The primary disadvantage of current design techniques for model predictive control (MPC) is their inability to deal explicitly with plant model uncertainty. In this paper, we present a new approach for robust MPC synthesis which allows explicit incorporation of the description of plant uncertainty in the problem formulation. The uncertainty is expressed both in the time domain and the frequency domain. The goal is to design, at each time step, a statefeedback control law which minimizes a "worst-case" infinite horizon objective function, subject to constraints on the control input and plant output. Using standard techniques, the problem of minimizing an upper bound on the "worst-case" objective function, subject to input and output constraints, is reduced to a convex optimization involving linear matrix inequalities (LMIs). It is shown that the feasible receding horizon state-feedback control design robustly stabilizes the set of uncertain plants under consideration. Several extensions...

The state space and input/output formulations of model predictive control are compared and preference is given to the former because of the industrial interest in multivariable constrained problems. Recently, by abandoning the assumption of a finite output horizon several researchers have derived powerful stability results for linear and nonlinear systems with and without constraints, for the nominal case and in the presence of model uncertainty. Some of these results are reviewed. Optimistic speculations about the future of MPC conclude the paper. 1 Introduction The objective of this paper is to review some major trends in model predictive control (MPC) research with emphasis on recent developments in North America. We will focus on the spirit rather than the details, i.e. we do not attempt to provide a complete list of all the relevant papers published during the last few years. 1 We will try to contrast the motivations driving the research in the different camps. There is...

Since 1984 there has been a concentrated e ort to develop e cient interior-point methods for linear programming (LP). In the last few years researchers have begun to appreciate a very important property of these interior-point methods (beyond their e ciency for LP): they extend gracefully to nonlinear convex optimization problems. New interior-point algorithms for problem classes such as semide nite programming (SDP) or second-order cone programming (SOCP) are now approaching the extreme e ciency of modern linear programming codes. In this paper we discuss three examples of areas of control where our ability to e ciently solve nonlinear convex optimization problems opens up new applications. In the rst example we show how SOCP can be used to solve robust open-loop optimal control problems. In the second example, we show how SOCP can be used to simultaneously design the set-point and feedback gains for a controller, and compare this method with the more standard approach. Our nal application concerns analysis and synthesis via linear matrix inequalities and SDP. Submitted to a special issue of Journal of Process Control, edited by Y. Arkun & S. Shah, for papers presented at the 1997 IFAC Conference onAdvanced Process Control, June 1997, Ban. This and related papers available via anonymous FTP at

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In order to ensure robust feasibility and stability of model predictive control (MPC) schemes, it is often necessary to optimise over feedback policies rather than open-loop trajectories. All specific proposals to date have required the solution of nonlinear programs and/or the solution of a large number of optimisation problems. In this paper we introduce a new stage cost and show that the use of this cost allows one to formulate a robustly stable MPC problem that can be solved using a single linear program. Furthermore, this is a multi-parametric linear program, which implies that the receding horizon control (RHC) law is piecewise affine, and can be explicitly pre-computed, so that the linear program does not have to be solved on-line. Two numerical examples are presented; one of these is taken from the literature, so that a direct comparison of solutions and computational complexity with earlier proposals is possible.

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This paper is concerned with the practical real-time implementability of robustly stable model predictive control (MPC) when constraints are present on the inputs and the states. We assume that the plant model is known, is discretetime and linear time-invariant, is subject to unknown but bounded state disturbances and that the states of the system are measured. In this paper we introduce a new stage cost and show that the use of this cost allows one to formulate a robustly stable MPC problem that can be solved using a single linear program. Furthermore, this is a multiparametric linear program, which implies that the receding horizon control (RHC) law is piecewise affine, and can be explicitly pre-computed, so that the linear program does not have to be solved on-line.

A framework for robustness analysis of input constrained finite receding horizon control is presented. Under the assumption of quadratic upper bounds on the finite horizon costs, we derive sufficient conditions for robust stability of the standard discrete-time linear-quadratic receding horizon control formulation. This is achieved by recasting conditions for nominal and robust stability as an implication between quadratic forms, lending itself to S-procedure tools which are used to convert robustness questions to tractable convex conditions. Robustness with respect to plant/model mismatch as well as for state measurement error is shown to reduce to the feasibility of linear matrix inequalities. Simple examples demonstrate the approach. Keywords: predictive control, optimal control, linear systems, robustness, S-procedure, LMI. 1 Introduction Receding horizon, moving horizon and model predictive control are names for a state feedback control technique where the control action is dete...