Reachability Analysis of Discrete-Time Systems With Disturbances

Abstract—This paper presents new results that allow one to com-pute the set of states that can be robustly steered in a finite number of steps, via state feedback control, to a given target set. The as-sumptions that are made in this paper are that the system is dis-crete-time, nonlinear and time-invariant and subject to mixed con-straints on the state and input. A persistent disturbance, depen-dent on the current state and input, acts on the system. Existing results are not able to address state- and input-dependent distur-bances and the results in this paper are, therefore, a generaliza-tion of previously published results. One of the key aims of this paper is to present results such that one can perform the relevant set computations using polyhedral algebra and computational ge-ometry software, provided the system is piecewise affine and the constraints are polygonal. Existing methods are only applicable to piecewise affine systems that either have no control inputs or no disturbances, whereas the results in this paper remove this limita-tion. Some simple examples are also given that show that, even if all the relevant sets are convex and the system is linear, convexity of the set of controllable states cannot be guaranteed. Index Terms—Controllability, control systems, nonlinear sys-tems, Piecewise affine systems, reachability analysis, robustness, set invariance. I