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Auto Tuning of Fuzzy PI Type Controller Using Fuzzy Logic Seema Chopra, R. Mitra and Vijay Kumar
Abstract— In this paper, a simple and effective method for tuning of fuzzy PI (proportional-integral) controller based on fuzzy logic is proposed. Here the input scaling factors are tuned online by gain updating factors whose values are determined by rule base with the error and change in error as inputs according to the required controlled process. The rules are designed for tuning the scaling factors based on the performance measures such as peak overshoot (OV), Rise time (RT) and Amplitude (AM). The performance comparison of conventional FLC with auto tuned fuzzy PI type controllers has been done in terms of several performance measures such as peak overshoot, settling time and rise time and integral square error (ISE). In addition to the responses due to step set-point change, a random noise is also added in some systems. Simulation results show the effectiveness and robustness of the proposed auto tuning mechanism. Furthermore, a clustering method is used to reduce the fuzzy inference rules of the three fuzzy reasoning blocks which reduces the computational time and memory. The clustering based Fuzzy Logic Controllers is compared with those of conventional Fuzzy Logic Controllers in both cases (with and without tuning). A simulation analysis of a wide range of linear and nonlinear processes is carried out and comparison of results shows computational time and memory is reduced to a great c 2008 Yang’s Scientific Research Institute, LLC. extent. Copyright ° All rights reserved. Index Terms— Fuzzy PI controller, fuzzy logic.
I. I NTRODUCTION
F
UZZY logic controllers (FLCs) are increasingly applied to many systems with nonlinearity and uncertainty and it is based on experience of a human operator. While controlling a plant a skilled human operator manipulates the output of the controller based on error and change in error with an aim to reduce the error with a shortest possible time. The two types of structure of FLC have been studied so far: one is position-type fuzzy controller which generates control input (u) from error (e) and change in error, and the other is velocity-type fuzzy controller which generates incremental control input (∆u) from error and change in error. The former is called PD type FLC and the latter is called PI type FLC according to the characteristics of information that they process. In the viewpoint that the FLC is based on the knowledge of human experts, and generally FLC’s applied to Manuscript received November 21, 2006 ; revised November 16, 2007. Seema Chopra, GE Global Research, Bangalore, India. R. Mitra and Vijay Kumar, Department of Electronics and Computer Engineering, Indian Institute of Technology, Roorkee, India. Email:
[email protected](S. Chopra),
[email protected](R. Mitra),
[email protected](V. Kumar) Publisher Item Identifier S 1542-5908(08)10103-8/$20.00 c Copyright °2008 Yang’s Scientific Research Institute, LLC. All rights reserved. The online version posted on July 09, 2008 at http://www.YangSky.com/ijcc/ijcc61.htm
unknown or partially known systems, PI type FLC is known to be more practical than PD type FLC. Other comparisons also can be seen in the facts that human, generally, are not so sensitive to absolute values of data in their sensing and actuation, and besides sometimes it is not possible to remove out steady state error with PD type controllers for large class of systems [1,2]. There is a lot of literature concerning fuzzy PI type controller to design its parameters and how to systematically determine those parameters. Among those efforts towards the parameter design, the Fuzzy Neural Network approach by Jang [24] was one of the practical and successful method to derive fuzzy rules and MFs. Although much effort has been devoted to the fuzzy scaling factors in the past decades, there is still no effective solution. Most of the research works on FLC have either neglected this issue by directly applying a set of scaling factors (SFs) without explanation or simply given some rough idea to guide the choice of the SFs to a specific problem, their adopted solutions are essentially empirical and with the trial-and-error nature [16]. Different types of adaptive FLC’s such as self-tuning and self-organizing controllers have also been developed [3–10] and implemented for various practical processes. Of the various tunable parameters, SF’s have the highest priority due to their global effect on the control performance. However, relative importance of the input and output SF’s to the performance of a fuzzy logic control system is yet to be fully established. A set of fuzzy rules often needs to be manually adjusted on a trial-and-error basis before it reaches the desired level of performance. This tuning process is non-trivial, and could be time consuming for a first-time FLC developer. This limitation of FLC is related to a more general problem in process control, namely, that changes in the operating conditions of a process plant are difficult to predict and adjust for. Hence, it is desirable to develop an auto tuning method that can improve its performance based on its experience, and to adapt its response in relation to variations in the process dynamics. In addition to being able to adapt automatically to a new operating environment, an auto-tuned fuzzy controller can further simplify the task of developing rules, for the designer only needs to come up with an initial set of rules which are roughly correct. The burden of manually tuning the rules is thus removed from the designers. Many researchers are working in this area of automated controller tuning using fuzzy logic [4-7], [17-20]. But most of them reported their work on the tuning of output SF only or for conventional PID controllers. There is a need to put little efforts on the tuning of input scaling factors for FLCs as
CHOPRA, MITRA & KUMAR, AUTO TUNING OF FUZZY PI TYPE CONTROLLER USING FUZZY LOGIC
they affect directly on performance measure and controller part [10,19]. Designing an auto tuning method using input scaling factor is not an easy task because both the parameters has opposite effect on the performance. Other reason for this is that changing the input SFs changes the normalized universes of discourse, the domains of the MFs of input/output variable of FLC. The input scaling factors affect the performance measure and the controller part while the output scaling factor affects only the output of the controller. Increasing the input scaling factors makes the performance measures more sensitive around the set-point and less sensitive during rise time. Decreasing or increasing the two parameters has the opposite effect. Both parameters should be defined on low limits below which the amount on tolerance on rise time, steady state error and oscillations around the set point become unacceptable. So the input SFs should be determined very carefully for the successful implementation of a FLC. Due to the above problems, we are motivated to design a simple and effective auto tuning method for fuzzy logic controllers. In this method, the input scaling factors are tuned online by some updating factors whose value are determined by rule base with the error and change in error as inputs according to the required controlled process. It should be noted that this tuning method does not train the rule base of FLC while it assumes a working set of rules and is used to tune the controller for a desirable response. The auto-tuning method is applied to PI type FLC for simulation experiments with various types of processes including well known example of coupled tank. A number of performance indices [13,14] such as peak overshoot (%ov), settling time (ts (s)), rise time (tr (s)) and integral square error (ISE) are computed for a detailed performance comparison of the auto tuned FLC with conventional FLC. Now it is observed that using the proposed tuning mechanism, the performance is better than the existing papers but the computational time and memory used is comparatively higher, because it has three fuzzy reasoning blocks, one is for generating for incremental change in control output ∆u and other two are for updating factors α and β for input scaling factors. To minimize the amount of memory used and computational time, we can put constraints on the type of fuzzy controller (e.g., membership functions) or limit the rules. But it will affect the performance of the system; hence we need the solution which exhibits good performance with smallest possible rule base. In our previous research [11,12,25,2729] for the reduction of rules for fuzzy controllers we used Fuzzy curve, Fuzzy subtracting clustering (FSC) and neural networks and neurofuzzy techniques. FSC [21]-[23], [29] is one of the fast and robust method of clustering to generate the suitable initial membership functions and shortest rule base from input/output data. It has also been proved in [11,25] that after reducing the rules (using the FSC approach) the performance of the reduced rule controller is similar to the original system. After simulation and comparison of results, it can be seen that the computational time and memory is reduced substantially and the performance is also identical to the original (system without clustering).
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II. P ROPOSED PI T YPE F UZZY L OGIC C ONTROLLER In this section, the detailed design of a PI type FLC is given whose input gains are tuned online by fuzzy logic itself. Here the input scaling factors (SFs) are tuned online by some updating factors whose value is determined by rule base with the error and change in error as inputs according to the required controlled process. No doubt the best performance of FLC can be obtained by tuning both input and output SFs simultaneously. But we are trying here to obtain the same performance by adjusting the input gains only. The block diagram of proposed auto tuned PI type FLC is shown in Fig 1. The error signal is defined as e(k)=Set point (kth sample time)- Output (kth sample time) (1) The change in error is defined as ∆e(k) = e(k) − e(k − 1)
(1)
The operation of PI type FLC can be described by u(k) = u(k − 1) + ∆u(k)
(2)
In Eq. (2), k is the sampling instant and ∆u is the incremental change in controller output.
A. Scaling Factors In the case of normalized universe, an appropriate choice of specific operating areas requires a scaling factor. An input SF transforms a crisp input into a normalized input in order to keep its value within the universe. An output SF provides a transformation of the defuzzified crisp output from the normalized universe of the controller output into an actual physical output. The scaling factors which describe the particular input normalization and output denormalization play a role similar to that of the gains of a conventional controller. Hence, they are very important with respect to controller stability and performance. The relationship between the SFs (Ge , G∆e , Gu ) are the input and output variables of the FLC is
eN = Ge × e, ∆eN = G∆e × ∆e, ∆u = G∆u × ∆uN
(3)
Selection of suitable values for Ge , G∆e and Gu are made based on the knowledge about the process to be controlled and sometimes through trial and error to achieve the best possible control performance. This is so because, unlike conventional nonfuzzy controllers to date, there is no well-defined method for good setting of SF’s for FLC’s. But the SFs are the main parameters used for tuning the FLC because changing the SFs changes the normalized universe of discourse, the domains, and the membership functions of input /output variables of FLC.
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B. Auto Tuning of Fuzzy PI Type Controller
1
An auto tuning PI type FLC is proposed for the tuning of input SFs by developing the adjustment rules defined in terms of e and ∆e for updating the SFs, in dependence on the performance of the closed loop system. Auto tuning mechanism simply means that the self tuning of input gains based on error and change in error. Based on this mechanism, the incremental change in e and ∆e is obtained by (4). eN = (α.Ge ) × e, ∆eN = (β.G∆e ) × ∆e
(4)
where α and β are the updating factors for incremental change in e and ∆e which are computed online based on fuzzy logic reasoning using the error and change in error at each sampling time. Thus the input gains of the auto tuning FLC does not remain fixed while the controller is in operating condition, infect it is updating at each sample by updating factors α and β.
NB
1
NM
NS
ZE
PS
PM
PB
M
B
VB
2
3
4
5
1 Fig. 4.
MFs for gain updating factor β.
D. Rule Bases The fuzzy rules may be extracted from operator’s expertise. For Fuzzy PI type controller, ∆u is the incremental change in controller output, which is determined by the rules of the form If e is E and ∆e is ∆E, then ∆u is ∆U. The rule base for computing is shown in Table I, which is a fairly standard one. TABLE I RULE BASE FOR ∆u. ∆e/e NB NM NS ZE PS PM PB
C. Membership Functions All membership functions (MFs) for controller inputs (i.e., e and ∆e) and incremental change in controller output (i.e., ∆u) are defined on the common normalized domain [-1,1]. The membership functions are shown in Fig 2. The MFs for α are defined on the range [-0.5, 1.5] but with two fuzzy sets small and big and the MFs of β corresponding to the singleton fuzzy sets and varies from [1,5] as shown in Fig 3 and 4. It is assumed that α is in the prescribed range and the appropriate range is determined by simulations.
S
NB NB NB NB NB NM NS ZE
NM NB NM NM NM NS ZE PS
NS NB NM NS NS ZE PS PS
ZE NM NM NS ZE PS PM PM
PS NS NS ZE PS PS PM PB
PM NS ZE PS PM PM PM PB
PB ZE PS PM PB PB PB PB
In the case of updating factors α and β, we drive the rules experimentally based on the step response of the process. The evaluation performance measures are peak overshoot (OV), Rise time (RT) and Amplitude (AM). Fig. 5 shows an example of a desired time response. 1.5
OV
-1
-0.5
0
0.5
1 1 AM
MFs for e, ∆e and ∆u. Output
Fig. 2.
1
0.5
Big
Small
RT
0
-0.5 Fig. 3.
0
0.5
MFs for gain updating factor α.
1
0
1.5 Fig. 5.
1
2
3
4
5 Time
6
7
8
9
10
Performance measure of step response.
For example, if the system response is slower than desired, i.e. ∆RT is positive, and then it really needs to increase
CHOPRA, MITRA & KUMAR, AUTO TUNING OF FUZZY PI TYPE CONTROLLER USING FUZZY LOGIC
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Rule-Base for β
Z-1
R +
∆eN
∆e
+ e
βG∆e e
αGe
y
Control Rule-Base
eN
∆uN
∆u +
u
Gu + Z-1
Rule-Base for α
y
Process Fig. 1.
Block diagram of FPIC.
the effect of error on the system and decrease the effect of derivative error. If e is +ve (PB, PM or PS) and ∆e is –ve (NB, NM or NS) then α is B and β is S. Then input scaling factors Ge, increases and G∆e decreases. Similarly, if the overshoot or amplitude of oscillation is higher, then decrease the effect of error and increase the effect of derivative of error on the controller. If e is NB and ∆e is ZE then α is S and β is VB. Thus the input scaling factor G∆e is increased in this case. The other rules could be explained similarly. The effectiveness of tuning based on scaling factors is sometimes bounded by the contradictory requirements in these factors resulting from different performance measures. For example if change in OV and RT are both negative, then rules say that input scaling factor G∆e (β) should be PB or NB. Such type of conflicts can be resolved by effective a correction based on the relative firing strengths of the conflicting rules. The rule base for α and β is shown in Table II and Table III. TABLE II RULE BASE FOR α. ∆e/e NB NM NS ZE PS PM PB
NB B S S S S S B
NM B B S S S B B
NS B B B S B B B
ZE B B B B B B B
PS B B B S B B B
PM B B S S S B B
PB B S S S S S B
TABLE III
III. F UZZY L OGIC C ONTROLLER U SING FSC A PPROACH To make the implementation possible with limited processor throughout, this section is focused on reducing the number of fuzzy rules. One of the most important problems while designing fuzzy controller is to derive the desired and shortest
TABLE III RULE BASE FOR β. ∆e/e NB NM NS ZE PS PM PB
NB S M B VB B M S
NM S M M B M M S
NS S S M M M S S
ZE S S S M S S S
PS S S M M M S S
PM S M M B M M S
PB S M B VB B M S
fuzzy rule base [27]. Trial and error has been a natural choice to design fuzzy controller in this case. The selection of fuzzy if-then rules often relies on a substantial amount of heuristic observations to express proper strategy. Obviously it is difficult for human experts to examine all the input-output data from a complex system to find the proper number of rules for a fuzzy system. To cope with this difficulty, the optimal way is to reduce the number of rules covering the whole input/output data using clustering techniques. Here a clustering technique FSC is used to reduce the number of rules [21]. It has been proved in this section that FSC approach also reduces the computational time and memory with reduced set of rules and gives similar performance. To identify the Fuzzy controller using FSC approach, some data is needed, i.e., a set of two-dimensional input vectors X={X1 , X2 ,. . . .. Xn } where X={e and ∆e} and the associated set of one-dimensional output vectors as Y={Y1 ,. . . . . . . Yn } where Y={u} is required. In the present work this data has been generated by sampling input variables e and ∆e uniformly at the step size of 0.1 from the above Fuzzy controllers, and computing the value of {u}, {α} and {β} for each sampled point. The Fuzzy toolbox of Matlab and programming in Matlab has been used to generate the data points. In all the above fuzzy controllers the membership functions for controller inputs (i.e., e and ∆e) and incremental change in controller output (i.e., ∆u) are all defined on the common normalized domain [-1, 1]. The number of data points thus generated is 441 in each case. The next task is to extract a smaller set of rules using FSC
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IV. S IMULATION R ESULTS In order to demonstrate the effectiveness and feasibility of the proposed auto tuning mechanism, by computer simulations we test time responses for the control of the variety of linear and nonlinear processes. For a clear comparison between the conventional and auto tuning FLC’s, several performance measures such as peak overshoot, settling time, rise time and integral square error (ISE) are used. The values of different performance indices are provided in are provided in tabular forms in Table IV for each process separately. In case of auto tuned system, Fuzzy PI type controller is denoted by ATFPIC and system without tuning (conventional FLC) is denoted by FPIC. The processes are described as follows. System 1: The process of coupled tank [15] is given by: 0.4219 (5) 1149.893s2 + 111.834s + 1 The proposed fuzzy controller is applied on coupled tank for controlling the level of fluid. The control input is the pump drive voltage. The sensed output is the water depth in tank 2. Initially the scaling factors are chosen as Ge, = 0.02, G∆e = 1 and Gu = 2. After that input scaling factors are modified at each sample by the gain updating factors α and β. There is no specific method to decide the initial scaling factors, but the suitable value of the scaling factors can be decided by the knowledge of the process to be controlled. Response characteristics for the system with auto-tuning and without tuning are shown in Fig 6. G1 (s) =
System 2: The transfer function is given by: 1 .e−0.1S (6) s+1 Secondly, the proposed fuzzy logic controller is applied on a well known example of first order delayed process. This form of transfer function is typically used to approximate process control systems (6) while computing Ziegler Nichol parameters for a PI controller. The process plant is taken as first order system with time delay and a random noise with maximum and minimum peak value of ±0.01 (which implies G2 (s) =
1.5 FPIC ATFPIC
1
Output
approach to do the same. To extract the rules, firstly data is separated into groups according to their respective classes. Subtractive clustering is then applied to the input space of each group of data individually for identifying each class of data [23]. Each cluster center may be translated into a fuzzy rule for identifying the class. The fuzzy subtractive clustering approach is shown to reduce 49 rules to 8 rules for ∆u, 7 rules for α and 9 rules for β maintaining almost the same level of performance as shown in section 4 for system 3. After reducing the rules the computation become fast and it also consumes less memory. The computational time is calculated using the Process Explorer –Sysinternals software. It has been shown clearly in the Table in section 4 and the values of computational time and memory using same simulation time with and without clustering for each system is given in tabular forms in Table V.
0.5
0
0
500
1000
1500
Time (sec.)
Fig. 6.
Unit step response of system 1.
2 % random noise w.r.t input signal) is also added using a block of random number from matlab simulink in the system. Initially the gains are Ge, =1, G∆e =1 and Gu =2. Response characteristics for the system with auto-tuning and without tuning are shown in Fig 7. 1.4 FPIC ATFPIC 1.2
1
0.8 Output
16
0.6
0.4
0.2
0
0
Fig. 7.
2
4
6
8
10 12 Time (sec.)
14
16
18
20
Unit step response of System 2.
System 3: This is a marginally stable system because one of its poles is at the origin [4] and presence of dead time makes the system further difficult to control. G3 (s) =
1 .e−LS s(s + 1)
(7)
Initially the scaling factors are chosen as Ge, = 0.9, G∆e = 15 and Gu = 0.35. Performance curves for the system with auto-tuning and without tuning with L=0.1 are shown in Fig 8. For comparison, the response curve of the self tuning scheme for output scaling factor proposed by [4] can be seen.
CHOPRA, MITRA & KUMAR, AUTO TUNING OF FUZZY PI TYPE CONTROLLER USING FUZZY LOGIC
After increasing the output scaling factor by three times as suggested by Mudi and Pal, the performance indices such as peak overshoot (%ov), settling time (ts (s)), rise time (tr (s)) can be compared. In each case, our scheme gives better results. The response with and without clustering is also shown in the Fig 9 from which it can be identified that the response using clustering is similar to the response of original system.
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The process has been tried for different values of dead time (L). Here the response characteristics are presented with L = 0.1, Ge, = 0.9, G∆e = 11 and Gu = 1 as shown in Fig 10. The overall performance of the ATFPIC can also be compared with existing paper [26] and it performs better in all cases. 1.6
1.8 FPIC ATFPIC
1.6
FPIC ATFPIC
1.4 1.2
1.4 1 Output
Output
1.2 1
0.6
0.8 0.6
0.4
0.4
0.2
0.2
0
0
0
Fig. 8.
10
20
30
40 50 Time (sec.)
60
70
80
Fig. 10.
1
0.8
0.6
0.4
0.2
Fig. 9.
10
15
20
25 30 Time (sec.)
35
40
45
50
Unit step response of System 4.
V. C ONCLUSION
1.2
0
5
Unit step response of system 3.
Without clustering With clustering
0
0
90
1.4
Output
0.8
5
10
15 Time (sec.)
20
25
30
Unit step response of System 3 with and without clustering.
System 4: Different nonlinear processes are also simulated for performance comparison of auto tuned system and conventional FLC. In all cases, our proposed scheme gives a remarkably improved performance. One such nonlinear process [26] is reproduced as y¨ + y˙ + 0.25y 2 = u(t − L)
(8)
Fuzzy logic controllers are believed to be robust to the system disturbances and non-linearity. The performance of FLC can be further improved by make it adaptive. A methodology was proposed to design a PI type fuzzy logic controller using auto tuning mechanism. The proposed controller was tuned online by gain updating factors for input scaling factors of FLC. The updating factors are based on fuzzy inference rules defined on error and change in error. Performances of auto tuned FLC were also compared with those of their corresponding conventional FLC with respect to several indexes such as peak overshoot, settling time and rise time and integral square error (ISE). By comparing their performance with existing paper [4], it is seen that auto tuned PI type FLC performed better than self tuned FLC but needs one more set of fuzzy inference rules for its updating factors for input scaling factors than self tuned FLC. Certainly, the improved performance is at the cost of increased computational time and memory. But in this paper, the FSC approach is used to reduce the rules which in turn reduce the computational time and memory considerably. A simulation analysis on a wide range of linear and nonlinear processes and comparison of results has shown that the proposed methodology gives better performance than existing schemes, fast, robust, reduces computational time and consumes less memory. The other feature of this scheme is it depends neither on the process being controlled nor on the controller used.
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TABLE IV VALUE OF PERFORMANCE MEASURES . System 1 2 3 4
FLC ATFPIC TFPIC ATFPIC TFPIC ATFPIC TFPIC ATFPIC TFPIC
%OV 25 40.9 7.6 23.5 8.7 72 20 51
Settling Time(sec) 570 723 4.1 5.5 8.4 61 12.2 19.4
Rise Time (sec) 170 140 2.5 2.1 6.25 4.15 3.8 2.8
ISE 831 898 9.14 12.25 496 979 19.49 22.38
TABLE V VALUES OF COMPUTATIONAL TIME AND MEMORY USING SAME SIMULATION TIME . System 1 2 3 4
Without Clustering With Clustering Without Clustering With Clustering Without Clustering With Clustering Without Clustering With Clustering
Computational Time 3 min 7 sec 656 ms 26 sec 62ms 38 sec 985 ms 750ms 1 min 9 sec 281 ms 2 sec 125ms 2 min 10 sec 423 ms 2 sec 31ms
R EFERENCES [1] Jihong Lee, “On Methods for Improving Performance of PI-Type Fuzzy Logic Controllers,” IEEE Transactions On Fuzzy Systems, Vol. 1. No. 4, November 1993. [2] Parasiliti, M.Tursini, D.Q.Zhang, “An Analytical Approach to the Derivation of Fuzzy PI Scaling Factors”. E-mail:
[email protected] [3] Petr Pivonka, “Comparative analysis of Fuzzy PI/PD/PID controller based on classical PID controller approach”. URL: http://www.feec.vutbr.cz/∼pivonka/ [4] R.K.Mudi and N.R.Pal, A robust self-tuning scheme for PI and PD type fuzzy controllers, IEEE trans. Fuzzy System., 7(1), 1999, 2-16. [5] Walter C. Daugherity, Balaji Rathakrishnan and John Yen, “Performance Evaluation of a Self-Tuning Fuzzy Controller,” in Proc. IEEE Int. Conference Fuzzy systems, San Diego, CA, March 1992, pp 389-387. [6] Kuldip S. Rattan and Dale Van Cleave, “Design and Implementation of a Reduced Rule Fuzzy Logic PID Controller,” URL: email:
[email protected]. [7] Yu Cheng, Fu-Rong Lei, Wen-Li Xu and Yi-Sheng Zhong, “Speed Control of Ultrasonic Motors by Auto-Tuning Fuzzy PI Control,” Proceedings of the 4th World Congress on Intelligent Control and Automation, Shanghai, China, June 10-14, 2002, PP- 1882-1886. [8] James Carvajal, Guanrong Chen, Haluk Ogmen, “Fuzzy PID controller: Design, performance evaluation, and stability analysis, Information Sciences, 123, 2000, 249- 270. [9] Driankov, H. Hellendorn, and M. Reinfrank, “An Introduction to Fuzzy Control”. New York: Springer-Verlag, 1993. [10] R.R. Yager and D.P. Filev, “Essentials of Fuzzy Modeling and Control”. Singapore: John-Wiley & Sons, 1994. [11] Seema Chopra, R. Mitra and Vijay Kumar, “Identification of Self-Tuning Fuzzy PI type controllers with reduced rule set,” Proc. of the IEEE International Conference on Networking, Sensing and Control, pp-537 March, 2005. [12] Seema Chopra, R Mitra, Vijay Kumar, “Analysis and Design of Fuzzy models using Fuzzy Curves”, Proceedings of the Second national Conference on Intelligent systems & networks, Page 65-71, Feb, 25-26, 2005. [13] K. Ogata, Modern Control Engineering. Englewood Cliffs, NJ: PrenticeHall, 1970. [14] M. Gopal, Control Systems Principles and Design. India: Tata McGrawHill, 1993. [15] Kevin M.Passino and Stephen Yurkovich, “Fuzzy Control”, Addison Wesley Longman, Inc., California, 1998. [16] C. C. Lee, “Fuzzy logic in control systems: Fuzzy logic controller—Parts I, II,” IEEE Trans. Syst., Man, Cybern., vol. 20, pp. 404–435, Mar./Apr. 1990. [17] S. Z. He, S. Tan, F. L. Xu, and P. Z. Wang, “Fuzzy self-tuning of PID controller,” Fuzzy Sets Syst., vol. 56, pp. 37–46, 1993.
Memory 3592k 1180k 1222k 124k 23688k 116k 400k 92k
Simulation Time 1500 sec 1500 sec 10 sec 10 sec 90 sec 90 sec 50 sec 50 sec
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