Simulation Of Adaptive Control Of A Tubular Chemical Reactor

The paper deals with continuous-time adaptive control of a tubular chemical reactor with the countercurrent cooling as a nonlinear single input – single output process. The nonlinear model of the reactor is approximated by an external linear model with parameters estimated via corresponding delta model. The control system structure with two feedback controllers is considered. The resulting controllers are derived using the polynomial approach. The method is tested on a mathematical model of the tubular chemical reactor. INTRODUCTION Tubular chemical reactor are units frequently used in chemical industry. From the system theory point of view, tubular chemical reactors belong to a class of nonlinear distributed parameter systems with mathematical models described by sets of nonlinear partial differential equations (NPDRs). The methods of modelling and simulation of such processes are described e.g. in (Luyben 1989), (Ingham et al. 1994) and (Dostál et al. 2008). It is well known that the control of chemical reactors, and, tubular reactors especially, often represents very complex problem. The control problems are due to the process nonlinearity, its distributed nature, and high sensitivity of the state and output variables to input changes. Evidently, the process with such properties is hardly controllable by conventional control methods, and, its effective control requires application some of advanced methods. One possible method to cope with this problem is using adaptive strategies based on an appropriate choice of a continuous-time external linear model (CT ELM) with recursively estimated parameters. These parameters are consequently used for parallel updating of controller‘s parameters. Some results obtained in this field were presented by authors of this paper e.g. in (Dostál et al. 2004). For the CT ELM parameter estimation, either the direct method or application of an external delta model with the same structure as the CT model can be used, e.g. (Middleton and Goodwin 1990) or (Mukhopadhyay et al. 1992). Although delta models belong into discrete models, they do not have such disadvantageous properties connected with shortening of a sampling period as discrete z-models. In addition, parameters of delta models can directly be estimated from sampled signals. Moreover, it can be easily proved that these parameters converge to parameters of CT models for a sufficiently small sampling period (compared to the dynamics of the controlled process), as shown in (Stericker and Sinha 1993). This paper deals with continuous-time adaptive control of a tubular chemical reactor with a countercurrent cooling. With respect to practical possibilities of a measurement and control, the mean reactant temperature temperature is chosen as the controlled output, and, the coolant flow rate as the control input. The nonlinear model of the reactor is approximated by a CT external linear model with a structure chosen on the basis of computed controlled output step responses. The control structure with two feedback controllers is considered, e.g. (Dostál et al. 2007). The resulting controllers are derived using the polynomial approach (Kučera 1993) and the pole assignment method, e.g. (Bobál et al. 2005). The method is tested on a mathematical model of a tubular chemical reactor. MODEL OF THE REACTOR An ideal plug-flow tubular chemical reactor with a simple exothermic consecutive reaction 1 2 k k A B C → → in the liquid phase and with the countercurrent cooling is considered. Heat losses and heat conduction along the metal walls of tubes are assumed to be negligible, but dynamics of the metal walls of tubes are significant. All densities, heat capacities, and heat transfer coefficients are assumed to be constant. Under above assumptions, the reactor model can be described by five PDRs in the form 1 A A r A c c v k c t z ∂ ∂ + = − ∂ ∂ (1) 1 2 B B r A B c c v k c k c t z ∂ ∂ + = − ∂ ∂ (2)