Control of discrete-time piecewise affine probabilistic systems using reachability analysis

This paper proposes an algorithmic approach to synthesize stabilizing control laws for discrete-time piecewise affine probabilistic (PWAP) systems based on computations of probabilistic reachable sets. The considered class of systems contains probabilistic components (with Gaussian distribution) modeling additive disturbances and state initialization. The probabilistic reachable state sets contain all states that are reachable with a given confidence level under the effect of time-variant control laws. The control synthesis uses principles of the ellipsoidal calculus, and it considers that the system parametrization depends on the partition of the state space. The proposed algorithm uses LMI-constrained semi-definite programming (SDP) problems to compute stabilizing controllers, while polytopic input constraints and transitions between regions of the state space are considered. The formulation of the SDP is adopted from a previous work in [1] for switched systems, in which the switching of the continuous dynamics is triggered by a discrete input variable. Here, as opposed to [1], the switching occurs autonomously and an algorithmic procedure is suggested to synthesis a stabilizing controller. An example for illustration is included.