**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.

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**CITE THIS PAPER AS**:

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.

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