H∞ positive filter-based control for positive linear systems

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Abstract
This paper deals with the design of a positive functional H∞ filter-based controller for positive linear systems subject to bounded energy disturbances. 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 filter-based controller is obtained into two steps. First, we search for positive state feedback gain for the design of a control law such that the closed-loop is positive, stable and ensures an H∞ performance requirement. This design of state-feedback gain is solved via constrained Linear Matrix Inequalities (LMIs). Then, we search in a second step for a positive functional filter-based controller permitting to reconstruct this control law, and so to estimate only a functional of the state useful for control purposes. The filter 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 filter-based controller and also, on the unbiasedness of the estimation error by solving a Sylvester equation. Then we derive conditions for the establishment of such filterbased controller in terms of an optimization problem, that can be solved via constrained LMIs. An algorithm that summarizes the different steps of the designed positive controller for positive linear systems is given. Finally, a numerical example is given to illustrate the effectiveness of the proposed method.

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