Slovak Society of Chemical Engineering Institute of Chemical and Environmental Engineering Slovak University of Technology in Bratislava PROCEEDINGS

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1 Slovak Society of Chemical Engineering Institute of Chemical and Environmental Engineering Slovak University of Technology in Bratislava PROCEEDINGS 41 st International Conference of Slovak Society of Chemical Engineering Hotel Hutník Tatranské Matliare, Slovakia May 6 30, 014 Editor: prof. Jozef Markoš ISBN: , EAN: Vasičkaninová, A., Bakošová, M., Kmeťová, J.: Fuzzy control of a heat exchanger using fuzzy c- means clustering algorithm, Editor: Markoš, J., In Proceedings of the 41st International Conference of Slovak Society of Chemical Engineering, Tatranské Matliare, Slovakia, , 014.

2 FUZZY CONTROL OF A HEAT EXCHANGER USING FUZZY C-MEANS CLUSTERING ALGORITHM ANNA VASIČKANINOVÁ, MONIKA BAKOŠOVÁ, JANA KMEŤOVÁ Slovak University of Technology in Bratislava, Faculty of Chemical and Food Technology, Institute of Information Engineering, Automation and Mathematics, Radlinského 9, Bratislava, Slovakia, {anna.vasickaninova, mona.bakosova, jana.kmetova}@stuba.sk Keywords: heat exchanger, PID control, Mamdani type fuzzy controller, Takagi-Sugeno type fuzzy controller. 1 INTRODUCTION Heat exchangers represent nonlinear processes (Janna, 009) and control of them is a complex problem due to the non-linear behaviour and complexity caused by many phenomena such as leakage, friction, temperature-dependent flow properties, contact resistance, unknown fluid properties, etc. (Serth, 007). Many factors enter into the design of heat exchangers, including thermal analysis, weight, size, structural strength, pressure drop and cost. Owing to the wide utilization of heat exchangers in industrial processes, their cost minimization is an important target for both, designers and users (Pan et al., 011). Cost evaluation is obviously an optimization process dependent upon the other design parameters. The method which is capable of utilizing the maximum allowable stream pressure drops is described in Panjeshahi et al. (010). The approach can result in minimum surface area requirements. Economics plays a key role in the design and selection of heat exchanger equipment. The weight and size of heat exchangers are significant parameters in the overall application and thus may still be considered as economic variables. A particular application will dictate the rules that one must follow to obtain the best design considering size, weight, economic criteria, etc. They all must be considered in practice (Holman, 009). Conventional control system design depends upon the development of a mathematical description of the system s behaviour. This usually involves assumptions being made in relation to the system dynamics and any non-linear behaviour that may occur. Fuzzy logic is the application of logic to imprecision and has found application in control system design in the form of Fuzzy Logic Controllers. Fuzzy logic controllers facilitate the application of human expert knowledge, gained through experience, intuition or experimentation, to a control problem. Fuzzy set theory has been developed for modelling of nonlinear, uncertain and complex systems (Ross, 004). The database of a rule-based system may contain imprecisions which appear in the description of the rules given by the expert. Because such an inference cannot be made by the methods which use classical two valued logic or many valued logic, Zadeh in (Zadeh, 1975) and Mamdani in (Mamdani, 1977) suggested an inference rule called "compositional rule of inference". Using this inference rule, several methods for fuzzy reasoning were proposed. Zadeh (Zadeh, 1979) extends the traditional Modus Ponens rule in order to work with fuzzy sets, obtaining the Generalized Modus Ponens rule. Fuzzy control has been suggested as an alternative approach to conventional control techniques in many situations. Analytical structure for a fuzzy PID controller is introduced in Mohan and Sinha, 008. In Sun and Er, 004 an approach toward optimal design of a hybrid fuzzy PID controller using GA is proposed. 917

3 Numerous methods have been developed in the literature to analyse and design a variety of fuzzy control systems (Feng, 006). Over the past two decades, researchers have investigated the analytical structure of various fuzzy controllers (Ying, 003; Haj-Ali and Ying, 004). A number of successful applications have been reported in the literature and these applications of the fuzzy control to industrial processes have often produced results superior to those of classical control. Salmasi classifies and overviews the state-of-the-art control strategies for hybrid electric vehicles (Salmasi, 007). The design of the controller based on the use of a finite-dimensional approximate model, of high order, derived by spatially lumping the infinite-dimensional model of the heat exchanger is described in (Maidi et al., 008). In (Denai et al., 007) a fuzzy logic-based decision support system for automating the decisionmaking strategies in cardiac intensive care units is presented. In (Hladek et al., 009), multi-agent control system based on a fuzzy inference system for a group of two wheeled mobile robots executing a common task is proposed. Wakabayashi describes procedures related to the application of PI fuzzy control in a semi-batch reactor for the production of nylon 6 (Wakabayashi et al., 009). Hayward and Davidson illustrate the power of fuzzy logic through a simple control example (Hayward and Davidson, 003). In (Peri and Simon, 005) a fuzzy logic control of the motion of differential drive mobile robots has been presented. In (Mendes et al., 014) a new method for automatic extracting all fuzzy parameters of a Fuzzy Logic Controller in order to control nonlinear industrial processes is proposed. A major contribution of fuzzy logic is its capability of representing vague data (González et al., 013). In (Markowski and Siuta, 013) a general framework for dealing with uncertainties in each stage of consequence modelling is presented. In this paper, a fuzzy c-means algorithm is used in the design of the fuzzy sets for the Mamdani and Takagi-Sugeno types of the fuzzy logic controllers for the temperature control in a shell-and-tube heat exchanger. The controlled heat exchanger is used for pre-heating of petroleum by hot water. The controlled output is the measured output temperature of the heated stream - petroleum and the control input is the volumetric flow rate of the heating stream - water. The problems of set point tracking and disturbance rejection are investigated. Simulations of control were done in the Matlab/Simulink environment. Simulation results obtained using designed controllers were compared by the integral performance indexes IAE and ISE. FUZZY CONTROL The fuzzy logic controller (FLC) is a set of linguistic control rules related by the dual concepts of fuzzy implication and the compositional rule of inference (Passino, 1998). The FLC provides algorithms which can convert the linguistic control strategy based on expert knowledge into an automatic control strategy. The FLC consists of a set of rules of the form if (a set of conditions are satisfied) then (a set of consequences can be inferred). Take e.g. a typical fuzzy controller (Figure 1): if error is negative and change in error is negative then output is negative big if error is negative and change in error is zero then output is negative medium. The collection of rules is called a rule base. The computer is able to execute the rules and compute a control signal depending on the measured inputs error and change in error. The inputs are most often hard or crisp measurements from some measuring equipment. A dynamic controller would have additional inputs, for example derivatives, integrals, or previous values of measurements backwards in time. The block fuzzification converts each piece of input data to degrees of membership by a lookup in one or several membership functions. The rules may use several variables both in the condition and the conclusion of the rules. Basically a linguistic controller contains rules in the if-then format, but they can be presented in different formats. The resulting fuzzy set must be converted to a number that can be sent to the process as a control signal. This operation is called defuzzification. There are several 918

4 defuzzification methods. Output scaling is also relevant. reference input fuzzification inference mechanism defuzzification inputs Process outputs rule base Fuzzy controller Figure 1. Fuzzy control..1 Mamdani controller Two typical fuzzy control systems are prevalent in the literature on fuzzy logic. These are known as Mamdani type and Takagi-Sugeno type. Mamdani type fuzzy controllers employ membership functions for both, antecedents and consequents of the rules. Based on Mamdani controller, Takagi-Sugeno controller is a modified controller whose rules contain membership functions for antecedents and output is defined as a function of the inputs instead of fuzzy sets, which means that in each fuzzy subspace a linear input-output relation is formed. Mamdani introduced the concept of FLC in 1974 (Mamdani, 1974), which was strongly motivated by the theory of fuzzy sets developed by Zadeh (Zadeh, 1973). The main idea in Mamdani controller is to use the linguistic terms to fuzzify the input variables which are put into use in the process stage based on the control rules. The computational core consists of a three-step process consisting of a determination of the degree of membership of the input in the ruleantecedent, a computation of the rule consequences and an aggregation of rule consequences to the fuzzy set control action. Suppose that the control system with k inputs and outputs is y = f(x i ), where x i are independent variables, y is the dependent variable, f is unknown, A i, B i are levels of fuzzy sets, according to the R: If x 1 is A 1 and... and x k is A k Then y 1 is B 1 and y is B (1) where i is the integer ranged from <a, b>.. Takagi-Sugeno controller Tomohiro Takagi and Michio Sugeno introduced a mathematical tool to build a fuzzy model of a system where fuzzy implication and reasoning are used in their paper in the year of 1985 (Takagi and Sugeno, 1985). Let us denote the membership function of a fuzzy set A as A(x), xx. All the membership functions associated to the fuzzy sets are linear. It is suggested that the format of a fuzzy implication R is written as R: If (x 1 is A 1,..., x k is A k ) Then y = g(x 1,..., x k ) () where y is consequence variable whose value is inferred, x 1,..., x k are premise variables that also appear in the part of consequence, A 1,..., A k are fuzzy sets with linear membership functions representing a fuzzy subspace in which the implication R can be implemented for reasoning, f is logical function connects the propositions in the premise, g is function that implies the value of y when x 1,..., x k satisfy 919

5 the premise. In the premise we shall only use logical and connectives and adopt a linear function in the consequence. So an implication is written as R: If x 1 is A 1 and... and x k is A k Then y= p 0 + p 1 x p k x k (3).3 Fuzzy c-means clustering algorithm Fuzzy c-means (FCM) is a method of clustering which allows one piece of data to belong to two or more clusters (Bezdek,1981). This is frequently used in pattern recognition (Zhang et al., 006). It is based on minimization of the objective function: where J m = n c i= 1 k= 1 u m d (4) k i d = x c (5) Here, m is any real number greater than 1, n is the number of data points, c is the number of clusters u is the degree of membership of x k in the cluster i, x k is the k th of d-dimensional measured data, c i is the d-dimension center of the cluster, and * is any norm expressing the similarity between any measured data and the center. With fuzzy c-means, the centroid of a cluster is computed as being the mean of all points, weighted by their degree of belonging to the cluster. The degree of being in a certain cluster is related to the inverse of the distance to the cluster. Fuzzy partitioning is carried out through an iterative optimization of the objective function shown above, with the update of membership u and the cluster centers c i by: 1 1 c d m u = (6) j= 1 d jk c i = n k= 1 n u k= 1 m u x m This iteration will stop when k ( j 1) ( j) max u u (8) i, k where is a termination criterion between 0 and 1, j is the iteration step. This procedure converges to a local minimum or a saddle point of J m. 3 SIMULATIONS AND RESULTS 3.1 Controlled process Consider a co-current tubular heat exchanger (Vasičkaninová et al., 011), where petroleum is heated by hot water through a copper tube (Figure ). (7) 90

6 Figure. Scheme of the tubular heat exchanger. The controlled variable is the outlet petroleum temperature T 1out. Among the input variables, the hot water flow rate q 3 (t) is selected as the control variable. The mathematical model of the heat exchanger is derived under several simplifying assumptions. The coordinate z measures the distance of a modelled section from the inlet. The fluids move in a plug velocity profile and the petroleum, tube and water temperatures T 1 (z,t), T (z,t) and T 3 (z,t) are functions of the axial coordinate z and the time t. The petroleum flow rate is constant, whereas the water flow rate q 3 (t) is a function of time t only. The petroleum, water and tube material densities i as well as the specific heat capacities C Pi, i = 1,, 3, are assumed constant. The simplified nonlinear dynamic mathematical model of the heat exchanger is described in (Vasičkaninová et al., 011) Parameters and steady-state inputs of the heat exchanger are given in Table 1, where the superscript s denotes the steady state and the subscript in denotes the inlet (Vasičkaninová and Bakošová, 01). Here, d is the tube diameter, is the density, C P is the specific heat capacity, h is the heat transfer coefficient, q is the volumetric flow rate. Table 1. Heat exchanger parameters and inputs Variable Unit Value Variable Unit Value n kg m l m 10 C P1 J kg -1 K d 3 m 0.05 C P J kg -1 K d 1 m 0.05 C P3 J kg -1 K d 3 m 0.08 q 1 m 3 s h J s -1 m - K s q 3in m 3 s h J s -1 m - K s T 1in K kg m s T in K kg m s T 3in K 34.8 For the identification, the step changes ±15%, ±30%, ±50% of the inlet mass flow-rate of the heating water were generated at the time t = 0. The step responses of the outlet temperature are shown in Figure 3, where step responses on the input changes ±15% are represented by the solid lines, on the input changes ±30% by the dashed lines, on the input changes ±50% by the dotted lines. 91

7 Figure 3. Step responses of the outlet temperature on the step changes of the inlet mass flow-rate of the heating water. According to these step changes, the heat exchanger is a time-delay nonlinear system with asymmetric dynamics. The model was identified using the Strejc method (Mleš and Far, 007) from the step in the form of the n th order plus time delay transfer function Ds K S = (9) n τs+ 1 Because the heat exchanger can be represented also as a system with interval parametric uncertainty, for various step responses were obtained intervals for values of the gain K, the time constant, the time delay D, the system order n=3 (Table ). The mean values of the parameters are considered to be nominal. Table. Identification of the process dynamics min max mean K min K max K mean D min D max D mean PI Control of the heat exchanger PI controllers described by the transfer function 1 C k p 1 (10) tis with k p the proportional gain and t i the integral time, were tuned using Cohen-Coon method (Ogunnae and Ray, 1994) and Strejc method (Mleš and Far, 007). The model was identified from the step response of the heat exchanger in the form (9). The PI controller parameters obtained using the Cohen-Coon formulas for the nominal values of the parameters are k p = 810-5, t i = 36 s and those obtained using the Strejc formulas are k p = , t i = 3.5 s. 3.3 Takagi-Sugeno fuzzy controller Sugeno-type fuzzy inference system was generated using FCM clustering in the form: If e is A i and e is B i Then f i = p i e + q i e+ r i, i=1, (11) 9

8 where e is the control error, q 3 (t) is the calculated control input and p i, q i, r i are consequent parameters. The symmetric Gaussian function are used for the fuzzification of inputs and it depends on two parameters and c as it is seen in (1) f x;σ,c = e xc σ The parameters and c for Gaussian membership function are listed in the Table 3. The consequent parameters in the control input rule (11) are listed in Table 4. (1) Table 3. Parameters of the Gaussian curve membership functions e e i c i i c i Table 4. Consequent parameters p i q i r i Mamdani fuzzy controller Mamdani-type fuzzy inference system was generated using FCM clustering in the form: If e is A i and e is B i Then u is C i, i=1,..., 6 (13) where e is the control error, u is the calculated control input and p i, q i, r i are consequent parameters. The symmetric Gaussian functions (1) are used for the fuzzification of inputs and outputs. The parameters and c for input membership functions are listed in the Table 5 and the parameters and c for output membership functions are listed in the Table 6. Table 5. Parameters of the Gaussian curve membership functions e e i c i i c i i Table 6. Consequent parameters c i Simulation results obtained using designed fuzzy controllers and two PI controllers are shown in Figure 4. The controlled outputs are compared in the task of set point tracking and in the case when disturbances affect the controlled process. The set point changes from 36.9 C to 39 C, then to 38 C at 400 s and then to 40 C at 800 s. Disturbances were represented by water temperature changes from 30 C to 35 C at 00 s, from 35 C to 31 C at 600 s and to 34 C at 1000 s. The control inputs are presented in Figure 5. The simulation results were compared also using integral criteria IAE (integrated absolute error) and ISE (integrated squared error) (Ogunnae and Ray, 1994). The results for different performance measures are compared in Table 7. The control response obtained by Mamdani fuzzy controller has the smallest values of IAE and ISE, but there are small overshoots and control error for a 93

9 long time. The control response obtained by Sugeno fuzzy controller is similar to those obtained using Cohen-Coon PI controller. Table 7. Values of IAE and ISE controller IAE ISE fuzzy Sugeno fuzzy Mamdani PI Cohen-Coon PI Strejc Figure 4. Comparison of the outlet petroleum temperature control. Figure 5. Comparison of the control inputs. 4 CONCLUSIONS In this paper, an application of the fuzzy c-means algorithm is presented. The algorithm is used in the design of the fuzzy sets for the Mamdani and Takagi-Sugeno types of the fuzzy logic controllers for the temperature control in the heat exchanger. The simulation results confirm that fuzzy control is one of the possibilities for successful control of heat exchangers. The advantage of this approach is that it is not linear-model-based strategy. Simulation results obtained using designed fuzzy controllers and PI controllers tuned using Cohen-Coon and Strejc methods were compared using integral quality criteria 94

10 IAE and ISE. Fuzzy and PI controllers were compared in the task of set-point tracking and in the task of disturbance rejection. Designed Mamdani fuzzy controller leads to the smallest values of IAE and ISE, but there are small overshoots and control error for a long time. The control response obtained by Sugeno fuzzy controller has smaller values of IAE and ISE than Cohen-Coon and Strejc PI controllers too and it has smaller overshoots and shorter settling times. Simulations confirmed that fuzzy controllers can be successfully used for control of the heat exchangers. All simulations were done using MATLAB. ACKNOWLEDGMENTS The authors gratefully acknowledge the contribution of the Scientific Grant Agency of the Slovak Republic under the grant 1/0973/1. References BEZDEK, J Pattern Recognition with Fuzzy Objective Function. Plenum Press, New York. DENAÏ, M.; MAHFOUF, M.; ROSS, J A Fuzzy Decision Support System for Therapy Administration in Cardiovascular Intensive Care Patients. IEEE International Conference on Fuzzy Systems, 1-6. ISSN FENG, G A survey on analysis and design of model-based fuzzy control systems. IEEE Trans. Fuzzy Syst., 14(5), GONZÁLEZ, J. R.; DARBRA, R. M.; ARNALDOS, J Using Fuzzy Logic to Introduce the Human Factor in the Failure Frequency Estimation of Storage Vessels in Chemical Plants, Chemical Engineering Transactions, 3, HAJ-ALI, A.; YING, H Structural analysis of fuzzy controllers with nonlinear input fuzzy sets in relation to nonlinear PID controlwith variable gains, Automatica, 40, HAYWARD, G.; DAVIDSON, V Fuzzy Logic Applications. Analyst, 18, HLADEK, D.; VAŠČÁK, J.; SINČÁK, P Multi-robot control system for pursuit-evasion problem, Journal of Electrical Engineering, 60(3), HOLMAN J. P Heat Transfer, McGraw-Hill, New York. JANNA, W. S Engineering Heat Transfer, 3 rd Edition, The University of Memphis, Tennessee. MAIDI, A.; DIAF, M.; CORRIOU, J. P. 008, Optimal linear PI fuzzy controller design of a heat exchanger, Chemical Engineering and Processing: Process Intensification, 47(5), MAMDANI, E. H Application of Fuzzy Algorithms for the Control of a Dynamic Plant, Proc. IEE,, MAMDANI, E. H Application of fuzzy logic to approximate reasoning using linguistic synthesis, IEEE Transactions on Computers, 6(1), MARKOWSKI, A. S.; SIUTA, D Application of Fuzzy Logic Approach to Consequence Modeling in Process Industries, Chemical Engineering Transactions, 31, MENDES, J.; ARAÚJO, R.; MATIAS, T.; SECO, R.; BELCHIOR, C Automatic extraction of the fuzzy control system by a hierarchical genetic algorithm, Engineering Applications of Artificial Intelligence, 9, MIKLEŠ, J.; FIKAR, M Process Modeling, Identification, and Control. Berlin, Springer. OGUNNAIKE, B. A.; RAY W. H Process Dynamics, Modelling, and Control. Oxford, New York. MOHAN, B. M.; SINHA, A Analytical structure and stability analysis of a fuzzy PID controller. Appl. Soft Comput. 8(1), ) PAN, M.; BULATOV, I.; SMITH, R.; KIM J. K Improving energy recovery in heat exchanger network with intensified tube-side heat transfer. Chemical Engineering Transactions, 5,

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