Efficient Nonlinear Model Predictive Control Rolf Findeisen ∗ andFrankAllg¨ower † Institute for Systems Theory in Engineering University of Stuttgart, Germany Moritz Diehl ‡ , H. Georg Bock § , and Johannes P. Schl¨ oder ¶ Interdisciplinary Center for Scientific Computing (IWR) University of Heidelberg, Germany Zoltan Nagy ‖ Faculty of Chemistry and Chemical Engineering “Babes-Bolyai” University of Cluj, Romania Abstract The growing interest in model predictive control for nonlinear systems, also called NMPC, is motivated by the fact that today’s processes need to be operated under tighter performance specifications to guarantee profitable and environmentally safe production. One of the remaining essential problems for NMPC is the high on-line computational load. At each sampling instant, a nonlinear optimal control problem must be solved. In this paper, we summarize recent results showing the practical applicability of NMPC for process control. We show how recent advances in NMPC theory and dynamic optimization can be used to make the real-time application of NMPC feasible even for high dimensional problems. As an application example the real-time control of a high purity distillation column is considered. Keywords Nonlinear model predictive control, Real-time optimization, Quasi-infinite horizon, Multiple shooting, Computational effort, Large scale, Distillation control Introduction Over the last two decades model predictive control (MPC), also referred to as moving horizon control or receding horizon control, has become an attractive feed- back strategy. Linear MPC approaches have found successful applications, especially in the process indus- try (Qin and Badgwell, 1996). Nowadays, tighter prod- uct quality specifications, increasing productivity de- mands and environmental regulations require systems to be operated closer to the boundary of the admissible op- erating region. To allow operation near the boundary, a linear model is often not adequate to describe the pro- cess dynamics. This motivates the use of nonlinear sys- tem models, non-quadratic cost functions and nonlinear constraints in the predictive framework, thus leading to nonlinear model predictive control (NMPC). Recently NMPC schemes with favorable properties in- cluding guaranteed closed-loop stability or reduced com- putational demand have been developed, see for exam- ple De Nicolao et al. (2000); Allg¨ower et al. (1999) for a review. Despite these advances concern has been raised that due to the high on-line computational load none of the available NMPC schemes can be used for real-time control in practice. This concern is based on the fact that at every sampling instant a high-dimensional non- linear, finite horizon optimal control problem has to be solved. In this paper we summarize results of an ongoing study (Nagy et al., 2000; Bock et al., 2000b; Allg¨ ower et al., 2000) showing the practical applicability of NMPC ∗ findeise@ist.uni-stuttgart.de † allgower@ist.uni-stuttgart.de ‡ moritz.diehl@iwr.uni-heidelberg.de § bock@iwr.uni-heidelberg.de ¶ schloeder@iwr.uni-heidelberg.de ‖ znagy@chem.ubbcluj.ro to medium/high dimensional processes. We consider the control of a high purity distillation column using NMPC. In contrast to (Nagy et al., 2000; Bock et al., 2000b) we consider the output feedback case in this paper. Our goal is to outline the key components for real- time application of NMPC. The conclusion is that a successful application of NMPC is possible even nowa- days, if a combination of special dynamic optimization strategies (Bock et al., 2000b; Biegler, 2000) and NMPC schemes with reduced online computational load (Chen and Allg¨ower, 1998; De Nicolao et al., 1996) is used. The paper is organized as follows: In the first section, we review NMPC strategies that require reduced computa- tional load. In the second section, one specially tailored dynamic optimization strategy for the solution of the oc- curring optimal control problems is described. Finally, the control of a high-purity distillation column is consid- ered. Nonlinear Model Predictive Control In Figure 1 the general principle of model predictive control is shown. For simplicity of exposition, we as- sume that the control and prediction horizon have the same length. Based on measurements obtained at time t, the controller predicts the future dynamic behav- ior of the system over a control horizon T c and de- termines the manipulated input such that a predeter- mined open-loop performance objective functional is op- timized. In order to incorporate some feedback mech- anism, the open-loop manipulated input function ob- tained is implemented only until the next measurement becomes available. We assume that this is the case ev- ery δ seconds (sampling time). Using the new measure- ment, at time t + δ, the whole procedure—prediction and optimization—is repeated to find a new input function. 374