Design of Robust Control Systems : From Classical to Modern Practical Approaches,9781575241432

Design of Robust Control Systems : From Classical to Modern Practical Approaches

by
ISBN13: 9781575241432
ISBN10: 1575241439
Format: Hardcover
Pub. Date: 2000-12-01
Publisher(s): Krieger Pub Co
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Table of Contents

Introduction
1(13)
The Structure and Components of a Feedback Controlled System
2(1)
The components of a feedback controlled system
2(1)
The signals in a feedback controlled system
3(1)
Why Feedback?
3(2)
Items to be Considered by Designers of Feedback Controlled Systems
5(2)
Design specifications
5(1)
Uncertainty of the plant parameters
5(1)
Saturation of the plant
6(1)
Sensor noise amplification
6(1)
Book Contents
7(6)
Basic Properties and Design of SISO Feedback Systems
13(54)
Introductory Definitions
13(3)
Mathematical Definition of Feedback
16(6)
The Origin of Feedback Theory
16(1)
Return Ratio, Return Difference and the Invariance of its Numerator Polynomial
17(4)
Asymptotic Stability and Internal Stability
21(1)
Asymptotic stability
21(1)
Internal stability
21(1)
Definition of Some Feedback Control Issues Using the Basic Feedback Equation
22(18)
Asymptotic and Relative Stability Considerations in the s-Plane and in the ω-Domain
23(1)
Nyquist stability criterion
23(1)
Relative stability (gain and phase margins)
24(1)
Design in the Frequency-Domain Using Bode Techniques
24(1)
Logarithmic magnitude
24(1)
Decibel
25(1)
Bode plots
25(1)
The Nichols Chart and its Special Characteristics
26(1)
Unity feedback constant gain contours on the Nichols chart
26(2)
Location of the stability critical point and identification of phase and gain margins
28(1)
Inverted Nichols chart for disturbance analysis and design
28(1)
Superiority of design based on Bode plots and on the Nichols chart over design based on the Nyquist plot in the complex plane
28(1)
Distinction between One and Two DOF Feedback Systems
29(2)
Frequency-Domain Design of ODOF Feedback Systems
31(1)
Crossover frequency ωCO, and GM frequency ωGM
31(1)
Conditional stability
31(1)
Closed-loop performance and bandwidth, ω-3dB
31(1)
Disturbance rejection
31(1)
Loopshaping
32(8)
Limitations of one-degree-of-freedom feedback systems
40(1)
Design in the Frequency-Domain of TDOF Systems
40(1)
Design Tradeoffs in Feedback systems
40(7)
The Sensor Noise Amplification Problem in Feedback Systems
41(2)
RMS Computation of |Tun(jω)|
43(1)
Optimization of the Loop Transmission Function L(jω)
44(3)
Loopshaping based on H∞-Norm Optimization
47(17)
Definition of the H∞ and H2 Norms
48(1)
Interpretation of H2 and H∞ norms in real physical systems
49(1)
Basic Relative Stability Performance Requirements
50(1)
Gain and phase margins in terms of |Tl(jω)|max
50(1)
Gain and phase margins in terms of |S(jω)|max
51(1)
Weighted Sensitivity
52(2)
Mixed Sensitivity
54(2)
The Standard H∞-Regulator Problem
56(3)
Weighting Function Selection
59(1)
H∞-Norm Solution of the Mixed Sensitivity Problem
59(5)
Classical versus Modern H∞-Norm Loopshaping
64(2)
Summary
66(1)
Practical Topics in the Design of SISO Feedback Systems
67(82)
Introduction
67(1)
Frequency and s-Domain to Time-Domain Translations
68(20)
s-Domain to Time-Domain Translation
68(1)
The domainant pole approach
68(3)
Some definitions of a step response
71(3)
Frequency-Domain to Time-Domain Translation: R(ω) to y(t) Translation
74(3)
Evaluation of the impulse time response from a real trapezoid standard part
77(1)
Evaluation of the unit-step time response from a real trapezoid standard part
78(3)
Overshoot of a step time response evaluated from frequency response characteristics
81(1)
Frequency-Domain to Time-Domain Translation: |T(jω)| into y(t) Translation
81(1)
Time-delay due to poles located at high frequencies
82(1)
Rise time tr
83(5)
Input-Output Time-Domain Characteristics of NMP SISO Feedback Systems
88(4)
Optimal L(jω) for Minimum-Phase Feedback Systems
92(10)
Formulation of the Problem
92(1)
The Ideal Bode Characteristic and its Derivation
93(2)
Computation of ω1
95(1)
Computation of ω2 (first approach)
96(1)
Computation of ω2 (second approach)
96(2)
Optimization of Lω for Conditionally Stable Systems
98(2)
Benefits of feedback and the number of integrators at s = 0
100(2)
Optimization of the loop-transmission function
102(1)
Limitations on L(jω) including RHP poles or zeros
102(12)
Limitations on the Sensitivity Function S(jω) for Feedback Systems with RHP Poles
104(2)
Bandwidth Limitations Due to RHP zeros, Maximum Achievable ωco
106(1)
Formulation of the problem
107(1)
Computation of relationships between gain margin, phase margin and ωco
107(4)
Equivalent NMP Zero Approximation to Multiple NMP Zeros
111(3)
Limitations on Unstable L(jω)-Minimal Achievable ωco
114(7)
Introduction
114(1)
Functional Relations
114(4)
Application of the Results in Practical Problems
118(1)
Equivalent RHP Pole Approximation to Several RHP Poles
119(2)
Stabilization
121(6)
Introduction
121(1)
Strongly Stabilizable Plants
121(1)
Parametrization of the Stabilizing Controller: Stable Plant
122(3)
Parametrization of the Stabilizing Controller: Unstable Plant
125(1)
Coprime factorization
125(1)
Parametrization of the stabilizing controller
126(1)
A Design Procedure for Maximizing Gain and Phase Margins of a Class of Unstable-NMP plants
127(5)
Formulation of the Problem
127(2)
Basic Stages in the Proposed Design Procedure
129(3)
Limitations on Feedback Systems Including an Unstable-NMP Plant
132(14)
Reformulation of the Design Problem
133(1)
Plant with an Unstable Pole Preceding a NMP Zero
133(7)
Plant with a NMP Zero Preceding an Unstable Pole
140(6)
Summary
146(3)
Synthesis of SISO LTI Uncertain Feedback Systems
149(100)
Introduction
149(1)
Statement of the Uncertain Plant Feedback Problem
150(3)
Tracking specifications
152(1)
Disturbance rejection specifications
152(1)
Peaking of the disturbance rejection transfer functions gains
152(1)
A Design Technique for Minimum-phase LTI Uncertain Plants
153(1)
Step 1: Translation of Time-Domain into Frequency-Domain Tolerances
153(22)
Direct time-domain into frequency-domain translation
154(1)
Time-domain into frequency-domain translation via s-domain
154(1)
Step 2: Preparation of Templates for the Uncertain Plant
155(2)
Step 3: Derivation of Bounds on the Loop Transmission Ln(jω) in the Nichols Chart
157(1)
Bounds on Ln(jω) for satisfying (input-output) tracking sensitivity specifications
158(1)
Bounds on Ln(jω) for satisfying disturbance rejection specifications
159(2)
Bounds on Ln(jω) for satisfying maximum peaking in |Tl(jω)ω
161(1)
Combined bounds for tracking and disturbance rejection specifications
162(1)
Step 4: Design of L(s) to Satisfy the Specification Bounds
163(1)
Definition and properties of optimal L(jω)
164(1)
Step 5: Derivation of the Control Network G(s)
165(1)
Step 6: Design of the Prefilter F(s) to Achieve |T(jω)| Specifications
165(2)
Step 7: Evaluation of the Design
167(1)
Main Example
168(7)
Design Technique for Unstable Uncertain Plants
175(13)
Design Technique for NMP Uncertain Plants
188(12)
Bounds on L(jω) in the Nichols chart for NMP systems
189(11)
Design Technique for Plants with Pure Time Delays
200(1)
Design Technique for Sampled-Data Feedback Systems with LTI Uncertain Plants
201(12)
Design of Sampled-Data Feedback Systems in the Frequency-Domain
201(3)
Time into Frequency-Domain Translation for Sampled-Data Systems
204(2)
Synthesis Technique for Sampled-Data Feedback Systems with Uncertain Plants
206(7)
Design of Continuous LTI Uncertain Feedback Systems by H∞-Norm Optimization
213(34)
Introduction
213(1)
Robust stability
214(2)
Plant Uncertainty Modeling
216(1)
Multiplicative uncertainty model
217(3)
Inverse multiplicative uncertainty model
220(1)
Additive uncertainty model
221(1)
Inverse additive uncertainty model
221(2)
Robust Stability for Different Kinds of Uncertainty Plant Models
223(4)
Robust Performance
227(1)
Algebraic derivation
228(1)
Graphical derivation
229(1)
Algebraic design constraints
230(1)
Design for robust performance specifications
231(3)
H∞-Norm Optimization Used in Design of TDOF Feedback System Structures
234(12)
Comparison between ``Classical'' and ``H∞'' Control Designs
246(1)
Summary
247(2)
Single-Input/Multi-Output Uncertain Feedback Systems
249(24)
Introduction
249(1)
Statement of the Problem
249(22)
Design Philosophy
250(2)
Bounds on L2
252(2)
Derivation of bounds on L2 in the frequency range R1, ω > ωa
254(2)
Universal bounds on L2(jω) in the frequency range R1, ω > ωa
256(1)
Bounds on L2 in the frequency range R2, ωa > ω > ωh
256(3)
Bounds on L2 in the frequency range R3, ω < ωh
259(2)
The Sensor Noise Amplification Problem
261(6)
Control network derivation
267(4)
Extension to Three Loops
271(1)
Summary
272(1)
MIMO Robust Feedback Systems Solved with Nyquist/Bode-Based Design Techniques
273(52)
Introduction
273(2)
Introductory Definitions for MIMO Feedback Systems
274(1)
Design of MIMO Feedback Systems with Certain Plants
275(26)
Complexity of the n x n Feedback System Problem
275(1)
Design by Direct Diagonalization of the Open-Loop Transfer Matrix
276(4)
Noninteraction by Inverse-Based Controller
280(2)
Sequential Loop Closing Design Technique
282(4)
The Generalized Nyquist Stability Theorem
286(1)
Characteristic transfer functions and corresponding eigenvector functions
286(2)
Characteristic loci
288(1)
Adaptation of the Nyquist stability criterion to MIMO systems
288(4)
The Characteristic Locus Design Method
292(3)
Internal Stability in MIMO Feedback Systems
295(1)
Well-possedness of feedback loops
295(1)
Internal stability
296(2)
Q-Parametrization
298(1)
Q-parametrization: stable plants
299(1)
Q-parametrization: unstable plants
299(2)
Uncertain MIMO Feedback Systems, Classical Approach-Statement of the Problem
301(22)
Complexity of the n x n Feedback System Problem
301(1)
Derivation of a Synthesis Technique Based on QFT
302(3)
The 2 x 2 Uncertain Feedback System Design Problem
305(1)
Modification of the tolerances
306(1)
Constraints at high frequencies
307(15)
The general interacting case
322(1)
The 3 x 3 Feedback System Design Problem
323(1)
Summary
323(2)
Introductory to Design Techniques in the State-Space Framework
325(40)
Introduction
325(1)
MIMO Feedback Control in the State-Space Framework
326(1)
The Linear-Quadratic-Regulator (LQR) Problem
327(25)
The General Case
329(1)
Solution of the Riccati equation
329(4)
The Steady-State LQR Problem
333(1)
Controllability
334(1)
Stabilizability
334(1)
Observability
334(1)
Detectability
334(1)
Selection of the Q, R and M Matrices
335(3)
Model Following LQR Oriented Design Techniques
338(1)
Implicit model-following
339(5)
Explicit model-following
344(5)
Frequency-Domain Characterization of Optimality in the LQR Oriented Design
349(1)
Kalman inequality
349(3)
Gain and Phase Margins for Optimal LQR Designed Feedback Systems
352(1)
The Linear-Quadratic-Gaussian (LQG) Problem
352(11)
The State Estimator Problem
353(1)
Kalman-Bucy Filter
354(2)
The Separation Principle and the Solution of the LQG Problem
356(3)
Loop Transfer Recovery (LTR)
359(4)
Summary
363(2)
MIMO Robust Feedback Systems Solved with H∞-Norm Optimization Technique
365(34)
Introduction
365(1)
Singular Values and their Use in MIMO Feedback Systems
365(7)
Singular Values as a Means to Express Transfer Function Matrix Size
366(1)
Singular values
367(1)
Principal gains
367(1)
Singular value decomposition of a matrix
367(3)
Singular Values as a Means to Define Performance in MIMO Feedback Systems
370(2)
Solution of the Standard H∞-Regulator Problem
372(7)
Introduction
372(1)
Definition of the Generalized plant
372(2)
State-space realization of the generalized plant
374(1)
Algorithms for the Solution of the Standard H∞-Control Problem
375(1)
General control problem formulation
376(1)
Assumptions on M(s) for feasibility of the optimal controller
376(1)
Interpretation of constraining assumptions on M(s)
377(1)
Solution of an H∞-optimization algorithm
377(1)
γ-iteration
378(1)
Loopshaping of MIMO Feedback Control Systems with Fixed and Known Plants
378(1)
Uncertainty, Robust Stability and Performance in MIMO Feedback Systems
379(7)
Introduction
379(1)
Uncertainty Modeling of MIMO plants
379(1)
Multiplicative perturbation modeling
379(1)
Output multiplicative uncertainty modeling
380(1)
Input multiplicative uncertainty modeling
381(1)
Inverse multiplicative uncertainty modeling
381(1)
Additive perturbation modeling
381(1)
Stability Considerations and the Small Gain Theorem
382(1)
Small gain theorem
383(1)
Robust Stability for Uncertain MIMO Feedback Systems
383(1)
Conditions for robust stability with output multiplicative uncertainty modeling
384(1)
Robust Performance of MIMO Uncertain Feedback Systems
385(1)
Design of TDOF Uncertain Feedback Systems
386(11)
Introduction
386(1)
Design Procedure for the TDOF MIMO Uncertain Feedback System
387(10)
Summary and conclusions
397(1)
Summary
397(2)
Appendix A. Signal Flow Graphs 399(4)
A.1 Introduction
399(1)
A.2 Definitions for Signal Flow Graphs
400(1)
A.3 Gain Formula for Signal Flow Graphs
401(2)
Appendix B. Mathematical Background Related to MIMO and to H∞ Analysis and Design 403(14)
B.1 Introduction
403(1)
B.1 Algebraic and Vector Norms
403(3)
B.2.1 Time Domain Scalar Functions
404(1)
B.2.2 Scalar System Functions
404(1)
B.2.3 Vector Norms
405(1)
B.3 Matrix Norms
406(3)
B.3.1 Singular Values and Principal Gains of Transfer Function Matrices
406(1)
Singular-value decomposition of a matrix
407(1)
Definition of matrix norms
407(1)
B.3.2 Singular Value Inequalities
408(1)
B.4 State-State Formulation of Linear Systems
409(4)
B.4.1 State-Space Formulation
409(1)
B.4.2 State-Space Relization and Minimal Realization
410(1)
B.4.3 Basic Properties and Concepts from State-Space Theory
410(1)
Controllability
411(1)
Observability
411(1)
Asymptotical stabilization by state-feedback
412(1)
Hidden modes
412(1)
B.4.4 Characteristic Polynomials in Feedback Control Systems
412(1)
B.5 Eigenvalues and Eigenvectors of Transfer Functions Matrices
413(2)
B.6 Linear Fractional Transformation (LFT)
415(2)
Appendix C. Control Networks for Loop Shaping 417(20)
C.1 Introduction
417(1)
C.2 Frequency Response of a Real Pole (zero)
418(1)
C.3 Frequency Response of a Complex Pole (zero)
418(5)
C.4 Frequency Response of a Lead-Lag Control Network
423(4)
C.5 Frequency Response of a Lead-Lag-Lag-Lead Control Network
427(6)
C.6 The Nichols chart
433(1)
C.7 The Inverted Nichols chart
434(3)
Appendix D. Facts About Fourier and Laplace Transforms 437(8)
D.1 Introduction
437(1)
D.2 Definitions and some General Properties of Fourier Transforms
437(2)
Real time functions
438(1)
Causal time functions
439(1)
D.3 Definitions and some General Properties of Laplace Transforms
439(2)
D.4 Solution of Linear Differential Equations
441(2)
Differentiation theorem
441(1)
Integration theorem
441(2)
D.5 Initial and Final Value Theorems
443(1)
D.6 Time Convolution Theorem
443(1)
D.7 Frequency Convolution Theorem
443(1)
D.8 Parceval's Formula
444(1)
Appendix E. Bode Formulas and Transform Relations 445(14)
E.1 Introduction
445(1)
E.2 Preliminary Definitions and Facts
445(2)
E.3 Some Restrictions on Physical Transmission Function at Real Frequencies
447(4)
E.3.1 Cauchy's Integral Theorems
447(1)
E.3.2 Integral of the Sensitivity Function
448(2)
E.3.3 Phase Integral
450(1)
E.4 Formulate Relating The Real and The Imaginary Parts of Transmission Functions
451(5)
E.4.1 Phase Characteristics Corresponding to a Prescribed Attenuation Characteristic
451(2)
Phase characteristics corresponding to a constant slope attenuation characteristic
453(1)
The semi-infinite constant slope characteristics
453(1)
E.4.2 Attenuation Characteristics Corresponding to a Prescribed Phase Characteristic
454(1)
Abrupt step change in phase characteristic
455(1)
Finite line phase segment
455(1)
E.5 A and B Prescribed in Different Frequency Ranges
456(3)
Appendix F. Order of Compensator for Pole-Placement 459(2)
Appendix G. Steady-State Error Coefficients 461(6)
G.1 Introduction
461(1)
G.2 The Steady State Error Equation
461(6)
Step input
463(1)
Velocity input
463(1)
Acceleration input
464(3)
References 467(8)
Index 475

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