A new test for checking the reachability (observability) of positive discrete-time linear systems is proposed. Conditions are established under which the unreachable pair (A,B) and the unobservable pair (A,C) of positive discrete-time system can be decomposed into reachable and unreachable parts and observable and unobservable parts, respectively. It is shown that the transfer matrix of the positive system is equal to the transfer matrix of its reachable (observable) part.
DECOMPOSITION OF THE PAIRS (A,B) AND (A,C)
OF POSITIVE DISCRETE-TIME SYSTEMS
In this paper, new criteria for examining the reachability (observability) of positive linear discrete-time systems are proposed. Conditions are provided under which an unreachable pair (A,B) and an unobservable pair (A,C) of a positive discrete-time system can be decomposed into a reachable and an unreachable part, and an observable and an unobservable part, respectively. It is shown that the transfer matrix of a positive system is equal to the transfer matrix of only its reachable part and, respectively, only its observable part.
1. INTRODUCTION
In positive systems, inputs, state variables, and outputs take only non-negative values.
Examples of positive systems include industrial processes involving chemical reactors, heat exchangers, and distillation columns, storage systems, compartmental systems, and water and atmospheric pollution models. A variety of models exhibiting positive linear behavior can be found in engineering, management science, economics, social sciences, biology, medicine, etc. An overview of the state of the art in positive linear theory is provided in the monographs [2, 3].
The notions of controllability and observability and the decomposition of linear systems were introduced by Kalman [7, 8]. These notions are the basic concepts of modern control theory [1, 6, 9, 10, 5]. They have also been extended to positive linear systems [2, 3].
In this paper, the idea of Kalman’s decomposition theorem will be extended to positive discrete-time linear systems. (…)
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Source: Logistyka Magazine