Analysis and Synthesis of Single-Input Single-Output Control Systems

Transcription

1 Lino Guzzella Analysis and Synthesis of Single-Input Single-Output Control Systems l+kja» \Uja>)W2(ja»\ um

2 Contents 1 Definitions and Problem Formulations Introduction Definitions Systems and Signals SISO/MIMO Systems Linear/Nonlinear Systems System Classification Models Control Systems Control System Design Problems General Remarks Systems Analysis Reference Tracking Disturbance Rejection TradeofF Robustness versus Performance Stabilization Noise Attenuation 13 2 Modeling of Dynamic Systems Introduction General Modeling Guidelines Some Examples Example Water Tank Example Stirred Reactor Example Cruise Control Example Loudspeaker Example Conveyor Belt Model Uncertainty 24

3 VIII Contents 3 System Representation and Transformation Introduction Loudspeaker, First-Order Equations Objectives Normalization Example Water Tank, Normalization Example Cruise Control, Normalization Linearization Example Water Tank, Linearization Example Cruise Control, Linearization Parametric Uncertainty General Remarks Example Cruise Control, Parametric Uncertainty Linear State Space Forms Example Loudspeaker System, State Space Form State Space and System Coordinates Plant and Control System Interface 39 4 Analysis of Linear Systems Part I Introduction Time Domain Behavior of Linear Systems Matrix Exponentials General Solution First-Order Systems Impulse Response Step Response Ramp Response Harmonie Response Stability Definitions Spectral Methods Stability of the Inverted Pendulum Non-Exponential Asymptotic Stability* 54 5 Analysis of Linear Systems Part II Introduction Reachability, Controllability and Stabilizability Introduction Reachability Conditions An Explicit Feedforward Control Signal* Example Controllability of a Geostationary Satellite Nonlinear Systems* Observability Conditions Observability of the Inverted Pendulum State Space Decomposition 64

4 Contents IX 5.5 Spectral Decompositions* I/O Descriptions and Canonical Realizations I/O Description State-Space Description State-Space -+ I/O Description* Coordinate Transformations* Concluding Remarks 74 Laplace Transformation Part I Introduction Properties of the Laplace Transformation Transfer Functions Introduction and Definitions Properties of Transfer Functions Transfer Functions and System Realizations Transfer Functions as Conformal Mappings* Overview System Representations and Transformations 86 Laplace Transformation Part II Introduction Solution of Low-Order ODE Example of a Inverse Laplace Transformation Poles and Zeros of Transfer Functions Poles and BIBO Stability Definition and Interpretation of System Zeros Influence of Poles and Zeros on System Dynamics Algebraic Stability Criteria* Introduction The Hurwitz Criterion The Kharitonov Theorem 104 Frequency Responses Introduction Frequency Responses Definition of Frequency Responses Representation of Frequency Responses Bode's Amplitude/Phase Law* Asymptotic System Properties System Identification Using Frequency Responses General Remarks Example System Identification Nonparametric Uncertainty Introduction Uncertainty Representations Uncertainty Estimation 122

5 X Contents 9 Analysis of Feedback Systems Introduction Definitions Closed-Loop System Stability Nyquist Theorem Nominal Closed-Loop System Stability Robust Closed-Loop System Stability Constraints on Closed-Loop Systems Spectral Properties of Noise and Disturbance Sensitivity Constraints Limitations on Crossover Frequencies Example Pendulum on a Cart* Summary Specifications for Feedback Systems Introduction Static Errors Specifications Based on Second-Order Systems Frequency-Domain Specifications Introduction Peaking Limitations Multiplicative Specifications of the Sensitivity* Summary Feedback Control Design Introduction PID Controllers Introduction PID Tuning Rules Loop Shaping by Parameter Tuning Classical Iterative Loop Shaping First-Order Lead/Lag Elements Second-Order Lead/Lag Elements Closed-Form Cross-Over Specification* Aström and Hägglund Rules* Predictive PI Control Systems* Predictive PI Control of Simple Plants Smith Predictors Numerical Optimization Feedback Control Design II Introduction Loop Shaping and Robustness Introduction Plant Inversion Methods 200

6 Contents XI 12.3 Loop Shaping and Non-Minimum Phase Zeros Introduction Example Loop Shaping for NMP Plants Loop Shaping for Unstable Systems Introduction Example Control of Inverted Pendulum I Feedback Control Design III Introduction Cascaded Control Loops General Remarks An Example Root-Locus Methods Introduction General Rules of Root Locus Designs Compensation in Root Locus Designs Control Systems Implementation* Introduction PID Controllers in Practical Applications Enhanced PID Structures Dealing with Actuator Saturation Bumpless Transfer and Gain Scheduling Realization with Analog Components Realization with Digital Computers Case Study* Introduction Modeling General Remarks Model Development Linearization and Controller development Estimation of the model uncertainties Specifications Controller Design 245 A Library of Standard Elements 251 A.l Integrator Element 252 A.2 Differentiator Element 253 A.3 First-Order Element 254 A.4 Realizable Derivative Element ("Dirty D") 255 A.5 Second-Order Element 256 A.6 Lag Element 257 A.7 Lead Element 258 A.8 PID Element 259

7 XII Contents A.9 First-Order All-Pass Element 260 A.10 Delay Element 261 B Some Mathematical Results 263 B.l Linear Algebra 263 B.l.l Notation 263 B.1.2 Geometry 264 B.l.3 Determinants and Traces 267 B.l.4 Linear Equations 269 B.l.5 Characteristic Polynomials 270 B.1.6 Eigenproblems 272 B.l.7 Structured Matrices 274 B.2 Complex Analysis 276 B.2.1 Signal Power Spectra 277 B.2.2 Some Theorems 278 B.3 Proof of the Nyquist Theorem 279 B.4 Proof of the Cross-Over Frequency Specification Method 281 B.5 Proof of the Apollonius Circle Condition 284 C Solutions to Quick Checks 285 C.l Definitions and Problem Formulations 285 C.l.l Introduction 285 C.1.2 Definitions 285 C.l.3 System Classification 287 C.1.4 Models 287 C.1.5 Control Systems 288 C.l.6 Control System Design Problems 289 C.2 Modeling of Dynamic Systems 290 C.2.1 Introduction 290 C.2.2 General Modeling Guidelines 290 C.2.3 Some Examples 290 C.3 System Representation and Transformation 291 C.3.1 Introduction 291 C.3.2 Normalization 291 C.3.3 Linearization 292 C.3.4 Linear State Space Forms 293 C.4 Analysis of Linear Systems Part I 293 C.4.1 Introduction 293 C.4.2 Time-Domain Behavior of Linear Systems 293 C.4.3 First-Order Systems 295 C.4.4 Stability 297 C.5 Analysis of Linear Systems Part II 299 C.5.1 Introduction 299 C.5.2 Reachability, Controllability and Stabilizability 299 C.5.3 Observability Conditions 301

8 Contents XIII C.5.4 State Space Decomposition 302 C.5.5 Spectral Decompositions 302 C.5.6 I/O Descriptions and Canonical Realizations 303 C.6 Laplace Transformation Part C.6.1 Introduction 306 C.6.2 Properties of the Laplace Transformation 306 C.6.3 Transfer Functions 307 C.7 Laplace Transformation Part II 310 C.7.1 Introduction 310 C.7.2 Solution of Low-Order ODE 310 C.7.3 Poles and Zeros of Transfer Functions 311 C.7.4 Algebraic Stability Criteria 313 C.8 Frequency Responses 315 C.8.1 Introduction 315 C.8.2 Frequency Responses 315 C.8.3 Asymptotic System Properties 318 C.8.4 System Identification Using Frequency Responses 319 C.8.5 Nonparametric Uncertainty 321 C.9 Analysis of Feedback Systems 322 C.9.1 Introduction 322 C.9.2 Definitions 322 C.9.3 Closed-Loop System Stability 323 C.9.4 Nyquist Theorem 323 C.9.5 Constraints on Closed-Loop Systems 327 C.10 Specifications for Feedback Systems 329 C.10.1 Introduction 329 C.10.2 Specs Based on Second-Order System Approximations. 330 C.10.3 Frequency Domain Specifications 330 C.ll Feedback Control Design I 333 C.ll.l Introduction 333 C.11.2 PID Controllers 333 C.ll.3 Classical Iterative Loop Shaping 336 C.ll.4 Closed-Form Cross Over Specification 336 C.ll.5 Predictive PI Control Systems 336 C.12 Feedback Control Design II 337 C.12.1 Introduction 337 C.12.2 Loop Shaping and Robustness 337 C.12.3 Loop Shaping and Non-Minimum Phase Zeros 338 C.12.4 Loop Shaping for Unstable Systems 338 C.13 Feedback Control Design III 340 C.13.1 Introduction 340 C.13.2 Root-Locus Methods 340 C.14 Control Systems Implementation 341 C.14.1 Introduction 341 C.14.2 PID Controllers in Practical Applications 341

9 XTV Contents C.14.3 Realization with Analog Components 342 C.14.4 Realization with Digital Computers 343 D Translation English to German 345 E List of Symbols 347 References 351