The Randomized Ellipsoid Algorithm for Constrained Robust Least Squares Problems

In this chapter a randomized ellipsoid algorithm is described that can be used for finding solutions to robust Linear Matrix Inequality (LMI) problems. The iterative algorithm enjoys the property that the convergence speed is independent on the number of uncertain parameters. Other advantages, as compared to the deterministic algorithms, are that the uncertain parameters can enter the LMIs in a general nonlinear way, and that very large systems of LMIs can be treated. Given an initial ellipsoid that contains the feasibility set, the proposed approach iteratively generates a sequence of ellipsoids with decreasing volumes, all containing the feasibility set. A method for finding an initial ellipsoid is also proposed based on convex optimization. For an important subclass of problems, namely for constrained robust least squares problems, analytic expressions are derived for the initial ellipsoid that could replace the convex optimization. The approach is finally applied to the problem of robust Kalman filtering.