Reach set computation and control synthesis for discrete-time dynamical systems with disturbances

Control problems with hard bounds on the control values, restrictions on the state trajectory over a finite time horizon, and guaranteed behavior despite the disturbances, are difficult to solve using frequency based design methods. Such problems have received much attention over the past decade. In order to address them one needs to study system evolution in the time domain. The central concept that emerges in such studies is that of the reach set. This paper is devoted to the formulation of the reachability problem for discrete-time dynamical systems with disturbances. The concept of maxmin and minmax forward and backward reach sets is addressed. Invariance of the backward reach set is discussed. The emphasis of the paper is on discrete-time linear systems, for which the ellipsoidal computational method is described. The synthesis of maxmin and minmax closed-loop control for steering the system to a given target set using ellipsoidal backward reach set approximations is explained. The ellipsoidal method covered in the paper is implemented in the Ellipsoidal Toolbox for MATLAB, a popular collection of ellipsoidal calculus routines freely available online.