Positive 1D and 2D Systems
By: by Tadeusz Kaczorek
Series: Communications and Control Engineering; Communications and control engineering
Publisher: Springer London; Imprint; Springer
Published: 2002
Place of publication: London
Language: eng
Number of pages: 431
ISBN-13: 9781447102212; 9781447110972
Subjects: Engineering; Systems theory
Description: 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.
Table of contents: title: 1. Positive matrices and graphs; title: 1.1 Generalised permutation matrix, nonnegative matrix, positive and strictly positive matrices; title: 1.2 Reducible and irreducible matrices; title: 1.3 The Collatz; title: Wielandt function; title: 1.4 Maximum eigenvalue of a nonnegative matrix; title: 1.5 Bounds on the maximal eigenvalue and eigenvector of a positive matrix; title: 1.6 Dominating positive matrices of complex matrices; title: 1.7 Oscillatory and primitive matrices; title: 1.8 The canonical Frobenius form of a cyclic matrix; title: 1.9 Metzler matrix; title: 1.10 M-matrices; title: 1.11 Totally nonnegative (positive) matrices; title: 1.12 Graphs of positive systems; title: 1.13 Graphs of reducible, irreducible, cyclic and primitive systems; title: Problems; title: References; title: 2. Continuous-ime and discrete-ime positive systems; title: 2.1 Externally positive systems; title: 2.2 Internally positive systemst; title: 2.3 Compartmental systems; title: 2.4 Stability of positive systems; title: 2.5 Input-output stability; title: 2.6 Weakly positive systems; title: 2.7 Componentwise asymptotic stability and exponental stability of positive systems; title: 2.8 Externally and internally positive singular systems; title: 2.9 Composite positive linear systems; title: 2.10 Eigenvalue assignment problem for positive linear systems; title: Problems; title: References; title: 3. Reachability, controllability and observability of positive systems; title: 3.1 discrete-time systems; title: 3.2 continuous-time systems; title: 3.3 Controllability of positive systems; title: 3.4 Minimum energy control of positive systems; title: 3.5 Reachability and controllability of weakly positive systems with state feedbacks; title: 3.6 Observability of discrete-time positive systems; title: 3.7 Reachability and controllability of weakly positive systems; title: Problems; title: References; title: 4. Realisation problem of positive 1D systems; title: 4.1 Basic notions and formulation of realisation problem; title: 4.2 Existence and computation of positive realisations; title: 4.3 Existence and computation of positive realisations of multi-input multi-output systems; title: 4.4 Existence and computation of positive realisations of weakly positive multi-input multi-output systems; title: 4.5 Positive realisations in canonical forms of singular linear; title: Problems; title: References; title: 5. 2D models of positive linear systems; title: 5.1 Internally positive Roesser model; title: 5.2 Externally positive Roesser model; title: 5.3 Internally positive general model; title: 5.4 Externally positive general model; title: 5.5 Positive Fornasini-Marchesini models and relationships between models; title: 5.6 Positive models of continuous-discrete systems; title: 5.7 Positive generalised Roesser model; title: Problems; title: References; title: 6 Controllability and minimum energy control of positive 2D systems; title: 6.1 Reachability, controllability and observability of positive Roesser model; title: 6.2 Reachability, controllability and observability of the positive general model; title: 6.3 Minimum energy control of positive 2D systems; title: 6.4 Reachability and minimum energy control of positive 2D continuous-discrete systems; title: Problems; title: References; title: 7. Realisation problem for positive 2D systems; title: 7.1 Formulation of realisation problem for positive Roesser model; title: 7.2 Existence of positive realisations; title: 7.3 Positive realisations in canonical form of the Roesser model; title: 7.4 Determination of the positive Roesser model by the use of state variables diagram; title: 7.5 Determination of a positive 2D general model for a given transfer matrix; title: 7.6 Positive realisation problem for singular 2D Roesser model; title: 7.7 Concluding remarks and open problems; title: Problems; title: References; title: Appendix A Oeterminantal Sylvester equality; title: Appendix B Computation of fundamental matrices of linear systems; title: Appendix C Solutions of 20 linear discrete models; title: Appendix D Transformations of matrices to their canonical forms and lemmas for 1D singular systems.