Robust State Feedback Stabilization of Positive LTI Systems with Polytopic Uncertainty

The paper contains some novel ideas in the field of direct search controller design The main paper of the attached data is available via http://dx.doi.org/10.1080/00207179.2022.2135025 <br> Summary: The paper deals with the robust state feedback stabilizability problem of LTI systems in the presence of polytopic uncertainty. This paper contributes two major results involving necessary and sufficient conditions to simultaneously check both positivity and robust stability of an uncertain LTI system and propose a design algorithm to solve the main stabilizability problem. It is shown that a polytopic uncertain LTI system is robustly stable and positive if and only if all its corner matrices are Metzler and all coefficients of its characteristic polynomial are nonnegative. The conditions for checking the non-negativity of the characteristic polynomials are not in terms of linear matrix inequalities because the characteristic polynomial is generally a non-affine function of the uncertain parameters. To cope with this issue, a design algorithm is proposed which checks the design space of the stabilizer parameters, divides it to smaller subspaces, checks the feasibility of the corner points of the design subspaces (via proposed equivalent conditions), removes the detected total infeasible design subspaces, and shrinks the remained parts to solve the design problem, iteratively. The proposed algorithm is applied to the cancer chemotherapy application and the results demonstrate that it is able to successively control the model considering the positivity condition. <br>