Controller Synthesis for Stochastic Systems with Persistent Noise via Semi-definite Programming

We propose convex controller synthesis algorithms for a class of stochastic differential equations (SDEs) with persistent noise. This includes SDEs in which the noise does not vanish at the equilibria of the system. Our performance criterion is Noise-to-State Stability (NSS) in the moments, which is a generalization of the input-to-state stability (ISS) for SDEs. We formulate synthesis algorithms that, in addition to guaranteeing asymptotic convergence in the case of zero input noise, ensure that an upper bound on the effect of input noise (defined by the Frobenius norm of the noise covariance) is minimized. In the case of linear SDEs, the algorithm is in terms of linear matrix inequalities and, in the case of polynomial data, the method is based on polynomial optimization. The method is illustrated by examples