Computing controlled invariant sets for hybrid systems with applications to model-predictive control

Abstract

In this paper, we develop a method for computing controlled invariant sets using Semidefinite Programming. We apply our method to the controller design problem for switching affine systems with polytopic safe sets. The task is reduced to a semidefinite programming problem by enforcing an invariance relation in the dual space of the geometric problem. The paper ends with an application to safety critical model predictive control.

A simple hierarchy for computing controlled invariant sets

“…Although their approach was proposed for a more general class of discrete-time hybrid systems, we adapt it to the discrete-time linear systems case. For the example above, in Figure 6, the incremental union of the white set with each of the green stripes represents the CIS obtained with our method using an FTS with 1 to 5 states respectively, and in orange is the ellipsoidal CIS obtained from [22]. This figure highlights one of the advantages of our method, which stems from the proposed hierarchy.…”

“…If the computed CIS is deemed insatisfactory, an incrementally larger CIS can be obtained by increasing the states of the FTS. In Figure 6 we can observe that by using L = 2 we already obtain a CIS larger than the one produced by the techniques in [22]. For the next experiment we consider the system of a truck with N trailers [30], which is shown in Fig.…”

“…To further demonstrate the performance and flexibility of our method, we compare with the approach proposed in [22]. There, the authors utilize Sum-Of-Squares (SOS) programming to compute ellipsoidal CISs.…”

“…In the same spirit, two methods [1,22] compute exact CISs and also guarantee finite termination. In [22], an efficient method was presented using SDPs that computes an ellipsoidal CIS for a class of hybrid systems.…”

“…In the same spirit, two methods [1,22] compute exact CISs and also guarantee finite termination. In [22], an efficient method was presented using SDPs that computes an ellipsoidal CIS for a class of hybrid systems. In turn, [1] proposed a novel method that computes CISs by lifting the problem to a higher dimensional space, in which invariant hyper-rectangles are computed in closed-form.…”