Positive 1D and 2D Systems : Communications and Control Engineering - T. Kaczorek

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Positive 1D and 2D Systems

By: T. Kaczorek

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Hardcover

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Hardcover
448 Pages

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24.1 x 16.51 x 3.81

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The presence of uncertainty in a system description has always been a critical issue in control. Moving on from earlier stochastic and robust control paradigms, the main objective of this book is to introduce the reader to the fundamentals of probabilistic methods in the analysis and design of uncertain systems. Using so-called randomized algorithms, this emerging area of research guarantees a reduction in the computational complexity of classical robust control algorithms and in the conservativeness of methods like H¥ control. Features: · self-contained treatment explaining the genesis of randomized algorithms in the principles of probability theory to their use for robust analysis and controller synthesis; · comprehensive treatment of sample generation, including consideration of the difficulties involved in obtaining identically and independently distributed samples; · applications of randomized algorithms in congestion control of high-speed communications networks and the stability of quantized sampled-data systems. Randomized Algorithms for Analysis and Control of Uncertain Systems will be of certain interest to control theorists concerned with robust and optimal control techniques and to all control engineers dealing with system uncertainties.

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Positive matrices and graphs	p. 1
Generalised permutation matrix, nonnegative matrix, positive and strictly positive matrices	p. 1
Reducible and irreducible matrices	p. 3
The Collatz - Wielandt function	p. 8
Maximum eigenvalue of a nonnegative matrix	p. 10
Bounds on the maximal eigenvalue and eigenvector of a positive matrix	p. 13
Dominating positive matrices of complex matrices	p. 18
Oscillatory and primitive matrices	p. 19
The canonical Frobenius form of a cyclic matrix	p. 23
Metzler matrix	p. 26
M-matrices	p. 27
Totally nonnegative (positive) matrices	p. 30
Graphs of positive systems	p. 34
Graphs of reducible, irreducible, cyclic and primitive systems	p. 41
Continuous-time and discrete-time positive systems	p. 51
Externally positive systems	p. 51
Internally positive systems	p. 55
Compartmental systems	p. 60
Stability of positive systems	p. 63
Input-output stability	p. 72
Weakly positive systems	p. 79
Componentwise asymptotic stability and exponental stability of positive systems	p. 97
Externally and internally positive singular systems	p. 104
Composite positive linear systems	p. 115
Eigenvalue assignment problem for positive linear systems	p. 119
Reachability, controllability and observability of positive systems	p. 127
Discrete-time systems	p. 127
Continuous-time systems	p. 136
Controllability of positive systems	p. 139
Minimum energy control of positive systems	p. 144
Reachability and controllability of weakly positive systems with state feedbacks	p. 151
Observability of discrete-time positive systems	p. 156
Reachability and controllability of weakly positive systems	p. 161
Realisation problem of positive 1D systems	p. 173
Basic notions and formulation of realisation problem	p. 173
Existence and computation of positive realisations	p. 175
Existence and computation of positive realisations of multi-input multi-output systems	p. 201
Existence and computation of positive realisations of weakly positive multi-input multi-output systems	p. 211
Positive realisations in canonical forms of singular linear	p. 222
2D models of positive linear systems	p. 241
Internally positive Roesser model	p. 241
Externally positive Roesser model	p. 243
Internally positive general model	p. 248
Externally positive general model	p. 250
Positive Fornasini-Marchesini models and relationships between models	p. 251
Positive models of continuous-discrete systems	p. 254
Positive generalised Roesser model	p. 264
Controllability and minimum energy control of positive 2D systems	p. 275
Reachability, controllability and observability of positive Roesser model	p. 275
Reachability, controllability and observability of the positive general model	p. 283
Minimum energy control of positive 2D systems	p. 291
Reachability and minimum energy control of positive 2D continuous-discrete systems	p. 297
Realisation problem for positive 2D systems	p. 311
Formulation of realisation problem for positive Roesser model	p. 311
Existence of positive realisations	p. 312
Positive realisations in canonical form of the Roesser model	p. 328
Determination of the positive Roesser model by the use of state variables diagram	p. 332
Determination of a positive 2D general model for a given transfer matrix	p. 337
Positive realisation problem for singular 2D Roesser model	p. 340
Concluding remarks and open problems	p. 359
Determinantal Sylvester equality	p. 367
Computation of fundamental matrices of linear systems	p. 387
Solutions of 2D linear discrete models	p. 403
Transformations of matrices to their canonical forms and lemmas for 1D singular systems	p. 411
Index	p. 429
Table of Contents provided by Blackwell. All Rights Reserved.

Positive 1D and 2D Systems

At a Glance

Hardcover

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