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

by Sidi, Marcel J.

• ISBN13: 9781575241432
• ISBN10: 1575241439
• Format: Hardcover
• Copyright: 2000-12-01
• Publisher: Krieger Pub Co

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