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Control of Robot

Master Thesis

Ioannis Manganas MCE4 - 1023

Aalborg UniversityDepartment of Energy Technology

Copyright c© Aalborg University 2018

LATEXhas been used for typesetting this document, using the TeXstudio IDE. All plots havebeen created using either Matlab or Inkscape. Simulink has been used for all simulations.Measurement data have been acquired using LabVIEW.

Title: Control of RobotSemester: 10Semester theme: Master’s thesisProject period: 01.02.2018 to 01.06.2018ECTS: 30Supervisor: Torben Ole Andersen & Lasse SchmidtProject group: MCE4-1023

Ioannis Manganas

SYNOPSIS:

The focus of this thesis is the development of a back-stepping controller that avoids the "explosion" of termsand for which boundedness results are based on Lya-punov stability theory. The controller is designed forapplication to symmetrical hydraulic cylinders, oper-ating as joint actuators for a 2DOF manipulator.Each electro-hydraulic system is subject to parame-ter variations and external disturbances. Initially, thestate of art is investigated for a simplified backstep-ping based design. The model of the system is de-veloped and validated from experimental data. Thecontroller selected is redesigned for the investigatedelectro-hydraulic system where it is found difficult toprove stability for the closed loop system.A different controller is designed, based on slidingmode disturbance observer, for which boundedness re-sults are acquired for the closed loop system. Again,the high complexity of the backstepping technique isavoided. Efficacy of the designed algorithm is inves-tigated by conducting simulation results and compar-ing selected performance indices with reference con-trollers. Sensitivity analysis suggests increased robust-ness of the proposed control algorithm.

Pages, total: 155Appendix: 55Supplements: -

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Contents

Preface vii

Summary ix

1 Introduction 11.1 Scope of the report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The backstepping control design procedure . . . . . . . . . . . . . . . 21.3 State Of Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3.1 Combination with adaptive techniques . . . . . . . . . . . . . 61.3.2 Combination with SMC . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Preliminary conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 System Modeling 112.1 Nonlinear model of the system . . . . . . . . . . . . . . . . . . . . . . 11

2.1.1 Model of the manipulator in joint space . . . . . . . . . . . . . 112.1.2 Model of the manipulator in the actuator space . . . . . . . . 122.1.3 Modeling of the hydraulic actuators . . . . . . . . . . . . . . . 122.1.4 Model of the system . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Validation of the nonlinear model . . . . . . . . . . . . . . . . . . . . 152.2.1 Verification of the gravitational terms . . . . . . . . . . . . . . 152.2.2 Verification of the dynamics . . . . . . . . . . . . . . . . . . . 16

2.3 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4 Chapter 2 summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Simplified backstepping control design 193.1 Backstepping controller design . . . . . . . . . . . . . . . . . . . . . . 193.2 Load pressure controller design . . . . . . . . . . . . . . . . . . . . . . 26

3.2.1 Analysis of the closed loop system stability in the Lyapunovframework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3 Chapter 3 summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

iii

iv Contents

4 Problem formulation and control design 374.1 Problem formulation - Hypothesis . . . . . . . . . . . . . . . . . . . . 374.2 Challenges of the design . . . . . . . . . . . . . . . . . . . . . . . . . . 374.3 Approach 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.3.1 Controller design and analysis . . . . . . . . . . . . . . . . . . 384.3.2 Summary of Approach 1 . . . . . . . . . . . . . . . . . . . . . 44

4.4 Approach 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.4.1 Summary of the first two steps of the adaptive backstepping

design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.4.2 Third step based on disturbance observer . . . . . . . . . . . . 48

4.5 Simulation of proposed algorithm in continuous time . . . . . . . . . 554.6 Chapter 4 summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5 Reference Controllers 595.1 Linearization and plant transfer functions . . . . . . . . . . . . . . . . 59

5.1.1 Analysis in frequency domain . . . . . . . . . . . . . . . . . . 605.1.2 Pressure feedback . . . . . . . . . . . . . . . . . . . . . . . . . 615.1.3 Disturbance rejection . . . . . . . . . . . . . . . . . . . . . . . . 61

5.2 Proportional-Integral controller design . . . . . . . . . . . . . . . . . 625.2.1 Anti-windup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.3 Proportional-Lead and Proportional compensator design . . . . . . . 655.4 Velocity feedforward . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.5 Chapter 5 summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6 Performance and robustness comparisons 716.1 Performance indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.1.1 Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.1.2 Velocity estimation . . . . . . . . . . . . . . . . . . . . . . . . . 746.1.3 Periodic trajectories . . . . . . . . . . . . . . . . . . . . . . . . 74

6.2 Performance under nominal conditions . . . . . . . . . . . . . . . . . 746.3 Robustness comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.3.1 Parameters that vary . . . . . . . . . . . . . . . . . . . . . . . . 806.3.2 Variation of the initial volumes . . . . . . . . . . . . . . . . . . 816.3.3 Variations in leakage coefficient and Coulomb friction . . . . 826.3.4 Variation of the viscous friction coefficient . . . . . . . . . . . 846.3.5 Variations in effective oil bulk modulus . . . . . . . . . . . . . 876.3.6 Variations in trajectory and load . . . . . . . . . . . . . . . . . 88

6.4 Summary of simulation results and comments . . . . . . . . . . . . . 936.5 Conclusions from simulations . . . . . . . . . . . . . . . . . . . . . . . 98

Contents v

7 Conclusions and Future Work 1017.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1017.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

Bibliography 103

A Manipulator kinematics 109A.1 Forward kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

A.1.1 Homogeneous transformations . . . . . . . . . . . . . . . . . . 109A.1.2 Tool tip frame {hxhyh} . . . . . . . . . . . . . . . . . . . . . . 111A.1.3 Center of mass frames {Rxcm1ycm1} and {Cxcm2ycm2} . . . . 111

A.2 Positions and Velocities of centers of mass . . . . . . . . . . . . . . . 111A.2.1 Position vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 111A.2.2 Velocity vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

A.3 Dynamic modeling of the mechanical system . . . . . . . . . . . . . . 112A.3.1 The Euler-Lagrange equation . . . . . . . . . . . . . . . . . . . 113A.3.2 Generalized coordinates and energy . . . . . . . . . . . . . . . 113

B Trajectory generation and inverse kinematics 115B.1 Workspace of the manipulator . . . . . . . . . . . . . . . . . . . . . . 115B.2 Trajectory in Cartesian space . . . . . . . . . . . . . . . . . . . . . . . 116B.3 Slow trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

B.3.1 Trajectory in joint space . . . . . . . . . . . . . . . . . . . . . . 118B.4 Fast trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

B.4.1 Fast trajectory in joint space . . . . . . . . . . . . . . . . . . . . 122B.4.2 Fast trajectory in actuator space . . . . . . . . . . . . . . . . . 123

B.5 Spectrum analysis of piston reference signals . . . . . . . . . . . . . . 124B.5.1 Periodic trajectories . . . . . . . . . . . . . . . . . . . . . . . . 124

B.6 Realizability of trajectories and PL −QL diagrams . . . . . . . . . . . 125B.6.1 Maximum power transfer . . . . . . . . . . . . . . . . . . . . . 126B.6.2 PL −QL diagrams for the trajectories . . . . . . . . . . . . . . 126B.6.3 PL −QL diagrams with load at the tool tip . . . . . . . . . . . 127

C Linearization 129C.1 Linearization of the valve flow equation . . . . . . . . . . . . . . . . . 129C.2 Linearization of the mechanical system . . . . . . . . . . . . . . . . . 130C.3 Block diagram representation of the linearized system . . . . . . . . 130C.4 Selection of the linearization point . . . . . . . . . . . . . . . . . . . . 131

D Velocity estimation 133D.1 Differentiator design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133D.2 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

vi Contents

E Discretization 137E.1 Discretization of the linear reference controllers . . . . . . . . . . . . 137

E.1.1 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137E.1.2 The bilinear transform . . . . . . . . . . . . . . . . . . . . . . . 138E.1.3 PI reference controller . . . . . . . . . . . . . . . . . . . . . . . 139E.1.4 P and P-Lead compensator . . . . . . . . . . . . . . . . . . . . 140E.1.5 Overview of reference control algorithms . . . . . . . . . . . . 140

E.2 Nonlinear controllers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141E.2.1 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142E.2.2 RABLIN - Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . 142E.2.3 Proposed Approach 2 - Section 4.4 . . . . . . . . . . . . . . . . 143

F Theorems 145F.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

F.1.1 Lipschitz continuity . . . . . . . . . . . . . . . . . . . . . . . . 146F.2 Lyapunov stability theorems and extensions . . . . . . . . . . . . . . 146

G Parameters 149G.1 Parameters used in the simulation model . . . . . . . . . . . . . . . . 149

G.1.1 Mechanical and hydraulic model . . . . . . . . . . . . . . . . . 149G.1.2 Parameters used in the hydraulic model . . . . . . . . . . . . 150

G.2 Linearization point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151G.3 Controller parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

G.3.1 Reference controllers . . . . . . . . . . . . . . . . . . . . . . . . 152G.3.2 RABLIN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153G.3.3 Approach 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

Preface

This thesis documents the 10th semester project of MCE4-1023. It investigatesthe possibility of designing a controller based on integrator backstepping, whileavoiding the high complexity of the last design step and on the same time derivingstability conclusions based on Lyapunov theory.The thesis is divided in 4 main parts. In the first part, the integrator backsteppingtechnique is studied and the state of art of application in electro-hydraulic servosystems is investigated. The works that present decreased complexity are selectedas candidates for application to the selected physical setup.In the second part, the model of the system is briefly developed. The model isvalidated with laboratory data. The selected method is redesigned, based on thegiven system.In the third part, a controller that preserves the low complexity of the selected con-troller is designed, based however on the possibility to prove stability or bounded-ness for the closed loop system.Finally, in the fourth part, the selected, designed and two reference controllers arecompared under nominal conditions and sensitivity analysis is performed. Theconclusions from the simulation results and the design process follow.

Aalborg University, June 1, 2018

Ioannis Manganas<[email protected]>

vii

Summary

Electro-hydraulic servo systems combine fast response with high power densityand are utilized in a plethora of industrial applications. However, their nonlineardynamics, as well as time varying parameters render the control design processrather complex, if high tracking performance is required.In this thesis, control design for electro-hydraulic servo systems in the form of sym-metrical cylinders is considered. Specifically, the integrator backstepping controldesign procedure is investigated for application in these servo systems. This tech-nique can be combined with adaptive and robust control laws, promising increasedperformance under parameter variations. In the last step, Lyapunov stability forthe closed loop system can be proven by proper selection of the control variable.The disadvantage of backstepping is the increased complexity, stemming from the"explosion" of terms. In this thesis, it is investigated whether this level of com-plexity can be decreased. A method found in the literature is adopter for the con-trol design for the system into consideration. This system is a 2DOF manipulatorwhose joints are actuated by the symmetrical cylinders. Apart from the decreasedcomplexity, proving stability using Lyapunov theory is required.Initially, in order to design the selected control algorithm and test proving stability,a model of the system is developed. The model is validated using experimentaldata. Afterwards, the selected reference controller is redesigned for application tothe system. This controller reduces the complexity by replacing the final step ofthe backstepping procedure by a linear controller for the load pressure loop. It ishowever proven difficult to prove stability for the whole closed loop system usinga Lyapunov function candidate that is widely used for the backstepping design.In order to keep the low complexity of the reference controller, but at the sametime derive stability results, the third backstepping step is designed using slidingmode control techniques. In the first approach, the term that induces complexityis directly canceled using a first order differentiator. However, selecting the pa-rameters for the adaptive parameters proves to be difficult and a second approachis made. In this second approach, a sliding mode disturbance observer is used.Boundedness region for the system parameters, apart from the adaptation errorterms, are estimated.

ix

x Summary

As a next step, two linear controllers are designed as reference and all four con-trollers are compared in simulations based on selected performance indices. Itis found that the proposed controller, while robust to parameter variations andshowing increased tracking performance, is sensitive to measurement noise. An-other issue is the successful implementation using velocity estimation.

Chapter 1

Introduction

Electro-hydraulic systems are widely used in industrial processes, owing to theirhigh power density. However, the hydraulic parameters can vary with operatingconditions and can be difficult to estimate. The range of variation can also be signif-icant. Furthermore, their dynamics are highly nonlinear, which poses a challengefor control design.Linear controllers have been found to achieve good performance, however whendealing with systems with uncertainties and varying parameters, other approachescan provide better tracking results. These uncertainties can stem from unknownor time varying parameters, such as the effective oil bulk modulus changing withtemperature, a time varying load and the initial control volumes. Another typeof uncertainty is due to neglected dynamics and nonlinearities that have not beenaccounted for, eg. friction forces.In [5], linear and nonlinear controllers were applied on a hydraulically actuatedmanipulator and compared in regard to tracking error. For the nonlinear con-trollers and especially the adaptive inverse dynamics controllers, the position track-ing performance was higher than that of the linear controllers. In [7] it was foundthat the nonlinear controllers yielded better accuracy results, with Variable struc-ture controllers presenting the highest accuracy, albeit complex to design. ModelReference Adaptive Control (MRAC) was found to be accurate when the initialtransients were absent, or when initial values of the estimated parameters could beestablished beforehand.

1.1 Scope of the report

In this report, the design of a control algorithm for two electro-hydraulic systemsis considered. Each of these systems represents an actuated link of a two degreeof freedom (2-DOF) manipulator, located at the Fluid Power and Mechatronic Sys-tems laboratory at Aalborg University.

1

2 Chapter 1. Introduction

Even though designing control algorithms for serial link manipulators is a wellstudied topic in literature, the specific application poses additional challenges. Themajority of the control design techniques, mostly for the nonlinear case, are basedon the selection of the links’ angles as generalized coordinates and consider asinput the torque applied at each joint. However, in the specific application theactuators are linear and as a result their linear positions provide a more intuitivechoice as state variables. Furthermore, the applied torque is substituted by theapplied force from each actuator. The aforementioned peculiarities change theform of the kinematic and dynamic description of the 2-DOF manipulator to amore complex form.The second difference of the specific system is that the actuator dynamics arehighly nonlinear, contrary to the dynamics of direct current (DC) motors, or thehighly researched dynamics of alternating current (AC) motors, especially of thepermanent magnet type.In order to take into consideration the actuator dynamics, alongside the dynamicsof the manipulator, the system can be considered as two subsystems. The backstep-ping approach is preferred in this report, since it provides a way of constructinga Lyapunov function for the whole system, that can be rendered ideally negativedefinite by a suitable selection of the control input. Another advantage of thistechnique is that it can be augmented with different control design approaches,overcoming their specific restrictions. An example of this would be the matchingcondition.The difficulty of applying the backstepping approach is the high complexity ofthe resulting control law. As a result, in this project the feasibility of designing anonlinear control law based on backstepping and applying it to the specific setup isstudied. The aim is to decrease the complexity in the highest degree possible thatstill guarantees crisp stability conclusions in the sense of Lyapunov. A secondaryobjective is to make the resulting control law more robust to parameter variations.In this chapter, the main idea of the backstepping design procedure is introduced.The state of art of backstepping based designs in electro-hydraulic systems follows,that acts as a guide on the selection of the simplest algorithm to consider for thespecific application of the report.

1.2 The backstepping control design procedure

A method to design nonlinear controllers for electro-hydraulic systems is the back-stepping approach [26]. The main advantages of the backstepping design are thatthe matching criterion is relaxed [24] and a Lyapunov function for the system isconstructed. Furthermore, using the backstepping approach it becomes somewhateasier to discern the nonlinear terms to be canceled in each step, leading to reducedcontrol effort. The disadvantages are the increased complexity, stemming from the

1.2. The backstepping control design procedure 3

differentiation of the intermediate control laws in each step.

Example design

The backstepping procedure is illustrated for a general system in strict feedbackform [26]. This example system has a similar form to a hydraulic servo system andis described by:

x1 = x2 (1.1)

x2 = f (x1, x2) + g(x1, x2)x3 (1.2)

x3 = h(x1, x2, x3) + k(x1, x2, x3)u (1.3)

It is assumed that no external disturbances exist and the parameters of the sys-tem are exactly known. Furthermore, the functions g(x1, x2) and k(x1, x2, x3) areassumed to be nonsingular. The purpose of the design is that x1 tracks a referencesignal r(t). Defining the tracking error variable z1 = x1 − r, the Equation 1.1 canbe rewritten as:

z1 = x2 − r (1.4)

The tracking error could be asymptotically stabilized by a state feedback law of theform α1(z1) plus a feedforward term, if the state variable x2 is seen as the inputvariable. However, since x2 is a state variable, another controller is designed sothat the error between the actual value of x2 and its desired value, α1(z1), becomeszero. To this end, the error variable z2 is defined:

z2 = x2 − α1(z1) (1.5)

In order to regulate z2 to zero, the state variable x3 is seen as the input. A controllaw of the form α2(z1, z2) is designed by viewing the state variable x3 as an inputvariable. The same reasoning leads to the definition of the error variable z3 as:

z3 = x3 − α2(z1, z2) (1.6)

As a result, the error dynamics of the system described by Equations 1.1 - 1.3 arewritten as:

z1 = z2 + α1(z1)− r (1.7)

z2 = f (x1, x2) + g(x1, x2)z3 + g(x1, x2)α2(z1, z2)− α1(z1) (1.8)

z3 = h(x1, x2, x3) + k(x1, x2, x3)u− α2(z1, z2) (1.9)

It is also assumed that the functions g(x1, x2) and k(x1, x2, x3) never become equalto zero and the full state vector is available for measurement. From Equation 1.9,the actual input variable is available and a controller can be designed so that for the


system described by the Equations 1.7 - 1.9 the origin is a Globally AsymptoticallyStable (GAS) equilibrium point.The procedure starts by initially making the origin GAS for the subsystem in Equa-tion 1.7 by selecting the desired stabilizing feedback law α1(z1), which is the de-sired value for the state variable x2. The selection can be facilitated by using thefollowing Control Lyapunov Function (CLF) 6 [26]:

V1(z1) =12

z21 (1.10)

V1(z1) = z1z1 = z1 [z2 + α1(z1)− r] (1.11)

By selecting α1(z1) as:

α1(z1) = −k1z1 + r (1.12)

The derivative of the Lyapunov function V1(z1) becomes:

V1(z1) = −k1z21 + z1z2 (1.13)

If z2 becomes equal to zero, V1(z1) < 0 and tracking error approaches zero asymp-totically, with a rate dependent on the value of k1. This motivates the selectionof a control law that drives z2 to zero, or in other words, makes the state variablex2 equal to its desired value α1(z1). To achieve this the subsystem described byEquations 1.7 - 1.8 is considered.A CLF for the second subsystem can be written as:

V2(z1, z2) = V1(z1) +12

z22 (1.14)

V2(z1, z2) = −k1z21 + z1z2 + z2 [ f (x1, x2) + g(x1, x2)z3 + g(x1, x2)α2(z1, z2)− α1(z1)]

(1.15)

By selecting:

α2(z1, z2) =1

g(x1, x2)(−z1 − f (x1, x2) + α1 − k2z2) (1.16)

Then, for V2(z1, z2):

V2(z1, z2) = −k1z21 − k2z2

2 + g(x1, x2)z2z3 (1.17)

To stabilize z3 to the origin, the whole system described by Equations 1.7 - 1.9 isconsidered. A CLF is selected as:

V3(z1, z2, z3) = V2(z1, z2) +12

z23 (1.18)

V3(z1, z2, z3) = −k1z21 − k2z2

2 + g(x1, x2)z2z3 + z3 [h(x1, x2, x3) + k(x1, x2, x3)u− α2(z1z2)](1.19)

1.3. State Of Art 5

Now the actual input variable can be used to design a control law so that for thewhole error dynamics system, the origin is rendered GAS. This can be achievedvia the control law:

u =1

k(x1, x2, x3)[α2(z1, z2)− h(x1, x2, x3)− g(x1, x2)z2 − k3z3] (1.20)

As a result, for the Lyapunov function V3(z1, z2, z3):

V3(z1, z2, z3) = −k1z21 − k2z2

2 − k3z23 (1.21)

and the origin is GAS. This means that the error variables z2, z3 approach zeroand as a result, the tracking error also approximately becomes equal to zero. Theprocedure is illustrated in Figure 1.1. The virtual control variables are used so thata Lyapunov function is designed for the whole system, which is rendered negativedefinite using the actual control input.

x1

f

gx2x3

h

kk1r z1

k1

dtd

α1

dtd

k2

f

α2

dtd

k3

z2

g

h

z3

SystemInt. Control 1 Int. Control 2

ug1

Figure 1.1: The backstepping procedure.

In Figure 1.1, the functions f (x1, x2), g(x1, x2), h(x1, x2, x3) and k(x1, x2, x3) are writ-ten without arguments and without all their signal paths. They are assumed to becompletely known and are canceled. The derivatives of the intermediate, or vir-tual, control laws α1(z1) and α2(z1, z2) can be analytically computed. They areexpressions of the system states and derivatives of the reference signal, which areassumed known. However, these expressions become more complex with everyrecursion of the control design.

1.3 State Of Art

Owing to the aforementioned advantages, nonlinear controllers based on back-stepping have been designed and applied to electro-hydraulic systems. To accountfor the parameter variations, different techniques have been applied alongside thebackstepping design.


1.3.1 Combination with adaptive techniques

In [45], a backstepping controller is designed for a symmetrical cylinder for forceand position control. Simulation results are performed to compare the trackingperformance to different controllers. In [28] a backstepping controller is designedfor pressure tracking of a symmetrical electro-hydraulic actuator. Full state feed-back is assumed. An adaptation algorithm has been designed based on the Lya-punov function to account for lack of accurate knowledge of the valve’s dischargecoefficient. In the first part of [42], exact backstepping controller design is ap-plied to an asymmetric cylinder for position reference tracking, while in the sec-ond part an adaptation algorithm for the estimation of the hydraulic parameters isestablished. This algorithm is based on the Lyapunov function. In the experimen-tal results, the proposed controller, albeit complex, outperforms a PD controller.However, the mechanical system’s parameters are assumed to be exactly knownand that full state feedback is available.The problem of unknown mechanical and hydraulic parameters is solved in [22]by an identification procedure based on the recursive least squares algorithm. Thisprocedure is realized before the control design with a sinusoidal input plus lowpower white noise. Afterwards, the exact, full state backstepping algorithm isdesigned and compared with a PID controller. The nonlinear controller is found torequire less control input power for better tracking performance, especially whenthe load is increased. The Lyapunov function is designed in a way that the statematrix of the resulting closed loop error dynamics is almost diagonal when thetuning parameters are selected appropriately. Then, the negative effects of themagnitude of the hydraulic elements is alleviated.In [49], the Adaptive Robust Control (ARC) method is applied to an asymmetricalcylinder. This method is based on backstepping and for each step, the virtual con-trol law is composed of adaptive and robust terms. In [50], ARC is applied on asymmetrical cylinder. The discontinuous projection method is used alongside tun-ing functions for parameter estimation. Both uncertain parameters and uncertainnonlinearities are dealt with. The resulting controller is rather complex and manytuning parameters are required.In all the aforementioned works, the use of adaptive laws assumes that the sys-tem is linear in the parameters. However, this assumption does not hold whenthe initial control volumes are unknown. This is tackled in [15], where adaptivebackstepping control is applied to a valve actuated asymmetrical cylinder. A mod-ified Lyapunov function is introduced so that the uncertain initial volumes can beestimated and used for control design, even though the system is not linear inparameters in regard to these terms. Furthermore, the overall controller is com-posed of many tuning parameters which can be difficult to select. Experimentalresults show that the proposed controller provides superior tracking performanceto a non-adaptive backstepping controller and to an adaptive controller assum-

1.3. State Of Art 7

ing known initial control volumes. The same approach was applied to a pumpcontrolled asymmetrical cylinder system in [1].In [9] and [8] an adaptive backstepping controller is designed for a symmetricalcylinder servo system. The tuning functions approach is used. Simulations showthat including the valve’s second order dynamics results in increased tracking per-formance. The full state vector is considered available and the resulting controllaw is complex and computationally heavy. This is deduced by the number ofassignments and mathematical manipulations of the algorithm.In order to overcome the increased complexity of the backstepping based con-trollers, mainly due to the differentiation of the virtual control laws, different ap-proaches have been made. In [10], the backstepping design stops at the first twosteps and the load pressure error is regulated by a PID controller. The simulationresults, comparing the developed scheme to the one proposed in [9] augmentedwith the LuGre friction model, present good tracking performance. The complex-ity of the control algorithm is reduced by a wide margin. However, stability is notproven for the whole system, with or without the inclusion of the valve dynamics.Another method, described in [41], is proposed to decrease the complexity. Thesystem is a 1DOF manipulator, actuated by an asymmetrical, servo valve controlledcylinder. An adaptive backstepping controller is designed similar to [15]. In orderfor the complexity to be reduced and ease implementation, the final step in thebackstepping procedure is omitted. The force controller is designed as a pressurefeedback loop and a proportional controller. Furthermore, the backstepping con-troller for the mechanical part is reduced to a PD controller, by noticing the result-ing virtual control law of the second step. The overall tuning of the PD controller isbased on linear techniques. The resulting controller presents increased end-pointtrajectory tracking performance, compared to a reference linear controller withflow feedforward.Instead of adapting to an unknown disturbance, such as the external force in [4], adisturbance observer is utilized. The backstepping design procedure then follows.The controller is compared experimentally to a Sliding Mode Controller (SMC) anda proportional linear controller. In [12], a backstepping observer is designed for theload pressure estimation. Uniform ultimate boundedness is shown for the closedloop system.

1.3.2 Combination with SMC

SMC is used in [16] for trajectory tracking of a symmetrical cylinder. An integraltype sliding surface is selected for the tracking error. However, to avoid chattering,a robust controller without a switching is designed. As a result, only boundednessof the tracking error can be proven. The adaptation laws for the unknown param-eters, which include the initial volumes of the system, are found via a modifiedLyapunov function, similar to the one used in [15]. In this work, the disturbance


forces include the not modeled nonlinearities, external forces and unmodeled fric-tion. This function has been assumed to be smooth and bounded, as well as its firstderivative. Furthermore, since the state vector does not contain the load pressure,boundedness of the load pressure is assumed. In [25], SMC is designed based onthe backstepping technique for position tracking control of a symmetrical cylinder.The parameters are considered known. The controller is designed on the linearizedsystem, and is compared to a PID and an SMC controller.SMC is also used in [37]. The bulk modulus is considered constant as well as therest of the parameters of the system, which is a servo valve controlled symmetricalcylinder. The backstepping approach is followed for the mechanical subsystem. Inthe second step, an SMC is introduced for robustness. The selected sliding surfaceis linear. Moreover, to avoid chattering, the sign function is replaced with a hy-perbolic tangent one. The inner, load pressure loop is designed based on makingthe derivative of the Lyapunov function of the error negative definite. The flowequation is avoided via an inverse model of the valve’s characteristics, acquiredexperimentally. A feedforward friction compensation term, as well as a reducedorder disturbance observer are designed. The controllers are tested on an experi-mental setup.In [44], control of a hydrostatic transmission system is investigated. A backstep-ping based controller is designed for velocity tracking of the hydraulic motor. Asecond order SMC is designed to tackle the unknown disturbances due to the loadtorque and a state observer, based on extended linearization is used. The observeralso features a robust, switching term. To attenuate the chattering effect, the signfunctions are substituted by the hyperbolic tangent function, for the experiment.The proposed scheme presents marginally increased tracking performance, com-pared to a backstepping controller without SMC.In [48], the position tracking of a hydraulic motor is investigated. The unmeasur-able states as well as the derivatives of the virtual control laws are acquired viaa differentiator. This differentiator is designed with singular perturbation tech-niques and is finite time convergent. Neither phase lag nor chattering is present.The square root function of the orifice flow equation, which contains the sign func-tion, is approximated by the upper and lower bounds of the chamber pressures.No adaptation algorithms are present. The added complexity stemming from thedesign parameters is used to adjust the tracking performance. However, this in-creases the control effort. The results of the backstepping controller with the twodifferentiators are validated in simulations with added measurement white noise.A different approach was made in [47] for the tracking control of a symmetricalcylinder. Initially, for the model formulation, the state vector contains the position,velocity and acceleration, instead of the position, velocity and load pressure. Thisassumes however that the disturbance terms can be differentiated. The dependenceof the control input gain on the load pressure is dealt with via a pre-compensator,

1.4. Preliminary conclusions 9

since the load pressure can be measured. Through the appropriate definition of theposition tracking error variable and three switching like error dynamics surfaces,the control input is designed as three different terms. The first term contains astabilizing and an error integral term, the second term a feedforward term for thecancellation of the linear-in-parameters terms and finally, a third control term. Thisfinal term is an asymptotic-like SMC with variable gain. The interesting point ofthe adaptive term is that for the parameter update law, only the reference valuesof trajectory are needed, thus the effect of noise is attenuated. The discontinuousprojection method is used for the parameter update law. The whole state vectoris assumed known and the velocity and acceleration measurements are acquiredvia the backward differentiation of the position measurement, passed through aButterworth filter. The designed controller is experimentally compared to a PI, andadaptive backstepping controller and a version of the proposed controller withoutadaptation. The average value and the standard deviation of the tracking error areshown to be reduced.

1.4 Preliminary conclusions

Given the time varying and nonlinear nature of the electro-hydraulic servo sys-tems, the backstepping controller design technique can be a viable approach thatensures a stable design. However, control algorithms of high complexity are de-rived. This puts a computational burden on the real time computer, as well asmakes debugging and tuning a difficult procedure. Moreover, combination withSMC can induce chattering for low complexity designs and adaptation laws giverise to more tuning parameters. It is therefore highly desirable that a stable con-troller can be designed without many computations, so that it is easily deployable.This aspect is also considered in [9]. Furthermore, the varying and unknown pa-rameters have to be considered, either by adaptation or by the controller beingrobust against them.A design procedure similar to the ones proposed in [41] or [10] seem to be promis-ing in that the step presenting the highest complexity of the backstepping designhas been replaced with a simpler algorithm. As a result, it is intended at this pointthat a similar design be applied for the studied 2-DOF manipulator system. Afterthe design is complete, stability in the sense of Lyapunov is considered.In order to design a control algorithm, the model of the system is developed. Thisis the topic of the following chapter.

Note about block diagrams In the entirety of the report, block diagrams are usedthat feature feedback loops. The feedback signals are provided by sensors, whoseoutput is voltage. It is considered that the bandwidth of all sensors is very wide,comparable to the sampling frequency. As a result, the sensors can be represented


by simple gains. Furthermore, since the gains have been adjusted in the dataacquisition algorithm, in the block diagram each sensor’s gain can be consideredto be equal to one. The following block diagram represents an ideal general case,where as an example a position feedback loop is considered. In Figure 1.2, thisexample system with measurement units is illustrated.

G (s)p

K=1

X(s) [m]

[m][V] [V][m]

X (s)ref [V] [V]

Figure 1.2: Block diagram of position feedback loop.

Using block diagram manipulation, the same system can be represented by theblock diagram of Figure 1.3:

G (s)pX(s) [m]

[m]

X (s)ref [V] [V]K=1

[V][m]

[m][V][m]1

K =1X (s)ref [m]

Figure 1.3: Manipulated block diagram of position feedback loop.

The plant is represented by the transfer function Gp(s), the control input is a volt-age signal for an electric valve. In the case a compensator was present in cascadewith the plant, its input can be in the unit of the physical variable, here in [m], andits output in [V]. This will be the case for all the block diagrams of the report. Theshaded part will be taken into consideration, meaning that the reference input andthe measured output of a system will be in the same units, those of the output.

Note regarding existence and uniqueness of solutions In this report, existenceand uniqueness of the solution that satisfy the initial conditions for the differentialequations of the form:

x = f(x) , x(t0) = x0

Existence and uniqueness of the solutions requires Lipschitz continuity [19] , [24] ofthe function f(x). It is assumed to hold for all the systems of nonlinear differentialequations in the report.

Chapter 2

System Modeling

The development of the selected control algorithms requires the development ofa model. The behavior of this model should represent as close as possible theactual system. As a result, in this chapter the model of the system is developed.Validation of the model using experimental data follows, a procedure that showswhether the model, and in extension the simulations, can be trusted.

2.1 Nonlinear model of the system

2.1.1 Model of the manipulator in joint space

Figure 2.1: Illustration of the rigid linkmanipulator.

uv

Ps

PT

xp

Figure 2.2: Diagram of the symmetrical actuator.

The nonlinear model of the rigid manipulator, illustrated in Figure 2.1, can be de-rived using the Euler-Lagrange equations, selecting the joint angles as coordinates.The derivation of the model for the planar 2DOF manipulator has been derivedin the literature in works such as [43] and [23]. The equations of motion have the

11

12 Chapter 2. System Modeling

form:

D(φ)φ + H(φ, φ)φ + G(φ) = τ (2.1)

where τ is the vector of applied joint torques and φ is the vector of the joint angles.

2.1.2 Model of the manipulator in the actuator space

Since the equations of motion are functions of the selected generalized coordinates,which are the joint angles, it is desirable that they are expressed using variablesdirectly related to the actuators, such as the actuator length or the piston position.The reason for this transformation is that these variables are measured and directlycontrolled.The joint angles can be related to the actuator lengths via the relationships:

φ1 = w1 − v1 + arccos

(L2

OQ + L2OH − x2

1

2LOQLOH

)(2.2)

φ2 = 2π − w2 + v2 − arccos

(L2

BW + L2BG − x2

22LBW LBG

)(2.3)

Furthermore, the angular velocities and acceleration of the joints are related to thepiston velocities and acceleration by:

φ = Js(x)x (2.4)

φ = Js(x)x + Js(x)x (2.5)

Js =

2x1√4L2

OQ L2OH−(L2

OQ+L2OH−x2

1)2

0

0 − 2x2√4L2

BG L2BW−(L2

BG+L2BW−x2

2)2

(2.6)

The joint torques, τ, can be related to the forces applied by the actuators, F, by therelationship:

τ =JD(x)F =(J−1

s (x))T

F (2.7)

JD(x) =[

LOQ sin(α1) 00 −LBG sin(α2)

](2.8)

α1 = arccos

(L2

OQ + x21 − L2

OH

2LOQx1

), α2 = arccos

(L2

BG + x22 − L2

BW2LBGx2

)(2.9)

2.1.3 Modeling of the hydraulic actuators

The hydraulic actuators are two symmetrical hydraulic cylinders. The position ofeach cylinder’s piston is controlled via a 4/3 flow control proportional servo valve.

2.1. Nonlinear model of the system 13

Furthermore, the supply and tank pressures present small variations throughoutoperation, so they are assumed to be almost constant. A diagram of one actuatoris presented in Figure 2.2. The two control volumes are highlighted in differentcolours.

Servo valve

The servo valve’s response can be described as that of a second order system, usingan approximation of the frequency response from the datasheet [33]:

Xv(s) =ω2

vs2 + 2ζvωvs + ω2

vU(s) (2.10)

Pressure dynamics

Flow continuity equations are used to model the pressure dynamics for the cham-bers of each cylinder. An assumption here is made regarding the effective oil bulkmodulus of each chamber. It is assumed that for each cylinder, both are equal.Furthermore, since the cylinder is symmetric and the valve is matched and sym-metric, the load pressure can be used instead of the each chamber’s pressure. Thisdecreases the order of the system. The load pressure dynamics for each cylinder isdescribed by [2]:

PL = βe f f

(1

VA(xp)+

1VB(xp)

)(QL − Ax− Cl PL) (2.11)

where the effective bulk modulus, considering constant operation temperature, isgiven by [20]:

βe f f = a1βe f f ,maxlog10

(a2

PPmax

+ a3

)(2.12)

Other models for effective oil bulk modulus, where the dissolved air in the fluidis taken directly into consideration are available [2]. However, knowledge of thepercentage of dissolved air is difficult to acquire.In Equation 2.11, cross-port or internal leakage flow is assumed to be laminar andis characterized by the leakage coefficient Cl . The volumes of each chamber, VA(xp)

and VB(xp) respectively, contain the volume of the hoses and the volume of eachchamber.

Flow through the valve

For the symmetric and matched servo valve orifices, the load flow equation is givenby [2]:

QL = Kvxv

√Ps − Pt − sign(xv)PL (2.13)


assuming that the flow is turbulent.

Motion of cylinder piston

Applying Newton’s second law for the cylinder piston, the equation of motion isacquired [20]:

mp x = APL − Bv x− FC − F (2.14)

The mass of the fluid in the cylinder’s chambers has been neglected as well as anygravitational effects. Furthermore, the mass of the piston, mp, is much smaller thanthe elements of the inertia matrix and can also be neglected.

2.1.4 Model of the system

Expressing Equation 2.1 in actuator space, using Equations 2.2 - 2.3 , 2.4 - 2.5 and2.7, the model of the manipulator in actuator space is derived:

M(x)x + C(x, x)x + G(x) = F (2.15)

where x =[x1 x2

]T, F =

[F1 F2

]Tand from Equation 2.7, for each actuator:

Fi = APL,i − Bv,i x− FC,i i = 1, 2 (2.16)

Finally, the relationship between the total actuator length, xi and the piston positionxp,i can be derived using Figure 2.3:

xi = xmin,i + HB + xp,i (2.17)

xi,min

Figure 2.3: Relationship between the total actuator length and the piston position.

Summary of the nonlinear model model

In actuator space, the model of the system can be written as:

M(x)x + [C(x, x) + Bv] x + [G(x) + FC] = APL (2.18)

2.2. Validation of the nonlinear model 15

and for each actuator:

PL = βe f f

(1

VA(xp)+

1VB(xp)

)(QL − Ax− Cl PL) (2.19)

QL = Kvxv

√Ps − Pt − sign(xv)PL (2.20)

Xv(s) =ω2

vs2 + 2ζvωvs + ω2

vUv(s) (2.21)

2.2 Validation of the nonlinear model

In order to proceed with the controller design, the derived model should be verifiedwith data from the laboratory setup. This is an essential step, because simulationusing a verified model can help with the selection of tuning parameters that arenot easy to systematically derive. Moreover, in the case of linear controller design,the designed controller can be applied on the actual system with minimal retuning.

2.2.1 Verification of the gravitational terms

To verify the gravitational terms, a small input voltage is applied on the servo valveof one servo system, while the other remains on a fixed position. Then the cylindermoves with a low velocity from end to end. By doing this test, the hydraulic modelis verified, as well as the gravitational term of the mechanical system, since owingto the low value of the velocity, only the gravitational term from the mechanicalmodel has value other than zero. The gravitational terms are verified through theload pressure for each servo system.In Figures 2.4 and 2.5, the load pressures of each servo system when it is beingretracted is shown. For both cases, the other servo system is in the extendedposition.

3 4 5 6 7 8 9 10 11Time [s]

-80

-60

-40

-20

0

20

40

60

Loa

d P

ress

ure

[b

ar]

Cylinder 1

MeasurementSimulation

Figure 2.4: Load pressure of servo system 1.

4 6 8 10 12 14Time [s]

-50

-40

-30

-20

-10

0

10

20

30

40

Loa

d P

ress

ure

[b

ar]

Cylinder 2


Figure 2.5: Load pressure when of servo system 2.

The discrepancies between the experimental values and the simulations are prob-ably due to the parameters selected to model the effective bulk modulus. Using


Equation 2.12, it is not easy to tune all parameters for an exact result. Other modelsfor the effective oil bulk modulus, describe the physical process in a more naturalway, however rely on parameters that are difficult to determine, such as the per-centage of the air dissolved in the fluid. Another factor that can lead to differencesfrom the actual system, is the angle of the center of mass of the second link, whosecalculation is not considered in this report. All in all, the behavior of the forceapplied from the cylinders to the links is of the same form for both the model andthe actual system.

2.2.2 Verification of the dynamics

Verifying the dynamics of the system is of crucial importance for the assessmentof the plant to be controlled, as well as for the evaluation of the control design.To proceed with the verification, a series of step voltage inputs was applied toeach servo valve. Then, the same inputs were applied to the simulation model atthe same time instances. The point of interest is the magnitude and frequency ofoscillations during the transients cause by the step inputs. The input voltages areshown in Figures 2.6 and 2.7.

0 5 10 15 20 25 30 35Time [s]

-3

-2

-1

0

1

2

3

Inp

ut

volt

age

[V]

Cylinder 1

Figure 2.6: Step voltage sequence for servovalve 1.

0 5 10 15 20Time [s]

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

3

Inp

ut

volt

age

[V]

Cylinder 2

Figure 2.7: Step voltage sequence for servovalve 2.

Regarding the first servo system, that drives the first link of the manipulator, theload pressure response can be seen in Figure 2.8The highest discrepancy between the response of the model and the laboratorysetup happens when the voltage is stepped and the cylinder is close to its endpositions, at 9.65 [s] and 22.75 [s] . At these time instances also, the input voltagewas stepped with a different sign, leading to change in the direction of motion.This leads to the initial conclusion that the damping ratio varies with the pistonposition. Similar results were acquired when Cylinder 2 was retracted.Regarding the second servo system, the step responses are shown in Figure 2.10:The same conclusions can be made regarding the damping of the system. Theconstant difference on the load pressure of the second servo system may be due

2.3. Linearization 17

5 10 15 20 25Time [s]

-50

0

50

100

Loa

d P

ress

ure

[b

ar]

Cylinder 1


Figure 2.8: Load pressure of servo system 1 duringstep voltage inputs. Cylinder 2 is extended.

9.5 10 10.5 11 11.5 12 12.5 13Time [s]

0

10

20

30

40

50

60

70

80

90

100

Loa

d P

ress

ure

[b

ar]

Cylinder 1


Figure 2.9: Enhanced part of Figure 2.8.

4 6 8 10 12 14 16 18 20Time [s]

-70

-60

-50

-40

-30

-20

-10

0

10

20

30

Loa

d P

ress

ure

[b

ar]

Cylinder 2


Figure 2.10: Load pressure of servo system 2 dur-ing step voltage inputs. Cylinder 1 is extended.

9 9.5 10 10.5 11 11.5 12Time [s]

-25

-20

-15

-10

-5

0

5

10

15

Loa

d P

ress

ure

[b

ar]

Cylinder 2


Figure 2.11: Enhanced part of Figure 2.10.

to the values of the moment of inertia around the center of mass, as well as theposition of the center of mass. Both topics were not considered for the report andthe values of the parameters used were derived in previous reports, such as [11].All things considered, the model can be trusted for simulations, however careshould be taken before applying a control law to the actual system, if in the sim-ulated model the magnitude of oscillation is high, since the model seems moredamped in specific positions.

2.3 Linearization

Linearizing a nonlinear model can provide information about the system proper-ties that can be important for the control design process, even if nonlinear con-trollers are to be applied. Information regarding damping ratio, amplitude peaksand eigenfrequencies can help explain phenomena observed during simulationsor during the testing of the controller on the system. Furthermore, this piece ofinformation can be used to explain in a more quantitative way intuitive observa-tions made by an initial overview of the nonlinear model. The linearized system is


derived in Appendix C.

2.4 Chapter 2 summary

In this chapter, the model of the system was derived using the Euler-Lagrangeequations. The hydraulic system model was also derived, based on the assump-tions of turbulent flow through the valves’ orifices, laminar leakage flow betweeneach chamber of the cylinders and equal effective oil bulk modulus in both cham-bers of each cylinder. The model was validated against experimental data. In thefollowing chapter, the selected control design are adapted to the 2DOF manipulatorsystem.

Chapter 3

Simplified backstepping control de-sign

In this chapter, the results of the designs of reduced complexity, are adapted tothe system in question. The design follows that of [41]. In the final section ofthe chapter, stability of the system in the sense of Lyapunov is investigated. It isdeemed difficult to estimate the stability margins for the system in a systematicway. Nonetheless, this is an approach that greatly reduces the complexity of thebackstepping controller, at the cost of global stability results.This controller will be denoted as RABLIN, from [41], standing for Robust Adap-tive Backstepping LINear controller.

3.1 Backstepping controller design

The goal of the control design is that each hydraulic servo system follows its pistonposition measurement. To this end each system is written as a SISO system andthe couplings due to the mechanical model are considered as disturbances. As aresult, the two mechanical systems can be written as:

M11(xp)xp1 + M12(xp)xp2 + (C11(xp, xp) + B1)xp1 + C12(xp, xp)xp2 + G1(xp) + FC,1(xp1) = APL,1

(3.1)

M22(xp)xp2 + M21(xp)xp1 + (C22(xp, xp) + B2)xp2 + C21(xp, xp)xp1 + G2(xp) + FC,2(xp2) = APL,2

(3.2)

PL,i = βe f f

(1

VA,i(xp,i)+

1VB,i(xp,i)

)(−Cl,iPL,i − Axi + Kv,iuv,i) (3.3)

i = 1, 2 for each servo system. The dynamics of the valve have been neglectedand furthermore, a pre-compensator has been used for the system since the load

19

20 Chapter 3. Simplified backstepping control design

pressure is available for measurement and no parameters are needed:

uv,i =

√2

Ps − Pt − sgn(uv,i)PL,iuv,i (3.4)

It is thus assumed that the model of the system, using the pre-compensator, isLipschitz continuous. This means that the solutions to the differential equationsexist and are unique.The controller design has the same objective for both servo systems, so the follow-ing applies equally to both. However, due to the differences in the parameters, anydesign parameters will not be equal. The backstepping controller design is shownfor the first servo system in the following. The subscript i is therefore dropped aswell as the arguments of the system parameters.Defining the state variables x1 = xp, x2 = xp, the hydraulic servo system is writtenin the state space form:

x1 = x2 (3.5)

x2 = −ϑ1x2 + ϑ2PL − d(t) (3.6)

PL = −ϑ3PL − ϑ4x2 + ϑ5u (3.7)

where the uncertain system parameters and disturbances due to coupling and thegravitational force:

ϑ1 =C11 + B1

M11, ϑ2 =

AM11

, d(t) =M12 xp2 + C12 xp2 + G1

M11(3.8)

ϑ3 = βe f f

(1

VA(xp)+

1VB(xp)

)Cl , ϑ4 = βe f f

(1

VA(xp)+

1VB(xp)

)A (3.9)

ϑ5 = βe f f

(1

VA(xp)+

1VB(xp)

)Kv (3.10)

The system described by Equations 3.5 to 3.7 is in strict feedback form. From [26],the error dynamics of the system are derived via the coordinate transformation:

z1 = x1 − r z2 = x2 − a1(z1) z3 = PL − a2(z1, z2, ϑ) (3.11)

where ϑ = ϑ + ϑ is the estimate of the unknown parameters, r the piston positionreference signal and aj, j = 1, 2 are the virtual control laws. The parameter estima-tion error is represented by the symbol ϑ. The derived error dynamics in the new

3.1. Backstepping controller design 21

coordinates z are:

z1 = x2 − r (3.12)

z2 = −ϑ1x2 + ϑ2PL − d(t)− ∂a1(z1)

∂z1z1 (3.13)

z3 = −ϑ3PL − ϑ4x2 + ϑ5u−

∂a2(z1, z2, ϑ)

∂z1z1 +

∂a2(z1, z2, ϑ)

∂z2z2 +

(∂a2(z1, z2, ϑ)

∂ϑ

)T˙ϑ

(3.14)

The backstepping controller is developed in the following steps:

Step 1 The error dynamics for the first subsystem, z1, with control input a1(z1)

can be written as:

z1 = x2 − r = z2 + a1(z1)− r (3.15)

A radially unbounded, decrescent Lyapunov function candidate is:

V1(z1) =12

z21 (3.16)

For

a1(z1) = −k1z1 + r k1 > 0 (3.17)

the derivative of V1(z1) becomes:

V1(z1) = −k1z21 + z1z2 (3.18)

which is negative definite when z2 = 0. The seconds step ensures that z2 is drivento zero, so that the first subsystem with Lyapunov function V1(z1) has the originas global asymptotic stable equilibrium point.

Step 2 The error dynamics of the first subsystem and the state variable x2 and itsdesired value, or the virtual control, a1(z1) become:

z1 = −k1z1 + z2 (3.19)

z2 = x2 − a1(z1) = −ϑ1x2 + ϑ2PL − d(t)− (r + k1z1) =

= −ϑ1x2 + ϑ2[z3 + a2(z1, z2, ϑ)

]− d(t)− r− k1r + k1x2 (3.20)

The virtual control input, a2 is multiplied by an uncertain term, that varies alongthe trajectory. An adaptive algorithm can be designed, however there will be a di-vision with the adaptive term ϑ2. This is unwanted, since the adaptive terms can beproven to be bound, but there is no guarantee that their value will not reach zero.


In [41], as well as in [15] and [50], the discontinuous projection algorithm is usedto guarantee that the virtual control input is not infinite. However, this approachis more suitable if the other steps of the backstepping procedure are considered.Here, since it is known that the last step will be designed in a simplified way, amethod to tackle the input multiplicative uncertainty is the following [26].Instead of estimating ϑ2 and dividing with its estimate, the reciprocal of ϑ2 isestimated. Selecting the virtual control input a2 as:

a2 = ρ2u2 , ρ2 =1ϑ2

(3.21)

then the term

ϑ2a2 = ϑ2ρ2u2 = ϑ2ρ2u2 + ϑ2ρ2u2 = u2 + ϑ2ρ2u2 (3.22)

Then, the dynamic equation of z2 becomes:

z2 = −ϑ1x2 + ϑ2z3 + u2 + ϑ2ρ2u2 − d(t)− a1(z1) (3.23)

where az1 = r + k1(x2 − r) is considered available, assuming the velocity is accu-rately measured or estimated. In order to select a virtual control law u2 and assessstability of the second subsystem, {z1, z2}, a Lyapunov function candidate can be:

V2(z1, z2, ρ2, ϑ1, u2) = V1(z1) +12

z22 +

12γ1

ϑ21 +|ϑ2|2γ2

ρ22 (3.24)

V2(z1, z2, ρ2, ϑ1, u2) is positive definite, radially unbounded and decrescent, since:

12||z||2 + 1

2γ1ϑ2

1 +|ϑ2,min|

2γ2ρ2

2 ≤ V2 ≤12||z||2 + 1

2γ1ϑ2

1 +|ϑ2,max|

2γ2ρ2

2 (3.25)

Taking the first derivative of V2, with the arguments dropped and using the resultsfrom Equation 3.18:

V2 =− k1z21 + ϑ2z2z3 + z2 [z1 − ϑ1x2 + u2 − d(t)− a1(z1)] +

1γ1

ϑ1˙ϑ1+

+ ρ2

[|ϑ2|γ2

˙ρ2 + z2ϑ2u2

](3.26)

By noting that the sign of ϑ2 is always positive and, furthermore, by selecting:

u2(z1, z2, ϑ1, a1) = −z1 + ϑ1x2 + a2r + a1 − k2z2 , k2 > 0 (3.27)

the derivative of V2 becomes:

V2 =− k1z21 − k2z2

2 + ϑ2z2z3 + z2 [a2r − d(t)] + ϑ1

[˙ϑ1

γ1+ z2x2

]+

+ ρ2|ϑ2|[ ˙ρ2

γ2+ z2u2

](3.28)

where it is assumed that the rate of variation of ϑ2 is much slower than that of ρ2.


Selection of the parameter update algorithm In order to render Equation 3.28negative semidefinite, to ensure boundedness, the parameter adaptation laws canbe selected as:

˙ϑ1 ≈ ˙ϑ1 = −γ1z2x2 ˙ρ2 ≈ ˙ρ2 = −γ2z2u2 (3.29)

The resulting Lyapunov function candidate derivative becomes:

V2 =− k1z21 − k2z2

2 + ϑ2z2z3 + z2 [a2r − d(t)] (3.30)

Design of the robust term The term a2r is a robust control input to compensatefor the disturbance term d(t). One way to select it, in order for the controller to besmooth, is according to [15] , [41]:

a2r = −z2Dδ

(3.31)

where D = max(|d(t)|) is the maximum value of the disturbance term d(t) and δ

is a positive design variable. From Equation 3.30:

V2(z1, z2, ϑ) = −k1z21 − k2z2

2 + ϑ2z2z3 +

[−z2

2Dδ− z2d(t)

]=

= −k1z21 − k2z2

2 + ϑ2z2z3 −Dδ

[z2 +

δ

2Dd(t)

]2

+δ

4Dd(t)2

≤ −k1z21 − k2z2

2 + ϑ2z2z3 +δ

4DD2 (3.32)

The second equality came from completing the squares of the bracketed term.Supposing that z3 = 0, in order to conclude on the stability and boundedness inthe presence of the disturbance d(t) and the robust control law a2r, the derivativeof V2(z1, z2, ϑ) should be negative. This happens when:

k1z21 + k2z2

2 >δD4⇒ z2

1α2 +

z22

β2 > 1 (3.33)

α =

√δD4k1

β =

√δD4k2

(3.34)

which denotes an ellipse in state space of z1, z2 with center on the origin, as shownin Figure 3.1 A more conservative estimate of the set where V2 < 0 can be foundfrom 3.32:

V2(z1, z2, ϑ) ≤ −k0||z||2 +δD4

(3.35)


z2

z1

β

α

V2 < 0

0

Figure 3.1: Boundedness set of the subsystem of z1, z2, for k2 > k1 The dotted circle presents a moreconservative estimate.

where k0 = min {k1, k2}, leading to the condition:

||z||2 = z21 + z2

2 >δD4k0

(3.36)

The above results are based on the fact that the disturbance d(t) is bounded, whichpre-assumes stability in the sense of Lyapunov for both servo systems. For themore conservative estimate, the set for which V2 ≤ 0 is:

R =

{z(t) : ||z(t)|| ≥

√δD4k0

}(3.37)

with the equality being satisfied on the circle circumference. Since everywherein the set R, outside the circle, V2 ≤ 0 and also from the fact that V2 is positivedefinite, radially unbounded [26]:

||z||∞ ≤{||z(0)||,

√δD4k0

}(3.38)

By selecting the design variable δ appropriately, the limit can be decreased. So, ifthe initial values of the error states trajectories:

• are located outside the set R, V2 is positive semi-definite as a worse estimateand the states tend to move away from the origin. Then, their trajectoriesencounter the bounds of the circle and stay trapped there, since outside thesebounds the derivative of V2 is negative semi definite.

• are located outside of the bounds, then the derivative of V2 is negative semidef-inite. In order to derive some additional assumptions about the behavior of


these trajectories, the LaSalle-Yoshizawa Theorem (5) can be used [26]. SinceV2 is positive definite, radially unbounded and, inside the set R:

V2 ≤ −W(z1, z2) (3.39)

W(z1, z2) = k0||z||2 −Dδ

4(3.40)

W is positive semi-definite. From the LaSalle-Yoshizawa Theorem 5 [26] then,

limt→∞

W(z1, z2) = 0⇒ ||z|| =

√δD4k0

(3.41)

As a result, the error states will lie on the bound of the circle. However, nothingcan be said about the parameter estimates convergence in the presence of the dis-turbance d(t), apart from the fact that they are bounded from V2 being negativesemi-definite.

Summary of the first two steps

Up to this point, a Lyapunov function for the subsystem of z1, z2 has been con-structed by selecting appropriate virtual control laws. The mechanical subsystemis characterized by two uncertain parameters and a disturbance term, which is as-sumed to be bounded. This term is the result of couplings between the two linksand the gravitational force of the respective link.The two uncertain terms have been dealt with a certainty equivalence control law,with parameter update algorithms derived by the Lyapunov function. Further-more, a robust control law was designed so that the boundedness of the state vec-tor can be ensured. However, this result is valid only when the third error variablez3 is equal to 0.The closed loop subsystem derived by substituting the virtual control law a2(z1, z2, ϑ)

from Equation 3.21,3.27 in 3.20, becomes:[z1

z2

]=

[−k1 1−1 −(k2 +

Dδ )

]︸︷︷︸

A

[z1

z2

]+

[01

]ϑt

Tφ1 +

[0−1

]d(t) +

[01

]ϑ2z3 (3.42)

where ϑt =[ϑ1 ρ2

]T. In absence of the disturbance term, assuming fast adapta-

tion and if the error variable z3 = 0, then the error variables z1 and z2 will go to 0if matrix A is Hurwitz. The eigenvalues of A are:

λ1 = −k1 < 0 λ2 = −(k2 +Dδ) < 0 (3.43)

that verify the Hurwitz criterion. D and δ are estimated for the worst case andselected, respectively. The values of k1, k2 can be selected. The higher, the wider


the bandwidth of the outer loop. This is not always desirable, due to actuatorsaturation and sensor noise. Tuning of these parameters will be done in a followingsection.The final step would be to complete the backstepping design for the load pressureloop. In the dynamic equation of z3 the actual control law, uv, is available fordesign. However, a high degree of complexity is required, due to the need for thetime derivative of the virtual control law a2(z1, z2, ϑ). Moreover, three uncertainparameters that need to be dealt with appear. This leads to more design variablesthat need tuning. Finally, if it is desirable that the initial control volumes are alsoestimated, since the system is not linear in these parameters, the method proposedin [15] leads to new parameter estimations and more complex expressions for theLyapunov function.A simplification of this more complex step was considered in [41]. A similar ap-proach will follow in the next section.

3.2 Load pressure controller design

The purpose of the inner, load pressure controller is to render the error variablez3 = PL − a2(z1, z2, ϑ) equal to 0. In other words, the load pressure needs to followits reference provided by the virtual control variable a2. This way, the completesystem will be bounded in a region around 0 in the state space. One approachto accomplish this, avoiding the complex final step of an adaptive backsteppingdesign, is to design a linear controller for the linearized system, with the loadpressure as output and valve voltage, uv as input. Linearization of the system isdeveloped in Appendix C.The block diagram of one of the two hydraulic servo systems is shown in Figure3.2.The block diagram is in the time domain, but Laplace notation is used forintegration and to also denote that the valve dynamics are present, just neglectedin the analysis here.

KqG (s)vuv

βeff

Vtot

VA BV1s

Cl Kqp

PL

A

A 1M Bs

1s

xp xp

d(t)

v

Figure 3.2: Block diagram of one of the two the hydraulic servo systems.

Since a pressure controller needs to be designed, the block diagram is brought inthe form shown in Figure 3.3, with the load pressure and output and the valvevoltage as input. The damping controller denoted by H(s) is designed in section

3.2. Load pressure controller design 27

3.2 and is not present in the actual system, being part of the controller.

KqG (s)vuv

βeff

Vtot

VA BV1s

Cl Kqp

PL

A A1M Bs

d(t)

H(s)

v

Figure 3.3: Block diagram of one of the twohydraulic servo systems.The shaded partis a damping controller, or pressure feedbackloop.

KqG (s)vuv

βeff

Vtot

VA BV1s

Cl Kqp

PL

A A1M Bs

d(t)

H(s)

v

Kq

uc

Figure 3.4: Implementation of pressure feedbackloop.

Finally, the block diagram is simplified in the form illustrated in Figure 3.5

KqG (s)vuv

PL

AM Bs

d(t)

GPL

v

'

Figure 3.5: Block diagram of the pressureloop with the disturbance term. The primein GPL means that Kq is not included in theblock diagram, but is shown at thesummation point.

Kquv

PL

AM Bs

d(t)

GPLGC

a (z , z ,θ)1 22

v

'

Figure 3.6: Block diagram of the closed loop loadpressure control system. The transfer function ofthe valve is not shown.

From Figure 3.5, the transfer function neglecting the valve dynamics can be writtenas:

GPL(s) =PL(s)Uv(s)

=Kq

Ms+BvBv(Cl−Kqp)+A2(

sωn

)2+ 2 ζ

ωns + 1

(3.44)

ωn =

√βe f f Vtot

[Bv(Cl − Kqp) + A2

]MVA,0VB,0

(3.45)

ζ =ωn

2BvVA,0VB,0 + M(Cl − Kqp)βe f f Vtot

βe f f Vtot[A2 + Bv(Cl − Kqp)

] (3.46)

One approach is to use feedback control for the load pressure to follow its referencea2(z1, z2, ϑ), as shown in Figure 3.6. In order to design linear controllers based onthe frequency domain analysis, the point of linearization must be first selected, sothat the Bode diagram of the system to be controlled can be plotted. The choiceof the point of linearization is considered in Appendix C.4. The linearization pointconstitutes one possible choice, that corresponds to a combined worst case possibleconfiguration of the mechanical system. The design could possibly be completedby using the absolute most challenging point for each isolated servo system, but the


design would be conservative. The reason for this is that the worst cases for eachservo system might not correspond to an actual configuration of the manipulator.A bode plot for the transfer function 3.44 is shown in Figure 3.7.

80

100

120

140

160

Mag

nitu

de (

dB)

Uncompensated System 1Uncompensated System 2Compensated System 1Compensated System 2

10-1 100 101 102 103 104-90

-45

0

45

90

Pha

se (

deg)

Bode Diagram

Frequency (rad/s)

Figure 3.7: Bode plots of the servo systems,with and without pressure feedback.

-150

-100

-50

0

50

Mag

nitu

de (

dB)

10-2 10-1 100 101 102 103 104 105-270

-180

-90

0

Pha

se (

deg)

Servo system 1Servo system 2

Bode Diagram

Frequency (rad/s)

Figure 3.8: Bode plots of the compensated servosystems, including valve dynamics.

Designing a load pressure controller would become easier in the case the resonantpeaks were absent. Furthermore, the magnitude due to the zero of the open looptransfer function leads to an increase in the magnitude of the closed loop transferfunction and should be avoided. This is done via the pressure feedback loop.

Design of the pressure feedback loop

The pressure feedback loop in the shaded part of Figure 3.3 denoted by the transferfunction H(s) is used to increase the damping of the system in order to improverelative stability margins. Depending on the way this transfer function is designed,the low frequency gain of the system is affected in different ways. However, since acompensator will close the loop, the low frequency gain will not be highly affectedif the pressure feedback is realized via a gain similar to "adding" leakage.Selecting the pressure feedback controller as a gain:

H(s) = Kp f (3.47)

the transfer function of GPL becomes:

GPL(s) =PL(s)Uv(s)

=Kq

Ms+BvBv(Cl+Kp f−Kqp)+A2(s

ωn

)2+ 2 ζ

ωns + 1

(3.48)

ωn =

√βe f f Vtot

[Bv(Cl + Kp f − Kqp) + A2

]MVA,0VB,0

(3.49)

ζ =ωn

2BvVA,0VB,0 + M(Cl + Kp f − Kqp)βe f f Vtot

βe f f Vtot[A2 + Bv(Cl + Kp f − Kqp)

] (3.50)


Solving Equation 3.50 for Kp f with the desired damping ratio yields:

zdes1,2 = 0.7

Kp f ,1 = 9.7791 · 10−12[

m3

sPa

]Kp f ,2 = 2.3894 · 10−11

[m3

sPa

](3.51)

The bode plots pressure feedback compensated servo systems are plotted in Figure3.7. The pressure feedback loop is realized through control input. Neglecting thedynamics of the valve, the realization is shown in Figure 3.4, where uc is a possiblecontrol input.Next step is the design of the closed loop load pressure system compensator. Thepresence of the zero makes the selection of the desired gain crossover frequencymore difficult, in that the low frequency gain is reduced but for high values of thecrossover frequency. It is not desired to have a closed loop bandwidth for the loadpressure loop comparable with the frequency of the valve. As a result, the zero canbe canceled by introducing a filter in cascade with the plant:

GF(s) =1 + τF,zs1 + τF,ps

(3.52)

The pole of the filter is selected to appear at frequency lower than the lowest fre-quency the zero due to the mechanical system appears:

ωF,p =1

τF,p≤ Bv,i

Mii[rad/s] (3.53)

The zero of the filter is placed in frequencies a little over than the poles can appear:

ωF,p,1 = 0.5 [rad/s] ωF,z,1 = 100 [rad/s]

ωF,p,2 = 1 [rad/s] ωF,z,2 = 200 [rad/s] (3.54)

The valve dynamics can be incorporated in the selection of the load pressure cas-cade controller. The controller can be chosen so that the closed loop load pressuresystem has a desired bandwidth, lower than the valve’s cutoff frequency. This canbe achieved with different ways. A Proportional, P, controller can be used, as wellas a Proportional-Integral, PI, that will provide increased low frequency gain sothat any disturbances at low frequencies are attenuated. In order to simplify thedesign, since by including the valve dynamics additional lag is included in thesystem, the P controller is used. This way, adding unnecessary negative phase isalso avoided.The fact that the inner loop that controls the load pressure has a large bandwidth,means that its settling time will also be as small. Then, this inner loop can beconsidered as a constant gain that has reached its reference very faster than thereference value is generated. This result however is valid only when the outer


loops that provide the reference have slower dynamics, or in other words, largertime constants.In this case, the selected gain crossover frequency for the pressure control loopis selected way below the frequency of the valve, at 100 [rad/s] for both servosystems. The proportional gains for the load pressure controllers are selected as:

Gc,1 = 8.3176 · 10−5 Gc,2 = 6.9984 · 10−5 (3.55)

The open loop bode plots for both servo systems, with the cascaded proportionalcontrollers are shown in Figure 3.8.In order to conclude on the inner, load pressure controller, a block diagram isshown in Figure 3.9

G (s)pca2 PL

d(t) AMs+Bv

G (s)pc

G (s)c

Figure 3.9: Load pressure controller.

In the block diagram of Figure 3.9, the reference input a2 comes from Equation3.21, Gpc is the pressure feedback compensated transfer function and Gc(s) is theproportional controller with the filter of Equation 3.52 in cascade.

Design of the controller for the whole system

In order to finalize the design, the tuning parameters for the backstepping con-troller should be selected. It can be noted that the reference value for the loadpressure, or the second virtual control law derived in Equation 3.21, can also bewritten as:

a2(e, e, ϑ1, ϑ2, r) =1ϑ2

[e(1 + k1k2) + e(k1 + k2) + ϑ1x2 + a2r + r

](3.56)

e = r− x1 (3.57)

e = r− x2 (3.58)

which looks similar to a Proportional-Derivative controller with acceleration feed-forward and adaptation terms. The gains of this Proportional-Derivative (PD) con-troller are given by:

KP = 1 + k1k2 KD = k1 + k2 (3.59)

k1 =KD

2∓

√K2

D − 4(KP − 1)

2k2 =

KD

2±

√K2

D − 4(KP − 1)

2(3.60)


on the condition that

K2D > 4(KP − 1) or KD = 2κ

√(KP − 1) , κ > 1 (3.61)

Selecting k1 and k2 parameters for the backstepping controller It was shown inEquation 3.56 that the reference for the load pressure loop looks similar to a PDcontroller with:

KD = k1 + k2 KP = 1 + k1k2 (3.62)

Before beginning with the selection of the gains, some assumptions are made:

• The parameter errors are zero, due to fast adaptation and persistence of ex-citation of the regressor vector. However this assumption is difficult to hold,due to the disturbance term d(t).

• The magnitude of the closed loop load pressure system is constant, and closeto 1, for a wide band of frequencies. This means that the response of the loadpressure to changes in its reference is fast.

Using Equations 3.8, 3.9 and 3.44, the block diagram of the open system is illus-trated in Figure 3.10

1s

1s

x2 x1

θ1

d(t)

θ2

PL

AMs+Bv

G (s)pc

G (s)pcG (s)c

a2

Figure 3.10: Block diagram of one servo system, representing one link of the manipulator.

By closing the position and velocity loops, with the control input a2 from Equation3.21, the block diagram is transformed in the one shown in Figure 3.11, with TPL

being the closed loop transfer function of the load pressure system:

TPL =PL(s)as(s)

=Gc(s)Gpc(s)

1 + Gc(s)Gpc(s)(3.63)

In order to show the cancellations of the adaptation terms, to reach the final system,the closed loop transfer function is not shown in the following and is assumed tobe close to 1. With this assumption, the closed loop block diagram is redrawn inFigure 3.12 as:


1s

1s

x2 x1

θ1

d(t)

θ2

PLa2 T (s)PL

θ2

1^

a2rθ1^

r

KD

KP

r

r

Figure 3.11: Block diagram of one closed loop servo system.

1s

1s

x2 x1

θ1

d(t)

θ2θ2

1^

a2rθ1^

r

KD

KP

r

r


1s

1s

x2 x1

θ1

d(t)

a2rθ1^

r

KD

KP

r

r


1s

1s

x2 x1

-θ1θ1^

r

KD

KP

r

r

Figure 3.14: Block diagram of one closedloop servo system.

1s

1s

x2 x1

KD

KPr TPL

s

Figure 3.15: Block diagram of one closed loopservo system.

Assuming that ϑ2 u ϑ2 the new block diagram is shown in Figure 3.13 Now as-suming that d(t) is "canceled" by a2r from Equation 3.31:and using the fact that the adaptation error ϑ1 = 0 the block diagram is redrawn inFigure 3.14 where the load pressure closed loop transfer function is taken into con-sideration, and the acceleration feedforward is neglected from the superpositionprinciple as another input.The transfer function of the plant to be controlled, considering the state variable x1


as an output, is:

G(s) =X1(s)Uv(s)

= TPL(s)1s2 (3.64)

The PD equivalent compensator is in cascade with the plant G(s) and has transferfunction:

PD(s) = KP + KDs

|PD(jω)|ωgc = KP

√1 +

(KD

KP

)2

ω2gc

∠PD(jω)|◦ωgc= arctan

(KD

KPωgc

)where ωgc [rad/s] is the desired gain crossover frequency. To design the controllerfor each servo system the desired gain crossover frequency and phase margin (PM)are defined. As a result, at the desired ωgc:

∠PD(jω)|◦ωgc= −180◦ + PM◦ −∠G(jω)|◦ωgc

(3.65)

|PD(jω)|ωgc =1

|G(jω)|ωgc

(3.66)

Taking into consideration the necessary condition of Equation 3.61, in order for thegains k1 and k2 to be real numbers:

|PD(jω)|ωgc = KP

√1 +

4κ2(KP − 1)K2

Pω2

gc ≈√

K2P + 4κ2KPω2

gc KP >> 1

∠PD(jω)|◦ωgc= arctan

(2κ√

KP − 1KP

ωgc

)≈ arctan

(2κ√KP

ωgc

)Favoring low derivative gains, KD, it is arbitrarily decided for the value of κ:

κ = 1.1 (3.67)

This is probably not the optimal value in regard to maximum attainable closedloop bandwidth or phase margin, but noise considerations led to this choice.This choice of κ leaves only one degree of freedom available for the design. Thegain KP can be selected so as to satisfy the desired gain crossover frequency cri-terion from Equation 3.66 and then this solution is used to calculate the phasemargin from Equation 3.65. Then, if the results are not satisfactory, the process canbe repeated for different gain crossover frequency or in the reverse order.For the first servo system, a gain crossover frequency of around 30 [rad/s] wasselected. As a result, the pressure feedback loop is around 3 times faster than


the position loop. This is conservative for the reference trajectory. Calculationsyielded:

KP,1 = 213.8447 KD,1 = 32.0962

PM = 67.5070◦ GM = 20.9 [dB]

k1,1 = 9.3625 k2,1 = 22.7337 (3.68)

For the second servo system, the same gain crossover frequency was selected, ie.ωgc = 30 [rad/s]. The PD compensator gains were calculated as:

KP,2 = 417.9481 KD,2 = 44.9225

PM = 57.7894◦ GM = 19.9 [dB]

k1,2 = 13.1039 k2,2 = 31.8185 (3.69)

The bode plots of the uncompensated and compensated transfer functions of G(s)for each servo system are presented in Figures 3.16 and 3.17 respectively.

-400

-300

-200

-100

0

100

Mag

nitu

de (

dB)

10-1 100 101 102 103 104 105-450

-360

-270

-180

-90

Pha

se (

deg)

UncompensatedCompensated

Bode Diagram

Frequency (rad/s)

Figure 3.16: Bode plots of the uncompensatedand compensated open loop position controlsystem 1.

-400

-300

-200

-100

0

100M

agni

tude

(dB

)

10-1 100 101 102 103 104 105-450

-360

-270

-180

-90

Pha

se (

deg)

UncompensatedCompensated

Bode Diagram

Frequency (rad/s)

Figure 3.17: Bode plots of the uncompensated andcompensated open loop position control system 2.

3.2.1 Analysis of the closed loop system stability in the Lyapunov frame-work

In order to investigate the stability of the closed loop system in the whole statespace, a Lyapunov function candidate is selected for the whole system, as in theprevious steps. Since a dynamic feedback controller has been used, the system isaugmented with a new state. Furthermore, since the output of the linear controlleris directly applied to the servo valves, the pre-compensator is not used and theparameter ϑ5 is now dependent on the load pressure. The resulting third errordynamic equation becomes:

z3 = −ϑ3PL − ϑ4x2 + ϑ5uv − a2(z1, z2, ϑ) (3.70)

3.3. Chapter 3 summary 35

The output of the linear controller is given by:

uv = −Kp f PL + ucl (3.71)

Ucl(s) =τF,zs + 1τF,ps + 1

[−Z3(s)] or (3.72)

ucl(t) = Gc

(τF,p − τF,z

τF,p

)x(t)− Gc

τF,z

τF,pz3(t) (3.73)

x(t) = − 1τF,p

x(t)− 1τF,p

z3(t) (3.74)

where x(t) is a state of the controller, used here for the stability proof.In order to investigate the stability properties of the origin as an equilibrium pointfor the error dynamics system, a Lyapunov function candidate is selected as:

V3(z1, z2, z3) = V2(z1, z2, ϑ) +12

z23 +

12

x2 (3.75)

Based on the positive definiteness and decrescence properties of V2, V3 is also pos-itive definite, radially unbounded and decrescent. The derivative of V3 is derivedusing also Equation 3.32 and using the same adaptation laws as for V2 in Equation3.29:

V3(z1, z2, z3) ≤ −k1z21 − k2z2

2 +δD4

+ ϑ2z2z3 + z3z3 + xx

≤ −k1z21 − k2z2

2 − ϑ5GcτF,z

τF,pz2

3 −1

τF,px2 +

δD4+

+ z3

[−(ϑ3 + ϑ5Kp f

)PL + ϑ2z2 −

1τF,p

x− ϑ4x2 + ϑ5GcτF,p − τF,z

τF,px− a2

]︸︷︷︸

≤0

(3.76)

In order to prove boundedness for the error variables of the whole system, the lastterm of the last inequality in 3.76 should be less or equal to zero. This approach willultimately need numerical calculation of the time derivative of a2, so that boundscan be set for all states and parameters.The disadvantage of this approach is that stability for linearized systems is onlyvalid locally and it is difficult to establish bounds of the states using Lyapunovanalysis. As a result, using the aforementioned approach, the stability of the closedloop system cannot be fully assessed for a larger portion of the state space and notat the region close to the linearization point.


In this chapter the backstepping-based control design of [41] was adapted for use inthe 2DOF manipulator system. The first two steps of the backstepping procedure,


of low complexity, are developed. The challenge in these first two steps is theselection of the adaptation gains, as well as the use of the discontinuous projectionalgorithm that ensures that there is no division by zero.In the last step, a linear control design technique was used so that the analyticalcalculation of the time derivative of the second virtual control law, a2, is avoided.The challenge in this step is the design of the damping controller, since apart fromthe resonant peak due to the low damping of the hydraulic system, the zero dueto the mechanical system is included. The limitation of the method is that anystability properties are local, around the point of linearization.The objective of the following chapter is that a control design, based on back-stepping, is designed that is robust to parameter variations and also avoids thedifferentiation of a2.

Chapter 4

Problem formulation and control de-sign

4.1 Problem formulation - Hypothesis

The simplified backstepping based design of Chapter 3 suffers from the difficulty ofassessing stability for the closed loop system. As a result, the problem formulationof the report can be defined based on the following hypothesis:

Hypothesis For the system described by Equations 3.8 - 3.10, it is possible to de-sign a nonlinear, backstepping based controller that provides robustness againstparameter variations and avoids high implementation complexity. Lyapunov sta-bility analysis can provide the stability bounds of the closed loop system.

4.2 Challenges of the design

The state transformation, as well as the error dynamics of the transformed systemare repeated at this point:

• Transformation:

z1 = x1 − r z2 = x2 − a1 z3 = PL − a2 (4.1)

• Error dynamics:

z1 = z2 + a1 − r

z2 = −ϑ1x2 − d(t) + ϑ2(a2 + z3)− a1

z3 = −ϑ3PL − ϑ4x2 + ϑ5u− a2

The challenges for the control design for the specific problem are the following:

37

38 Chapter 4. Problem formulation and control design

• The multiplicative uncertain terms ϑ2 and ϑ5 that are multiplied with thesecond virtual and the actual control input variable respectively.

• The time varying, additive disturbance term d(t).

• The high complexity of selecting the actual control law, in the final step ofthe backstepping procedure. This is due to the fact that even after findingthe derivative of a2, the uncertain terms in the second equation of the errordynamics enter the third equation.

• The uncertain parameters ϑ1, ϑ3 and ϑ4.

• Finally, the last challenge is acquiring signals that cannot be measured, suchas the piston velocity, especially in the presence of noisy sensor signals.

It is stated at this point that the approach considered up to now only fails todetermine stability for the whole closed loop system. This happens due to thesimplification reducing the complexity of third step of the backstepping procedure.It is natural at this point to consider, almost unchanged, the successful two firststeps and then search for a solution to the complexity problem due to the differen-tiation of the second virtual control.

4.3 Approach 1

In this subsection, the first approach for a simplified solution is developed. Thisdesign is based on using adaptive control with a robust part for the second step,almost exactly similar to the already discussed approach. The only difference isthe slight modification of the adaptation algorithm for the parameter ϑ2, so thatthe projection algorithm is avoided. For the third step, an adaptive controller isdesigned and the derivative of the second virtual control input is estimated bya differentiator based on a second order sliding mode technique using the supertwisting algorithm [39], [51].

4.3.1 Controller design and analysis

First step

For the first subsystem

z1 = z2 + a1 − r (4.2)

with radially unbounded and decrescent Lyapunov function candidate

V1(z1) =12

z21 (4.3)

4.3. Approach 1 39

the virtual control input

a1(z1, r) = −K1z1 + r (4.4)

leads to

V1(z1) = −K1z21 + z1z2 (4.5)

Second step

A similar adaptive and robust controller is repeated in the second step for thesubsystem:

z1 = −K1z1 + z2 (4.6)

z2 = −ϑ1x2 − d(t)− a1(z1, r) + ϑ2a2 + ϑ2z3 (4.7)

The term a1(z1, r) has no uncertainty and is easy to acquire, supposing the velocitysignal is known. The uncertain term ϑ2 is approached as [26]:

ρ2 =1ϑ2

, ρ2 =1ϑ2

= ρ2 + ρ2 (4.8)

If the virtual control input a2 is selected as:

a2 = ρ2u2 ⇒ ϑ2a2 = u2 + ϑ2ρ2u2 (4.9)

then Equation 4.7 can be written as:

z2 = −ϑ1x2 − d(t)− a1(z1, r) + u2 + ϑ2ρ2u2 + ϑ2z3 (4.10)

Selecting the radially unbounded decrescent Lyapunov function candidate

V2(z1, z2, ϑ1, ρ2) = V1(z1) +12

z22 +

12γ1

ϑ12+|ϑ2|2γ2

ρ22 (4.11)

its derivative along the solutions of the subsystem defined by Equations 4.6 - 4.7can be found to be:

V2(z1, z2, ϑ1, ρ2) =− K1z21 + z1z2 + z2 [ϑ2z3 − ϑ1x2 − d(t) + u2 + ϑ2ρ2u2 − a1(z1, r)] +

+1

γ1ϑ1

˙ϑ1 +|ϑ2|γ2

ρ2 ˙ρ2 (4.12)

under the assumption that ϑ2 varies slowly. Now the virtual control input u2 can beselected in order to render V2(z1, z2) at least negative (semi-)definite in the absenceof the state variable z3. One choice of u2 can be:

u2 = a1(z1, r) + ϑ1x2 − z1 + u2r − K2z2 (4.13)

u2r = −z2Dδ

(4.14)


Using this virtual control input, the derivative of V2 becomes:

V2(z1, z2, ϑ1, ρ2) =− K1z21 − K2z2

2 + ϑ2z2z3 + z2

(−z2

Dδ− d(t)

)+

+ ϑ1

[z2x2 +

1γ1

˙ϑ1

]+ ρ2|ϑ2|

[z2u2sign(ϑ2) +

1γ2

˙ρ2

](4.15)

From the definition of the unknown parameter ϑ2 in Equation 3.8, it is known thatit will always be positive. Moreover, assuming also that the actual parameters donot vary very fast, the following parameter adaptation laws can be derived:

˙ϑ1 ≈ ˙ϑ1 = −γ1z2x2 , ˙ρ2 ≈ ˙ρ2 = −γ2z2u2sign(ϑ2) = −γ2z2u2 (4.16)

Using the above adaptation laws:

V2(z1, z2, ϑ1, ρ2) ≤ −K1z21 − K2z2

2 + ϑ2z2z3 +δD4

(4.17)

Again, the same boundedness results can be derived for the error state variablesz1, z2, as in the case of the initial approach. As a result they are not repeated here.The virtual control law for the second step of the backstepping algorithm becomes:

a2(z1, z2, ϑ1, ρ2) = ρ2u2 (4.18)

u2 = a1(z1, r) + ϑ1x2 − z1 + u2r − K2z2 , u2r = −z2Dδ

(4.19)

˙ϑ1 = −γ1z2x2 (4.20)˙ρ2 = −γ2z2u2 (4.21)

a1(z1, r) = a1(x2, r, r) = r + K1r− K1x2 (4.22)

Third step

The third step involves the design of the controller that stabilizes the whole system.The error dynamics system can be written using the virtual control laws as:

z1 = −K1z1 + z2 (4.23)

z2 = −z1 −(

K2 +Dδ

)z2 − d(t) + ϑ2z3 + ϑ1x2 + ϑ2ρ2u2 (4.24)

z3 = −ϑ3PL − ϑ4x2 + ϑ5u− a2(z1, z2, ϑ1, ρ2) (4.25)

The difficulty in repeating the adaptive controller designed in step 2 stems fromthe need to find the time derivative of the second virtual control law a2. Of course,this would also introduce the uncertain terms of the second equation. However,their estimates cannot be used and new adaptation laws should be designed forthe already estimated parameters, leading to over-parameterization. A remedy for

4.3. Approach 1 41

this would be to use tuning functions [26], but the use of the term ρ2 makes thedesign more complicated.It would be desirable to somehow estimate the derivative of a2 and cancel in thedesign of the actual control input u. Then, a similar adaptive controller can beused, with only one over-parameterization for the parameter ϑ2.One way to calculate a2 using a non analytical way, is to use a second order slidingmode differentiator that utilizes the super twisting algorithm.

Design of the differentiator The differentiator is designed based on the conceptof second order sliding mode control. As a result, to sliding variables σ and σ aredriven to zero. These sliding variables can be designed as [51] , [39]:

σ = a2(z1, z2, ϑ1, ρ2)− a2(z1, z2, ϑ1, ρ2) (4.26)

σ = ˙a2(z1, z2, ϑ1, ρ2)− a2(z1, z2, ϑ1, ρ2) (4.27)

In the following, the arguments of a2 will be dropped for economy of space and toease the analysis. The estimate of a2, ˙a2, can be selected as:

˙a2 = −L1

√|σ|sign(σ) + q (4.28)

q = −L0sign(σ) (4.29)

L1 = 1.5L12 L0 = 1.1L

with L being the upper bound of a2 [39] , [51]:

L ≥ max|a2|

Then, the closed loop system of the differentiator, described by Equations 4.28 and4.29, becomes:

σ = −a2 − L1

√|σ|sign(σ) + q = −L1

√|σ|sign(σ) + w (4.30)

w = −L0sign(σ)− a2 ≤ L− L0sign(σ) (4.31)

The existence of the discontinuous functions in the equations of σ, w denote asliding mode for the variables w, σ, meaning that the latter will be driven to zero infinite time, as long as L0 ≥ L = |a2|max. Since σ = 0→ σ = 0 and the differentiationis almost exact with very small sampling period.At this point, the closed loop system has been augmented with two additional statevariables, namely σ and σ. Figure 4.1 presents the process of the differentiator inblock diagram form. The finite time convergence of σ, σ to the origin proof is pre-sented in [51]. In section D the same differentiator is used for velocity estimation.


a2^

a2^

a2

qq( 4.28 )( 4.29 )

σ

Figure 4.1: Block diagram of the second order, super twisting sliding mode differentiator.

Design of the stabilizing controller Having an estimate of the term inducingthe high complexity, an approach similar to the one used for the second step isfollowed. Consequently, the parameter ρ5 is defined:

ρ5 =1ϑ5

(4.32)

ϑ5u = ϑ5ρ5u3 = ϑ5ρ5u3 = u3 + ϑ5ρ5u3 (4.33)

As a result, using the previously designed virtual control laws, the third equationof the system to be stabilized is written as:

z3 = −ϑ3PL − ϑ4x2 + u3 + ϑ5ρ5u3 − a2 (4.34)

A Lyapunov function candidate for the system described by Equations 4.23 , 4.24 ,4.34 , 4.26 and 4.27 can be selected as [46]:

V3(z1, z2, z3, ϑ1, ρ2, σ1, σ1, ϑ2, ρ5, ϑ3, ϑ4) = V2(z1, z2, ϑ1, ρ2)

+12

z23 +

12γ3

ϑ23 +

12γ4

ϑ24 +

12γ6

ϑ22 ++

|ϑ5|2γ5

ρ25+

+

√12

σ2 + L0|σ| (4.35)

The arguments will be again dropped in the following. The derivative of V3 alongthe trajectories of the system can be calculated as:

V3 ≤− K1z21 − K2z2

2 +δD4

+ z3 [ϑ2z2 − ϑ3PL − ϑ4x2 + u3 + ϑ5ρ5u3 − a2] +

+1

γ3ϑ3

˙ϑ3 +1

γ4ϑ4

˙ϑ4 +1

γ6ϑ2

˙ϑ2 +|ϑ5|γ5

ρ5 ˙ρ5+

+− L1

2σ2

|σ| − a2σ√|σ|√

12

σ2

|σ| + L0

(4.36)

4.3. Approach 1 43

At this point, the selection of the actual control input variable u3 becomes moreapparent. One way to select it can be:

u3 = −ϑ2z2 + ϑ3PL + ϑ4x2 + ˙a2 − K3z3 (4.37)

Consequently, V3 is written with this selection for u3:

V3 ≤ −K1z21 − K2z2

2 − K3z23 +

δD4

+ z3[−ϑ2z2 + ϑ3PL + ϑ4x2 + ϑ5ρ5u3 + σ

]+

+1

γ3ϑ3

˙ϑ3 +1

γ4ϑ4

˙ϑ4 +1

γ6ϑ2

˙ϑ2 +|ϑ5|γ5

ρ5 ˙ρ5+

+− L1

2σ2

|σ| − a2σ√|σ|√

12

σ2

|σ| + L0

≤ −K1z21 − K2z2

2 − K3z23 +

δD4

+− L1

2σ2

|σ| − a2σ√|σ|√

12

σ2

|σ| + L0

+ z3σ (4.38)

with the selection of parameter adaptation laws, on top of the laws of Equations4.32 and 4.33, as:

˙ϑ3 = −γ3z3PL˙ϑ4 = −γ4z3x2 (4.39)

˙ϑ2 = γ6z3z2 ˙ρ5 = −γ5z3u3sign(ϑ5) = −γ5z3u3 (4.40)

under the assumption of slowly varying unknown parameters and since the signof the parameter ϑ5 is always known.Regarding the fifth term of the inequality 4.38, under the assumptions [46]:

• |a2| < A2 , A2 > 0 , a condition that pre-assumes stability,

• σ2

|σ| > ε2 , ε > 0

• −a2 < 12 L1εγ , 0 < γ < 1

then

V3 ≤ −K1z21 − K2z2

2 − K3z23 +

δD4

+ z3σ− L1

2(1− γ)

ε2√12 ε2 + L0

≤ −K1z21 − K2z2

2 − K3z23 +

δD4

+ z3σ(0) (4.41)

In the case where the fourth and fifth terms were absent, V3 would be negative-semidefinite and stability in the sense of Lyapunov of the closed loop system wouldbe assured. Another note is that the last term is free of the parameters σ and σ,meaning that the rate of decrease of these states is constant. Since the rate ofconvergence of both σ and σ is constant, σ ≤ σ(0). If the first three terms arelarger in magnitude than the third and fourth terms, then the states are bounded.


4.3.2 Summary of Approach 1

The actual control law, derived via the first approach and before using the pre-compensator of Equation 3.4, is summarized below:

u = ρ5u3 = ρ5(−ϑ2z2 + ϑ3PL + ϑ4x2 + ˙a2 − K3z3)

σ = a2 − a2 σ = ˙a2 − a2

a2 = ρ2

(a1 + ϑ1x2 − z1 − z2

Dδ− K2z2

)a1 = r + K1r− K1x1

˙a2 = −L1

√|σ|sign(σ) + q q = −L0sign(σ)

with parameter update laws:

˙ϑ1 = −γ1z2x2 ˙ρ2 = −γ2z2u2˙ϑ3 = −γ3z3PL

˙ϑ4 = −γ4z3x2˙ϑ2 = γ6z3z2 ˙ρ5 = −γ5z3u3

A drawback discovered in simulations

It is not initially evident how to select the gains for the adaptation laws, as well as toassess the robustness of the control algorithm against parameter variations and thepresence of measurement noise. Simulation can help qualitatively evaluate someselections of tuning parameters. To this end, the closed loop system is simulatedin Matlab Simulink [30]. Using initial conditions only for the estimations of theparameters ρ2 and ρ5.However, when the adaptation law is active for the parameter ρ5, very large, chat-tering input signals occur, which is unacceptable. Even with very fine tuning thisalgorithm cannot be trusted for use on the laboratory setup.

A trade-off A direct modification of the first approach considered, is to not useadaptation for the specific parameter. As long as bounds for stability can be esti-mated, all the other adaptation parameters will, eventually, reach trajectories thatguarantee that the states remain bounded. However, an auxiliary control input toat least bound the uncertainty coming from the rough estimate of ϑ5 could be used.In this case, the control law for u becomes:

u =1ϑ5

u3 + vaux (4.42)

where the ϑ5 is the estimate for the parameter ϑ5 made off-line and does not changedynamically and u3 the same as in Equation 4.37. Substituting the control input of

4.4. Approach 2 45

Equation 4.42 in 4.25:

z3 = −K3z3 − ϑ2z2 +

(ϑ5

ϑ5− 1)(ϑ3PL + ϑ4x2 + ˙a2 − K3z3 − ϑ2z2) + ϑ5vaux (4.43)

where an oversimplification has been made that the parameter adaptation errorsare zero, as well as the states σ and σ. It is not easy to find an upper bound for theterm multiplied by the uncertainty, since the parameter estimates ϑ2, ϑ3 and ϑ4 areonly bounded and no assumption is made regarding their upper bound.

4.4 Approach 2

A second approach is deemed necessary in order to tackle more effectively theshortcomings of the aforementioned first approach. The main shortcoming of thefirst approach is the large number of parameters that need tuning. In this secondapproach, it is intended that this number is kept to a minimum. In simulationsthere was no difficulty in selecting the adaptation gains for the uncertain parame-ters of the mechanical system, during the second step of the backstepping design.As a result, it is decided that these two first steps are designed the same way. Thethird step, which involves the design of the actual control input signal to the servovalve involved the parameter whose adaptation gain was difficult to tune. As aresult, the focus of the second approach is to use a robust approach, instead of anadaptive one, to counteract the effect of uncertainty due the unknown terms of thepressure dynamics subsystem.

4.4.1 Summary of the first two steps of the adaptive backstepping design

The error dynamics in the transformed state variables are:

z1 = z2 + a1 − r

z2 = −ϑ1x2 − d(t) + ϑ2(a2 + z3)− a1

z3 = −ϑ3PL − ϑ4x2 + ϑ5u− a2

Step 1 For the subsystem

z1 = z2 + a1 − r (4.44)

with radially unbounded and decrescent Lyapunov function candidate

V1(z1) =12

z22 (4.45)

selecting the first virtual control input a1(z1) as:

a1(z1) = −k1z1 + r , k1 > 0 (4.46)


leads to the closed loop subsystem:

z1 = −k1z1 + z2 (4.47)

which is exponentially stable when z2 = 0, as can be also seen by the derivative ofV1(z1):

V1(z1) = −k1z21 + z1z2

z2=0==⇒ V1(z1) ≤ −2k1V1(z1) (4.48)

and using the Comparison Lemma [19],[24] it can be derived that:

z1(t) ≤ |z1(0)|e−k1t (4.49)

which means that if z2 = 0, z1 will converge exponentially to the origin. Thiscondition is ensured in the second step.

Step 2 For the second step, the error dynamics subsystem to be stabilized is:

z1 = −k1z1 + z2 (4.50)

z2 = −ϑ1x2 − d(t) + ϑ2a2 + ϑ2z3 − a1(z1) (4.51)

The parametric uncertainties stem from the terms:

ϑ1 =Cii + Bv

Miiϑ2 =

AMii

(4.52)

and the disturbance term is:

d(t) =Cij xp,2 + Mij xp,2 + Gi

Mii(4.53)

with i = 1, 2 and j = 2, 1 for each servo system.A Lyapunov function candidate for the second subsystem can be:

V2(t, z1, z2, ϑ1, ρ2) = V1(z1) +12

z22 +

12γ1

ϑ21 +|ϑ2|2γ2

ρ22

=12

z21 +

12

z22 +

12γ1

ϑ21 +|ϑ2|2γ2

ρ22 (4.54)

which is also decrescent and radially unbounded, since:

12||z||2 + 1

2γ1ϑ2

1 +|ϑ2|min

2γ2ρ2

2 ≤ V2 ≤12||z||2 + 1

2γ1ϑ2

1 +|ϑ2|max

2γ2ρ2

2 (4.55)

where z =[z1 z2

]Tfor the second subsystem only and || • || denotes the Euclidean

norm.

4.4. Approach 2 47

Repeating the design of the first approach, which is similar to the reference con-troller of Chapter 3, the second virtual control law a2 is selected as:

a2(z1, z2, ϑ1, ρ2) = ρ2u2 (4.56)

u2 = a1(z1)− z1 + ϑ1x2 + u2r − k2z2 , k2 > 0 (4.57)

u2r = −z2Dδ

, D, δ > 0 (4.58)

The time derivative of V2 becomes:

V2 = −k1z21 − k2z2

2 + ϑ2z3 + z2 [u2r − d(t)] + ϑ1

(˙ϑ1

γ1+ z2x2

)+ |ϑ2|ρ2

( ˙ρ2

γ2+ z2u2

)(4.59)

under the assumption that the rate of variation of parameter ϑ2 is very small, ormore precisely:

˙ρ2 >> ρ2ϑ2

2ϑ2(4.60)

The parameter update laws are selected as:

˙ϑ1 ≈ ˙ϑ1 = −γ1z2x2 (4.61)˙ρ2 ≈ ˙ρ2 = −γ2z2u2 (4.62)

and furthermore, by writing u2r as in Equation 4.58, for the time derivative of V2:

V2 = −k1z21 − k2z2

2 + ϑ2z2z3 + z2

[−z2

Dδ− d(t)

]≤ −k1z2

1 − k2z22 + ϑ2z2z3 +

δD4

(4.63)

by completing the squares in the bracketed term and using D as the upper boundof the disturbance term d(t). If z3 = 0, then

V2 ≤ −min {k1, k2}︸︷︷︸k0

||z||2 + δD4

≤ −(

k0||z||2 −δD4

)(4.64)

The term in brackets in the last inequality is positive or zero only when

||z|| ≥

√δD4k0

(4.65)


However, at this point only the fact that the states z1, z2 are bounded in this set canbe stated, because the adaptation terms are absent from the derivative of V2.The second adaptive backstepping control design step results in the closed loopsubsystem:

z1 = −k1z1 + z2 (4.66)

z2 = −z1 − k2z2 + ϑ2z3 + ϑ1x2 + ϑ2ρ2u2 + [u2r − d(t)] (4.67)˙ϑ1 = −γ1z2x2 (4.68)˙ρ2 = −γ2z2u2 (4.69)

with u2, u2r given by Equations 4.58 and 4.58 respectively. The time derivative ofthe radially unbounded, decrescent Lyapunov function V2 can be written:

V2 ≤ −k1z21 − k2z2

2 + ϑ2z2z3 +δD4

(4.70)

4.4.2 Third step based on disturbance observer

The subsystem to be stabilized by the third step is the whole system, since theactual control input is present. The system is rewritten:

z1 = −k1z1 + z2 (4.71)

z2 = −z1 − k2z2 + ϑ2z3 + ϑ1x2 + ϑ2ρ2u2 + [u2r − d(t)] (4.72)

z3 = −ϑ3PL − ϑ4x2 + ϑ5u− a2 (4.73)

The main difficulty is the time derivative of the virtual control law a2, but alsothe uncertain parameters ϑ3, ϑ4, ϑ5 as well as the cross term with the uncertainparameter ϑ2.To proceed with the design, it is assumed that the unknown parameters are boundedand that the maximum and minimum values of these bounds are known. As a re-sult, an estimate for the parameter ϑ5 is ϑ5. For

u =1ϑ5

u3 (4.74)

the third equation of the system, Equation 4.73, becomes:

z3 = −ϑ3PL − ϑ4x2 − a2 +

(ϑ5

ϑ5− 1)

u3︸︷︷︸g(PL,x2,a2,ϑ5,u3)

+u3 (4.75)

and a Lyapunov function candidate for the system could be:

V3(t, z1, z2, z3, ϑ1, ρ2) = V2(t, z1, z2, ϑ1, ρ2) +12

z23 (4.76)

4.4. Approach 2 49

V3 is radially unbounded and decrescent, using an almost identical argument thatwas used for V2 in Inequality 4.55. The time derivative of V3 is:

V3 ≤ −k1z21 − k2z2

2 +δD4

+ z3

ϑ2z2−ϑ3PL − ϑ4x2 − a2 +

(ϑ5

ϑ5− 1)

u3︸︷︷︸g(PL,x2,a2,ϑ5,u3)

+u3

(4.77)

The input term u3 can be designed as:

u3 = −k3z3 + u3r + uC , k3 > 0 (4.78)

The term u3r is a robust term to counteract the disturbance coming from ϑ2z2z3 andthe term uC is a term that can ideally cancel the disturbance g. It is thus assumedthat there exists an input term uc so that:

uC = −g(PL, x2, a2, ϑ5, u3) (4.79)

With this assumption, the time derivative V3 can be rewritten:

V3 ≤ −k1z21 − k2z2

2 − k3z23 +

δD4

+ ϑ2z2z3 + z3u3r + z3 (g + uC) (4.80)

where the parameter adaptation laws of Equations 4.61 and 4.62 have been used.The last bracketed term of Equation 4.80 is equal to zero under the aforementionedassumption and using Young’s inequality:

ϑ2z2z3 ≤ |ϑ2,max||z2||z3| ≤ϑ2

2,maxz22z2

3

2ε+

ε

2, ε > 0 (4.81)

Replacing the term used in the above inequality, V3 becomes:

V3 ≤ −k1z21 − k2z2

2 − k3z23 +

Dδ

4+ z3u3r +

ϑ22,maxz2

2z23

2ε+

ε

2(4.82)

The robust input u3r can be selected to cancel the second to last term:

u3r = −z3ϑ2

2,maxz22

2ε(4.83)

And substituting the selected u3r in V3 results in:

V3 ≤ −k1z21 − k2z2

2 − k3z23 +

δD4

+ε

2

≤ −ko||z||2 +δD + 2ε

4(4.84)


where, now for the whole system, z =[z1 z2 z3

]Tand ko = min {k1, k2, k3}.

Only boundedness can be deduced for the states of the system z1, z2, z3, ϑ1, ρ2 andthat for the region where:

||z|| ≥

√δD + 2ε

4ko(4.85)

However, when the state vector z has initial values outside this region, then V3 > 0and the trajectories are diverging. When they reach the surface of the sphere with

radius√

δD+2ε4ko

they cannot escape to infinity and are trapped on that surface.Selecting the design parameters ε and δ can reduce this radius.The above analysis is based on the assumption that there exists an input termuC = −g and that it is available. The next section will describe how this term canbe designed.

Design of the disturbance observer input uC

From the previous analysis, it was deduced that exact knowledge of the distur-bance term g, that contains uncertain parameters as well as the complicated terma2, that also includes uncertain parameters and the disturbance term d(t), is of cru-cial importance. Sliding Mode Control can provide finite time convergence to theunknown term. To this end, a sliding mode disturbance observer will be utilizedto estimate the unknown term gThe state equation that contains the disturbance term is rewritten from Equation4.76:

z3 = −ϑ3PL − ϑ4x2 − a2 +

(ϑ5

ϑ5− 1)

u3︸︷︷︸g(PL,x2,a2,ϑ5,u3)

+u3

Selecting a sliding variable as [39] :

S = z3 + w (4.86)

with w a function to be determined. The derivative of S can be written as:

S = z3 + w = g + u3 + w (4.87)

A radially unbounded Lyapunov function candidate for the variable S is:

VS(S) =12

S2 (4.88)

4.4. Approach 2 51

with

VS(S) = S [g + u3 + w]

≤ |S|G + S (u3 + w) (4.89)

where G > 0 is the upper bound of g. Replacing w with:

w = vS − u3 (4.90)

leads to:

VS(S) ≤ |S|G + vS = |S|(G− ρ) = − α√2|S| = −αV1/2

S (S) (4.91)

vS = −ρsign(S) (4.92)

Choosing

ρ = G +α√2

(4.93)

makes S → 0 in finite time, and more specifically in tr =√

2|S(0)|α [s]. This means

that also S→ 0 leading to:

S = 0⇒ g + u3 + w = 0

⇒ g + u3 + vS,eq − u3 = 0⇒ vS,eq = −g (4.94)

The equivalent control vS,eq is used because the sign function is not defined whilesliding on the surface S, or in other words when S = 0.As a result, a suitable selection for the term uC would be:

uC = vS,eq (4.95)

However, in reality this variable vS,eq is not available, but constitutes the time aver-age of the switching function vS = −ρsign(S).A way to approach the desired value for uC, a first order filter can be used toapproximate the average value of the switching control input vS, and consequentlyvS,eq. As a result, the control input term uC can be realized by low pass filteringthe switching input vS [39]:

uC = LPF(vS) = LPF [−ρsign(S)]⇒

uC = − 1τ

uC +1τ[−ρsign(S)] (4.96)

If the time constant of the first order low pass filter, τ, is sufficiently small, thenthe approximation is rather accurate. Since the maximum frequency of the systemis below 120 [rad/s] and the sampling frequency is 2000 [Hz] or 12566 [rad/s], thetime constant can be selected arbitrarily low, such as τ = 1

400 [s], or even smaller. Itis however not desirable that the cutoff frequency of the filter become so high thatcontrol input chatters.


Selection of the upper bound G The upper bound of the disturbance terms canbe selected as [19]:

G = |ϑ3|max|PL|+ |ϑ4|max|x2|+ | ˙a2|+ |β5 − 1|K3|z3|+ |β5 − 1||u3| (4.97)

β5 =

√|ϑ5,max||ϑ5,min|

, ϑ5 =√|ϑ5,max||ϑ5,min| (4.98)

The measured state variables can be used in the above equations for increasedrobustness [39], but can also be assigned an upper bound by assuming stability.The estimate of the derivative of a2 is acquired from the second order differentiatorthat is designed in section 4.3.1.Since by using the first order low pass filter, the convergence of uC to the desiredvalue vS,eq = −g is asymptotic, there will exist an exponentially vanishing residualterm ures = uC − vS,eq. The convergence of S and consequently S to 0 in finite timeensures that vS,eq = −g. Then, depending on the filter time constant τ, uC ≈ vS,eq.The modified Lyapunov function candidate for the whole system becomes:

V3m = V3 +12

S2 (4.99)

Vm ≤ −k1z21 − k2z2

2 − k3z23 −

α√2|S|+ δD

4+

ε

2+ z3ures (4.100)

Assuming that the residual term ures is small in magnitude, the system states areagain bounded.

Summary of the control approach 2

The closed loop error dynamics system, in the z transformed variables becomes:

z1 = −k1z1 + z2

z2 = −z1 − k2z2 + ϑ2z3 + ϑ1x2 + ϑ2ρ2u2 + u2r − d(t) , u2r = −z2Dδ

z3 = −k3z3 + u3r , u3r = −z3ϑ2

2,maxz22

2ε

In reality, the actual system is the one described by Equations 3.5 - 3.7, repeatedhere:

x1 = x2

x2 = −ϑ1x2 + ϑ2PL − d(t)

PL = −ϑ3PL − ϑ4x2 + ϑ5u

4.4. Approach 2 53

The control input voltage to the servo valve is:

uv =

√2

Ps − Pt − sign(u)PLu (4.101)

u =1ϑ5

u3 , ϑ5 =√

ϑ5,maxϑ5,min

u3 = −k3z3 + u3r + uC , uC = − 1τ

uC +1τ[−ρsign(S)]

S = z3 + w , w = vS − u3 , vS = −ρsign(S)

z3 = PL − a2 , a2 = ρ2

(a1 − z1 + ϑ1x2 − z2

Dδ− k2z2

)a1 = −k1z1 + r

z1 = x1 − r z2 = x2 − a1 z3 = PL − a2

with x1 being the total actuator length and x2 the velocity of the piston. Finally ρ

is given by Equation 4.97.

Selection of gains k1,2,3 Observing the resulting closed loop dynamics of thetransformed states system, in z:z1

z2

z3

=

−k1 1 0−1 −(k2 +

Dδ ) ϑ2

0 0 −(k3 +ϑ2

2,maxz22

2ε )

︸︷︷︸

A1

z1

z2

z3

+

010

(ϑ1x2 + ϑ2ρ2u2) +

0−10

d(t)

(4.102)

and assuming that the disturbance and parameter estimation error terms are ne-glected, a non-autonomous nonlinear system is derived. Selection of the parame-ters k1,2,3 should first and foremost guarantee stability of the system, at least in theabsence of any disturbance inputs. The element of the A1(2, 3) is dependent bothon systems states, but also on the position of the second cylinder, since

ϑ2 =A

Mii(xp,1, xp,2): nonsingular , positive

In order to assess stability for this system, a Lyapunov function candidate can be[19]:

W = zTz (4.103)

with time derivative:

W = zTz + zT z = zT(

AT1 + A1

)z ≤ λmax

(AT

1 + A1

)zTz

≤ λmax

(AT

1 + A1

)W (4.104)


The origin will be exponentially stable, in the absence of other disturbance inputs,if all the eigenvalues of the matrix AT

1 + A1 have negative real parts. The eigenval-ues of the matrix were calculated as:

λ1 = −2k1

λ2,3 =−δϑ2

2,maxz22 − [D + (k2 + k3) δ] ε± o

δε

o =

√{ε [δ (k2 − k3) + D]− δz2

2ϑ22,max

}2+ ϑ2

2δ2ε2

The term under the square root cannot be negative and as a result, no eigenvalueswith complex part can occur. The most important part now is to establish condi-tions that the eigenvalues will always be negative. The third eigenvalue, with theminus sign in the o term, is always negative for all values of z2 and ϑ2. Conse-quently, investigating the numerator of the eigenvalue λ2, it is desired that:

−{

δϑ22,maxz2

2 + ε [D + δ (k2 + k3)]}+

√{ε [δ (k2 − k3) + D]− δz2

2ϑ22,max

}2+ ϑ2

2δ2ε2 < 0

(4.105)

A rough estimate can be given by selecting k1 and k2 in a way that the square rootterm is minimized. This happens when:

ε [δ (k2 − k3) + D] = δz22ϑ2

2,max or when

k2 = k3 −Dδ+

z22ϑ2

2,max

ε> 0 (4.106)

Substituting in the criterion inequality 4.105 the selection for k2 from Equation4.106, the criterion for the negativity of λ2 transforms to:

δϑ22,maxz2

2 + [D + δ (k2 + k3)] ε > ϑ2δε (4.107)

leading to the condition:

k3 >ϑ2

2−

z22ϑ2

2,max

εor k3 >

ϑ2,max

2(4.108)

Comparing inequalities 4.108 and 4.106, for the worst case:

k3 >ϑ2,max

2

k3 >Dδ

>>ϑ2,max

2(4.109)

As a result, the resulting eigenvalues are:

λ1 = −2k1 , λ2,3 << −(ϑ2,max ∓ ϑ2) (4.110)

4.5. Simulation of proposed algorithm in continuous time 55

This analysis regarding the worst case for the eigenvalues does not take into con-sideration the disturbance input term d(t) or the parameter estimation errors.However, the robust terms have been used as guides to dominate the disturbanceterms and have been included in the analysis. As a result, these selection valuesfor k2 and k3 can provide a point to begin the selection of these gains.

4.5 Simulation of proposed algorithm in continuous time

Since a stability analysis has been developed for the proposed controller, undercertain assumptions, the next step is to evaluate its performance through simu-lations. These simulations are performed using the continuous time controllers.Furthermore, they show whether there are unwanted phenomena such as controlinput chattering and parameter drift. Finally, they allow for an initial tuning of theadaptation gains. No noise was added to the measured signals, since this will beconsidered in Chapter 6. The piston velocity is considered to be measured.For this initial simulation the reference trajectory is repeated 5 times. The selectedgains are:

• Servo system 1:

k1 = 100 k2 = 25 k3 = 50 γ1 = 104 γ2 = 10

δ = 0.05 D = 1.2245 (4.111)

• Servo system 2:

k1 = 100 k2 = 94 k3 = 180 γ1 = 104 γ2 = 10

δ = 0.025 D = 2.3456 (4.112)

Regarding the filter time constant for the disturbance observer, it was initially set atτ = 0.0025 [s] for both servo systems. However, since chattering was present in thecontrol input, it was increased to τ = 0.05 [s] and τ = 0.1 [s] for servo system 1 and2 respectively. This greatly decreased the chattering effect without observing highlag effect in the simulation, compared to the mean value of the filtered switchingsignal with the initial filter time constant. Nonetheless, the stability proof stillassumes that the residual of the time average of the switching disturbance observerand the filtered feedforward input uC is still small.In Figures 4.2 and 4.3 the input voltage to the servo valves is plotted. The chatteringeffect has been reduced.


0 5 10 15 20 25

Time [s]

-4

-3

-2

-1

0

1

2

3

4

5

Vo

ltag

e [V

]

Input voltage - Servo system 1

Figure 4.2: Input voltage to servo valve 1.

0 5 10 15 20 25

Time [s]

-8

-6

-4

-2

0

2

4

6

Vo

ltag

e [V

]

Input voltage - Servo system 2

Figure 4.3: Input voltage to servo valve 2.

The tracking errors are presented in Figures 4.4 and 4.5:

0 5 10 15 20 25

Time [s]

-0.35

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

Err

or

[mm

]

Position tracking error - Servo system 1

Figure 4.4: Tracking error servo system 1.

0 5 10 15 20 25

Time [s]

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3E

rro

r [m

m]

Position tracking error - Servo system 2

Figure 4.5: Tracking error servo system 2.

The estimated parameters are presented in Figures 4.6 , 4.7 , 4.8 and 4.9.

4.5. Simulation of proposed algorithm in continuous time 57

0 5 10 15 20 25

Time [s]

-6

-4

-2

0

2

4

6

8

1[k

g/s

]

Estimation of parameter 1

- Servo system 1

EstimateActual

Figure 4.6: Estimation of parameter ϑ1 inservo system 1.

0 5 10 15 20 25

Time [s]

0

2

4

6

8

10

12

14

16

18

20

1[k

g/s

]

Estimation of parameter 1

- Servo system 2

EstimateActual

Figure 4.7: Estimation of parameter ϑ1 inservo system 2.

Since the control input does not chatter highly, the tracking error is acceptable inthe presence of disturbance and the estimated parameter ϑ1 does not drift

0 5 10 15 20 25

Time [s]

2.7

2.8

2.9

3

3.1

3.2

3.3

3.4

3.5

3.6

3.7

2[m

2/k

g]

106 Estimation of 2

- Servo system 1EstimateActual

Figure 4.8: Estimation of parameter ρ2 inservo system 1.

0 5 10 15 20 25

Time [s]

4.5

5

5.5

6

6.5

7

7.5

2[m

2/k

g]

105 Estimation pf parameter 2

- Servo system 2

EstimateActual

Figure 4.9: Estimation of parameter ρ2 inservo system 2.


0 5 10 15 20 25

Time [s]

3.194236

3.194237

3.194238

3.194239

3.19424

3.194241

3.1942422

[m2/k

g]

106 Estimation of 2

- Servo system 1EstimateActual

Figure 4.10: Detail of Figure 4.8.

0 5 10 15 20 25

Time [s]

5.94279

5.9428

5.94281

5.94282

5.94283

5.94284

5.94285

5.94286

5.94287

5.94288

2[m

2/k

g]

105 Estimation pf parameter 2

- Servo system 2

EstimateActual

Figure 4.11: Detail of Figure 4.9.

Since the control input does not highly chatter, the tracking error is acceptable inthe presence of disturbance and the estimated parameter ϑ1 does not drift, this ap-proach can look promising for further simulations and robustness tests. Regardingthe estimation of parameter ρ2, a possibility is that 5 cycles of the trajectory is nota sufficient number with the specified adaptation gain, for the parameter to showboundedness. It will be investigated if a different gain will yield better results.


In this chapter, it was attempted to design an algorithm that keeps the reducedcomplexity of the controller in Chapter 3, but stability or boundedness of the closedloop system can be established.The adaptive algorithm of the second backstepping step was conserved and thechanges were made in the last step. An initial approach based on an adaptiveinput and a second order sliding mode differentiator was proven to bound the errorstates of the system to an area around the origin, however selecting the adaptationgains was found to be challenging.A second approach, based on a sliding mode disturbance observer was found toprovide the same boundedness results under given assumptions, but avoid usingany adaptive terms for the third step. Simulation results showed that this approachto be viable in the simulations.

Chapter 5

Reference Controllers

In order for the evaluation of the performance of the proposed controllers to becomplete, specified performance indices should be compared to the indices frommore "industry-standard" linear controllers. In this chapter, the reference linearcontrollers for the servo systems’ position tracking problem will be designed usingfrequency domain techniques. The major advantage of using these controllers isthat there is no need to have velocity measurement.

5.1 Linearization and plant transfer functions

In order to design linear controllers, the system should be described by a systemof linear differential equations. To this end, the nonlinear system described byEquations 2.15 and 2.16 is linearized using first order Taylor approximation abouta point of operation. This process is considered in Appendix C.The plant to be controlled has as output the piston position Xp(s) and the valveinput voltage for input Uv(s). Instead of the resulting linear system being describedby a system of linear differential equations, coupled between them, it was decidedthat two SISO systems are used for the design of the reference controllers, with thecouplings considered as disturbance inputs. A block diagram, equivalent for bothservo systems, is repeated here form Appendix C.

G (s)v K qX (s)v,i β

effVtot

VA,0VB,0

1s

P (s)L,i AM s+Bv

U (s)v,i

C - Kl qp

A

sX (s)p,i 1s

X (s)p,i

ii

(M s X +G )/Aji2

ip,j

Figure 5.1: Block diagram of one of the two the hydraulic servo systems.

59

60 Chapter 5. Reference Controllers

The design procedure is similar for both servo systems and the only differencescome from design choices, such as bandwidth and stability margins. These consid-erations may lead to different selections of the controllers for each servo system.

5.1.1 Analysis in frequency domain

The transfer function that describes the input output relationship in the Laplacedomain, neglecting the valve dynamics, is:

Xp(s)Uv(s)

= Gp(s) =G(

sωl

)2+ 2 ζl

ωls + 1

1s

(5.1)

G =AKq

Bv(Cl − Kqp) + A2 (5.2)

ωl =

√βe f f Vtot

[A2 + Bv(Cl − Kqp)

]MVA,0VB,0

(5.3)

ζl =ωl

2

√Mβe f f Vtot(Cl − Kqp) + VA,0VB,0Bv

βe f f Vtot[Bv(Cl − Kqp) + A2

] (5.4)

The bode plots for the two servo systems are plotted in dashed lines in Figure 5.2.

-200

-150

-100

-50

0

Mag

nitu

de (

dB)

Servo system 1 UncompensatedServo system 2 UncompensatedServo system 1 CompensatedServo system 2 Compensated

100 101 102 103 104-270

-225

-180

-135

-90

Pha

se (

deg)

Bode Diagram

Frequency (rad/s)

Figure 5.2: Bode plots of the position control servo systems. Uncompensated with dashed lines. Useof pressure feedback in solid lines.

The resonant peaks for the servo system either limit the maximum attainable band-width, or can amplify specific frequencies of the input. Both are undesirable andan active damping loop, using pressure feedback is designed.

5.1. Linearization and plant transfer functions 61

5.1.2 Pressure feedback

Design of the pressure feedback loop can be performed in the same fashion asin section 3.2 and shown in the block diagram of Figure 3.3. However, since aProportional Integral controller is to be used for the position control loop, the highpass filter can be substituted by a simple gain. The integral part of the PI controllerwill compensate for the reduced low frequency gain due to the pressure feedbackloop.Since a simple gain Kp f ,i, i = 1, 2 is decided to be used for the pressure feedbackloop, this gain can be considered as an artificial leakage term in parallel with theterm Cl − Kqp of the block diagram of Figure 5.1. Then, the gain can be selected toyield the desired damping ratio for the open loop position control system. Choos-ing desired damping ratios for both servo systems ζ = 0.707, the gains are calcu-lated as:

Kp f ,1 = 1.0147e− 11[

m3

sPa

]Kp f ,2 = 2.6356e− 11

[m3

sPa

]For implementation, the pressure feedback gains are divided by the valve flowcoefficient of the respective servo system. The bode plots of the pressure feedbackcompensated, open loop systems is shown in the solid lines of Figure 5.2.

5.1.3 Disturbance rejection

Since the objective of the control design procedure is tracking, the disturbanceterm is not explicitly taken into consideration. However, since this term can be-come larger in magnitude, it is useful to also investigate whether the gravity andcouplings affect the tracking performance.It is noted that the disturbance term has the same dynamics as the plant, meaningthat they have the same characteristic polynomial. The only things that change arethe steady state gain and high frequency gain, or the numerator. This can be seenfrom Equation C.10. When a compensator is placed in cascade with the plant toclose the loop with the position reference, the block diagram with the disturbanceterm is shown in Figure 5.3:

G (s)p

G (s)d

G (s)c

D(s)

X (s)p

ref U (s)v X (s)p

Figure 5.3: Block diagram of the cascade compensated system with the disturbance term.


The output piston position in the Laplace domain can be written as:

Xp(s) =Gc(s)Gp(s)

1 + Gc(s)Gp(s)Uv(s) +

Gd(s)1 + Gc(s)Gp(s)

D(s) (5.5)

It can be seen that a sufficient condition for disturbance rejection is that for the fre-quencies of interest, up to the gain crossover frequency, the product Gc(s)Gp(s) >>

Gd(s). In reality, only the magnitude of the compensator matters, since disturbancetransfer function can be written in terms of the plant transfer function.

5.2 Proportional-Integral controller design

The PI compensator is probably the most used controller in the industry [27]. Thisis the reason this compensator is selected to be used as a reference controller. Sincethe plant transfer function of the position control system approximately includes afree integrator, there is finite steady state error to ramp inputs. This is acceptablefor tracking, but by increasing the free integrators by one, using a PI controller incascade with the plant, this error becomes zero. Apart from the increase in the typeof the system, there is an increase in the low frequency gain, providing robustnessagainst disturbance terms of low frequencies.The disadvantage of using the PI compensator comes from the restriction of thebandwidth selection. Due to the free integrator of the plant, the bode plot beginswith an angle of −90◦. The PI controller makes the phase diagram of the bode plotbegin from −180◦. As a result, care must be taken so that at the gain crossoverfrequency, the magnitude of the open loop transfer function is not larger than 0[dB]. At the same time, a sufficient phase and gain margin should be attained.The following selection of the PI parameters is by no means optimal and favorslow frequency gain over closed loop bandwidth, for selected phase margin.The block diagram for the closed loop position control system is shown in Figure5.4. The plant transfer function is the pressure feedback compensated plant shownin the solid lines of the bode plots in Figure 5.2.

PIG (s) G (s)p

X (s)pX (s)pref

U (s)vE(s) V (s)v

Figure 5.4: Block diagram of the PI compensated servo system. The saturation of the valves is takeninto consideration via the saturated output Vv(s).

5.2. Proportional-Integral controller design 63

The transfer function of the PI controller is given by:

GPI(s) =kPs + k I

s= k I

kPkI

s + 1

s= k I

αPIs + 1s

, αPI =kP

k I(5.6)

∠GPI(jω) = −90◦ + arctan(αPIω) , |GPI(jω)| = k I

√1 + α2

PIω2

ω(5.7)

The frequency of the compensator’s zero will determine the maximum gain crossoverfrequency for a given phase margin. Since a feedforward input will be used andthe tracking performance will be increased, a higher phase margin is preferredagainst a high gain crossover frequency. The conditions that need to hold at thegain crossover frequency are:

∠GPI(jωgc) +∠Gp(jωgc) = −180◦ + PM⇒

αPI = −1

ωgc

tan(40◦) + tan(∠Gp(jωgc))

1− tan(40◦) tan(∠Gp(jωgc))= − 1

ωgc

f1(ωgc)

f2(ωgc)> 0

(5.8)

|GPI(jωgc)||Gp(jωgc)| = 1⇒

k I =1

|Gp(jωgc)|ωgc√

1 + α2PIω

2gc

(5.9)

The phase margin PM is selected to be close to 50◦. From the last inequality of 5.8,and since ωgc, αPI > 0 the following options arise:

f1(ωgc) > 0 and f2(ωgc) < 0 or f1(ωgc) < 0 and f2(ωgc) > 0 (5.10)

The functions f1(ωgc) and f2(ωgc) are plotted with regard to the parameter ∠Gp(jωgc)

in Figure 5.5. These values are valid for both servo systems. What changes are thevalues of the magnitude of the plant transfer function at the selected frequency, aswell as the frequency where the desired plant angle is located.In order to keep the closed loop system stable with reasonable values for the gains,but at the same time having the most wide bandwidth possible, it was decidedthat the angle if Gp = −128◦ at the gain crossover frequency. From the pressurefeedback compensated bode plots of Figure 5.2, it can be deduced that:

∠Gp,1(jω) = −128◦ at ω ≈ 15.2 [rad/s] (5.11)

∠Gp,2(jω) = −128◦ at ω ≈ 39.3 [rad/s] (5.12)

As a result, for the two servo systems, these are the selected gain crossover fre-quencies. For these frequencies, use of Equations 5.8 and 5.9 yields:

αPI,1 = 1.8840 , k I,1 = 226.3004 , kP,1 = 426.3500 (5.13)

αPI,2 = 0.7287 , k I,2 = 1766.8 , kP,2 = 1287.5 (5.14)


-280 -260 -240 -220 -200 -180 -160 -140 -120 -100 -80

Gp(j

gc)

-60

-40

-20

0

20

40

60

f1( gc)

f2( gc)

Figure 5.5: Values of f1(ωgc) and f2(ωgc) withvarying ∠Gp(jωgc).

-260 -240 -220 -200 -180 -160 -140 -120 -100

Gp(j

gc)

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

f1( gc)

f2( gc)

data1

X: -128Y: -0.074

Figure 5.6: Enhanced version of Figure 5.5.

with relative stability margins:

PM1 = 50◦ , GM1 = 10.1 [dB] (5.15)

PM2 = 50◦ , GM2 = 9.66 [dB] (5.16)

A final check ensures that the designed PI controllers satisfy the M-1.3 criterion[20]. The bode plots of both systems’ compensated open loop transfer functionsare shown in Figure 5.7.

-200

-100

0

100

200

Mag

nitu

de (

dB)

10-2 10-1 100 101 102 103 104-270

-225

-180

-135

-90

Pha

se (

deg)


Bode Diagram

Frequency (rad/s)

Figure 5.7: Bode plots of the PI compensatedservo systems.

-100

-50

0

50

Mag

nitu

de (

dB)


101 102 103 104-270

-225

-180

-135

-90

-45

Pha

se (

deg)

Bode Diagram

Frequency (rad/s)

Figure 5.8: Bode plots of the PLead and P com-pensated servo systems.

5.2.1 Anti-windup

A PI compensator was designed in the preceding section. However, the actuatorhas limits, that have to be accounted for. The input voltage to the servo valvescannot surpass 10 [V] or become lower than -10 [V]. If this happens, while the

5.3. Proportional-Lead and Proportional compensator design 65

compensator gives command for increased control input voltage, the the integratoraccumulates a large value, making its response slower. In fact, if the control inputis outside the set of the actuator limits, there exists no more feedback loop and theplant is in open loop configuration with a constant control input value [3], [13].This is another shortcoming of using an integral term in the linear controller, thefirst being the addition of negative phase.In this case of trajectory tracking, since no large step changes in the referenceposition can occur, saturation of the actuator could happen in the event of largedisturbances or if the desired trajectory momentarily goes out of the limits of thePL − QL plot. This in turn could happen if a load was abruptly attached to thetool tip of the manipulator, or if a very fast trajectory is commanded, leading toan increase in the velocity dependent Coriolis terms. In the case of the specificsystem, only the input voltages to the valves are measured while their outputs, thespool position, are not.Since the controller is digitally implemented and the saturation limits are known,it is simple to design a model of the actuator as an "if loop" in the following way[13], in the discrete time domain:

uv(t) =

uv,max, if uv(t) > uv,max

uv,min, if uv(t) < uv,min

kPe(t) + k I

∫ t

0e(τ)dτ otherwide

(5.17)

It is desired that the integrator is reset when the actuator is saturated. A feed-back loop can be placed around the integrating part and drive its input to zerofor the duration the actuator is saturated. Since the actuator’s output is not di-rectly measured, a model of it can be included. In Figure 5.9 the block diagramof the PI controller using an anti-windup loop is illustrated. The saturation blockcorresponds to the model of the actuator. In this case, the servo valve with maxi-mum and minimum values of +10 [V] and -10 [V] respectively. The output of thecontroller in the Laplace domain now becomes:

Uv(s) = kPE(s) +1s[k I E(s) + kTEi(s)] (5.18)

The gain kT can be chosen as [6]:

kT = 10k I

kP(5.19)

5.3 Proportional-Lead and Proportional compensator design

Since tracking of a trajectory signal is required instead of regulation at a set point,a different controller can be designed without integral action. This way, achievinga wider closed loop bandwidth with the same phase margin is feasible.


kP

1s

G (s)p

kT

E(s) U (s)v V (s)vkI

E (s)i

Figure 5.9: Block diagram of the PI controller with anti-windup loop.

The reference trajectory signal contains frequencies of significant magnitude for upto around 12 [rad/s], as shown in section B.5. As a result, a closed loop bandwidthof around 40 [rad/s] is more than satisfactory, especially when the feedforwardsignal will be included. The design of the feedforward signal is the topic of thenext paragraph.Observing the bode plots of the pressure feedback compensated servo systems ofFigure 5.2, it can be seen that for a phase margin of 50◦ for both servo systems:

• The second servo system can achieve the desired value of gain crossoverfrequency with only proportional feedback.

• The first servo system can achieve this value of gain crossover frequency if alead compensator is placed in cascade with the plant.

Since the design for the second servo system is simpler, this design follows first.

Design of proportional compensator for servo system 2 From the bode plot, itis observed that for PM= 50◦, ∠Gp,2(jω) = −130◦ at ωgc = 41 [rad/s]. At thisfrequency |Gp,2(jωgc)| = −62.6 [dB]. As a result, a gain of

Kp,2 = 1062.620 = 1348.9628 (5.20)

is selected.

Design of proportional-lead compensator for servo system 1 Use of a lead com-pensator is essential to attain a wider bandwidth and ensure stability for the closedloop system of servo system 1. The lead compensator is used to add positive phaseand increase the stability margins. Its transfer function is given by:

Gld(s) =1 + αldTs

1 + Ts, αld > 1 (5.21)

or in the frequency domain

Gld(jω) =1 + αldTωj

1 + Tωj=

1 + αlduj1 + uj

, u = Tω (5.22)

5.3. Proportional-Lead and Proportional compensator design 67

Due to noise considerations, it is decided that the lead effect is active for one anda half decade, or that ωld,p = 15ωld,z, leading to

αld = 15 (5.23)

This limitation in the selection of the zero also limits the attainable closed loopbandwidth for a given phase margin. For the angle of the lead controller holdsthat

sinφm =1− u2

1 + u2 when αld =1u2 ⇒ T =

1ω√

αld(5.24)

For the desired phase margin of 50◦, at the gain crossover frequency and for phasemargin of 50◦:

∠Gld(jωgc) +∠Gp(jω) = −180◦ + PM = −130◦ (5.25)

The selection of ald due to noise considerations effectively limits the maximumangle the lead compensator can add to the system. This is quantified as:

αld = 15 5.24=⇒ u2 =

115⇒

⇒ φm = arcsin1− 1

15

1 + 115

= arcsin1416

= 61.045◦ (5.26)

Now that the maximum phase the compensator can add is defined, the frequencywill be pinpointed. From Equation 5.25, using the result for the maximum angle,at the gain crossover frequency:

∠Gp(jωgc) = −130◦ − 61.045◦ = −191.045◦ (5.27)

which from the bode plot can be observed that is when ωgc ≈ 40.1 [rad/s]. Asa result, at this frequency the maximum phase of the lead compensator will beadded. Also, from Equation 5.24, the tuning parameter T is calculated:

αld = 15 ω = ωgc = 40.1 [rad/s] (5.28)

T =1

ω√

αld= 0.0064 (5.29)

To finalize the design, a proportional gain is cascaded with the lead compensator,so that at the gain crossover frequency of 40.1 [rad/s] the magnitude of the com-bined gain-compensator-plant open loop transfer function is equal to 1, in absoluteunits. The magnitude of Gld(jωgc)Gp(jωgc) = −53.7 [dB]. A gain of

Kp,1 = 1053.720 = 484.1724 (5.30)


will be used. The resulting bode plots of the Proportional-Lead compensated servosystem 1 and the Proportional compensated servo system 2 are plotted in Figure5.8. The relative stability margins are:

PM1 = 50◦ GM1 = 12.2 [dB]

PM2 = 50◦ GM2 = 9.43 [dB]

Comparison of the two linear designs The two linear reference controllers havesimilar stability margins, however, for servo system 1, the gain crossover frequencyis much higher for the Proportional-Lead case. This would mean a faster response,or rise time. However, the low frequency gain of the PI controller is much higher,leading to increased disturbance rejection at these frequencies.

5.4 Velocity feedforward

Tracking performance of the reference controllers can be greatly improved by in-cluding information from the trajectory as a feedforward signal. Feedforward ismade possible with position feedback, by having the reference velocity, obtainedusing the inverse kinematics of the manipulator. The idea for using feedforwardin the control design process, is illustrated using the block diagram of a positionfeedback loop in Figure 5.10: Now the input to the actuator/plant is composed of

F(s)

G (s)c G (s)pX (s)pX (s)p

ref

U (s)ff

U (s)vU (s)fb

Figure 5.10: Position control loop using feedfor-ward.

K

G (s)c G (s)pX (s)pX (s)p

ref

U (s)ff

U (s)vU (s)fb

ffX (s)p

ref

Figure 5.11: Position control loop using velocityfeedforward with static gain.

two inputs, one from the feedback compensator, u f b(t), and one from the feedfor-ward term, u f f (t). The input output relationship of the measured position to itsreference can be written as:

Xp(s) =Gc(s)Gp(s)

1 + Gc(s)Gp(s)Xre f

p (s) +F(s)Gp(s)

1 + Gc(s)Gp(s)Xre f

p (s) =

=Gc(s)Gp(s) + F(s)Gp(s)

1 + Gc(s)Gp(s)Xre f

p (s) (5.31)

It can be discerned that if

F(s) =1

Gp(s)(5.32)

5.5. Chapter 5 summary 69

then Xp(s) = Xre fp (s), meaning that perfect tracking is achieved. To this end,

inverting the plant described by Equation 5.1 leads to:

F(s) =1

Gp(s)= s

sωl

2 + 2 ζlωl

s + 1

G=

s3

ω2l+ 2 ζl

ωls2 + s

G(5.33)

which is non causal, since the plant Gp(s) is strictly proper. To implement such adesign, that promises perfect tracking, differentiation should be used:

U f f (s) =1G

sXre fp (s) +

2 ζlωl

Gs2Xre f

p (s) +1

ω2l

Gs3Xre f

p (s) (5.34)

where sXre fp (s) = xre f

p (t) and in an equivalent way for the other terms. However,instead of differentiating the position reference, the reference velocity is alreadyavailable from the inverse kinematics of the manipulator. As long as the referencetrajectory for the acceleration and the jerk are available, the other terms can alsobe used.For this case, only the velocity information will be fed forward. The inverse ofthe plant’s DC gain can be used as gain. This decision is taken since simulationsonly show marginal decrease in the position error by using acceleration and jerkfeedforward. At the same time, these references should be loaded in the memoryof the real time controller. As a result, they are not used, and K f f ,i =

1Gi

:

K f f ,1 = 25.3924 [s] K f f ,2 = 27.1304 [s] (5.35)


In this chapter, reference controllers were designed. The reason for this is to pro-vide a basis to compare the performance of the proposed controller against. Thebandwidth of the controllers is designed to be as wide as possible, respecting therelative stability margins. Nonetheless, the bandwidth is wider than the frequencyspectrum of the reference signal.What follows is a comparison of the tracking and robustness performance of theproposed and reference controllers. This step is done in the same simulation en-vironment. However, since all controllers are to be applied to an actual system,they should be expressed in discrete time. Taking the discretization into accountcan provide more accurate results for the simulations and is thus considered in thenext chapter.

Chapter 6

Performance and robustness com-parisons

In this chapter, simulation results from the developed controllers will be comparedunder nominal conditions. Afterwards, their performance under variations in thesystem parameters, changes in the reference trajectory and the addition of mea-surement noise will be considered.

6.1 Performance indices

The main objective of the designed controller is trajectory tracking of the tool tipposition. Assuming that the length measurements of the links, as well as the pistonstrokes are exact, the tool tip position reference can be expressed as piston positionreference. As a result, the focus of the indices is piston position tracking error.Regarding performance indices, the maximum and root mean square (RMS) of thetracking error will be used. The maximum error provides information regardinginstantaneous disturbances or behavior that induces overshoots. However, smallmaximum error does not guarantee perfect tracking or even acceptable perfor-mance. If the response is slow, the tracking will not be exact. The RMS error indexcan account for both maximum error, but also the controller response time.Another index is the variance, or standard deviation of the tracking error. If thisindex is high, then the error deviates from its mean value, showing that the perfor-mance of the controller is not consistent. This can be due to disturbances, or fromthe specific configuration of the manipulator.Other indices can be used to evaluate the tracking performance of the controllers.In [31], the ρ performance index is used, that normalizes the absolute value of theposition tracking error with the absolute maximum velocity. A similar index willbe used here, but reffering to the piston position error and velocity. Since velocityis not directly measured, its simulation value will be used.

71

72 Chapter 6. Performance and robustness comparisons

The aforementioned indices only penalize the tracking performance. However, thecontrol effort should not be overlooked. In some instances, low control effort couldbe preferable to very accurate tracking performance, provided that the error stayswithin tolerable limits. As a result, the maximum value of the voltage applied toeach servo valve will also be accounted for. Another criterion is the control effort,since a controller can require more effort along the reference trajectory comparedto another, that only shows some peaks.To conclude, the criteria against which the performance of the controllers will becompared are:

• Maximum piston position tracking error:

emax = max{|xp − xp,re f |

}(6.1)

• RMS tracking error:

eRMS =

√√√√ 1N

N

∑k=1

[xp(k)− xp,re f (k)

]2 (6.2)

where N is the total number of samples.

• Standard deviation:

σ2 = E{

e2}− E {e}2 (6.3)

with E {} denoting the mean value of the bracketed term.

• ρ index:

ρ =emax

|xp|max(6.4)

The lower the value of ρ, the better the performance of the controller. This isexplained by the fact that small tracking error can be achieved at high velocity[31].

• Maximum input voltage:

u = |uv|max (6.5)

• Control effort:

Ju =N

∑k=1

u(k)2 (6.6)

6.1. Performance indices 73

6.1.1 Noise

The presence of noise in the system is unavoidable. Noise can be categorized inprocess noise and measurement noise [14]. Process noise is due to external dis-turbances and here have been considered as the couplings due to Coriolis forcesand the inertia matrix elements having considerable magnitude. Other than that,the system is isolated from the environment, since it is operating in an internal, en-closed space. Consequently, only measurement noise has to be explicitly accountedfor.Since no other data than measurements are available, an initial approach to as-sessing the effect of noise is to acquire measurements from the sensors at a givenconstant and certain value of the variable in question. For example, the positioncan be measured when the piston is at the end limit of the cylinder. A pressuremeasurement can be taken when the system is not pressurized. This assumes thatthe variable signal and noise are uncorrelated, which is assumed to hold. After themeasurement has been acquired, the known value of the variable is deducted anda signal containing noise is what remains.This noisy signal contains noise from different sources:

• Noise that is the result of a white noise driven process. It is assumed thatwhite noise is present at the input of the sensor. However, the resulting noisesignal does not have a constant flat spectrum, but decreases with frequency.

• Noise due to aliasing. The sampling frequency may have been selected lowerthat two times the sensor bandwidth. This leads to the high frequency whitenoise signal being seen at low frequencies in the data sampled from the sen-sor.

Since it is difficult to discern each source from the acquired signal, as well asexpress its spectral density [14] as a rational function of ω2, it is decided that thenoise will be modeled as an additive signal that takes values following a normaldistribution, with zero mean and variance found by the measured signals.

Position sensor noise

The noise signal in Figure 6.1 is acquired by measuring the piston position ofcylinder 1 when it is fully extended. Since the value of the position there is knownexactly, it is subtracted from the measurement and the noise signal is what remains.It is assumed that this signal follows a normal distribution with zero mean andvariance σ2

xp,1 = 5.6644 · 10−4[mm2] for the position sensor of the first cylinder andσ2

xp,2 = 3.24 · 10−4[mm2] for the sensor of the second cylinder.


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time [s]

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

Pis

ton

po

siti

on

[m

m]

Noise of position sensor signal

Figure 6.1: Noise on the position signal.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time [s]

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Pre

ssu

re [

bar

]

Pressure sensor signal noise

Figure 6.2: Noise in the pressure signal.

Pressure sensor noise

Similar statements can be made regarding the noise in the pressure sensor signal,shown Figure 6.2. The variance for the pressure sensor of the first cylinder isσPL,1 = 2.7 · 10−3[Pa2] and for the second it is calculated as σPL,2 = 2.7 · 10−3[Pa2].

6.1.2 Velocity estimation

The velocity is not physically measured in the laboratory setup. An estimator/d-ifferentiator based on the super twisting algorithm is used. This differentiator isdiscussed in section D.

6.1.3 Periodic trajectories

Instead of simulating the trajectory only once, it is decided to repeat it for a multi-tude of periods. The reason is twofold; the tracking error that is due to the initialposition is penalized less and the behavior of the adaptive terms can be seen. Theselected trajectory might not contain a large number of sinusoidal signals and asa result, the adaptation might be slow. This leads to an increased tracking error.Should the parameters converge to the suitable values, this error is expected toincrease. Furthermore, it is of high importance to investigate whether the adaptiveparameters follow the actual time varying unknown parameters and do not drift.As a result, the trajectories will be repeated 20 times.

6.2 Performance under nominal conditions

Under nominal conditions the disturbance terms d(t) as defined in Equation 3.8for each servo system along one period of the trajectory can be seen in Figure 6.3.The disturbance terms as defined for the linearized system design are shown inFigure 6.4.

6.2. Performance under nominal conditions 75

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time [s]

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

Acc

eler

atio

n [

m/s

2]

Disturbance terms - Nonlinear system


Figure 6.3: Disturbance terms for nonlinearsystem.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time [s]

-1500

-1000

-500

0

500

1000

1500

2000

2500

3000

Fo

rce

[N]

Disturbance terms - Linearized system


Figure 6.4: Disturbance terms for linearized sys-tem.

It can be seen that for the nonlinear system, the disturbance term is larger in ab-solute terms for the second servo system, regarding the peak and mean value. Forthe linearized systems, conclusions cannot be made based on Figure 6.4 only, butthe bode plot should be used in order to compare the effect the disturbance has oneach servo system with a specific controller.The first comparison of the benchmark and designed controllers’ performance isunder nominal conditions. The parameters of the systems are the ones the con-trollers were designed for, no additional load is present at the tool tip and thetrajectory is the nominal one, of 5 [s], as designed in Appendix B.It was observed that the measurement error had a significant impact on the systemscontrolled by the RABLIN and the proposed controllers. This is observed in theinput voltages. To elaborate, the input voltages to the servo valve for the first servosystem are shown in Figures 6.5, 6.6, 6.7 and 6.8.

95 95.5 96 96.5 97 97.5 98 98.5 99 99.5 100

Time [s]

-4

-3

-2

-1

0

1

2

3

4

Vo

ltag

e[V

]

Servo system 1 - PI

Figure 6.5: Input voltage servo system 1, PI.

98 98.5 99 99.5 100 100.5 101 101.5 102 102.5 103

Time [s]

-4

-3

-2

-1

0

1

2

3

4

5

Vo

ltag

e[V

]

Servo system 1 - RABLIN

Figure 6.6: Input voltage servo system 1, RABLIN.


95 95.5 96 96.5 97 97.5 98 98.5 99 99.5 100

Time [s]

-5

-4

-3

-2

-1

0

1

2

3

4

5

Vo

ltag

e[V

]

Servo system 1 - P-Lead

Figure 6.7: Input voltage servo system 1,P-Lead.

98 98.5 99 99.5 100 100.5 101 101.5 102 102.5 103

Time [s]

-5

-4

-3

-2

-1

0

1

2

3

4

5

Vo

ltag

e[V

]

Servo system 1

Figure 6.8: Input voltage servo system 1, Ap-proach 2.

Table 6.1: Performance indices under nominal conditions.

Servo System 1Controller emax

[mm]eRMS[mm]

σ [mm] ρ [s] umax [V] Ju · 105 [V2]

PI 3.7901 1.4617 1.4614 0.0258 4.329 7.5659P-Lead 3.63 1.7168 1.1458 0.0261 4.6045 7.1869RABLIN 1.4574 0.9056 0.4361 0.0107 4.7830 33.257Approach 2 0.3672 0.1495 0.1322 0.0027 4.9014 36.193

Table 6.2: Performance indices under nominal conditions.

Servo System 2Controller emax

[mm]eRMS[mm]

σ [mm] ρ [s] umax [V] Ju · 106 [V2]

PI 1.2694 0.4921 0.4921 0.0078 5.1118 1.3819P 1.828 0.74 0.5495 0.0113 5.0897 1.3663RABLIN 2.0916 0.9712 0.5392 0.0127 5.2024 6.4072Approach 2 0.5485 0.1855 0.1597 0.0034 6.0705 7.4482

The PI compensated system is expected to be less affected by noise, since the PIcontroller has increased gain only at low frequencies. On the contrary, the Leadcompensated system is affected by noise, since the compensator amplifies highfrequency signals at its input. This can be reduced by using a lower value of αldfrom Equation 5.23. This will lead however to decreased gain crossover frequencyresulting in a system with slower rise time. The RABLIN controller is not affectedas much as the proposed controller, mainly due to the relatively low gains k1 andk2 that are multiplied with the noisy position feedback signal.


The reason why the proposed controller is affected by noise are the tuning pa-rameters k1,2. The position signal is multiplied by a relatively high gain k1 for thestabilization of the first subsystem, as shown in Equation 4.46. Afterwards, thevirtual control input, with amplified noise, is used for the definition of the errorvariable z2, from Equation 4.1. For the definition of the second virtual control,a2, the variable z2 is further multiplied with k2 for the stabilization of control law,as well as with D

δ for the robust part. Compared to the RABLIN controller, thegains k1, k2 have significantly increased values. Finally, the noisy signal a2 is alsoused for the definition of the error variable z3,which is used in the sliding modedisturbance observer design. However, due to the low cutoff frequency of the lowpass filter in Equation 4.96, this last step does not seem to highly contribute on thenoise propagation. To conclude, the main reason the noise affects the control inputis the high gains of the stabilizing controllers, k1 and k2, as well as the value of δ.In order to counter the effect of noise, that yields a practically unusable controller,the stabilizing gains are reduced, alongside the tracking accuracy. If the parametersare now selected as:

k1,1 = 30 k2,1 = 30 k3,1 = 55 δ1 = 0.05

k1,2 = 36 k2,2 = 36 k3,2 = 130 δ2 = 0.025

and the performance indices are now changed to the ones on Table 6.3.

Table 6.3: Performance indices under nominal conditions for the proposed controller, under the newtuning parameters.

Reduced gainsApproach 2 emax

[mm]eRMS[mm]

σ [mm] ρ [s] umax [V] Ju [V2]

System 1 1.1956 0.6359 0.5498 0.0078 5.2472 36.194 ·105

System 2 1.6776 0.5989 0.5269 0.0068 7.7554 7.2754 ·106

The last 5 seconds of the input voltage now have the form shown in Figure 6.9.


95 95.5 96 96.5 97 97.5 98 98.5 99 99.5 100

Time [s]

-5

-4

-3

-2

-1

0

1

2

3

4

5

Vo

ltag

e[V

]

Servo system 1- Approach 2

Figure 6.9: Input voltage servo system 1, Approach 2, reduced gains.

It is noted that the combination of the velocity estimator with the RABLIN and theproposed controller, even in the case of the reduced gains for the latter, produceinput voltages that seem to be practically unusable. Maybe the model used forthe measurement noise is not realistic, in that the noise power is constant for thewhole frequency spectrum. However, up to this point, it is not easy to find a wayto reduce the effect of the noise, at least in the simulation model. The PI controlleris the most suitable for application, but the P-Lead controller is easily tuned to notbe affected as much.

Parameter adaptation

For the RABLIN and Approach 2 controller that utilize adaptation algorithms, theactual and estimated parameters are shown in the following figures.


0 10 20 30 40 50 60 70 80 90 100

Time [s]

2.8

3

3.2

3.4

3.6

3.8

2[k

g/m

2]

106 Servo system 1 - Parameter 2

EstimateActual

Figure 6.10: Adaptation of parameter ρ2,1under Approach 2.

0 10 20 30 40 50 60 70 80 90 100

Time [s]

4.5

5

5.5

6

6.5

7

7.5

2[k

g/m

2]

105 Servo system 2 - Parameter 2

EstimateActual

Figure 6.11: Adaptation of parameter ρ2,2under Approach 2.

0 10 20 30 40 50 60 70 80 90 100

Time [s]

-2

0

2

4

6

8

10

12

14

16

1[s

N/m

]

Servo system 1 - Parameter 1

EstimateActual

Figure 6.12: Adaptation of parameter ϑ1,1under Approach 2.

0 10 20 30 40 50 60 70 80 90 100

Time [s]

0

5

10

15

20

25

30

35

40

2[s

N/m

]


EstimateActual

Figure 6.13: Adaptation of parameter ϑ1,2under Approach 2.

0 10 20 30 40 50 60 70 80 90 100

Time [s]

-0.5

0

0.5

1

1.5

2

2.5

3

1[s

N/m

]


RABLIN

EstimateActual

Figure 6.14: Adaptation of parameter ϑ1,1under RABLIN.

0 10 20 30 40 50 60 70 80 90 100

Time [s]

-2

0

2

4

6

8

10

12

14

1[s

N/m

]


RABLIN

EstimateActual

Figure 6.15: Adaptation of parameter ϑ1,2under RABLIN.


The estimated value of ρ2 is almost the same for both controllers. Even for highadaptation gain, equal to 1000, it varies very lowly in small increments.

Comments on the performance under nominal conditions

Regarding the first servo system, the performance of the proposed controller interms of error indices is increased compared to the others, except for the standarddeviation. This shows that the error of the RABLIN controller is mostly constantalong the trajectory and is not affected by the disturbances due to different con-figurations of the manipulator. In terms of the control effort criteria, the resultsare almost uniform dor the nonlinear controllers, due to noise. However, the PIcontroller is almost not affected by measurement noise and is thus most suitablefor implementation due to valve wear considerations.Regarding the second servo system, the performance of the PI controller is deemedthe most satisfactory. The proposed controller is better than the proportional andRABLIN controllers in terms of the error indices, but has the highest control effort.

6.3 Robustness comparisons

The parameters of the system vary during operation. This not only due to the vary-ing positions of the actuators alongside the trajectory, but also due to exogenousreasons. Reasons the hydraulic parameters can change are the operating temper-ature of the oil, the air dissolved in the oil, the leakage coefficient and Coulombfriction of the cylinders and their estimation, as well as the initial volumes of thehoses. These parameters can vary compared to the ones the design was realizeddue to poor estimation or because the actual setup need to be moved or the actu-ators modified. The parameters of the mechanical system can vary due to a loadattached to the tool tip, or because a different, realizable, trajectory is required. Theperformance of the control system under parameter variations should not deviatemuch from that under nominal conditions.

6.3.1 Parameters that vary

In the following sections, the performance indices of the already designed con-trollers will be compared under the following cases:

• Change in the initial volumes, which can occur if the setup is moved in aplace further than the pump and tank.

• Changes in the leakage coefficient and Coulomb friction. This can occur ifdifferent cylinders are used as actuators. Commercial cylinders have almostnegligible leakage coefficient and the Coulomb friction is also increased.

6.3. Robustness comparisons 81

• Variation of the viscous friction coefficient. These could occur due to inaccu-rate estimation.

• Change in the effective oil bulk modulus. This can occur if there is a consid-erable percentage of dissolved air in the oil and if the working temperatureof the oil is varied.

• Change in the requested trajectory and/or load of the manipulator.

No changes will be made to the parameters and the only terms that will changeare the ones that involve some parameter adaptation.

6.3.2 Variation of the initial volumes

In the following simulations, all the other parameters are nominal apart from theinitial volumes of the hoses. Again the trajectory is repeated 20 times and theresults regard the performance form the beginning of the trajectory.

Table 6.4: Performance indices under increased initial volumes by 400%.

Servo System 1Controller emax [mm] eRMS [mm] σ [mm] ρ [s] umax [V] Ju · 105 [V2]

PI 3.8918 1.462 1.4617 0.0263 4.2855 7.3648P-Lead 3.5951 1.7124 1.1409 0.0257 4.5359 6.9948

RABLIN 1.4656 0.9038 0.4348 0.0108 4.871 32.03Approach 2 1.4704 0.6355 0.5507 0.0108 4.1387 34.8

Table 6.5: Performance indices under increased initial volumes by 400%.


PI 1.2678 0.4935 0.4935 0.0078 5.1204 1.3808P 1.8301 0.7404 0.5503 0.0113 5.0946 1.3651

RABLIN 2.0946 0.9732 0.5392 0.0128 5.177 6.377Approach 2 1.6419 0.5975 0.5253 0.01 5.2255 6.884

Increased initial volumes by 400% The increased initial volumes lead to a moredamped system. However the eigenfrequency of the system is also decreased. Inthe case of the P-Lead controller of servo system 1, the error indices are decreasedeven though the gain crossover frequency is decreased. This can be explained bythe fact that using the nominal pressure feedback gain the damping ratio is notequal to the desired value. This leads to the existence of a small resonant peak that


keeps the magnitude of the open loop transfer function higher than the nominalone for some frequency range. This leads to better disturbance attenuation, asnoted in section 5.1.3. This is shown in the bode plot of Figure 6.16.

-10

0

10

20

Mag

nitu

de (

dB)

Nominal400% increase in initial volume

100 101 102-225

-180

-135

-90

-45

Pha

se (

deg)

Bode Diagram

Frequency (rad/s)

Figure 6.16: Bode plot of the open loop transfer function of servo system 1, with nominal andincreased initial volumes, compensated by P-Lead control.

Note regarding the velocity estimator Up to now, it is seen that the noise effectis severe and the tracking results, though acceptable, cannot be achieved in a realsetup. This may be due to the way the noise has been modeled. The most importantproblem however comes from the velocity estimator. As described in Appendix D,the procedure is to first wait at the initialization position until the velocity, andposition, estimation error becomes zero and then start tracking using the desiredcontroller. However, this change of controllers in the majority of the sensitivitytests that follow, made the system unstable. This was remedied by decreasingthe controller gains. However, tracking performance was decreased. In all thefollowing tests the velocity is acquired from the simulation model.

6.3.3 Variations in leakage coefficient and Coulomb friction

In this test, the Coulomb friction coefficient was increased by 1000% and the leak-age coefficient decreased by 99%. The resulting performance indices are presentedin Tables 6.6 and 6.7.


Table 6.6: Performance indices under increased Coulomb friction coefficient and decreased leakage.


PI 5.9912 2.4639 2.4636 0.0405 4.9314 9.5293P-Lead 5.1859 2.416 2.0151 0.0378 5.0939 8.6975

RABLIN 2.1985 0.9991 0.6178 0.016 4.7348 8.829Approach 2 1.764 0.593 0.4999 0.0129 4.8363 8.8073

Table 6.7: Performance indices under increased Coulomb friction coefficient and decreased leakage.


PI 2.2978 0.9691 0.9691 0.014 5.4485 1.5761P 2.6859 1.2814 1.1925 0.0166 5.4117 1.5384

RABLIN 3.5364 1.2176 0.9284 0.0214 5.3918 1.564Approach 2 1.9783 0.5714 0.4975 0.0121 5.567 1.5668

While the effect of decreasing the leakage coefficient leads to increased Coulombfriction, the effect on the system can be seen in two distinct regions. Initially,the increase in the Coulomb friction coefficient increases in absolute terms thedisturbances for each servo system. These increases are shown in Figures 6.17 and6.18.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time [s]

-0.5

0

0.5

1

1.5

2

Acc

eler

atio

n [

m/s

2]

Disturbance term - Servo system 1

Nominal1000% increased Coulomb coefficient

Figure 6.17: Increased disturbance term ofservo system 1.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time [s]

-5

-4

-3

-2

-1

0

1

2

Acc

eler

atio

n [

m/s

2]

Disturbance term - Servo system 2

Nominal1000% increased Coulomb coefficient

Figure 6.18: Increased disturbance term ofservo system 1.

A second effect can be seen from Equations C.6 or 3.44. The steady state gainis increased, the eigenfrequency is decreased. Furthermore, the damping factoris also decreased. However, due to the effect of pressure feedback and by thefact that the valve pressure coefficient has higher value, the effects of the leakage


coefficient decrease are not observed. The major performance decrease comes fromthe increase in the disturbance term.

6.3.4 Variation of the viscous friction coefficient

Changes in the value of viscous friction coefficient can occur mainly due to poorestimation. Initially, the increase of the viscous friction coefficient will be investi-gated. It can be observed from Equation C.6 that an increase in the friction coeffi-cient leads to decreased steady state gain, but also increased eigenfrequency. Howmuch the damping ratio is affected is not as straightforward and will be calculated.For a 700% increase in viscous friction coefficient:

Table 6.8: Linearized system parameters after a 700% increase of Bv.

Servo system 1Nominal Varied Bv % Change

ωl [rad/s] 33.666 34.0704 +1.2ζl 0.0597 0.2277 +281.4

G [m/V] 0.0392 0.0383 -2.296Servo system 2

Nominal Varied Bv % Changeωl [rad/s] 81.1191 82.6498 +1.887

ζl 0.0818 0.4831 +490.5868G [m/V] 0.0353 0.0367 -3.8147

However, the most significant effect of the increase of the friction coefficient is seenfrom the PL −QL diagram of Figure 6.19:

-100 -80 -60 -40 -20 0 20 40 60 80 100

PL

[bar]

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

QL

[l/s

]

PL

- QL

diagram - 700% increase in Bv


Figure 6.19: PL −QL plot for increased viscous friction coefficient.

Since the along the nominal trajectory the second servo system surpassed the max-


imum flow, it is expected to saturate. This can be seen from the linearized loadflow equation of the valve:

QL ≈ Kquv + KqpPL (6.7)

If the required flow for a given load pressure is higher than the maximum, then itcan only be provided if uv > uv,max, which is not possible. The situation worsensas the load pressure moves to the critical point PL ≈ 2

3 Ps, because then also Kq

decreases and saturation occurs for lower desired load flow. It is evident that forthe linear controllers the increase in the error indices will happen due to saturationfrom the second servo system. For the PI controller, having an anti-windup loopwill reduce the error since the integral term will not have increased when thetrajectory returns inside the limits.For the case of the nonlinear controllers, the increase in Bv can be more easilyobserved via the uncertain term ϑ1. The performance indices are presented inTables 6.9 and 6.10.

Table 6.9: Performance indices under increased viscous friction coefficient by 700%.


PI 11.9337 4.8785 4.8784 0.0921 6.8093 16.04P-Lead 10.1009 4.07 3.7298 0.0813 7.0297 13.89

RABLIN 3.5569 1.0164 0.6524 0.0260 10 20.289Approach 2 2.4734 0.6964 0.6191 0.0181 10 20.03

Table 6.10: Performance indices under increased viscous friction coefficient by 700%.


PI 8.2183 3.8645 3.8604 0.0512 10 4.8194P 9.2771 4.0848 4.0312 0.0608 10 4.0585

RABLIN 8.6967 1.7113 1.484 0.0539 10 5.1Approach 2 5.0825 0.9579 0.9093 0.0315 10 4.9389

Since the uncertain term ϑ1 is directly connected to the viscous friction coefficient,for the last two controllers that utilize parameter adaptation the error indices arealso shown for the last 40 seconds of the trajectory, where the term ϑ1 presents amore settled behavior.


Table 6.11: Performance indices under increased viscous friction coefficient by 70%, last 40 [s].

Servo System 1Controller emax [mm] eRMS [mm] σ [mm] ρ [s] umax [V]RABLIN 1.644 0.9206 0.4896 0.012 10

Approach 2 1.4646 0.6521 0.5667 0.0107 10

Table 6.12: Performance indices under increased viscous friction coefficient by 70%, last 40 [s].

Servo System 2Controller emax [mm] eRMS [mm] σ [mm] ρ [s] umax [V]RABLIN 3.6821 1.1825 0.8582 0.0228 10

Approach 2 3.8536 0.8141 0.7632 0.0239 10

The adaptive term ϑ1 for each servo system and controller is shown in Figures 6.20- 6.23. It can be seen that it has reached a value that reduces the error indices afterabout 60 [s].

0 10 20 30 40 50 60 70 80 90 100

Time [s]

0

2

4

6

8

10

12

14

16

18

20

1[s

N/m

]


RABLIN

EstimateActual

Figure 6.20: Adaptation of term ϑ1,RABLIN, servo system 1.

0 10 20 30 40 50 60 70 80 90 100

Time [s]

-20

0

20

40

60

80

100

120

1[s

N/m

]


RABLIN

EstimateActual

Figure 6.21: Adaptation of term ϑ1,RABLIN, servo system 2.


0 10 20 30 40 50 60 70 80 90 100

Time [s]

-5

0

5

10

15

20

25

30

1[s

N/m

]Servo system 1 - Parameter

1Approach 2

EstimateActual

Figure 6.22: Adaptation of term ϑ1,Approach 2, servo system 1.

0 10 20 30 40 50 60 70 80 90 100

Time [s]

-20

0

20

40

60

80

100

120

140

1[s

N/m

]


Approach 2

EstimateActual

Figure 6.23: Adaptation of term ϑ1,Approach 2, servo system 2.

A decrease in the viscous friction coefficient would only lead to a decrease inthe damping ratio in the case of the linearized system, but the pressure feedbackloop is tuned already conservatively. At the same time, the flow demand wouldbe decreased, making it easier for the controller to achieve the desired trajectorytracking objective.

6.3.5 Variations in effective oil bulk modulus

The effective oil bulk modulus varies due to load pressure, working temperatureof the oil as well as due to the presence of free air trapped in the fluid [2]. Theeffective oil bulk modulus affects the dynamics of the system. In the case of thenonlinear system, the bulk modulus is present in all the uncertain terms of theload pressure dynamics, as seen from Equations 3.7 and 3.9 - 3.10. For the first twolinear controllers, reduced effective oil bulk modulus leads to lower eigenfrequencyand a slight decrease in the damping ratio. The last conclusion is derived throughcalculation of the damping ratio for lower values of effective bulk modulus at thelinearization point.Since an empirical model has been used for the effective oil bulk modulus in Equa-tion 2.12, one thing that can be done is to change the tuning parameters of themodel and observe in simulations if the value has dropped adequately for the sen-sitivity tests. After selecting for the first servo system the maximum oil stiffnessequal to 3500 [bar] and for the second servo system equal to 1500 [bar], for simu-lation purposes, the resulting peak effective oil bulk modulus was for both servosystems around 4500 [bar]. The resulting indices are shown in Tables 6.13 and 6.14.


Table 6.13: Performance indices under decreased effective oil bulk modulus.


PI 3.8238 1.4613 1.4610 0.0259 4.2928 7.4237P-Lead 3.6064 1.7135 1.1421 0.0258 4.5508 7.0505

RABLIN 1.5980 0.8941 0.4477 0.0118 4.3525 6.8289Approach 2 1.4752 0.6377 0.5527 0.0109 4.181 6.8665

Table 6.14: Performance indices under decreased effective oil bulk modulus.


PI 1.3539 0.4939 0.4939 0.0083 5.1214 1.3806P 1.8289 0.7408 0.5508 0.0113 5.0958 1.3649

RABLIN 2.555 0.9754 0.5518 0.0157 4.9803 1.3689Approach 2 1.6422 0.5977 0.5256 0.01 5.2367 1.3835

The performance of the last two controllers improved in the last 40 [s], but onlymarginally. It is observed that for the P-Lead controller, the performance slightlyimproves. This is due to a slight increase in the gain crossover frequency of theopen loop transfer function with the controller.

6.3.6 Variations in trajectory and load

In the following sensitivity tests, the parameters that will vary are the trajectoryand the load of the manipulator, both of which are in general known and can beselected by the operator. However, the controllers are left with the parameters theywere tuned for.

Nominal trajectory with load

In this section, the nominal, 5 [s] trajectory is still tracked, but now the manipulatorhas attached to its tool tip position a point mass of 35 [kg]. The nominal trajectoryis still realizable, as shown in section B.6.3.The added mass changes the dynamics of the system, as well as the disturbanceterms. The added mass is present in the elements of the inertia matrix M, Coriolismatrix C and in the gravitational terms G. The disturbance terms are shown inFigure 6.24.


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time [s]

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

Acc

eler

atio

n [

m/s

2]

Disturbance terms - Nominal Trajectory - 35 [kg] Load


Figure 6.24: Disturbance terms under nominaltrajectory and added load of 35 [kg].

0 0.5 1 1.5 2 2.5 3 3.5

Time [s]

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

Acc

eler

atio

n [

m/s

2]

Disturbance terms - Fast trajectory - No load


Figure 6.25: Disturbance terms under fastertrajectory and no added load.

In terms of the linearized system, the eigenfrequency is decreased and the dampingratio slightly increased, after calculations.The resulting error indices are presented in Tables 6.15 and 6.16.

Table 6.15: Performance indices under added load and nominal trajectory.


PI 6.9368 2.0854 2.0846 0.0457 5.3372 9.2543P-Lead 6.1157 3.2536 1.5266 0.0432 5.4056 8.5443

RABLIN 2.6677 1.7913 0.5613 0.0195 4.9946 8.3768Approach 2 1.9108 0.9221 0.6063 0.014 5.1405 8.3828

Table 6.16: Performance indices under added load and nominal trajectory.


PI 1.5994 0.6483 0.6482 0.0097 5.7556 1.5067P 3.2667 1.6775 0.6975 0.0201 5.7431 1.4872

RABLIN 4.0784 2.4775 0.5995 0.0248 5.6806 1.4981Approach 2 1.9108 0.9221 0.6063 0.014 5.1405 8.3828

Especially for the last controller, the error indices for the last 40 [s] are reformedas:

emax,1 = 1.646 [mm] emax,2 = 1.7 [mm]

It is expected that the controllers that utilize parameter adaptation are more robusttowards variations in the parameters that contain elements of the inertia and Cori-olis matrices. The adaptation gains or algorithm can be varied to achieve faster


adaptation. However, it is evident that the tracking performance is not heavilydeteriorated.

Faster trajectory without load

It might be required that a different trajectory is to be tracked. A simple exampleis to track the same trajectory in shorter time, meaning that the velocity and ac-celeration commands are larger in magnitude. No load is attached to the tool tipposition. This trajectory is presented in section B.4. The differences will lie in thedisturbance terms and in the uncertain parameter ϑ1. Furthermore, the Coriolismatrix elements might not be negligible for the case of the linearized systems. Thedisturbance terms are shown in Figure 6.25. The performance indices are presentedin the following tables.

Table 6.17: Performance indices for faster trajectory and no load.


PI 7.3644 2.2037 2.2034 0.0346 7.1851 12.345P-Lead 5.3855 2.0192 1.4589 0.0271 7.2891 10.993

RABLIN 1.6124 0.9349 0.4363 0.0081 6.7155 10.676Approach 2 2.1407 0.6745 0.5923 0.0108 6.8897 10.75

Table 6.18: Performance indices for faster trajectory and no load.


PI 2.3226 0.6826 0.6826 0.0094 7.5929 2.0672P 2.3163 0.86 0.7125 0.0095 7.4911 2.0414

RABLIN 2.9738 1.081 0.6881 0.0121 7.5206 2.0566Approach 2 2.5627 0.7138 0.6487 0.0104 7.6897 2.0602

After 60 [s] , the error indices for the RABLIN and Approach 2 controllers become:

Table 6.19: Performance indices for faster trajectory and no load, for the last 40 [s].

Servo System 1Controller emax [mm] eRMS [mm]RABLIN 1.5078 0.933

Approach 2 1.3826 0.6583


Table 6.20: Performance indices for faster trajectory and no load for the last 40 [s].


Approach 2 1.7844 0.6948

The parameter ϑ1 for the two servo systems and controllers is shown in Figures6.26 - 6.29 .

0 10 20 30 40 50 60 70 80 90

Time [s]

-0.5

0

0.5

1

1.5

2

2.5

3

1[s

N/m

]

Parameter 1

- Servo system 1 - RABLIN - Fast trajectory

EstimateActual

Figure 6.26: Adaptation to parameter ϑ1 underthe fast trajectory.

0 10 20 30 40 50 60 70 80 90

Time [s]

-2

0

2

4

6

8

10

12

14

1[s

N/m

]

Parameter 1

- Servo system 2 - RABLIN - Fast trajectory

EstimateActual


0 10 20 30 40 50 60 70 80 90

Time [s]

-2

0

2

4

6

8

10

12

14

16

18

1[s

N/m

]

Parameter1

- Servo system 1 - Approach 2 - Fast trajectory

EstimateActual


0 10 20 30 40 50 60 70 80 90

Time [s]

-5

0

5

10

15

20

25

30

35

40

45

1[s

N/m

]

Parameter1

- Servo system 2 - Approach 2 - Fast trajectory

EstimateActual


It can be seen that the estimated parameter achieves a non increasing trend fasterin the case of the faster trajectory.


Faster trajectory with load

As a final test, the faster trajectory alongside a load at the tool tip position willbe simulated. The load is elected to be equal to 15 [kg], so that the required loadpressure along the trajectory never surpasses the critical point of maximum powertransfer. This is seen in Figure B.42. The resulting performance indices are shownin the following tables.

Table 6.21: Performance indices for faster trajectory and with load.


PI 7.8513 2.692 2.6916 0.047 7.9728 13.761P-Lead 7.2307 2.7073 1.7198 0.0369 7.998 11.993

RABLIN 2.0933 1.3138 0.5095 0.0105 7.7054 11.818Approach 2 2.3491 0.7863 0.624 0.0118 7.8038 11.831

Table 6.22: Performance indices for faster trajectory and with load.


PI 2.5648 0.7963 0.7963 0.047 7.943 2.1401P 2.9465 1.2187 0.8043 0.012 7.7306 2.1082

RABLIN 3.6157 1.6766 0.7624 0.0146 7.691 2.1269Approach 2 2.4042 0.8536 0.6703 0.0097 7.7059 2.1155

Table 6.23: Performance indices for faster trajectory and with load, for the last 40 [s].


Approach 2 1.6014 0.7737

Table 6.24: Performance indices for faster trajectory and with load for the last 40 [s].


Approach 2 1.6385 0.8382

6.4. Summary of simulation results and comments 93

6.4 Summary of simulation results and comments

In this section, selected indices from the presented tables will be aggregated andthe results commented on. Moreover, the cases where the performance was highlyaffected will be considered. Since the focus is on tracking performance, the eRMSwill be shown in the graphs.

Coulomb friction and leakage variations

Coulomb friction +1000% , Leakage coefficient - 99% - Servo system 1

Approach 2 P-Lead PI RABLIN0

1

2

3

4

5

6

em

ax[m

m]

VariedNominal

+18.6 %

+42.9 % +58.1 %

+37.4 %

Figure 6.30: Bar graph of emax for servosystem 1.

Coulomb friction +1000% , Leakage coefficient -99% Servo system 2

Approach 2 P PI RABLIN0

0.5

1

1.5

2

2.5

3

3.5

4

em

ax[m

m]

VariedNominal

+20.5 %

+47 %+81 %

+38.3 %

Figure 6.31: Bar graph of emax for servosystem 2.

Coulomb friction +1000% , Leakage coefficient -99% - Servo system 1


0.5

1

1.5

2

2.5

eR

MS

[mm

]

VariedNominal

- 7.5 %

+40.7 % +68.6 %

+11.7 %

Figure 6.32: Bar graph of eRMS for servosystem 1.

Coulomb friction +1000% , Leakage coefficient -99% Servo system 2


0.2

0.4

0.6

0.8

1

1.2

1.4

eR

MS

[mm

]

VariedNominal

-4.3 %

+97 %

+73.2 %

+24 %


As a first note, the eRMS is decreased in the simulations for the proposed controller.A possible explanation can be found in the disturbance observer and specificallyin the determination of the gain G from Equation 4.97. The value of ϑ3 from 3.9is decreased by 99%, due to the decrease in Cl . Assuming the momentary value


of the load pressure to be equal for the nominal and varied case, this increases thevalue of G for the nominal system. This leads to the disturbance being "tracked"faster, since the sliding surface S, 4.86 reaching the value zero in faster time thanthe nominal case. The increase in the peak value of the error however, is prob-ably due to the increase in the peak value of the disturbance terms by 53% and94.1% for the first and second servo system respectively. The average values onlymarginally increased. The difference in the RABLIN controller that deteriorates theperformance comes from the design of the inner load pressure loop. The distur-bance terms have the same effect as they have on the proposed control system, butalso affect the inner loop. In order to show how they affect it, the transfer functionof the load pressure to the disturbance term are investigated for the nominal andvaried parameter system. From the block diagram of Figure 3.9, the disturbancetransfer function is written:

PL(s)D(s)

= −Gpc(s) A

Mii(s)+Bv,i

1 + Gpc(s)Gc(s)(6.8)

The only thing that changes in the transfer function is the value of the leakagecoefficient, that is negligible when using pressure feedback. As a result, only themagnitude of the disturbances affects the tracking performance of the RABLINcontroller. Regarding the reference linear controllers that utilize pressure feed-back, the effect of the leakage coefficient change does not change the form of thedisturbance transfer function, but they do not include parameter adaptation, so thetracking error is increased.

Viscous friction coefficient

The most obvious way the increased viscous friction coefficient Bv affects the sys-tem is the required flow to realize the trajectory. This leads to increased controleffort.For the proposed controller, the term Bv affects the parameter ϑ1, as seen in Equa-tion 3.8. This is the case for the RABLIN controller, but now the viscous frictioncoefficient affects also the design of the load pressure loop, as can be seen fromEquation 3.48. In the same fashion, the reference controllers are affected, this canalso be seen from the respective bode plots.First, the increase in eRMS is shown in Figure 6.34 and 6.35 for each servo system.


Viscous friction coefficient +700% , Servo system 1


0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

eR

MS

[mm

]

VariedNominal

+ 8.5%

+ 137%+ 233%

+ 13.6%


Viscous friction coefficient +700% , Servo system 2


0.5

1

1.5

2

2.5

3

3.5

4

4.5

eR

MS

[mm

]

VariedNominal

+ 60.4%

+ 452%+ 685.3%

+ 75.4%


Regarding the proposed controller, the controller is affected mainly by the in-creased value of ϑ1, for which an adaptive term is utilized in the controller. Thisis the reason the tracking error is kept low. The RMS tracking error is decreasedafter 60 [s], when the adaptation algorithm has a reached a specific trajectory. Theresults presented here consider the whole time history of the 20 reference trajecto-ries.The RABLIN controller includes an adaptive term for the same parameter. How-ever, the gain crossover frequency of the compensated system is affected. This isshown in the bode plot for the second servo system, where this reduction is moreevident, in Figure 6.36. The new gain crossover frequency reduced to 74.7 [rad/s].This reduction leads to decreased response time.

10-2 10-1 100 101 102

-20

-10

0

10

20

30

40

Mag

nitu

de (

dB)

Nominal+700% B

v

Bode Diagram

Frequency (rad/s)

Figure 6.36: Bode plot of Gpc(s)Gc(s) forservo system 2, RABLIN.

10-2 10-1 100 101 102

-20

0

20

40

60

80

100

120

Mag

nitu

de (

dB)

Nominal+700% B_v

Bode Diagram

Frequency (rad/s)

Figure 6.37: Bode plot of PI compensatedservo system 2.

Regarding the reference controllers, the error indices increase due to the decreasein the steady state gain. This decrease would require an inversely proportionalincrease in the feedforward gain, so that the tracking performance is increased,


as seen from Equation 5.34. Furthermore, from the bode plot of Figure 6.37, thegain crossover frequency of the open loop compensated transfer function is alsodecreased, reducing the response time of the controller to variations in the distur-bance term. The same holds for the P-Lead and P compensated servo systems.Especially for the proportionally compensated servo system the gain crossover fre-quency is reduced from 41 [rad/s] to 20.8 [rad/s].

Variations in reference trajectory and load.

For the simultaneous case when the reference trajectory completion time is de-creased a load is added to the tool tip position, the RMS error variation is shownin Figures 6.38 and 6.39.

Faster trajectory , 15 [kg] load - Servo system 1


0.5

1

1.5

2

2.5

3

eR

MS

[mm

]

VariedNominal

+ 22.5%

+ 57.7% + 84.2%

+ 46.9%


Faster trajectory , 15 [kg] load - Servo system 2


0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

eR

MS

[mm

]

VariedNominal

+ 42.9%

+ 64.7%

+ 61.8%

+ 71.9%


The change in the load affects the inertia matrix elements and the gravitationalterms. This means that the magnitude of the uncertain terms ϑ1 and ϑ2, as wellas d(t) from Equation 3.8 will be affected. The magnitude of the Coriolis matrixelements will also be affected by the load, as well as the faster reference trajec-tory. The adaptive algorithm in both RABLIN and the proposed controllers willtackle the uncertain terms. However, due to the variation in the magnitude of thedisturbance term, boundedness of the error variables for the second subsystemis not guaranteed. The condition in order for the second subsystem to be stablecan be written, using the same Lyapunov function candidate from Equation 3.24.Using the same adaptation terms and input u2 as in Equations 3.27 and 3.29, thederivative of the Lyapunov function candidate for the second subsystem becomes:

V2 = −k1z21 − k2z2

2 + ϑ2z2z3 + z2[−d′(t) + a2r

]=

= −k1z21 − k2z2

2 + ϑ2z2z3 + z2 {− [d(t) + ∆d(t)] + a2r} (6.9)


Selecting a2r = −z2Dδ as in Equation 3.31 and assuming that z3 = 0:

V2 ≤ −k1z21 − k2z2

2 +Dδ

4− z2∆d(t) ≤ −k0||z||2 +

Dδ

4+ ||z||∆d(t) (6.10)

with the term ∆d(t) denoting the variation of the disturbance term due to thechange in the Coriolis and inertia matrix elements, and k0 = min {k1, k2}. In orderto find a condition for stability or a bound, the time derivative of V2 is written as:

V2 ≤ −k0(1− θ)||z||2 + δD4− k0θ||z||2 + ∆d(t)||z|| (6.11)

0 < θ < 1

Completing the squares for the last two terms:

V2 ≤ −k0(1− θ)||z||2 + δD4− k0θ

[||z|| − ∆d(t)

2k0θ

]2

+∆d2(t)4k0θ

≤ −k0(1− θ)||z||2 + δD4

+|∆d(t)|2max

4k0θ

≤ −[

k0(1− θ)||z||2 − δD4− |∆d(t)|2max

4k0θ

]︸︷︷︸

≥0

(6.12)

The condition is satisfied in the set

U =

{||z|| ≥

√δD

4k0(1− θ)+|∆d(t)|2max

4k20θ(1− θ)

}(6.13)

Following the same procedure as in section 3.1, it is deduced that the limit the errorvariables z1 and z2 converge to is increased. What this result means is that a partof k0 is used to dominate the part originating from the variation of the disturbanceterm. This requires that the gain k0 be sufficiently high. It is assumed that for thesystem to be stable, θ exists. It is thus shown how the variation of the Coriolisterms affects the tracking error of the RABLIN and the proposed controller.Regarding the load pressure loop of the RABLIN and the reference controllers, theyare based on a linearized model, that neglects the Coriolis term in the mechani-cal system. This variation also deteriorates the tracking error performance but ismore difficult to visualize, since the Coriolis matrix elements depend not only onposition, but also on velocity that has to be selected at the point of linearization.Especially for the reference controllers, it is easier to see from Equation C.7 thatthe steady state gain is decreased along the faster trajectory, because the load pres-sure moves towards the point of maximum power transfer point, decreasing thevalve flow gain Kq. This would require retuning of the velocity feedforward gain,


however this effect is not the most deteriorating factor. At the same time, the dis-turbance terms, defined as forces for the linearized systems, present the followingvariation alongside the faster trajectory with load. These changes are:

Table 6.25: Increase in disturbance terms for the linearized system.

d(t)[N] |d(t)|max % d(t)RMS %d1(t) +26 +32d2(t) +98 +66

The loop gain of the compensated reference systems are shown in the bode plotsof Figures 6.40 and 6.41 .

10-2 10-1 100 101 102

-10

0

10

20

30

40

50

60

70

80

90

100

Mag

nitu

de (

dB)

PI , NominalP-Lead , NominalPI , VariedP-Lead , Varied

Bode Diagram

Frequency (rad/s)

Figure 6.40: Bode plot of loop gain ofservo system 1.

10-2 10-1 100 101 102

0

20

40

60

80

100

120

Mag

nitu

de (

dB)

PI , NominalP , NominalPI , VariedP , Varied

Bode Diagram

Frequency (rad/s)

Figure 6.41: Bode plot of loop gain ofservo system 2.

The magnitude of the loop gain is slightly decreased for the whole frequency rangeup to the respective gain crossover frequency, which is also slightly reduced. How-ever, the magnitude of the disturbances is increased, leading to poor disturbancerejection, that reduces tracking performance.

6.5 Conclusions from simulations

In this chapter, selected performance indices were compared for the controllers de-signed in this report. Initially, simulations were performed under nominal designconditions. Afterwards, variations were made to system parameters that are timevarying during operation, are difficult to estimate, or are prone to change due tooperation conditions or operator requirements.Simulation results proved the increased robustness gained from the nonlinear de-sign of the RABLIN and the proposed controller. The sliding mode disturbance

6.5. Conclusions from simulations 99

observer, that substitutes the linear design of the load pressure loop of the RABLINcontroller leads to increased performance. However, it induces chattering for lowfilter time constants.Under nominal conditions, for the second servo system the PI compensator leadsto the best tracking performance, compared to control algorithm complexity andease of implementation. Furthermore, the PI compensator is not sensitive to mea-surement noise, which highly affects the proposed controller.The RABLIN controller provides increased robustness and tracking performancethanks to the adaptive algorithms and ease of selection of tuning parameters. Inthis report, the inner loop has also been designed conservatively, compared to theeigenfrequency of the valve.The proposed controller is the most robust to parameter variations in the simula-tion sensitivity tests, providing increased performance at the same time. However,in order to decrease its sensitivity to measurement noise, the feedback gains needto be reduced, at the cost of tracking accuracy.All the aforementioned conclusions are based on the fact that the velocity esti-mation is exact. However, it was noticed in simulations that the actual processof including the differentiator and changing controllers was challenging. This pro-cess could be made easier by decreasing the gains of the RABLIN and the proposedcontroller. However, the performance indices will be negatively affected.

Chapter 7

Conclusions and Future Work

7.1 Conclusions

The proposed controller provides a solution to the "explosion" of terms that rid-dles the backstepping procedure. Under the assumption that the low pass filteraccurately tracks the value of the disturbance, provided by the average value ofthe sliding mode observer switching function, the proposed algorithm is proven tobound the closed loop tracking error to a limit that can be estimated. Furthermore,the different variations in the performance in the sensitivity tests are more intuitiveto explain.The RABLIN controller proves to be easier to implement, modify and tune, whileproviding increased robustness. In this report, it can be tuned to provide widerclosed loop bandwidth for the load pressure loop, enhancing the disturbance rejec-tion capabilities and response time. The only downside is the difficulty at provingthe boundedness or stability of the closed loop system, under Lyapunov theory.Both nonlinear controllers utilize parameter adaptation algorithms. These increasethe tracking performance under parameter variations, but are difficult to tune.Only limited knowledge of these parameters is however required. Furthermore,both controllers make use of a robust term in the second backstepping step. It isconsisted of one tuning parameter and the estimated maximum absolute value ofthe disturbance term. This last term is relatively easy to calculate for a system, ifthe requirements are known in advance. Finally, the proposed controller uses moreworst case estimates in the design of the disturbance observer, as well as an esti-mate for the maximum value of the second derivative of the second virtual controllaw. The last term is difficult to acquire. It is thus selected to be of sufficiently largevalue, increasing the switching effort.The linear reference controllers, especially the PI, perform satisfactorily undernominal conditions. However, if it is known that some parameters will change,the design has to be repeated. However, for the proposed controller, only a maxi-

101

102 Chapter 7. Conclusions and Future Work

mum value will have to be changed in the algorithm.Concluding, a robust controller has been designed following the backstepping pro-cedure that is proven to keep the tracking error bounded, using Lyapunov theory.The disadvantage of the controller is sensitivity to measurement noise and the factthat velocity measurement is required. Both issues can be addressed by using asliding mode differentiator and at the same time decreasing the tuning parametervalues.

7.2 Future Work

A part of the proposed controller that hinders assumptions regarding stability orfinding an ultimate bound for the tracking errors, is the inclusion of the adaptiveterms. In the way they are designed, they render the Lyapunov function negative-semidefinite. As a result, only boundedness can be assumed for the parameterestimation error. The selection of the adaptive gains is also not systematic.Another part that could be more investigated is the selection of the tuning gainsfor the proposed controller. In the present report, in order to estimate a startingpoint for the selection, assumptions regarding the parameter estimation error andthe disturbance terms were made. However, even using these assumptions, inthe absence of measurement noise, a stable and well performing system could bedesigned. It was however found in the simulation tests that the effect of noise issignificant. Maybe the model of the noise used for the simulations is not realistic,however in order to get acceptable voltage inputs for the servo valves, the controllergains have to be decreased by a wide margin. As a result, the tracking performanceis deteriorated.Moreover, the time constant of the low pass filter that is used for the sliding modedisturbance observer has not been included in the analysis. Only the residual termbetween the input and the output of the filter is considered to be small in mag-nitude. However this does not give any information regarding how to select thistime constant, if it should be taken into consideration when selecting the tuninggains, or if it affects the stability of the closed loop system.Last but not least, the way the velocity estimator can be used, resulting in a stablesystem needs to be more investigated. The procedure of initial positioning usinga PI controller seems to be viable in the sense that the estimation error becomeszero and the differentiation is almost exact. The difficulty comes when switchingbetween the initialization PI and the tracking controller. The decrease in the track-ing controller’s gains showed immediate improvement, however the tracking errorincreased.

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Appendix A

Manipulator kinematics

In this chapter, the position of the tool tip position and each link’s center of massis referred to a common reference frame as a function of joint angles. This way,the equations of motion for the manipulator can be derived via the Euler-Lagrangeequations.

A.1 Forward kinematics

Each center of mass, joint and the toll tip positions are assigned reference frames.These reference frames are then referred to the base reference frame. A sketchshowing how these reference frames are defined is presented in Figure A.1. Thebase reference frame is denoted as {Ox0y0}, those of the first and second jointas {Ox1y1} and {Bx2y2} respectively, those of the first and second link’s centerof mass as {Rxcm1ycm1} and {Cxcm2ycm2} and the reference frame of the tool tipposition as {hxhyh}. The tool tip’s local reference frame is considered to be aproduct of pure translation of the local reference frame {Bx2y2}.

A.1.1 Homogeneous transformations

As a first step, all local reference frames are referred to the base reference frame{Ox0y0}. This is done via homogeneous transformation matrices [43]. The homo-geneous matrix is of the form

T =

[R2x2 p2x101x2 1

](A.1)

where the matrix R2x2 describes the rotation of the loclas frame’s axes in regard tothe axes of the base reference frame {Ox0y0} and the vector p2x1 the position ofthe local reference frame’s origin in regard to the base reference frame’s origin.

109

110 Appendix A. Manipulator kinematics

x0

y0x1

y1

x2

y2

O

R

B Cφ2

φ1

δ2

δ1

φh

h

Figure A.1: Sketch of the manipulator with reference frames.

Joint frames {Ox1y1} and {Bx2y2}

The homogeneous transformation matrices are defined for each joint referenceframe [29].

T01 =

cos(φ1) − sin(φ1) 0sin(φ1) cos(φ1) 0

0 0 1

(A.2)

T02 = T01 ·

1 0 LOB0 1 00 0 1

·cos(φ2) − sin(φ2) 0

sin(φ2) cos(φ2) 00 0 1

=

=

cos (φ1 + φ2) − sin (φ1 + φ2) cos (φ1) LOB

sin (φ1 + φ2) cos (φ1 + φ2) sin (φ1) LOB0 0 1

(A.3)

A.2. Positions and Velocities of centers of mass 111

A.1.2 Tool tip frame {hxhyh}

T0h = T02 ·

1 0 LBh cos(φh)

0 1 LBh sin(φh)

0 0 1

=

=

cos (φ1 + φ2) − sin (φ1 + φ2) LBh cos (φ1 + φ2 + φh) + LOB cos (φ1)

sin (φ1 + φ2) cos (φ1 + φ2) LBh sin (φ1 + φ2 + φh) + LOB sin (φ1)0 0 1

(A.4)

A.1.3 Center of mass frames {Rxcm1ycm1} and {Cxcm2ycm2}

T0cm1 =

cos (φ1 + δ1) − sin (φ1 + δ1) LOR cos (φ1 + δ1)

sin (φ1 + δ1) cos (φ1 + δ1) LOR sin (φ1 + δ1)

0 0 1

(A.5)

T0cm2 = T02

cos (δ2) − sin (δ2) LBC cos (δ2)

sin (δ2) cos (δ2) LBC sin (δ2)

0 0 1

= (A.6)

=

cos (φ1 + φ2 + δ2) − sin (φ1 + φ2 + δ2) LBC cos (φ1 + φ2 + δ2) + cos (φ1) LOB

sin (φ1 + φ2 + δ2) cos (φ1 + φ2 + δ2) LBC sin (φ1 + φ2 + δ2) + sin (φ1) LOB

0 0 1

A.2 Positions and Velocities of centers of mass

Now that the homogeneous transformations have been derived, it is easy to referany point of interest, such as the centers of mass of the links, to the base referenceframe. This way, the expressions of their positions and velocities as functions of thejoint angles can be derived. These expressions will be used in the Euler-Lagrangeequations.

A.2.1 Position vectors

xcm1

ycm1

1

= T0cm1 ·

001

=

LOR cos (φ1 + δ1)

LOR sin (φ1 + δ1)

1

(A.7)


xcm2

ycm2

1

= T0cm2 ·

001

=

LBC cos (φ1 + φ2 + δ2) + cos (φ1) LOB

LBC sin (φ1 + φ2 + δ2) + sin (φ1) LOB

1

(A.8)

A.2.2 Velocity vectors

Linear Velocities

vcm1 =ddt

[xcm1

ycm1

]=

[xcm1

ycm1

]=

[−LOR sin(φ1 + δ1)φ1

LOR cos(φ1 + δ1)φ1

](A.9)

vcm2 =

[xcm2

ycm2

]=

[−LOB sin(φ1)φ1 − LBC sin(φ1 + φ2 + δ2)(φ1 + φ2)

LOB cos(φ1)φ1 + LBC cos(φ1 + φ2 + δ2)(φ1 + φ2)

](A.10)

Angular velocities

The angular velocities of each center of mass can be extracted from the skew-symmetric matrix form of [43]:

[ω] = S(ω) = Rcm·R−1cm· (A.11)

where

[ω] = S(ω) =

[0 −ω

ω 0

](A.12)

leading toωcm1 = φ1k (A.13)

ωcm2 = (φ1 + φ2)k (A.14)

where k is a unit vector pointing out of the page.

A.3 Dynamic modeling of the mechanical system

In this section the dynamic model of the manipulator is derived with the jointangles as generalized coordinates.

A.3. Dynamic modeling of the mechanical system 113

A.3.1 The Euler-Lagrange equation

To model the mechanical system, the Euler Lagrange equation is used, since ex-plicit knowledge of the constraint forces of the links is not needed [43]. An as-sumption is made at this point, regarding the possible load mass at the tool tipposition. It is assumed that the load mass is modeled as a point and that it doesnor change the position of the center of mass for the joints.

ddt(

ϑLϑqi

)− ϑLϑqi

= Qi (A.15)

L = K−U (A.16)

K = K1 + K2 + Kh (A.17)

U = U1 + U2 + Uh (A.18)

A.3.2 Generalized coordinates and energy

The joint angles are chosen as generalized coordinates for the system:

q =[q1 q2

]T=[φ1 φT

2

](A.19)

The kinetic and dynamic energy of each link can be written as:

K1 =12

vTcm1m1vcm1 +

12

Jcm1ω2cm1 =

12(m1L2

OR + Jcm1)φ21 (A.20)

K2 =12

vTcm2m2vcm2 +

12

Jcm2ω2cm2

=12(m2L2

BC + Jcm2)(φ1 + φ2)2 +

12

m2L2OBφ2

1 + m2LOBLBC cos(φ2 + δ2)(φ21 + φ1φ2)

(A.21)

Kh =12

mhL2Bhω2

cm2 =12

mhL2Bh(φ1 + φ2)

2 (A.22)

U1 =m1gLOR sin(φ1 + δ1) (A.23)

U2 =m2gLBC sin(φ1 + φ2 + δ2) + m2gLOB sin(φ1) (A.24)

Uh =mhg [LBh sin (φ1 + φ2 + φh) + LOB sin (φ1)] (A.25)

Using the derived expressions for the potential and kinetic energy in the Euler-Lagrange equations, the dynamic model of the mechanical system, with the jointangles as state variables and joint torques as inputs, becomes:


τ = D(φ)φ + C(φ, φ)φ + H(φ) (A.26)

with:

τ =[τ1 τ2

]T(A.27)

φ =[φ1 φ2

]T(A.28)

D(φ) =

[D1,1 D1,2

D2,1 D2,2

](A.29)

D1,1 = m1L2OR + Jcm1 + Jcm2 + m2

[L2

OB + L2BC + 2LOBLBC cos(φ2 + δ2)

]+

+ mh[L2

OB + L2Bh + 2LOBLBh cos(φ2 + φh)

]D1,2 = Jcm2 + m2LBC [LBC + LOB cos(φ2 + δ2)] + mhLBh [LBh + LOB cos(φ2 + φh)]

D2,1 = Jcm2 + m2LBC [LBC + LOB cos(φ2 + δ2)] + mhLBh [LBh + LOB cos(φ2 + φh)]

D2,2 = Jcm2 + m2L2BC + mhL2

Bh

C(φ, φ) =

[C1,1 C1,2

C2,1 C2,2

](A.30)

C1,1 = − [m2LOBLBC sin(φ2 + δ2) + mhLOBLBh sin(φ2 + φh)] φ2

C1,2 = −LOB [LBC sin(φ2 + δ2)m2 + LBh sin(φ2 + φh)mh] (φ1 + φ2)

C2,1 = [m2LOBLBC sin(φ2 + δ2) + mhLOBLBh sin(φ2 + φh)] φ1

C2,2 = 0

H(φ) =

[H1

H2

](A.31)

H1 = m1gLOR cos(φ1 + δ1) + m2gLBC cos(φ1 + φ2 + δ2) + (m2 + mh)gLOB cos(φ1)+

+ mhgLBh cos(φ1 + φ2 + φh)

H2 = m2gLBC cos(φ1 + φ2 + δ2) + mhgLBh cos(φ1 + φ2 + φh)

Appendix B

Trajectory generation and inverse kine-matics

In this section, the reference trajectory for the tool tip is derived. First, the trajectoryis defined in the Cartesian space for the tool tip. However, since it is the pistonpositions that are directly controlled, this trajectory is translated from the Cartesianspace to the joint space. Afterwards, from the joint reference trajectory, using therelationship that connects the actuator length to the joint angle, the trajectory istranslated as piston reference trajectory.

B.1 Workspace of the manipulator

In this initial section, the workspace of the manipulator, in the Cartesian space isdefined. This first step is essential for the definition of the reference trajectory, sinceit must be reachable by the tool tip position. Furthermore, each cylinder’s motionin a region near the end of the strokes is dampened for safety reasons. As a result,the characteristics of the motion are different in these regions. In order to makethe analysis and the design simpler, instead of taking these damping effects intoconsideration, these regions are completely avoided in the design of the referencetrajectory.The positions where the motion of the cylinder is dampened, are found via movingeach cylinder’s piston from end to end and observing the position measurements.These experiments showed that the movement should be confined for both cylin-ders between ±150 [mm], with 0 [mm] being the mid position. This allows somemargins before the motion starts to become dampened.In Figure B.1, all the reachable points of the tool tip are enclosed by the blue line. Ifthe confined movement of the pistons is taken into consideration, then the pointsreduce to those enclosed by the red line. The trajectory will be designed insidethese red limits and is the topic of the following section.

115

116 Appendix B. Trajectory generation and inverse kinematics

-0.5 0 0.5 1 1.5 2 2.5

X axis [m]

-1

-0.5

0

0.5

1

1.5

2

2.5

Y a

xis

[m]

Manipulator Workspace

Full piston stroke

Limited piston stroke

Figure B.1: Workspace and reference tool tip trajectory of the manipulator. The cross denotes thestarting position and the direction is clockwise.

B.2 Trajectory in Cartesian space

The desired trajectory [29] for the tool tip to follow is a rectangular shape. In theCartesian space each corner of the rectangular can be described as [18]:

ri = xi i + yi j [m] , i = [0, 1, 2, 3] (B.1)

The rectangular trajectory can be described by four straight line segments. Eachsegment can be further parameterized by a time polynomial whose parameters arecalculated based on given constraints. The constraints are, apart from the initialand final position, a sufficiently smooth movement. For a movement like thisto happen, the velocity, acceleration and its derivative, or jerk, should ideally bezero at the starting and end positions. In total, eight constraints are present in amovement like this on a straight line. As a result, the resulting polynomial of timeshould have eight parameters. Such a polynomial can be described by: [18],[43]

s(t) = a0 + a1t + a2t2 + a3t + a4t4 + a5t5 + a6t6 + a7t7 [m] (B.2)

s(t) = a1 + 2a2t + 3a3t2 + 4a4t3 + 5a5t4 + 6a6t5 + 7a7t6 [m/s] (B.3)

s(t) = 2a2 + 6a3t + 12a4t2 + 20a5t3 + 30a6t4 + 42a7t5 [m/s2] (B.4)

...s (t) = 6a3 + 24a4t + 60a5t2 + 120a6t3 + 210a7t4 [

m/s3] (B.5)

The constraints for a movement between two points are:

s(t0) = q0, s(t0) = 0, s(t0) = 0,...s (t0) = 0 (B.6)

s(t f ) = q f , s(t f ) = 0, s(t f ) = 0,...s (t f ) = 0 (B.7)

B.3. Slow trajectory 117

The parameters to be determined are ai, i = 0, 1, .., 7 and q(t) is the coordinate thatchanges. For example, if the tool tip should move on a straight vertical line wherethe tool tip’s x axis coordinate is constant, q is the y axis coordinate and, morespecifically, q(t0) is y axis coordinate in the beginning of the movement and q(t f )

is the y axis coordinate at the destination point. The polynomial’s parameters, foreach line segment, can be determined by solving:

a = A−1b (B.8)

a =[a0 a1 a2 a3 a4 a5 a6 a7

]T(B.9)

A =

1 t0 t20 t3

0 t40 t5

0 t60 t7

00 1 2t0 3t2

0 4t30 5t4

0 6t50 7t6

00 0 2 6t0 12t2

0 20t30 30t4

0 42t50

0 0 0 6 24t0 60t20 120t3

0 210t40

1 t f t2f t3

f t4f t5

f t6f t7

f0 1 2t f 3t2

f 4t3f 5t4

f 6t5f 7t6

f0 0 2 6t f 12t2

f 20t3f 30t4

f 42t5f

0 0 0 6 24t f 60t2f 120t3

f 210t4f

b =

[q(t0) 0 0 0 q(t f ) 0 0 0

]T

The described procedure is followed four times, one for each linear segment of therectangular trajectory. The results are the toll tip’s position vector coordinates inthe Cartesian space r, as described by Equation B.1, for different time instants.

B.3 Slow trajectory

Two different reference trajectories are designed. The first one is completed inT = 5 [s] and the second one is completed in T = 3.5 [s]. For the slow trajectory,each vertical part is completed in 1.5 [s] and each horizontal one in 1 [s]. Forthe faster trajectory each vertical part is completed in 1 [s] and each horizontalis completed in 0.75 [s]. The trajectory profile in the Cartesian space is shown inFigures B.2 to B.5.


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time [s]

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Po

siti

on

[m

]

Tool tip position in Cartesian space

x axisy axis

Figure B.2: Tool tip position trajectory.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time [s]

-1.5

-1

-0.5

0

0.5

1

1.5

Vel

oci

ty [

m/s

]

Tool tip position velocity

x axisy axis

Figure B.3: Tool tip velocity trajectory.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time [s]

-3

-2

-1

0

1

2

3

Acc

eler

atio

n [

m/s

2]

Tool tip acceleration

x axisy axis

Figure B.4: Tool tip acceleration trajectory.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time [s]

-20

-15

-10

-5

0

5

10

15

20

Jerk

[m

/s3]

Jerk profile of tool tip

x axisy axis

Figure B.5: Tool tip jerk trajectory.

B.3.1 Trajectory in joint space

Since it the each cylinder’s piston position that is controlled, the reference tra-jectory must be expressed as piston position reference. This is accomplished viathe inverse kinematics, that are acquired via the geometrical characteristics of themanipulator. Asa first step, the tool tip’s reference trajectory is translated in jointspace reference. In the next step, the trajectory is expressed from the joint anglesinto the piston position reference.The geometric characteristics of the manipulator can be derived via the sketch ofFigure B.6:

B.3. Slow trajectory 119

φh

β2φ

1

β1

Ο

c

Bφ2

β3

h (x,y)

Figure B.6: Geometry of the manipulator.

From Figure B.6:

c =√

x2 + y2 (B.10)

φ2 = π + β3 − φh (B.11)

φ1 = β + β2 (B.12)

β2 = atan2(y, x) = tan−1(yx) (B.13)

β1 = cos−1(c2 + L2

OB − L2Oh

2cLOB) (B.14)

β3 = cos−1(L2

OB + L2Bh − c2

2LOBLBh) (B.15)

Due to the limits of the cylinders, there can only be an ’elbow-up’ solution.[43]Equations B.11 and B.12 give the relationship between the tool tip coordinates andjoint angles. In Equation B.13 the command atan2(y, x) calculates the solutionfor the four quadrants, taking into consideration the signs of its arguments. Theoutput of atan(y, x) with solution between −π

2 and π2 [43].

The Jacobian matrix which connects tool tip velocities and joint velocities, can bederived by finding the from the forward kinematics the tool tip position and takingits derivative. From Equation A.4, which describes the homogeneous transforma-tion matrix of the tool tip, its position with regard to joint angles can be foundas:

r =[

xy

]=

[LOBcos(φ1) + LBhcos(φ1 + φ2 + φh)

LOBsin(φ1) + LBhsin(φ1 + φ2 + φh)

](B.16)


The velocities can be related by the time derivative:

r =[

xy

]= J

[φ1

φ2

](B.17)

J =[− [LOBsin(φ1) + LBhsin(φ1 + φ2 + φh)] −LBhsin(φ1 + φ2 + φh)

[LOBcos(φ1) + LBhcos(φ1 + φ2 + φh)] LBhcos(φ1 + φ2 + φh)

](B.18)

The Jacobian B.18 is invertible when its determinant is different than zero, whichis the case for the manipulator. The determinant of J is always different than 0 forall values of permissible joint angles:

det(J) = LBhsin(φ2 + φh) (B.19)

As a result, the velocity and acceleration of the joints can be found by:[φ1

φ2

]= J−1

[xy

](B.20)[

φ1

φ2

]= −J−1J

[φ1

φ2

]+ J−1

[xy

](B.21)[...

φ1...φ2

]= J−1

[...x...y

]− J−1 J

[φ1

φ2

]− 2J−1 J

[φ1

φ2

](B.22)

The profiles of the trajectory in joint space can be seen in the following figures:

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time [s]

0.9

1

1.1

1.2

1.3

1.4

1.5

Ang

le [r

ad]

Figure B.7: Reference angle forjoint 1.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time [s]

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Ang

ular

vel

ocity

[rad

s-1

]

Figure B.8: Reference velocityfor joint 1.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time [s]

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Ang

ular

acc

eler

atio

n [r

ads-2

]

Figure B.9: Reference accelera-tion for joint 1.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 54

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

5


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time [s]

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Ang

ular

vel

ocity

[rad

s-1

]


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time [s]

-3

-2

-1

0

1

2

3

Ang

ular

acc

eler

atio

n [r

ads-2

]


B.4. Fast trajectory 121

Trajectory in actuator space

Having the trajectory profiles expressed in joint space, the reference positions andvelocities for the two actuators can be derived using Equations 2.2, 2.3, 2.4, 2.5 and2.17. The following figures present the trajectory profiles in actuator space.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time [s]

-60

-40

-20

0

20

40

60

80

Pis

ton

posi

tion

[mm

]

Servo System 1

Figure B.13: Reference positionfor cylinder 1.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time [s]

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Pis

ton

velo

city

[m s

-1]

Figure B.14: Reference velocityfor cylinder 1.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time [s]

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Pis

ton

acce

lera

tion

[m s

-2]

Figure B.15: Reference accelera-tion for cylinder 1.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time [s]

-60

-40

-20

0

20

40

60

80

100

120

Pis

ton

posi

tion

[mm

]

Servo System 2

Figure B.16: Reference positionfor cylinder 2.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time [s]

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Pis

ton

velo

city

[m s

-1]

Figure B.17: Reference velocityfor cylinder 2.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time [s]

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

Pis

ton

acce

lera

tion

[m s

-2]

Figure B.18: Reference accelera-tion for cylinder 2.

B.4 Fast trajectory

For the case of the faster, 3.5 [s] trajectory, the same profiles are shown in Figures


0 0.5 1 1.5 2 2.5 3 3.5

Time [s]

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Po

siti

on

[m

]

Tool tip position - Faster

x axisy axis

Figure B.19: Tool tip position trajectory.

0 0.5 1 1.5 2 2.5 3 3.5

Time [s]

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Vel

oci

ty [

m/s

]

Tool tip velocity - Faster

x axisy axis

Figure B.20: Tool tip velocity trajectory.

0 0.5 1 1.5 2 2.5 3 3.5

Time [s]

-8

-6

-4

-2

0

2

4

6

8

Acc

eler

atio

n [

m/s

2]

Tool tip acceleration - Faster

x axisy axis

Figure B.21: Tool tip acceleration trajectory.

0 0.5 1 1.5 2 2.5 3 3.5

Time [s]

-50

-40

-30

-20

-10

0

10

20

30

40

50

Jerk

[m

/s3]

Tool tip jerk - Faster

x axisy axis

Figure B.22: Tool tip jerk trajectory.

B.4.1 Fast trajectory in joint space

0 0.5 1 1.5 2 2.5 3 3.5

Time [s]

0.9

1

1.1

1.2

1.3

1.4

1.5

An

gle

[ra

d]

Joint 1 angle reference - Faster


0 0.5 1 1.5 2 2.5 3 3.5

Time [s]

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

An

gu

lar

velo

city

[ra

d/s

]

Joint 1 angular velocity reference - Faster


0 0.5 1 1.5 2 2.5 3 3.5

Time [s]

-4

-3

-2

-1

0

1

2

3

4

An

gu

lar

acce

lera

tio

n [

rad

/s2]

Joint 1 angular acceleration reference - Faster


B.4. Fast trajectory 123

0 0.5 1 1.5 2 2.5 3 3.5

Time [s]

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

An

gle

[ra

d]

Joint 2 reference angle - Faster


0 0.5 1 1.5 2 2.5 3 3.5

Time [s]

-1.5

-1

-0.5

0

0.5

1

1.5

An

gu

lar

velo

city

[m

/s]

Joint 2 angular velocity reference - Faster


0 0.5 1 1.5 2 2.5 3 3.5

Time [s]

-5

-4

-3

-2

-1

0

1

2

3

4

5

An

gu

lar

acce

lera

tio

n [

rad

/s2]

Joint 2 angular acceleration reference - Faster


B.4.2 Fast trajectory in actuator space

0 0.5 1 1.5 2 2.5 3 3.5

Time [s]

-60

-40

-20

0

20

40

60

80

Po

siti

on

[m

m]

Cylinder 1 piston position reference - Faster

Figure B.29: Reference positionfor cylinder 1 piston.

0 0.5 1 1.5 2 2.5 3 3.5

Time [s]

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Vel

oci

ty [

m/s

]

Cylinder 1 piston velocity reference - Faster

Figure B.30: Reference velocityfor cylinder 1 piston.

0 0.5 1 1.5 2 2.5 3 3.5

Time [s]

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Acc

eler

atio

n [

m/s

2]

Cylinder 1 piston acceleration reference - Faster

Figure B.31: Reference accelera-tion for cylinder 1 piston.

0 0.5 1 1.5 2 2.5 3 3.5

Time [s]

-60

-40

-20

0

20

40

60

80

100

120

Po

siti

on

[m

m]

Cylinder 2 piston position reference - Faster

Figure B.32: Reference positionfor cylinder 2 piston.

0 0.5 1 1.5 2 2.5 3 3.5

Time [s]

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Vel

oci

ty [

m/s

]

Cylinder 2 piston velocity reference - Faster

Figure B.33: Reference velocityfor cylinder 2 piston.

0 0.5 1 1.5 2 2.5 3 3.5

Time [s]

-1

-0.5

0

0.5

1

1.5

Acc

eler

atio

n [

m/s

2]

Cylinder 2 piston acceleration reference - Faster



Jerk reference in the actuator space

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Time [s]

-4

-3

-2

-1

0

1

2

3

4

Jerk

[m

/s3]

Jerk reference for the pistons

Cylinder 1Cylinder 2

Figure B.35: Piston jerk reference.

0 0.5 1 1.5 2 2.5 3 3.5

Time [s]

-10

-8

-6

-4

-2

0

2

4

6

8

10

Jerk

[m

/s3]

Jerk reference for pistons - Faster

Cylinder 1Cylinder 2

Figure B.36: Piston jerk reference, faster trajectory.

B.5 Spectrum analysis of piston reference signals

The frequency spectrum of the position reference signals is of high importance,especially for the linear controllers design. When the frequency content of thereference input signal is known, the closed loop bandwidth choice can be selectedaccordingly. It is thus desired for the closed loop transfer function of each servosystem to have magnitude equal to 1 for the frequencies where the majority of thefrequency content is located.Since the reference trajectory which the design is based upon is the slower one, ofthe 5 [s] duration, this is the signal whose frequency content is to be analyzed.

B.5.1 Periodic trajectories

The reference trajectory is selected to be repeated. This is to investigate the behav-ior of the adaptive terms, as well as to penalize steady state errors. However, thisrepetition helps in the analysis of the frequency content, since it renders the refer-ence signal periodic. However, since the number of samples affects the efficiencyof the Discrete Time Fourier transform in that leakage around the frequencies ofinterest is avoided, a rectangular window of magnitude 1 is assumed [21]. Inreality, however, the main reason for the analysis is to determine the highest fre-quency with significant content and not the magnitude of the signal at the specificfrequency.By repeating each reference signal 5 times and using a rectangular window oflength 216 = 65536 samples to add zeros increasing the sample size, the magnitudeplot of the Fast Fourier Transform (FFT) is plotted in Figure B.37. The windowfunction is equal to 1 for as many samples as the signal sample number and therest of it is zeros, increasing the sample size.

B.6. Realizability of trajectories and PL −QL diagrams 125

Since the sampling frequency used, fs = 2000 [Hz] is sufficiently high, the fre-quency content of the window function is approximately nonzero only at frequency0 [rad/s]. So, in order to get the actual magnitude of the signal, the magnitudeof the signal’s FFT is divided by the value of the windows function’s FFT at 0[rad/s] [21]. Furthermore, only the positive frequencies have been plotted, sincethe negative ones are mirrored after the frequency fs

2 .

0 2 4 6 8 10 12

Frequency [rad/s]

0

10

20

30

40

50

60

70

Mag

nit

ud

e [m

m]

FFT of position reference signal


Figure B.37: FFT of position reference signal.

0 2 4 6 8 10 12 14 16

Frequency [rad/s]

0

10

20

30

40

50

60

70

Mag

nit

ud

e [m

m]

FFT of position reference signal - Faster


Figure B.38: FFT of position reference signal of thefaster trajectory.

It can be seen that the frequency content of the reference trajectories lies below 12[rad/s]. For comparison, the faster trajectory has significant frequency content upto around 16 [rad/s]. As a result, the selected closed loop bandwidths of the linearcontrollers cover these frequencies and good tracking should be achieved for bothcases.The plotted FFT magnitudes are shown as continuous lines only to give a clearindication of the maximum frequency where the magnitude is important. Further-more, a window with sufficiently more zeros could have been used to decrease theleakage effect. However, the important frequencies can still be discerned.

B.6 Realizability of trajectories and PL −QL diagrams

The actuators are driven by servo valves. The first condition to be fulfilled is thatthe power demanded by the load, the manipulator in the different configurations,is never higher than the power that can be provided by the servo valve, assumingthat the supply pressure is constant. However, this is not the only restriction thatapplies. It is shown that the load pressure for each servo system should alsonot exceed a specific pressure value, so that the design of the control system stillapplies and the load can complete the trajectory, without stalling. As a result,along the trajectory, two things are of importance:


• That the load pressure does not exceed the pressure at the maximum powertransfer point, and

• The power demanded by the trajectory does not exceed the maximum powerthat can be provided by the servo valve at a given, constant supply pressure.This can be verified by the fact that for each load pressure value, below themaximum power transfer pressure, the flow demand does not exceed themaximum flow provided.

These considerations can be visualized with the help of PL −QL diagrams for bothtrajectories. This will also help determine for each trajectory which is the maximumallowed load the tool tip can carry, so that the desired trajectory is completed andthe load never stalls.

B.6.1 Maximum power transfer

The power transferred to the load, in steady state, can be written as [17]:

P = PLQL = PLKvxv

√Ps − Pt − sign(xv)PL

2= Kvxv

√P2

L(Ps − Pt)− sign(xv)P3L

2[W]

(B.23)

The load pressure at the point of maximum power transfer can be found via:

∂P∂PL

= 0⇒ PL =23(Ps − Pt)sign(xv) ≈

23

Pssign(xv) [Pa] (B.24)

According to [32], since the load pressure varies along the trajectory while thesupply pressure does not, maximum power transfer is not feasible. However, it isdesirable that the load pressure is below 2

3 Ps. This is due to the fact that if PL isincreased and approaches Ps, then the flow to the load will become almost zeroand the motion will stop. This can also be seen via the value the flow gain Kq

takes when PL ≈ Ps as can be seen from Equation C.2. When Kq ≈ 0, control isessentially lost. But also in the case where PL is over 2

3 Ps the value of the static gainof the plant is decreased by a wide margin that should be taken into considerationin the control design procedure. All things considered, the instantaneous value ofthe load pressure should be lower than the maximum power transfer pressure atthe majority of the trajectory. Furthermore, the demanded load flow should be lessthan the supplied value. In the following section, how these values are found isshown and the PL −QL diagrams plotted for both trajectories.

B.6.2 PL −QL diagrams for the trajectories

The first step in plotting the power diagrams, is calculating the load flow and loadpressure demand along the trajectories.

B.6. Realizability of trajectories and PL −QL diagrams 127

The maximum supply flow is defined for varying load pressure values from 0 toPs [bar] and when the valve is fully opened, uv = 10 [V]. Then, the value of theload pressure at the point of maximum power transfer is also plotted. The supplyand pressures have been found to be almost constant, with values:

Ps = 100 [bar] Pt = 0.8 [bar] (B.25)

Along each trajectory, the demanded load flow is approximately calculated insteady state by the flow continuity equation 2.11:

QL = Axp (B.26)

considering the leakage coefficient almost zero. The load pressure PL is calculatedfrom the mechanical model from Equation 2.18 for each servo system:

PL,i =Mii xp,i + (Cii + Bv,i) xp,i + Gi + Mij xp,j + Cij xp,j + Fc,i

Ai = 1, 2 j = 2, 1

(B.27)

The resulting diagrams, for the slow and fast trajectory, are shown in Figures B.39and B.40 respectively.

-100 -80 -60 -40 -20 0 20 40 60 80 100

Load pressure [bar]

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Lo

ad f

low

[l/s

]

PL

- QL

diagram


Figure B.39: PL −QL diagram for the slowertrajectory.

-100 -80 -60 -40 -20 0 20 40 60 80 100

Load pressure [bar]

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Lo

ad f

low

[l/s

]

PL

- QL

diagram - Faster


Figure B.40: PL −QL diagram for the fastertrajectory.

The brown vertical lines denote the value of load pressure at the point of maxi-mum power transfer. It is deemed that for zero tool tip load, both trajectories arerealizable.

B.6.3 PL −QL diagrams with load at the tool tip

In the case the tool tip is loaded, both trajectories might not be realizable under thesame load and the same supply pressure. In the diagrams B.41 and B.42, the same


PL − QL plots are shown for loads that bring each trajectory to the load pressurelimit:

-100 -80 -60 -40 -20 0 20 40 60 80 100

Load pressure [bar]

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Lo

ad f

low

[l/s

]

PL-Q

Ldiagram - m

h=35 [kg]


Figure B.41: PL −QL diagram for the slowertrajectory with tool tip load.

-100 -80 -60 -40 -20 0 20 40 60 80 100

Load pressure [bar]

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Lo

ad f

low

[l/s

]

PL-Q

Ldiagram - Faster - m

h=15 [kg]


Figure B.42: PL −QL diagram for the fastertrajectory with tool tip load..

It can be seen that if both trajectories should be tested under load, in order to avoidactuator saturation, the load of mh = 15 [kg] should be used.

Appendix C

Linearization

C.1 Linearization of the valve flow equation

Linearization of Equation 2.20 shows how variations in the load pressure, PL, andvalve input voltage, uv, affect the flow through the valve [34]. The valve dynamicshave been neglected.First order Taylor series approximation of the flow equation yields:

∆QL = Kq∆uv + Kqp∆PL (C.1)

Kq = Kv√

Ps − Pt − PL,0 , Kqp = − Kvuv,0

2√

2√

Ps − Pt − PL(C.2)

The ∆ notation stands for "variation of the variable in a neighborhood around thepoint of linearization" and is dropped in the following.The point of linearization for the valve input voltage is selected to be equal to asmall value, for example uv = 0.01 [V], so that the valve is just open. This realisticscenario corresponds to the least possible damping of the system.The resulting pressure dynamics, using the linearized equation of the valve flowcan be written as:

PL =βe f f Vtot

VA,0VB,0

[−(CL − Kqp)PL − Ax + Kqpuv

](C.3)

where VA,0, VB,0 correspond to the initial volumes of the cylinder at the point oflinearization. The variations of the volumes is considered very small around thepoint of linearization, compared to the initial volumes.The load pressure at the point of linearization can be found from Equation 2.18 insteady state, at the point of linearization, for each servo system:

PL,0 =Gi,0

A(C.4)

129

130 Appendix C. Linearization

C.2 Linearization of the mechanical system

The mechanical system can be linearized about the selected point of operationand be represented as two Single Input Single Output (SISO) systems, with thecouplings between them represented as disturbance inputs. These couplings arisedue to the Inertia and Coriolis matrices. A further simplification is that the elementof the Coriolis matrix corresponding to each servo system, ie. Cii i = 1, 2, are smallin magnitude, compared to the gravitational forces and the inertia matrix elements,as well as the viscous friction coefficient term. They will thus be neglected. Finally,the elements of the inertia matrix and the gravitational terms were not linearized,even though they are consisted of the positions of the cylinders’ pistons. This isjustified because when the linearization point is selected, they are substituted bytheir value at that specific point.For each hydraulic servo system, the mechanical part around the point of lineariza-tion can be written as:

Mii,0 xi + (Bv,i + Cii,0) xi +

di(t)︷︸︸︷(Gi,0 + Mij,0 xj + Cij,0 xj) = APL,i

simplifications=======⇒

Mii,0 xi + Bv,i xi + (Gi,0 + Mij,0 xj) = APL,i (C.5)

i = 1, 2 and j = 2, 1. The bracketed terms constitute the disturbance inputs toeach servo system. In the case the trajectory is faster, these simplified terms willbe taken into consideration, since the disturbance term is increased. Furthermore,since the Coriolis terms can take also negative values, in the case of small viscousfriction damping coefficient the term should be considered.

C.3 Block diagram representation of the linearized system

Equations C.3 and C.5 describe two coupled servo systems. They can be repre-sented by block diagrams since they are linear. The couplings will be representedas disturbance inputs, as mentioned earlier. In Figure C.1, the block diagram ofone of the two hydraulic servo systems is illustrated.

G (s)v K qX (s)v,i β

effVtot

VA,0VB,0

1s

P (s)L,i AM s+Bv

U (s)v,i

C - Kl qp

A

sX (s)p,i 1s

X (s)p,i

ii

(M s X +G )/Aji2

ip,j

Figure C.1: Block diagram of one of the two the hydraulic servo systems.

C.4. Selection of the linearization point 131

Using the block diagram of Figure C.1, the following transfer function can be de-rived for each servo system i = 1, 2:

Xp(s)Uv(s)

= Gp(s) =G(

sωl

)2+ 2 ζl

ωls + 1

1s

(C.6)

G =AKq

Bv(Cl − Kqp) + A2 (C.7)

ωl =

√βe f f Vtot

[A2 + Bv(Cl − Kqp)

]MVA,0VB,0

(C.8)

ζl =ωl

2

√Mβe f f Vtot(Cl − Kqp) + VA,0VB,0Bv

βe f f Vtot[Bv(Cl − Kqp) + A2

] (C.9)

One way to estimate the critical points, which can be used as points of linearization,is to find the conditions that are more adverse for control design. These points canbe the position of each cylinder where the eigenfrequency is lower, the dampingratio is lowest, or the magnitude of the transfer function is highest.The transfer function of the piston position to the disturbance input can be written:

Xp(s)D(s)

= Gd(s) = −Gdist(

sωl

)2+ 2 ζl

ωls + 1

1s

(C.10)

Gdist =VA,0VB,0s− βe f f (Kqp − Cl)

βe f f Vtot[A2 + Bv(Cl − Kqp)

] (C.11)

The block diagram of the system including the disturbance term is shown in theblock diagram of Figure C.2.

G (s)p

G (s)d

U (s)v

D(s)

X (s)p

Figure C.2: Block diagram of the linearised servo system.

C.4 Selection of the linearization point

The linearization point can be selected as a trade-off that corresponds to the worstpossible case in terms of damping ratio and lowest eigenfrequency for each servosystem. The damping ratio and eigenfrequency for each servo system, for varying

132 Appendix C. Linearization

-200 -150 -100 -50 0 50 100 150 200

Piston 1 position [mm]

24

26

28

30

32

34

36

38

40

42

Eig

enfr

equ

ency

[ra

d/s

]

Servo System 1 - Eigenfrequency

-175-100-5050100175

Piston 2Position[mm]

Figure C.3: Eigenfrequencies of servo system 1.

-200 -150 -100 -50 0 50 100 150 200


60

65

70

75

80

85

90

95

Eig

enfr

equ

ency

[ra

d/s

]

Servo System 2 - Eigenfrequency

Figure C.4: Eigenfrequencies of servo system 2.

-200 -150 -100 -50 0 50 100 150 200


0.014

0.016

0.018

0.02

0.022

0.024

0.026

0.028

0.03

0.032

0.034

Damping ratio [-]

Servo System 1 - Damping ratio

-175-100-5050100175

Piston 2Position [mm]

Figure C.5: Damping ratio of servo system 1.

-200 -150 -100 -50 0 50 100 150 200


0.035

0.04

0.045

0.05

0.055

0.06

0.065

0.07

0.075

Dam

pin

g r

atio

[-]

Servo System 2 - Damping ratio

Figure C.6: Damping ratio of servo system 2.

values of each piston’s position, are shown in Figures C.3 , C.4 and Figures ?? , C.6respectively for varying positions of both pistons within their range of motion.Observing the plots for the eigenfrequencies and the damping ratio for each servosystem, a fair worst case condition trade-off could be to select the point of lin-earization to be at the piston positions:

xp,1 = xp,2 = 100 [mm] (C.12)

Appendix D

Velocity estimation

D.1 Differentiator design

In this section, a way to estimate the velocity of the piston is described. A firstorder differentiator based on the super twisting algorithm is used [39],[51]. Thesame differentiator has been used for the estimation of the time derivative thesecond virtual control law of the backstepping procedure in section 4.3.1.The differentiator acts as an observer. The dynamic system that describes the ob-server is the dynamics of the estimation error. This estimation error needs to bedriven to zero and is written as:

σv = xp − xp (D.1)

where xp is the estimation of the piston position xp. It is needed however that theestimation error of the velocity become zero. This estimation error is written as:

σv = ˙xp − xp (D.2)

These two equations form a first order dynamic system. If σv = 0, then the esti-mation of the velocity is equal to the actual velocity. To this end, the only variablethat can be manipulated is the estimate of the velocity. Using the super twistingalgorithm is justified because it creates a sliding mode for the states of second or-der systems, but of relative degree one. This means that both states of the secondorder system converge to zero in finite time. The system is of relative degree one,because the input to the observer system, ˙xp, appears to the equation of the firstderivative of the output, σv.The input is selected according to [51]:

˙xp = −C1

√|σv|sign(σv) + qv (D.3)

qv = −C0sign(σv) (D.4)

133

134 Appendix D. Velocity estimation

Using this input, the closed loop system of the observer/differentiator becomes:

σv = −C1

√|σv|sign(σv) + qv − xp (D.5)

qv = −C0sign(σv) (D.6)

which can be rewritten as:

σv = −C1

√|σv|sign(σv) + wv (D.7)

wv = −C0sign(σv)− xp ≤ −C0sign(σv) + |xp|max (D.8)

The second order sliding mode is imposed on the system D.7 and D.8, meaningthat [σv wv]

T will reach [0 0]T in finite time. Since σv = 0⇒ σv = 0 and the velocityestimation error will be zero. According to [51], the parameters of the differentiatorcan be selected as:

C0 = 1.1C C1 = 1.5√

C C ≥ |xp|max (D.9)

Convergence of σv and σ can be shown by writing the closed loop system insteadof the form shown in D.7 and D.8, as:

σv = −C1

2|σv|−

12 σv − C0sign(σv)− xp

≤ −C1

2|σv|−

12 σv − C0sign(σv) + C (D.10)

Phase plane analysis can be used [40], [39], and a sketch is shown in Figure D.1.

l1

σ

σ0

l2

-l3

v

v

σ =0v

σ >0v

σ <0v

Figure D.1: Phase plane plot for the phase variables of D.10.

If it can be shown that l1 > l3, or that l3l1< 1, then it is sufficient for the algorithm

convergence [51].

D.1. Differentiator design 135

For the first quartile, where both σv and σv are positive, the curve σv can be written:

σv = −C1

2|σv|−

12 σv − (C0 − C) (D.11)

Writing σv = σvdσvdσv

:

σvdσv

dσv= −C1

2|σv|−

12 σv − C0 + C ≤ −(C0 − C) (D.12)

since σv, σv > 0 since the trajectory still lies in the first quartile. From the inequalityof D.12, a relationship between l1 and l2 can be found by:∫ 0

l1σvdσv ≤ −(C0 − C)

∫ l2

0dσv ⇒

l2 ≤l21

2(C0 − C)(D.13)

A conservative selection for l1 leads to

l2 =l21

2(C0 − C)(D.14)

The next step is to find the conservative intersection point (0,−l3). For simplicity[39], when σv = 0, the trajectory D.11 becomes:

−C1

2|σv|−

12 σv = C0 − C (D.15)

For the point (0,−l3), using also D.14:

−C1

2

√2(C0 − C)

l1(−l3) = C0 − C ⇒

l3l1

=

√2(C0 − C)

C1< 1 (D.16)

where the last inequality comes form the selection of C0 and C1 in Equation D.9. Itis thus proven that l1 > l3, which is sufficient for the convergence of σv and σv tothe origin.Another way to show boundedness of σv, σv is through the Lyapunov functioncandidate:

V = 2

√12

σ2v + C0|σv| (D.17)

V D.10= σv

−C12

σv√|σv|− xp√

12 σ2

v + C0|σv|=−C1

2σ2

v|σv| − xp

σv√|σv|√

σ2v

2|σv| + C0

(D.18)

136 Appendix D. Velocity estimation

If σ2v|σv| ≥ ε2 , ε > 0, [46]

V ≤−C1

2 ε2 + Cε√ε2

2 + C0

≤ −3√

C4

εε− 4

√C

3√ε2

2 + C0

(D.19)

which is negative definite for ε > 4√

C3 . Finite time convergence is shown in [46].

D.2 Application

The estimation error of the differentiator is only equal to zero on the sliding man-ifold. This is reached in finite time, but not from the beginning of the trajectorynonetheless. As a result, in order to avoid any adverse effects due to large velocityestimation errors, the following procedure is followed.

1 A PI controller initializes the position of each actuator to the first elementof the reference trajectory position vector. This initial position is kept forabout 1 [s]. It has been verfieid by simulations, with a specific value of Cthat in this time period the estimation error becomes zero. All this time, thedifferentiator algorithm is run in parallel.

2 After the estimation error has converged to zero, the tracking of the referencetrajectory can start, using a desired controller and the velocity is provided bythe estimate of the differentiator.

The selection of C ≥ |xp|max is important, especially when a faster trajectory is tobe tracked. The estimation error should be zero before starting any trajectory. Ifhowever this gain is selected large, when the trajectory starts, a large variation inthe input voltage is observed. On the other hand, the tracking error is reduced.Since this is unwanted, it is desired that the value of C is kept as small as possibleto achieve a sliding mode. It can be possible that the value of C can change alongthe trajectory, depending on the acceleration. If a technique like this is used, it wasobserved that when the tracking began, due to the rapid increase in the accelera-tion, the input voltage had an abrupt large value. As a result, the gain C is selectedconstant and equal to 1.1 times the value of |xre f

p |max of the faster trajectory.

Appendix E

Discretization

In this chapter, the implementation of the proposed controllers is considered. Sincea real time computer and an FPGA are to be used, the controllers must be expressedin discrete time. However, the sampled system is a fundamentally different sys-tem than the continuous time one. As a result, the effects of sampling will beadditionally considered.In the following, it is assumed that there is no time delay due to computations.

E.1 Discretization of the linear reference controllers

E.1.1 Sampling

A block diagram of a closed loop sampled time system is shown in Figure E.1: In

G (s)pzohG (z)c

U(z) U(s) X (s)pX (z)pref

Figure E.1: Block diagram os sampled time control system.

Figure E.1, the continuous time system is denoted by its transfer function in theLaplace domain. All discrete time systems are denoted by their z transform andthey are discrete, meaning that they have value only on the sampling instances.Finally, the output of the zero order hold, which is a part of the sample and holdsystem, is continuous in time and is denoted by U(s). The discrete time transferfunction Gc(z) denotes the controller transfer function and shares only the samename with its continuous counterpart. The discrete time signals and transfer func-tions can be denoted in continuous time by their star transform [36].The zero order hold in continuous time can be denoted by the transfer function in

137

138 Appendix E. Discretization

the Laplace and the frequency domains [36]:

Gzoh(s) =1− e−Tss

sor Gzoh(jω) = Ts

sin(πωωs

)πωωs

e−j πωωs (E.1)

with Ts [s] the sampling time and ωs [rad/s] the sampling frequency.The fact that the sampled system is different than the continuous time systemcan be seen by the fact that the plant to be controlled is described by the transferfunction:

G(s) = Gzoh(s)Gp(s) (E.2)

The effect of the sampling, represented by the zero order hold transfer function, inthe frequency domain can be seen by:

|Gzoh(jω)| = Ts|sin(πω

ωs)

πωωs

| (E.3)

∠Gzoh(jω) = −πω

ωs, for sin(

πω

ωs) > 0 (E.4)

The most important effect of the sampling is the addition of negative phase as thefrequency of the system increases and reaches half the sampling frequency. Thesampling frequency is predetermined by the data acquisition software, the effecton the critical frequencies of the system can be evaluated.

E.1.2 The bilinear transform

Since controllers in the frequency domain have already been designed for the plantGp(s), from Equation 5.1, it is desirable that they are used to control also the sam-pled system. As a first step, a way to transform the continuous time designedcontrollers into the discrete time domain needs to be found. Afterwards, the con-trollers in the z domain are expressed as difference equations.A transformation from the Laplace to the z domain that preserves stability anddoes not cause large distortion for high sampling frequencies is the bilinear, orTustin, transform. The mapping from the Laplace to the z domain is described by:

sz =2Ts

1− z−1

1 + z−1 (E.5)

ωz =2Ts

tan(Ts

2ω) =

ωs

πtan(

π

ωsω) (E.6)

where sz denotes the complex frequency domain equivalent of the discretized con-troller, ωz is the frequency of the mapped domain on the frequency of the Laplacedomain ω [36]. The primary range of this domain spans for frequencies up to

E.1. Discretization of the linear reference controllers 139

ωs2 [rad/s]. The transform of Equation E.5 will be applied to the designed linear

controllers to yield their discrete equivalent. A necessary condition for the twotransfer functions, the transformed, continuous one in the Laplace domain, andits discrete equivalent in the z domain, to be equivalent is that their frequency re-sponse is equivalent for a wide band of frequencies. The condition under whichthe characteristics of the frequency response of the original system are preservedfor a specific range of the frequency response of the discrete equivalent can befound by Equation E.5 and can be written as [36]:

ω <ωs

10(E.7)

which means that the for frequencies of the Laplace domain ω satisfying the con-dition, the mapping of the frequency domain of the discrete equivalent is almostexact on the Laplace domain, and consequently sz ≈ s.The sampling frequency is set equal to 2000 [Hz] or 12566 [rad/s]. For this sam-pling frequency, for frequencies up to a maximum bandwidth of ωmax = 700[rad/s], the difference in the mapping of the z-equivalent frequency domain andthe Laplace domain can be calculated by using Equation E.6.

ωmax = 0.0557ωs or ωs = 17.9514ωmax (E.8)ωz

ωmax=

17.9514π

tan(π

17.9514) = 1.0103 (E.9)

meaning that the distortion of the mapping is a little higher than 1%. As a result,for the sampling frequency of 2000 [Hz] and frequencies up to 700 [rad/s], it canbe assumed that s = sz and the bilinear transform can be used to acquire a discreteequivalent from the continuous controllers.

E.1.3 PI reference controller

For the PI compensator designed in section 5.2, the gain crossover frequenciesof the open loop compensated plant were ωgc,1 = 15.2 [rad/s] and ωgc,2 = 39.3[rad/s]. From the expression of the phase of the zero order hold in Equation E.4,the change in the phase at these frequencies due to sampling are calculated as:

φzoh,1 = −0.2177◦ φzoh,2 = −0.5629◦ (E.10)

As a result, the relative stability margins are not expected to vary considerably.The output of the PI compensator with an anti-windup loop in the Laplace domainis:

UPI(s) = P(s) + I(s) (E.11)

P(s) = kPE(s) (E.12)

I(s) =1s[k I E(s) + kTEi(s)] (E.13)


Applying the bilinear transform of Equation E.5:

UPI(z) = P(z) + I(z) (E.14)

P(z) = kPE(z) (E.15)

I(z) = z−1 I(z) +Ts

2[k I E(z) + kTEi(z)] +

Ts

2

[k Iz−1E(z) + kTz−1Ei(z)

](E.16)

Finally, using the real translation property of the z transform [36], the algorithmthat describes the control output uv(kTs) of each PI controller using anti-winduploop is:

uPI(k) = p(k) + i(k) (E.17)

p(k) = kPe(k) (E.18)

i(k) = i(k− 1) +Ts

2[k Ie(k) + kTei(k)] +

Ts

2[k Ie(k− 1) + kTei(k− 1)] (E.19)

with e(k) = xp,re f (k)− xp(k) and ei(k) = uv(k)− vv(k) where

vv(k) =

uv,max , if uv(k) > uv,max

uv,min , if uv(k) < uv,min

uPI(k) otherwide

(E.20)

E.1.4 P and P-Lead compensator

With the given sampling frequency, no major negative phase addition will hap-pen due to the hold effect and the mapping of the bilinear transform is accurate.Repeating the same procedure:

uld(k) =2T − Ts

2T + Tsuld(k− 1) + Kp,1

2αldT + Ts

2T + Tse(k) + Kp,1

Ts − 2αldT2T + Ts

e(k− 1) (E.21)

uprp(k) = Kp,2e(k) (E.22)

E.1.5 Overview of reference control algorithms

In the aforementioned controller expressions, the input from the velocity feed-forward and pressure feedback loops should be also added. However, since nodynamics are included in both of these loops, their discrete time expression isstraightforward.

E.2. Nonlinear controllers 141

PI controller

Equivalent expression for both servo systems:

uv,j(k) = uPI,j(k) + K f f ,j xp,re f ,j(k)−Kp f ,j

Kq,jPL,j(k) (E.23)

uPI,j(k) = pj(k) + ij(k) (E.24)

pj(k) = kP,jej(k) (E.25)

ij(k) = ij(k− 1) +Ts

2[k I,jej(k) + kT,jei,j(k)

]+

Ts

2[k I,jej(k− 1) + kTei,j(k− 1)

](E.26)

j = 1, 2 for each servo system. The controller parameters are:

kP,1 = 426.35, k I,1 = 226.3004, kT,1 = 5.3079, Kp f ,1 = 1.0147 · 10−11, K f f ,1 = 25.3924(E.27)

kP,2 = 1287.5, k I,2 = 1766.8, kT,2 = 13.7227, Kp f ,2 = 2.6356 · 10−11, K f f ,2 = 27.1304(E.28)

P and P-Lead compensator

Servo system 1:

uv,1(k) = uld(k) + K f f ,1 xp,re f ,1(k)−Kp f ,1

Kq,1PL,1(k) (E.29)

uld(k) =2T − Ts

2T + Tsuld(k− 1) + Kp,1

2αldT + Ts

2T + Tse1(k) + Kp,2

Ts − 2αldT2T + Ts

e1(k− 1)

(E.30)

Servo system 2:

uv,2(k) = uprp(k) + K f f ,2 xp,re f ,2(k)−Kp f ,2

Kq,2PL,2(k) (E.31)

uprp(k) = Kp,2e2(k) (E.32)

with additional tuning parameters:

T = 0.0064, αld = 15, Kp,1 = 484.1724 (E.33)

Kp,2 = 1349 (E.34)

E.2 Nonlinear controllers

Emulation is to be used to discretize the nonlinear controllers, developed for con-tinuous time systems. The effect of sampling is more difficult to quantify in thiscase. Furthermore, since the z transform is defined only for linear systems, thediscrete integration is defined in the following section.


E.2.1 Integration

The nonlinear controllers contain adaptation terms that come from integration.Using the trapezoidal rule, an expression for the signal that is integrated is [35]:

o(k) = o(k− 1) +Ts

2[i(k) + i(k− 1)] (E.35)

The signal o(kTs) is the output of the integrator and the signal i(kTs) is the input.

E.2.2 RABLIN - Chapter 3

When the load pressure is not filtered in the pressure feedback loop:

uv,i(k) = Gc,iuF,i(k) [PL,i(k)− a2,i(k)]−Kp f ,i

Kq,iPL,i(k) (E.36)

uF,i(k) =2τF,p − Ts

2τF,p + TsuF,i(k− 1) +

2τF,z + Ts

2τF,p + Tsei(k)−

2τF,z − Ts

2τF,p + Tsei(k− 1) (E.37)

ei(k) = a2,i(k)− PL,i(k) (E.38)

a2,i(k) =1

ϑ2,i(k)

[r(k) + ϑ1,ix2,i(k) +

(1 + k1,ik2,i +

Di

δik1,i

)ei(k) +

(k1,i + k2,i −

Di

δi

)ei(k)

](E.39)

i = 1, 2 for each servo system, e(kTs) = xi(kTs)− r(kTs) the total actuator lengtherror and e(kTs) = x2,i(kTs)− r(kTs) = xi(kTs)− r(kTs) the velocity tracking error.It is assumed here that the velocity is measured.The parameter estimations are the output of integrating filters:

ϑ1,i(k) = ϑ1,i(k− 1) +Ts

2γ1,i {x2,i(k) [e1(k) + k1,iei(k)] + x2,i(k− 1) [e1(k− 1) + k1,iei(k− 1)]}

(E.40)

ρ2,i(k) = ρ2,i(k− 1)− Ts

2γ2,i {u2,i(k) [e1(k) + k1,iei(k)] + u2,i(k− 1) [e1(k− 1) + k1,iei(k− 1)]}

(E.41)

The tuning parameters are:

k1,1 = 9.3625 k2,1 = 22.7337 Gc,1 = 8.3176 · 10−5 γ1,1 = 100 γ2,1 = 10(E.42)

k1,2 = 13.1039 k2,2 = 31.8185 Gc,2 = 6.9984 · 10−5 γ1,2 = 100 γ2,2 = 10(E.43)

τF,p,1 = 2 τF,z,1 = 0.01 τF,p,2 = 1 τF,z,2 = 0.005 [s] (E.44)

and the pressure feedback gains :

Kp f ,1 = 9.7791 · 10−12 Kp f ,2 = 2.3894 · 10−11 (E.45)

E.2. Nonlinear controllers 143

E.2.3 Proposed Approach 2 - Section 4.4

For each servo system, i = 1, 2:

uv,i(k) =

√2

Ps(k)− Pt(k)− sign(u(k))PL,i(k)ui(k) (E.46)

ui(k) =1

ϑ5,i[−k3,iz3,i(k) + u3r,i(k) + uC,i(k)] (E.47)

uC,i(k) =2τ − Ts

2τ − TsuC,i(k− 1) +

Ts

2τ + Ts[−ρisign [Si(k)]− ρisign [Si(k− 1)]] (E.48)

Si(k) = z3,i(k) + wi(k) (E.49)

wi(k) = w(k− 1) +Ts

2[−ρisign [Si(k)]− u3,i(k)− ρisign [Si(k− 1)]− u3,i(k− 1)]

(E.50)

z3,i(k) = PL,i(k)− a2,i(k) (E.51)

a2,i(k) = ρ2,i

[ri(k)− k1(xi(k)− ri(k))− xi(k) + ri(k) + ϑ1(k)x2(k)− z2,i(k)

Di

δi− k2,iz2,i(k)

](E.52)

a1,i(k) = −k1,iz1,i(k) + ri(k) (E.53)

z1,i(k) = xi(k)− ri(k), z2,i(k) = xi(k)− a1,i(k), z3,i = PL,i − a2,i(k) (E.54)

where xi and xi are the total length and velocity of each actuator respectively.Selection of the tuning parameters k1,i, k2,i, k3,i is investigate in section 4.4.2. Theadaptation parameters are calculated via:

ϑ1,i(k) = ϑ1,i(k− 1)− γ1,iTs

2[z2,i(k)xi(k) + z2,i(k− 1)xi(k− 1)] (E.55)

ρ2,i(k) = ρ2,i(k− 1)− γ2,iTs

2[z2,i(k)u2,i(k) + z2,i(k− 1)u2,i(k− 1)] (E.56)

u2,i(k) = ri(k)− k1(xi(k)− ri(k))− xi(k) + ri(k) + ϑ1(k)− z2,i(k)Di

δi− k2,iz2,i(k)

(E.57)

Appendix F

Theorems

This report considers stability of equilibrium points in the sense of Lyapunov, forsystems of differential equations. The solutions of these systems are the trajec-tories of the state variables in the state space. What is conceptually investigatedis whether he trajectories, having initial conditions, or starting , within a regionaround the equilibrium point, remain in a (different) region close to it as time goesto infinity. In this section, the theorems used in the report to investigate stabilityare rewritten from the cited books.The system of differential equations has the form:

x(t) = f(t, x(t)) (F.1)

The time argument of the state vector is omitted. The system is non-autonomous,due to parameter variations. The state vector for the actual system is

x =[x1 x2 x3

]T=[xp xp PL

]T(F.2)

It is also assumed that x ∈ Rn, even though the position of the piston is limited tothe stroke of the cylinder and that check valves do not allow the chamber pressureto become higher than the supply pressure. The input variable is not in explicitlyincluded in Equation F.1, because it is a function of the states of the system.In this report the backstepping procedure is used for control design. As a result, asystem of similar form to Equation F.1 needs to be stabilized:

z = g(z, t) (F.3)

z = h(x) (F.4)

Equation F.4 shows that the new state vector z is a transform of the state vectorx. Again, this is a system without input, assuming that the input variable is a stetfeedback law. This is because Lyapunov theory deals with stability of equilibriumpoints of system without inputs. However, since the objective is first select the

145

146 Appendix F. Theorems

control input based on stability and not vice versa, the definition of the ControlLyapunov Function will also be stated, since it has been used extensively in thisreport. Then it will be possible to design the control input variable for the system:

z = f′(z, u, t) (F.5)

where the prime is used to denote that this is a different mapping.

F.1 Assumptions

Before writing the theorems, some assumptions are made regarding the systems ofdifferential equations that apply to the whole report.

F.1.1 Lipschitz continuity

Since stabilization of the system described by the system of differential equationF.5 is required, it is therefore needed that the origin is an equilibrium point andthat the states trajectories are driven at this equilibrium. In order for a solution ofthe system to exist and be unique, for f′ the following Theorem can be used.

Theorem 1 (Theorem 4.16 [19] ,p.152 ) If the function f(x, t) is continuous in t, and ifthere exists a strictly positive constant L such that

||f(x2, t)− f(x1, t)|| ≤ L||x2 − x1|| (F.6)

for all x1 and x2 in a finite neighborhood of the origin and all t in the interval [t0,t0 + T](with T being a strictly positive constant), then x = f(x, t) has a unique solution x(t) forsufficiently small initial states and in a sufficiently short time interval.

Condition F.6 is called Lipschitz condition. It is assumed that the Lipschitz con-dition is satisfied for the system in question. Then the above theorem is used toprove uniqueness of the solution. A function f is Lipschitz if the following Lemmaholds.

Theorem 2 (Lemma 3.2 [24] ,p.90 ) If f(x, t) and[

∂f∂x (x, t)

]are continuous on [a,b]×D

for some domain D⊂ Rn, then f is locally Lipschitz in x on [a,b]×D.

Lipschitz continuity is always assumed in the report.

F.2 Lyapunov stability theorems and extensions

For the case of autonomous (sub-)systems, the following theorem is used to provestability.

F.2. Lyapunov stability theorems and extensions 147

Theorem 3 ( Theorem 3.2 Local stability [19] ,p.62) If, in a ball BR0 , there exists ascalar function V(x) with continuous first partial derivatives such that

• V(x) is positive definite (locally in BR0)

• V(x) is negative semi-definite (locally in BR0)

then the equilibrium point 0 is stable. If, actually, the derivative V(x) is locally negativedefinite in BR0 , then the stability is asymptotic.

For the case of non-autonomous (sub-)systems, the following theorem is used toprove stability. Non-autonomous systems arise due to terms that vary with time,like the time varying reference, or vary according to variables exogenous to thesystem, eg. the velocity of the second link, while the system describing the firstservo system is considered only.

Theorem 4 ( Theorem 4.1 Lyapunov theorem for non-autonomous systems [19] ,p.107)Stability: If, in a ball BR0 around the equilibrium point 0, there exists a scalar functionV(x,t) with continuous partial derivatives such that

• V is positive definite

• V is negative semi-definite

then the equilibrium point 0 is stable in the sense of Lyapunov.Uniform stability: If, furthermore,

• V is decrescent

then the origin is uniformly stable.

Sometimes, it is not easy from the time derivative of the Lyapunov function todeduce useful boundedness results, or it is wanted to further investigate asymp-totic stability when it is only negative semi-definite. Especially for the latter, theLyapunov-like theorem based on Barbalat’s Lemma [19] can be used. However,for the boundedness results derived in this report, it is easier to use the LaSalle-Yoshizawa theorem. This is due to the fact that the inequality 3.32 can be formedthat provides more intuitive results.

Theorem 5 (Theorem 2.1 (LaSalle - Yoshizawa) [26] ,p.24) Let x = 0 be an equilib-rium point of x = f(x, t) and suppose f is locally Lipschitz in x uniformly in t. LetV:Rn → R+ be a continuously differentiable , positive definite and radially unboundedfunction V(x) such that

V =∂V∂x

(x)f(x, t) ≤ −W(x) ≤ 0 , ∀t ≥ 0 , ∀x ∈ Rn (F.7)

148 Appendix F. Theorems

where W is a continuous function. Then, all solutions of x = f(x, t) are globally uniformlybounded and satisfy

limt→∞

W(x(t)) = 0 (F.8)

Again, it is assumed that f is Lipschitz. The Lyapunov-like theorem requires uni-form continuity of V(x) so that

limt→∞

V(x) = 0 (F.9)

which does not offer as much information as the use of Theorem 5 in the inequality3.32. On the other hand, the Lipschitz assumption is not needed.As a final note, the control Lyapunov function (clf) definition will be written. Themajority of Lyapunov function candidates used in the report are in fact clf’s. Thisis because first a Lyapunov function candidate is selected and its time derivativeis rendered negative (semi-)definite using the control input variable. Instead, Lya-punov theorems provide stability results for the closed loop system and by them-selves provide little intuition of the selection of this input, or in the selection ofthe Lyapunov function candidate itself. In other words, clf’s are used for controldesign and Lyapunov function candidates are used for analysis of a given system[26].

Theorem 6 ( Definition 2.4 Control Lyapunov Function [26] ,p.26) A smooth posi-tive definite and radially unbounded function V:Rn → R+ is called a control Lyapunovfunction (clf) for

x = f(x, u) , x ∈ Rn , u ∈ R , f(0, 0) = 0

if

infu∈R

{∂V∂x

(x)f(x, u) < 0}

, ∀x 6= 0 (F.10)

A slightly different definition of the clf is the following:

Theorem 7 ( Definition 3.41 Control Lyapunov Function [38] ,p.113) A smooth, pos-itive definite, and radially unbounded function V(x) is called a control Lyapunov function(CLF) for the system

x = f(x) + g(x)u

if for all x 6= 0,

∂V∂x

(x)g(x) = 0⇒ ∂V∂x

(x)f(x) < 0 (F.11)

Appendix G

Parameters

G.1 Parameters used in the simulation model

G.1.1 Mechanical and hydraulic model

Table G.1: Parameters of mechanical model.

Symbol Value Units

LOB 1.206 mLBG 1.022 mLBW 0.232 mLOQ 1.077 mLOH 0.2576 mLOR 0.483 mLBC 0.376 mLBh 1.09 mδ1 0.2808 radδ2 0.3526 radv1 0.6510 radv2 0.0698 radw1 0.3142 radw2 0.2330 radφh 0.1311 radg 9.82 m·s−2

m1 191.8 kgm2 45.8 kgJcm1 40.78 kg ·m2

Jcm2 18.6 kg ·m2

149

150 Appendix G. Parameters

G.1.2 Parameters used in the hydraulic model

Table G.2: Parameters of hydraulic model.

Symbol Value Units

Ps 100 barPt 0.8 barHB 0.175 m

x1,min 0.92 mx2,min 0.846 mCl,1 4.984 ·10−13 m3s−1Pa−1

Cl,2 8 ·10−13 m3s−1Pa−1

Bv,1 6000 N · s ·m−1

Bv,2 6000 N · s ·m−1

FC,1 160 NFC,2 105 NA 9.46 ·10−4 m2

V1,init 1.65 ·10−4 m3

V2,init 1.65 ·10−4 m3

Kv,1 1.62 ·10−8 m3s−1Pa−1V−1

Kv,2 1.5 ·10−8 m3s−1Pa−1V−1

ζv 0.6ωv 1.0933 ·10−3 rad/s

Pmax 180 barβe f f ,max,1 8500 bar

α1,1 1α2,1 30α3,1 12

βe f f ,max,2 6000 barα1,2 1.7375α2,2 80α3,2 14

G.2. Linearization point 151

G.2 Linearization point

Table G.3: Parameters of model at the point of linearization.

Symbol Value Units

Ps 100 barPt 0.8 bar

xp,1 0.1 mxp,2 0.1 mβe f f 7000 baruv,0 0.01 V

VA,0,1 4.2515 ·10−4 m3

VB,0,1 2.3595 ·10−4 m3

VA,0,2 4.2515 ·10−4 m3

VB,0,2 2.3595 ·10−4 m3

M11 3.655 ·103 kgM22 630.8245 kgG1 0.0392 NG2 0.0367 NPL,1 -6.5458 barPL,2 -8.9061 barKq,1 3.725 ·10−5 m3s−1V−1

Kqp,1 -1.7613 ·10−14 m3s−1Pa−1

Kq,2 3.4874 ·10−5 m3s−1V−1

Kqp,2 -1.6130 ·10−14 m3s−1Pa−1

ωn,1 33.6666 rad s−1

ζ1 0.0597Gxp,1 0.0392 m V−1

ωn,2 81.1191 rad s−1

ζ2 0.0818Gxp,2 0.0367 m V−1


G.3 Controller parameters

G.3.1 Reference controllers

Pressure feedback

Table G.4: Parameters of pressure feedback loop.

Symbol Value

Kp f ,1 1.0147 ·10−11

Kp f ,2 2.6356 ·10−11

Velocity feedforward

Table G.5: Parameters of velocity feedforward.

Symbol Value

K f f ,1 25.3924K f f ,2 27.1304

Proportional and Proportional-Lead

Table G.6: Parameters of P and P-Lead controller.

Symbol Value

Kp,1 484.1724Kp,2 1349αld 15T 0.0064

G.3. Controller parameters 153

Proportional Integral

Table G.7: Parameters of PI controllers.

Symbol Value

KP,1 426.35KI,1 226.3004KT,1 5.3089KP,2 1287.5KI,2 1766.8KT,2 13.7227

G.3.2 RABLIN

Table G.8: Parameters of RABLIN controller.

Symbol Value

k1,1 9.3625k2,1 22.7337k1,2 13.1039k2,2 31.8185Gc,1 8.3176 ·10−5

Gc,2 6.9984 ·10−5

D1 1.2245D2 2.3456δ1 0.0250δ2 0.0500

γ1,1 1000γ1,2 1000γ2,1 10γ2,2 10

τF,p,1 2τF,z,1 0.01τF,p,2 1τF,z,2 0.005Kp f ,1 9.7791 ·10−12

Kp f ,2 2.3894 ·10−11


G.3.3 Approach 2

Table G.9: Parameters of the controller designed in Section 4.4.

Symbol Value

k1,1 100 (30)k2,1 25 (30)k3,1 50 (55)k1,2 100 (36)k2,2 94 (36)k3,2 180 (130)D1 1.2245D2 2.3456δ1 0.0250δ2 0.0500ε1 200ε2 200

γ1,1 1000γ1,2 1000γ2,1 10γ2,2 10

Table G.10: Parameters used in determination of gain for sliding mode disturbance observer.

Symbol Value

β5,1 4.4105β5,2 4.4105

ϑ2,max,1 3.6180 ·10−7

ϑ2,max,2 2.0306 ·10−6

ϑ5,1 6.2601 ·104

ϑ5,2 5.7964 ·104

ϑ3,max,1 72.4945ϑ3,max,2 116.3636ϑ4,max,1 1.376 ·1010

ϑ4,max,2 1.3760 ·1010

α1 141.4214α2 141.4214

τlp f ,1 0.5τlp f ,2 0.5

G.3. Controller parameters 155

Table G.11: Sliding mode differentiator parameter, maximum value of |a2|.

Symbol Value

L 5 ·108

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