Design of a positive observer-based controller for~positive linear systems

Abstract

In this paper, we investigate the design of a positive functional observer-based controller for positive linear systems subject to unknown inputs. We propose a new approach to numerically compute the controller, which is obtained via a function of the state to be estimated with the same order as the controller. The positive observer-based controller, is obtained into two steps. First, we search for a positive state feedback gain for the design of a control law such that the closed-loop is positive and asymptotically stable. This synthesis of state-feedback gain is solved using Linear Programming (LP) technique. Then, we search in a second step for a positive functional observer-based controller permitting to reconstruct this control law, and so to estimate only a functional of the state useful for control purposes. The observer is positive, that is, it ensures the nonnegativity of the estimated states. The proposed procedure is based on the positivity of an augmented system composed of dynamics of both considered system and proposed observer-based controller and also, on the unbiasedness of the estimation error by the resolution of Sylvester equation. Then existence conditions of such observer-based controller are formulated in terms of linear programming (LP) problem, where we use the Perron-Frobenius theorem applied to Metzler matrices. An algorithm that summarizes the different steps of the designed positive controller for positive linear systems is given. Finally, a numerical example and simulation results are given to illustrate the effectiveness of the proposed method.