INTERNATIONAL CONFERENCE ON CONTROL, AUTOMATION, COMMUNICATION AND ENERGY CONSERVATION 2009, KEC/INCACEC/708 Design and Development of an Optimized Fuzzy Proportional-Integral-Derivative Controller using Genetic Algorithm K. T. K. Teo, G. Sainarayanan, and C. S. X. Loh 1 Abstract The polymerization of resin adhesives, which is highly nonlinear, is chosen as a control process in this paper. When phenol and formaldehyde are mixed, a sudden and unpredictable heat is produced due to an exothermic reaction, which could cause the deviation of process temperature and hence diminish the product quality. Therefore, temperature control is necessitated for exothermic batch reactor. The most common approach used in industries is proportional-integral-derivative (PID) controller, which is tuned by the operators based on their experience and knowledge on the process, and hence, may not be precise in control since the process dynamic is changing rapidly. Therefore, this paper aims to design an auto-tuning PID controller for nonlinear exothermic process control. This paper presents fuzzy PID (FPID) and Genetic Algorithm-optimized fuzzy PID (GAFPID) controllers. In FPID controller, PID gains are tuned by fuzzy system. In GAFPID controller, the inner loop is a FPID controller and the outer loop is GA controller, which is used to optimize the fuzzy membership functions. The performances of these controllers are investigated in MATLAB-SIMULINK. From the results, it can be concluded that GAFPID controller provides a better control in the nonlinear exothermic process due to its robustness against the variable time delay and inconsistent exothermic heat. Index Terms Exothermic, Fuzzy Logic, Genetic Algorithm, PID control 1 INTRODUCTION T HE polymerization of resin adhesives, which is a highly nonlinear process, is chosen for simulation studies in this paper. Resin adhesives are produced by mixing phenol and formaldehyde together with the use of catalyst. The mixture has to go through a specific heating process to achieve the good quality of resin. However, when these chemicals are mixed together, a sudden and unpredictable heat is produced due to a nonlinear exothermic reaction [1]. Each time the reactor temperature changes from the desired temperature, the quality of resin diminishes, the appearance of undesired solid product increases, and the overall performance of the reaction decreases which directly affects the economic performance [2]. Therefore, the temperature control plays a decisive role in the production of resin adhesives. The most commonly used strategy in polymer industries to control the process temperature is PID controller. The PID gains are usually determined by the operators based on their experience and knowledge on the process. Therefore, it is difficult to determine the best values to produce the desired output especially when the time delay of the process is varying within a wide range all the time. Furthermore, as the process dynamic is rapidly K. T. K. Teo is with the School of Engineering and Information Technology, Universiti Malaysia Sabah, 88999, Kota Kinabalu, Sabah, Malaysia. E-mail: kenteo@ums.edu.my. G.Sainarayanan is with the Department of Electrical and Electronics, New Horizon College of Engineering, Bangalore, India. E-mail: drsai@ieee.org. C. S. X. Loh is with the School of Engineering and Information Technology, Universiti Malaysia Sabah, 88999, Kota Kinabalu, Sabah, Malaysia. E-mail: chrissxloh@gmail.com. changing throughout the process, it is very time consuming for the operator to determine suitable values through trial and error method. As a result, the conventional PID controller is no longer suitable for temperature control of exothermic batch reactors since it is slow and has inherent lack of efficiency in handling nonlinear systems [1]. Therefore, this paper aims to design and develop an autotuning PID controller to control the temperature of nonlinear exothermic process. 2 METHODOLOGY 2.1 Modeling of Nonlinear Process Plant It is necessary to have an appropriate model of the nonlinear exothermic process plant together with a PID controller before an auto-tuning technique can be applied on the PID controller. Industrial practice in the polymerization of resin adhesives has shown that by controlling the reactor temperature, it is possible to obtain products with specified quality. Hence, the reference values of the process temperature should first be determined. The process requires two stages of temperature control, as shown in Fig. 1. In the first stage, the reactant temperature has to be increased at a specific rate from a room temperature (25 C) to a target value (65 C); while in the second stage, this final target temperature has to be perfectly regulated for a selected length of time. This temperature schedule has to be maintained irrespective of varied exothermic reaction [1].
2 Fig. 3. Plant model in MATLAB-SIMULINK. Fig. 1. Reference temperature profile. During the polymerization of resin adhesives, the heat due to exothermic reaction is introduced into the process. This exothermic heat causes a sudden change in reactant temperature. Besides that, the rising rate, the peak value and also the decaying rate of the heat generated during exothermic reaction may not be the same every time. Therefore, two sets of exothermic heat are modeled in MATLAB-SIMULINK as shown in Fig. 2. Modeling of the plant requires the determination of the plant transfer function. It is common to characterizing the dynamic response of the plant to be controlled with a first-order model with time delay, which can be described by the following transfer function: K D s G( s ) = e (1) τ s 1 where K is the process gain, τ is the time constant and D is the time delay. This model is obtained by applying a step input to the plant and measuring the three parameters at the output [3]. The process gain and the time constant are found to be 0.011 C/kW and 787 s respectively, whereas the time delay is approximately 20 s. Due to dynamic in nature, the time delay of the process may not be constant all the time. This condition has been considered by modeling a variable time delay for the process. Fig. 3 shows the plant model in MATLAB-SIMULINK. The modeling of PID controller is based on the PID control algorithm given in (2). t de( t ) u( t ) = K pe( t ) Ki e( τ )dτ K d (2) 0 dt where u is the control variable, e is the control error, K p is the proportional gain, K i is the integral gain, and K d is the derivative gain. The Ziegler-Nichols method has been employed to determine the parameters of PID controller. Then, the plant model and the PID controller model are combined to obtain a process plant with PID controller in MATLAB-SIMULINK, as shown in Fig. 4. 2.2 Design of Fuzzy PID Controller Fuzzy logic has been chosen as the auto-tuning methods for PID controller in this paper. Fuzzy logic is easy to be used since fuzzy rules can be set in terms of linguistic rules using human expertise in tuning the PID parameters. The PID controller that is tuned by fuzzy logic is called fuzzy PID (FPID) controller. The fuzzy tuning involves four basic operations, which are fuzzification, rule base, inference engine and defuzzification [4], as shown in Fig. 5. The fuzzification procedure comprises of finding appropriate membership functions to describe the crisp data [4]. A membership function is a curve that defines how each point in the input space is mapped to a membership value. The input space is referred to as the universe of Fig. 4. Process plant with PID controller. Fuzzification Rule base, Inference engine (Decision making) Defuzzification PID Fig. 2. Exothermic heat models. Fig. 5. Basic operations of fuzzy tuning method.
3 discourse. Fuzzy system in FPID controller has two inputs: error and change in error of the process, and three outputs: proportional (P), integral (I), and derivative (D) gains. The universe of discourse for both inputs is [ 2 2] and the universe of discourse for the three outputs, which are P, I and D gains, are [1 2500], [0 16] and [0 6200] respectively. Fig. 6 shows the membership functions of the inputs, while Fig. 7 shows the membership functions of the outputs. Fuzzy rule base is a collection of if-then rules. These rules contain all the information about the controller parameters tuning [4]. For example, If (E is hot) and (CE is hotter) then (P is highest) and (I is low) and (D is high) is one of the fuzzy rule where hot, hotter, highest, low, and high are linguistic values defined by fuzzy inputs and outputs membership functions. The function of the inference engine is to deduce a logical conclusion using the rule base. Inference engine formulates the mapping from the given inputs to an output using fuzzy logic [4]. The function of defuzzifcation is to resolve a single output value from the fuzzy set. The defuzzification method used for the FPID controller is centroid method, which returns the center of area under the curve [5]. The block diagram of process plant with FPID controller is shown in Fig. 8. 2.3 Design of GA-Optimized Fuzzy PID Controller A GAFPID controller is designed in which fuzzy parameters in FPID controller is optimized by Genetic Algorithm (GA) controller. The GAFPID controller has two control loops, which are inner loop and outer loop. The inner loop comprises of a FPID controller, while the outer loop consists of a GA controller, which is used to optimize the membership functions of the fuzzy system in the inner loop at a time interval of 10 s. GA is a stochastic global search method based on natural selection and genetics. The flow chart of the GA is shown in Fig. 9. First, an initial population of solutions must be randomly generated. The initial population size is chosen as 50. Since the outputs of this GA function are optimum fuzzy inputs and outputs membership functions, the solutions for GA are strings of just five parameters, which can completely characterize all the input and output membership functions, as shown in Fig. 10. Fig. 6. Membership functions of fuzzy inputs. Setpoint Error FPID controller Control signal Plant Exothermic heat Output Fig. 8 Block diagram of process plant with FPID controller. Initial Population Measure Fitness Selection Crossover Mutation Check Stopping Criterion Optimum Fuzzy Membership Functions Fig. 7. Membership functions of fuzzy outputs. Fig. 9. Flow chart of Genetic Algorithm.
4 Fig. 10. Fuzzy membership functions represented by GA solution string. Then, the fitness of each solution is measured using fitness function, which interprets the solutions in terms of physical representation and evaluates its fitness based on desired characteristics. The definition of fitness function is very important since GA choose the optimal solution based on the evaluation of its fitness [6]. In this paper, the fitness function of GA is based on the predicted output response, which is calculated using the transfer function of the process plant. As discussed before, the transfer function of the plant is 0.011 G ( s ) = (3) 787s 1 This continuous transfer function is discretized into discrete transfer function and the sample time is chosen as 10 s. The discrete transfer function becomes 0.00014 G ( z ) = (4) z 0.99 By rewriting (4), the output response for the next 10 s can be predicted. The equation then becomes y k = 0.00014 u k y k 1 (5) [] [] [ ] where y[k] is the predicted output response for the next 10 s, u[k] is the control signal for the next 10 s that is sent to the process plant and y[k 1] is the current output response. The control signal for each GA solution must be obtained before the output response can be predicted. First, the preset fuzzy system is loaded into the GA function. Then, the fuzzy inputs and outputs membership functions are built using the values in the GA solution strings. The fuzzy inputs, which are error and change in error of the process, are input into these characterized fuzzy systems to evaluate the outputs, which are P, I and D gains. The control signal for each set of PID gains can then be calculated by the equation u = K e K se K ce (6) p i where e is the error, se is the sum of error and ce is the change in error. In order to compute the control signal, the e, se and ce are needed to be input into GA. By using the predicted output response, the predicted error can be computed by obtaining the difference between the predicted output response and the reference temperature. Finally, the fitness of each GA solution can be calculated by taking the reciprocal of the square predicted error, since smaller predicted error indicates that the solution is fitter. The equation of fitness function is 1 Fitness = (7) r y d ( ) 2 where r is the reference temperature and y is the predicted output response. After the fitness of the solutions are evaluated, the population of solutions is passed to GA operators, which are selection, crossover and mutation. The selection operator is to emphasize the fitter solutions in the population by making duplicates of those solutions in hopes that their offspring will in turn have even higher fitness, while keeping the population size constant. Method chosen for selection operation in this paper is ranking selection. First, the N solutions are sorted according to their fitness, from the worst (rank 1) to the best (rank N). Each member in the sorted list is assigned a new fitness, which is equal to the rank of the solution in the list. The implementation of this selection operator can be thought of as a roulette-wheel mechanism [7]. The cumulative fitness of each solution is first calculated by adding the individual ranked fitness from the top of the list. The roulette-wheel concept can be simulated by realizing that the i-th solution in the population represents the cumulative fitness in the range [C i-1, C i ] where C i is the cumulative fitness of that i-th solution. N random numbers are then created to choose N solutions. If a random number is within the cumulative fitness range of a solution, then the solution is copied to the mating pool. The solution with a higher fitness value represents a larger range of cumulative fitness and therefore has a higher probability of being copied into the mating pool. Crossover operator is applied next to the solutions of the mating pool. In crossover operator, two solutions (called parent solutions) are picked from the mating pool at random and some portions of the solutions are exchanged between the solutions to create two new solutions (called offspring) [6]. In this paper, blending method is used for crossover operation. This method combines variable values from the two parents into new variable values in the offspring. The first offspring variable value comes from a combination of two corresponding offspring variable values. The second offspring is merely the complement of the first. The equations used to create offspring solutions are x = β x ( 1 β ) x (8) x n1 n2 p1 p1 p2 = ( 1 β )x β x (9) p2
5 where β is a random number in [0,1], x p1 and x p2 are the parent solutions and x n1 and x n2 are the offspring solutions [8]. In order to preserve some fit solutions selected during the selection operator, not all solutions in the population are used in the crossover. In this paper, the crossover rate is chosen as 0.9. That means only 90% of solutions in the population are used in the crossover operation and 10% of the solutions are simply copied to the new population. The mutation operator helps in randomly searching other areas of the solution spaces that may be unexplored and might containing global maxima. However, the probability of mutation must be kept low to prevent the loss of too many fit solutions and affect the convergence of solutions [9]. Hence, the mutation rate is set to 0.01 in this paper. The method used in this paper is random mutation, which is the simplest mutation operator that chooses a solution randomly from the entire solution spaces to be mutated. The mutated solution is simply replaced by a new random number, which is independent of the parent solution [8]. The stopping criterion of this GA function is whenever the maximum number of generation is reached. The maximum number of generation is set to 10 in order to reduce the computation time. Therefore, the GA will stop after 10 generations and returns the optimal fuzzy parameters to FPID controller. A S-function is written so that the programmed GA function can be implemented in SIMULINK. S-function is a computer language description of a SIMULINK block. S-functions can be written in MATLAB m-file, C, C, Ada, or Fortran. S-functions use a special calling syntax that enables user to interact with Simulink s equation solvers [10]. In this paper, the S-function is written in MATLAB m-file. A M-file S-function block is then used to incorporate the written S-function into the SIMULINK model. The block diagram of process plant with GAFPID controller is shown in Fig. 11. Fig. 12. Results of PID controller. 3 RESULTS The simulation results, which are the output response of the processes with variable time delay and two sets of exothermic heat, of the PID, FPID, and GAFPID controllers are shown in Fig. 12, Fig. 13, and Fig. 14. GA Fuzzy parameters FPID Output Plant Setpoint Error controller Control signal Exothermic heat Fig. 11. Block diagram of process plant with GAFPID controller. Fig. 13. Results of FPID controller.
6 smallest undershoot, have no oscillation and remains unchanged in the changing exothermic heat. In other words, the GAFPID controller possesses the highest efficiency in controlling the temperature of the process with variable time delay and inconsistent exothermic heat. 5 CONCLUSION In this paper, a process plant model with PID controller has been developed based on the characteristics of polymerization of resin adhesives. Fuzzy logic has been employed to auto-tune the PID parameters in FPID controller. Besides that, GA has also been used to optimize the fuzzy parameters in GAFPID controller. The performances of these controllers have been studied using the simulation results from MATLAB-SIMULINK. From the simulation results, it can be concluded that the GAFPID controller provides a better temperature control in nonlinear exothermic process due to its robustness against the variable time delay and inconsistent exothermic heat. REFERENCES Fig. 14. Results of GAFPID controller. 4 DISCUSSION From the simulation results, it can be noticed that the output responses of process with PID controller have greatest undershoot. The undershoot is further worsened by the rise in the exothermic heat. Hence, the PID controller with only one set of parameters throughout the process is not robust in nonlinear exothermic process control. It can be observed that both output responses of process with FPID controller in Fig. 13 have fairly great undershoot. This shows that the FPID controller cannot effectively control the process with variable time delay. By the way, the FPID controller shows the robustness against the changing of exothermic heat since the increase in exothermic heat does not worsen the undershoot in the output response. In fuzzy system, the rules and membership functions have to be predefined precisely based on the knowledge on process control. If the rules or membership functions have been defined incorrectly, fuzzy system may not return the optimum parameters. By employing GA to fine-tune the fuzzy parameters, the fuzzy system with optimum membership functions can be obtained. Besides that, the fuzzy parameters can be continuously and automatically updated by GA throughout the process according to the process dynamic changes. The effectiveness of GA in optimizing fuzzy parameters has been proven by the GAFPID controller, which has the best simulation results since the output responses of process with the GAFPID controller have the [1] T. T. K. Kenneth, S. Yaacob, R. Nagarajan, and G. Sainarayanan, Certain Studies on Sample Time for a Predictive Fuzzy Logic Controller through Real Time Implementation of Phenol- Formaldehyde Manufacturing, TENCON 2004 IEEE Region 10 Conference, pp. 514-517, 2004. [2] E. S. Cuellar, J. L. Coronado, C. G. Moreno, and J. M. C. Izquierdo, Neuro-Fuzzy Architecture in a Batch Reactor Temperature Control System, Proceedings of the 1994 IEEE International Symposium on Intelligent Control, pp. 11-15, 1994. [3] G. J. Silva, A. Datta, and S. P. Bhattacharyya, PID Controllers for Time-Delay Systems, USA: Birkhauser Boston, pp. 223-224, 2004. [4] P. N. Paraskevopoulos, Modern Control Engineering, New York: Marcel Dekker Inc., pp. 680-692, 2002. [5] Fuzzy Logic Toolbox User s Guide, The MathWorks, Inc., 2002. [6] W. A. Chang, Advances in Evolutionary Algorithms: Theory, Design and Practice, New York: Springer Berlin Heidelberg, pp. 11-12, 2005. [7] M. Melanie, An Introduction to Genetic Algorithm, UK: MIT Press, pp. 124-127, 1996. [8] R. L. Haupt and S. E. Haupt, Practical Genetic Algorithms, New Jersey: John Wiley & Sons, Inc., pp. 57-62, 2004. [9] K. O Donoghue, Real-time Adaptive PID control of a Nonlinear Process Based on Genetic Optimization, Master dissertation, School of Electronic Engineering, Dublin City Univ., Ireland, 2002. [10] Writing S-Functions, The MathWorks, Inc., 2000.