Hybrid Adaptive LQ Control Of Chemical Reactor

Abstract

The paper shows one approach to control of a nonlinear process represented by an isothermal continuous stirredtank reactor. The hybrid adaptive controller designed with the use of a polynomial synthesis, LinearQuadratic approach and spectral factorization was used for controlling of the product‘s concentration via change of the volumetric flow rate of the reactant. An adaptivity of the system is satisfied by the on-line recursive identification of a external linear delta model of the system. The proposed control method satisfies basic control requirements such as a stability, a disturbance attenuation and a reference signal tracking. All methods are tested by the simulation on the mathematical software MATLAB. INTRODUCTION A major group of systems in the nature not only in the industry has a nonlinear behavior. The chemical rector is a typical member of nonlinear processes widely used in the chemical or the biochemical industry. The behavior of such processes could be observed by experiments on the real system or its smaller real model (Vojtesek and Dostal 2008). This method produce more realistic results but it could be dangerous or time and money demanding. The other approach uses modeling techniques for creating of a mathematical model as an abstract representation of the system. The mathematical model in the form of the set of Ordinary Differential Equations (ODE) is then subjected to simulations which show the static and the dynamic behavior of the system. The role of the simulation grows nowadays with the increasing speed and the decreasing price of computers. The control of these processes with the conventional controllers with fixed parameters could lead to the unstable, inaccurate or unwanted output response when the state of the system change or the disturbance occurs. The adaptive control (Astrom and Wittenmark 1989) is one way how we can solve these problems. This control method uses idea from the nature where plants or animals “adapt” their behavior to the actual state or environmental conditions. The adaptive controller adapts parameters or the structure to parameters of the controlled plant according to he selected criterion (Bobal et al. 2005). The adaptive approach here is based on the choice of the External Linear Model (ELM) as a linear approximation of the originally nonlinear system, parameters of which are identified recursively and parameters of the controller are recomputed according to identified ones. The choice and the order of the ELM come from the dynamic analysis. The δ-models (Middleton and Goodwin 2004) used here are special type of discrete-time (DT) models parameters of which are related to the sampling period. It was proofed, that parameters of the δ-model approach to parameters of the continuous-time (CT) model for the small sampling period (Stericker and Sinha 1993). The polynomial synthesis (Kucera 1993) together with the spectral factorization and the Linear-Quadratic (LQ) approach were used for designing of the controller. The product of this synthesis is the continuous-time controller which satisfies basic control requirements such as the stability, the reference signal tracking and the disturbance attenuation. The resulted controller is called “hybrid” because it works in continuous-time but its parameters are recomputed in discrete time intervals together with the δ-ELM identification. The control technique was tested on the mathematical model of the isothermal Continuous Stirred-Tank Reactor (CSTR) the mathematical model of which is described by the set of five ordinary differential equations (Ingham et al. 2000). All results shown in this contribution come from the simulation on the mathematical model and they were done on the mathematical simulation software Matlab. ISOTHERMAL CHEMICAL REACTOR The nonlinear system under the consideration is an isothermal Continuous Stirred-Tank Reactor (CSTR). The schematic representation of this reactor is in Figure 1. The reactions inside the reactor could be described by the scheme: