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 poleplacement 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 stirredtank reactor with cooling in the jacket as a typical nonlinear system.
The control of the chemical processes in the industry is always challenging because of the nonlinearity of the major group of systems. The continuous stirredtank reactor (CSTR) is one of the most common used types of chemical reactors because of easy controllability [
The adaptive control [
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 [
The controlled system, CSTR, with originally nonlinear behavior could be mathematically described for the control purposes by the external linear model (ELM) [
The results are also compared with classical adaptive control which uses only ELM as a linear representation of the originally nonlinear controller [
The proposed control strategies were verified by simulations on the mathematical model of CSTR with cooling in the jacket [
The system under the consideration is a continuous stirredtank reactor (CSTR) with the socalled
Continuous stirredtank reactor with cooling in the jacket.
If we introduce common simplifications like the perfect mixture of the reactant, all densities, transfer coefficients, heat capacities, and the volume of the reactant are constant throughout the reaction, and the mathematical model developed with the use of material and heat balances inside has form of the set of ordinary differential equations (ODEs) [
The variable
Equations (
Fixed parameters of the CSTR.
Name of the parameter  Symbol and value of the parameter 

Volume of the reactant 

Density of the reactant 

Heat capacity of the reactant 

Weight of the coolant 

Heat capacity of the coolant 

Surface of the cooling jacket 

Heat transfer coefficient 

Preexponential factor for reaction 1 

Preexponential factor for reaction 2 

Preexponential factor for reaction 3 

Activation energy of reaction 1 to 

Activation energy of reaction 2 to 

Activation energy of reaction 3 to 

Enthalpy of reaction 1 

Enthalpy of reaction 2 

Enthalpy of reaction 3 

Input concentration of compound 

Input temperature of the reactant 

It is good to know behavior of the system before the design of the controller. This behavior is usually obtained from the steadystate and dynamic analyses of the system which will be described in the next subchapters.
This analysis observes the behavior of the system in the steadystate, that is, in time
Results of steadystate analyses for different volumetric flow rate of the reactant
Steadystate characteristics of the product’s concentration
Both graphs show highly nonlinear steadystate behavior of this system.
The second, dynamic, analysis shows the response of the system to the step change of the input quantity. Although there could be theoretically four input quantities, the volumetric flow rate of the reactant,
Results of dynamic analysis for the step changes of the heat removal of the cooling,
Results of dynamic analysis for the step changes of the volumetric flow rate of the reactant,
The control strategy here is based on the factorization of controller into the static nonlinear part (SNP) and the dynamic linear part (DLP); see Figure
The scheme of the nonlinear controller.
As written in the previous part, there are theoretically four input and four output variables. In this case, the change of the output concentration,
The dynamic part DLP in Figure
The schematic representation of the control system can be found in Figure
Control scheme of the nonlinear adaptive control.
The nonlinear part uses properties of the system in the steadystate described above.
If we do the steadystate characteristic for the volumetric flow rate of the reactant,
The steadystate characteristic (a) and noised data in new coordinates
Due to later approximation and better unification of the variables, the new
It is common that the measured data on the real system are affected by the measurement errors—see Figure
The difference of the input volumetric flow rate of the coolant is from (
The values of
The procedure for computing of the value of this derivative for the specific value of products concentration,
The simulated and approximated steadystate characteristic in new coordinates (a) and the course of the derivative of
For example, the exponential function in the general form
As there is the derivative
The course of this function is shown in Figure
The dynamic behavior of the controlled system, in our case CSTR, together with the SNP derived above is observed for the step responses of the input
Results of dynamic analyses for the changes of input
The gain of the system SNP+CSTR is computed as
Although the system has nonlinear behavior, presented output dynamic responses could be described by the first order continuoustime transfer function
The online identification of the continuoustime ELM (
The deltamodel introduces a new complex variable
The continuous model (
Equation (
The last part from Figure
The scheme of this control configuration is shown in Figure
One degreeoffreedom (1DOF) control configuration.
On the other hand, polynomials of the ELM
There are several ways to construct this optional polynomial, for example, the poleplacement method, LQ approach, and so forth. The choice here combines the poleplacement method with spectral factorization of the identified polynomial
The control synthesis presented above is derived in the continuoustime, but identification and recomputation of the controllers parameters run in discretetime (
It is good to show how the nonlinear adaptive control could improve classic adaptive control described, for example, in [
Let us consider the control configuration displayed in Figure
Control scheme of the classic adaptive control.
This means that system is controlled only with the use of adaptive controller based on the ELM without the knowledge about static behavior of the system. The design and computation of the controller are the same as what is described in Sections
Results of this control are displayed and commented on in the next section.
The goal of this last section is to verify proposed classic and nonlinear adaptive controllers by simulations on the mathematical model (
Figure
The course of the output variable
Outputs from the LDP
The course of identified parameters during the control is shown in Figure
The course of identified parameters
The task of this contribution was also to show improvement of the nonlinear adaptive approach compared with the classic adaptive control described in Section
The course of the output variable
The courses of the computed input variable,
Compared control results for
Comparison of resulted courses of output variable
The paper deals with the adaptive control of the CSTR as a typical member of the nonlinear system with lumped parameters. The mathematical model of such system is described by the set of four nonlinear ordinary differential equations and simulation is in this case related to the numerical solution of this set of ODEs. The static and dynamic analysis have shown high nonlinearity of this system which means that controlling of such process with conventional control methods could lead to suboptimal or even very bad control results. The adaptive control is one way to how we can overcome this problem. The adaptive approach here was based on the choice of the delta external linear model of the originally nonlinear system, parameters of which are identified recursively during the control, and the parameters of the controller are also recomputed according to these identified ones. This method satisfies appropriate reaction of the controller to the change of the state of the system or the random disturbance. The control synthesis employs polynomial theory together with the poleplacement method and spectral factorization. These methods satisfy basic control requirements such as stability, reference signal tracking, and disturbance attenuation. The contribution shows also the improvement of this socalled classic adaptive control by the nonlinear theory which is based on the Wiener system where the controller is divided into the dynamic linear part and the static nonlinear part. The dynamic linear part is the same as in classic adaptive control but the static nonlinear part uses simulated or measured steadystate characteristics of the mathematical model to describe the relation between controlled concentration of the product and the change of the reactants volumetric flow rate as an input variable. Both controllers could be tuned by the choice of the parameter
The authors declare that there is no conflict of interests regarding the publication of this paper.