Saturday , June 23 2018

Simulation of Adaptive LQ Control of Nonlinear Process

Jiri Vojtesek
Department of Process Control, Faculty of Applied Informatics,
Tomas Bata University in Zlin,
nam. T. G. Masaryka 5555, Zlin, 760 01, Czech Republic

Petr Dostal
Department of Process Control, Faculty of Applied Informatics,
Tomas Bata University in Zlin
nam. T. G. Masaryka 5555, Zlin, 760 01, Czech Republic

Abstract: The contribution is focused on the adaptive control of the nonlinear system represented by the continuous stirred-tank reactor with the spiral cooling in the jacket. The mathematical model of this reactor is described by two nonlinear ordinary differential equations which were solved numerically. The adaptive controller is based on the choice of the external linear delta model of the originally nonlinear process, parameters of which are identified recursively and parameters of the controller are recomputed too. The structure of the controller was constructed with the use polynomial synthesis together with linear-quadratic approach. The resulted controller fulfills basic control requirements and it can be used for system with negative control properties. All experiments were made by simulations in the MATLAB environment.

Keywords: Adaptive control, External linear model, Recursive identification, Polynomial synthesis, Linear quadratic theory, Continuous stirred-tank reactor, Nonlinear system.

>>Full text
Jiri VOJTESEK, Petr DOSTAL, Simulation of Adaptive LQ Control of Nonlinear Process, Studies in Informatics and Control, ISSN 1220-1766, vol. 21 (3), pp. 315-324, 2012.

1. Introduction

The use of the computer simulation not only in the control engineering grows rapidly nowadays with the increasing speed of the computers and low prices of the hardware. Furthermore, the simulation is very often used at present as it has many advantages over an experiment on a real system, which is not feasible and can be dangerous, time and money demanding. A modelling of the system usually precedes the simulation [1]. The mathematical model is a kind of abstract representation of the process which uses input, state or output variables, relations between these variables collected in the set of mathematical equations [1] and [2]. Some simulation and modelling examples can be found also in [3] and [4].

It is known, that almost all processes in the nature have a nonlinear behaviour [1], [5] and our goal is to cope with this nonlinearity. Typical examples of nonlinear systems are chemical reactors. A chemical reactor is a vessel or pipe which is used for the production of chemicals used in chemical, biochemical, drug and other industries through a specific reaction inside [6]. The controlled system here is represented by a Continuous Stirred-Tank Reactor (CSTR) as a typical member of a group of nonlinear systems used not only in the chemical industry. The mathematical model of the plant is described by the set of two nonlinear Ordinary Differential Equations (ODE) [7].

The thorough analysis of the system usually precedes the controller design. Steady-state and dynamic analyses as a typical simulation tools gives overview of system’s behavior especially for chemical reactors [5], [8], [9] etc. The methods used here was a Simple iteration method [10] and a Runge-Kutta’s standard method for the numerical solving of set of ODE. Big advantage of both methods is that both are easily programmable or even build-in functions in popular mathematical software, such as MATLAB [11], Mathematica etc.

The idea of an adaptive control [12] comes from the nature where every organism even humans try to “adapt” for the current environment. Transformed to the control theory, the controller also adapts parameters, structure etc. to the actual state of the controlled plant or desired course of the output signal [13]. The adaptive approach here is based on the approximation of the nonlinear system by the appropriate linear model, parameters of which are estimated online.

The structure of the controller uses the polynomial synthesis [14] with Linear Quadratic (LQ) theory [15]. Resulted controller fulfills basic requests for the control loop such as stability, reference signal tracking and disturbance attenuation – [14] and [16].

Although there could be found a lot of contributions dealing with the simulation of control, the goal of this contribution is to describe the procedure from the steady-state and dynamic analyses to the design of the hybrid adaptive controller for temperature control inside the CSTR as a typical member of the nonlinear processes. This method could be applied to similar nonlinear processes which are described also by the mathematical model.

The contribution is divided into six main parts. The second section after this introduction describes the mathematical model of the controlled plant (CSTR) and the results of steady-state and dynamic analyses. Then, the third part describes theoretical background to the adaptive control with recursive identification and the LQ approach and polynomial synthesis of the controller. The fourth part is dedicated to various simulation experiments of the proposed controller on the mathematical model. After that, the last part before conclusion presents usability of the controller to the real plant followed by the final conclusion and future work.

All experiments were done by the simulations on the mathematical software MATLAB, version 7.0.3.


  1. INGHAM, J., I. J. DUNN, E. HEINZLE, J. E. PRENOSIL, Chemical Engineering Dynamics. An Introduction to Modelling and Computer Simulation. Second, Completely Revised Edition, VCH Verlagsgesellshaft, Weinheim, 2000.
  2. LUYBEN, W. L., Process Modelling, Simulation and Control for Chemical Engineers. McGraw-Hill, New York, 1989.
  3. LEBRUN, M., D. VASILIU, N. VASILIU, Numerical Simulation of the Fluid Control Systems by AMESim. Studies in Informatics and Control, vol. 18(2), 2009, pp. 111-118.
  4. CHORUKOVA, E., I. SIMEONOV, Neural and Hybrid Modelling of Biotechnological Process, Studies in Informatics and Control, vol. 17(3), 2008, pp. 305-31.
  5. CORRIOU, J.-P., Process Control. Theory and Applications. Springer-Verlag London. 2004.
  6. SCHMIDT, L. D., The Engineering of Chemical Reactions. Oxford University Press, 1997.
  7. GAO, R., A. O’DYWER, E. COYLE, A Non-linear PID Controller for CSTR Using Local Model Networks. Proceedings of the 4th World Congress on Intelligent Control and Automation. Shanghai, China, 2002, pp. 3278-3282.
  8. GAZDOS, F., L. MACKU, Control-Oriented Simulation Analysis of a Semi-Batch Reactor, International Review of Automatic Control, vol. 2(5), 2009, pp. 584-591. ISSN 1974-6059.
  9. MACKU, L., D. SAMEK, Two Step, PID and Model Predictive Control using Artificial Neural Network Applied on Semi-batch Reactor, WSEAS Transactions on Systems, vol. 9 (10), 2010, ISSN 1109-2777.
  10. JOHNSTON, R. L., Numerical Methods. John Wiley & Sons, 1982.
  11. MATHEWS, J. H., K. K. FINK, Numerical Methods Using Matlab, Prentice-Hall, 2004.
  12. ASTROM, K. J., WITTENMARK, B., Adaptive Control, 2nd edition, Prentice Hall, 1994.
  13. BOBAL, V., J. BOHM, J. FESSL, J. MACHACEK, Digital Self-tuning Controllers. Algorithms, Implementation and Applications, Springer, 2005.
  14. KUCERA, V., Diophantine Equations in Control – A Survey. Automatica, vol. 29, 1993, ISSN 1361-1375.
  15. HUNT, K. J., V. KUCERA, M. SEBEK, Optimal Regulation using Measurement Feedback. A Polynomial Approach. IEEE Transactions on Automation Control, vol. 37(5), pp. 682-685.
  16. KUCERA, V., Analysis and Design of Discrete Linear Control Systems. Prentice-Hall, London 1991.
  17. STERICKER, D. L., N. K. SINHA, Identification of Continuous-time Systems from Samples of Input-Output Data using the -operator. Control-Theory and Advanced Technology, vol. 9, 1993, pp. 113-125.
  18. MUKHOPADHYAY, S., A. G. PATRA, G. P. RAO, New Class of Discrete-time Models for Continuous-time Systems. International Journal of Control, vol. 55, pp. 1161-1187.
  19. VOJTESEK, J., P. DOSTAL, Adaptive Control of Continuous-Stirred Tank Reactor in Two Stable Steady-States, in Proceedings of the IFAC Workshop Adaptation and Learning in Control and Signal Processing (ALCOSP 2010), Antalya, Turkey, 2010.
  20. VOJTESEK, J., P. DOSTAL, Adaptive LQ Approach Used in Conductivity Control inside Continuous-Stirred Tank Reactor, in Proceedings of the 17th IFAC World Congress, Seoul, 2008, pp. 12929-12934.