

e of

Cha

nge

of A

zim

uth

Azimuth Phase Portrait

Figure 2-15: Phase portrait of Azimuth

2.3.5 Controllability

A control system is said to be controllable if, for all initial times and all initial

states, there exists some input function that drives the state vector to any final state at

some finite time [19]. The controllability matrix of the LTI system is defined by the pair

(A,B) as follows:

2 1( , ) nC A B B AB A B A B (2.37)

The LTI system in Eq. (2.25) is said to be controllable if C matrix has rank ‘n’. Therefore

for helicopter model if we place the A,B values from Eq.(2.29), the rank of C matrix

comes out to be ‘7’, which concludes that our model is completely controllable.

2.3.6 Observability

A control system is said to be observable if, for all initial times, the state vector can be

determined from the output function [19]. The observability matrix of the LTI system is

defined by the pair (A,C) as follows

___________________________________- 27 -_______________________________

2 1( , )TnO A C C CA CA CA (2.38)

The LTI system defined in Eq.(2.25) is said to be observable if O matrix has rank ‘n’.

Helicopter model defined in Eq.(2.29) delivers an observable matrix ‘O’ with rank ‘7’.

Therefore this test verifies that our model is completely observable.

2.4 Control Challenges

The major challenges determined from the above analysis are tracking control and

reduction of coupling with in the system dynamics. The tracking problem can be seen

from singular values shown in Figure 2-13, the lower singular values have poor gains at

lower frequencies and the upper and lower singular values have slope around 2 at zero

crossing, these two characteristics of model ensures the poor tracking response of

helicopter model. The stability constraints are visible in root locus i.e. we cannot have

high gains for controllers. Similarly the non-minimum phase behavior is also to be kept

in mind while designing and testing controllers of CE 150 helicopter model.

The major challenge residing in the model is interstate coupling. Eq. (2.19) in non-

linear model shows the effect of horizontal plane dynamics in elevation caused by

gyroscopic torque. This coupling is not as significant as the coupling is induced from 1st

input to azimuth. Eq. (2.22) in non-linear model shows the clear effect of it. In fact this

coupling has been indentified in detail in the helicopter manual [7]. This coupling affects

the performance of model especially in horizontal plane dynamics.

After discussing the helicopter model in detail and figuring out its control objectives,

now the control and decoupling techniques will be explored. Several attempts have been

made to achieve the control objectives which will be discussed in the coming chapters.

Secondly, to over come the coupling problem various decoupling techniques are

discussed to handle the interstate coupling which affects the helicopter model

performance.

___________________________________- 28 -_______________________________

Chapter 3

Control Algorithms

&

Decoupling Techniques

Chapter Objectives

Linear/Nonlinear Controllers Review

Decoupling Techniques Review

___________________________________- 29 -_______________________________

The objective of this chapter is to indicate the control algorithms that have been

already utilized by the control community to meet the goals formulated in analysis

process for helicopter model. Helicopter control is one of the challenging problems faced

by control engineers. The cross-couplings in its dynamics and parametric uncertainties

lead to foundation of designing robust controllers. Furthermore, this chapter will explore

some decoupling techniques that will be later on utilized to attain decoupling in

helicopter dynamics. The CE 150 helicopter model has been used for validation of

control algorithms, which has MATLAB as working environment. The advantage of this

environment is that the designed controllers are easy to implement but the processing rate

cannot be achieved beyond certain limit. However, one can manipulate the code for

implementation.

3.1 Existing Controllers

Controllers ranging from linear to nonlinear domain have been validated on this

model. The educational manual accompanied with the model has some classical control

design techniques. Moreover, several other control engineers have validated their

designed control algorithms on this model. This section will discuss an overview of some

of the previously designed controllers for this model.

3.1.1 Classical Controllers

The classical controllers, the control techniques invented in early 1960s are

termed as classical control techniques. Few of these techniques have been implemented

on this model which includes PID and state feedback controllers. Although these

techniques give good performance but under uncertain conditions these techniques fail to

cope up with desired performance.

3.1.1.1 PID Controller

The gains of proportional, integral and derivative constant in PID controller are

extracted from the root locus, shown in Figure 2-11. The root locus from main motor as

actuator and elevation as measured output will give the limits of PID gains which will

ensure the system’s eigenvalues in the left half plane similarly the PID gains for side

motor as actuator and azimuth as output can be obtained from its root locus. More details

___________________________________- 30 -_______________________________

can be found in [7]. Following the practice discussed above the control law for the

helicopter model can be written as

( ) ( ) ( )ip d

KU s K K s E s

s (3.1)

Although the control action from the PID controller is sufficient enough to deliver the

required response but its rise time and settling time is high and one cannot decrease the

response times due to limits applied by movement of poles from left half plane to right

half plane which will cause instability. Similarly this controller is not robust in nature and

is unable to withstand coupling effect when introduced in the system.

3.1.1.2 State feedback control

The second classical control technique which has been applied for the helicopter

tracking control is pole placement using state feedback control. The main idea of this

method is that the location of the closed loop poles of a linear system determines its

dynamics. In this problem the poles of a closed loop system are placed according to the

position specified by the designer. This may be done only by the state feedback. Eq.(3.2)

delivers the mathematical concept of pole placement using state feedback, where K is the

feedback controller which places the closed loop system poles at desired location to get

the preferred performance.

( )

X AX BU

U KX

X A BK X

(3.2)

Although this control action provides the satisfactory performance results but still the

intentionally introduced coupling was still unhandled. Further details for pole placement

using feedback can be found in [7].

3.1.2 Non-Linear Predictive Control

The model predictive control [20] problem is formulated as solving on-line a

finite horizon open-loop optimal control problem subject to system dynamics and

constraints involving states and controls. Based on the measurements obtained at certain

time, the controller predicts the future dynamic behavior of the system over a prediction

horizon and determines the input such that a predetermined open-loop performance

___________________________________- 31 -_______________________________

objective functional is optimized. Dutka et al [21] have implemented nonlinear predictive

control for tracking control of helicopter model. The nonlinear algorithm is based on

state-space generalized predictive control. The non-linearity is handled by converting the

state-dependent state-space representation into the linear time-varying representation.

The predictions of the future controls are used to calculate predictions of the future states

and future time varying system parameters. Applied to the helicopter model, the

algorithm performs well. It is capable of the stabilizing the system for maneuvers for

which its linear counterpart fails. The authors have only addressed the tracking

performance which is very slow as its settling time is high and secondly the coupling

effect on its performance is not considered.

3.1.3 Feedback linearization

Feedback linearization [6] offers the user to cancel out the nonlinearities and

introduce Hurwitz polynomial that will be responsible for control of system. Based on

system properties one has to decide whether to use exact feedback linearization, input

output linearization or input state linearization. M.Lopez et al [9], [10], [11] have

presented control of helicopter model using feedback linearization. The feedback

linearization techniques are carried out in elevation dynamics and azimuth dynamics are

kept at zero. The approximate feedback linearization of elevation dynamics is performed

by means of over parameterization of elevation model. After the linearization, LQR

robust control algorithm has been implemented to deliver desired results. In the second

attempt, the authors have used input output linearization technique to linearize elevation

dynamics; it was presented as the input state linearization was not suitable when the

velocity of the rotors were next to zero, since the control law saturated. The fore

discussed techniques have presented tracking control of the helicopter elevation dynamics

and coupling effect was ignored as the azimuth dynamics were kept at zero.

3.1.4 Time Optimal and Robust control

Te-Wei et al [12] proposed time optimal control for helicopter model. The MIMO

system is first decomposed in two SISO systems and coupling effect is taken as

disturbances or change of system parameters. For each SISO system optimal control has

been designed which can tolerate 50% changes in the system parameters; however the

___________________________________- 32 -_______________________________

results show good tracking response but the coupling issue is not addressed in simulation

results. Jun et al [13] presented robust stabilization and H∞ control for class of uncertain

systems. The quadratic stabilizing controllers for uncertain systems are designed by

solving standard H∞ control problem. This method was verified by implementing it on

helicopter model. The analytical analysis of the algorithm after implementing it on

helicopter model tells that tracking has been achieved but the coupling is unaddressed in

this effort. M.Lopez et al [14] suggested H∞ controller for helicopter dynamics. First

feedback linearization was used for decoupling the inputs and outputs, and then the

system was indentified at higher frequencies, as the relative uncertainty increases at

higher frequencies. The controller was designed for the system identified at higher

frequencies which was unable to cater coupling in the final results as shown in the results

given in the paper. M.Lopez et al [15] delivered the non linear H∞ approach for handling

the coupling taken as disturbance. This approach considers a nonlinear H∞ disturbance

rejection procedure on the reduced dynamics of the rotors, which includes integral terms

on the tracking error to cope with persistent disturbances. The resulting controller

exhibits the structure of non-linear PID with time varying constants according to system

dynamics. The implementation of this controller on the model shows remarkable decrease

in coupling caused by vertical plane dynamics on horizontal plane dynamics along with

good tracking results.

3.1.5 Sliding Mode Control

In sliding-mode control [6] design a hyper-plane is defined as a sliding-surface.

This design approach comprises of two stages; first is the reaching phase and second is

the sliding phase. In the reaching phase, states are driven to a stable manifold by the help

of appropriate equivalent control law and in the sliding phase states slide to an

equilibrium point. One advantage of this design approach is that the effect of nonlinear

terms which may be construed as a disturbance or uncertainty in the nominal plant has

been completely rejected. Another benefit accruing from this situation is that the system

is forced to behave as a reduced order system; this guarantees absence of overshoot while

attempting to regulate the system from an arbitrary initial condition to the designed

equilibrium point. Koudela et al [22] designed sliding mode control for the regulation

___________________________________- 33 -_______________________________

control of helicopter angular positions in vertical and horizontal planes. The sliding mode

controller delivers excellent regulation results, after the authors resolved the chattering

issue for the implementation by introducing the saturation and hyperbolic tangent

function which smooth out the chattering effect. The control law used for the regulation

is derived from the sliding surface which includes the sum of proportional, derivation and

integral components of the error dynamics as shown in Eq. (3.3)

1 1 1 1 121 1 0

2 2 3 3 322 2 0

2 1

2 1

t

t

u e e e dtT T

u e e e dtT T

(3.3)

The results shown in are [22] are acceptable but still authors didn’t considered coupling

effects on the performance parameters. Gwo R. Yu et al [16] considered sliding mode

control of helicopter model via LQR. The LQR was first applied to control the elevation

and azimuth dynamics and then sliding mode controller is employed to guarantee the

robustness. The results demonstrated are optimal in performance and robust but the

coupling problem is still visible in the results displayed. J.P. Su et al [23] designed

procedure that involves primarily finding an ideal inverse complementary sliding mode

control law for the mechanical subsystem with asserted good tracking performance.

Then, a terminal sliding mode control law is derived for the electrical subsystem to

diminish the error arises from the deviation of the practical inverse control from the idea

of inverse control for the mechanical subsystem

3.1.6 Higher Order sliding mode (HOSM) control

Higher order sliding mode control [24], [25], [26]is a recent approach that allows

the removal of all standard sliding mode restrictions, while preserving the main sliding

mode feature and improving its accuracy. Traditional SMC has some intrinsic problems,

such as discontinuous control that often yields chattering [6]. To cope with problem and

achieve higher accuracy, HOSM is proposed [24], [25]. Obviously kth order HOSM

stabilizes the sliding variable at zero as well as its derivatives. On the other hand, since

the high-frequency switching is hidden in the higher derivative of the sliding variable, the

___________________________________- 34 -_______________________________

effect of chattering will be reduced. In other words, HOSM has two important features

that make it a better choice in designing the controller. It improves the accuracy of the

design, which is a very important issue, and may provide a continuous control. This

thesis also contributes towards the 2-SMC design for helicopter model. More details can

be found in Section 5.3.

3.2 Decoupling Techniques

Decoupling techniques involves a procedure which helps us to overcome couplings

residing in the system dynamics. This cross-couplings effect desired performance of the

system. Decoupling in control theory can be achieved implicitly by declaring coupling as

disturbance and handle it with robust controller or explicitly by introducing decouplers

which negate the couplings affects.

The coupling in the helicopter model has discussed in detail in Section 2.3 and

Section 2.4. The control techniques that have been implemented on the model have the

objective to extract the tracking performance from the helicopter model without handling

the coupling effects on the system dynamics. Therefore in order to get maximum tracking

performance and to overcome the undesirable coupling dynamics several decoupling

techniques have been explored. All of the following discussed techniques have been

attempted to decouple our system but few of the techniques could deliver desired results.

Other algorithms were unsuccessful to deliver results at certain stage.

3.2.1 Multi Variable Decouple control

Another popular approach for dealing with control loop interactions is to design

on interacting or decoupling control schemes. Here, the objective is to eliminate

completely the effects of loop interactions. This is achieved via the specification of

compensation networks known as decouplers. Essentially, the role of decouplers is to

decompose a multivariable process into a series of independent single-loop sub-systems.

If such a situation can be achieved, then complete or ideal decoupling occurs and the

multivariable process can be controlled using independent loop controllers [28]. The

Figure 3-1 shows the general decoupling control structure.

___________________________________- 35 -_______________________________

Figure 3-1: General Structure to decouple the system

The idea discussed can be implemented in the following two techniques

3.2.1.1 Boksenbom & Hood Decoupling Technique

The Boksenbom and Hood technique [28] takes the decoupler as shown in Figure

3-2. The system will be decoupled if the off diagonal terms are forced to zero and only

diagonal terms are there in the pseudo plant.

Figure 3-2: Decoupling control system (Boksenbom and Hood)

This can be achieved as

*1 2( , )cGG diag q q (3.4)

The matrix product of the two multivariable transfer functions will as

___________________________________- 36 -_______________________________

* *

11 12 1,11 ,12** *

21 22 2,21 ,22

0

0c c

cc c

G G qG GGG

G G qG G

(3.5)

* * * *

111 ,11 12 ,21 11 ,12 12 ,22** * * *

221 ,11 22 ,21 21 ,12 22 ,22

0

0c c c c

cc c c c

qG G G G G G G GGG

qG G G G G G G G

(3.6)

From Eq. (3.6) we can say

* *1 11 ,11 12 ,21c cq G G G G (3.7)

* *11 ,12 12 ,220 c cG G G G (3.8)

* *21 ,11 22 ,210 c cG G G G (3.9)

* *2 21 ,12 22 ,22c cq G G G G (3.10)

The diagonal component of *cG are the simple controllers designed for the outputs but the

off diagonal terms of *cG can be calculated by Eq.(3.8) and Eq. (3.9) respectively as

*

12 ,22*,12

11

cc

G GG

G

(3.11)

*

21 ,11*,21

22

cc

G GG

G (3.12)

Now the every single is described in the forward path, finally we can say that our over all

system is decoupled for servo control problem as online tuning of the controllers will

change the off diagonal to be recalculated to avoid this retuning we will discuss other

scheme.

3.2.1.2 Zalkin & Lyben Decoupling Technique

The second scheme for decoupling multi variable systems is shown in Figure 3-3,

which is known as Zalkin and Lyben decoupling technique [28]. Here, in addition to the

decoupling network, there are two extra blocks which represent the forward path

controllers. In contrast to the previous strategy, the decoupling network forms the

secondary post-compensation block and allows more flexibility in the implementation

and commissioning of the non-interacting control scheme.

The decoupler values in the forward can be found by using linear algebra

techniques as

___________________________________- 37 -_______________________________

*1 2( , )cX GG diag x x (3.13)

* 1cG G X (3.14)

Where

22 121

21 1111 22 12 21

1 G GG

G GG G G G

(3.15)

Since 1 2( , )X diag x x , then

22 1 12 2* 1

21 1 11 211 22 12 21

1c

G x G xG G X

G x G xG G G G

(3.16)

The simplest form of this decoupling matrix has unity diagonal elements, i.e.

* *,11 ,22 1c cG G (3.17)

This leads to the off diagonal terms as

* 12,12

11c

GG

G

(3.18)

* 21,21

22c

GG

G

(3.19)

The above equations show that with this method, the decoupling elements are

independent of the forward path controllers. On-line tuning of the controllers, therefore,

does not require redesign of the decoupling elements; controller modes may be changed

say from PI to PID and either of the forward path controllers may be placed in manual

without loss of decoupling. Note also, that decoupling occurs between the forward path

control signals and the process outputs, and not between set-points and process outputs.

This technique is therefore not restricted to the servo problem [7].

___________________________________- 38 -_______________________________

Figure 3-3: Non-interacting decoupling control structure (Zalkind &Luyben)

The above discussed techniques are not applicable on the helicopter model as they

require access to each and every intermediate state of the system however in our system

we have only access to the two inputs and two outputs of the system, which restricts the

application of these techniques to helicopter model.

3.2.2 State Space Approach for Decoupling

Consider the linear system

x Ax Bu

y Cx Du

(3.20)

Where nx R , , mu y R , n nA R , n mB R , m nC R , m mD R . The transfer function of

system in Eq. (3.20) can be written as

1( ) ( )G s C sI A B D (3.21)

The system is said to be completely decoupled if G(s) is diagonal matrix and non-

singular. In order to achieve such a matrix from state-variable feedback control the

following two techniques are considered

3.2.2.1 Static Decoupling

A system is said to be statically decoupled if it is stable and its static gain matrix

G(0) is diagonal and non-singular. The control law for system in Eq. (3.20) can be taken

as

___________________________________- 39 -_______________________________

u Kx Fr (3.22)

Such that the close loop transfer function

1( ) [( )( ) ]H s C DK sI A BK B D F (3.23)

The static decoupling problem by state feedback is solvable if and only if (A,B) is

stabilizable and rank of A B

n mC D

where ‘n’ is the number of states and ‘m’ is the

number of inputs. Assuming these conditions are satisfied the problem is solvable by

proceeding with the following steps.

i. Design ‘K’ which ensures the roots of ( )sI A BK in stable region.

ii. Obtain F as 1 1[( )( ) ]F C DK sI A BK B D

iii. Design control law as given in Eq.(3.22).

This technique has been implemented in [7] where A,B,C,D are given in Eq. (2.29) and

more details of the technique can be found in [29].

3.2.2.2 Dynamic Decoupling

The system in Eq.(3.20) is said to be dynamically decoupled by the control law in

Eq. (3.22) if and only if the matrix B* exist and it is non-singular. If this is the case, by

choosing

1( *)F B (3.24)

1( *) *K B C (3.25)

The resultant feedback system has the transfer function in Eq.(3.23) as diagonal matrix.

Where

01

1* 2

1

T

T

T mm

c A B

c A BB

c A B

(3.26)

and

___________________________________- 40 -_______________________________

1

2

1

* 2

m

T

T

Tm

c A

c AC

c A

(3.27)

and

1

2

T

T

Tm

c

cC

c

(3.28)

This technique can be implemented on our system by taking A,B,C,D as given in Eq.

(2.29). More details about this technique can be found in [29].

3.2.3 Near Decoupling Techniques

Several approaches for decoupling control have been discussed in the previous

sections. Robust stability and plant uncertainties are not addressed explicitly in design. It

is noted that exact decoupling may not be necessary even in the nominal case for many

applications and usually impossible in the uncertain case. What is really needed instead is

to limit the loop interaction to an acceptable level. In this section the concept of near-

decoupling and development, an approach to near-decoupling controller design for both

nominal and uncertain systems is discussed. Two techniques are discussed here in brief.

Details and rest of the techniques can be found in [29].

Consider the system defined in Eq. (3.20) where the B,C,D can be split into ‘p’ partitions.

1 11 12 1

2 21 22 2

1 2

1 2

... , ,

p

p

p

p p p pp

C D D D

C D D DB B B B C D

C D D D

(3.29)

Correspondingly, the closed loop transfer function defined in Eq.(3.23) can also split as

___________________________________- 41 -_______________________________

11 12 1

21 22 2

1 2

( ) ( ) ( )

( ) ( ) ( )( )

( ) ( ) ( )

p

p

p p pp

H s H s H s

H s H s H sH s

H s H s H s

(3.30)

The described system is said to be nearly decoupled for input and output pairs ( , )i iu y if

the system (3.20) is stable and

,( ( )) {1,2..... }, [ , ]cl ii iH j i p

, , , {1, 2,..... },cl ijH i j p i j

holds for given numbers of 0i , 1, 2.... 0i p and

3.2.3.1 Near-Decoupling: State Feedback

Consider the system defined by Eq.(3.29), this system is said to be near state-

feedback decouplable if there exists matrices 0Q and F such that the following LMIs

hold

2

( )0

( )

{1, 2..... }

T T T T T Ti i i ii

T T T T T T Ti ii i i i ii ii

QA AQ F B BF B QC F D D

B D QC F D I D D

i m

(3.31)

0

( )

, {1,2..... },

T T T T T Tj i i

T Tj ij

T T T Ti i ij

QA AQ F B BF B QC F D

B I D

QC F D D I

i j m i j

(3.32)

The state feedback law is given by

u Kx (3.33)

1K FQ (3.34)

And the close loop performance indices satisfy

___________________________________- 42 -_______________________________

,( ( )) {1,2..... }, [ , ]cl ii iH j i p (3.35)

, , , {1, 2,..... },cl ijH i j p i j (3.36)

The robust near decoupling and output feedback decoupling algorithms can be found with

detail in [29].

3.2.4 Hadamard Weights in LSDP for Robust Decoupling

Loop shaping procedure involves the reshaping of system dynamics by

introducing additional transfer functions i.e. weights in the loop. Loop shaping design

procedure (LSDP) allows the user to modify the system by increasing the system gains at

lower frequencies, decreasing the gain at higher frequencies and making the slope around

1 at cross-over frequency, thus making the system to work according to user desires.

These weights later on help us in formulating H∞ controller and these weights are merged

with controller in the implementation. The weights are dependent on the system inputs

and outputs for example if we have two inputs and two outputs system the weight

dimensions will be [2 2] . Normally, these weights are introduced to the system through

matrix multiplication which relates each component of the weights to each component of

plant dynamics as

11 12

21 22

W WW

W W

(3.37)

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

21 22 21 22 21 11 22 21 21 11 22 21

G G W W G W G W G W G WG W

G G W W G W G W G W G W

(3.38)

Even, if we chose W as diagonal matrix, the matrix multiplication procedure will relate

each weight term with each system dynamics. To avoid this relation and to introduce

decoupling in the system through these weights, F.Van Diggelen and K. Glover [5]

proposed element by element weighting which instead to relating weight components

shown in Eq. (3.38) relates the weight components element by element as

11 12 11 12 11 11 12 12

21 22 21 22 21 21 22 22

G G W W G W G WW G

G G W W G W G W

(3.39)

This procedure gives the liberty to handle each component of our plant independently.

Coupling can be catered by this technique by introducing off diagonal terms as zero or

___________________________________- 43 -_______________________________

very small positive numbers and diagonal terms can be introduced to reshape the plant

dynamics as required. This technique has been applied on helicopter model for

decoupling using LSDP which is explained in Section 4.3 with the design procedure in

detail and its results.

3.3 Conclusion

In view of the above literature survey, the already existing controllers to meet control

demands have been explored in detail. The coupling issue has been addressed by both

methods, one is to handle the coupling directly by introducing decoupling technique and

the second approach is to declare coupling as disturbance and design such a robust

controller that will cope up with desired performance even in the presence of coupling.

The suggested controller design procedures are discussed in next chapters.

___________________________________- 44 -_______________________________

Chapter 4

H∞

Controller Design

Chapter Objectives

Mixed Sensitivity procedure

Loop Shaping Design procedure

Hadamard Weights

Controllers Evaluations

___________________________________- 45 -_______________________________

A control system is robust if it remains stable and achieves certain performance

criteria in the presence of possible uncertainties. The robust design is to find a controller,

for a given system, such that the closed-loop system is robust. The H∞ optimization

approach, being developed in the last two decades has been shown to be an effective and

efficient robust design method for linear, time invariant control systems. Various robust

stability considerations and nominal performance requirements were formulated as a

minimization problem of the infinitive norm of a closed-loop transfer function matrix.

Hence H∞ optimization approach solves, in general, the robust stabilization problems and

nominal performance designs [17]. In this chapter, we shall discuss formulation of a

robust design problem for our helicopter model.

4.1 General Control Problem Formulation for H∞ Control

There are many ways in which feedback design problems can be cast as H∞

optimization problems. It is very useful therefore to have a standard problem formulation

[19] shown in Figure 4-1 into which any particular problem may be manipulated.

Figure 4-1 General control configuration for H∞ control

The system described in Figure 4-1 can be written as

11 12

21 22

( ) ( )( )

( ) ( )

P s P sz w wP s

P s P sv u u

(4.1)

The state space realization of the generalized plant [19] can be given as

___________________________________- 46 -_______________________________

1 2

1 11 12

2 21 22

sA B B

P C D D

C D D

(4.2)

The input signals are ‘u’ the control variable and ‘w’ the exogenous inputs including the

disturbances and commands. The outputs include ‘v’ the measured output and exogenous

outputs ‘z’ which includes error or the signals which are to be minimized. The closed

loop transfer function from ‘w’ to ‘z’ can be formulated as

( , )lz F P K w (4.3)

Where

111 12 22 21( , ) ( )lF P K P P K I P K P

The H∞ control involves the minimization of infinity norm of ( , )lF P K which means to

find the stabilizing controllers K which minimizes

( , ) max ( ( , )( ))l lF P K F P K j

(4.4)

This can interpreted as the performance parameter which involves minimization of

maximum singular value of ( , )lF P K [19].Practically, the sub optimal controller for H∞

problem is simpler to design therefore, if we assume min the minimum value of

( , )lF P K

over all stabilizing controllers K, then the sub-optimal control problem is

given by a min and the sub-optimal problem can be formulated as

( , )lF P K (4.5)

This can be efficiently solved using the algorithm discussed in [4].

4.2 Mixed sensitivity procedure

Mixed sensitivity [19] is the name given to transfer function shaping problems in

which the sensitivity function 1–S I GK

is shaped along with one or more other

closed loop transfer functions such as KS or the complementary sensitivity

function –T I S .For the given closed loop configuration shown in Figure 4-2 in

which ‘d’ is disturbance, ‘n’ is noise in the measure sensors and ‘r’ is the reference input.

In order to impose performance and robustness conditions the following close loop

___________________________________- 47 -_______________________________

relationships between output and the error, the control signal and the generalized

disturbances acting on the system must be taken in account.

o o oy T r S d T n (4.6)

( )o oe S r d T n (4.7)

( )ou KS r n d (4.8)

Where oS and oT are the output sensitivity function and complementary sensitivity

function respectively.

From the above equations, we can conclude that shaping oT is desirable for

tracking problems, noise attenuation and robust stability with respect to multiplicative

uncertainty. On the other hand, shaping oS will allow to control performance of the

system. In addition the control signal should be attenuated along the frequency.

Figure 4-2: One degree of freedom configuration

The stated problem can be solved out by means of mixed sensitivity problem S/KS [[4]].

The optimal control [19] is formulated as finding stabilizing controller K(s) such that

expression in Eq.(4.9) is minimized.

___________________________________- 48 -_______________________________

1

2

( ) ( )

( ) ( ) ( )o

o

W s S s

W s K s S s

(4.9)

In the above expression, the W1 and W2 are the weighting function, which are employed

to shape the close loop transfer function oS and oKS respectively. Figure 4-3 shows the

new augmented plant configuration.

Figure 4-3: S/KS mixed sensitivity Plant configuration for tracking control

After we have formulated the control problem the generalized plant for helicopter model

can be written as

11 12

21 22

P PP

P P

(4.10)

Where

111 0

WP

112

2

W GP

W G

21P I 22P G

Where the values for transfer function ‘G(s)’ can be taken from Eq. (2.30) and W1 and

W2 are formulated in the coming sections.

4.2.1 Choice of Weights and controller design

The weighting function W1 is used to impose performance specifications to the

system. According to Eq. (4.6) it is necessary that oS should be having smaller gains at

___________________________________- 49 -_______________________________

lower frequency to reject disturbances at the output and to reduce error tracking.

Therefore to achieve this aim singular values should have high gain at lower frequencies.

Figure 4-4 shows the selected W1 for our system.

10

-310

-210

-110

010

110

210

3-100

-90

-80

-70

-60

-50

-40

-30

-20Singular Values

Frequency (rad/sec)

Sin

gul

ar V

alue

s (

dB)

Figure 4-4 W1, Weighting function for S

The objective of W2 is to reduce overshoots of the temporal response without changing

too much rise time. W2 is typically scalar high pass filter with the cross over frequency

approximately equal to required bandwidth. Figure 4-5 shows the selected W2 for mixed

sensitivity H∞ solution.

10-2

10-1

100

101

102

103

104

105

106

107

108

-180

-160

-140

-120

-100

-80

-60Singular Values

Frequency (rad/sec)

Sin

gul

ar V

alue

s (

dB)

Figure 4-5 W2, Weighting function for KS

___________________________________- 50 -_______________________________

The H∞ controller has been designed using algorithm implemented in [19], which yields a

suboptimal controller. The controller designed delivered the results discussed in next

section.

4.2.2 Simulation Results

The suboptimal controller applied for tracking control on helicopter model

delivered the step responses shown in Figure 4-6. Although, the tracking has been

achieved in 2 seconds but still the coupling effect can be seen in the start for half a

second which serves as barrier to achieve the desired controlled response. The respective

control effort is shown in Figure 4-7.

0

0.5

1

1.5From: In(1)

To: O

ut(1

)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

To:

Out

(2)

From: In(2)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Step response

Time (sec)

Ang

le (

deg)

Figure 4-6 Step response (Mixed sensitivity)

___________________________________- 51 -_______________________________

-5

0

5

10

15From: In(1)

To: O

ut(1

)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-0.01

-0.005

0

0.005

0.01

0.015T

o: O

ut(2

)

From: In(2)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Control input

Time (sec)

Vol

tage

(V

)

Figure 4-7 Control Signal (Mixed sensitivity)

4.3 Loop shaping design procedure (LSDP)

The second design procedure for H∞ is LSDP. Loop shaping is essentially a two

stage design procedure. First, the open-loop plant is augmented by pre and post-

compensator to give a desired shape to the singular values of the open-loop frequency

response. Then the resulting shaped plant is robustly stabilized with respect to co-prime

factor uncertainty using H∞ optimization [19].

For loop shaping design procedure, the stabilization of the plant G which has a

normalized left co-prime factorization [19].

1G M N (4.11)

Figure 4-8: H∞ Robust Stabilization Problem

The perturbed plant is shown in Figure 4-8. The objective of the robust stabilization is to

stabilize not only the nominal model ‘G’ but also the perturbed model given as

___________________________________- 52 -_______________________________

1( ) ( )G M M N N (4.12)

[ ]N M

(4.13)

The maximum value of can be obtained from

1

21 1/22min max

1 (1 ( ))H

N M XZ

(4.14)

Where .H

denotes Hankel norm, max denotes maximum stability margin [4 and

denotes the spectral radius. For a minimal state space realization (A,B,C,D) of G, Z is the

unique positive definite solution to the algebraic Riccati equation [19].

1 1 1 1( ) ( ) 0T T T T TA BS D C Z Z A BS D C ZC R CZ BS B (4.15)

Where

,T TR I DD S I D D

And X is the unique positive definite solution of the following algebraic Riccati equation.

1 1 1 1( ) ( ) 0T T T TA BS D C TX X A BS D C C R C XBS B X (4.16)

The plant should be strictly proper in nature to satisfy the above equations. The

controller is designed by the procedure defined in [19] and the controller K which

guarantees

1 1( )K

I GK MI

(4.17)

will give the optimal/suboptimal controller for loop shaping design.

4.3.1 Choice of Weights and Controller Design

The singular values of the plant are shown in Figure 2-13. To align the singular

values in the frequency range of interest, pre-compensator and post-compensator are

design requirements. Specifications on the closed loop system are indirectly imposed by

selecting the weighting function W1 and W2 and appropriate adjusting the open-loop

frequency response [19]. High gain at low frequencies of the shaped plant implies good

tracking ability. Similarly low gain at higher frequencies implies better noise rejection.

The slope of singular values at crossover frequency also affects the tracking; if the slope

is not ‘1’ then the tracking performance is not guaranteed.

___________________________________- 53 -_______________________________

The controller is implemented in the way as shown in the Figure 4-9 with pre and

post weighting functions in the series connection. In helicopter model, we have designed

pre weight function and chose post weight function to be identity, as we have simple

digital encoder as sensor and we cannot modify the sensor dynamics like we desire. The

weights are merged with the designed controller at the time of implementation. The new

controller comprises of both the weights and the controller obtained from the procedure

described in [19].

Figure 4-9 LSDP implementation

The pre weight function W1 designed for shaping the helicopter dynamics is

2

1

2

2

2

2.2 0.4

1000 1

2.2

0

00.4

1000 1

s s

s s

s s

s

W

s

(4.18)

The weights designing procedure described in Eq. (3.38) is the standard practice adopted

in loop shape design procedure. This weight allows us to modify model dynamics by

improving its tracking by increasing the gains at lower frequencies and changing the

slope at cross over frequency to ‘1’ as shown in Figure 4-10.

___________________________________- 54 -_______________________________

10-3

10-2

10-1

100

101

102

103

-200

-150

-100

-50

0

50

100

150

Singular Values

Frequency (rad/sec)

Sin

gul

ar V

alue

s (

dB)

Gs

G

Figure 4-10: Modified Singular Values using Traditional Weights in LSDP

The re-shaped model of helicopter can be utilized for controller synthesis as described in

[19]. In this practice the coupling is term as disturbance, the controller designed should

be so robust that it handles the coupling very efficiently.

4.3.2 Hadamard weight

Hadamard weights are used in loop shaping design procedure for element by

element weighting in multivariable system which helps in decoupling the plant behavior.

The procedure of integrating weight dynamics to change the system dynamics described

in Eq.(3.39). The weight transfer function used for helicopter model is

21

2

2

2

1980 4356 792

1000 1

1920 4224 768

10

0

000 1

s s

s s

s s

s

W

s

(4.19)

This pre weight function modified the plant singular values as shown in Figure 4-11.

___________________________________- 55 -_______________________________

10-3

10-2

10-1

100

101

102

103

-150

-100

-50

0

50

100

150

Singular Values

Frequency (rad/sec)

Sin

gul

ar V

alue

s (

dB)

Gs

G

Figure 4-11: Modified Singular Values with Hadamard Weights in LSDP

The modified helicopter dynamics are used for controller synthesis as in the previous

case. The element by element weighting gives us the liberty to handle the coupling

directly by making the off diagonal elements of transfer function in Eq. (2.30) equal to

zero by the designed weight, which ensures the decoupling in the system; however this

affects the tracking performance in the consequence. So, we have to trade off between the

decoupling and tracking performance to get optimum results.

4.3.3 Simulations Results

In the first phase, Matlab based simulations are carried out. The sub optimal

controller with min 2.5914 , designed using weighting procedure as described in Eq.

(3.38) yields acceptable results as shown in Figure 4-12 and control effort required to

obtain these results is shown in Figure 4-13. It can be observed in results that the desired

elevation and azimuth angles are achieved in finite time but in the same time coupling is

also noticeable for few seconds in the start of simulation. So to reduce the coupling we

will have to increase the robustness, the redesigning of the weight will provide more

robustness but the performance will deteriorate. So the designed weight in Eq.(4.18)

delivers the optimum results which reduces coupling to a certain level at the same time

generates the optimal performance results.

___________________________________- 56 -_______________________________

0

5

5From: In(1)

0 5 10 15 205

5

0

5

5

From: In(2)

0 5 10 15

Step response

Figure 4-12: Step response of the system with Traditional Weighted H∞ controller

0

0

0

0

0From: In(1)

0 5 10 15 200

0

0

0

0

From: In(2)

0 5 10 15

Control input

Figure 4-13: Control Effort of Traditional Weighted H∞ controller

Hadamard weights are also examined first on Matlab based simulations. The sub

optimal controller with min 2.3992 designed using the weighting procedure described

in Eq.(3.39). yields results shown in Figure 4-14 and the control effort is shown in Figure

4-15. It can be observed in the results that elevation and azimuth angles achieve the

desired value with in a finite time but the coupling is totally eliminated from system

dynamics. These simulations delivered the ideal results and the controller is now ready

for implementation on actual helicopter model.

___________________________________- 57 -_______________________________

-1

-0.5

0

0.5

1

1.5From: In(1)

To: O

ut(1

)

0 2 4 6 8 10 12 14 16 18 20-1

-0.5

0

0.5

1

1.5T

o: O

ut(2

)

From: In(2)

0 2 4 6 8 10 12 14 16 18 20

Step response

Time (sec)

Ang

le (

deg)

Figure 4-14: Step response of the system with Hadamard Weighted H∞ controller

-100

-50

0

50

100

150

200

250

300From: In(1)

To: O

ut(1

)

0 2 4 6 8 10 12 14 16 18 20-100

-50

0

50

100

150

200

250

300

To:

Out

(2)

From: In(2)

0 2 4 6 8 10 12 14 16 18 20

Control input

Time (sec)

Vol

tage

(V

)

Figure 4-15: Control Effort of Hadamard Weighted H∞ controller

4.4 Experimental test Results

After the simulations have been carried out, the controller is now implemented at

the actual model. The weights designed in Eq. (4.18) and Eq. (4.19) have been attained

after number of attempts because there were many other weights which delivered ideal

simulation but when it comes implementation the controllers designed on those weights

failed to deliver desired results. The rid tests are carried out to verify various results.

Figure 4-16 shows the step response of helicopter model to acquire equilibrium position,

when implemented with H∞ controller having high robustness i.e. min 2.5914 and

___________________________________- 58 -_______________________________

Traditional weighting technique is employed. Figure 4-17 gives the idea about the control

effort. It can be observed that rise time is almost 15 to 17 seconds and there is no over

shoot in the response. However when the system settles down at equilibrium state and

second degree of freedom is introduced at 32 second, the coupling is visible both in

elevation and azimuth and with in 5 seconds the controller copes up with coupling and

maintains the equilibrium position. The controller effort to maintain the equilibrium state

can be seen in Figure 4-17 after 32 seconds. Figure 4-18 shows controller response in

when operated in nonlinear domain.

10 15 20 25 30 35 40 45-10

-5

0

5

10

15

20

25

Ele

vatio

n (d

eg)

H inf controller (Normal Weights)

10 15 20 25 30 35 40 45-20

-10

0

10

20

Time (sec)

Azi

mut

h (d

eg)

Figure 4-16: Actual System response with Traditional Weighted H∞ controller, when exposed to

coupling at 32 sec.

___________________________________- 59 -_______________________________

10 15 20 25 30 35 40 450

0.2

0.4

0.6

0.8

1

Ele

vatio

n

Control Effort (Normal Weights)

10 15 20 25 30 35 40 45-1

-0.5

0

0.5

1

Time (sec)

Azi

mut

h

Figure 4-17: Traditional Weighted H∞ controller effort to over come coupling effects introduced at 32

sec in azimuth plane.

5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

Ele

vatio

n (d

eg)

Output States Response Initialized in Nonlinear Domain

5 10 15 20 25 30 35 40 45 50-40

-20

0

20

40

Time (sec)

Azi

mut

h (d

eg)

Figure 4-18: Actual System response with Traditional weighted H∞ controller to attain equilibrium

position when initialized in nonlinear range

___________________________________- 60 -_______________________________

5 10 15 20 25 30 35 40 45 500

0.2

0.4

0.6

0.8

1Controller Effort

Ele

vatio

n

5 10 15 20 25 30 35 40 45 50-1

-0.5

0

0.5

1

1.5

2

Time (sec)

Azi

mut

h

Figure 4-19: Traditional Weighted H∞ controller effort to acquire equilibrium position when actual

system was initialized in nonlinear range.

The rise time in this case is very high which makes the systems response sluggish,

however if we reduce the robustness of the controller to achieve better performance, the

H∞ controller with min 3.914 implemented on the helicopter yields the results shown

in Figure 4-20 which shows that the rise time has been significantly decreased but

overshoot can be observed. The second degree of freedom is introduced at 31 seconds,

the controller is unable to handle coupling and in azimuth we can see that the helicopter

reaches -40 degrees and then after 20 seconds the controller is able to transport back

helicopter to equilibrium position.

10 20 30 40 50 60 700

5

10

15

20

25

Ele

vatio

n (d

eg)

Step resposne (Normal Wt)

10 20 30 40 50 60 70

-40

-20

0

20

40

Time (sec)

Azi

mut

h (d

eg)

Figure 4-20: Traditional Weighted H∞ controller response to coupling when robustness was

compromised with performance

___________________________________- 61 -_______________________________

In the second attempt the H∞ with Hadamard weighting technique is implemented on the

helicopter model. Figure 4-21 shows the response given by the helicopter to reach at

equilibrium level and Figure 4-22 shows its respective controller effort. It can be clearly

observed that the rise time is around 5 seconds and there is no over shoot. At the time

when second degree of freedom i.e. azimuth is introduced, the controller successfully

copes up with coupling and with in 2 to 3 seconds it assists the helicopter to gain its

equilibrium position. In the introduction of helicopter dynamics it was said that this

system is minimum phase system and its zero dynamics are unstable. The effect of right

half plane zeros make the helicopter to respond opposite to the command issued which

produces undershoots as a result. This typical response of minimum phase systems can be

seen in Figure 4-23. The H∞ controller with Hadamard weighting technique is employed

for multi step command issued to reach the equilibrium position which delivers the

response shown in Figure 4-23.

5 10 15 20 25 30 35 40 45-10

-5

0

5

10

15

20

25

Ele

vatio

n (d

eg)

H inf Control (Hadamard weights)

5 10 15 20 25 30 35 40 45-40

-20

0

20

40

Time (sec)

Azi

mtu

h (d

eg)

Figure 4-21: Actual System response with Hadamard Weighted H∞ controller, when exposed to

coupling at 32 sec

___________________________________- 62 -_______________________________

5 10 15 20 25 30 35 40 450

0.2

0.4

0.6

0.8

1

Con

trol

Eff

ort

(Ele

vatio

n)

Control effort (Hadamard Weights)

5 10 15 20 25 30 35 40 45-1

-0.5

0

0.5

1

Time (sec)

Con

trol

Eff

ort

(Azi

mut

h)

Figure 4-22: Hadamard Weighted H∞ controller effort to over come coupling effects introduced at 32

sec in azimuth plane

5 10 15 20 25 30 35 40 45 50 55 600

5

10

15

20

25

Time

Ele

vatio

n (d

eg)

Multi Step Response (Hadamard Wt)

Figure 4-23: Undershoots in the multi step response (Minimum phase behavior) of helicopter system

with 2-DOF

4.5 Performance Evaluation

The analytical performance evaluation of controller based on the output can be

carried out by using performance indices [30]. A performance index is the quantitative

measure of the performance of the system. The four performance indices are defined

based on error e y r dynamics.

___________________________________- 63 -_______________________________

2

0

T

ISE e dt (4.20)

Integral of square of error (ISE) is total sum of square of error that is mostly used to

evaluate practical implementations. Similarly, to reduce the contribution of large initial

error total sum of time multiple of error square (ITSE) or total sum of time multiple of

absolute error (ITAE) are used.

2

0

T

ITSE te dt (4.21)

0

T

ITAE t edt (4.22)

Total sum of absolute of error (IAE) is generally used for the evaluation of computational

simulations

0

T

IAE e dt (4.23)

These performances indices give the idea about the performance calculated from the

error, the more the indices value the more will be low graded performance. The Table 4-1

shows the numeric values of the four performance indices of both horizontal and vertical

plane dynamics with both controllers. Table 4-1 shows indices of H∞ with Traditional

weights and with Hadamard weights. These performance indices give us the idea that

elevation dynamics are well controlled by the H∞ with Hadamard, however the

decoupling at the cost of performance [5] is well depicted as the azimuth dynamics loose

performance.

Performance

Indices

Elevation

(Traditional

Wt)

Elevation

(Hadamard Wt)

Azimuth

(Traditional

Wt)

Azimuth

(Hadamard Wt)

ISE 46.14 24.85 0.4031 3.127

ITSE 287.7 82.97 14.59 107.7

IAE 4.387 2.334 0.3114 0.8696

ITAE 44.92 18.01 11.61 31.51

Table 4-1: Performance Indices comparison between Traditionally and Hadamard Weighted H∞

___________________________________- 64 -_______________________________

4.6 Conclusion

The simulation results and the experimental tests lead to some very interesting

conclusions. The H∞ controller design with mixed sensitivity procedure is not preferred

due to controller effort which is not as optimized as in LSDP and this technique has not

been explored for decoupling i.e. loop shaping design procedure. Therefore, LSDP

technique is selected for further controller design procedures and implementation.

Loop shaping design procedure provides the liberty to muddle through the

coupling by either ways. In the first attempt, the coupling is taken as disturbance and the

robustness of the controller provides the guarantee to over come its effect but at the cost

of performance which can be seen in the experimental tests results i.e. the more we

reduce the min the more robustness we get but at the same time the response time

increases. However in the LSDP with Hadamard weighting technique optimum

robustness and performance is achieved at the same time. The same conclusion can be

drawn from the controller efforts; the controller in Figure 4-17 has to exert more effort

after the coupling has been introduced as compared to in Figure 4-22 but controller in

vertical plane dynamics has to exert more effort in H∞ control design with Traditional

weighting technique.

The robust control algorithms from linear domain discussed above deliver acceptable

results but the drawbacks of linear range still stick with them like these controllers should

be applied in linear range as the system is linearized around a set point. The linearization

forces to ignore dynamics of the system described non-linearly, which causes to have

loose control over those ignored dynamics. The performance indices showed that vertical

plane dynamics are well handled by H∞ with Hadamard weights and the horizontal plane

dynamics are well handled by H∞ with Traditional weights. So we are not yet having a

single controller that delivers desired results. Therefore we will now attempt for the

robust control algorithms from nonlinear domain to deliver the desired response.

___________________________________- 65 -_______________________________

Chapter 5

Nonlinear

Control Algorithms

Chapter Objectives

Lyapunov Based Control Design

Sliding Mode Control Design

Higher order Sliding Mode Control

Controllers Evaluations

___________________________________- 66 -_______________________________

The shortcomings in control laws generated from linear control guide us to engineer

control laws from nonlinear control algorithms. The nonlinear theory addresses each and

every dynamics of the systems which linear theory fails to address as the linear methods

rely on the key assumption of small range of operation for the linear model to be valid.

The required operating region for twin rotor system is beyond the capacity of linear range

especially in horizontal plane dynamics, which serves as basis to design nonlinear control

for helicopter model. These control algorithms are simpler to design, deeply rooted into

the system physics and are more intuitive than their linear counterparts. The nonlinear

controllers can handle the uncertainties or change in parameter much efficiently which

makes them also robust in nature. Keeping in mind the fore mentioned motivations

various nonlinear control algorithms have been attempted to deliver the desired objective

from the twin rotor system.

5.1 Lyapunov Theory

Lyapunov theory [6] helps us to develop control algorithms from the equations that

give the idea about the energy of the system. The rate of change of energy, if negative

definite, ensures the system stability. Mathematically, we can say that if energy function

is positive definite ( ) 0V x and rate of change of energy function is negative definite

( ) 0V x then the system stability is ensured [6].

Based on this theory we can develop control law from energy equation which will

make helicopter model asymptotically stable. The helicopter model can be decomposed

in elevation and azimuth planes thus allowing us to model two energy functions for the

whole system. Therefore, we can choose the energy function based on state vector in Eq.

(2.16) for elevation dynamics as

2 2 21 2 3

1( ) ( )

2V x x x x (5.1)

And for azimuth dynamics the Lyapunov function can be written as

2 2 24 5 6

1( ) ( )

2V x x x x (5.2)

The respective states are defined in Section 2.2. These energy functions are positive

definite and their negative rate of change will help us to calculate respective control laws.

___________________________________- 67 -_______________________________

The control law for elevation dynamics to ensure its stability from Eq. (5.1) can be

derived as

1 1 2 2 3 3( )V x x x x x x x (5.3)

Plugging Eq.(2.18) and Eq. (2.19) in Eq. (5.3) we will get

21 1 1 2 3 3 1 1 1 1 1 3 2 1 6 2

1 1

1 1( ) ( ( )) ( (( ) sin cos ))g gyroV x x x u x x x a x b x B x T x K u x x

T I (5.4)

The control law in vertical plane dynamics comes out to be

2

2311 2 3 1 1 1 1 1 3 2

1 1 16 2

1

1( [( ) sin ])

( cos )g

gyro

xxu x x a x b x B x T x

x T IK x xT

(5.5)

From Eq. (5.5) we reach at singularity in the control law as motor angular speed 1x and

rate of change of azimuth 6x will be initially at zero.

Similarly for azimuth the control law can be derived as

4 4 5 5 6 6( )V x x x x x x x (5.6)

The control law for horizontal plane dynamics, after plugging Eq.(2.20) and Eq. (2.21),

comes out to be

24 4 2 5 6 6 2 4 2 4 2 6 1 7 2 1

2 2

1 1( ) ( ( )) ( [( ) - - ])V x x x u x x x a x b x B x k x k u

T I (5.7)

2 222 4 5 6 6 2 4 2 4 2 6 1 7 2 1

4 2 2

1 1( ( [( ) - - ]))

Tu x x x x a x b x B x k x k u

x T I (5.8)

Eq.(5.8) leads to the same conclusion as in elevation dynamics case. The singularity due

to side motor speed 4 0x initially forbids applying this control law.

The same conclusions are drawn from the Lyapunov functions defined as

2 22 3

1( ) ( )

2V x x x (5.9)

2 25 6

1( ) ( )

2V x x x (5.10)

Finally, we can conclude that various Lyapunov functions delivered number of control

laws but due to singularities in the design the laws are not applicable for implementation

on the twin rotor system. Moreover, the system is prone to uncertainties and disturbances,

the control laws achieved as a consequence of Lyapunov functions do not offer

___________________________________- 68 -_______________________________

robustness against them, thus formulating the basis to proceed for robust nonlinear

control synthesis procedures.

5.2 Sliding mode control

In control theory, sliding mode control, or SMC, is a form of variable structure

control (VSC). It is a nonlinear control method that alters the dynamics of a nonlinear

system by application of a high-frequency switching control. The state-feedback control

law is not a continuous function of time; it switches from one smooth condition to

another. That is, the structure of the control law changes based on the position of the state

trajectory; hence, sliding mode control is a variable structure control method because it

switches from one smooth control law to another. The multiple control structures are

designed so that trajectories always move toward a switching condition, and so the

ultimate trajectory will not exist entirely within one control structure. Instead, the

ultimate trajectory will slide along defined manifolds. The motion of the system as it

slides along these boundaries is called a sliding mode [6].

Intuitively, sliding mode control uses practically high gain to force the trajectories

of a dynamic system to slide along the restricted sliding mode subspace. Trajectories

from this reduced-order sliding mode have desirable properties (e.g., the system naturally

slides along it until it comes to rest at a desired equilibrium). The main strength of sliding

mode control is its robustness. Because the control can be as simple as a switching

between two states (e.g., "on"/"off" or "forward"/"reverse"), it need not be precise and

will not be sensitive to parameter variations that enter into the control channel.

Additionally, as the control law is not a continuous function, the sliding mode can be

reached in finite time (i.e., better than asymptotic behavior). In particular, because

actuators have delays and other imperfections, the hard sliding-mode-control action can

lead to chatter, energy loss, plant damage, and excitation of unmodeled dynamics.

The design process of sliding-mode controller involves a hyper-plane defined as a

sliding-surface. This design approach comprises of two stages; first is the reaching phase

and second is the sliding phase. In the reaching phase, states are driven to a stable

manifold by the help of appropriate equivalent control law and in the sliding phase states

slide to an equilibrium point. One advantage of this design approach is that the effect of

___________________________________- 69 -_______________________________

nonlinear terms which may be construed as a disturbance or uncertainty in the nominal

plant has been completely rejected. Another benefit accruing from this situation is that

the system is forced to behave as a reduced order system; this guarantees absence of

overshoot while attempting to regulate the system from an arbitrary initial condition to

the designed equilibrium point.

5.2.1 Sliding Surface Design 1

The sliding manifold based on elevation states in Eq. (2.16) can be chosen as

1 3 2( )s x x x (5.11)

The positive definite Lyapunov function for sliding manifold can be chosen as

21

1( )

2V s s (5.12)

The time derivative of the Lyapunov function can be written as

1 1( )V s s s (5.13)

where

1 2 3( )s x x x (5.14)

Plugging the values of Eq.(2.19) in Eq.(5.14), we get

21 3 1 1 1 1 1 3 2 1 6 2

1

1( ) ( (( ) sin cos )g gyros x x a x b x B x T x K u x x

I (5.15)

Putting Eq.(5.15) in Eq. (5.13) we get

21 3 1 1 1 1 1 3 2 1 6 2

1

1( ) ( ( (( ) sin cos ))g gyroV s s x a x b x B x T x K u x x

I (5.16)

From Eq.(5.15) we can derive the equivalent control law equ as

23 1 1 1 1 1 3 2

6 2 1

1 1( ( (( ) sin )

coseq ggyro

u x a x b x B x T xK x x I

(5.17)

And the control law that will ensure the rate of change of Lyapunov function in Eq.(5.12)

to be negative definite can be written as

1 1( )equ u Ksign s (5.18)

Eq.(5.13) that ensures the system stability, comes out to be

___________________________________- 70 -_______________________________

1 1( ) ( )V s s Ksign s (5.19)

The above designed control law poses singularity which serves as an obstacle to apply on

the helicopter model. The rate of change of azimuth ( 6x ) is at zero initially which will

make the equivalent control in Eq.(5.17) as undefined.

Similarly for azimuth dynamics the same practice fails to yield control law as in

Eq.(5.24) the control input is not appearing which helps in calculating the equivalent

control. To access the control input we will have to take another derivative of sliding

surface which is out of scope of first order sliding mode control.

2 6 2 5( )s x x x (5.20)

22

1( )

2V s s (5.21)

2 2( )V s s s (5.22)

2 6 2 5( )s x x x (5.23)

22 2 4 2 4 2 6 1 7 2 1 2 6

2

1( ) ( [( ) - - ])s x a x b x B x k x k u x

I (5.24)

5.2.2 Sliding Surface Design 2

In the second attempt to design the sliding mode control for twin rotor system is

carried out by splitting the MIMO system into two SISO systems, for each SISO system

the sliding manifolds are designed based on their error dynamics defined in Eq. (5.25)

eqE X X (5.25)

where eqX is the vector of desired values of the system states at equilibrium position. The

sliding manifolds in Eq. (5.26) are taken as Hurwitz polynomial of the states defined in

Eq. (2.16)

3 1 21

6 4 52

( , )

( , )

e f e esS

e f e es

(5.26)

Where 1 2 1 1 2 2

4 5 4 4 5 5

( , )

( , )

f e e c e c e

f e e c e c e

___________________________________- 71 -_______________________________

The above system in Eq. (5.26) can be rewritten as

3 1 21

6 4 52

( , )

( , )

e f e es

e f e es

(5.27)

The system in Eq. (5.27) will be stable if 0S and the rate of convergence will be

governed by the manifold dynamics. The Lyapunov function for surfaces defined in Eq.

(5.26) can be written as

21 1

22 2

1

21

2

V s

V s

(5.28)

which are positive definite functions and their time derivative can be written as

1

2

1 1

2 2

V s s

V s s

(5.29)

The equivalent control 1equ for elevation dynamics and 2equ for azimuth dynamics on the

manifold 1 1 1 2 2 3 0s c e c e e and 2 4 4 5 5 6 0s c e c e e respectively can be seen in

Eq. (5.30) and Eq. (5.31).

21 11 1 1 1 1 1 1 3 2 2 3

1 1 1

1[ ( ) ([ (( ) sin )] )]eq g

T cu x a x b x B x T x c x

c T I (5.30)

22 42 4 5 6 2 4 2 4 2 6

4 2 2

1[ (- ) ( (( ) - ))]eq

T cu x c x a x b x B x

c T I (5.31)

The control input vector ‘U’ that will make the system to converge at 0S can be

written as in Eq.(5.32), this control law will ensure the system convergence to sliding

manifold along with robustness against the cross-couplings.

1 1 11

2 2 22

( )

( )eq

eq

u K sign suU

u K sign su

(5.32)

To avoid high frequency switching i.e. chattering, implementation of the control laws

have been performed by employing saturation function ( )sat S defined as

___________________________________- 72 -_______________________________

1 11

1 11

sat(s ) = sign( ); abs( )>1

sat(s ) = ; abs( )<1

s sif

s sif

(5.33)

The chattering reduction depends on value of ‘ 1 ’. The chattering reduction depends

on value of ‘ 1 ’ but at the cost of robustness, the more the value of , the lesser will

be chattering but at the same time robustness will be reduced. As we know, angular

positions and velocities in the dynamical model always remain bounded due to the

mechanical structure limitations therefore system uncertainty always remains bounded.

Owing to the factor described above, bounded uncertainties and perturbations in the

elevation dynamics can be introduced as

1 1 1

1 1 1 1 1 1

g g gT T T B B B

a a a b b b

For azimuth dynamics the perturbation and bounded uncertainties are taken as

2 2 2 2 2 2

2 2 2

b b b B B B

a a a

The cross-coupling affects in the elevation and azimuth dynamics can be taken as

3 3 1x x (5.34)

6 6 2x x (5.35)

where 1 1 6 2cosgyroK u x x is azimuth affect on elevation and 2 1 7 2 1-k x k u is the

elevation affect on azimuth dynamics caused by the gyroscopic torque. Now by replacing

control laws (Eq. (5.32)) in Eq. (5.29) we get

21

1 3 1 11 1 1 1 2 1 1 11

1 )( ( sin ( ))g

cx B x a x b x T x K sat s

TV s

(5.36)

2 42 2 4 2 4 2 6 4 2 2 2

22 ( ( ) ( ))

cb x a x B x x K sat s

TV s

(5.37)

Now if

___________________________________- 73 -_______________________________

211 1 3 1 1 2 11 1 1

1

) sin( gK T xc x B x a x b xT

(5.38)

2 42 4 2 4 2 6 4 2

22 ( )

cK b x a x B x x

T

(5.39)

From the bounds of the system states in Eq. (5.40)

1

2

3

4

5

6

0 0.5560 0.4363

0 0.10 1.1120.7 0.70 0.35

xx

xx

xx

(5.40)

and with 50% perturbation in parameters defined in Table 2-1, we can compute that

1 0.0011K for elevation dynamics and 2 0.0018K for azimuth dynamics, that will

ensure

1

21V s (5.41)

and

2

22V s (5.42)

1V in (5.41) and 2V in Eq. (5.42) will always be negative definite for non-zero

manifolds. The above conditions in Eq. (5.38) and Eq. (5.39) assures that the sliding

surface variables reach zero in finite time and once the trajectories are on the sliding

surface they remain on the surface, and approaches to the equilibrium points

asymptotically.

5.2.2.1 Simulation Results

The controller designed in Eq.(5.32) delivers the following simulation results.

Figure 5-1 shows the regulation control of the helicopter output states. Figure 5-2 and

Figure 5-3 are having the phase portraits of elevation and azimuth respectively. The

controller effort to deliver the desired results is shown Figure 5-4. The sliding surface

designed to generate the control laws show their convergence in Figure 5-5.

___________________________________- 74 -_______________________________

0 1 2 3 4 5 6 7 8 9 10-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time (sec)

Ang

ular

Pos

ition

(ra

d)

Sliding Mode Control

Elevation

Azimuth

Figure 5-1: Regulation control of Helicopter outputs

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

Elevation

Rat

e of

Cha

nge

of E

leva

tion

Phase Portrait

Figure 5-2: Phase Portrait for Elevation dynamics

___________________________________- 75 -_______________________________

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1Phase Portait II

Azimuth

Rat

e O

f C

hang

e of

Azi

mut

h

Figure 5-3: Phase portrait for Azimuth Dynamics

0 1 2 3 4 5 6 7 8 9 10-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Time (sec)

Con

trol

ler

Eff

ort

Sliding Mode Controller Effort

Elevation

Azimuth

Figure 5-4: Sliding mode controller effort for regulation

Elevation

Azimuth

___________________________________- 76 -_______________________________

0 1 2 3 4 5 6 7 8 9 10-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Time (sec)

Slid

ing

surf

ace

Sliding Surfaces

Elevation

Azimuth

Figure 5-5: Sliding manifolds convergence which ensures states convergence

5.2.2.2 Experimental Test Results

The implementation of the sliding mode controller designed in Eq. (5.32)

delivered the output states response as shown in Figure 5-6 and respective effort can be

seen in Figure 5-7. It can be observed that the equilibrium state in vertical plane is

achieved in 7 sec. The coupling in horizontal plane is introduced at 22 sec. It can be seen

that equilibrium position in horizontal plane is maintained by the controller with the

divergence of 1 degree for about 5 sec after the coupling has been introduced.

The controller guides the system both in vertical and horizontal planes to

equilibrium positions as shown in Figure 5-8 with effort shown in Figure 5-9. The results

show that the system when released at 40 deg. initially in horizontal plane, the SMC

controller governs the system, released in nonlinear range of horizontal plane, to

equilibrium point with in 5 sec with effort shown in Figure 5-9.

Elevation

Azimuth

___________________________________- 77 -_______________________________

5 10 15 20 25 30 35 40 45 50-5

0

5

10

15

20

25Sliding Mode Control

Ele

vatio

n (d

eg)

5 10 15 20 25 30 35 40 45 50

-20

-10

0

10

20

Time (sec)

Azi

mut

h (d

eg)

Figure 5-6: Response of Helicopter system with sliding mode controller when exposed to coupling at

22 seconds

5 10 15 20 25 30 35 40 45 500.2

0.4

0.6

0.8

1

1.2

Time (sec)

Azi

mut

h

5 10 15 20 25 30 35 40 45 500.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Ele

vatio

n

Controller Effort

Figure 5-7: Sliding mode Controller effort to decouple when exposed to coupling at 22 seconds.

___________________________________- 78 -_______________________________

5 10 15 20 25 30 35 40 45 50-5

0

5

10

15

20

25

Ele

vatio

n (d

eg)

Sliding Mode Control

5 10 15 20 25 30 35 40 45 50

-20

-10

0

10

20

30

40

50

Time (sec)

Azi

mut

h (d

eg)

Figure 5-8: Response of Actual system with sliding mode control when initialized in nonlinear range

5 10 15 20 25 30 35 40 45 500.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Ele

vatio

n

Controller Effort

5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

Time (sec)

Azi

mut

h

Figure 5-9: Sliding mode Controller effort to reach equilibrium position when released in nonlinear

range.

5.2.2.3 Performance evaluation

The performance indices for elevation dynamics in Table 5-1 and for azimuth

dynamics in Table 5-2 quantitatively certify the improved performance of the twin rotor

system dynamics with sliding mode control after the introduction of cross-coupling. The

sliding mode controller has improved performance as equilibrium position is well

maintained with lesser errors.

___________________________________- 79 -_______________________________

Performance

Indices

Elevation

(H∞ with

Traditional Wt)

Elevation

(H∞ with

Hadamard Wt)

Elevation

(SMC)

ISE 46.14 24.85 24.05

ITSE 287.7 82.97 75.35

IAE 4.387 2.334 2.012

ITAE 44.92 18.01 10.56

Table 5-1: Performance indices of Elevation Dynamics

Performance

Indices

Azimuth

(H∞ with

Traditional Wt)

Azimuth

(H∞ with

Hadamard Wt)

Azimuth

(SMC)

ISE 0.4031 3.127 0.1956

ITSE 14.59 107.7 5.4

IAE 0.3114 0.8696 0.2671

ITAE 11.61 31.51 8.232

Table 5-2: Performance indices of Azimuth Dynamics

5.2.3 Sliding Surface Design via LMIs

Linear matrix inequalities (LMI) based sliding manifold has been proposed in

[32] The method described in [32] allows us to design a linear sliding manifold for twin

rotor system which assures the system performance and the control costs required to

maintain sliding, in an optimal way. The convex optimization problem has been

developed based on structural assumption on the Lyapunov matrix for the closed-loop

system. The solution of the convex problem allows us to attain optimized sliding surface

which helps in designing feedback controller.

Consider the system in Eq.(2.25) that can be reformulated as

x Ax Bu (5.43)

Where

___________________________________- 80 -_______________________________

11 12

21 22

( ) ( )11 22

0

m

n m n m m m

A AA B

IA A

A A

The A and B for helicopter model in Eq.(2.29) can be reformulated for Eq.(5.43) as

-0.9874 -7.1760 12.4544 4.0089 -0.2337 -7.1334 2.9235

0.8188 0 0 -0.4696 0.0209 -0.3296 0

0 0 -1.3330 0 0 0 0

-0.6376 4.1157 14.5719 3.1223 -0.2015 -6.6450 -1.6767

0.5483 -0.1836 0.3187 0.8323 0.0061 0.2033 0.0748

-0.4637 2.8887 10.19

A

74 4.9631 -0.2663 -8.6611 -1.1768

0 0 0 0 0 0 -3.3330

(5.44)

0 0

0 0

0 0

0 0

0 0

1 0

0 1

B

(5.45)

0 0.4094 0 0 0 0 0

0.0105 0 0 -0.0060 -0.4091 0.0086 0

C

(5.46)

The sliding surface for the above described system can be defined as

11 2 1 1 2 2

2

( ) [ S ]x

s x Sx S S x S xx

(5.47)

If

0s Sx SAx SBu (5.48)

Then equivalent control comes out to be

1( )equ SB SAx (5.49)

The feedback controller ‘ L ’ from sliding mode control can be designed as

1( )L SB SA (5.50)

Under given coordinate system, the switching function will be as

___________________________________- 81 -_______________________________

2 mS S M I (5.51)

Where

( )

2

m n m

m m

M

S

The partition of the system is now in the form of 1 2x , x associated with system

represented in canonical form.

We can now say, when

0s (5.52)

2 1x Mx (5.53)

Thus

1 11 1 12 2 11 12 1( )x A x A x A A M x (5.54)

2 21 1 22 2 11 12 1( )x A x A x Bu A A M x Bu (5.55)

The new coordinate system can be introduced as

( )nx I A x (5.56)

Where does not belong to current eigenvalues of the system. Our system in new

coordinates will be given as

1( ) ( )n nA I A A I A (5.57)

( )nB I A B (5.58)

1( )( )m n nL M I I A I A (5.59)

1( )m nS M I I A (5.60)

The close loop system resulting in the form of S will have to minimize

0

( ) ( ) ( ) ( )T TJ x Qx u Ru d

(5.61)

Where

n n

m m

Q

R

The Lyapunov stability that will guarantee the system stability in new coordinates can

written as

___________________________________- 82 -_______________________________

(( ) ( )) ( )T T Tq qX A BL A BL X X Q Q L RL X (5.62)

( ) ( ) 0T T Tq qX A BL A BL X XQ Q X XL RLX (5.63)

The Eq.(5.63) can be reformulated in linear matrix inequalities as

1/2

1/2

( ) ( )

0 0

0

T T Tq

q q

m

A BL X X A BL XQ XL R

Q X I

R LX I

(5.64)

And our minimization problem can be formulated as

1(( ) ( ))Tn nMinimize Trace I A X I A (5.65)

Subject to LMI in Eq.(5.64). The decision variables are L and X where

1

2

00

0

XX

X

(5.66)

And

( ) ( )1

2

n m n m

m m

X

X

The LMI in Eq.(5.64) has two solution variables therefore we will have to formulated two

LMIs that will deliver solution in two variables. So the new problem LMI can be

formulated as

( )Minimize Trace Z (5.67)

Subject to

1/2

1/2

( )

0 0

0

T T T Tq

q q

m

AX XA BN BN XQ N R

Q X I

R N I

(5.68)

( )

0( )

Tn

n

Z I A

I A X

(5.69)

Where

1 2N LX N X (5.70)

The feedback controller based on sliding surface can be calculated as

___________________________________- 83 -_______________________________

1 11 1[ ]mL NX N X I (5.71)

11 1M N X (5.72)

And in original coordinates the feedback controller can be written as

( )m nL M I I A (5.73)

5.2.3.1 Simulation Results

The feedback controller based on sliding surface delivers the results for regulation

control as shown in Figure 5-10. The parameter values are taken as

-3

30

m

Q C

R I

The controller effort to meet the desired regulation results is shown in Figure 5-11 and

the sliding surface convergence is shown in Figure 5-12.

1 2 3 4 5 6 7 8 9 10-0.06

-0.04

-0.02

0

0.02

0.04

0.06

Time (sec)

Ang

ular

Pos

ition

(ra

d)

Helicopter

Elevation

Azimuth

Figure 5-10: Output States response for LMI based Sliding mode control

___________________________________- 84 -_______________________________

0 1 2 3 4 5 6 7 8 9 10

0

5

10

15Controller Effort

Time (sec)

Con

trol

Eff

ort

Figure 5-11: LMI based sliding mode controller effort

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

70Sliding Surfaces

Time (sec)

Slid

ing

Sur

face

Elevation

Azimuth

Figure 5-12: LMI based sliding surface convergence

The controller has to exert more effort in order to attain the desired results. This is

because the designed controller is based on input and output energy of the system which

relates to H2 norm and the controller based on H2 norm have exert more effort to deliver

results. This reason forbids us to apply the designed controller on actual system and

restricts us to simulations only.

5.3 Higher order sliding mode control

Higher order sliding mode control [24], [25], [26] is a recent approach that allows

the removal of all standard sliding mode restrictions, while preserving the main sliding

___________________________________- 85 -_______________________________

mode feature and improving its accuracy. Traditional SMC has some intrinsic problems,

such as discontinuous control that often yields chattering [6]. To cope with problem and

achieve higher accuracy, HOSM is proposed [24], [25]. Obviously, kth order HOSM

stabilizes the sliding variable at zero as well as its derivatives. On the other hand, since

the high-frequency switching is hidden in the higher derivative of the sliding variable, the

effect of chattering will be reduced. In other words, HOSM has two important features

that make it a better choice in designing the controller. It improves the accuracy of the

design, which is a very important issue, and may provide a continuous control.

5.3.1 Super Twisting Algorithm

The 2-SMC super twisting algorithm [31] delivers control law for the sliding

surface with relative degree, 1r . The continuous nature of the control law helps to

overcome chattering caused by sign function in sliding mode control. The designed

control law ensures the system convergence in finite time along with robustness against

the external disturbances and parameter uncertainties.

The controller design procedure first involves the reduction of twin rotor MIMO

system in two SISO systems and then two controllers can be designed for each system

based on their error dynamics defined as

eqE X X (5.74)

where eqX are the desired values of the system states at equilibrium position The sliding

surface with 1r for elevation and azimuth can be designed as in Eq.(5.75) and

Eq.(5.76) respectively.

1 2 1 2 0 2e e e (5.75)

2 5 3 5 2 5e e e (5.76)

Their respective time derivatives are

1 2 1 2 0 2e e e (5.77)

2 5 3 5 2 5e e e (5.78)

The new local coordinates can be defined as

___________________________________- 86 -_______________________________

2 11 1 1 2;

TY Y (5.79)

2 12 2 1 2;

TY Y (5.80)

The finite time stabilization problem for the uncertain second order system equivalent to

second order sliding mode control problem which can be written as [].

1 2

2 ( , ) ( , )

Y Y

Y t X t X U

(5.81)

Where 2 11 2;

T

2 11 2;

T

2 11 2;

TU U u u

From Eq. (2.18) to Eq. (2.21), we can say

1 1 2 3 1 3( , ) ( , )x t x t c e c e (5.82)

3

1

21 11 1 1 1 1 1 1 3 3 2 22

1

( )1( , ) [{ 4 2 } ( )( cos sin )]g g

x ux t a x a u b B x T x x x x

I T

(5.83)

1 1 11

1 1 1

21( , ) ( )

a x bx t

I T T (5.84)

and

2 2 2 6 1 6( , ) ( , )x t x t c e c e (5.85)

2 2 2

2 4 2 22 6

2

2 2 4 22 2 2

4 21({ )( , ) }(- )

a x a bB x

Ix t u x u

T T T (5.86)

2 22 4

2 2

2( , ) ( )

a bx t x

T T (5.87)

As the states are bounded and their range can be defined as

1

2

3

4

5

6

0 0.556

0 0.4363

0 0.1

0 1.112

0.7 0.7

0 0.35

x

x

x

x

x

x

(5.88)

___________________________________- 87 -_______________________________

and all other constants have bounded values which can be found in [7]. Therefore, we can

claim that there exists 1,2 0 and m1,2 1,20 M such that 1,2 1,2 and

m1,2 1,2 1,20 ( , ) Mx t are satisfied [31]. The computed bounds of helicopter model

for elevation are 1 0.75 , m1 7.13 , 1 96.20M and for azimuth

dynamics 2 0.62 , m2 28.40 , 2 99.31M .

These bounds computation for helicopter model certify that there exist sliding

mode control and the 2-SMC super twisting controller can be designed for twin rotor

system.

The super twisting continuous control law [31] constitutes of two terms. The first

term is defined by means of its discontinuous time derivatives, while the other is

continuous function of the available sliding variable. Consider the control algorithm

( ) '( ) ''( ) 1, 2i i iu t u t u t i (5.89)

where

'( ) ( ) 1, 2

( ) 1''( )

( ) 1

i i i i

i ii

i i i

u t sign i

u t if uu t

sign if u

The constants ( ,i i ) values [31] are computed keeping in mind the bounds defined as

22

( )4

( )

0 0.5

i

i

m ii

ii

m

M i iii

m i i

(5.90)

Finally for elevation we have the values as 1 2.31 , 1 0.105 and for azimuth we

have 2 0.053 , 2 0.021 .

The designed control law in Eq.(5.89) and constants bounds computed in

Eq.(5.90) , the sliding variables 1 and 2 will stabilize at zero in finite time assuming

that the uncertainties ( g ) in the parameters are bounded. The chattering effect in the

control law has been filtered out by integrating discontinuous portion caused by the sign

term making control law continuous in nature.

___________________________________- 88 -_______________________________

5.3.1.1 Simulation Results

The simulations are carried out with the control laws in Eq.(5.89). The regulation

of the output states is shown in Figure 5-13 and their respective phase portrait is shown in

Figure 5-14. The crossings in the phase trajectories depict involvement of third state i.e.

motors state. The successful simulations validated the control laws for implementation on

actual experimental setup.

0 5 10 150

0.05

0.1

0.15

0.2

0.25

Time (sec)

Ang

ular

Pos

ition

(ra

d)

Output States Response

Elevation

Azimuth

Figure 5-13: Regulation response of output states of dynamical model of helicopter

-0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Angular Position

Rat

e of

Cha

nge

of A

ngul

ar P

ositi

on

Phase Portrait

Elevation

Azimuth

Figure 5-14: Phase portraits of elevation and azimuth dynamics

___________________________________- 89 -_______________________________

5.3.1.2 Experimental Results

The implementation of the controller in Eq.(5.89) with α1=1.5, β1=0.9 on

elevation dynamics delivered results shown in Figure 5-15. These parameters were

selected because system response was with lesser rise and settling time thus delivering

desired performance. The variation in CG is shown in Figure 5-16 which is serving as

uncertain parameter creating disturbance torque. The respective controller effort to

overcome the uncertainties and maintain the equilibrium position is shown in Figure

5-17. The sliding surface convergence can be seen in Figure 5-18. It can be observed that

the helicopter model acquires the referred position in 10 sec. and maintains the position

even with perturbed CG. In the first case, when the CG is perturbed 50% ahead of its

current value at 35 sec. the torque produced compels the helicopter model to hover with

nose bend down for 7 sec. and it deviates 9% from its referred path. When the CG attains

its normal position again the resulting torque makes the model to hover with nose tilted

up and with 7 sec. the controller nullifies the effect and attains the given path.

10 20 30 40 50 60 70 80 90 100 1100

5

10

15

20

25Elevation

Time (sec)

Ang

ular

Pos

ition

(de

g)

Reference

Elevation

Figure 5-15: Response of helicopter model in Elevation when exposed uncertainty in center of gravity

___________________________________- 90 -_______________________________

10 20 30 40 50 60 70 80 90 100 110-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time (sec)

CoG

Center of Gravity

Figure 5-16: Variations in center of gravity serving as parametric uncertainty

In the second case when the CG is perturbed 70% behind of its normal position at 69 sec.

the resultant torque forces helicopter model to fly with nose tilted up and the 17%

deviation is nullified by the designed controller in 7 sec. and vice versa when CG again

attains its normal position.

10 20 30 40 50 60 70 80 90 100 110

0

0.5

1

1.5Elevation

Time (sec)

Con

trol

ler

Eff

ort

Figure 5-17: 2-SMCController effort in voltage to acquire and maintain equilibrium position in

elevation plane

___________________________________- 91 -_______________________________

10 20 30 40 50 60 70 80 90 100 110-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time (sec)

Slid

ing

Sur

face

Sliding Surface Response

Figure 5-18: Elevation dynamics sliding manifold convergence

The control law in Eq.(5.89) with α2=1.1, β2=0.5 for azimuth dynamics delivered results

in horizontal plane shown in Figure 5-19 with the respective controller effort in Figure

5-20. The azimuth dynamics sliding surface convergence is shown in Figure 5-21. It can

be observed that the helicopter when released from 35 deg. in azimuth converges to

referred position in 15 sec. and maintains the position afterwards. The horizontal plane

dynamics remain unaffected by the perturbation in center of gravity.

The control inputs for both the elevation and azimuth dynamics are having

chattering in their response, however this phenomena is not really “dangerous” for the

system as the magnitude are well under the maximum limits of the inputs. However these

responses of control laws signify that the 2-SMC super twisting algorithm certainly

reduces the chattering but fails to remove it. Although, the reduction in chattering can be

achieved by altering in Eq.(5.89) or by boundary layer in the discontinuous control

part [] but as a consequence the performance of the system gets reduced.

___________________________________- 92 -_______________________________

10 20 30 40 50 60 70-10

-5

0

5

10

15

20

25

30

35

Time (sec)

Ang

ular

Pos

ition

(de

g)

Azimuth

Reference

Azimuth

Figure 5-19 Response of helicopter model with 2-SMC controller in horizontal plane

10 20 30 40 50 60 70 80-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8Azimuth

Time (sec)

Con

trol

ler

Eff

ort

Figure 5-20: 2-SMC Controller effort to maintain equilibrium position in azimuth

___________________________________- 93 -_______________________________

10 20 30 40 50 60 70-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6Azimuth Sliding Surface

Time (sec)

Slid

ing

Sur

face

Figure 5-21: Azimuth dynamics sliding surface convergence

This section presented successful solution for the precise helicopter maneuvers in the

presence of unwanted moments caused by variations in the on board loaded equipments

in the fuselage. The simulations and practical implementation of the control algorithm

guarantees the smooth helicopter flight even with unbalance dynamics caused by the

perturbed CG.

___________________________________- 94 -_______________________________

Chapter 6

Conclusion

&

Future Work

___________________________________- 95 -_______________________________

This thesis contained several attempts to design and implement robust controllers

for helicopter systems which can handle cross-coupling along with parametric uncertainty

and guarantee smooth helicopter flight. In the first phase, H∞ controllers were designed

based on traditional and Hadamard weights with loop shaping design techniques. The

main issue encountered during the design procedure was the selection of weights for loop

shaping of singular values of helicopter system. After number of attempts suitable

weights were selected, which transformed the loop gains such that the singular values at

lower frequencies were with high gains and with low gains at higher frequencies. The

main problem was faced in reduction of the slope of singular values at crossover

frequencies to 1 db/decade. After number of attempts the weights were chosen, but the

validation of the selected weights was carried through practical implementation although

the simulation results were not supportive in terms of performance parameters. The

controller was implemented in SIMULINK environment where the state space model of

the controller was engaged with system to achieve desired results. The controller

provided robustness against the cross-couplings and uncertainty in the CG of the

helicopter model; however the performance declined as a result. Hadamard weighted H∞

controller failed to operate in nonlinear range, conversely traditional weighted H∞

successfully guided the helicopter model to equilibrium position when released in

nonlinear range.

The limitations of linear theory founded the base for nonlinear control design,

sliding mode control and its advance algorithms were considered to overcome limitation

of linear domain. Sliding manifold designing involved the availability of rotor speeds,

which was extracted from Luenberger observer. The second reservation for this technique

resides in defining required rotor speeds in control law, as a consequence the coupling

reduction was achieved but robustness against CG uncertainty vanished. This drawback

in the 1-SMC made to move towards higher sliding mode control i.e. 2-SMC, where the

manifold’s convergence guaranteed the robustness against CG uncertainty.

The implementation of controllers was carried out in SIMULINK, which made it

easy to design and implement all controllers. As a consequence some of the practical

issues were overshadowed like sampling time issue in SMC and implementation of

difference equations of H∞ controller.

___________________________________- 96 -_______________________________

Controllers Decoupling Robust

Against

CG

Variation

Nonlinear

Range

(Azimuth)

Rise Time

(sec)

Overshoots

(≤10%)

Settling

Time (sec)

Ele Az Ele Az Ele Az

PID No Yes Yes 25 5 Yes No 30 25

Traditional

Weighted H∞

Yes Yes Yes 20 2 Yes No 25 23

Hadamard

Weighted H∞

Yes Yes No 10 N/A Yes N/A 15 N/A

1-SMC Yes No Yes 10 10 Yes Yes 10 10

2-SMC No Yes Yes 10 20 Yes Yes 26 26

Table 6-1: Comparison between proposed controllers in this thesis

After going through the efforts to produce this thesis several other areas also

require attention in future. The areas from controller design may include decoupled

controller using Higher Order Sliding mode control. Although we have attempted to

solve cross-coupling using this technique but the results show that more effort is needed

to put in to get to the solution.

The other area that needs consideration is parameter estimation using nonlinear

techniques. Although, the accompanied manual has these parameters estimated for the

mathematical model using optimization techniques, but these parameters can be

estimated more accurately using techniques from nonlinear control theory like sliding

mode observers, higher order sliding mode observers.

Similarly, the concepts of fault diagnostics can also be validated on this model. The

fault estimation involves the system output and controller effort, therefore fault

diagnostics is an area that can be explored further on the basis of current efforts.

___________________________________- 97 -_______________________________

___________________________________- 98 -_______________________________

Chapter 7

Appendix

___________________________________- 99 -_______________________________

7.1 MATLAB code of modeling of Helicopter Model

clc clear all close all

% System parameter values;

t1=0.3;

b1=0.00936; I1=4.37e-3;

B1=1.84e-3; Tg=3.83e-2;

t2=0.25;

b2=0.0294;

Tor=2.7; Tpr=0.75;

Kr=0.00162;

I2=4.14e-3;

B2=8.69e-3;

Kg=0.015;

% System State Space Matrices A=[-3.333 0 0 0 0 0 0;

0 0 1 0 0 0 0;

b1*1.667/I1 -Tg/I1 -B1/I1 0 0 0 0; 0 0 0 -1.3333 0 0 0;

0 0 0 0 -4 0 0;

0 0 0 0 0 0 1; 0 0 0 0.0899/I2 b2*2/I2 0 -B2/I2];

B=[2 0;

0 0; 0 0; 0.0625 0;

0 2;

0 0; -0.0058/I2 0];

C=[0 1 0 0 0 0 0; 0 0 0 0 0 1 0]; D=[0 0;

___________________________________- 100 -_______________________________

0 0]; Gss=ss(A,B,C,D); % Conversion to Transfer function from State Space model

[NUM1,DEN1]=ss2tf(A,B,C,D,1); [NUM2,DEN2]=ss2tf(A,B,C,D,2);

[r11 c11]=size(NUM1);

[r12 c12]=size(DEN1); num11=NUM1(1,1:c11);

den11=DEN1(1,1:c12);

num21=NUM1(2,1:c11);

den21=DEN1(1,1:c12);

[r21 c21]=size(NUM2);

[r22 c22]=size(DEN2); num12=NUM2(1,1:c21);

den12=DEN2(1,1:c22);

num22=NUM2(2,1:c21);

den22=DEN1(1,1:c22);

g11=tf(num11,den11);

g21=tf(num21,den21);

g12=tf(num12,den12); g22=tf(num22,den22);

G=[g11 g12;g21 g22];

7.2 MATLAB code for H∞ with Traditional weights

model_linear

%%%%%%%%%%%%%Scaling

Win=[12 0;0 6];

G=series(Gss,Win);

T=feedback(G,eye(2));

S=1-T;

figure(2),sigma(G,{1e-3,1e3});

%%%%%%%%Weights%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

___________________________________- 101 -_______________________________

W1=01*[9000*tf(conv([1 2],[1 0.2]),conv([1 0.001],[1 1000])) 0;0

10000*tf(conv([1 2],[1 0.2]),conv([1 .001],[1 1000]))];

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Gs=G*W1;

%%%%%%%%%%%%%%frequency alignment

Gf=freqresp(Gs,30);

Ka=align(Gf);

W1=W1*Ka;

Gs=G*W1;

% Conversion to Digital Domain W1d=c2d(W1,0.01,'tustin');

W1ss=ss(W1);

W1ssd=ss(W1d);

[Wa,Wb,Wc,Wd]=ssdata(W1ss);

[Wad,Wbd,Wcd,Wdd]=ssdata(W1ssd);

figure(4),sigma(Gs,'r',G,'k',{1e-3,1e3});grid;

legend('Gs','G',1)

% Controller Designing

gammarel=1.1;

[Ks,gammin]=coprimeunc(Gs,gammarel);

Ks=-Ks;

Ksd=c2d(Ks,0.01,'tustin');

[Ac,Bc,Cc,Dc]=ssdata(Ks);

[Acd,Bcd,Ccd,Dcd]=ssdata(Ksd);

% For Implementation

Kco=series(W1,Ks);

Kcod=c2d(Kco,0.01,'tustin');

[Aco,Bco,Cco,Dco]=ssdata(Kco);

[Acdo,Bcdo,Ccdo,Dcdo]=ssdata(Kcod);

% Simulations T2=feedback(series(G,Kco),eye(2));

S2=1-T2;

figure(5),step(T2,20);grid;title('Step

response');xlabel('Time');ylabel('Angle (deg)');

figure(6),step(Kco*S2,20);grid;title('Control

input');xlabel('Time');ylabel('Voltage (V)');

___________________________________- 102 -_______________________________

7.3 MATLAB code for H∞ with Hadamard weights

model_linear

% Scaling Weight Win=[12 0;0 6];

% Hadamard Weighting of Scaling Weighting Function

G=Win.*G;

figure(2),sigma(G,{1e-3,1e3});

% Input Weighting Function

W1=01*[165*tf(conv([1 2],[1 0.2]),conv([1 0.001],[1 1000])) 0;0

160*tf(conv([1 2],[1 0.2]),conv([1 .001],[1 1000]))];

alpha=12;

s=2;

W1=W1.*[alpha s*alpha;s*alpha alpha];

% Hadamard Multiplying Gs=G.*W1;

Gs=ss(Gs,'min');

figure(4),sigma(Gs,'r',G,'k',{1e-3,1e3});grid;

legend('Gs','G',1)

% Controller Designing

gammarel=1.1;

[Ks,gammin]=coprimeunc(Gs,gammarel);

Ks=-Ks;

Ksd=c2d(Ks,0.01,'tustin');

[Ac,Bc,Cc,Dc]=ssdata(Ks);

[Acd,Bcd,Ccd,Dcd]=ssdata(Ksd);

% Conversion to frequency domain

[NUM1c,DEN1c]=ss2tf(Ac,Bc,Cc,Dc,1);

[NUM2c,DEN2c]=ss2tf(Ac,Bc,Cc,Dc,2);

[r11c c11c]=size(NUM1c);

[r12c c12c]=size(DEN1c);

num11c=NUM1c(1,1:c11c);

den11c=DEN1c(1,1:c12c);

num21c=NUM1c(2,1:c11c);

den21c=DEN1c(1,1:c12c);

___________________________________- 103 -_______________________________

[r21c c21c]=size(NUM2c);

[r22c c22c]=size(DEN2c);

num12c=NUM2c(1,1:c21c);

den12c=DEN2c(1,1:c22c);

num22c=NUM2c(2,1:c21c);

den22c=DEN1c(1,1:c22c);

g11c=tf(num11c,den11c);

g21c=tf(num21c,den21c);

g12c=tf(num12c,den12c);

g22c=tf(num22c,den22c);

Gc=[g11c g12c;g21c g22c];

% Controller for Implementation

Kco=Gc.*W1;

Kcod=c2d(Kco,0.01,'tustin');

[Aco,Bco,Cco,Dco]=ssdata(Kco);

[Acdo,Bcdo,Ccdo,Dcdo]=ssdata(Kcod);

T2=feedback(Kco.*G,eye(2));

S2=1-T2;

figure(5),step(T2,20);grid;title('Step

response');xlabel('Time');ylabel('Angle (deg)');

figure(6),step(Kco*S2,20);grid;title('Control

input');xlabel('Time');ylabel('Voltage (V)');

___________________________________- 104 -_______________________________

7.4 SIMULINK Diagram for H∞ Implementation on Helicopter Model

-K-

r to

d

-K-

r 2

d

t

To

Wo

rksp

ace

4E

le1

To

Wo

rksp

ace

3

Az1

To

Wo

rksp

ace

2

Ele

To

Wo

rksp

ace

1

Az

To

Wo

rksp

ace

Sco

pe

3

Sco

pe

2

Sco

pe

1

Sco

pe

Ou

t1

Ou

t2

Ou

t3

Ini

He

lico

pte

r

1 G2

0

G

14 E

le

In1

Ou

t1

Dis

turb

an

ce

(n)=

Cx(

n)+

Du

(nn

+1

)=A

x(n

)+B

u(

Co

ntr

oll

er1

Clo

ck

0 CG

3

1 CG

2-4

0 Az

pi/

18

0

1

pi/

18

0

Figure 7-1 SIMULINK Diagram for H∞ Implementation on Helicopter Model

___________________________________- 105 -_______________________________

7.5 SIMULINK Diagram for H∞ Implementation on Helicopter Model

-K-

r to

d

-K-

r 2

d

t

To

Wo

rksp

ace

4

Az1

To

Wo

rksp

ace

3

Ele

1

To

Wo

rksp

ace

2

Ele

To

Wo

rksp

ace

1

Az

To

Wo

rksp

ace

Sco

pe

3

Sco

pe

2

Sco

pe

1

Sco

pe

Out

1

Out

2

Out

3

Ini

He

lico

pte

r

1 G2

0

G

In1

Out

1

Dis

turb

an

ce

x2 x6

u1 u2

Co

ntr

oll

er

Clo

ck

1 CG

3

1 CG

2

Figure 7-2 SIMULINK Diagram for H∞ Implementation on Helicopter Model

___________________________________- 106 -_______________________________

2

u2

1

u1

x2

x7

u1

Elevation

x6

u1

u2

x7

Azimuth

2

x6

1

x2

Figure 7-3 SMC Controller Block

ss/e

x1

x2

x3

v1

u1x1

1

u1

u2

x1^2

sin

sin(x2)

K Ts

z-1int

K (z-1)Ts z

der

Switch

Sign

Product

-K-

Gain1 -K-

Gain

f(u)

Fcn

0.34

Constant

-K-

B1

|u|

Abs

-K-

3.33

2

2

-K-

1/I4

-K-

1/I2

K-

1/I11

1/2

1 0.001

-K-

+3.33

2

x7

1

x2

Figure 7-4 Elevation Controller Block

___________________________________- 107 -_______________________________

x5u2

v1

x7

x6

x5

s/es

x4

2

x7

1

u2

u2

x1^2

K Ts

z-1int1

K Ts

z-1int

K (z-1)Ts z

der

Switch

Sign

-K- Gain1

-K-

Gain

.44

Constant

-K-

B2

-K-

B1

|u|

Abs

-K-

3.33

2

2

-K-

1/I2

K-

1/I1-K-

1/2

-K-

-4

-K-

-3-K-

-2

-K-

-1

2

u1

1

x6

Figure 7-5 Azimuth Controller Block

___________________________________- 108 -_______________________________

7.6 SIMULINK Block Diagram for 2-SMC

-K-

r to

d

-K-

r 2

d

t

To

Wo

rksp

ace

4

Ele

1

To

Wo

rksp

ace

3 Az1

To

Wo

rksp

ace

2

Ele

To

Wo

rksp

ace

1

Az

To

Wo

rksp

ace

Sco

pe

3

Sco

pe

2

Sco

pe

1

Sco

pe

Out

1

Out

2

Out

3

Ini

He

lico

pte

r

1 G2

0

G

In1

Ou

t1

Dis

turb

an

ce

x2 x6

u

Co

ntr

oll

er

Clo

ck

0 CG

3

1 CG

2

Figure 7-6 SIMULINK Block Diagram for 2-SMC

___________________________________- 109 -_______________________________

x6

x7

x2

x3

-0.5

-0.15

1

u

sqrt(abs(s1))*sign(s1)1

sqrt(abs(s1))*sign(s1)

-K-

lambda2

-K-

lambda1

K Ts

z-1

int1

K Ts

z-1

int

K (z-1)Ts z

der1

K (z-1)Ts z

der

Switch2

Switch1

Sign1

Sign

sqrt

MathFunction1

sqrt

MathFunction

Add4

Add1

|u|

Abs2

|u|

Abs1

-K-

-1

-K- -0.3

-K- -0.2

-K-

-0.1

-K-

-0.01

-K-

-.1

2

x6

1

x2

Figure 7-7 2-SMC controller Block

___________________________________- 110 -_______________________________

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