Stochastic Differential Contraction in Nonlinear System Analysis

By adopting on a notion of stochastic differential contraction, the paper presents new results on the incremental mean squared gain (IMSG) analysis of nonlinear systems with stochastic inputs. The relative power between two stochastic processes is defined as the asymptotic average (over time) of the second moment of the point-wise distance (in some metric) between the two processes. The IMSG of a system is then defined as the relative power between two output trajectories driven by two independent instances of i.i.d. inputs with unit relative power. While contracting metrics have been previously used for analysis of nonlinear systems, the formulation and analysis method in this paper lead to new conditions that yield a more accurate upper bound on the system gain. The idea is to introduce a notion of stochastic differential contraction which does not explicitly embed an exponential rate of contraction. This approach is more suitable for analysis of systems with stochastic inputs. In particular, and unlike previous approaches, the standard H2-norm analysis results for linear systems can be recovered as a special case in this setting.