Design of Different Fuzzy Controllers for Delayed Systems

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1 Design of Different Fuzzy Controllers for Delayed Systems Kapil Dev Sharma 1, Shailika Sharma 2, M.Ayyub 3 1, 3 Electrical Engg. Dept., Aligarh Muslim University, Aligarh, 2 Electronics& Com. Engg. Dept., DGI, Gr. Noida kdsharma.en@gmail.com Abstract Controller may be conventional or intelligent. In the intelligent controller human intelligence is embed with the help of certain soft computing algorithms, or otherwise. This paper addresses the performance evaluation of different conventional and intelligent controllers implemented with an objective to have a good time response. Different types of fuzzy logic controllers were considered along with the example systems having a time delay. The fuzzy logic controllers considered are: PI-like fuzzy logic controller (FLC), gain scheduled PI-like FLC and a self-tuning PI-like FLC. In this work the tuning parameters for the FLC based control system were Ke, Kce, Ku. The optimized values of tuning parameters were achieved from the optimization of Integral of Square of Error (ISE) by running a PSO (Particle Swarm Optimization) program. The effectiveness of the proposed fuzzy logic controllers was demonstrated through simulation of the methods applied to different example systems. A of these controllers is also given. The performance assessment of the studied controllers is based on transient response. By comparison the performances, it was found that the self-tuning PI-like FLC performed better than the gain scheduled PI-like FLC, but needs an extra set of inference rules for the online tuning of updating factor.analysed. earlier. For this reason, dead time is recognized as the most difficult dynamic element to be controlled naturally occurring in physical systems. In order to have a satisfactory performance, the controller output or process input should be a non-linear function of the process state (e and Δe). FLCs try to incorporate this nonlinearity by a limited number of IF-THEN rules which may not always be enough to produce a good approximation to the controller output required for the optimum performance. In such a situation only fixed valued SFs and predefined MFs may not be sufficient to eliminate this drawback. To overcome these lots of research work on tuning of FLCs has been reported where either the input-output SFs or the definitions of fuzzy sets are tuned on-line to match the current plant characteristics [3. 5, 6, 7 and 9]. The basic fuzzy inference system may take fuzzy inputs or crisp inputs depending upon the process and its outputs, in most of the cases, are fuzzy sets. Index Terms Fuzzy logic controller (FLC), Gain scheduled FLC, Self-tuning FLC, Scaling factor (SF). I. INTRODUCTION FLCs are proved to be more robust and their performances are less sensitive to parametric variations [1-3] than conventional controllers. Of course, FLCs are sometimes comparable in performance to the conventional PID controllers for small parametric changes. Most of the practical processes are non-linear higher order systems and may have considerable dead time. Sometimes their parameters may be randomly changed with changes in ambient conditions or with time. Control action is unavoidably delayed in a process with dead time. In such situations, a feedback control system tries to correct for conditions not as they are now, but as they existed sometime Fig. 1: A pure fuzzy system The fuzzy inference system shown in Fig. 1 takes fuzzy sets as input and produces fuzzy output so known as a pure fuzzy system. The fuzzy rule base is the part of the fuzzy system responsible for storing all the rules of the system and hence it can also be called as the knowledge base of the fuzzy system. Fuzzy inference system (FIS) is responsible for necessary decision making to produce a required output. II. FUZZY LOGIC CONTROLLER The fuzzy control systems are rule-based systems in which a set of fuzzy rules represent a control decision 2103

2 mechanism for adjusting the effects of certain system stimuli. With an effective rule base, we can replace a skilled human operator by the fuzzy control systems. The rule base reflects the human expert knowledge, expressed as linguistic variables, while the membership functions represent expert interpretation of those variables. Linguistic variables in membership functions are:- NB stands for Negative Big, NM - Negative Medium, NS - Negative Small, ZE - Zero, PS - Positive Small, PM - Positive Medium, and PB - Positive Big. The Mamdani based FIS uses linear membership function for both inputs and outputs. The ranges of the values are normalized between -1 to 1. Fig. 3, 4 and 5 show these membership functions for the input error, the input change in error and output respectively. Fig. 5: Membership function for output (du) In Mamdani based FIS the IF-THEN rules are generated from the knowledge gained by the experienced operator. The IF-THEN rules of Mamdani type fuzzy inference system is summarized in Table 1. Fig. 6 shows the rule surface of rules defined in table 1. Fig. 2: Mamdani fuzzy inference system for fuzzy controller Fig. 3: Membership function for first input (e) Fig. 6: Rule surface for FLC Rule base Pre processing Crisp Input Fuzzification Fuzzified input Fuzzy Inference System Fuzzified output Defuzzification Post processing Crisp Output Fig.7: Block diagram of fuzzy control system Fig. 4: Membership function for second input (ce) 2104

3 u(t) e(t) Table 1: IF-THEN rules for fuzzy inference system e(t) NB NM NS ZE PS NM PB NB NB NB NB NM NS NS ZE NM NB NM NM NM NS ZE PS NS NB NM NS NS ZE PS PM ZE NB NM NS ZE PS PM PB PS NM NS ZE PS PS PM PB PM NS ZE PS PM PM PM PB PB ZE PS PS PM PM PB PB analytical plant models. In particular, linearization scheduling can be used when plant information is limited to a few equilibriums and the corresponding plant linearization. In our work, we tuned the output scaling factor Ku by gain scheduling it at different set points [3]. The design algorithm uses a coefficient γ to adjust Ku as follows: Ku,sp = γ. Ku,0 (1) Where sp stands for set point and Ku,0 is some reference value of Ku. The value of γ is determined by γ = 1/(1+0.1*sp), (2) IV. SELF-TUNING FUZZY LOGIC CONTROLLER III. GAIN-SCHEDULING OF FUZZY LOGIC CONTROLLER A technique where controller parameters (gains) are changed on-line in a predefined way is known as gain scheduling. It is not always counted as an adaptive control, but it enlarges the operation area of a linear controller to perform well even if the system is nonlinear. Gain Scheduling is a linear parameter varying feedback regulator whose parameters are changed as a function of operating conditions [11]. In gain scheduling, the controller parameters schedule according to predefined parameter values with respect to changing operating conditions. The signal which tells about the current state of operation conditions should be used as scheduling variable. Gain-scheduling is a well-known technique of industrial control. When a plant is subjected to large changes in its operating state, a situation that is typical in industry, gain scheduling is used. Large changes in the operating state lead to corresponding variations in the parameters of the linearised models of the plant about these operating states. It is not possible to design a controller to operate satisfactorily at each operating state and expect it to perform well. Gain scheduling enables a controller to respond rapidly to changing operating conditions. For this it is important that the selected scheduling variables reflect changes in plant dynamics as operating conditions change [6]. In conventional gain-scheduling controllers the parameters of the controller are varied usually as a function of some exogenous variable in an attempt to compensate for the changes in the operating state of the plant through stepwise changes in the controller parameters. In fuzzy gain scheduling, fuzzy rules based system are used to establish the required control policy. A self tuning fuzzy controller is designed by self-adjusting either the control rules, or the membership function, or the scaling factors. Among them, the control rules and the scaling factors play more important roles [9]. Here, we used a real time tuning for output scaling factors. Its main advantages over a general FLC are stronger control capability, increased flexibility and robustness shorter developing cycle and combining fuzzy and non-fuzzy inference. Since we are using the PI-type fuzzy controller, the steady-state performance is certainly better. But for obtaining good performance at the transient state, we have to improve the shortcoming of the PI-type fuzzy controller. For this we follow a strategy.the strategy is that the system has a positive large acceleration at the beginning, means, we have to increase the scaling factor such that the rise time and settling time are reduced on the other hand, the system has a negative acceleration when the output signal is near the set-point, means, we have to reduce the scaling factors such that the overshoot is reduced or eliminated [9]. where α (k) = f (e(k), ce(k)) (3) Where f is a nonlinear function of e and ce, which is described by the rule base shown in Table 2 and the associated inferencing scheme. Fig. 8 & 9 shows the MFs for input & output respectively. Thus, when the controller is in operation, the gain of the self-tuning FLC will not remain fixed rather it will modify at each sampling time by the gain updating factor α, where α is obtained online based on fuzzy logic reasoning using the error and change of error at each sampling time. Gain scheduling employs powerful linear design tools on difficult nonlinear problems. Gain scheduling does not require severe structural assumptions on the plant model, and the approach can be used in the absence of complete, 2105

4 may be due to the delay offered by sensor and/or an actuator installed. G1(s) = (4) Example 2: A second order process with a time delay of 0.02 was considered as a second example. The plant transfer function is given by: Fig. 8: Membership functions for the input error of the STFLC G2(s) = (5) Example 3: It is also a second order process with time delay but in this the time delay is increased to 0.2 sec. The plant transfer function is: G3(s) = (6) The simulink model for closed loop control of the system with FLC, GSFLC, STFLC are shown in fig. 10, 11, 12 respectively. Fig. 9: Membership functions for the output a of the STFLC Table 2: Fuzzy Rules for Computation of α e(k) / ce(k) N ZE P N B M S ZE M S M P S M B Fig. 10: Simulink Model of Plant with FLC. V. SIMULATION Normally hit and trail method is adopted to get reasonably good values of the parameters Ke, Kce and Ku; however an optimized value is a better choice. In this work the tuning parameters Ke, Kce and Ku for the FLC based control system was achieved from the optimization of Integral of Square of Error (ISE) by running a PSO (Particle Swarm Optimization) program. The simulation run was carried out in MATLAB / SIMULINK environment with commonly used parameters of the simulation as available in the menu driven SIMULINK software package. To demonstrate the viability of FLC, GSFLC, and STFLC simulation was conducted with examples taken, as discussed below: Example 1: First example system considered was a First-Order Process with Time Delay: The transfer function of a process can be approximated with that given by G1(s). The delay time (taken as 0.5 sec.) represents the overall delay of the response for the actual process. The delay time occurs Fig. 11: Simulink Model of Plant with Gain Scheduling FLC. Fig. 12: Simulink Model of Plant with Self-Tuning FLC. 2106

5 response response response International Electrical Engineering Journal (IEEJ) VI. RESULTS AND DISCUSSIONS Simulation results using MATLAB /SIMULINK are discussed for the different type of Fuzzy logic controller. Comparisons between step responses of different systems using FLC of different types (FLC, GSFLC, and STFLC) are shown in fig. 13, 14 and 15 and in Tables 3, 4 and 5 for systems described by equ. 4, 5 and 6 respectively TABLE- 3 Performance Comparison in System 1 FLC GSFLC STFLC Ke, Kce, 0.082,6.84, 0.27,22.52, Ku ,4, 0.37 Tr (sec.) Ts (sec.) Mp (%) TABLE- 4 Performance Comparison in System 2 FLC GSFLC STFLC Ke, Kce, 0.96, 6.147, 0.934, 0.631, 0.32, Ku , Tr (sec.) Ts (sec.) Mp (%) step response STFLC GSFLC FLC step response FLC GSFLC STFLC time(sec.) Fig. 14: Step Response of Example System 2 with FLC, GSFLC, and STFLC step response FLC GSFLC STFLC time(sec.) Fig. 15: Step Response of Example System 3 with FLC, GSFLC, and STFLC time(sec.) Fig. 13: Step Response of Example System 1 with FLC, GSFLC, STFLC. TABLE- 5 Performance Comparison in System 3 FLC GSFLC STFLC Ke, Kce, Ku 0.286,2.967, ,4.06, ,3.83, Tr (sec.) Ts (sec.) Mp (%) In this paper we use three types of FLCs. In which one is conventional FLC (PI- like FLC), others are gain scheduling FLC (GSFLC), and self tuning FLC (STFLC). When we compare these FLCs conventional FLC gave worst performance. On comparing the tuning mechanisms of the different kinds of FLCs used in this work i.e., GSFLC, STFLC the gain scheduled FLC is the simpler. It has one fuzzy reasoning block generating the incremental change in the control command u, and one tuning coefficient γ adjusting the output scaling factor. Self tuning FLC has two fuzzy reasoning blocks, one with output u and the other with output α. The performance of the self-tuning PI-like FLC is better than that of gain scheduled PI-like FLC. Certainly, the improved performance is at the cost of increased implementation effort i.e., the cost of two FLCs. VII. CONCLUSIONS Different configurations of the PI-type FLCs were implemented for different examples and the experimental results demonstrated the effectiveness of the PI-type FLCs. By comparing their performance, it is seen that the self-tuning PI-like FLC performed better than the gain 2107

6 scheduled PI-like FLC, but needs an extra set of inference rules for the online tuning of updating factor. This requires significantly more implementation effort than the gain scheduling PI-like FLC. This demonstrates that as more user knowledge is incorporated into the controller design, the performance of the FLC increases. REFERENCES [1] Jan Jantzeni, Foundations of Fuzzy Control, John Wiley & Sons, [2] J.-S.R. Jang, C.-T. Sun, E. Mizutani, Neuro - fuzzy and soft computing, Prentice Hall, [3] M. Ayyub, Application of Genetic Algorithm for Optimal Design of Fuzzy Logic Controller, Proc. National Conf. on Control, Communication and Information Systems, Goa, Jan ,2004, pp [4] Yanan Zhao and Emmanuel G. Collins, Jr, Fuzzy PI Control Design for an Industrial Weigh Belt Feeder, IEEE Transactions on Fuzzy Systems, VOL. 11, NO. 3, pp , Jun [5] R. K. Mudi and N. K. Pal, A self-tuning fuzzy PI controller, Fuzzy Sets Syst., vol. 115, no. 2, pp , [6] Wilson J. Rugh, Jeff S. Shamma, "Survey Paper Research on gain scheduling, Automatica 36, pp , [7] Luca Cammarata, Leena Yliniemi, Development of a Self-Tuning Fuzzy Logic Controller (STFLC) for a Rotary Dryer,. Report A No 10, University of Oulu, Control Engineering Laboratory. December [8] R. K. Mudi and N. K. Pal, Robust self-tuning scheme for PI- and PDtype fuzzy controller, IEEE Transactions on Fuzzy System, vol. 7, no. 1, pp. 2-16, [9] Hung-Yuan Chung, Bor C. Chen,Jin J. Lin A PI-type fuzzy controller with self-tuning scaling factors, Fuzzy Sets and systems, pp , [10] Zheng J, Guo P & Wang J D STFC self-tuning fuzzy controller, Proc. IEEE International Conference on Systems, Man and Cybernetics, 2 pp , [11] Maamoon F. Al-Kababji, Ahmed N. B. Alsammak Reactive Power Compensation using Fuzzy Gain Scheduling (FGS) based PID Controller of Synchronous Machine, Al-Rafidain Engineering, Vol.17, No.1, pp. 1-24, [12] Kapil Dev sharma, Shailika sharma, M. Ayyub, "PSO Tuned FLC for Better Performance", International Journal for automatic Control system, vol. 1, No. 2, 1-7, [13] Kapil Dev sharma, M. Ayyub, Sumit Saroha, Ahmad Fara's, "Advanced controller using FLC for Performance Improvement", International Elrctrical Engineering Journal, vol. 5, No. 6, , [14] Sunil Kumar, Kapil Dev Sharma, "Adaptive Predictive Feedback Techniques for Network Control system, IRJECE, Vol. 1, no. 1, 1-5,