Necessary and sufficient conditions for constraint satisfaction in switched systems using switch‐robust control invariant sets

Abstract

Summary This paper studies the control of constrained systems whose dynamics and constraints switch between a finite set of modes over time according to an exogenous input signal. We define a new type of control invariant sets for switched constrained systems, called switch–robust control invariant (switch‐RCI) sets, that are robust to unknown mode switching and exploit available information on minimum dwell‐time and admissible mode transitions. These switch‐RCI sets are used to derive novel necessary and sufficient conditions for the existence of a control‐law that guarantees constraint satisfaction in the presence of unknown mode switching with known minimum dwell‐time. The switch‐RCI sets are also used to design a recursively feasible model predictive controller (MPC) that enforces closed‐loop constraint satisfaction for switched constrained systems. We show that our controller is nonconservative in the sense that it enforces constraints on the largest possible domain, ie, constraints can be recursively satisfied if and only if our controller is feasible. The MPC and switch‐RCI sets are demonstrated on a vehicle lane‐changing case study.

An Efficient MPC Algorithm For Switched Systems with Minimum Dwell Time Constraints

“…Remark 4 Definition 2 is similar to the SRCI set defined in [37] in the way that the latter defines different feasible sets under different modes. The SRCI set notion defined in [37] is different and requires one step invariance for one mode and l-step invariance for others. A similar algorithm was presented in [37] for computing corresponding 1-step SRCI sets.…”

“…The SRCI set notion defined in [37] is different and requires one step invariance for one mode and l-step invariance for others. A similar algorithm was presented in [37] for computing corresponding 1-step SRCI sets. The adopted l-step SRCI definition in this paper requires l-step invariance under all modes and reduces to a subset of the SRCI definition in [37] only when l = 1.…”

“…A similar algorithm was presented in [37] for computing corresponding 1-step SRCI sets. The adopted l-step SRCI definition in this paper requires l-step invariance under all modes and reduces to a subset of the SRCI definition in [37] only when l = 1.…”

Set‐invariance‐based interpretations for the L1 performance of nonlinear systems with non‐unique solutions

“…Motivated by this advantage, there have been a number of studies on the viability property of discrete time systems. [15][16][17][18] On the other hand, in the case of continuous-time systems, the weak and strong invariances do not coincide with each other as discussed in Reference 12 since nonlinear differential equations do not have unique solutions in general. In connection with this, aiming at employing some useful properties in the strong invariance, the arguments in the previous studies [19][20][21] are confined themselves to (non)linear differential equations with unique solutions.…”

“…For discrete‐time systems, these two invariance properties coincide with each other because the solution for the corresponding difference equation is uniquely determined through a sort of algebraic computations. Motivated by this advantage, there have been a number of studies on the viability property of discrete time systems 15‐18 . On the other hand, in the case of continuous‐time systems, the weak and strong invariances do not coincide with each other as discussed in Reference 12 since nonlinear differential equations do not have unique solutions in general.…”