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Hybrid Adaptive Control Of CSTR Using Polynomial Synthesis And Pole-Placement Method

This paper deals with simulation studies of the adaptive control on the continuous stirred tank reactor (CSTR) as a typical chemical equipment with nonlinear behaviour and continuously distributed parameters. Mathematical model of this reactor is described by the set of two nonlinear ordinary differential equations (ODE). The simulation of the steady-state and dynamics results in optimal working point and external linear model (ELM) which is used in adaptive control. Adaptive approach here is based on the recursive identification of ELM and parameters of the controller are recomputed in each step too. The polynomial approach together with the poleplacement method and spectral factorization satisfies basic control requirements although the system has negative control properties. INTRODUCTION The most of the proceses in technical praxis has nonlinear behaviour and conventional control methods could lead to unoptimal or unwanted output responses. However, adaptive control (Astrom 1989) is one way how to deal with the negative control properties such as non-minimum phase behaviour, nonlinearity, time delays etc. There are several adaptive approaches. The one used in this work is based on the choice of an External Linear Model (ELM) parameters of which are recomputed recursively during the control (Bobal et al. 2005). It means that the steady-state and dynamic analyses need to be done for contructing of optimal ELM. The ELM could be chosen from the range of the continuous-time or discrete-time models. Disadvantage of the continuous-time ELM is difficult on-line recursive identification. On the other hand, external delta models (Middleton and Goodwin 2004) used here for parameter estimation belong to the range of discrete models but parameters of these models approach up to some assumptions to their continuous-time counterparts (Stericker and Sinha 1993). Well known and widely used Ordinary recursive least squares method (Fikar and Mikles 1999) was used for parameter estimation during the control. The polynomial method (Kucera 1993) used for controller synthesis together with the pole-placement method ensures basic control requirements such as stability, reference signal tracking and disturbance attenuation. Two control system configurations (Grimble 1994) were used – the first with one degreeof-freedom (1DOF) which has regulator only in feedback part and the second with two degrees-offreedom (2DOF) with feedback and feedforward parts. The resulting controllers are hybrid because polynomial synthesis is made for continuous-time but recursive identification runs on the delta-model, which belongs to the class of discrete-time models. The Continuous Stired Tank Reactor (CSTR) which is an equipment widely used in chemical industry for its good control properties. Mathematical model of this plant is described by the set of two nonlinear ordinary differential equations which are solved numerically (Ingham et al. 2000). The system has two suitable input variables – volumetric flow rates of reactant and cooling and two outputs – product’s temperature and concentration. The article provides several control simulation studies on the CSTR, all of them were done on mathematical simulation software Matlab, version 6.5.1. MODEL OF THE PLANT The controlled process here is represented by the continuous stirred tank reactor (CSTR) and its scheme can be found in Figure 1. Due to the simplification we expect that reactant is perfectly mixed and reacts to the final product with the concentration cA(t). The heat produced by the reaction is represented by the temperature of the reactant T(t). Furthermore we also expect that volume, heat capacities and densities are constant during the control. A mathematical model of this system is derived from the material and heat balances of the reactant and cooling. The resulted model is then a set of two Ordinary Differential Equations (ODEs) (Gao et al. 2002):