Control of a non-isothermal continuous stirred tank reactor by a feedback–feedforward structure using type-2 fuzzy logic controllers
Introduction
Systems characterized by high nonlinearities are difficult to control by controllers developed using linearized models, like traditional PID controllers. Although these controllers may be tuned in order to be effective at certain conditions, they are not very robust and may also destabilize the whole system if some parameters change. This is particularly true for systems which present bifurcations. These non linear systems are in fact dependent upon one or more parameters and their operative conditions are stable only if the values of these parameters remain within a limited range [18]. This is the case for the non-isothermal continuous stirred tank reactor considered in this work. If the reactor parameters, that behave as bifurcation parameters, go out of this range, and this can happen also for very small changes, then the initial equilibrium point may become unstable, or the reactor may also reach new equilibrium points that although stable are unacceptable as operative conditions of the reactor. Nonlinear controllers, like fuzzy logic controllers, are used to control such systems because they are much more robust and can handle the system parameter changes.
Two types of fuzzy logic controllers have been so far considered: type-1 [46] and type-2 [23], [33], [45]. Past works [7], [11], [19], [31], [42], [44] have shown the superiority of type-2 fuzzy logic controllers over their type-1 counter-parts. This is because type-2 fuzzy logic controllers can also handle uncertainties [26], [34] present in the system and in the input data to the controller [15]. Type-2 fuzzy logic controllers [16], [17] have been already applied in the field of process control: liquid level process control [44], wheeled mobile robot control [32], autonomous mobile robot [10], micro-robot [1], anaesthesia control [7], DC motor control [6], Kundur Test System [40], biochemical reactor [13], nth order nonlinear system [27], cable-driven parallel mechanism [4], inverted pendulum system [28], chaotic systems [29], multivariable nonlinear systems [30].
This paper presents the application of type-2 fuzzy controllers, with a mixed feedback–feedforward structure, to the control of an isothermal continuous stirred tank reactor (CSTR) that presents bifurcations. The simple feedback control is not adequate for the control of the reactor due to of the presence of several bifurcation parameters. Furthermore the uncertainty of parameters does not permit to have a robust controller.
A mixed feedback–feedforward control structure that makes use also of type-2 fuzzy controllers is proposed. It is a new approach that uses the feedforward control to cope with the presence of bifurcations that may arise from measurable disturbances and type-2 fuzzy sets to make the control system more robust in particular if there is parameter uncertainty.
The approach is based on the knowledge of the continuation diagrams of the process model for choosing the control strategy. The model is obviously different from the real process but if the parameter uncertainties can be taken into account by the control system an optimal use of the knowledge of the process is achieved.
The paper is organised as follows: in Section 2, the mathematical model of the CSTR is introduced; in Section 3, the dynamic behaviour of the uncontrolled reactor is analysed; Section 4 discusses the two different types of controllers used in the simulation and their implementation; simulation results and discussion are given in Section 5; conclusions are presented in Section 6.
Section snippets
CSTR model
The case of a simple non-isothermal CSTR [25], [41] is considered in this paper. The reactor is the one presented in various works by Perez and Albertos [38], [39] in which the exothermic reaction A → B is assumed to take place. The heat of reaction is removed via the cooling jacket that surrounds the reactor. The jacket cooling water is assumed to be perfectly mixed and the mass of the metal walls is considered negligible, so that the thermal inertia of the metal is not considered. The reactor
Analysis of the reactor dynamics
It is well known that an exothermic CSTR without control can have multiple steady states and bifurcations [2], [39]. This can lead to difficulties in the design of a controller. Using the dimensionless Eqs. (5), (6) a thorough analysis of CSTR dynamics was carried out with the aim of allowing to choose a suitable control configuration and to design the controllers. For the system under study the bifurcation parameter is the coolant flow rate of the CSTR jacket (dimensionless parameter x5). In
Type-2 fuzzy sets
Here only the essential part of type-2 fuzzy sets and logic is presented. A more detailed introduction can be found in [14], [23].
A type-2 fuzzy set [22] is defined as:in which is a type-2 membership function, x ∈ X and u ∈ Jx ⊆ [0, 1], while the primary membership of x is the domain of the secondary membership function.
In this paper only a particular case of type-2 fuzzy sets is treated: the interval type-2 fuzzy sets (IT2FS) [5]. An interval type-2
Control configuration
The control objective for the CSTR under study is to keep the reactor temperature at a desired value despite the presence of disturbances like load changes or parameter variations.
Two main problems must be dealt with: the existence of bifurcation points and the uncertainty in the knowledge of some parameters. The solution of the first problem can be found in the choice of a suitable control configuration like the feedback–feedforward control described in Section 3, while the use of type-2 FLCs
Results and discussion
Simulations were firstly carried out using only feedback controllers and keeping all the parameters of the reactor model constant. The response of the reactor temperature to a step change in the inlet temperature (x30) at τ = 5, while maintaining a constant temperature (x3) set-point, is shown Fig. 11. It can be seen that the response obtained with the two controllers is very similar, in terms of overshooting and response time.
Maintaining the same set-point and introducing a step change of x30 in
Conclusions
In this paper a mixed feedback–feedforward control configuration and type-2 fuzzy logic controllers were considered for the temperature control of a non-isothermal CSTR, presenting bifurcations, parameter variations and uncertainty in variable measurements.
The proposed control configuration is based on the knowledge of the complex dynamics of the uncontrolled system. The results of simulations show that the non-linearities present in the system can be better handled rather than using the
References (47)
- et al.
Type-2 fuzzy sliding mode control without reaching phase for nonlinear system
Eng. Appl. Artif. Intell.
(2011) - et al.
Fuzzy logic control with genetic membership function parameters optimization for the output regulation of a servomechanism with nonlinear backlash
Expert Syst. Appl.
(2010) - et al.
Systematic design of a stable type-2 fuzzy logic controller
J. Appl. Soft Comput. Inform. Control
(2008) - et al.
Control of a nonlinear continuous bioreactor with bifurcation, by a type-2 fuzzy logic controller
Comput. Chem. Eng.
(2008) - et al.
Control of the biodegradation of mixed wastes in a continuous bioreactor by a type-2 fuzzy logic controller
Comput. Chem. Eng.
(2009) - et al.
Stability analysis of fuzzy control systems
Fuzzy Sets Syst.
(1999) - et al.
Inverse control of cable-driven parallel mechanism using type-2 fuzzy neural network
Acta Autom. Sin.
(2010) Based on interval type-2 fuzzy-neural network direct adaptive sliding mode control for SISO nonlinear systems
Commun. Nonlinear Sci. Numer. Simul.
(2010)- et al.
Synchronization of uncertain chaotic systems based on adaptive type-2 fuzzy sliding mode control
Eng. Appl. Artif. Intell.
(2011) - et al.
Direct adaptive interval type-2 fuzzy control of multivariable nonlinear systems
Eng. Appl. Artif. Intell.
(2009)