Stability and Stabilization of Linear Systems with Saturating Actuators	3
	Foreword	6
	Preface	8
	Contents	11
	Notation	17
	Part I: Generalities	20
	Chapter 1: Linear Systems Subject to Control Saturation-Problems and Modeling	21
	1.1 Introduction	21
	1.2 The Open-Loop System	22
	1.3 The Closed-Loop System	23
	1.4 The Region of Linearity	30
	1.5 The Region of Attraction	31
	1.6 Problems Considered	32
	1.6.1 Asymptotic Stability Analysis	33
	1.6.1.1 Approaches Considering the RAS Included in the Region of Linearity	34
	Polyhedral RAS	34
	Ellipsoidal RAS	35
	1.6.1.2 Approaches Considering the RAS not Included in the Region of Linearity	35
	1.6.2 Asymptotic Stabilization	36
	1.6.2.1 Global Stabilization	37
	1.6.2.2 Semi-global Stabilization	39
	1.6.2.3 Local (Regional) Stabilization	42
	Saturation Avoidance	42
	Saturation Allowance	44
	1.6.3 External Stability Analysis	46
	1.6.4 External Stabilization	48
	1.6.5 The Anti-windup Problem	49
	1.7 Models for the Saturation Nonlinearity	50
	1.7.1 Polytopic Models	51
	1.7.1.1 Polytopic Model I	51
	1.7.1.2 Polytopic Model II	53
	1.7.1.3 Polytopic Model III	55
	1.7.2 Sector Nonlinearity Models	58
	1.7.2.1 Classical Sector Condition	59
	1.7.2.2 Generalized Sector Condition	61
	1.7.3 Regions of Saturation Models	62
	1.8 Equilibrium Points	64
	1.9 Conclusion	65
	Part II: Stability Analysis and Stabilization	67
	Chapter 2: Stability Analysis and Stabilization-Polytopic Representation Approach	68
	2.1 Introduction	68
	2.2 Asymptotic Stability Analysis	69
	2.2.1 Ellipsoidal Sets of Stability	69
	2.2.2 Polytopic Approach I	71
	2.2.3 Polytopic Approach II	73
	2.2.4 Polytopic Approach III	75
	2.2.5 Optimization Problems	76
	2.2.5.1 Size Criteria	77
	2.2.5.2 BMI x LMI Problems	79
	2.3 External Stability	82
	2.3.1 Amplitude Bounded Exogenous Signals	83
	2.3.2 Energy Bounded Exogenous Signals	88
	2.4 Stabilization	93
	2.4.1 State Feedback Stabilization	93
	2.4.1.1 Regional Asymptotic Stabilization	93
	A. Regional Guaranteed Stability	95
	B. Maximization of an Estimate of the Region of Attraction	95
	C. Optimization of the Actuator Size	96
	2.4.1.2 Performance Issues	97
	2.4.1.3 Trade-off Between Saturation, Size of the Stability Region and Time-Domain Performance	99
	2.4.1.4 Piecewise Linear State Feedback	101
	2.4.1.5 Regional External Stabilization	103
	(A) Amplitude Bounded Exogenous Signals	103
	(B) Energy Bounded Exogenous Signals	104
	2.4.2 Observer-Based Feedback Stabilization	108
	2.4.3 Dynamic Output Feedback Stabilization	112
	2.4.4 Global Stabilization	118
	2.5 Uncertain Systems	120
	2.5.1 Stability Analysis	120
	2.5.2 Extensions	125
	2.6 Discrete-Time Case	126
	2.6.1 Ellipsoidal Regions of Asymptotic Stability	127
	2.6.2 Polyhedral Regions of Asymptotic Stability	129
	2.6.2.1 Preliminaries	129
	2.6.2.2 Maximal Positively Invariant Set in S(|K|,u0alpha)	132
	2.6.3 Extensions	136
	2.6.3.1 External Stability	136
	2.6.3.2 Stabilization	137
	2.7 Conclusion	137
	Chapter 3: Stability Analysis and Stabilization-Sector Nonlinearity Model Approach	139
	3.1 Introduction	139
	3.2 Asymptotic Stability Analysis	140
	3.2.1 Quadratic Lyapunov Function	141
	3.2.2 Lure Lyapunov Function	145
	3.2.3 Computational Burden	148
	3.2.4 Optimization Problems	149
	3.2.4.1 Quadratic Case	149
	3.2.4.2 Lure Case	150
	3.3 External Stability	152
	3.3.1 Amplitude Bounded Exogenous Signals	152
	3.3.2 Energy Bounded Exogenous Signals	157
	3.3.3 Optimization Issues	161
	3.3.3.1 Amplitude Bounded Exogenous Signals	161
	3.3.3.2 Energy Bounded Exogenous Signals	162
	3.4 Stabilization	162
	3.4.1 State Feedback Stabilization	162
	3.4.1.1 Asymptotic Stabilization	162
	Optimization Issues	164
	3.4.1.2 External Stabilization	164
	Amplitude Bounded Exogenous Signals	164
	Energy Bounded Exogenous Signals	165
	Optimization Issues	167
	3.4.2 Observer-Based Feedback Stabilization	171
	3.4.3 Dynamic Output Feedback Stabilization	177
	Amplitude Bounded Exogenous Signals	178
	Energy Bounded Exogenous Signals	182
	Optimization Issues	185
	3.5 Uncertain Systems	187
	Norm-Bounded Uncertainty	188
	Polytopic Uncertainty	189
	Gain Scheduling Approach	190
	3.6 Discrete-Time Case	194
	3.7 Extensions	196
	3.7.1 Nested Saturations	196
	3.7.2 Nested Nonlinearities	197
	3.7.3 Nonlinear and/or Hybrid Systems	198
	3.8 Conclusion	199
	Chapter 4: Analysis via the Regions of Saturation Model	200
	4.1 Introduction	200
	4.2 Polyhedral Regions of Stability	201
	4.2.1 Positive Invariance	201
	4.2.2 Contractivity-Compact Case	204
	4.2.3 Determination of Stability Regions	208
	4.3 Ellipsoidal Regions	211
	4.3.1 Test Condition 1	212
	4.3.2 Test Condition 2	215
	4.4 Discrete-Time Case	217
	4.4.1 Positive Invariance	217
	4.4.2 Contractivity-Compact Case	218
	4.5 Unbounded Sets of Stability	219
	4.5.1 Unbounded Polyhedra	219
	4.5.2 Unbounded Ellipsoidal Sets	221
	4.6 Conclusion	222
	Chapter 5: Synthesis via a Parameterized ARE Approach or a Parameterized LMI Approach	224
	5.1 Introduction	224
	5.2 The Parameterized ARE Approach	225
	5.2.1 Preliminaries	226
	5.2.2 The Single-Input Case	228
	5.2.3 The Multi-variable Case	231
	5.2.4 Control Computation and Implementation	234
	5.3 A Parameterized LMI Approach	239
	5.4 Multi-objective Control: Eigenvalues Placement and Guaranteed Cost	245
	5.4.1 Preliminaries	245
	5.4.2 The Single-Input Case	248
	5.4.3 The Multi-variable Case	253
	5.5 Disturbance Tolerance	258
	5.5.1 Preliminaries	260
	5.5.2 tau-Parameterized Control	262
	5.5.3 Control Law Computation and Implementation	266
	5.6 Nonlinear Bounded Control for Time-Delay Systems	270
	5.6.1 Problem Statement	270
	5.6.2 Preliminaries	271
	5.6.3 Riccati Equation Approach	272
	5.6.4 LMI Approach	275
	5.7 Conclusion	279
	Part III: Anti-windup	280
	Chapter 6: An Overview of Anti-windup Techniques	281
	6.1 Introduction-Philosophy	281
	6.2 General Anti-windup Architecture	282
	6.3 A Bit of History	285
	Early Academic Study	287
	Constrained Input Control	287
	Modern Anti-windup	287
	6.4 Formulation of Problems	288
	6.5 Regional (Local) Versus Global Strategies	292
	6.5.1 A Quick Overview in the Regional (Local) Context	292
	6.5.2 A Quick Overview in the Global Context	293
	6.6 Mismatch-Based Anti-windup Synthesis	294
	6.7 Conclusion	295
	Chapter 7: Anti-windup Compensator Synthesis	296
	7.1 Introduction	296
	7.2 Problems Setup	296
	7.2.1 Direct Linear Anti-windup	298
	7.2.2 Model Recovery Anti-windup	299
	7.3 Direct Linear Anti-windup Design	300
	7.3.1 Preliminary Elements	301
	7.3.2 DLAW Schemes with Global Stability Guarantees	303
	7.3.3 DLAW Schemes with Regional Stability Guarantees	307
	7.3.4 Some Algorithms	310
	7.4 Model Recovery Anti-windup Design	313
	7.4.1 Preliminary Elements on the Architecture	313
	7.4.2 Some Algorithms	315
	7.5 Anti-windup Algorithms Summary	317
	7.6 Some Extensions	318
	7.6.1 Rate Saturation	319
	7.6.2 Sensor Saturations	319
	7.6.3 Nested Saturations	320
	7.6.4 Time-Delay Systems	321
	7.6.5 Anti-windup for Nonlinear Systems	322
	7.7 Conclusion	322
	Chapter 8: Applications of Anti-windup Techniques	323
	8.1 Introduction	323
	8.2 Static Anti-windup Examples	323
	8.3 Dynamic Anti-windup Examples	327
	Enlargement of the Domain of Admissible Initial States	332
	Maximization of Disturbance Rejection	334
	8.4 Toward More Complex Nonlinear Actuators	336
	8.4.1 Actuator with Position and Rate Saturations	337
	8.4.2 Dynamics Restricted Actuator	342
	8.4.3 Electro-Hydraulic Actuator	347
	8.5 Pseudo Anti-windup Strategies when the Dead-Zone Element is not Accessible	351
	8.5.1 Anti-windup Scheme Using a Fictitious Linear Element	351
	8.5.2 Observer-Based Anti-windup Scheme	355
	8.6 Conclusion	363
	Appendix A: Some Concepts Related to Stability Theory	365
	A.1 Introduction	365
	A.2 Stability of Linear Autonomous Systems	366
	A.3 Stability for Nonlinear Systems	367
	A.3.1 Stability Definition (Lyapunov Stability)	367
	A.3.2 Lyapunov's Stability Theorem	369
	A.3.3 The Invariance Principle	371
	A.3.4 Region of Attraction	372
	A.3.5 Set of Equilibrium Points	373
	A.4 Application of Lyapunov Stability	373
	A.4.1 Absolute Stability	374
	A.4.2 Positive Real Function Concept	375
	A.4.3 Circle Criterion	376
	A.4.4 Popov Criterion	377
	A.4.5 Quadratic Stability	378
	Appendix B: Quadratic Approach for Robust Control	379
	B.1 Introduction	379
	B.2 Uncertain Models and Quadratic Stability	380
	B.2.1 Uncertain Models	380
	B.2.1.1 Norm-Bounded Uncertainty (NB-Uncertainty)	380
	B.2.1.2 Bounded-Real Uncertainty (BR-Uncertainty)	381
	B.2.1.3 Positive Real Uncertainty (PR-Uncertainty)	381
	B.2.1.4 Structured Uncertainty	382
	B.2.1.5 Polytopic Uncertainty (P-Uncertainty)	382
	B.2.2 Quadratic Stability	382
	B.2.2.1 P-Uncertainty	383
	B.2.2.2 NB-Uncertainty	384
	B.3 Quadratic Stabilizability	385
	B.3.1 P-Uncertainty	386
	B.3.2 NB-Uncertainty	387
	B.4 Guaranteed Cost Control	389
	B.4.1 P-Uncertainty	391
	B.4.2 NB-Uncertainty	392
	B.5 Pole Placement	394
	B.5.1 Pole Placement in a Disk	394
	B.5.1.1 d-Stabilizability	394
	B.5.1.2 P-Uncertainty	395
	B.5.1.3 NB-Uncertainty	395
	B.5.2 Pole Placement in LMI Regions	397
	B.5.2.1 Quadratic R-Stabilizability	398
	B.5.2.2 P-Uncertainty	398
	B.5.2.3 NB-Uncertainty	399
	Appendix C: Linear Matrix Inequalities (LMI) and Riccati Equations	401
	C.1 Introduction	401
	C.2 Algebraic Lyapunov Equation (ALE)	401
	C.2.1 Continuous Algebraic Lyapunov Equation (CALE)	402
	C.2.2 Discrete Algebraic Lyapunov Equation (DALE)	403
	C.3 Algebraic Riccati Equation (ARE)	404
	C.3.1 Continuous Algebraic Riccati Equation (CARE)	404
	C.3.1.1 Characterization of Solutions of CARE	405
	C.3.1.2 Stabilizing Solution-Riccati Operator	406
	C.3.2 Discrete Algebraic Riccati Equation (DARE)	407
	C.3.2.1 Stabilizing Solution-Riccati Operator	408
	C.4 Linear Matrix Inequalities (LMI)	409
	C.5 Schur's Complement	410
	C.6 Bilinear Matrix Inequalities (BMI)	412
	C.6.1 Eigenvalues and Generalized Eigenvalues Problems	413
	C.6.1.1 EigenValue Problems (EVP)	413
	C.6.1.2 Generalized EigenValue Problems (GEVP)	414
	C.7 The Elimination Lemma and the S-Procedure	415
	C.7.1 The Elimination Lemma	415
	C.7.2 The S-Procedure	416
	C.7.2.1 The S-Procedure for Quadratic Functions and Nonstrict Inequalities	416
	C.7.2.2 The S-Procedure for Quadratic Forms and Strict Inequalities	417
	C.8 Ellipsoids and Polyhedral Sets	417
	C.8.1 Ellipsoid and Invariant Ellipsoid	417
	C.8.2 Convex Polyhedron and Invariant Polyhedron	418
	C.8.3 Maximum Volume Ellipsoid Contained in a Symmetric Polytope	419
	C.8.4 Smallest Volume Ellipsoid Containing a Symmetric Polytope	420
	References	422
	Index	440