Robust Model Predictive Control of Cyber-Physical Systems

Abstract

Cyber-physical systems (CPS) are integrations of computation, communications and physical processes. These systems inherently require an effective interaction between the physical and digital elements. Design of CPS involves both capturing the integrated dynamics with models that can provide accurate predictions, and effective controllers which can guarantee performance, particularly when dealing with safety critical applications. Model Predictive Control (MPC) is frequently used for the control of CPS since it can provide online decision making, thanks to its ability to handle dynamic and static constraints. Since MPC is a model-based technique, the presence of uncertainty when generating predictions could result in trajectories with significant variations. This becomes particularly challenging when generating methods for CPS controller design, since CPS model approximations are hard to synthesize, and often lead to non-negligible modeling errors. This motivates the study of methods that can provide guaranteed performance under the effects of uncertainty and dynamic system constraints. In this work, we consider systems evolving in discrete-time (which could come from the discretization of a CPS model), where the state variable is set-valued. The solutions to these systems are given by sequences of sets that are explicitly generated by a set-valued map and a constraint set. This representation is beneficial since it can easily encode constraints, uncertainty and system dynamics that exhibit non-determinism. Under this approach, we can also analyze the effects of inputs generated from anytime optimization routines, where, for instance, input variability can be represented as a set and propagated to analyze reachability, in order to guarantee safety. For these systems, to which we refer as set dynamical systems, we provide several results which can be used to synthesize controllers which achieve robust performance, embedded in the set-based formulation. First, we develop tools to locate omega limit sets by using Lyapunov-like conditions and an invariance principle for set dynamical systems. The latter uses non-strict Lyapunov functions to determine convergence properties of set-valued solutions and, conveniently, is similar in spirit to the invariance principles available for continuous and discrete-time systems. Next, we provide tools to certify asymptotic stability of sets for set dynamical systems. We develop such tools by following a similar path to the development of the continuous and discrete-time counterpart for classical dynamical systems. In this way, we exploit the invariance principle for set dynamical systems, and generate sufficient conditions for asymptotic stability that resemble those by Krasovskii and Lyapunov. These results are then used to provide - though not constructive - sufficient conditions for the design of stabilizing state-feedback controllers for such systems, which, unlike the classical approach, allow for feedback laws that involve sets rather than points. Finally, we use this set-valued state representation to generate prediction models used in a robust MPC formulation. In our proposed setting, since the state trajectory is set-valued, the cost functional uses set-to-points maps to characterize the cost associated to each set-valued trajectory. For this formulation, we propose conditions that, analogous to the classical MPC formulation, lead to feasible set-valued predictive controller formulations.