GPC Control of a Fractional–Order Plant: Improving Stability and Robustness Miguel Romero,* Blas M. Vinagre,** Ángel P. de Madrid*** *Departamento de Sistemas de Comunicación y Control, UNED, Madrid, Spain (Tel: +34 91 398 71 47; e–mail: mromero@bec.uned.es). **Departamento de Ingeniería Eléctrica, Electrónica y Automática, Universidad de Extremadura, Badajoz, Spain (e–mail:bvinagre@unex.es) *** Departamento de Sistemas de Comunicación y Control, UNED, Madrid, Spain, (e–mail: angel@scc.uned.es)} Abstract: This work deals with the use of Generalized Predictive Control (GPC) with fractional order plants. Low integer–order discrete approximations will be used as models to design the controllers. The stability and robustness of the closed loop system will be studied with the Nyquist criterion. Three techniques will be proposed to enhance robustness: the improvement of the model response at low frequencies, the use of the prefilter T(z –1 ), and a new recommendation to choose two of the parameters (the control horizon N u and the error weighting sequence λ) of the GPC controller. 1. INTRODUCTION Fractional Calculus can be defined as integration and differentiation of noninteger order. Fractional differentiation (integration) is the generalization of the derivative (integral) operator D n (D –n ) using real or even complex values for the ordinary integer value n (Oldham and Spanier, 1974; Podlubny, 1999a). Fractional integro–differential calculus generally uses two definitions: (1) Grünwald–Letnikov (GL) and (2) Riemann– Liouville (RL): ( ) 0 0 () lim ( 1) k j t kh h j D ft h f kh jh j α α α - = → = ⎛ ⎞ = - - ⎜ ⎟ ⎝ ⎠ ∑ (1) 1 0 1 () ( ) () ( ) n t n n d D ft t f d n dt α α τ τ τ α - - = - Γ - ∫ (2) with α > 0 for derivation and α < 0 for integration. The Laplace domain is frequently used to describe the fractional operations. Expression (3) is given as Laplace transform of the Riemann–Liouville derivative/integral (2) under zero initial conditions (Oldham and Spanier, 1974): { } () () LD ft s Fs α α ± ± = (3) For a wide class of functions, which appear in real physical and engineering applications, the two definitions GL and RL are equivalent. For this reason, RL is usually used for algebraic manipulations and GL (together with the short memory principle), for numerical integration and simulation (Podlubny, 1999a). Fractional order controllers have been used to enhance system performance. Typical fractional order controllers include the CRONE control (Oustaloup, et al., 1995) and the PI Ȝ D ȝ controller (Petráš, 1999; Podlubny, 1999b). More control applications are described in (Vinagre and Chen, 2002; Oustaloup, 2006). Model–Based Predictive Control (MPC) has been proposed to control plants with fractional dynamics (Romero, et al., 2007). Predictive control has been in use in the process industries during the last 30 years, where it has become an industry standard due to its intrinsic ability to handle input and state constraints for large scale multivariable plants (Maciejowski, 2002; Rossiter, 2003; Camacho and Bordóns, 2004). In this paper low integer–order discrete approximations will be used as models to design the controllers, so a model– process mismatch will appear. For this reason the stability and robustness of a fractional order plant with a predictive control law will be studied and some methods to improve them will be proposed. This paper is organized as follows: In section 2 GPC, one of the most representative predictive controllers, is introduced. Section 3 describes how to study the stability and robustness of a GPC control loop with a fractional order plant. In section 4 this study is illustrated with some examples. In section 5 some techniques to improve the robustness are proposed. Finally, section 6 draws the main conclusions of this work. 2. GENERALIZED PREDICTIVE CONTROL GPC stands for Generalized Predictive Control (Clarke, et al., 1987a, 1987b), one of the most representative predictive controllers due to its success in industrial and academic applications (Clarke, 1988). All predictive controllers share a common methodology: at each “present” instant t, future process outputs y(t+k|t) are Proceedings of the 17th World Congress The International Federation of Automatic Control Seoul, Korea, July 6-11, 2008 978-1-1234-7890-2/08/$20.00 © 2008 IFAC 14266 10.3182/20080706-5-KR-1001.3100