By Lorenzo Farina, Sergio Rinaldi
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Positive Linear Systems: Theory and Applications by Lorenzo Farina and Sergio Rinaldi Copyright © 2000 John Wiley & Sons, Inc. Part II Properties Positive Linear Systems: Theory and Applications by Lorenzo Farina and Sergio Rinaldi Copyright © 2000 John Wiley & Sons, Inc. 5 Stability This chapter and Chapter 6 are devoted to the stability of positive systems. The results we discuss in this chapter are quite general and do not refer to the structure of the system. In contrast, Chapter 6 presents some remarkable properties (known since 1912) of the spectrum of the A matrix of irreducible systems (cyclic and primitive).
The reader can easily extend these proofs to the continuous-time case. a and b imply c. This has already been proved (see Theorem 20). b and c imply a. The asymptotic stability of x(t + 1) = Ax(i) + bü guarantees that x(t) tends, as t —» oo, toward x for every x(0). , An~lb would be zero. Thus, from the Cayley-Hamilton theorem (An + a\An~l + a2An~2 -t. . -)- anI) = 0 the ith component of the vector Anb = - o i r ' i - a2A"-2b anb would be zero, as well as that of all the vectors Alb with t > n+1.
If all the <5a¿¿ are < a¿j, the influence graph remains unchanged, that is, Gx = Gx. On the contrary, if for some (i,j) δα^ = a,ij, the influence graph Gx is obtained PROPERTIES 43 by deleting from Gx the arc (j, i). It is therefore possible to consider perturbations that transform a given system into a collection of noninteracting subsystems. By contrast, it is impossible to add new interconnections since the conditions ay = O andO < Süij < ay imply ¿ay = 0. This class of perturbations describes pretty well what happens in a real system with components subject to ageing, malfunctioning, or breaking down.