Uniting bounded control and MPC for stabilization of constrained linear systems

Abstract

In this work, a hybrid control scheme, uniting bounded control with model predictive control (MPC), is proposed for the stabilization of linear time-invariant systems with input constraints. The scheme is predicated upon the idea of switching between a model predictive controller, that minimizes a given performance objective subject to constraints, and a bounded controller, for which the region of constrained closed-loop stability is explicitly characterized. Switching laws, implemented by a logic-based supervisor that constantly monitors the plant, are derived to orchestrate the transition between the two controllers in a way that safeguards against any possible instability or infeasibility under MPC, reconciles the stability and optimality properties of both controllers, and guarantees asymptotic closed-loop stability for all initial conditions within the stability region of the bounded controller. The hybrid control scheme is shown to provide, irrespective of the chosen MPC formulation, a safety net for the practical implementation of MPC, for open-loop unstable plants, by providing a priori knowledge, through off-line computations, of a large set of initial conditions for which closed-loop stability is guaranteed. The implementation of the proposed approach is illustrated, through numerical simulations, for an exponentially unstable linear system.

Introduction

Virtually all practical control systems are subject to hard constraints on their manipulated inputs. One of the key limitations imposed by input constraints is the restriction on the set of initial states of the closed-loop system that can be steered to the origin with the available control action. For a control policy to be effective in dealing with the problem of input constraints, it needs to provide not only the stabilizing feedback control law, but also an explicit characterization of the set of initial conditions starting from where constrained closed-loop stability is guaranteed. The absence of an a priori explicit characterization of this set can make an impact on the practical implementation of the given control policy by requiring extensive closed-loop simulations over the whole set of possible initial conditions, to check for stability, or by limiting the operation within an unnecessarily small and conservative neighborhood of the operating point. These considerations have motivated significant work on the design of stabilizing bounded control laws that guarantee explicitly defined, large regions of attraction for the closed-loop system (e.g., Lin & Sontag, 1991; Teel, 1992). Recently, in El-Farra and Christofides 2001, El-Farra and Christofides 2003, a class of bounded robust Lyapunov-based nonlinear controllers, inspired in part by the results on bounded control originally presented in Lin and Sontag (1991), was developed. The controllers enforce robust stability in the closed-loop system and provide, at the same time, an explicit characterization of the region of guaranteed closed-loop stability. Despite their well-characterized stability and constraint-handling properties, the above controllers are not necessarily optimal with respect to an arbitrary performance criterion (in El-Farra & Christofides, 2001, it is shown that these controllers are inverse optimal with respect to meaningful costs).

Currently, model predictive control (MPC), also known as receding horizon control (RHC), is one of the few control methods for handling constraints within an optimal control setting. Numerous research investigations into the stability properties of MPC have led to a plethora of MPC formulations that focus on closed-loop stability (see, for example, Allgower & Chen, 1998; Morari & Lee, 1999; Mayne, Rawlings, Rao, & Scokaert, 2000) for extensive surveys of these developments). For open-loop stable and integrating linear systems with bounded inputs, infinite horizon formulations have been developed that are globally stabilizing (e.g., Rawlings & Muske, 1993; Zheng & Morari, 1995). This progress notwithstanding, the issue of obtaining, a priori (i.e. before controller implementation), an analytic characterization of the region of constrained closed-loop stability for MPC controllers remains to be adequately addressed, particularly for open-loop (exponentially) unstable plants where, unlike stable plants, stabilization is the overriding requirement. Part of the difficulty in this regard owes to the fact that, unlike in analytical bounded control, the stability of MPC feedback loops depends on a complex interplay between several factors such as the stabilizability of the initial condition, the penalties in the performance index, and the choice of the control horizon. A priori knowledge of the stability region of MPC requires an explicit characterization of these interplays. This difficulty can have an impact on the implementation of MPC on unstable plants by requiring extensive closed-loop simulations over the whole set of possible initial conditions, to check for stability.

Motivated by the above, we propose in this paper a controller switching strategy that merges the bounded control approach with MPC in a way that allows both approaches to complement the stability and optimality properties of each other. The guiding principle in realizing this strategy is the idea of decoupling optimality from the characterization of the region of constrained closed-loop stability. Specifically, by relaxing the optimality requirement, an explicit bounded feedback control law is designed and an explicit large estimate of the region of constrained closed-loop stability, which is not unnecessarily conservative, is computed. An optimal MPC controller that minimizes a given cost functional subject to the same constraints is then designed and implemented within the stability region of the bounded controller. Switching laws, that place appropriate restrictions on the evolution of the closed-loop trajectory under MPC within the stability region are then constructed to orchestrate the transition between the two controllers in a way that guarantees closed-loop stability for all initial conditions within the stability region. The switching scheme is shown to provide a safe mechanism for the practical implementation of MPC, especially for open-loop unstable systems, by providing, through off-line computations, a priori knowledge of a large set of initial conditions for which closed-loop stability is guaranteed.

The idea of switching, between different controllers (or models), for the purpose of achieving some objective that either cannot be achieved or is more difficult to achieve using a single controller has been widely used in the literature, and in a variety of contexts (e.g., Rugh & Shamma, 2000; Hespanha et al., 2001; Bemporad & Morari, 1999; Banerjee & Arkun, 1998; Aufderheide, Prasad, & Bequette, 2001). In this work, switching is employed between two structurally different, though complementary, control approaches as a tool for reconciling the objectives of optimal stabilization of the constrained closed-loop system (through MPC) and the a priori (off-line) determination of set of initial conditions for which closed-loop stability is guaranteed (through bounded control). The rest of the paper is organized as follows. In Section 2, we present the class of systems considered and review briefly how the constrained control problem is addressed in both bounded control and MPC. We then proceed in Section 3 to formulate the controller switching problem and propose a number of switching schemes that, with varying degrees of flexibility, address the problem. Finally, the implementation of the proposed switching schemes is demonstrated through numerical simulations.