Positive 1D and 2D Systems

Moving on from earlier stochastic and robust control paradigms, this book introduces the reader to the fundamentals of probabilistic methods in the analysis and design of uncertain systems. It significantly reduces the computational cost of high-quality control and the complexity of the algorithms involved.

Contenu

1. Positive matrices and graphs.- 1.1 Generalised permutation matrix, nonnegative matrix, positive and strictly positive matrices.- 1.2 Reducible and irreducible matrices.- 1.3 The Collatz - Wielandt function.- 1.4 Maximum eigenvalue of a nonnegative matrix.- 1.5 Bounds on the maximal eigenvalue and eigenvector of a positive matrix.- 1.6 Dominating positive matrices of complex matrices.- 1.7 Oscillatory and primitive matrices.- 1.8 The canonical Frobenius form of a cyclic matrix.- 1.9 Metzler matrix.- 1.10 M-matrices.- 1.11 Totally nonnegative (positive) matrices.- 1.12 Graphs of positive systems.- 1.13 Graphs of reducible, irreducible, cyclic and primitive systems.- Problems.- References.- 2. Continuous-ime and discrete-ime positive systems.- 2.1 Externally positive systems.- 2.1.1 continuous-time systems.- 2.1.2 discrete-time system.- 2.2 Internally positive systemst.- 2.2.1 continuous-time systems.- 2.2.2 discrete-time systems.- 2.3 Compartmental systems.- 2.3.1 continuous-time systems.- 2.3.2 discrete-time systems.- 2.4 Stability of positive systems.- 2.4.1 Asymptotic stability of continuous-time systems.- 2.4.2 Asymptotic stability of discrete-time systems.- 2.5 Input-output stability.- 2.5.1 BIBO stability of positive continuous-time systems.- 2.5.2 BIBO stability of internally positive discrete-time systems.- 2.6 Weakly positive systems.- 2.6.1 Weakly positive continuous-time systems.- 2.6.2 Equivalent standard systems for singular systems.- 2.6.3 Reduction of weakly positive systems to their standard forms.- 2.6.4 Weakly positive discrete-time systems.- 2.6.5 Reduction of weakly positive systems to standard positive systems.- 2.7 Componentwise asymptotic stability and exponental stability of positive systems.- 2.7.1 continuous-time systems.- 2.7.2 discrete-time systems.- 2.8 Externally and internally positive singular systems.- 2.8.1 continuous-time systems.- 2.8.2 discrete-time systems.- 2.9 Composite positive linear systems.- 2.9.1 Discrete-ime systems.- 2.9.2 continuous-time systems.- 2.10 Eigenvalue assignment problem for positive linear systems.- 2.10.1 Problem formulation.- 2.10.2 Problem solution.- 2.11.2 Positive systems with nonnegative feedbacks.- Problems.- References.- 3. Reachability, controllability and observability of positive systems.- 3.1 discrete-time systems.- 3.1.1 Basic definitions and cone of reachable states.- 3.1.2 Necessary and sufficient conditions of the reachability of positive systems.- 3.1.3 Application of graphs to testing the reachability of positive systems.- 3.2 continuous-time systems.- 3.2.1 Basic definitions and reachability cone.- 3.3 Controllability of positive systems.- 3.3.1 Basic definitions and tests of controllability of discrete-time systems.- 3.3.2 Basic definitions and controllability tests of continuous-time systems.- 3.4 Minimum energy control of positive systems.- 3.4.1 discrete-time systems.- 3.4.2 continuous-time systems.- 3.5 Reachability and controllability of weakly positive systems with state feedbacks.- 3.5.1 Reachability.- 3.5.2 Controllability.- 3.6 Observability of discrete-time positive systems.- 3.6.1 Cone of positive initial conditions.- 3.6.2 Necessary and sufficient conditions of observability.- 3.6.3 Dual positive systems and relationships between reachability and observability.- 3.7 Reachability and controllability of weakly positive systems.- 3.7.1 Reachability.- 3.7.2 Controllability.- Problems.- References.- 4. Realisation problem of positive 1D systems.- 4.1 Basic notions and formulation of realisation problem.- 4.1.1 Standard discrete-time systems.- 4.1.2 Standard continuous-time systems.- 4.2 Existence and computation of positive realisations.- 4.2.1 Computation of matrix D of a given proper rational matrix.- 4.2.2 Existence and computation of positive realisations of discrete-time single-input single-output systems.- 4.2.3 Existence and computation of positive realisations of continuous-time single-input single-output systems.- 4.2.4 Necessary and sufficient conditions for the existence of reachable positive realisations.- 4.2.5 Determination of an internally positive electrical circuit for a given internally nonpositive one.- 4.3 Existence and computation of positive realisations of multi-input multi-output systems.- 4.3.1 Discrete-time systems.- 4.4 Existence and computation of positive realisations of weakly positive multi-input multi-output systems.- 4.4.1 Problem formulation.- 4.4.2 Existence of WCF positive realisations.- 4.4.3 Computation of WCF positive realisations.- 4.4.4 Computation of positive realisations of complete singular systems.- 4.5 Positive realisations in canonical forms of singular linear.- 4.5.1 Problem formulation.- 4.5.2 Methods of determination of realisations.- Problems.- References.- 5. 2D models of positive linear systems.- 5.1 Internally positive Roesser model.- 5.2 Externally positive Roesser model.- 5.3 Internally positive general model.- 5.4 Externally positive general model.- 5.5 Positive Fornasini-Marchesini models and relationships between models.- 5.6 Positive models of continuous-discrete systems.- 5.6.1 Positive general continuous-discrete model.- 5.6.2 Positive Fornasini-Marchesini type models of continuous-discrete systems.- 5.6.3 Positive Roesser continuous-discrete type model.- 5.6.4 Derivation of solution to the Roesser continuous-discrete model.- 5.7 Positive generalised Roesser model.- Problems.- References.- 6 Controllability and minimum energy control of positive 2D systems.- 6.1 Reachability, controllability and observability of positive Roesser model.- 6.1.1 Reachability.- 6.1.2 Controllability.- 6.1.3 Observability.- 6.2 Reachability, controllability and observability of the positive general model.- 6.2.1 Reachability.- 6.2.2 Controllability.- 6.2.3 Observability.- 6.3 Minimum energy control of positive 2D systems.- 6.3.1 Positive Roesser model.- 6.3.2 Positive general model.- 6.4 Reachability and minimum energy control of positive 2D continuous-discrete systems.- 6.4.1 Positive 2D continuous-discrete systems.- 6.4.2 Positive 2D continuous-discrete Roesser model.- Problems.- References.- 7. Realisation problem for positive 2D systems.- 7.1 Formulation of realisation problem for positive Roesser model.- 7.2 Existence of positive realisations.- 7.2.1 Lemmas.- 7.2.2 Method 1..- 7.2.3 Method 2..- 7.2.4 Method 3..- 7.3 Positive realisations in canonical form of the Roesser model.- 7.3.1 Problem formulation.- 7.3.2 Existence and computation of positive realisations in the Roesser canonical form.- 7.4 Determination of the positive Roesser model by the use of state variables diagram.- 7.5 Determination of a positive 2D general model for a given transfer matrix.- 7.6 Positive realisation problem for singular 2D Roesser model.- 7.6.2 Problem solution.- 7.7 Concluding remarks and open problems.- Problems.- References.- Appendix A Oeterminantal Sylvester equality.- Appendix B Computation of fundamental matrices of linear systems.- Appendix C Solutions of 20 linear discrete models.- Appendix D Transformations of matrices to their canonical forms and lemmas for 1D singular systems.