Stochastic Sliding Mode Control and State Estimation

Abstract

In this chapter we show that the Sliding Mode Control (SMC) technique can be successfully applied to stochastic systems governed by the stochastic differential equations of the Ito type which contain additive as well as multiplicative stochastic unbounded white noise perturbations. The existence of a strong solution to the corresponding stochastic differential inclusion is discussed. To do this approach workable the gain control parameter is suggested to be done state-dependent on norms of system states. It is demonstrated that under such modification of the conventional SMC we can guarantee the exponential convergence of the averaged squared norm of the sliding variable to a zone (around the sliding surface) which is proportional to the diffusion parameter in the model description and inversely depending on the gain parameter. Then the behavior of a standard super-twist controller under stochastic perturbations is studied. The suggested analysis is based on the Lyapunov functions, designed for the stability analysis of the deterministic version of super-twist controllers. The major finding is that under stochastic (in fact, unbounded) perturbations the special selection of a gain-parameter of such controller, making it depending on this Lyapunov function and its gradient, provides the controller with an adaptivity property and guarantees the means square convergence of this function into the prespecified zone around the origin. Finally, The chapter deals with the problem of state estimation of “two-component” systems where the second vector - component may have unknown nonlinear Lipschitz-type dynamics and is subjected to stochastic perturbations of both additive and multiplicative types. Both vector components governed by a system of stochastic differential equations with state dependent diffusion. The system is supposed to be quadratically stable in the mean-squared sense. We consider a sliding mode observer with the gain parameter linearly depending on the norm of the output estimation error which is available during the process. It has the same structure as deterministic observer based on “the Equivalent Control Method”. The workability of the suggested observer is guaranteed for the group of trajectories with the probabilistic measure closed to one. All theoretical results are supported by numerical simulations.