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E-raamat: Linear Multivariable Control Systems

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  • Ilmumisaeg: 13-Jan-2022
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9781108899277
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Linear Multivariable Control SystemsSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 13-Jan-2022
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9781108899277
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This unique graduate text provides broad and systematic coverage of linear multivariable control systems, including several new approaches to design. Supported by MATLAB and SIMULINK examples and end-of-chapter problems, it is ideal for the first two graduate courses on the subject in aerospace, electrical and mechanical engineering.

This rigorous yet accessible textbook provides broad and systematic coverage of linear multivariable control systems, including several new approaches to design. In addition to standard state space theory, it provides a new measurement-based approach to linear systems, including a generalization of Thevenin's Theorem, a new single-input single-output approach to multivariable control, and analytical design of PID controllers developed by the authors. Each result is rigorously proved and combined with specific control systems applications, such as the servomechanism problem, the fragility of high order controllers, multivariable control, and PID controllers. Illustrative examples solved using MATLAB and SIMULINK, with easily reusable programming scripts, are included throughout. Numerous end-of-chapter homework problems enhance understanding. Based on course-tested material, this textbook is ideal for a single or two-semester graduate course on linear multivariable control systems in aerospace, chemical, electrical and mechanical engineering.

Arvustused

'High standards of linear control were established in the 1960s in pioneering works by R. Bellman, R. Kalman, and L. Pontryagin. However, this field of research remains active and is permanently enriched with new results and new applications, as the book by S. P. Bhattacharyya and L. H. Keel vividly demonstrates.' Boris Polyak, IFAC Fellow, Institute for Control Science, Moscow 'One of the most valuable features of this book is that it integrates the pillars of control theory with recent advancements, resulting in an up-to-date treatment of the topic.' Antonio Visioli, University of Brescia

Muu info

A graduate text providing broad coverage of linear multivariable control systems, including several new results and recent approaches.
Preface xv
1 A Measurement-Based Approach to Linear Systems
1(45)
1.1 Output Control of Unknown Systems Using Prescribed Inputs
1(3)
1.2 Current and Voltage Assignment Using Resistors in DC Circuits
4(20)
1.2.1 Current Control Using a Single Resistor
7(6)
1.2.2 Control Using Two Resistors
13(2)
1.2.3 Control Using m Resistors
15(9)
1.3 AC Circuits
24(7)
1.3.1 Control Using a Single Impedance
24(2)
1.3.2 Control Using Two Impedances
26(1)
1.3.3 Current Control Using Gyrator Resistance
26(1)
1.3.4 Control Using m Independent Sources
27(4)
1.4 Mass--Spring Systems
31(3)
1.5 Hydraulic Networks
34(7)
1.5.1 Flow Rate Control Using a Single Pipe Resistance
36(2)
1.5.2 Flow Rate Control Using Two Pipe Resistances
38(3)
1.6 Exercises
41(3)
1.7 Notes and References
44(2)
2 Classical Control Theory: A Brief Overview
46(69)
2.1 Introduction
46(1)
2.2 Reliable Systems Using Unreliable Components: Black's Amplifier
46(2)
2.3 Op-amp Circuits
48(2)
2.4 Robust Feedback Linearization
50(1)
2.5 High Gain in Control Systems
51(1)
2.6 Unity Feedback Servomechanisms
52(1)
2.7 Closed-Loop Stability
53(1)
2.8 Classes of Reference and Disturbance Signals
54(1)
2.9 Asymptotic Tracking and Disturbance Rejection
55(2)
2.10 Integral Control: PI and PID Controllers
57(3)
2.11 The Nyquist Criterion
60(3)
2.12 Counting Encirclements by the Nyquist Plot Using Bode Frequency Response
63(2)
2.13 Stabilizing Gains from the Nyquist Plot
65(1)
2.14 Gain and Phase Margin
66(1)
2.15 A Frequency Response Parametrization of All Stabilizing Controllers
67(10)
2.15.1 A Bode Equivalent of the Nyquist Criterion
68(3)
2.15.2 Arbitrary Order Controllers
71(2)
2.15.3 Performance Measures
73(4)
2.16 Gain, Phase, and Time-Delay Margins
77(3)
2.17 Stability Margin-Based Design of PI Controllers
80(4)
2.18 Stability Theory for Polynomials
84(21)
2.18.1 The Boundary Crossing Theorem
84(6)
2.18.2 The Hermite--Biehler Theorem
90(15)
2.19 Root Counting and the Routh Table
105(1)
2.20 The Gauss--Lucas Theorem
106(4)
2.20.1 Application to Control Systems
107(3)
2.21 Exercises
110(4)
2.22 Notes and References
114(1)
3 The [ A, B, C, D] State-Variable Model
115(52)
3.1 Introduction
115(7)
3.2 State-Variable Models from Differential Equations
122(5)
3.2.1 Single-Input Single-Output Systems
123(3)
3.2.2 Multi-Input Multi-Output Systems
126(1)
3.3 Solution of Continuous-Time State Equations
127(7)
3.4 Realization of Systems from State-Variable Models
134(2)
3.4.1 Integrators or Differentiators
134(1)
3.4.2 Synthesis of State-Space System Using Integrators, Adders, and Multipliers
135(1)
3.5 A Formula for the Characteristic Equation
136(3)
3.6 Calculation of (sI -- A)-1: Faddeev--LeVerrier Algorithm
139(2)
3.7 Block Diagram Algebra
141(4)
3.7.1 A Simplified Approach
142(1)
3.7.2 Examples
143(2)
3.8 State-Space Models for Interconnected Systems
145(6)
3.8.1 Series Connection
146(1)
3.8.2 Parallel Connection
146(1)
3.8.3 Feedback Connection
147(4)
3.9 Discrete-Time Systems
151(6)
3.9.1 Solution of Discrete-Time Systems
152(1)
3.9.2 Transfer Function
153(1)
3.9.3 Discretization of Continuous-Time Systems
153(4)
3.10 Stability of State-Space Systems
157(2)
3.11 Exercises
159(7)
3.12 Notes and References
166(1)
4 Modal Structure of State-Space Systems
167(34)
4.1 Similarity Transformation and the Jordan Form
167(18)
4.1.1 Similarity Transformation
167(2)
4.1.2 Diagonalization: Distinct Eigenvalues
169(2)
4.1.3 Jordan Form: Repeated Eigenvalues
171(6)
4.1.4 Finding the Transformation Matrix T: Generalized Eigen-vector Chains
177(3)
4.1.5 Jordan Form of Companion Matrices
180(5)
4.2 Jordan Form Using Polynomial Matrix Theory
185(1)
4.3 Examples
185(5)
4.4 Matrix Exponentials Evaluated Using Jordan Forms
190(1)
4.5 Modal Decomposition: Zero-Input Response
191(2)
4.6 Modal Decomposition: Zero-State Response
193(2)
4.7 Modal Decomposition of the State Space
195(2)
4.8 Exercises
197(3)
4.9 Notes and References
200(1)
5 Controllability, Observability, and Realization Theory
201(61)
5.1 Introduction
201(1)
5.2 Invariant Subspaces and the Kalman Decomposition
202(7)
5.2.1 Invariant Subspaces
202(3)
5.2.2 Controllability and Observability Reductions
205(1)
5.2.3 Kalman Canonical Decomposition
206(3)
5.3 Controllability: Steering to a Target State
209(5)
5.4 Observability: Extracting Internal States from Output Measurements
214(2)
5.5 Minimal Realizations
216(7)
5.5.1 Controllability, Observability, and Minimality
216(2)
5.5.2 Minimal Realization from the Kalman Canonical Decomposition
218(3)
5.5.3 Alternative Tests for Controllability and Observability
221(1)
5.5.4 Stabilizability and Detectability
222(1)
5.6 Realizations via Lyapunov Equations
223(10)
5.6.1 A Lyapunov Equation
223(3)
5.6.2 Singular Value Decomposition
226(1)
5.6.3 Controllability and Observability Reductions via Lyapunov Equations
227(2)
5.6.4 Balanced Realizations
229(4)
5.7 Some Common Realizations
233(10)
5.7.1 Companion Form Realizations
233(7)
5.7.2 Gilbert's Realization
240(3)
5.8 Minimal Realizations from MFDs: Wolovich's Realization
243(12)
5.8.1 Matrix Fraction Description
244(1)
5.8.2 Matrix Divisors
245(1)
5.8.3 Unimodular Matrices
245(1)
5.8.4 Computation of a GCRD by Row Compression
245(2)
5.8.5 Coprimeness of MFDs
247(1)
5.8.6 Extraction of a GCRD (GCLD) from an MFD
247(1)
5.8.7 Column (Row) Properness
247(1)
5.8.8 Wolovich's Realization
248(7)
5.9 Transfer Function Conditions for Stabilizability and Detectability
255(2)
5.10 Exercises
257(4)
5.11 Notes and References
261(1)
6 State Feedback, Observers, and Regulators
262(47)
6.1 Introduction
262(1)
6.2 State Feedback versus Output Feedback
263(2)
6.3 Eigenvalue Assignment by State Feedback
265(8)
6.4 Pole Assignment via Sylvester's Equation
273(2)
6.5 Partial Pole Assignment by State Feedback
275(2)
6.6 Output Feedback: Dynamic Compensators
277(2)
6.7 State Observers
279(19)
6.7.1 Full-Order Observers
279(4)
6.7.2 Reduced-Order Observers
283(4)
6.7.3 Closed-Loop Eigenvalues under Observer-Based Feedback
287(6)
6.7.4 Dual Observers and Controllers
293(1)
6.7.5 Structure of Robust Observers
293(5)
6.8 Minimal or Maximal Realizations?
298(8)
6.8.1 Motivation
298(1)
6.8.2 Nominal and Parametrized Models
299(2)
6.8.3 Structurally Stable Stabilization
301(5)
6.9 Exercises
306(2)
6.10 Notes and References
308(1)
7 The Optimal Linear Quadratic Regulator
309(56)
7.1 An Optimal Control Problem
310(3)
7.1.1 Principle of Optimality
310(1)
7.1.2 Hamilton--Jacobi--Bellman Equation
311(2)
7.2 The Finite-Time LQR Problem
313(3)
7.2.1 Solution of the Matrix Ricatti Differential Equation
315(1)
7.2.2 Accommodating Cross-Product Terms
315(1)
7.3 The Infinite-Horizon LQR Problem
316(3)
7.3.1 General Conditions for Optimality
316(2)
7.3.2 The Infinite-Horizon LQR Problem
318(1)
7.4 Solution of the Algebraic Riccati Equation
319(6)
7.5 The LQR as an Output Zeroing Problem
325(2)
7.6 Return Difference Relations
327(1)
7.7 Guaranteed Stability Margins for the LQR
328(4)
7.7.1 Gain Margin
330(1)
7.7.2 Phase Margin
330(1)
7.7.3 Single-Input Case
331(1)
7.8 Eigenvalues of the Optimal Closed-Loop System
332(1)
7.8.1 Closed-Loop Spectrum
332(1)
7.9 A Summary of LQR Theory
333(1)
7.10 LQR Computations Using MATLAB
334(2)
7.11 A Design Example: Control of a Rotary Double-Inverted Pendulum Experiment
336(3)
7.12 Observer-Based Optimal State Feedback: Loss of Guaranteed Margins
339(4)
7.13 Optimal Dynamic Compensators
343(7)
7.14 Dual Compensators
350(4)
7.15 Quadratic Optimal Discrete-Time Control
354(5)
7.15.1 Finite-Horizon Optimization Problem
354(1)
7.15.2 Infinite-Horizon Optimal Control
355(2)
7.15.3 Solution of the Discrete-Time ARE
357(2)
7.16 Exercises
359(4)
7.17 Notes and References
363(2)
8 H∞ and H2 Optimal Control
365(44)
8.1 Norms for Signals and Systems
365(10)
8.1.1 Norms of Signals
367(1)
8.1.2 Norms for Linear Systems
367(8)
8.2 Control Problems in a Generalized Plant Framework
375(5)
8.3 State-Space Solution of the Multivariable H∞ Optimal Control Problem
380(13)
8.3.1 H∞ Solution
380(5)
8.3.2 Observer-Reconstructed State Feedback H∞ Optimal Control
385(8)
8.4 H2 Optimal Solution
393(7)
8.4.1 Full State Feedback H2 Optimal Control
393(4)
8.4.2 Observer-Reconstructed State Feedback H2 Optimal Control
397(3)
8.5 Casting Control Problems into a Generalized Plant Framework
400(5)
8.5.1 H∞ Optimal Controller
402(2)
8.5.2 H2 Controller
404(1)
8.6 Exercises
405(3)
8.7 Notes and References
408(1)
9 The Linear Multivariable Servomechanism
409(33)
9.1 Introduction to Servomechanisms
409(1)
9.2 Signal Generators and the Servocontroller
410(3)
9.2.1 Signal Generators
410(1)
9.2.2 The Servocontroller
410(3)
9.3 SISO Pole Placement Servomechanism Controller
413(7)
9.4 Multivariable Servomechanism: State-Space Formulation
420(1)
9.5 Reference and Disturbance Signal Classes
420(1)
9.6 Solution of the Servomechanism Problem
421(6)
9.7 Multivariable Servomechanisms: Transfer Function Analysis
427(3)
9.8 Examples
430(9)
9.9 Exercises
439(2)
9.10 Notes and References
441(1)
10 Robustness and Fragility of Control Systems
442(90)
10.1 Introduction
442(3)
10.1.1 Classical and Stability Margins
443(1)
10.1.2 Parametric Stability Margins
444(1)
10.2 Unstructured Perturbations
445(1)
10.2.1 Singular Value Decomposition
445(1)
10.3 Stability Margins Against Unstructured Perturbations
446(6)
10.3.1 Small Gain Theorem
446(3)
10.3.2 Additive Perturbations
449(1)
10.3.3 Multiplicative Perturbations
450(2)
10.3.4 Remarks
452(1)
10.4 The Stability Ball in Parameter Space
452(11)
10.4.1 The Image Set Approach: Zero Exclusion
454(2)
10.4.2 Stability Margin Computation in the Affine Case
456(2)
10.4.3 L2 Stability Margin
458(5)
10.4.4 Possible Discontinuity of the Stability Margin
463(1)
10.5 Fragility of High-Order Controllers
463(9)
10.6 Polytopic Uncertainty: Robust Stability of a Polytope
472(3)
10.7 The Segment Lemma
475(20)
10.7.1 Bounded Phase Condition
475(6)
10.7.2 Hurwitz Case
481(3)
10.7.3 Schur Segment Lemma via Tchebyshev Representation
484(2)
10.7.4 Some Fundamental Phase Relations
486(6)
10.7.5 Phase Relations for a Segment
492(3)
10.8 The Vertex Lemma
495(4)
10.9 Interval Polynomials: Kharitonov's Theorem
499(2)
10.10 Generalized Kharitonov Theorem
501(7)
10.10.1 Motivation
501(2)
10.10.2 Problem Formulation and Notation
503(3)
10.10.3 The Generalized Kharitonov Theorem
506(2)
10.11 Edge Theorem: Capturing the Root Space of a Polytope
508(10)
10.11.1 Edge Theorem
508(5)
10.11.2 Exposed Edges
513(1)
10.11.3 Examples
514(4)
10.12 The Mapping Theorem
518(6)
10.12.1 Multilinear Dependency
518(3)
10.12.2 Robust Stability via the Mapping Theorem
521(3)
10.13 Exercises
524(7)
10.14 Notes and References
531(1)
11 Analytical Design of PID Controllers
532(81)
11.1 Introduction
532(1)
11.2 Principle of Operation
532(1)
11.3 Computation of Stabilizing Sets for Continuous-Time Systems
533(8)
11.3.1 Signature Formulas
534(3)
11.3.2 Computation of the Stabilizing Set with PID Controllers
537(4)
11.4 Computation of PID Stabilizing Sets for Discrete-Time Plants
541(12)
11.4.1 Tchebyshev Representation and Root Clustering
542(4)
11.4.2 Root-Counting Formulas
546(3)
11.4.3 Stabilizing Sets of Discrete-Time PID Controllers
549(4)
11.5 Stabilizing Sets from Frequency Response Data
553(17)
11.5.1 Phase, Signature, Poles, Zeros, and Bode Plots
556(3)
11.5.2 PID Controller Synthesis for Continuous-Time Systems
559(8)
11.5.3 Data-Based Design versus Model-Based Design
567(3)
11.6 Design with Gain and Phase Margin Requirements
570(21)
11.6.1 Magnitude and Phase Loci
570(2)
11.6.2 PI and PID Controllers
572(2)
11.6.3 Design with Classical Performance Specifications
574(17)
11.7 Design with H∞ Norm Constraints
591(13)
11.7.1 H∞ Optimal Control and Stability Margins
592(3)
11.7.2 Computation of γ for PI Controllers
595(5)
11.7.3 Computation of γ for PID Controllers
600(4)
11.8 Exercises
604(7)
11.9 Notes and References
611(2)
12 Multivariable Control Using Single-Input Single-Output Methods
613(36)
12.1 A Scalar Equivalent of a MIMO System
613(10)
12.1.1 Illustrative Examples
616(3)
12.1.2 Stability Margin Calculations
619(3)
12.1.3 Discussion
622(1)
12.2 SLSO Design Methods for MIMO Systems
623(5)
12.3 Multivariable Servomechanism Design Using SISO Methods
628(4)
12.3.1 Nonsquare Plants
631(1)
12.4 Illustrative Examples
632(4)
12.5 MIMO Stability Margin Calculations
636(3)
12.6 Example: Multivariable PI Controller Design
639(6)
12.7 Conclusions and Summary
645(1)
12.8 Exercises
646(2)
12.9 Notes and References
648(1)
Epilogue
649(3)
Appendix
652(19)
A.1 Cramer's Rule
652(1)
A.2 Proof of Lemma 3.1
653(2)
A.3 Vector and Matrix Norms
655(1)
A.4 Proof of the Cayley-Hamilton Theorem (Theorem 4.1)
656(2)
A.5 Singular Value Decomposition
658(1)
A.6 Positive or Negative-Definite Matrices
658(2)
A.7 Orthogonal, Orthonormal, and Unitary Matrices
660(1)
A.8 Polynomial Matrices: Basic Theory
660(6)
A.8.1 Elementary Operations on Polynomial Matrices
660(1)
A.8.2 Equivalence of Polynomial Matrices
661(1)
A.8.3 Hermite Forms: Row (Column) Compression Algorithm
661(1)
A.8.4 Smith Form, Invariant Polynomials, and Elementary Divisors
662(4)
A.9 Smith--McMillan Form
666(1)
A.10 Proof of the Faddeev--LeVerrier Algorithm (Section 3.6)
667(4)
References 671(6)
Index 677
Shankar P. Bhattacharyya is a professor of Electrical Engineering at Texas A&M University. He is a Fellow of the IEEE and IFAC, and a Foreign Member of The Brazilian Academy of Sciences and The Brazilian National Academy of Engineering. His contributions to control theory include the first solution of the multivariable servomechanism problem, an algorithm for eigenvalue assignment, robustness and fragility, and a modern analytical approach to the design of PID controllers. Lee H. Keel is a professor of Electrical and Computer Engineering at Tennessee State University. His areas of expertise include linear systems, parametric robust control, computational methods, and control applications.
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