Positive 1D and 2D Systems, Softcover reprint of the original 1st ed. 2002 Communications and Control Engineering Series

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.