Designing Linear and Nonlinear Controllers for Wheeled Pendulum

Contents

Preface xv

Acknowledgments xxiii

I Linear Control

1 Some mathematical basis for control of linear systems

1.1 Linear systems vs nonlinear systems .

1.1.1 Superposition principle for static functions

1.1.2 Superposition principle for differential equations .

1.1.3 Stability and transient response of linear systems .

1.1.4 The rationale behind control design based on time

response

1.1.5 Response to sinusoidal excitations

1.1.6 The rationale behind control design based on frequency

response

1.1.7 Frequency response in nonlinear differential equations

1.1.8 Laplace transform and nonlinear systems

1.2 Classical control

1.2.1 Root locus method .

1.2.2 Frequency response method .
1.2.3 The nonminimum phase systems approach

1.3 Sensitivity

1.4 Controllability .

1.5 Linear state feedback control

1.6 LQR control

1.7 Differential flatness and the state space-transfer function relationship

1.8 Linear approximation of nonlinear state equations

Bibliography

2 Wheeled pendulum model

2.1 Robot description

2.1.1 The path following task

2.2 Mathematical model of wheeled pendulum

2.2.1 Unconstrained dynamics

2.2.2 Mathematical model subject to nonholonomic constraints

2.2.3 Computing torques to be applied

2.3 The experimental prototype

2.3.1 Robot mechanical subsystem

2.3.2 Hardware used for controller implementation

2.4 Linear approximate model

2.4.1 Orientation control

2.4.2 The undisturbed wheeled pendulum model
2.4.3 Pole(eigenvalue)-placement design criterion

2.5 Differential flatness-based model

2.6 Conclusions

2.7 Exercises

Bibliography

3 Root locus based controller design

3.1 Design procedure

3.2 Controller gains selection

3.3 Experimental results

3.4 Simulation results

3.5 Effect of Cz on the open-loop pole

3.6 One-degree-of-freedom root-locus-based design

3.7 Conclusions

3.8 Exercises

Bibliography

4 Frequency response-based design

4.1 The proposed design procedure

4.2 Controller gains selection

4.3 Experimental results

4.4 Simulation results

4.5 Conclusions

4.6 Exercises
Bibliography

5 The nonminimum-phase systems approach 145

5.1 The design procedure

5.2 Controller gains selection

5.3 Experimental results

5.4 Simulation results

5.5 Conclusions

5.6 Exercises

Bibliography

6 LQR control 167

6.1 The design procedure

6.2 Controller gains selection

6.3 Experimental results

6.4 Simulation results

6.5 Conclusions

6.6 Exercises

Bibliography

7 Effects of diverse model parameters

7.1 The disturbed wheeled pendulum model

7.2 Effects of Cz, R, and Mp in closed-loop system performance

7.2.1 Robot control using different values for Cz

7.2.2 Robot control using different values for wheel radius
7.2.3 Increasing pendulum body mass

7.3 Induced limit cycles as control gains change

7.3.1 Study through experiments

7.4 Coupling between ¿¿ and (¿, ¿)¿subsystems

7.5 Discrete-time model of wheeled pendulum

7.5.1 Closed-loop stability when sampling period and wheel

radius change

7.6 Conclusions

7.7 Exercises

Bibliography

II Nonlinear Control 229

8 Some mathematical basis for control of nonlinear systems 231

8.1 Lyapunov stability

8.2 Positive and negative definiteness

8.3 Stability theorems

8.4 Passivity

8.5 Hamiltonian formulation

8.6 Linear systems and exponential stability

8.7 A particular second order differential equation

Bibliograph

9 Hamiltonian formulation 257

9.1 Hamiltonian formulation of arbitrary Euler-Lagrange systems
9.2 Hamiltonian formulation of arbitrary nonholonomic Euler-Lagrange

systems

9.2.1 Nonholonomic constraints

9.2.2 Hamiltonian modeling (Van Der Schaft [2000])

9.2.3 Obtaining the nonholonomic Hamiltonian model in

(9.33)

9.3 Hamiltonian representation of the error equation

9.4 Hamiltonian representation of the error equation subject to

nonholonomic constraints

9.5 Conclusions

Bibliography

10 Constrained Hamiltonian representation of wheeled pendulum

10.1 The constrained Hamiltonian model

10.2 Obtaining Hec(¿q + q¿, ¿p1e

) from ¿He(¿q + q¿, ¿pe)

10.3 Simplifying the Hamiltonian model for control design

10.4 Conclusions

11 Path following control

11.1 Hamiltonian approach

11.2 Feedback linearization control

11.2.1 A first proposal

11.2.2 A successful feedback linearization controller

11.3 Energy-shaping is equivalent to feedback linearization
11.4 Experimental results

11.4.1 Linear simplification of controller in Proposition

11.4.2 Linear simplification of controller in Proposition

11.4.3 Linear simplification of controller in Proposition

11.4.4 Controller gains selection

11.4.5 Experimental tests

11.5 Simulation results

11.6 Exercises

11.7 Conclusions

Bibliography

A Program Codes

A.1 Robot parameters

A.2 C code for microcontroller programming

A.3 Matlab code used to draw figs. 2.6, 2.7, and 2.8

A.4 Matlab code used to compute controller gains in Section

A.5 Simulation of linear control schemes

A.5.1 Straight line

A.5.2 8¿shaped trajectory

A.6 Matlab code used to compute controller gains in Section

A.7 Matlab code used to compute controller gains found in Section

5.2

A.8 Matlab code used to compute LQR controller gains

A.9 Matlab code used to compute data shown in Tables 7.1 and 7.2
A.10 Matlab code used for limit cycle analysis

A.11 Programs for discrete-time analysis

A.11.1 Matlab code used for Z¿transform computation

A.11.2 Matlab-Simulink block diagram for discrete-time simulations

A.12 Matlab code used to compute controller gains in Section 11.4.4

A.13 Matlab-Simulink block diagram used for simulations in Chapter

11

Bibliography

Index