Constrained Control of Linear Systems Subject to Combinations of Intersections and Unions of Concave Constraints

The explicit reference governor (ERG) is an add-on unit that provides constraint handling capabilities to pre-stabilized systems. In particular, the ERG acts on the reference of the pre-stabilized system in such a way that, even for large transients, constraints are satisfied. A standard way to build an ERG is to translate state and input constraints into a constraint on the value of the Lyapunov function associated to the currently applied reference. The main challenge of this approach is the determination of the largest reference-dependent Lyapunov level set that ensures constraint satisfaction. In general, the optimization problem to compute such a level set is non-convex. This letter proposes a novel systematic approach for designing an ERG for linear systems subject to combinations of intersections and unions of concave constraints. More precisely, first, it is shown that the solution of the stated non-convex problem can be approximated by solving multiple convex problems (one convex optimization problem for each constraint) and then suitably combining their solutions. Then, since these convex problems do not admit a closed-form solution, a virtual continuous-time system is proposed to estimate their solutions. Finally, an upper-bound for the estimation error is provided analytically, and a procedure is proposed to increase the robustness of the ERG against estimation errors. The effectiveness of the proposed scheme is demonstrated on a simulated case study.