Communications and Control Engineering

Filtering and Control

By: Sergio Bittanti, Patrizio Colaneri

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Hardcover | 26 November 2008

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448 Pages

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Periodic Systems gives a comprehensive treatment of the theory of periodic systems, including the problems of filtering and control. Topics covered include: basic issues, including Floquet theory, controllability and observability, canonical decomposition, system norms and Lyapunov and robust stability; the problem of state estimation in its various forms, filtering, prediction and smoothing; control design methods, particularly optimal and robust control. The text focuses on discrete-time signals and systems; however, an overview of the entire field, including the continuous-time case, is provided in the first chapter. The authors' presentation of the theory and results is mathematically rigorous while maintaining a readable style, avoiding excessive formalism. This makes the book accessible to graduate students and researchers from the fields of engineering, physics, economics and mathematics.

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Published: 19th October 2010

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Introduction	p. 1
Periodic Models	p. 1
Periodicity Induced by Linearization	p. 4
Periodicity Induced by Multirate Sampling	p. 4
Floquet-Lyapunov Theory	p. 8
Floquet-Lyapunov Transformation	p. 9
Characteristic Exponents	p. 12
Stability	p. 15
Lagrange Formula and External Stability	p. 17
Relative Degrees and Zeros	p. 19
Relative Degrees and Normal Form	p. 19
Zeros	p. 23
Periodic Systems in Frequency Domain	p. 26
Time-invariant Representations	p. 30
Sample and Hold	p. 30
Lifting	p. 31
Periodic Control of Time-invariant Systems	p. 32
Output Stabilization of Linear Time-invariant Systems	p. 33
The Chocolate Problem	p. 35
Periodic Optimization	p. 41
Periodic Control of Non-linear Time-invariant Systems	p. 47
Periodic Control of Periodic Systems	p. 47
Stabilization via State-feedback	p. 48
LQ Optimal Control	p. 49
Receding Horizon Periodic Control	p. 51
H∞ Periodic Control	p. 53
Periodic Control of Non-linear Periodic Systems	p. 55
Stochastic Periodic Signals	p. 57
Linear Periodic Systems	p. 61
The Periodic Transfer Operator	p. 61
PARMA Model	p. 65
The Transfer Operator in Frequency Domain	p. 69
State-Space Description	p. 73
Adjoint Operator and Its State-Space Description	p. 79
Bibliographical Notes	p. 80
Floquet Theory and Stability	p. 81
Monodromy Matrix	p. 81
The Characteristic Polynomial of a Monodromy Matrix	p. 82
Invariance of the Characteristic Multipliers	p. 83
The Minimal Polynomial of a Monodromy Matrix	p. 84
Floquet Theory	p. 86
Discrete-time Floquet Theory for the Reversible Case	p. 87
Discrete-time Floquet Theory for the Non-reversible Case	p. 87
Stability	p. 90
Stability and Characteristic Multipliers	p. 90
Time-freezing and Stability	p. 91
The Lyapunov Paradigm	p. 94
Lyapunov Stability Condition	p. 98
Lyapunov Function	p. 100
Robust Stability	p. 100
Quadratic Stability for Norm-bounded Uncertainty	p. 101
Robust and Quadratic Stability for Polytopic Uncertainty	p. 103
Companion Forms	p. 105
Bibliographical Notes	p. 108
Structural Properties	p. 109
Reachability and Controllability	p. 109
Basic Definitions	p. 109
Reachability	p. 110
Reachability Canonical Form	p. 114
Controllability	p. 116
Reachability and Controllability Grammians	p. 119
Modal Characterization of Reachability and Controllability	p. 121
Canonical Controllability Decomposition	p. 123
Observability and Reconstructability	p. 126
Basic Definitions	p. 126
Duality	p. 128
Observability	p. 129
Observability Canonical Form	p. 131
Reconstructability	p. 132
Observability and Reconstructability Grammians	p. 133
Modal Observability and Reconstructability	p. 134
Canonical Reconstructability Decomposition	p. 135
Canonical Decomposition	p. 136
Extended Structural Properties	p. 139
Stabilizability	p. 139
Detectability	p. 141
The Periodic Lyapunov Equation	p. 142
Bibliographical Notes	p. 154
Periodic Transfer Function and BIBO Stability	p. 155
Elementary Algebra of Periodic Polynomials	p. 155
Lifted Periodic Polynomials and Zeroes	p. 157
Weak and Strong Coprimeness	p. 162
Rational Periodic Operators	p. 164
BIBO Stability and Finite Impulse Response	p. 168
Bibliographical Notes	p. 170
Time-invariant Reformulations	p. 171
Sampling in the z-Domain	p. 172
Time-lifted Reformulation	p. 173
Lifting a Time-invariant System	p. 174
Lifting a Periodic System in the Input-Output Framework	p. 175
Lifting a Periodic System in the State-Space Framework	p. 178
Cyclic Reformulation	p. 180
Cycled Signal in the z-Domain	p. 181
Cycling a Time-invariant System	p. 182
Cycling a Periodic System in the Input-Output Framework	p. 183
Cycling a Periodic System in the State-Space Framework	p. 185
The Frequency-lifted Reformulation	p. 187
Frequency-lifting of a Periodic System	p. 188
Frequency-lifting in State-Space: the Fourier Reformulation	p. 189
Bibliographical Notes	p. 192
State-Space Realization	p. 193
Periodic Realization from Lifted Models	p. 195
Minimal, Quasi-minimal, and Uniform Periodic Realizations	p. 195
A Necessary Condition	p. 196
Periodic Realization from Markov Parameters	p. 197
Algorithm for the Derivation of a Minimal Periodic Realization	p. 201
Periodic Realization via Orthogonal Transformations	p. 202
Realization from a Fractional Representation	p. 205
Balanced Realization	p. 210
Bibliographical Notes	p. 212
Zeros, Poles, and the Delay Structure	p. 215
Cyclic Poles and Zeros	p. 215
Lifted Poles and Zeros	p. 218
Zeros via a Geometric Approach	p. 223
Delay Structure and Interactor Matrix	p. 228
Bibliographical Notes	p. 234
Norms of Periodic Systems	p. 235
L₂ Norm	p. 235
Lyapunov Equation Interpretation	p. 239
Impulse Response Interpretation	p. 241
Stochastic Interpretation	p. 243
L_∞ Norm	p. 244
Input-Output Interpretation	p. 245
Riccati Equation Interpretation	p. 247
Entropy	p. 255
Hankel Norm and Order Reduction	p. 258
Time-domain Specifications	p. 261
Bibliographical Note	p. 264
Factorization and Parametrization	p. 265
Periodic Symplectic Pencil	p. 265
Spectral Factorization	p. 269
From the Spectrum to the Minimal Spectral Factor	p. 270
From a Spectral Factor to the Minimal Spectral Factor	p. 273
J-Spectral Factorization	p. 276
Parametrization of Stabilizing Controllers	p. 278
Polynomial Formulation	p. 278
Rational Formulation	p. 284
Bibliographical Notes	p. 289
Stochastic Periodic Systems	p. 291
Cyclostationary Processes	p. 292
Spectral Representation	p. 295
Spectrum from the Transfer Operator	p. 296
Periodic State-Space Stochastic Models	p. 301
Stochastic Realization	p. 304
Problem Formulation	p. 305
Realization for SISO Periodic Systems	p. 306
Realization from Normalized Data	p. 308
Bibliographical Notes	p. 310
Filtering and Deconvolution	p. 311
Filtering and Deconvolution in the H₂ Setting	p. 313
Kalman Filtering	p. 318
A Factorization View	p. 320
Wiener Filtering	p. 322
H2 Prediction	p. 324
A Factorization View	p. 327
H2 Smoothing	p. 329
A Factorization View	p. 332
Filtering and Deconvolution in the H_∞ Setting	p. 334
H_∞ 1-step Prediction	p. 342
H_∞ Smoothing	p. 345
Bibliographical Notes	p. 352
Stabilization and Control	p. 353
The Class of State-feedback Stabilizing Controllers	p. 353
Robust Stabilization	p. 355
Pole Assignment for Control	p. 358
Pole Assignment via Sampled Feedback	p. 358
Pole Assignment via Instantaneous Feedback	p. 360
Exact Model Matching	p. 362
H2 Full-information Optimal Control	p. 368
H2 Optimal State-feedback Control	p. 370
State-feedback H∞ Control	p. 372
Static Output-feedback	p. 375
Sampled-data Approach	p. 375
A Riccati Equation Approach	p. 378
Dynamic Output-feedback	p. 382
Dynamic Polynomial Assignment	p. 382
Dynamic Robust Polynomial Assignment	p. 386
Stabilization with Input Constraints and Uncertain Initial State	p. 390
LQG Control	p. 397
Output-feedback H∞ Control	p. 402
Bibliographical Notes	p. 405
References	p. 407
Index	p. 421
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Communications and Control Engineering

Filtering and Control

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Hardcover

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