The ellipsoid algorithm for probabilistic robust controller design

This paper presents a new iterative approach to probabilistic robust controller design, which is an alternative to the recently proposed Subgradient Iteration Algorithm (SIA). In its original version (Polyak and Tempo, (2001)) the SIA possesses the useful property of guaranteed convergence in a finite number of iterations, but requires that the radius of a non-empty ball contained in the solution set is known a-priori. This rather restrictive assumption was later on released in Calafiore and Polyak, (2001) but only at the expense of an increased number of iterations. The approach in this paper does also not require the knowledge of such a radius, and offers a significant improvement even over the original SIA in terms of the maximum number of possible correction steps that can be executed before a feasible solution is reached. Given an initial ellipsoid that contains the solution set, the approach iteratively generates a sequence of ellipsoids with decreasing volumes, all containing the solution set. A method for finding an initial ellipsoid containing the solution set is also proposed. The approach is illustrated on a real-life diesel actuator benchmark model.