Online Rule Weighting of Fuzzy PID Controllers

Onur Karasakal, Mujde Guzelkaya, Ibrahim Eksin, Engin Yesil Istanbul Technical University, Faculty of Electrical and Electronics Engineering, Control Engineering Department,

Maslak, TR-34469, Istanbul, Turkey onur _ karasakal@yahoo.com guzelkaya@itu.edu.tr eksin@itu.edu.tr yesileng@itu.edu.tr

Abstract-In this study, a new weighting method is proposed for the fuzzy rules of the fuzzy PID controllers in an on-line

manner. First, the transient phase of the unit response of the closed loop system is taken into consideration and the response is divided into certain regions which are assigned in accordance with the number of membership functions defined for the error input of the fuzzy logic controller. Secondly, the relative

importance or influence of the fired fuzzy rules of the fuzzy logic controller are determined for each region and the meta-rules are derived for the adjustment of corresponding fuzzy rule weight values to obtain an efficient and appropriate control signal that

will achieve a desired system response. For this purpose, two simple functions based on the absolute value of the normalized system error are used for the assignment of the rule weights by an adequate arrangement in accordance with the meta-rules

derived. The effectiveness of the proposed self tuning method is demonstrated on various processes by simulations. The proposed new fuzzy rule weighting method improves the transient response in terms of overshoots, oscillations and settling time.

Keywords-Fuzzy PID controller, fuzzy rule weighting, self­tuning.

I. INTRODUCTION

Most of the industrial processes are still using the conventional PID controllers because of their simple control structure, ease of design and effectiveness for linear systems [1]. On the other hand, most of the real systems are of higher order, have time delays or nonlinearities or without of precise mathematical model. Due to their linear structure, conventional PID controllers are usually not effective for this kind of real systems. Thus, fuzzy logic is extensively used in processes where system dynamics is either very complex or exhibit a highly non-linear character. The frrst fuzzy logic control (FLC) algorithm, implemented by Mamdani [2], was constructed to synthesize the linguistic control protocol of a skilled human operator.

In literature, various types of fuzzy PID (including PI and PD) controllers have been proposed [3-13]. The design parameters of the fuzzy PID controllers can be summarized within two groups [14]:

a) structural parameters, and

b) tuning parameters.

Structural parameters include input/output (I/O) variables to fuzzy inference, fuzzy linguistic sets, membership functions (MF), fuzzy rules, inference mechanism and defuzzification mechanism. Tuning parameters include input/output (I/O) scaling factors (SF) and parameters of membership functions.

978-1-4244-6588-0/10/$25.00 ©201 0 IEEE

The structural parameters are generally determined during off­line design where the tuning parameters can be recalculated during on-line adjustments of the controller to enhance the process performance, as well as to accommodate the adaptive capability to system uncertainty and process disturbance.

In the case of the real systems have nonlinearities, parameter changes, modeling errors, disturbances, etc., the usage of fixed value scaling factors may not be sufficient to achieve the desired system performance. Therefore to overcome these kinds of disadvantages, a lot of heuristic and non-heuristic tuning algorithms for the adjustment of scaling factors of fuzzy controllers have been presented in literature [9, 15-19].

In addition to scaling factor tuning algorithms, although there is no systematic method to design membership functions and examine the number of fuzzy rules, the literature also includes some methods to tune the membership functions and fuzzy rules of the fuzzy controllers. Juang et al. [20] tuned the membership functions by means of genetic algorithm (GA) using a fitness function to improve the system performance. Ahn and Truong [21] used a robust extended Kahnan filter to tune the input membership functions and the weight of the controller output during the system operation process. Teng et al. [22] proposed a genetic weighted fuzzy rule based system in which the parameters of membership functions including position and shape of the fuzzy rule set and weights of the rules are evolved using a genetic algorithm in an off-line manner.

Alternatively, in [23] a fuzzy rule base shifting scheme for systems with time to improve system performance is proposed. The shifting scheme of the fuzzy rules is tabulated with respect to the normalized dead time, therefore, in some way, the structural parameters of the FLC was tuned in an on-line manner.

In this study, a new method is proposed for the tuning of the fuzzy rule weights of the Fuzzy PID (FPID) controllers in an on-line manner. Although the number of the membership functions defined for the error input of the FLC is assumed to be three, this is not a limitation for the proposed method. The transient phase of the response is frrstly assumed to be divided into certain regions which are assigned in accordance with the number of membership functions defmed for the error input of the FLC. Then, the relative importance or influence of the frred rules is determined for each region and the meta-rules are derived for the adjustment of their weight values to obtain an efficient and appropriate control signal. The weight tuning is accomplished using the system error value which is available for each sampling time during the transient system response.

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Fuzzy U Controller G(s) y

Figure l. The closed-loop control structure for fuzzy PID controller.

The outline of the paper can be swnmarized as follows: Section 2 includes the FPID controller structure without a rule weight tuning mechanism, the interpretation of rule weights and the proposed on-line rule weighting method for FPID controllers, Sections 3 present the simulation examples and results for various processes, and Section 4 provides the discussions and the conclusions.

II. THE PROPOSED Fuzzy RULE WEIGHTING METHOD

A. Fuzzy PID Controllers

In this study, a two input fuzzy PID controller given in Fig.I, which is formed using a fuzzy PD controller with an integrator and a summation unit at the output, is used.

The output of the fuzzy PID controller is given by

u =a,U + M Udt (1)

The error (e) and the derivative of error ( e) are used as the inputs and the change of the control signal (U) is used as the output of the FLC. The input scaling factors K., for error and Kd for the derivative of error normalize the inputs to the range in which the membership functions of the inputs are defmed. The output scaling factors a and � normalize the output of FLC to an applicable value. The universes of discourse are chosen to be [-I, 1] for the inputs e, e and output U.

N

NBNM

-I -0.8

� Z P

(a) e, e

� Z PM PB

o (b)

0.8 1 u

Figure 2. The membership functions for (a) inputs e and e, (b) output U

In this study, three membership functions are assigned to both inputs e and e. The membership functions for the inputs are chosen as uniformly distributed triangular functions and for the output singleton membership functions are used. The membership functions for the inputs and the output have been illustrated in Fig. 2a, and Fig. 2b, respectively. The diagonally symmetrical rule base, which is given in Table 1, has been used for the fuzzy PID controller.

TABLEt. THE RULE BASE OF THE FUZZY PID CONTROLLER

ele N Z P N NB (w/) NM(W2) Z(W3) z NM (W4) z (Wj) PM (W6) p z (w,) PM (ws) PB (W9)

B. The Weights o/Fuzzy Rule

A comprehensive analysis on the fuzzy rule weights can be found in [24]. The weight (w) is a real value variable that can change for each rule and it is generally added to a rule with the phrase "with w". The weight values can be considered as a structural parameter of a fuzzy PID controller. The weight of the rule shows the importance or influence of that fuzzy rule for the inputs at a specified time. The fuzzy rules of FLCs with weights are generally written as;

Rk: IF the error ( e) is Nk and the derivative of error ( e ) is Mk

THEN the change of control signal (U) is Ck with Wk where Nk and Mk are the fuzzy sets of the antecedent, Ck is the consequent fuzzy set.

Generally the weights can be applied to the rules in two different ways. In this study, only, the case where the weights are applied to complete rules has been taken into consideration. When the singleton membership functions are used at the consequent part of the rules, the rule weights occur in the nwnerator and the denominator. The output of FLC is calculated as;

m IfkCkwk

U = ...,k.;:,=!'--__

m Ifkwk k=!

(2)

where fk' Wk and Ck are the firing strength, weight and the output singleton membership value ofkth rule, respectively.

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Since the truth degree of a fuzzy rule changes in the interval of [0, I], the weights should be chosen also in the interval of [0, 1]. When the weight of the rule is equal to one, this rule will have a great importance, consequently, when the weight is zero then the rule will loose the importance. For any weight value in the interval of [0, I], will relatively defme the importance of the rule to the other active rules. Therefore, defining the weights within range of [0, I] enables us to compute and tune the output of FLC in the usual way.

C. Online Rule Weighting of a Fuzzy PID Controller

For the fuzzy PID controller structure shown in Fig. I, the step response of the closed loop system can be divided into four main regions as illustrated in Fig. 3, since three membership functions are defined for the error as illustrated in Fig. 2a.

y(t} G

.'..

1= G

0 N r

_G

� 0 Time

Figure 3. The partitioning of the step response into regions.

The calculated value of the FLC output takes different values at each region to achieve the desired system performance. The meta-rules derived to achieve the desired performance can be given as follows:

i. When the system error value is positive and the system response approaches the set-point, the control signal must be decreased in time to obtain the required fastness of the system response sufficiently.

ii. When the system error value is negative and the system response drifts apart from the set-point, a maximum negative big control signal is needed to prevent the overshoot.

iii. When the system error value is negative and the system response approaches the set-point, the control signal value must be increased in time to obtain the required fastness of the system response sufficiently.

iv. When the system error value is positive and the system response drifts apart from the set-point, a maximum positive big control signal is needed to prevent the undershoot.

These meta-rules can be associated with the importance of the fuzzy rules in order to obtain the required control signal for satisfactory system performance. For example; when a positive big control signal is needed, the importance of the fuzzy rules which have positive rule consequent will be greater than the importance of the fuzzy rules that have negative rule

consequent. Since the importance of a fuzzy rule is related to its weight value, the change of the importance of a fuzzy rule at any region can be expressed as changing the weight value of that rule.

The weight tuning can be accomplished using the value of the system error. Since the interval for rule weight values should lie within the range [0, I], the interval of the normalized system error [-I, I] is mapped to the interval [0, I] using the absolute value function. Then, the error functions

f) ( e )=a.Abs( e )

f2( e)= I-Abs( e)

(4)

(5)

are directly assigned as the rule weight values by an adequate arrangement in accordance with the meta-rules given above. By these assignments the error value becomes the tuning variable of the rule weights. The arrangement of the functions f) and f2 as the rule weight values is shown in Fig. 4 according to the membership functions of error and derivative of error.

According to meta-rules, the weight tuning at each region can be accomplished as given below:

(a) Region I: At this region, the absolute value of error decreases from 1 to 0. When the value of error decreases from 1 to 0, the importance of the fuzzy rules Rs and R7 decreases while the importance of fuzzy rules Rs and � increases in order to obtain the required fastness of the system response sufficiently. The value of "a" in function f) is taken to be 1 for all the regions except the first region. Since the error value possesses its extreme value at the first region the value of "a" is taken to be 0.5 so as to prevent the excessive acceleration of the system response and thus the possible overshoots. Thus, the weights can be tuned as;

w4=I-Abs(e), ws=I-Abs(e), wrAbs(e), ws=O.5Abs(e).

(b) Region 2: At this region, the absolute value of error increases from ° to maximum overshoot. In order to prevent overshoot, the importance of the fuzzy rules Rio R2, � and Rs should be kept at their maximum. So the weights can be tuned as;

w)=I-Abs(e), w2=I-Abs(e), w4=I-Abs(e), ws=I-Abs(e).

(c) Region 3: At this region, the absolute value of error decreases from maximum overshoot to zero. So the importance of the fuzzy rules R2 and R3 decreases while the importance of fuzzy rules Rs and � increases in order to obtain the required fastness of the system response sufficiently. So the weights can be tuned as;

w2=Abs(e), w3=Abs(e), ws=I-Abs(e), w6=I-Abs(e).

(d) Region 4: At this region, the absolute value of error increases from ° to I. In order to prevent undershoot, the importance of the fuzzy rules Rs, �, Rs and R9 should be kept maximum. So the weights can be tuned as;

ws=I-Abs(e), w6=I-Abs(e), ws=I-Abs(e), w9=I-Abs(e).

The self tuning rule weight table for nine fuzzy rules is given in Fig. 4.

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� I Z

N>=

N

Z

P

-1

N

f2

f2

fl

o

Z P f2 ! fl fl . f2 � f2 f2 . fl l f2 f2

Figure 4. The fuzzy rule weighting table for nine fuzzy rules.

III. SIMILATIONS EXAMPLES

In this study, many simulations are perfonned, but in this paper, following four benchmark systems are used for analysis of the proposed method for Fuzzy PID controller:

1) Time delay and double lag [25]:

G(s) = 1

e-s (1 + 0.5S)2 (6)

This system is similar to fIrst order plus time delay (FOPTD) systems, but it has more high frequency roll-off.

2) Marginally Stable System [16]:

G(s) = _1_e-o. 3s

s(s + 1) (7)

This is a marginally stable system because one of its poles is at the origin and presence of dead time makes the system further diffIcult to control.

3) Right halfplane zero [25]:

I-s G(s)=-­

(s + 1)3 (8)

This system has three equal poles and a right half plane zero. The difficulty of control increases as the location of the zero approaches to origin.

4) Unstable pole [25]:

1 G(s)=-­

S2 -1 (9)

This is a simple model of an inverted pendulum, or, an unstable batch reactor is an example from industry.

Matlab is used for all of the simulations. As a fIrst step, Matlab codes are developed for fuzzy PID controller (FPID) and online rule weighted FPID controller (ORW-FPID) using the membership functions illustrated in Fig. 2, and fuzzy rule based given Table I.

Secondly, for each process the input and scaling factors are detennined using trail-error methods just to obtain a quick stable system response with a small rise time. The scaling factors used in the simulations are summarized in Table II.

TABLE II. THE SCALINF FACTORS OF FPIO CONTROLLERS

Process Scaling Factors K. Kd a 0

1 1 0.2 0 0.75 2 0.8 2.5 0.22 0 3 0.7 0.2 0.55 0.6 4 0.85 1.05 2.2 2.9

In order to compare the perfonnance of transient responses of the proposed FPID controller with an online rule weight tuning method and the conventional FPID controller, fIve different perfonnance measures are considered. The two of these perfonnance measures are selected from the classical transient system response criteria; namely the maximum overshoot (%OS), and the settling time (Ts). The next three perfonnance measures are considered as follows:

i) Integral Absolute Error (IAE) which is defmed as

lAE = J; Ir(t) -y(t)1 dt (10)

ii) Integral Time Absolute Error (ITAE) which is defmed as

ITAE= J; t lr(t)-y(t)1 dt (11)

iii) Total Variation (TV) [26] of the control input u(t), which is defIned as

'" TV = L IUi+1 -ud (12) i=1

The unit step output responses and the control signals of four processes are illustrated in Fig. 5,6, 7 and 8, respectively. In addition, the perfonnance analysis of the FPID controllers according to the used perfonnance indices is given in Table III, IV, V and VI for each process.

The proposed online rule weight based fuzzy PID (ORW­FPID) controller reduces the overshoot value as it is expected when it is compared to fuzzy PID controller. As it can easily seen from the simulation results, FPID controllers with and without the rule weighting have almost the same rise time, but the settling time of the proposed ORW-FPID controller is signifIcantly smaller than the others. The IT AE value of EBW A based fuzzy controller is drastically less than the other FLCs since the oscillations vanish. Finally, the proposed ORW­FPID has a low value of TV that shows that it has the smoothest control signal.

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1.5 -"r -,--------r------r;::=====:::J======::::l k\ :,

J ---- - ---- ' :I�-FPID I ,\ /-, , �---�\�, -/����+J ��--+-----�------­\, I \, I

'r"

0.5 - _.J ------------,

o J o 10 20 30 40 50 Time [s) (a)

1.4 ,-------::-,-----,-----r:=====c::==::::::;l 1'\, I

1.2 I �\ , , -----, FPID ,'- \-----, ---- -, -- ORW-FPID " " 1.,- ... , I 1 ---t- - ,-- I .... #�

I ..... _/ : I 0.8 >: 0.6

0.4 0.2

------ �----- � ------

______ L _____ � _____ _ , , ______ L _____ � _____ _ , , ,

� ___ _ L ____ _ , ,

O �-----�----�------�-----L----� o 10 20 30 40 50 Time [s) (b)

Figure 5. Simulation results for the first process: (a) System outputs, and (b) control signals.

TABLE III. COMPARASION OF TWO FPID CONTROLLERS FOR THE FIRST PROCESS

Criteria Controller FPID ORW-FPID

O/OOS 49 34 Ts 8. 2 16.5 IAE 5.711 3.966 ITAE 33.45 ILl6 TV 0.3267 0.1874

IV. CONCLUSION

A new on-line method is presented for the adjustment of the fuzzy rule weights of the fuzzy PIO controller that has diagonal rule base. The adjustments of rule weight values are performed in an online way by using the absolute value function of the normalized system error directly as the weight values. Normalized system error can also be used in scaling factor adjustment of FLCs in a more complex way but in this case the weight values of the fired rules are equally tuned and the relative importance of the fIred rules cannot be considered. Since the proposed weighting method just changes the weight values of the fuzzy rules, it does not bring an additional computational complexity as compared with the fuzzy PIO controllers.

2

1.5

'5

0.5

0 0

0.5 0.4 0.3

'5 0.1 0

-0.1 -0.2 0

'[�-- F�ID I -- ORW-F�

\ I \ I '.../

�� , ,

10 20 30 40 Time[s) (a)

. , : ----- FPID ----, ------ ,------- -- ORW-FPID _ _ ___ � ______ L _____ � ______ � ____ _ 1\ , I \ \

_ \ __ � ______ L _____ � ______ � ____ _

\ ' \

__ _ \ _ � ______ � _____ � ______ L ____ _

. , ........ , , ," I , __ ..

\ I I I I - - - - -''';'J-l- - - - - � - - - - - � -- - - - - � - ----

10 20 30 Time[s) (b)

40

50

50

Figure 6. Simulation results for the second process: (a) System outputs, and (b) control signals.

TABLE IV. COMPARASION OF TWO FPID CONTROLLERS FOR THE SECOND PROCESS

Criteria Controller FPID ORW-FPID

%OS 34 7.5 Ts 24 10,3 IAE 6. 742 4.779 ITAE 47.64 16.26 TV 0.1113 0.0623

For the comparison between the FLCs, various performance indices such as maximum overshoot (% OS), settling time (Ts), integral absolute error (IAE), integral time absolute error (ITAE) and total variation (TV) of the control signal are used. The effectiveness of the proposed method is shown by several simulation examples. The results obtained from the simulations show the drastic improvements in favor of the proposed method over various performance indices.

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1.5 i--, ------, ------r;=======;_]

%: 0.5

I _____ , FPID --ORW-FPID

-----�------r-----�------ r--- - -I

---- � ------ � ----- � ------ � -----

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Figure 7, Simulation results for the third process: (a) System outputs, and (b) control signals.

TABLE V. COMPARASION OF TWO FPID CONTROLLERS FOR THE THIRD PROCESS

Criteria Controller FPID ORW-FPID

%OS 33 ---Ts 19,5 7. 4 IAE 6. 764 4.778 ITAE 39.75 12.47 TV 0.225 0.1138

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:J

1.5 i-:,=-, -,----,----r;=======;_]

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0.5 - ---- ------ ----- ______ L ____ _ I

0 '-----'-----'-----'------'----' o

0

-1

-2

-3 0

I I

10

I r

r

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I T

- T

I ,

30

-)r:---�-------�----� I I

5 10 Time [51 (b)

40 50

----- FPID --ORW-FPID -r I

__ .1. ______ _

15 20

Figure 8. Simulation results for the fourth process: (a) System outputs, and (b) control signals.

TABLE VI. COMPARASION OF TWO FPID CONTROLLERS FOR THE FOURTH PROCESS

Criteria Controller FPID ORW-FPID

O/OOS 30.5 15.5 Ts 7.1 3.5 IAE 1.792 1.428 ITAE 3.968 2.409 TV 0.8718 0.5906

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