Brief Communication - Some GPC stability results

Model based predictive control (MBPC) has attracted much attention since the 1990s (see, for example, Garcia et al. (1989) and Richalet (1993)). The early MBPC algorithms, such as MAC (Richalet et al. 1978), DMC (Cutler and Ramaker 1980), EHAC (Ydstie 1984) and GPC (Clarke et al. 1987), are mainly based on input/ output models. Each has a few di€ erent features which distinguish it from the others in the form of plant model, cost function, etc. Despite their popularity and success, they all su€ er from a lack of clear theory dealing with closed-loop stability. A few theorems have been presented (Clarke and Mohtadi 1989, Scattolini and Bittanti 1990, Rawlings and Muske 1993) but the problem still remains inadequately examined. In recent years, using fruitful results in state-space theory, some new types of MBPC algorithms, such as RHTC (Kwon and Byun 1989), SIORHC (Mosca and Zhang 1992), and CRHPC (Clarke and Scattolini 1991) have emerged. State-feedback control laws capable of stabilizing linear plants (Kleinman 1970, 1974, Kwon and Pearson 1975, 1978) have equipped these new algorithms with guaranteed stability properties under certain conditions. Simultaneously, extensions to basic MBPC schemes have also been developed with improved closed-loop stability (Demircioglu and Clarke 1993, Yoon and Clarke 1992). Among the various kinds of MBPC methods, GPC can be regarded as a representative, for it is very general and is regarded as a superset of many others (Clarke and Mohtadi 1989). This paper deals with the GPC stability issue in its original form. Other than the methods presented above, we derived an explicit expression of a GPC closed-loop transfer function. This makes it possible to investigate the GPC stability directly via a closed-loop eigen polynomial. This work is organized as follows. Section 2 formulates the closed-loop description of GPC, which underlies the following theoretical research. In section 3, stability conditions for a GPC closedloop system and limiting cases are studied. Section 4 outlines design guidelines for plants with monotonic or convex step responses parameters. Finally, our conclusions are given in section 5.