Robust Model Predictive Controller with Terminal Weighting for Multivariable Dead-Time Processes ⋆ Tito Lu´ ıs Maia Santos ∗ Julio Elias Normey-Rico ∗ Daniel Lim ´ on Marruedo ∗∗ ∗ Departamento de Automac ¸˜ ao e Sistemas, Universidade Federal de Santa Catarina, Caixa Postal 476, CEP 88040-900, Florian´ opolis, Santa Catarina, Brasil (e-mail:tito,julio@das.ufsc.br) ∗∗ Departamento de Ingeniar´ ıa de Sistemas y Autom´ atica, Avda Camino de los Descubrimientos s/n, 41092 Sevilla, Espa˜ na (e-mail:limon@cartuja.us.es) Abstract: This paper presents a robust model predictive controller with terminal cost weighting based on a filtered Smith predictor structure. A special state-space representation is applied in order that the states are composed by past control actions and predicted outputs up to the nominal delay. This strategy can be used to control multiple dead-time multivariable processes is such way that: (i) a terminal cost is applied to ensure nominal stability and (ii) the filtered Smith predictor structure is used to improve robustness. The proposed formulation avoids the high order model representation of long dead-time processes. A simulation example is used to illustrate the usefulness of the proposed approach. Copyright c  IFAC 2009 Keywords: Dead-Time Compensator, Smith Predictor, Robustness, Stability 1. INTRODUCTION Multivariable systems with multiple dead times are often dif- ficult to control because process variable interaction and dead time effect are added imposing harder limits on the closed-loop performance (Normey-Rico and Camacho, 2007). Amongst a number of advantages, model predictive control (MPC) strat- egy is interesting because multivariable processes and systems with long dead times can be easily dealt with (Camacho and Bordons, 1999). Despite the advantages of the MPC strategies, it is not simple to deal with robustness issues mainly in industrial practice. As shown in Normey-Rico and Camacho (1999) for generalized predictive controller (GPC) and in Normey-Rico and Camacho (2007) for dynamic matrix controller (DMC), MPC strategies may be very sensitive to dead-time uncertainty. Moreover, ro- bustness can be related to MPC algorithm predictor structures used to compute the output predictions up to the dead time. To overcome the robust tuning limitations, it was proposed a modified GPC algorithm based on the filtered Smith predic- tor called SPGPC (Normey-Rico and Camacho, 1999, 2000). Initially, the multi-input multi-output (MIMO) SPGPC version, which was presented in Normey-Rico and Camacho (2000), could be applied to control open loop stable process. Later, it was shown in Normey-Rico and Camacho (2007) that this approach can be extended to control unstable MIMO processes using a general formulation called DTC-GPC. Unfortunately, due to receding horizon principle, optimality does not imply in stability (Mayne et al., 2000). By using the GPC, if the predic- tion and control horizon are not properly chosen, the closed- ⋆ The authors were funded by Conselho Nacional de Desenvolvimento Cient´ ıfico e Tecnol´ ogico (CNPq) and Agˆ encia Nacional do Petr´ oleo, G´ as Natural e Biocombustiveis (ANP), Brazil, under project aciPG-PRH No 34 ANP/MCT. loop system may be unstable. In Garcia-Gabin and Camacho (2002), for instance, it was discussed that the GPC may present internal stability problems to control processes with right-half plane transmission zeros (RHPZ) which cannot be noticed by simple inspection. Moreover, open loop unstable process may also be unstable if the objective function is ill posed (small prediction horizon) (Rossiter, 2003). Similarly to the GPC, nominal stability is not guaranteed if the DTCGPC is used. Thus, a constrained horizon filtered Smith predictive structure (CHSPPC) for SISO systems was presented in Torrico and Normey-Rico (2005). The CHSPPC, which is based on the CRHPC proposed in Clarke and Scattolini (1991), links the terminal constraint concept (stability) with filtered Smith predictor idea (robustness). The main advantage in use this concept comes from the fact that control horizon, the prediction horizon and control weighting can be chosen arbitrarily without worry about nominal stability. Although this was an innovative idea, the CHSPPC was formulated for SISO systems and it is known that the terminal constraint also called dead-beat terminal condition has some weaknesses (Rossiter, 2003): (i) dead-beat control is known to use very active input signals; (ii) it would tend to give poor performance, measured by way of typical quadratic performance indexes; (iii) it usually has poor robustness. Moreover, the terminal weighting concept, also called end-point weighting, is more general (Demircioglu and Clarke, 1993). Conceptually the terminal weighting idea can be directly ap- plied to MIMO dead-time process. However, due to the dead- time characteristic, the conventional state-space representation dimension depends on the discrete delay order which may cause implementation problems for long dead-time processes. More- over, all the advantages of the DTCGPC could not be used.