Title:
Optimal control systems
Personal Author:
Publication Information:
Boca Raton, Fla. : CRC Press, 2003
ISBN:
9780849308925
Available:*
Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
|---|---|---|---|---|---|
Searching... | 33000000000033 | TJ213 N27 2003 | Open Access Book | Book | Searching... |
Searching... | 30000010029202 | TJ213 N27 2003 | Open Access Book | Book | Searching... |
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Summary
Summary
The theory of optimal control systems has grown and flourished since the 1960's. Many texts, written on varying levels of sophistication, have been published on the subject. Yet even those purportedly designed for beginners in the field are often riddled with complex theorems, and many treatments fail to include topics that are essential to a thorough grounding in the various aspects of and approaches to optimal control.
Optimal Control Systems provides a comprehensive but accessible treatment of the subject with just the right degree of mathematical rigor to be complete but practical. It provides a solid bridge between "traditional" optimization using the calculus of variations and what is called "modern" optimal control. It also treats both continuous-time and discrete-time optimal control systems, giving students a firm grasp on both methods. Among this book's most outstanding features is a summary table that accompanies each topic or problem and includes a statement of the problem with a step-by-step solution. Students will also gain valuable experience in using industry-standard MATLAB and SIMULINK software, including the Control System and Symbolic Math Toolboxes.
Diverse applications across fields from power engineering to medicine make a foundation in optimal control systems an essential part of an engineer's background. This clear, streamlined presentation is ideal for a graduate level course on control systems and as a quick reference for working engineers.
Table of Contents
| 1 Introduction | p. 1 |
| 1.1 Classical and Modern Control | p. 1 |
| 1.2 Optimization | p. 4 |
| 1.3 Optimal Control | p. 6 |
| 1.3.1 Plant | p. 6 |
| 1.3.2 Performance Index | p. 6 |
| 1.3.3 Constraints | p. 9 |
| 1.3.4 Formal Statement of Optimal Control System | p. 9 |
| 1.4 Historical Tour | p. 11 |
| 1.4.1 Calculus of Variations | p. 11 |
| 1.4.2 Optimal Control Theory | p. 13 |
| 1.5 About This Book | p. 15 |
| 1.6 Chapter Overview | p. 16 |
| 1.7 Problems | p. 17 |
| 2 Calculus of Variations and Optimal Control | p. 19 |
| 2.1 Basic Concepts | p. 19 |
| 2.1.1 Function and Functional | p. 19 |
| 2.1.2 Increment | p. 20 |
| 2.1.3 Differential and Variation | p. 22 |
| 2.2 Optimum of a Function and a Functional | p. 25 |
| 2.3 The Basic Variational Problem | p. 27 |
| 2.3.1 Fixed-End Time and Fixed-End State System | p. 27 |
| 2.3.2 Discussion on Euler-Lagrange Equation | p. 33 |
| 2.3.3 Different Cases for Euler-Lagrange Equation | p. 35 |
| 2.4 The Second Variation | p. 39 |
| 2.5 Extrema of Functions with Conditions | p. 41 |
| 2.5.1 Direct Method | p. 43 |
| 2.5.2 Lagrange Multiplier Method | p. 45 |
| 2.6 Extrema of Functionals with Conditions | p. 48 |
| 2.7 Variational Approach to Optimal Control Systems | p. 57 |
| 2.7.1 Terminal Cost Problem | p. 57 |
| 2.7.2 Different Types of Systems | p. 65 |
| 2.7.3 Sufficient Condition | p. 67 |
| 2.7.4 Summary of Pontryagin Procedure | p. 68 |
| 2.8 Summary of Variational Approach | p. 84 |
| 2.8.1 Stage I: Optimization of a Functional | p. 85 |
| 2.8.2 Stage II: Optimization of a Functional with Condition | p. 86 |
| 2.8.3 Stage III: Optimal Control System with Lagrangian Formalism | p. 87 |
| 2.8.4 Stage IV: Optimal Control System with Hamiltonian Formalism: Pontryagin Principle | p. 88 |
| 2.8.5 Salient Features | p. 91 |
| 2.9 Problems | p. 96 |
| 3 Linear Quadratic Optimal Control Systems I | p. 101 |
| 3.1 Problem Formulation | p. 101 |
| 3.2 Finite-Time Linear Quadratic Regulator | p. 104 |
| 3.2.1 Symmetric Property of the Riccati Coefficient Matrix | p. 109 |
| 3.2.2 Optimal Control | p. 110 |
| 3.2.3 Optimal Performance Index | p. 110 |
| 3.2.4 Finite-Time Linear Quadratic Regulator: Time-Varying Case: Summary | p. 112 |
| 3.2.5 Salient Features | p. 114 |
| 3.2.6 LQR System for General Performance Index | p. 118 |
| 3.3 Analytical Solution to the Matrix Differential Riccati Equation | p. 119 |
| 3.3.1 MATLAB Implementation of Analytical Solution to Matrix DRE | p. 122 |
| 3.4 Infinite-Time LQR System I | p. 125 |
| 3.4.1 Infinite-Time Linear Quadratic Regulator: Time-Varying Case: Summary | p. 128 |
| 3.5 Infinite-Time LQR System II | p. 129 |
| 3.5.1 Meaningful Interpretation of Riccati Coefficient | p. 132 |
| 3.5.2 Analytical Solution of the Algebraic Riccati Equation | p. 133 |
| 3.5.3 Infinite-Interval Regulator System: Time-Invariant Case: Summary | p. 134 |
| 3.5.4 Stability Issues of Time-Invariant Regulator | p. 139 |
| 3.5.5 Equivalence of Open-Loop and Closed-Loop Optimal Controls | p. 141 |
| 3.6 Notes and Discussion | p. 144 |
| 3.7 Problems | p. 147 |
| 4 Linear Quadratic Optimal Control Systems II | p. 151 |
| 4.1 Linear Quadratic Tracking System: Finite-Time Case | p. 152 |
| 4.1.1 Linear Quadratic Tracking System: Summary | p. 157 |
| 4.1.2 Salient Features of Tracking System | p. 158 |
| 4.2 LQT System: Infinite-Time Case | p. 166 |
| 4.3 Fixed-End-Point Regulator System | p. 169 |
| 4.4 LQR with a Specified Degree of Stability | p. 175 |
| 4.4.1 Regulator System with Prescribed Degree of Stability: Summary | p. 177 |
| 4.5 Frequency-Domain Interpretation | p. 179 |
| 4.5.1 Gain Margin and Phase Margin | p. 181 |
| 4.6 Problems | p. 188 |
| 5 Discrete-Time Optimal Control Systems | p. 191 |
| 5.1 Variational Calculus for Discrete-Time Systems | p. 191 |
| 5.1.1 Extremization of a Functional | p. 192 |
| 5.1.2 Functional with Terminal Cost | p. 197 |
| 5.2 Discrete-Time Optimal Control Systems | p. 199 |
| 5.2.1 Fixed-Final State and Open-Loop Optimal Control | p. 203 |
| 5.2.2 Free-Final State and Open-Loop Optimal Control | p. 207 |
| 5.3 Discrete-Time Linear State Regulator System | p. 207 |
| 5.3.1 Closed-Loop Optimal Control: Matrix Difference Riccati Equation | p. 209 |
| 5.3.2 Optimal Cost Function | p. 213 |
| 5.4 Steady-State Regulator System | p. 219 |
| 5.4.1 Analytical Solution to the Riccati Equation | p. 225 |
| 5.5 Discrete-Time Linear Quadratic Tracking System | p. 232 |
| 5.6 Frequency-Domain Interpretation | p. 239 |
| 5.7 Problems | p. 245 |
| 6 Pontryagin Minimum Principle | p. 249 |
| 6.1 Constrained System | p. 249 |
| 6.2 Pontryagin Minimum Principle | p. 252 |
| 6.2.1 Summary of Pontryagin Principle | p. 256 |
| 6.2.2 Additional Necessary Conditions | p. 259 |
| 6.3 Dynamic Programming | p. 261 |
| 6.3.1 Principle of Optimality | p. 261 |
| 6.3.2 Optimal Control Using Dynamic Programming | p. 266 |
| 6.3.3 Optimal Control of Discrete-Time Systems | p. 272 |
| 6.3.4 Optimal Control of Continuous-Time Systems | p. 275 |
| 6.4 The Hamilton-Jacobi-Bellman Equation | p. 277 |
| 6.5 LQR System Using H-J-B Equation | p. 283 |
| 6.6 Notes and Discussion | p. 288 |
| 7 Constrained Optimal Control Systems | p. 293 |
| 7.1 Constrained Optimal Control | p. 293 |
| 7.1.1 Time-Optimal Control of LTI System | p. 295 |
| 7.1.2 Problem Formulation and Statement | p. 295 |
| 7.1.3 Solution of the TOC System | p. 296 |
| 7.1.4 Structure of Time-Optimal Control System | p. 303 |
| 7.2 TOC of a Double Integral System | p. 305 |
| 7.2.1 Problem Formulation and Statement | p. 306 |
| 7.2.2 Problem Solution | p. 307 |
| 7.2.3 Engineering Implementation of Control Law | p. 314 |
| 7.2.4 SIMULINK Implementation of Control Law | p. 315 |
| 7.3 Fuel-Optimal Control Systems | p. 315 |
| 7.3.1 Fuel-Optimal Control of a Double Integral System | p. 316 |
| 7.3.2 Problem Formulation and Statement | p. 319 |
| 7.3.3 Problem Solution | p. 319 |
| 7.4 Minimum-Fuel System: LTI System | p. 328 |
| 7.4.1 Problem Statement | p. 328 |
| 7.4.2 Problem Solution | p. 329 |
| 7.4.3 SIMULINK Implementation of Control Law | p. 333 |
| 7.5 Energy-Optimal Control Systems | p. 335 |
| 7.5.1 Problem Formulation and Statement | p. 335 |
| 7.5.2 Problem Solution | p. 339 |
| 7.6 Optimal Control Systems with State Constraints | p. 351 |
| 7.6.1 Penalty Function Method | p. 352 |
| 7.6.2 Slack Variable Method | p. 358 |
| 7.7 Problems | p. 361 |
| Appendix A Vectors and Matrices | p. 365 |
| A.1 Vectors | p. 365 |
| A.2 Matrices | p. 367 |
| A.3 Quadratic Forms and Definiteness | p. 376 |
| Appendix B State Space Analysis | p. 379 |
| B.1 State Space Form for Continuous-Time Systems | p. 379 |
| B.2 Linear Matrix Equations | p. 381 |
| B.3 State Space Form for Discrete-Time Systems | p. 381 |
| B.4 Controllability and Observability | p. 383 |
| B.5 Stabilizability, Reachability and Detectability | p. 383 |
| Appendix C MATLAB Files | p. 385 |
| C.1 MATLAB for Matrix Differential Riccati Equation | p. 385 |
| C.1.1 MATLAB File lqrnss.m | p. 386 |
| C.1.2 MATLAB File lqrnssf.m | p. 393 |
| C.2 MATLAB for Continuous-Time Tracking System | p. 394 |
| C.2.1 MATLAB File for Example 4.1(example4_1.m) | p. 394 |
| C.2.2 MATLAB File for Example 4.1(example4_1p.m) | p. 397 |
| C.2.3 MATLAB File for Example 4.1(example4_1g.m) | p. 397 |
| C.2.4 MATLAB File for Example 4.1(example4_1x.m) | p. 397 |
| C.2.5 MATLAB File for Example 4.2(example4_1.m) | p. 398 |
| C.2.6 MATLAB File for Example 4.2(example4_2p.m) | p. 400 |
| C.2.7 MATLAB File for Example 4.2(example4_2g.m) | p. 400 |
| C.2.8 MATLAB File for Example 4.2(example4_2x.m) | p. 401 |
| C.3 MATLAB for Matrix Difference Riccati Equation | p. 401 |
| C.3.1 MATLAB File lqrdnss.m | p. 401 |
| C.4 MATLAB for Discrete-Time Tracking System | p. 409 |
| References | p. 415 |
| Index | p. 425 |
