Observer Design for Discrete-Time Positive Systems with Delays

Abstract

This paper addresses positive observer design problem for discrete-time positive systems with multidelays. Both the positive observer and the positive systems share the nonnegativity of their states. The observer is of Luenberguer-type, which is an extension of the classical linear observer. A set of equivalent necessary and sufficient conditions is proposed for the existence of the Luenberguer-type positive observer. All the results are formulated as linear programming (LP) and linear matrix inequality (LMI) problems and therefore can be easily verified or implemented. The obtained theoretical results are demonstrated by a numerical example.

Positive $$L_1 $$ L 1 Observer Design for Positive Switched Systems

“…Therefore, error system (4) with w(t) = 0 is exponentially stable for any switching signal with average dwell time (10). In the sequel, we will consider the L 1 -gain performance.…”

“…Taking α = 0.3, β = 0.5, γ = 0.5, λ = 0.15, u(t) = 2 |sin t|, and w(t) = 0.05e −0.05t , and solving (7)- (8) According to (10,11), we can get μ = 1.1400, and T * a = 0.8737. Figure 1 shows the switching signal with the average dwell time T a = 0.9.…”

“…Conditions (8,9) guarantee that the increase rate of the error subsystem p with unobservable pair (A p , C p ) is bounded. [8,10,17,35], the system considered in this section satisfies Assumption 1, which is universal in the application. Also, the L 1 observer is proposed for the first time to guarantee the robustness with regard to exogenous disturbance w(t).…”

“…However, for positive switched system (1), the positivity of the estimated statex(t) of system (2) or (3) should also be guaranteed according to [10,27,29,32,35]. Finally, according to Lemma 1, it is naturally required…”

“…Thus, imposing a positive restriction on the designed observer for positive dynamical systems is often necessary. Recently, some problems of positive observer design for positive linear systems have been investigated in [8,10,17,27,29,32,35]. It is worth pointing out that the aforementioned literatures do not consider the effect of disturbances.…”