Nonlinear Adaptive Control Of CSTR With Spiral Cooling In The Jacket

Abstract

The most of the processes not only in the industry has nonlinear behavior and control of such processes could be difficult. The controller here consists of linear and nonlinear part where the nonlinear part is derived from the static analysis and the linear part describes nonlinear elements in the loop by the External Linear Model (ELM), parameters of which are estimated recursively with the use of delta (δ-) models. The control synthesis employs polynomial approach with the pole assignment method. The proposed control method satisfies basic control requirements and it was tested by the simulations on the mathematical model of Continuous Stirred Tank Reactor (CSTR) with spiral cooling in the jacket as a typical member of the nonlinear system with lumped parameters. INTRODUCTION Chemical reactors are tools widely used not only in the chemical industry for production of various products. Although there are several types from the construction point of view such as batch, semi-bath etc. (Ingham et al. 2000), Continuous Stirred-Tank Reactors (CSTR) and tubular reactors are the most suitable for control purposes. The Continuous Stirred-Tank Reactor used in this work represents typical nonlinear plant described mathematically by the set of two nonlinear ordinary differential equations (ODE) (Gao et al. 2002). As it is described in (Vojtesek and Dostal 2010), this system has two stable and one unstable steady-state which could lead to very unstable or unoptimal output responses with the use of conventional control methods. One way how to overcome this inconvience is the use of the adaptive control (Astrom and Wittenmark 1989) which adopts parameters of the controller to the actual state of the system via recursive identification of the External Linear Model (ELM) as a linear representation of the originally nonlinear system (Bobal et al. 2005). The results of the adaptive control on this concrete mathematical model can be found for example in (Vojtesek et al. 2011). The control method used here is based on the combination of the adaptive control and nonlinear control. Theory of nonlinear control (NC) can be found for example in (Astolfi et al. 2008) and (Vincent and Grantham 1997), the factorization of nonlinear models of the plants on linear and nonlinear parts is described in (Nakamura et al. 2002) and (Sung and Lee 2004). The controller consists of a static nonlinear part (SNP) and a dynamic linear part (DLP). The static part is obtained from the steady-state characteristic of the system, its inversion, suitable approximation and its derivative. As a result of this nonlinear description, the linear part is then described by the external linear model with the use of delta (δ-) models (Middleton and Goodwin 2004) as a special type of discrete-time models which parameters approaches to the continuous ones for the small sampling period (Stericker and Sinha 1993). The polynomial approach (Kucera 1993) in the control synthesis can be used for systems with negative properties from the control point of view such as nonlinear systems, non-minimum phase systems or systems with time delays. Moreover, the pole-placed method with spectral factorization satisfies basic control requirements such as disturbance attenuation, stability and reference signal tracking. All graphs shown in this contribution come from the simulation on the mathematical model and they were done on the mathematical simulation software Matlab, version 6.5.1. CONTINUOUS STIRRED TANK REACTOR The controlled process under the consideration is the continuous stirred tank reactor (CSTR) with the spiral cooling in the jacket. The scheme of the system can be found in Figure 1. The complete mathematical description of the process is very complex and we must introduce some simplifications. At first, 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 Proceedings 26th European Conference on Modelling and Simulation ©ECMS Klaus G. Troitzsch, Michael Mohring, Ulf Lotzmann (Editors) ISBN: 978-0-9564944-4-3 / ISBN: 978-0-9564944-5-0 (CD) Ordinary Differential Equations (ODEs) (Gao et al. 2002):