Stability of Stationary Sets in Control Systems With Discontinuous Nonlinearities (Series on Stability, Vibration and Control of Systems, Series a, Vol. 14)

Preface......Page 6
List of Notations......Page 10
Contents......Page 12
1. Foundations of Theory of Differential Equations with Discontinuous Right-Hand Sides......Page 17
1.1.1 Difficulties encountered in the definition of a solution. Sliding modes......Page 18
1.1.2 The concept of a solution of a system with discontinuous nonlinearities accepted in this book. Connection with the theory of differential equations with multiple-valued right-hand sides......Page 22
1.1.3 Relation to some other definitions of a solution to a system with discontinuous right-hand side......Page 30
1.1.4 Sliding modes. Extended nonlinearity. Example......Page 36
1.2 Systems of Differential Equations with Multiple-Valued Right-Hand Sides (Differential Inclusions)......Page 42
1.2.1 Concept of a solution of a system of differential equations with a multivalued right-hand side the local existence theorem the theorems on continuation of solutions and continuous dependence on initial values......Page 43
1.2.2 "Extended" nonlinearities......Page 53
1.2.3 Sliding modes......Page 60
1.3.1 Basic definitions......Page 71
1.3.2 Lyapunov-type lemmas......Page 73
2. Auxiliary Algebraic Statements on Solutions of Matrix Inequalities of a Special Type......Page 77
2.1 Algebraic Problems that Occur when Finding Conditions for the Existence of Lyapunov Functions from Some Multiparameter Functional Class. Circle Criterion. Popov Criterion......Page 78
2.1.1 Equations of the system. Linear and nonlinear partsof the system. Transfer function and frequency response......Page 79
2.1.2 Existence of a Lyapunov function from the class of quadratic forms. S-procedure......Page 80
2.1.3 Existence of a Lyapunov function in the class of quadratic forms (continued). Frequency-domain theorem......Page 85
2.1.4 The circle criterion......Page 87
2.1.5 A system with a stationary nonlinearity. Existence of a Lyapunov function in the class "a quadratic form plus an integral of the nonlinearity"......Page 91
2.1.6 Popov criterion......Page 95
2.2.1 Controllability observability and stabilizability......Page 100
2.2.2 Frequency-domain theorem on solutions of some matrix inequalities......Page 107
2.2.3 Additional auxiliary lemmas......Page 117
2.2.4 The S-procedure theorem......Page 122
2.2.5 On the method of linear matrix inequalities in control theory......Page 125
3. Dichotomy and Stability of Nonlinear Systems with Multiple Equilibria......Page 127
3.1.1 Systems with several nonlinearities. Frequencydomain conditions for quasi-gradient-like behavior and pointwise global stability. Free gyroscope with dry friction......Page 128
3.1.2 The case of a single nonlinearity and det P#0 . Theorem 3.4 on gradient-like behavior and pointwise global stability of the segment of rest. Examples......Page 136
3.1.3 The case of a single nonlinearity and one zero pole of the transfer function. Theorem 3.6 on quasi-gradientlike behavior and pointwise global stability. The Bulgakov problem......Page 140
3.1.4 The case of a single nonlinearity and double zero pole of the transfer function. Theorem 3.8 on global stability of the segment of rest. Gyroscopic roll equalizer. The problem of Lur'e and Postnikov. Control system for a turbine. Problem of an autopilot......Page 146
3.2 Systems with Monotone Piecewise Single-Valued Nonlinearities......Page 157
3.2.1 Systems with a single nonlinearity. Frequency-domain conditions for dichotomy and global stability. Corrected gyrostabilizer with dry friction. The problem of Vyshnegradskii......Page 158
3.2.2 Systems with several nonlinearities. Frequencydomain criteria for dichotomy. Noncorrectable gyrostabilizer with dry friction......Page 176
3.3.1 Dichotomy and quasi-gradient-likeness of systems with gradient nonlinearities......Page 183
3.3.2 Dichotomy and quasi-gradient-like behavior of nonlinear electrical circuits and of cellular neural networks......Page 187
4.1.1 Special features of the dynamics of pendulum-like systems. The structure of their equilibria sets......Page 191
4.1.2 Canonical forms of pendulum-like systems with a single scalar nonlinearity......Page 199
4.1.3 Dichotomy. Gradient-like behavior in a class of nonlinearities with zero mean value......Page 205
4.2.1 Theorem on gradient-like behavior......Page 208
4.2.2 Phase-locked loops with first- and second-order lowpass filters......Page 217
4.3 An Analogue of the Circle Criterion for Pendulum-Like Systems......Page 219
4.3.1 Criterion for boundedness of solutions of pendulumlike systems......Page 220
4.3.2 Lemma on pointwise dichotomy......Page 226
4.3.3 Stability of two- and three-dimensional pendulum-like systems. Examples......Page 228
4.3.4 Phase-locked loops with a band amplifier......Page 232
4.4 The Method of Non-Local Reduction......Page 234
4.4.1 The properties of separatrices of a two-dimensional dynamical system......Page 235
4.4.2 The theorem on nonlocal reduction......Page 238
4.4.3 Theorem on boundedness of solutions and on gradient-like behavior......Page 239
4.4.4 Generalized Bohm-Hayes theorem......Page 244
4.4.5 Approximation of the acquisition bands of phaselocked loops with various low-pass filters......Page 245
4.5 Necessary Conditions for Gradient-Like Behavior of Pendulum-Like Systems......Page 251
4.5.1 Conditions for the existence of circular solutions and cycles of the second kind......Page 252
4.5.2 Generalized Hayes theorem......Page 260
4.5.3 Estimation of the instability regions in searching PLL systems and PLL systems with 1/2 filter......Page 261
4.6 Stability of the Dynamical Systems Describing the Synchronous Machines......Page 267
4.6.1 Formulation of the problem......Page 268
4.6.2 The case of zero load......Page 269
4.6.3 The case of a nonzero load......Page 274
5.1.1 Proof of the equivalence of controllability to properties (i)-(iv) of Theorem 2.6......Page 285
5.1.2 Proof of the Theorem 2.7......Page 289
5.1.3 Completion of the proof of Theorem 2.6......Page 290
5.1.5 Proof of Theorem 2.9 in the scalar case m = l = 1......Page 291
5.1.6 Proof of Theorem 2.9 for the case when either m > 1 or l > 1 and proof of Theorem 2.10......Page 293
5.1.7 Proof of Lemma 2.4......Page 295
5.1.8 Proof of Lemma 2.7......Page 297
5.2.1 Two lemmas. A detailed version of frequency-domain theorem for the nonsingular case......Page 299
5.2.2 Proof of Theorem 5.1. The theorem on solvability of the Lur'e equation......Page 305
5.2.3 Lemma on J-orthogonality of the root subspaces of a Hamiltonian matrix......Page 311
5.3.1 Proof of Lemma 5.1......Page 313
5.3.2 Proof of Theorem 2.13......Page 314
5.4.1 Proof of Theorem 2.12......Page 317
5.4.2 Necessity of the hypotheses of Theorem 2.14......Page 322
5.4.3 Sufficiency of the hypotheses of Theorem 2.14......Page 325
5.5.1 The Dines theorem......Page 332
5.5.2 Proofs of the theorems on the losslessness of the S-procedure for quadratic forms and one constraint......Page 334
Bibliography......Page 339
Index......Page 349