Positive 1D and 2D Systems

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 Necessar...