A new design of positive functional filters for positive linear time-delay systems

Abstract

This paper solves the problem of designing positive functional $H_{\mathrm{\infty }}$ filters for positive linear time-delay systems. In fact, we propose a new positive reduced order $H_{\mathrm{\infty }}$ filter for positive linear constant state time-delay systems for which the states remain in the nonnegative orthant of the state space, subject to disturbances and unknown inputs. This paper is a first attempt to design positive unknown inputs functional $H_{\mathrm{\infty }}$ filter for such positive linear delayed systems. The proposed approach is based on the positivity of an augmented system composed of dynamics of both considered system and proposed filter, on the unbiasedness of the estimation error by the resolution of Sylvester equation and also on Lyapunov-Krasovskii stability theory; then conditions for the establishment of such filters are formulated in terms of an optimization problem. An algorithm that summarizes the different steps of the proposed positive functional $H_{\mathrm{\infty }}$ filter design is given. Finally, numerical example and simulation results are given to illustrate the effectiveness of the proposed design method.

Keywords:

• Positive systems
• positive functional $H_{\mathrm{\infty }}$ filters
• linear systems
• time-delay system
• unknown inputs
• optimisation problem

Disclosure statement

No potential conflict of interest was reported by the author(s).