Stable Adaptive Systems

by Narendra, Kumpati S.; Annaswamy, Anuradha M.

• ISBN13: 9780486442266
• ISBN10: 0486442268
• Format: Paperback
• Copyright: 2005-05-31
• Publisher: Dover Publications

Summary

This graduate-level text focuses on the stability of adaptive systems, and offers a thorough understanding of the global stability properties essential to designing adaptive systems. Its self-contained, unified presentation of well-known results establishes the close connections between seemingly independent developments in the field. Prerequisites include a knowledge of linear algebra and differential equations, as well as a familiarity with basic concepts in linear systems theory.

Table of Contents

Preface	xv
1 Introduction	1	(41)
1.1 Introduction	1	(1)
1.2 Systems Theory	2	(3)
1.3 Adaptive Systems	5	(8)
1.3.1 What Is an Adaptive System?,	9	(3)
1.3.2 Parameter Adaptive and Structurally Adaptive Systems,	12	(1)
1.4 Direct and Indirect Control	13	(8)
1.4.1 Parameter Perturbation Method,	14	(4)
1.4.2 Sensitivity Method,	18	(3)
1.5 Model Reference Adaptive Systems and Self-Tuning Regulators	21	(7)
1.5.1 Identification Model,	22	(1)
1.5.2 Reference Model,	22	(2)
1.5.3 Reference Model with Inputs,	24	(1)
1.5.4 Model Reference Adaptive Control,	25	(3)
1.5.5 Self-Tuning Regulators,	28	(1)
1.6 Stable Adaptive Systems	28	(3)
1.6.1 Error Models,	30	(1)
1.7 Applications	31	(1)
1.8 Summary	32	(2)
1.9 Scope of the Book	34	(1)
Problems	35	(4)
References	39	(3)
2 Stability Theory	42	(57)
2.1 Introduction	42	(2)
2.2 Notation	44	(1)
2.3 Linear Systems	45	(5)
2.3.1 Linear Time-Invariant Systems,	46	(1)
2.3.2 Almost Time-Invariant Systems,	47	(3)
2.4 Lyapunov Stability	50	(13)
2.4.1 Definitions,	50	(3)
2.4.2 Lyapunov's Direct Method,	53	(7)
2.4.3 V(x, t) > 0, V(x, t) less than or equal to 0, 56
2.4.4 LTI Systems and Lyapunov Stability,	60	(1)
2.4.5 Linear Time-Varying Systems,	61	(2)
2.5 Positive Real Functions	63	(2)
2.5.1 Positive Real Transfer Matrices,	65	(1)
2.6 Kalman-Yakubovich Lemma	65	(2)
2.7 Input-Output Stability	67	(4)
2.8 Stability of Adaptive Systems	71	(17)
2.8.1 Two Differential Equations That Arise in Adaptive Systems,	71	(6)
2.8.2 Growth Rates of Signals in Dynamical Systems,	77	(9)
2.8.3 Stability in Terms of a Parameter Vector theta,	86	(2)
2.9 Other Stability Concepts	88	(6)
2.9.1 Total Stability,	89	(3)
2.9.2 BIBO Stability,	92	(2)
2.10 Summary	94	(1)
Problems	94	(2)
References	96	(3)
3 Simple Adaptive Systems	99	(41)
3.1 Introduction	99	(1)
3.2 Algebraic Systems	100	(2)
3.2.1 Identification,	100	(2)
3.3 Dynamical Systems	102	(24)
3.3.1 The Identification Problem: Scalar Case,	102	(7)
3.3.2 The Control Problem: Scalar Case (Direct Control),	109	(8)
3.3.3 The Control Problem: Scalar Case (Indirect Control),	117	(4)
3.3.4 Indirect Control: Dynamic Adjustment of Control Parameters,	121	(2)
3.3.5 Examples,	123	(3)
3.4 State Variables Accessible	126	(10)
3.4.1 The Identification Problem: Vector Case,	126	(2)
3.4.2 The Control Problem,	128	(8)
3.5 Summary	136	(1)
Problems	136	(3)
References	139	(1)
4 Adaptive Observers	140	(41)
4.1 Introduction	140	(1)
4.2 The Luenberger Observer	141	(3)
4.3 Adaptive Observers	144	(18)
4.3.1 Statement of the Problem,	144	(1)
4.3.2 Minimal Realization of Adaptive Observers,	145	(6)
4.3.3 Nonminimal Realization of Adaptive Observers,	151	(7)
4.3.4 Modified Adaptive Observer,	158	(4)
4.4 Design of Adaptive Observers	162	(15)
4.4.1 Representation 1,	163	(2)
4.4.2 Representation 2,	165	(1)
4.4.3 Hybrid Algorithms,	166	(3)
4.4.4 Time-Varying Adaptive Gains,	169	(1)
4.4.5 Integral Algorithm,	170	(1)
4.4.6 Multiple Models,	171	(2)
4.4.7 Adaptive Algorithms: An Overview,	173	(4)
4.5 Summary	177	(1)
Problems	177	(2)
References	179	(2)
5 The Control Problem	181	(57)
5.1 Introduction	181	(1)
5.2 Statement of the Problem	182	(2)
5.3 Adaptive Control of Plants with Relative Degree Unity	184	(15)
5.3.1 Case (i) kp, Unknown,	186	(1)
5.3.2 Case (ii) Unknown Zeros,	187	(3)
5.3.3 Case (iii) Unknown Poles,	190	(2)
5.3.4 General Case (n* = 1),	192	(7)
5.4 Adaptive Control of Plants with Relative Degree greater than 2	199	(13)
5.4.1 Bezout Identity,	199	(3)
5.4.2 n* > or equal to 2: Algebraic Part,	202	(1)
5.4.3 n* = 2: Analytic Part,	203	(1)
5.4.4 n* > or equal to 2: Analytic Part,	204	(2)
5.4.5 kp Known,	206	(3)
5.4.6 kp Unknown,	209	(3)
5.5 Proof of Global Stability of the Adaptive System	212	(10)
5.5.1 kp Known,	213	(5)
5.5.2 kp Unknown,	218	(4)
5.6 Comments on the Control Problem	222	(2)
5.7 Some Remarks on the Controller Structure	224	(2)
5.8 Stable Adaptive Control Using the Indirect Approach	226	(5)
5.8.1 n* = 1,	227	(3)
5.8.2 n* > or equal to 2,	230	(1)
5.9 Combined Direct and Indirect Approach	231	(2)
5.10 Summary	233	(1)
Problems	234	(2)
References	236	(2)
6 Persistent Excitation	238	(35)
6.1 Introduction	238	(1)
6.2 Persistent Excitation in Adaptive Systems	239	(7)
6.3 Definitions	246	(3)
6.3.1 Examples,	248	(1)
6.4 Properties of Persistently Exciting Functions	249	(9)
6.4.1 Algebraic Transformations,	249	(2)
6.4.2 Dynamic Transformations,	251	(5)
6.4.3 Almost Time-Invariant Systems,	256	(1)
6.4.4 Persistent Excitation of Unbounded Signals,	257	(1)
6.5 Applications to Adaptive Systems	258	(8)
6.5.1 Parameter Convergence in Adaptive Observers,	258	(2)
6.5.2 Parameter Convergence in Adaptive Control,	260	(3)
6.5.3 Design Considerations,	263	(3)
6.6 Persistent Excitation and Uniform Asymptotic Stability	266	(2)
6.7 Summary	268	(1)
Problems	269	(2)
References	271	(2)
7 Error Models	273	(18)
7.1 Introduction	273	(1)
7.2 Error Model 1	274	(4)
7.3 Error Model 2	278	(1)
7.4 Error Model 3	279	(3)
7.5 Error Model 4	282	(4)
7.6 Further Comments on Error Models 1-4	286	(1)
7.7 Use of Error Models in Adaptive Systems	287	(1)
Problems	288	(2)
References	290	(1)
8 Robust Adaptive Control	291	(68)
8.1 Introduction	291	(2)
8.2 Adaptive Observers in the Presence of Disturbances	293	(4)
8.2.1 Identifier,	293	(1)
8.2.2 Adaptive Observers,	294	(1)
8.2.3 Error Model,	295	(2)
8.3 Adaptive Control of a First-Order Plant in the Presence of Bounded Disturbances	297	(21)
8.3.1 Exact Cancellation of Disturbance,	297	(4)
8.3.2 Bounded Disturbances: Statement of the Problem,	301	(5)
8.3.3 Category 1: Modifications of the Adaptive Law,	306	(6)
8.3.4 Category 2: Persistent Excitation of the Reference Input,	312	(4)
8.3.5 Performance of the Adaptive System,	316	(2)
8.4 Adaptive Control of an nth Order Plant	318	(13)
8.4.1 Robustness without Persistent Excitation;	322	(5)
8.4.2 Robustness with Persistent Excitation,	327	(3)
8.4.3 Comments on Adaptive Algorithms for Bounded Disturbances,	330	(1)
8.5 Hybrid Adaptive Control	331	(6)
8.5.1 Hybrid Error Models,	331	(3)
8.5.2 Hybrid Adaptive Control,	334	(2)
8.5.3 Hybrid Adaptive Control in the Presence of Bounded Disturbances,	336	(1)
8.6 Adaptive Control of Time-Varying Plants	337	(8)
8.6.1 Adaptive Control of a First-Order Plant,	338	(4)
8.6.2 Time-Varying Plants of Higher Order,	342	(3)
8.7 Reduced-Order Controllers	345	(8)
8.7.1 Instability with Reduced Order Controllers: An Example,	346	(1)
8.7.2 Statement of the Problem,	347	(1)
8.7.3 Small Parameter μ in P,	348	(2)
8.7.4 Transfer Function Wm(s),	350	(3)
8.8 Summary	353	(1)
Problems	354	(2)
References	356	(3)
9 The Control Problem: Relaxation of Assumptions	359	(33)
9.1 Introduction	359	(1)
9.2 Assumption (i): Sign of kp	360	(6)
9.3 Assumption (iii): Upper Bound on the Order of the Plant	366	(8)
9.3.1 Regulation of Plant with Relative Degree n* = 1,	366	(2)
9.3.2 n* = 2,	368	(1)
9.3.3 Arbitrary n*,	369	(4)
9.3.4 Universal Controller,	373	(1)
9.4 Assumption (iv): Zeros of Wp(s) in C-	374	(9)
9.4.1 Direct Control,	375	(3)
9.4.2 Indirect Control,	378	(5)
9.5 Command Generator Tracking Method	383	(4)
9.5.1 Statement of the Problem,	383	(4)
9.6 Summary	387	(1)
Problems	388	(1)
References	389	(3)
10 Multivariable Adaptive Systems	392	(28)
10.1 Introduction	392	(1)
10.2 Mathematical Preliminaries	393	(6)
10.3 Multivariable Adaptive Identification	399	(3)
10.3.1 Statement of the Problem,	399	(1)
10.3.2 The Adaptive Observer: Minimal Realization,	400	(1)
10.3.3 The Adaptive Observer: Nonminimal Realization,	401	(1)
10.4 Adaptive Control Problem	402	(4)
10.5 Adaptive Control of Two Input-Two Output Plants	406	(2)
10.6 Dynamic Precompensator for a General Multivariable Plant	408	(9)
10.6.1 Design of the Adaptive Controller,	410	(7)
10.7 Summary	417	(1)
Problems	417	(2)
References	419	(1)
11 Applications of Adaptive Control	420	(46)
11.1 Introduction	420	(1)
11.2 Ship Dynamics	421	(8)
11.3 Process Control	429	(11)
11.3.1 Distillation Columns,	430	(10)
11.4 Power Systems	440	(3)
11.5 Robot Manipulators	443	(11)
11.6 Adaptive Control in Bioengineering	454	(10)
11.6.1 Blood Pressure Control,	455	(1)
11.6.2 Adaptive Control for Ventricular Assist Devices,	456	(4)
11.6.3 Muscular Control,	460	(4)
11.7 Comments and Conclusions	464	(2)
References	466	(7)
A Kalman-Yakubovich Lemma	473	(3)
B Growth Rates of Unbounded Signals	476	(4)
C Properties of the Adaptive System with Bounded Parameters	480	(4)
D Proof of Boundedness in the Presence of Bounded Disturbances	484	(5)
E Some Facts About Transfer Matrices	489	(2)
Index	491	(4)
Errata List	495