Positive observation problem for linear time-delay positive systems

This paper deals with the problem of positive observation for linear time-delay systems for which the states take nonnegative values whenever the initial conditions are nonnegative. We focus on the design of positive observers (possibly with time-delay) which guarantee nonnegative estimates of the current states. We derive necessary and sufficient conditions for the existence of a positive observer (extended Luemberger-type) and show that the solvability of the problem can be decided via standard linear programming techniques. Moreover, on the negative side, it is shown that one cannot stabilize any unstable positive time-delay system by using extended Luenberger type positive observers. In other words, the separation principle does not hold.

1 Introduction

Differential delay systems known also as hereditary or systems with aftereffects, represent a class of infinite-dimensional systems which model propagation phe-nomena, population dynamics and many physical and chemical processes. As matter of fact, the reaction of real world systems to exogenous signals is never instantaneous and always infected by certain time delays. Such pathological phenomena can be adequately described by a mathematical model in which the behavior of the rate of the state is described by an equation including some information on the past evolution of the system. In general, for linear time-delay systems and independently of the representation type, the delay effects on the stability and control of dynamical systems (delays in the state and/or in the input) are problems of a great interest since the delay presence may induce complex behaviors (oscillations, instability, bad performances) for the closed-loop system. Also, small delays may destabilize some systems, however large delays may stabilize others. In contrast, for positive linear time-delay systems, the presence of delays does not affect the stability performance of the system [13], [13], [16], [9]. In particular, this paper shows that the convergence of the estimated state to actual state is insensitive to constant delays.