The Equivalence between Model Predictive Control and Anti-Windup Control Schemes

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Introduction
Controller design for linear or nonlinear processes with actuator saturation nonlinearities has long been studied within various contexts. Kothare (1997a) provides a good discussion on the various approaches that have appeared in literature. There are two distinct classes of structures that handle input saturation nonlinearities: (a) On-line optimization based control structures, such as Model Predictive Control (MPC), and (b) anti-windup bumpless transfer (AWBT) controllers that have a closed form and do not perform online optimization. Controllers in the first class rely on a process model and on-line solution of a constrained optimization problem that minimizes an objective over a future horizon. If properly designed, such controllers can provide optimality, robustness, and other desirable properties. However, because of the time needed to perform the on-line optimization, controllers in the first class are usually implemented on relatively slow processes. Controllers in the second class completely bypass on-line optimization, therefore they are inherently faster and can be used on faster processes. The AWBT controller design approach is based on the following two-step design paradigm: Firstly, a linear controller is designed ignoring input constraints. In the next step, an anti-windup scheme is added to compensate for the adverse effects of the constraints on closed-loop performance. There have been many heuristic techniques for the design of AWBT controllers, virtually all of which can be summarized into a structure that includes a saturation nonlinearity in the forward path and a linear transfer function in the feedback path. Campo (1997) and Kothare et al. (1994) unified all the existing AWBT schemes and developed a general framework for studying stability and robustness issues. The importance of that work lies in that model uncertainty can be taken into account systematically and theory exists to analyze the closed-loop system for stability and robustness. However, their analysis is also based on the standard conic sector nonlinear stability theory. Therefore, the results could be potentially conservative. The design of AWBT controllers for SISO systems relies on a mix of intuitive and rigorous arguments, which become difficult to use in the MIMO case. As pointed by Doyle et al. (1987), for MIMO controllers, the saturation may cause a change in the plant input direction resulting in disastrous consequences. Through an example, Doyle et al. (1987) showed that all anti-windup schemes of the time failed to work on MIMO systems. Recently, Kothare et al. (1997b) described three performance requirements that should be incorporated in a multi-objective multivariable AWBT controller synthesis framework. Although promising lines for designing an AWBT controller using dynamic output feedback and one-step design were outlined, many of the details like "recovery of linear performance" need to be worked out. Model Predictive Control (MPC) is a discrete-time technique in which the control action is obtained by repeatedly solving an on-line optimization problem at each time step. The flexibility of MPC has been useful in addressing various implementation issues that traditionally have been problematic. From a practical viewpoint, the most important feature of MPC is its ability to naturally and explicitly handle multivariable input and output constraints by direct incorporation into the on-line optimization. The issues of stability and robustness of MPC are now a fairly well understood topics (Rawlings and Muske, 1993; Nikolaou, 1998). For linear plants the MPC problem is usually reduced to a quadratic program (QP) which can be solved efficiently (Cutler et al., 1980; García et al., 1986). Nonlinear plants, however, lead to nonlinear programs (NLP) which are in general non-convex and very demanding computationally (Biegler et al., 1991; Mayne et al., 1991; Mayne et al., 1990). Evidence of the difficulty and effort involved in controlling a real experimental system by solving the general NLP is shown in (Kreshenbaum et al., 1994). The MPC strategy was first exploited and successfully employed on linear plants, especially in the process industries, where relatively slow sampling times made extensive on-line intersample computation feasible. But for a fast process the implementation of MPC may be quite difficult and sometime infeasible. Both MPC and AWBT controllers have their own advantages and disadvantages. However, to the best of our knowledge, no clear relationship between these two control schemes has established. In this work, we rigorously show that there is a direct relationship between MPC and AWBT control. In fact, we show that the general structure of AWBT controllers emerges naturally from the structure of MPC with quadratic objective, input constraints, and plant model structure affine in the input variables. This realization is important for a number of reasons, including the following: • It provides theoretical justification for the empirical realization that virtually all heuristically developed AWBT control structures follow a similar pattern involving a nonlinear (saturation) block and linear transfer functions (Kothare, 1997a).