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E-grāmata: Linear Control Theory: Structure, Robustness, and Optimization

(Texas A&M University, College Station, USA), (Texas A&M University, College Station, USA), (Tennessee State University, Nashville, USA)
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Linear Control Theory: Structure, Robustness, and OptimizationPalielināt
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Successfully classroom-tested at the graduate level, Linear Control Theory: Structure, Robustness, and Optimization covers three major areas of control engineering (PID control, robust control, and optimal control). It provides balanced coverage of elegant mathematical theory and useful engineering-oriented results.

The first part of the book develops results relating to the design of PID and first-order controllers for continuous and discrete-time linear systems with possible delays. The second section deals with the robust stability and performance of systems under parametric and unstructured uncertainty. This section describes several elegant and sharp results, such as Kharitonovs theorem and its extensions, the edge theorem, and the mapping theorem. Focusing on the optimal control of linear systems, the third part discusses the standard theories of the linear quadratic regulator, Hinfinity and l1 optimal control, and associated results.

Written by recognized leaders in the field, this book explains how control theory can be applied to the design of real-world systems. It shows that the techniques of three term controllers, along with the results on robust and optimal control, are invaluable to developing and solving research problems in many areas of engineering.
Preface xvii
I THREE TERM CONTROLLERS
1(290)
Pid Controllers: An Overview of Classical Theory
3(22)
Introduction to Control
3(2)
The Magic of Integral Control
5(3)
PID Controllers
8(2)
Classical PID Controller Design
10(9)
The Ziegler-Nichols Step Response Method
10(1)
The Ziegler-Nichols Frequency Response Method
11(4)
PID Settings Using the Internal Model Controller Design Technique
15(2)
Dominant Pole Design: The Cohen-Coon Method
17(1)
New Tuning Approaches
17(2)
Integrator Windup
19(2)
Setpoint Limitation
20(1)
Back-Calculation and Tracking
20(1)
Conditional Integration
21(1)
Exercises
21(3)
Notes and References
24(1)
PID Controllers for Delay-Free LTI Systems
25(30)
Introduction
25(2)
Stabilizing Set
27(2)
Signature Formulas
29(4)
Computation of σ(p)
30(2)
Alternative Signature Expression
32(1)
Computation of the PID Stabilizing Set
33(5)
PID Design with Performance Requirements
38(14)
Signature Formulas for Complex Polynomials
40(2)
Complex PID Stabilization Algorithm
42(2)
PID Design with Guaranteed Gain and Phase Margins
44(1)
Synthesis of PID Controllers with an H∞ Criterion
44(5)
PID Controller Design for H∞ Robust Performance
49(3)
Exercises
52(2)
Notes and References
54(1)
PID Controllers for Systems With Time Delay
55(98)
Introduction
55(2)
Characteristic Equations for Delay Systems
57(3)
The Pade Approximation and Its Limitations
60(9)
First Order Fade Approximation
62(3)
Higher Order Pade Approximations
65(4)
The Hermite-Biehler Theorem for Quasi-polynomials
69(8)
Applications to Control Theory
71(6)
Stability of Systems with a Single Delay
77(8)
PID Stabilization of First Order Systems with Time Delay
85(31)
The PID Stabilization Problem
86(1)
Open-Loop Stable Plant
87(18)
Open-Loop Unstable Plant
105(11)
PID Stabilization of Arbitrary LTI Systems with a Single Time Delay
116(19)
Connection between Pontryagin's Theory and the Nyquist Criterion
117(4)
Problem Formulation and Solution Approach
121(2)
Proportional Controllers
123(3)
PI Controllers
126(2)
PID Controllers for an Arbitrary LTI Plant with Delay
128(7)
Proofs of Lemmas 3.3, 3.4, and 3.5
135(9)
Preliminary Results
135(4)
Proof of Lemma 3.3
139(2)
Proof of Lemma 3.4
141(1)
Proof of Lemma 3.5
141(3)
Proofs of Lemmas 3.7 and 3.9
144(4)
Proof of Lemma 3.7
144(1)
Proof of Lemma 3.9
145(3)
An Example of Computing the Stabilizing Set
148(2)
Exercises
150(1)
Notes and References
151(2)
Digital PID Controller Design
153(28)
Introduction
153(2)
Preliminaries
155(1)
Tchebyshev Representation and Root Clustering
156(4)
Tchebyshev Representation of Real Polynomials
156(2)
Interlacing Conditions for Root Clustering and Schur Stability
158(1)
Tchebyshev Representation of Rational Functions
159(1)
Root Counting Formulas
160(3)
Phase Unwrapping and Root Distribution
160(1)
Root Counting and Tchebyshev Representation
161(2)
Digital PI, PD, and PID Controllers
163(2)
Computation of the Stabilizing Set
165(5)
Constant Gain Stabilization
165(3)
Stabilization with PI Controllers
168(1)
Stabilization with PD Controllers
169(1)
Stabilization with PID Controllers
170(9)
Maximally Deadbeat Control
173(2)
Maximal Delay Tolerance Design
175(4)
Exercises
179(1)
Notes and References
179(2)
First Order Controllers for LTI Systems
181(26)
Root Invariant Regions
181(4)
An Example
185(4)
Robust Stabilization by First Order Controllers
189(1)
H∞ Design with First Order Controllers
190(5)
First Order Discrete-Time Controllers
195(10)
Computation of Root Distribution Invariant Regions
196(5)
Delay Tolerance
201(4)
Exercises
205(1)
Notes and References
206(1)
Controller Synthesis Free of Analytical Models
207(52)
Introduction
208(2)
Mathematical Preliminaries
210(5)
Phase, Signature, Poles, Zeros, and Bode Plots
215(3)
PID Synthesis for Delay Free Continuous-Time Systems
218(4)
PID Synthesis for Systems with Delay
222(2)
PID Synthesis for Performance
224(3)
An Illustrative Example: PID Synthesis
227(5)
Model Free Synthesis for First Order Controllers
232(5)
Model Free Synthesis of First Order Controllers for Performance
237(3)
Data Based Design vs. Model Based Design
240(3)
Data-Robust Design via Interval Linear Programming
243(8)
Data Robust PID Design
244(7)
Computer-Aided Design
251(4)
Exercises
255(2)
Notes and References
257(2)
Data Driven Synthesis of Three Term Digital Controllers
259(32)
Introduction
259(1)
Notation and Preliminaries
260(2)
PID Controllers for Discrete-Time Systems
262(8)
Data Based Design: Impulse Response Data
270(8)
Example: Stabilizing Set from Impulse Response
273(3)
Sets Satisfying Performance Requirements
276(2)
First Order Controllers for Discrete-Time Systems
278(4)
Computer-Aided Design
282(6)
Exercises
288(1)
Notes and References
289(2)
II ROBUST PARAMETRIC CONTROL
291(348)
Stability Theory for Polynomials
293(40)
Introduction
293(1)
The Boundary Crossing Theorem
294(8)
Zero Exclusion Principle
301(1)
The Hermite-Biehler Theorem
302(17)
Hurwitz Stability
302(8)
Hurwitz Stability for Complex Polynomials
310(2)
Schur Stability
312(7)
General Stability Regions
319(1)
Schur Stability Test
319(3)
Hurwitz Stability Test
322(6)
Root Counting and the Routh Table
326(1)
Complex Polynomials
327(1)
Exercises
328(4)
Notes and References
332(1)
Stability of A Line Segment
333(46)
Introduction
333(1)
Bounded Phase Conditions
334(7)
Segment Lemma
341(4)
Hurwitz Case
341(4)
Schur Segment Lemma via Tchebyshev Representation
345(3)
Some Fundamental Phase Relations
348(11)
Phase Properties of Hurwitz Polynomials
348(8)
Phase Relations for a Segment
356(3)
Convex Directions
359(10)
The Vertex Lemma
369(5)
Exercises
374(4)
Notes and References
378(1)
Stability Margin Computation
379(64)
Introduction
379(1)
The Parametric Stability Margin
380(4)
The Stability Ball in Parameter Space
380(2)
The Image Set Approach
382(2)
Stability Margin Computation
384(12)
l2 Stability Margin
387(4)
Discontinuity of the Stability Margin
391(1)
l2 Stability Margin for Time-Delay Systems
392(3)
l∞ and l1 Stability Margins
395(1)
The Mapping Theorem
396(9)
Robust Stability via the Mapping Theorem
399(3)
Refinement of the Convex Hull Approximation
402(3)
Stability Margins of Multilinear Interval Systems
405(11)
Examples
407(9)
Robust Stability of Interval Matrices
416(7)
Unity Rank Perturbation Structure
416(1)
Interval Matrix Stability via the Mapping Theorem
417(1)
Numerical Examples
418(5)
Robustness Using a Lyapunov Approach
423(9)
Robustification Procedure
427(5)
Exercises
432(9)
Notes and References
441(2)
Stability of A Polytope
443(106)
Introduction
443(1)
Stability of Polytopic Families
444(11)
Exposed Edges and Vertices
445(3)
Bounded Phase Conditions for Checking Robust Stability of Polytopes
448(4)
Extremal Properties of Edges and Vertices
452(3)
The Edge Theorem
455(15)
Edge Theorem
456(8)
Examples
464(6)
Stability of Interval Polynomials
470(28)
Kharitonov's Theorem for Real Polynomials
470(8)
Kharitonov's Theorem for Complex Polynomials
478(3)
Interlacing and Image Set
481(3)
Image Set Based Proof of Kharitonov's Theorem
484(1)
Image Set Edge Generators and Exposed Edges
485(2)
Extremal Properties of the Kharitonov Polynomials
487(5)
Robust State Feedback Stabilization
492(6)
Stability of Interval Systems
498(24)
Problem Formulation and Notation
500(4)
The Generalized Kharitonov Theorem
504(10)
Comparison with the Edge Theorem
514(1)
Examples
515(6)
Image Set Interpretation
521(1)
Polynomic Interval Families
522(14)
Robust Positivity
524(4)
Robust Stability
528(4)
Application to Controller Synthesis
532(4)
Exercises
536(10)
Notes and References
546(3)
Robust Control Design
549(90)
Introduction
549(2)
Interval Control Systems
551(2)
Frequency Domain Properties
553(10)
Nyquist, Bode, and Nichols Envelopes
563(10)
Extremal Stability Margins
573(4)
Guaranteed Gain and Phase Margins
574(1)
Worst Case Parametric Stability Margin
574(3)
Robust Parametric Classical Design
577(14)
Guaranteed Classical Design
577(11)
Optimal Controller Parameter Selection
588(3)
Robustness Under Mixed Perturbations
591(11)
Small Gain Theorem
592(1)
Small Gain Theorem for Interval Systems
593(9)
Robust Small Gain Theorem
602(4)
Robust Performance
606(3)
The Absolute Stability Problem
609(6)
Characterization of the SPR Property
615(10)
SPR Conditions for Interval Systems
617(8)
The Robust Absolute Stability Problem
625(7)
Exercises
632(5)
Notes and References
637(2)
III OPTIMAL AND ROBUST CONTROL
639(168)
The Linear Quadratic Regulator
641(48)
An Optimal Control Problem
641(4)
Principle of Optimality
642(1)
Hamilton-Jacobi-Bellman Equation
642(3)
The Finite-Time LQR Problem
645(3)
Solution of the Matrix Ricatti Differential Equation
647(1)
Cross Product Terms
647(1)
The Infinite Horizon LQR Problem
648(3)
General Conditions for Optimality
648(2)
The Infinite Horizon LQR Problem
650(1)
Solution of the Algebraic Riccati Equation
651(7)
The LQR as an Output Zeroing Problem
658(2)
Return Difference Relations
660(1)
Guaranteed Stability Margins for the LQR
661(4)
Gain Margin
663(1)
Phase Margin
663(1)
Single Input Case
664(1)
Eigenvalues of the Optimal Closed Loop System
665(2)
Closed-Loop Spectrum
665(2)
Optimal Dynamic Compensators
667(5)
Dual Compensators
670(2)
Servomechanisms and Regulators
672(8)
Notation and Problem Formulation
672(1)
Reference and Disturbance Signal Classes
673(1)
Solution of the Servomechanism Problem
673(7)
Exercises
680(7)
Notes and References
687(2)
SISO H∞ and l1 Optimal Control
689(70)
Introduction
689(4)
The Small Gain Theorem
693(11)
L-Stability and Robustness via the Small Gain Theorem
704(5)
YJBK Parametrization of All Stabilizing Compensators (Scalar Case)
709(7)
Control Problems in the H∞ Framework
716(9)
H∞ Optimal Control: SISO Case
725(22)
Dual Spaces
729(4)
Inner Product Spaces
733(3)
Orthogonality and Alignment in Noninner Product Spaces
736(1)
The All-Pass Property of H∞ Optimal Controllers
737(5)
The Single-Input Single-Output Solution
742(5)
l1 Optimal Control: SISO Case
747(8)
Exercises
755(2)
Notes and References
757(2)
H∞ Optimal Multivariable Control
759(48)
H∞ Optimal Control Using Hankel Theory
759(25)
H∞ and Hankel Operators
759(4)
State Space Computations of the Hankel Norm
763(6)
State Space Computation of an All-Pass Extension
769(2)
H∞ Optimal Control Based on the YJBK Parametrization and Hankel Approximation Theory
771(5)
LQ Return Difference Equality
776(4)
State Space Formulas for Coprime Factorizations
780(4)
The State Space Solution of H∞ Optimal Control
784(20)
The H∞ Solution
784(11)
The H2 Solution
795(9)
Exercises
804(2)
Notes and References
806(1)
SIGNAL SPACES
807(26)
Vector Spaces and Norms
807(12)
Metric Spaces
819(6)
Equivalent Norms and Convergence
825(3)
Relations between Normed Spaces
828(4)
Notes and References
832(1)
NORMS FOR LINEAR SYSTEMS
833(34)
Induced Norms for Linear Maps
833(11)
Properties of Fourier and Laplace Transforms
844(5)
Fourier Transforms
846(2)
Laplace Transforms
848(1)
Lp/lp Norms of Convolutions of Signals
849(3)
L1 Theory
849(1)
Lp Theory
850(2)
Induced Norms of Convolution Maps
852(13)
Notes and References
865(2)
EPILOGUE
867(2)
ROBUSTNESS AND FRAGILITY
869(18)
Feedback, Robustness, and Fragility
869(2)
Examples
871(12)
Discussion
883(2)
Notes and References
885(2)
References 887(18)
Index 905
Shankar P. Bhattacharyya, Aniruddha Datta, Lee H. Keel
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