Application of nonlinear system transformations to control design for a chemical reactor. J. Alvarez-Gallegos. Indexing term: Control theory. Abstract: The design ...
Application of nonlinear system transformations to control design for a chemical reactor J. Alvarez-Gallegos
Indexing term: Control theory
Abstract: The design of a nonlinear regulator for the control of a nonisothermal continuous stirred tank reactor (CSTR) is derived using the technique of nonlinear system transformation. It is shown that recent developments in the area of nonlinear transformation theory have direct application to the problem of controlling nonisothermal CSTRs. It is established that the transformation method leads to the design of robust and stable nonlinear regulators for these systems. List of symbols
V C T Tj Fo F Fj Co To Tj0 a E R X p Cp U A Vj pj Cj 1
= volume in the reactor = product concentration = reactor temperature = temperature of cooling water in jacket = input flow = output flow = cooling water flow = feed concentration = feed temperature = inlet cooling water temperature = pre-exponential factor = activation energy = perfect gas constant = heat of reaction = density of process liquid = heat capacity of process liquid = overall heat transfer coefficient = heat transfer area = volume of jacket = density of cooling water = heat capacity of cooling water Introduction
Nowadays, the processes constituting the chemical processing industry are very complex systems, connected to each other and interacting strongly among themselves. One of these systems, the chemical reactor, is one of the main examples of this set because chemical transformations of the processed materials are carried out in it. Paper 5903D (C8/C9) first received 18th March 1986 and in revised form 16th October 1987 Mr. Alvarez-Gallegos was formerly with the Department of Electrical Engineering, Imperial College of Science and Technology, Exhibition Road, London SW7 2BT, UK, and is now with the Centro de Investigation y de Estudios Avanzados del IPN, Departamento de Ingenieria Electrica, Apartado Postal 14-740,07000 Mexico 90
Thus, it is important to handle adequately this part of the system. There are several types of reactors, but in this work only one of them will be considered: the continuous stirred tank reactor (CSTR). A CSTR is used to convert reactants into products. The reactant is fed continuously into a vessel. There, a chemical reaction takes place and yields the desired product. Usually the chemical reaction generates heat. When the heat generated cannot be removed by the flow of material, and to keep the reactor temperature within safety and operating specifications, the heat removal is effected by extracting heat through a reactor diathermal wall. This is achieved by exchanging heat with a coolant medium that is circulated through a jacket. It is well known in the chemical engineering literature [1] that in some cases the reactor possesses a high state sensitivity to external disturbances. When this is the case and a disturbance is present, the ability to remove heat with a conventional PID controller may be surpassed by the process heat generation. This fact may force the process designer to sacrifice efficiency to assure adequate safety and operational levels. Moreover, this type of reactor may exhibit multiple stationary points, and it has been suggested [2] that in some cases it would be desirable to operate a reactor at an unstable stationary point, or to operate it in an induced periodic transient regime. These observations suggest that more powerful nonlinear control strategies could find application in the control of CSTRs. Most industrial reactor control is done by linear PID strategies. On the other hand, the usual approach to designing a controller based on state-space multivariable control theory is to linearise, with a truncated Taylor series approximation, the reactor model so that linear control techniques may be applied. These approaches perform well when the process is intrinsically robust and stable. However, some problems arise if the operating point is changed or internal, and if internal disturbances are present. In recent years, a variety of approaches have been used in the study of the synthesis of estimation and control algorithms for CSTRs. Applications of modern estimation and control techniques to CSTRs have been extensively reported in the last few years. For example, Lynch and Ramirez [8] have designed a time optimal controller with a Kalman filter for state estimation in a CSTR; Huang, Chao and Cheng [4] have used an adaptive control approach, also in a CSTR; Cebuhar and Costanza [3] applied a bilinear control, also to a CSTR. These authors compared the performance of the bilinear controller with conventional controllers, and concluded that moving from linear to nonlinear reactor control is feasible and convenient. Nevertheless, the bilinear technique is still based on an approximation to the nonlinear IEE PROCEEDINGS, Vol. 135, Pt. D, No. 2, MARCH 1988
model. Furthermore, the dimension of the resulting bilinear control model is doubled. The use of the concept of system transformation to linear controllable systems, in order to design nonlinear regulators for some class of nonlinear systems, was published some years ago [9, 10, 11]. The purpose of this work is to describe methods for designing nonlinear regulators using systems transformation techniques which allow regulation of the main variables of a nonisothermal CSTR. The design of the regulator will show how some theorems recently developed in the area of geometric nonlinear control theory approach a CSTR problem, offering an interesting solution. 2
as the perturbation variables. Then, the incremental model used for the synthesis of the control law is given by: - 9xx -
