Jiri Vojtesek

Simulation Of Adaptive Control Of A Continuous Stirred Tank Reactor.

The paper deals with simulation of continuous-time adaptive control of a continuous stirred tank reactor (CSTR). The control design is based on approximation of a nonlinear model of the process by a continuoustime external linear model.with parameters estimated using a corresponding delta model. The control system with two feedback controllers is considered. The controller design is based on the polynomial approach and the pole assignment method. The adaptive control is tested by simulations on the nonlinear model of the CSTR with a consecutive exothermic reaction. INTRODUCTION Continuous stirred tank reactors (CSTRs) belong to a class of nonlinear systems where both steady-state and dynamic behaviour are nonlinear. Their models are derived and described in e.g. (Ogunnaike and Ray 1994; Schmidt 2005; Corriou 2004). The process nonlinearities may cause difficulties when controlling using conventional controllers with fixed parameters. One possible method to cope with this problem is using adaptive strategies based on an appropriate choice of an external linear model (ELM) with recursively estimated parameters. These parameters are consequently used for parallel updating of controller‘s parameters. The control itself can be either continuous-time or discrete. Experiences of authors in the field of control of nonlinear technological processes indicate that the continuoustime (CT) approach gives better results when controlling processes with strong nonlinearities. For the CT ELM parameters estimation, either the direct method or application of an external delta model with the same structure as the CT model can be used (Dostal et al. 2004). In this paper, an approach based on the delta model was applied. The basics of delta models have been described in e.g. (Middleton and Goodwin 1990; Mukhopadhyay et al. 1992; Goodwin et al. 2001). Although delta models belong into discrete models, they do not have such disadvantages 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). Complete description and experimental verification can be found in (Stericker and Sinha 1993). This contribution presents in brief a procedure of a CSTR control design and, consequently, the control simulation results. The parameters of the CT ELM are obtained via corresponding delta model parameter estimation. The control system with two feedback controllers is used according to (Ortega and Kelly 1984; Dostal et al. 2007). This set-up gives better control results for the reference tracking than the use of only a feedback controller. Input signals for the control system are considered as step functions. The resulting continuous-time controllers derived using polynomial and pole assignment methods (Kucera 1993) guarantee stability of the control system, asymptotic tracking of step references and step load disturbances attenuation. The simulations are performed on a nonlinear model of the CSTR with a consecutive exothermic reaction.

Adaptive LQ Approach Used in Conductivity Control inside Continuous-Stirred Tank Reactor

This paper deals adaptive control of the real model represented by the Continuous-stirred tank reactor (CSTR). This type of reactor belongs to the class of systems with lumped-parameters. The chemical process inside the reactor is dilution of the salt with the clean water. The resulted mixture has specific conductivity, which is about to be controlled, depends on the content of the salt inside the reactant. Used adaptive approach is based on recursive identification of the systems parameters during the control. A polynomial approach used for the controller synthesis has satisfied control requirements and moreover, it could be used for systems with negative properties such as nonlinearity, non-minimum phase etc.

Adaptive Control of Continuous-Stirred Tank Reactor in Two Stable Steady-States

Abstract The paper deals with the adaptive control of the nonlinear process represented by the Continuous Stirred Tank Reactor (CSTR). This type of reactor belongs to the class of systems with lumped-parameters. Mathematical model of this system consist of two nonlinear Ordinary Differential Equations (ODEs). The steady-state analysis uncovers two stable and one unstable operating points. Used adaptive approach is based on recursive identification of the systems parameters during the control. A polynomial approach used for the controller synthesis has satisfied control requirements and moreover, it could be used for systems with negative properties such as nonlinearity, non-minimum phase etc. The goal of this work is to compare control for two stable steady-states.

Modelling And Simulation Of Water Tank.

The modelling and simulation play a very important role in the industry where it can help with the description of the system and the choice of the optimal control strategy. This contribution is focused on the modelling and simulation procedure which usually precedes the design of the controller. The mathematical model is derived with the use of material balance and produces nonlinear Ordinary Differential Equation (ODE). The static analysis provides optimal working point and the dynamic analysis gives an overview about the behavior of the system. Mentioned procedure is tested on the real model of the water tank as a part of the process control teaching system PCT40 from Armfield. Results have shown that proposed mathematical model is accurate and can be used for the design of the appropriate controller.

Effect of Weighting Factors in Adaptive LQ Control

An adaptive control is a technique with strong theoretical background and lots of applications to the abstract and real systems. The big advantage can be found in usability of this control method for systems with negative control properties such as nonlinearity, time-delay, non-minimum behavior etc. The adaptive approach here is based on the choice of the external linear model of the originally nonlinear system parameters of which are updated in defined time moments via recursive identification. The control synthesis employs polynomial approach with linear-quadratic approach and spectral factorization. Resulted controller has two weighting factors as tuning parameters. This paper explores the effect these factors to the control. All proposed approaches were tested by simulations on the mathematical model of the continuous stirred-tank reactor as a typical member of the nonlinear lumped-parameters systems.

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. (Dostal et al. 2007; Dostal 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; Banyasz 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; Voros 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. (Kucera 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

Maze Navigation on Ball & Plate Model

Today’s CCD or CMOS image sensors are advanced enough to satisfy the need for accurate object detection and tracking. This leads to implementation of computer vision into industry, transportation, medicine, robotics and other sectors. The aim of this paper is to present steps needed to determine correct path through the maze constructed on a plate and navigate a ball along this path. Image processing techniques used here are simple enough to understand, so students can easily implement them to further extend educational capabilities of Ball & Plate model. The paper also shows the use of watershed transform, which can be extended for similar problems. The added maze thus provides excellent application for the model and simulates real-world issues in research and development.

Cascade control of a tubular chemical reactor

The paper presents the cascade control design of a tubular chemical reactor with an exothermic consecutive reaction. The control is performed in primary and secondary control-loops where the primary controlled output of the reactor is the concentration of the main reaction product and the secondary output is the mean temperature of the reactant. A common control input is the coolant flow rate. The controller in the primary control-loop is a P-controller with the gain calculated using simulated or measured steady-state characteristics of the reactor. The controller in the secondary control-loop is an adaptive controller. The proposed method is verified by control simulations.

Nonlinear versus Ordinary Adaptive Control of Continuous Stirred-Tank Reactor

Unfortunately, the major group of the systems in industry has nonlinear behavior and control of such processes with conventional control approaches with fixed parameters causes problems and suboptimal or unstable control results. An adaptive control is one way to how we can cope with nonlinearity of the system. This contribution compares classic adaptive control and its modification with Wiener system. This configuration divides nonlinear controller into the dynamic linear part and the static nonlinear part. The dynamic linear part is constructed with the use of polynomial synthesis together with the pole-placement method and the spectral factorization. The static nonlinear part uses static analysis of the controlled plant for introducing the mathematical nonlinear description of the relation between the controlled output and the change of the control input. Proposed controller is tested by the simulations on the mathematical model of the continuous stirred-tank reactor with cooling in the jacket as a typical nonlinear system.

Numerical Solution of Ordinary Differential Equations Using Mathematical Software

The differential equation is mathematical tool widely used for description various linear or nonlinear systems and behaviour in the nature not only in the industry. The numerical solution of the differential equation is basic tool of the modelling and simulation procedure. There are various types of numerical methods, the ones described in this contribution comes from the Taylor’s series and big advantage of all of them is in easy programmability or even more some of them are included as a build-in functions in mathematical softwares such as Mathematica or MATLAB. The goal of this contribution is to show how proposed Euler and Runge-Kutta’s methods could be programmed and implemented into MATLAB and examine these methods on various examples. The comparable parameters are accuracy and also speed of the computation.