Stability Analysis for Stochastic Piecewise Affine Systems with Verification via Generalized Fokker Planck Equations

In this article, stability analysis for Stochastic Piecewise Affine Systems which are a subclass of stochastic hybrid systems is investigated. Here, additive noise signals are considered that does not vanish at equilibrium points. These noises will prohibit the exponential stochastic stability discussed widely in literature. Also, the jumps between the subsystems in this class of stochastic hybrid systems are state-dependent which make stability analysis more complex. The presented theorem considering both additive noise and state-dependent jumps, gives upper bounds for the second stochastic moment or variance of Stochastic Piecewise Nonlinear Systems trajectories and guarantees that stable systems have a steady state probability density function. Then, linear case of such systems is studied where the stability criterion is obtained in terms of Linear Matrix Inequality (LMI) and an upper bound on state covariance is obtained for them. Next, to validate the proposed theorem, solving the Fokker Plank equations which describes the evolution of probability density function, is addressed. A solution for the problem of boundary conditions that arises from jumps in this class of systems is given and then with finite volume method the corresponding partial differential equations are solved for a case study to check the results of the presented theorem numerically.