Stochastic optimization of large-scale parametrized dynamical systems

Abstract

Many problems in systems and control, such as controller synthesis and observer design, can be viewed as optimization problems involving dynamical systems: For instance, maximizing closed-loop performance in the controller synthesis setting. When the system includes large-scale, sparse state–space models, the optimization becomes computationally challenging. Existing methods in literature lack computational scalability or only solve an approximate version of the problem. We propose a method to locally minimize the $H_2$ norm of a differentiable parametrized dynamical system that resolves these issues. We do this by estimating the gradient of the $H_2$ norm using samples of the frequency response function, which can be obtained efficiently for large-scale, sparse state–space models. We prove that the scheme is guaranteed to preserve stability with high probability under boundedness conditions on the step size used in the optimization. We also obtain probabilistic guarantees that our method converges to a local minimizer. The method is applicable to problems involving non-realizable or infinite-dimensional dynamics. We demonstrate the effectiveness of the approach on two numerical examples.