CONTROLLER DESIGN FOR MIMO PROCESS USING OPTIMIZATION ALGORITHM

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1 Volume 7 No. 2 27, 63-7 ISSN: 3-88 (printed version); ISSN: (on-line version) url: CONTROLLER DESIGN FOR MIMO PROCESS USING OPTIMIZATION ALGORITHM Aashika Suresh, S. Meena 2, K. Chitra 3 PG Scholar, Department of Control and Instrumentation Engineering, St. Joseph s College of Engineering, OMR, Chennai 69, Tamilnadu, India. 2 Research Scholar, Department of Electronics & Control, Sathyabama University, OMR, Chennai 69, Tamilnadu, India. 3 Professor, SENSE, VIT University, Chennai 627, Abstract: In process industries, PI/PID controllers are widely implemented to achieve better outcomes. In this work, PID controller design is proposed for benchmark Multi Input and Multi Output (MIMO) distillation column models, such as Wardle and Wood (WW) and Vinante and Luyben (VL) existing in the literature. This paper implements a de-centralised controller for both the top and bottom product and the controller is designed with heuristic method known as Modified Firefly Algorithm (MFA). In the FA based optimal search, minimization of objective function is considered to discover the optimal PID values. The merit of the PID controllers is then tested for the reference tracking and the disturbance rejection operations and the performance measure, like the time domain values and the error values are then computed. Further, the superiority of the FA tuned PID is validated with the controllers designed using the GA, PSO, BFO and traditional FA. The experimental results confirm that, proposed approach offered better result on the considered process models compared with the alternatives. Keywords: MIMO system, Distillation column, PID controller, Modified firefly algorithm, Performance evaluation criteria.. Introduction In chemical/process industries, classical and modern form of PI/PID controllers[23] are extensively used to achieve better results, such as maximized production rate and minimized wastage. In the literature, a plethora of controller design procedures are existing to design PI/PID controllers for the SISO [2] and MIMO [3,7,3,9] process models. Traditional controller design procedures for the MIMO process models are complex and time consuming. Proper tuning of a PID controller is possible if and only if the proper model information is available. Tuning of PID may be cumbersome for certain systems while using ancient methods such as Zeigler and Nichols, Cohen and Coon and other tuning methods in the literature. Hence, in recent years, heuristic algorithms are widely implemented by the researchers to attain optimal results for the MIMO controller design problems [2,5]. The MIMO systems are difficult to be controlled as the change or disturbance occurring in one loop causes a change in the other loops. Heuristic algorithms do not require any kind of mathematical analysis and also works for systems with deviations to certain extent. Several heuristic algorithms have been proposed for optimal tuning methods and are GA, PSO, BFO, FA, ACO and few other [,4]. Among the existing algorithms Firefly algorithm proposed by Yang s evidenced to outweigh the performance of GA and PSO [8]. For MIMO systems, the required controllers can be designed based on the following procedures; (i) Centralised approach [2], (ii) De-centralised approach [3,5,7] and (iii) Optimal controller with decouplers [,8,,7]. Designing and implementing a controller with centralised approach is tedious compared to the decentralised approach. In this paper, optimal PID controller design for the benchmark MIMO distillation column models, such as Wardle and Wood (WW) and Vinante and Luyben (VL) are carried out using a Modified Firefly Algorithm (MFA). The entire paper is organised as follows: Section 2 outlines the methodology adopted in this work, section 3 presents the experimental results and its discussions and the conclusion of the present research is discussed in section 4. ijpam.eu 63

2 2. Methodology This section of the paper presents the methodology implemented in this paper. PID controller design based of the Modified Firefly Algorithm (MFA) is discussed in detail. 2. Related previous works Earlier studies show that controlling a MIMO system using centralised control procedure is quite complicated compared to the decentralised approach [7]. Thus, the MIMO system can be decomposed into a set of SISO systems and PID controller is designed for each SISO loop separately. In the proposed work, heuristic algorithm assisted PID controller design is considered for the considered MIMO systems. 2.2 Problem Formulation Multi-loop PIDs are widely adopted for MIMO control problems. Figure. represents MIMO structure considered in this paper. R _ PID U Process model Y e G G 2 G G 2 22 R 2 e 2 U 2 Y 2 PID 2 _ Figure.. Closed loop structure of MIMO system below: This structure consists two controllers, such as PID and PID 2 and the mathematical expression is given PID Ki Kds K p () s T s f PID 2 K p2 K s i2 Kd 2s T s f (2) where T f = T d/n f; T d = derivative controller time constant; and N f =. During the controller design task, the considered MIMO process is divided in to a group of SISO processes and the MFA approach is allowed to find the best possible values of Kp, Ki, Kd for each PID. 2.3 Firefly algorithm The Firefly algorithm was initially proposed by Yang, X.S (28). It s working is based on the flashing illuminances of the flies. The Firefly algorithm share similarities with the Bacterial foraging optimisation algorithm as in BFOA the attractiveness depends on the fitness of the bacteria and the distance among them. Whereas in FA the attractiveness depends on the brightness of the fly and the distance among them. There are three elementary assumptions made in the firefly algorithm follows, i) all fireflies are unisex so that one firefly will be attracted to other fireflies irrespective of their sex; ii) The attractiveness is proportional to the brightness and decrease as their distance increases. If there is no brighter fly than a particular firefly, it moves randomly; 3) The brightness of a firefly is aff ected or determined by the landscape of the objective function. For the specified problem the brightness which is determined by β is related with the objective function. Other forms of brightness can be defined in a similar way to the fitness function in genetic algorithms [8,9]. The proposed algorithm functions based on the following steps. Step : Initialize the population of firefly 64

3 Step 2: Generating the initial locations of n fireflies Step 3: Define the objective function. Step 4: Multiple iterations are carried out. Step 5: Update fireflies and the attractiveness parameter beta(β). Step 6: Display the best result. The movement of a firefly i is attracted to another more attractive (brighter) firefly j is determined [8] by X i t+ = X i t + β e -γ d ij 2 (X jt -X it ) + α (rand-/2) (2) In the proposed algorithm the fitness is defined by the relationship between the weighing factors (alpha and beta) and the impact of tuning parameters over the objective function. In the existing algorithm the fitness is defined by considering the overall impact, whereas in the proposed approach the fitness is calculated by considering the orientation individually. 2.4 Execution The MFA assisted approach is allowed to search the optimal values of the PID based on the cost function minimization concept. The structure of the execution procedure is depicted in Fig 2. The Decentralised PID values are arbitrarily varied till J min is arrived. Initialization Heuristic Algorithm J min K p K i K d R _ PID Y U Process model e G G 2 G G 2 22 J min W. ISE W2. IAE Figure 2. PID tuning for SISO System (4) where the weights are assigned as W = W2 = The error values considered in the proposed cost functions are presented below: IAE e (t)dt r( t ) y( t )dt (5) T T T 2 T ISE e ( t )dt [ r( t ) y (t)]dt (6) where T is the assigned simulation time (sec) The MFA continuously explores the search space until J min is reached. For MFA, the population size is assigned as 2 and maximum iteration number is chosen as 5 and the stopping criterion is chosen as J min. 3. Results and Discussion The proposed work is implemented using the MATLAB software and the experimental results obtained in this work are clearly presented in this section. Initially, the top product PID is designed using the MFA and the similar procedure is repeated ten times in order to attain the better controller values. The optimized controller values are recorded clearly for all the ten repetitions. In this work, the well-known MIMO systems presented below is considered: 65

4 VL distillation column model Eqn. 7 shows the model of the system. The delay time of the system is small; hence the simulation time is fixed as 3sec. s 2.2e y( s ) 7s y2( s ) 2.8e 9.5s.8 s.3s.3e 7s u ( s ).35s 4.3e u ( s ) 2 9.2s A disturbance signal of unity magnitude is applied at 5sec and its corresponding results are presented in Fig 3 (a) and (b) respectively. WW distillation column model Eqn. 8 presents the model of WW system extensively used by the researchers [2]. Throughout simulation, T is chosen as 25 sec. 6s.26e y ( s ) 6s y ( s ) 8s 2.94e 38s 2s.e ( 48s )( 45s ) u ( s ) 8s.2e u2( s ) 35s A unity step change for the process is also applied at 25sec as shown in Fig. 4. Fig 4 (a) presents the response top product SISO system and Fig 4 (b) depicts the bottom product result. (7) (8).5 (VL) top product.5 (VL) bottom product Process Variable.5 MFA FA setpoint GA PSO BFO time Process Variable.5 MFA FA setpoint GA PSO BFO time (a) Process response of Top-Loop (b) Process response of Bottom-Loop Figure 3 PID controller response for the top and bottom product of VL model.6 (WW)top product 2 (ww)bottom product Process Variable time (a) Process response of Top-Loop MFA FA setpoint GA PSO BFO Process Variable.5.5 BFO time (b) Process response of Bottom-Loop MFA FA setpoint GA PSO Figure 4 PID controller response for top and bottom product of WW model Initially as discussed in section 2.4, the experimental work is implemented for both the SISO loops. The best possible values of the top and bottom product PID values are obtained with the experiment and later the overall result of the MIMO system is recorded for the set-point tracking and disturbance rejection 66

5 operations. Fig. 3 and 4 depicts the results graphically. Table presents the optimal PID values obtained from the proposed approach and also the PID values collected from the literature []. Table 2 illustrates the performance measures obtained for the set-point tracking and Table 3 shows the error values for disturbance rejection operations. From Table 2 and 3 values, one can observe that, the PID values obtained with the MFA are better compared with other methods considered in this paper. This experimental result also confirms that, proposed approach offers better results, hence in future, this procedure can be adopted to design the optimal PI/PID controller for the MIMO systems. Table. The optimal controller values for VL and WW model Process Method Top product Bottom product K P K I K D K P K I K D MFA VL FA GA BFO PSO MFA WW FA GA BFO PSO Table 2Performance values of VL and WW models before disturbance Process Method Top product Bottom product ITAE ISE Mp Ts ITAE ISE Mp Ts MFA FA VL GA BFO PSO MFA FA WW GA BFO PSO

6 Table 3 Error values after disturbance Process VL WW Method Top product Bottom product ITAE ISE ITAE ISE MFA FA GA BFO PSO MFA FA GA BFO PSO Conclusion The PID controllers for the top and the bottom loops of the well-known benchmark models such as WW and VL model distillation column are separately designed based on the objective function minimization concept. During this procedure, a multi-objective function is considered to guide the FA based heuristic search. The merit of the designed controller is tested for the reference tracking performance and the disturbance operations. The performance of the proposed FA is confirmed against other heuristic approaches, like traditional FA, GA, PSO and BFO. The experimental results confirm that, proposed approach offers better result for both the reference tracking and the disturbance rejection operations. References. V.D. Angeline and N. Devarajan: Design of optimized PI controller with ideal decoupler for a nonlinear multivariable system using particle swarm optimization technique. International Journal of Innovative Computing, Information and Control, (), (24), I. Hassanzadeh and S. Mobayen: Controller design for rotary inverted pendulum system using evolutionary algorithms. Mathematical Problems in Engineering, vol. 2, Article ID , (2), 7 pages 3. M.J. Lengare, R.H. Chile and L.M. Waghmare: Design of decentralized controllers for MIMO processes. Computers and Electrical Engineering, 38(), (22), F. Vázquez and F. Morilla: Tuning decentralized PID controllers for MIMO systems with decoupling. Proceeding of the 5th IFAC World Congress, (22), Lones, Michael A. (24). "Metaheuristics in Nature-Inspired Algorithms" (PDF). GECCO '4. doi:.45/ Coelho, L.S and V.C. Mariani, 22. Firefly algorithm approach based on chaotic Tinkerbell map applied to multivariable PID controller tuning. Computers and Mathematics with Applications, 64(8): B. Labibi, H.J. Marquez and T. Chen: Decentralized robust PI controller design for an industrial boiler. Journal of Process Control, 9(2), (29), Yang, X.S., 29. Firefly algorithms for multimodal optimization. Stochastic Algorithms: Foundations and Applications, vol of Lecture Notes in Computer Science, pp: Yang, X-S., 28. Nature-Inspired Metaheuristic Algorithms, Luniver Press, Frome, UK. 68

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