Simulation Of The 2DOF Nonlinear Adaptive Control Of A Chemical Reactor

The paper deals with continuous-time nonlinear adaptive control of a continuous stirred tank reactor. The control strategy is based on an application of the controller consisting of a linear and nonlinear part. The static nonlinear part is derived in the way of an inversion and exponential approximation of measured or simulated input-output data. The design of the two degrees of freedom (2DOF) dynamic linear part is based on approximation of nonlinear elements in the control loop by a continuous-time external linear model with directly estimated parameters. In the control design procedure, the polynomial approach with the pole assignment method is used. The nonlinear adaptive control is tested by simulations on the nonlinear model of the CSTR with a consecutive exothermic reaction. INTRODUCTION Continuous stirred tank reactors (CSTRs) are units frequently used in chemical and biochemical industry. From the system theory point of view, CSTRs belong to a class of nonlinear systems with mathematical models described by sets of nonlinear differential equations. Their models are derived and described in e.g. (Corriou 2004; Ogunnaike and Ray 1994; Schmidt 2005). It is well known that the control of chemical reactors often represents very complex problem. The control problems are due to the process nonlinearity and high sensitivity of the state and output variables to input changes. In addition, the dynamic characteristics may exhibit a varying sign of the gain in various operating points as well as non-minimum phase behaviour. 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 exploits a linear adaptive controller with parameters computed and readjusted on the basis of recursively estimated parameters of an appropriate chosen continuous-time external linear model (CT ELM) of the process. Some results obtained by this method can be found in e.g. (Dostál et al. 2007; Dostál et al. 2009). An effective approach to the control of CSTRs and similar processes utilizes various methods of the nonlinear control (NC). Several modifications of the NC theory are described in e.g. (Astolfi et al. 2008; Vincent and Grantham 1997; Ioannou and Fidan 2006; Zhang et al. 2000). Especially, a large class of the NC methods exploits linearization of nonlinear plants, e.g. (Huba and Ondera 2009), an application of PID controllers, e.g. (Tan et al. 2002; Bányász and Keviczky 2002) or factorization of nonlinear models of the plants on linear and nonlinear parts, e.g. (Nakamura et al. 2002; Vallery et al. 2009; Chyi-Tsong Chen1 et al. 2006; Vörös 2008; Sung and Lee 2004). In this paper, the CSTR control strategy is based on an application of the controller consisting of a static nonlinear part (SNP) and dynamic linear part (DLP). The static nonlinear part is obtained from simulated or measured steady-state characteristic of the CSTR, its inversion, exponential approximation, and, subsequently, its differentiation. On behalf of development of the linear part, the SNP including the nonlinear model of the CSTR is approximated by a continuous-time external linear model (CT ELM). For the CT ELM parameter estimation, the direct estimation in terms of filtered variables is used, see e.g. (Rao and Unbehauen 2005; Garnier and Wang 2008). The method is based on filtration of continuous-time input and output signals where the filtered variables have in the s-domain the same properties as their nonfiltered counterparts. Then, the resulting 2DOF CT controller is derived using the polynomial approach and pole assignment method, e.g. (Kučera 1993). The simulations are performed on a nonlinear model of the CSTR with a consecutive exothermic reaction. MODEL OF THE CSTR Consider a CSTR with the first order consecutive exothermic reaction according to the scheme 1 2 A B C k k ⎯⎯→ ⎯⎯→ and with a perfectly mixed cooling jacket. Using the usual simplifications, the model of the CSTR is described by four nonlinear differential equations A r r 1 A Af r r d c q q k c c dt V V ⎛ ⎞ = − + + ⎜ ⎟ ⎝ ⎠ (1) Proceedings 25th European Conference on Modelling and Simulation ©ECMS Tadeusz Burczynski, Joanna Kolodziej Aleksander Byrski, Marco Carvalho (Editors) ISBN: 978-0-9564944-2-9 / ISBN: 978-0-9564944-3-6 (CD) B r r 2 B 1 A Bf r r d c q q k c k c c dt V V ⎛ ⎞ = − + + + ⎜ ⎟ ⎝ ⎠ (2) r r r h rf r c r r r r r ( ) ( ) ( ) ( ) p p dT h q A U T T T T dt c V V c ρ ρ = + − + − (3) c c h f c r c c c c ( ) ( ) ( ) c p dT q A U T T T T dt V V c ρ = − + − (4) with initial conditions s A A (0) c c = , s B B (0) c c = , s r r (0) T T = and s c c (0) T T = . Here, t is the time, c are concentrations, T are temperatures, V are volumes, ρ are densities, cp are specific heat capacities, q are volumetric flow rates, Ah is the heat exchange surface area and U is the heat transfer coefficient. The subscripts are denoted ()r for the reactant mixture, ()c for the coolant, ()f for steady-state inputs and the superscript () for initial conditions. The reaction rates and the reaction heat are expressed as 0 r exp , 1,2 j j j E k k j RT − ⎛ ⎞ = = ⎜ ⎟ ⎝ ⎠ (5) r 1 1 A 2 2 B h h k c h k c = + (6) where k0 are pre-exponential factors, E are activation energies and h are reaction entalpies. The values of all parameters, inlet values and steady-state values with used units are given in Tab. 1. Table 1: Parameters, Steady-State Inputs and Initial Conditions. Vr = 1.2 m Vc = 0.64 m ρr = 985 kg m ρc = 998 kg m k10 = 5.616 × 10 min k20 = 1.128 × 10 min h1 = 4.8 × 10 kJ kmol cpr = 4.05 kJ kgK cpc = 4.18 kJ kgK Ah = 5.5 m U = 43.5 kJ mminK E1/ R = 13477 K E2/ R = 15290 K h2 = 2.2 × 10 kJ kmol s A c = 1.5796 kmol m -3 s r T = 324.80 K s B c = 1.1975 kmol m -3