Design of Nonlinear Control Systems with the Highest Derivative in Feedback
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1 SERIES ON STAB1UTY, VIBRATION AND CONTROL OF SYSTEMS SeriesA Volume 16 Founder & Editor: Ardeshir Guran Co-Editors: M. Cloud & W. B. Zimmerman Design of Nonlinear Control Systems with the Highest Derivative in Feedback Valery D. Yurkevich Concoräia University, Canada Y * World Scientific NEWJERSEY LONDON 5INGAP0RE BEIJING SHANGHAI HONG KONG TAIPEI CHENNAi
2 Contents Preface vii 1. Regulär ly and singularly perturbed Systems Regularly perturbed Systems Nonlinear nominal System Linear nominal System Vanishing perturbation Nonvanishing perturbation Singularly perturbed Systems Singular perturbation Two-time-scale motions...; Boundary-layer System Stability analysis Fast and slow-motion Subsystems Degree of time-scale Separation Discrete-time singularly perturbed Systems Fast and slow-motion Subsystems Degree of time-scale Separation Notes Exercises Design goal and reference model Design goal Basic step response parameters Reference model, Notes 30 xiii
3 xiv Design of nordinear control Systems with the highest derivative in feedback 2.5 Exercises 31 Methods of control System design under uncertainty Desired vector field in the state space of plant model Solution of nonlinear inverse dynamics The highest derivative and high gain in feedback loop Differentiating füter and high-gain observer Influence of noise in control system with the highest derivative Desired manifold in the State space of plant model State vector and high gain in feedback loop Control Systems with sliding motions Example Notes Exercises 55 Design of SISO continuous-time control Systems Controller design for plant model of the Ist order Control problem Insensitivity condition Control law with the Ist derivative in feedback loop Closed-loop system properties Controller design for an nth-order plant model Control problem Insensitivity condition Control law with the nth derivative in the feedback loop Fast-motion Subsystem Slow-motion Subsystem Influence of small parameter Geometrie interpretation of control problem Solution Example Notes Exercises 77 Advanced design of SISO continuous-time control Systems Control aecuraey Steady State of fast-motion subsystem Steady State of slow-motion subsystem 80
4 Contents xv Velocity error due to external disturbance Velocity error due to reference input Control law in the form of forward compensator Root placement of FMS characteristic polynomial Degree of time-scale Separation Selection of Controller parameters Root placement based on normalized polynomials Bode amplitude diagram assignment of closed-loop FMS Block diagram of closed-loop System Bode amplitude diagram of closed-loop FMS Desired Bode amplitude diagram of closed-loop FMS Selection of Controller parameters Influence of high-frequency sensor noise Closed-loop System in presence of sensor noise Controller with infinite bandwidth Controller with fmite bandwidth Influence of varying parameters Influence of varying parameters on FMS and SMS Michailov hodograph for FMS Variation of FMS bandwidth Degree of control law differential equation Root placement of FMS characteristic polynomial Bode amplitude diagram assignment of open-loop FMS Relation with PD, PI, and PID Controllers Example Notes Exercises 112 Influence of unmodeled dynamics Pure time delay Plant model with pure time delay in control.. > Closed-loop System with delay in feedback loop Fast motions in presence of delay Stability of FMS with delay Phase margin of FMS with delay Control with compensation of delay Velocity error with respect to external disturbance Example Regulär perturbances 126
5 Design of nonlinear control Systems with the highest derivative in feedback Regularly perturbed plant model Fast motions in presence of regulär perturbances Selection of Controller parameters Control with compensation of regulär perturbances Example Singular perturbances Singularly perturbed plant model Fast motions in presence of singular perturbances Selection of Controller parameters Nonsmooth nonlinearity in control loop System preceded by nonsmooth nonlinearity Describing function analysis of limit cycle in FMS Effect of chattering on control accuracy Example Notes Exercises Realizability of desired output behavior Control problem Statement for MIMO control System MIMO plant model Control problem I Invertibility of dynamical Systems Role of invertibility of dynamical Systems Definition of invertibility of dynamic control System Invertibility condition for nonlinear Systems Insensitivity condition for MIMO control system Desired dynamics equations Insensitivity condition Internal stability Boundedness of jv7.d-control function Concept of internal stability Normal form of the plant model Internal stability of linear Systems Internal stability of nonlinear Systems Degenerated motions and zero-dynamics Example Output regulation of SISO Systems Realizability of desired output behavior Closed-loop System analysis 174
6 Contents Example 7.6 Switching regulator for boost DC-to-DC Converter Boost DC-to-DC Converter circuit model Model with continuous control variable Switching regulator External disturbance attenuation 7.7 Notes 7.8 Exercises Design of MIMO continuous-time control Systems 8.1 MIMO system without internal dynamics MIMO system with identical relative degrees MIMO system with different relative degrees MIMO control system design (identical relative degrees) Lnsensitivity condition Control system with the relative highest derivatives in feedback Fast-motion Subsystem Slow-motion Subsystem Control system design with zero steady-state error Example 8.3 MIMO control system design (different relative degrees) lnsensitivity condition and control law structure Closed-loop system analysis Control accuracy 8.4 MIMO control system in presence of internal dynamics Fast-motion Subsystem Slow-motion Subsystem Example. 8.5 Decentralized Output feedback Controller 8.6 Notes 8.7 Exercises Stabilization of internal dynamics 9.1 Zero placement by redundant control 9.2 Internal dynamics stabilization (particular case) 9.3 Internal dynamics stabilization (generalized case) 9.4 Stabilization of degenerated mode and zero dynamics...
7 Design of nonlinear control Systems with the highest derivative in feedback 9.5 Methods of internal dynamics stabilization Example Notes Exercises 232 Digital Controller design based on pseudo-continuous approach Continuous System preceded by zero-order hold Control problem Pseudo-continuous-time model with pure delay Digital Controller design Insensitivity condition Pseudo-continuous closed-loop System Influence of sampling period Digital realization of continuous Controller Example Digital Controller design with compensation of delay Control law structure Closed-loop System analysis Digital realization of continuous Controller Example Notes Exercises 250 Design of discrete-time control Systems SISO two-time-scale discrete-time control Systems Discrete-time Systems Control problem and insensitivity condition Discrete-time control law Two-time-scale motion analysis Robustness of closed-loop System properties Control accuracy Example SISO discrete-time control Systems with small parameter System with small parameter Two-time-scale motion analysis Interrelationship with fixed point theorem Root placement of FMS characteristic polynomial FMS design based on frequency-domain methods. 274
8 Contents xix 11.3 MIMO two-time-scale discrete-time control Systems MIMO discrete-time Systems Control law Two-time-scale motion analysis Example Notes Exercises Design of sampled-data control Systems SISO sampled-data control Systems Reduced Order pulse transfer function Input-output approximate model of linear system Control law Closed-loop system analysis Selection of Controller parameters Nonlinear sampled-data Systems Example MIMO sampled-data control Systems Control problem MIMO contirmous-time System preceded by ZOH Control law Fast-motion Subsystem Selection of Controller parameters Slow-motion Subsystem Example Notes Exercises Control of distributed parameter Systems One-dimensional heat System with distributed control Heat system with finite-dimensional control Degenerated motions Estimation of modes Notes Exercises 323 Appendix A Proofs 325 A.l Proof of expression (8.29) 325
9 xx Design of nonlinear control Systems with the highest derivative in feedback A.2 Proof of expression (8.42) 325 A.3 Proof of expression (8.65) 32^ A.4 Proof of expression (11.37) 327 A.5 Proof of expressions (11.40)-(11.41) 328 A.6 Proof of expression (11.47) 328 A.7 Proof of expression (11.51) 33 A.8 Proof of expression (12.56) 332 A.9 Proof of expression (12.57) 333 Appendix B Notation system 335 Bibliography Index 349
Contents. PART I METHODS AND CONCEPTS 2. Transfer Function Approach Frequency Domain Representations... 42
Contents Preface.............................................. xiii 1. Introduction......................................... 1 1.1 Continuous and Discrete Control Systems................. 4 1.2 Open-Loop
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