LMI-based robust model predictive control and its application to an industrial CSTR problem Fen Wu * Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC 27695, USA Received 10 February 2000; received in revised form 13 September 2000; accepted 14 September 2000 Abstract In this paper, robust model predictive control MPC) is studied for a class of uncertain linear systems with structured time- varying uncertainties. This general class of uncertain systems is useful for nonlinear plant modeling in many circumstances. The controller design is characterizing as an optimization problem of the ``worst-case'' objective function over in®nite moving horizon, subject to input and output constraints. A sucient state-feedback synthesis condition is provided in the form of linear matrix inequality LMI) optimizations, and will be solved on-line. The stability of such a control scheme is determined by the feasibility of the optimization problem. To demonstrate its usefulness, this robust MPC technique is applied to an industrial continuous stirred tank reactor CSTR) problem with explicit input and output constraints. Its relative merits to conventional MPC approaches are also discussed. # 2001 Elsevier Science Ltd. All rights reserved. Keywords: Model predictive control; Uncertain linear system; Structured uncertainty; Linear matrix inequality; Industrial application 1. Introduction Model predictive control MPC) techniques are widely used in industrial process control practice [19,10]. MPC solves an on-line optimization problem at each step to compute an optimal control pro®le over ®nite horizon of future time. Typically a sequence of predicted control moves will be calculated, but only the ®rst one is implemented. At the next sampling time, the optimization problem is solved again with new mea- surements, and control input is updated. The key advantage of such a methodology compared with many other control techniques is that it can handle input/output constraints directly. Although it is popular in industrial applications, most MPC techniques lack a guarantee of stability except for some special cases. The reason is that MPC is in principle a computational approach, and an analytic expression for the controller is generally not available. This limits further study of closed-loop stability properties which is based on such information. Standard MPC schemes virtually have no guaranteed robustness because they use nominal models and perform ®nite horizon optimization. In the presence of model mismatch, this type of algorithm could behave poorly. It is conjectured that the lack of robust stability on existing applications relates to the use of low per- formance controllers on open-loop stable processes. On the other hand, robust control design techniques were developed to address modeling uncertainty expli- citly, but there is no straightforward extension to deal with input/output constraints. Indirectly such constraints can be treated by transforming them from time domain to frequency domain, which often requires some insight of the real problem and expertise to choose appropriate weighting functions. Therefore, it is desirable to com- bine existing MPC and robust control techniques to develop a control synthesis methodology which can handle both modeling inaccuracy and hard input/output constraints simultaneously. The robust stability properties of MPC have been studied by many researchers from dierent aspects. Morari and Za®riou [16] discussed how to improve the robust stability of unconstrained MPC in the frame- work of internal model control IMC) by tuning IMC ®lters. By modifying the optimization problem to a ``min-max'' problem, Campo and Morari [4] studied 0959-1524/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0959-152400)00052-4 Journal of Process Control 11 2001) 649±659 www.elsevier.com/locate/jprocont * Tel.: +1-919-515-5268; fax; +1-919-515-7968. E-mail address: fwu@eos.ncsu.edu