A Scalable Convex Min-Max Optimization Algorithm for Robust Control of Polytopic Positive Systems

In this paper, we investigate the robust control of a class of bilinear positive systems. The main objective is to minimize the H-2 norm of the controlled system while dealing with polytopic uncertainties. In contrast to traditional Lyapunov-based robust control problems with polytopic uncertainties, we establish that minimizing the objective function over the vertices of the polytopic set, rather than considering its infinite elements, is adequate for achieving the globally optimal solution. Following this, we present a customized algorithm that relies on partial cutting plane to address the min-max problem. The introduced robust control strategy exhibits global convergence to the optimal solution and demonstrates scalability with the number of states and the number of vertices in the uncertain family. Simulation results validate the effectiveness of the proposed strategy in handling uncertainties.