�

Abstract—Fuzzy PID is a nonlinear controller with the advantages of simplicity, acceptance, and easy implementation. However, the fuzzy PID controller still lacks in standardized tuning method, and its performance can not be guaranteed. This paper presents a phase plane based fuzzy PID tuning method, and describes in detail designing of fuzzy PID rule base. Andfinally the effectiveness of the method is verified by Matlab simulation.

I. INTRODUCTION

PID controller is widely used in industrial processes.

Constant-coefficient PID controller has the satisfactory effect for most SISO linear processes, but as a linear controller, it is powerless for the complex processes. In order to improve the PID control effect, a variety of modified PID have been appeared. Variable-coefficient PID is one of the important forms, in which fuzzy reasoning provide a method of smooth handoff for the variable-coefficient PID's coefficients. The fuzzy PID [1]-[5] controllers of combining fuzzy reasoning and PID are widely used in practice.

The tuning of fuzzy PID often uses the trial and error method, it's tuning process is slow, and the control performance can not be guaranteed. The fuzzy PID tuning is divided into two stages including non-linear and linear tuning in the literature. The non-linear stage is the tuning of fuzzy rule base, and the linear tuning is for the quantization factor. The fuzzy rule base determines the nonlinear performance of controller, this article focuses on the rule base that based on the phase plane method. According to the dynamic response characteristics of control object, the phase plane is divided into different areas, different regions will be mapped to the rule base, thus the contents of rule base can be tuned. The tuning method is intuitive, versatile, and easy implementation.

II. THE STRUCTURE OF FUZZY PID

The structure of fuzzy PID is shown in figure 1, the discrete form is:

u kp e ki Ts e kd Ts e� � � � � � ��� (1)

kp, ki, kd each has the following linear mapping:

This work was supported by the Fundamental Research Funds for the Central Universities (2011QN144).

Yunfei Lv is with Second Ship Design institute,Wuhan 430064,China .Hui Luo is with Huazhong University of Science and Technology, Wuhan

430074, China (corresponding author to provide phone: 15387130919; e-mail: keyluo@yahoo.com.cn).

Yong Cai is with Second Ship Design institute,Wuhan 430064,China.

ˆ ˆ ˆ( , )ˆ ˆ ˆ( , )ˆ ˆ ˆ( , )

min p max min

min i max min

min d max min

kp=kp +k e e (kp -kp )

ki=ki +k e e (ki -ki )

kd=kd +k e e (kd -kd )

� �

� �

� �

(2)

y

fuzzyreasoning

PID Control object

d/dt ee�

refkp ki kd

u

Figure 1: the structure of fuzzy PID

ˆ ˆ,e Ge e e Gc e� � � � � � , Ge,Gc are respectively the quantizaiton factor of error e and error change rate Δ e , ˆ ˆ,e e�change in [-1, 1] after quantified; [ min maxkp kp ],

[ min maxki ki ], [ min maxkd kd ] are respectively the value range

of kp, ki, kd , ˆ ˆ ˆ( , )pk e e� , ˆ ˆ ˆ( , )ik e e� , ˆ ˆ ˆ( , )dk e e� are the

mapping functions that described in the following n n�fuzzy reasoning statement.

ˆ ˆ

ˆ ˆ

ˆ ˆ

1 1

11 11 11

1 2

12 12 12

n n

nn nn

if e is E , and e is DEthen kp is KP , ki is KI and kd is KDif e is E , and e is DEthen kp is KP , ki is KI and kd is KD

......if e is E , and e is DEthen kp is KP , ki is KI

�

�

�

nn and kd is KDIf the fuzzy reasoning system with product inference

engine single-valued fuzzy control and center average defuzzification control is used in the above fuzzy reasoning

process, the analytical expression of ˆ ˆ ˆ( , )pk e e� , ˆ ˆ ˆ( , )ik e e� ,

ˆ ˆ ˆ( , )dk e e� are shown as follows. In which,

ˆ ˆ( ), ( )E e DE e � is the membership function of ˆ ˆ,e e� ,KPij,KIij,KDij are the fuzzy values of kp, ki, kdcorresponded to the single-valued fuzzy control, ie fuzzy PID rule base. The process of setting for the rules table will be discussed in the following.

1 1

1 1

ˆ ˆ( ) ( )ˆ ˆ ˆ( , )

ˆ ˆ( ) ( )

n n

ij i ji j

p n n

i ji j

KP E e DE ek e e

E e DE e

� �

� �

� � �� �

� �

��

�� (3)

Research on Tuning Method for Fuzzy PID Yunfei Lv, Hui Luo, and Yong Cai

Fourth International Workshop on Advanced Computational Intelligence Wuhan, Hubei, China; October 19-21, 2011

978-1-61284-375-9/11/$26.00 @2011 IEEE 334

1 1

1 1

ˆ ˆ( ) ( )ˆ ˆ ˆ( , )

ˆ ˆ( ) ( )

n n

ij i ji j

i n n

i ji j

KI E e DE ek e e

E e DE e

� �

� �

� � �� �

� �

��

�� (4)

1 1

1 1

ˆ ˆ( ) ( )ˆ ˆ ˆ( , )

ˆ ˆ( ) ( )

n n

ij i ji j

d n n

i ji j

KD E e DE ek e e

E e DE e

� �

� �

� � �� �

� �

��

�� (5)

III. NON-LINEAR LEVEL SETTING THE TABLE OF FUZZY RULES

The typical step response curves of the control object is shown in figure 2.

t

ref

A B C D E

e

e�

y

Figure 2: the step response of control object The locus of step response on the phase plane is shown in

figure 3. According to the quadrant of �,e e� on the phase plane, the step response curve is divided into five districts including A B C D E. The values of the linguistic variables ˆ ˆ,e e� are {NB NS ZE PS PB}, the membership function takes symmetric trigonometric, the centers are uniformly distributed in [-1,1], The values of the linguistic

variables ˆ ˆ ˆ/ /p i dk k k are {B S Z}, use the single-valued fuzzy,

the centers are respectively in [0 0.5 1], its membership function is shown in figure 3. The phase plane is divided into areas of 5 5� by the language variable, it can be also divided into four regions including starting area speed zonebraking zone and the callback area, as shown in figure 4.

ˆ ˆ/e e�

ˆ ˆ( ) / ( )i jE e DE e �

0 0.5 1-1 -0.5

0 0.5 1

ZE PS PBNB NS

Z S B

1

1/ /ij ij ijKP KI KD

ˆ ˆ ˆ/ /p i dk k k

Figure 3: the membership functions of ˆ ˆ,e e� , ˆ ˆ ˆ/ /p i dk k k

B

C D

E

NB NS ZE PS PB

NB

NS

ZEPS

PB

A

starting area

speedzonebraking

area

callbackzone

brakingarea

e

e�

Figure 4: the language phase plane of 5 5�The roles of kp, ki, kd are respectively as follows:

bigger the kp, faster the speed of system response, but it is easy to overshoot, or even cause the system's instability; bigger the ki, faster the system to eliminate the static error, but if ki is too large, it will produce the integral saturation phenomenon in the early stages of the response process, which will lead to a larger overshoot of the response process. Kd inhibits the deviation to any side in the response process, but if kd is too large, it will make the response process brake in advance, thereby prolonging the adjustment time, and will reduce the system's auti-interference capacity.

Let the control object as large inertia no self-regulation system P(s)

0.02( )(50 1)

P ss s

��

To make the system step response a high dynamic performance small overshoot, and to eliminate steady state error, corresponding to the phase plane in figure 4, respectively, in all regions:

In the starting area, in order to speed up the system response speed, make the value of kp coefficient B; in order to prevent the integral saturation, make the value of ki coefficient Z; Δ e is negative, inhibiting the acceleration process, make the value of kd coefficient Z;

In the braking area, to prevent overshoot, make the value of kp coefficient Z;the value of kd coefficient B;

In the speed zone, in order to make the PID coefficient transits smoothly, make the value of kp coefficient S; the kd coefficient S;

In the callback zone, to make the output back to the reference value, make kp B, and ki B;

In the stable area, in order to reduce the sensitivity to noise, make kp s and kd S, increases ki to eliminate static error, make ki B. Based on the performance requirements, the stable region should be valued flexibly, if it's "tight control", make kp B, and kd B; if it's "similar control", make kp Z, and kd Z, only the integral works;

335

In the upper right and lower right area, for the path of

�,e e� is far away from the given direction, make the value of

kp B, make ki B, and make kd B. Thus, set the fuzzy rules of KP KI KD in table 1, table 2, and table 3.

TABLE I: FUZZY RULE TABLE OF KP

ee� NB NS ZE PS PB

PBPSZENSNB

The corresponding surface is as shown in figure 5.

Figure 5: the surface of ˆ ˆ ˆ( , )pk e e�

TABLE II: FUZZY RULE TABLE OF KI

ee� NB NS ZE PS PB

PBPSZENSNB

The corresponding surface is as shown in figure 6. Rewrite the integral term as �Ts ki e� �� , with the role of

separating integral.

Figure 6: surface of ˆ ˆ ˆ( , )ik e e�

TABLE III: FUZZY RULE TABLE OF KD

ee� NB NS ZE PS PB

PBPSZENSNB

The corresponding surface is as shown in figure 7.

Figure 7: surface of ˆ ˆ ˆ( , )dk e e�

IV. LINEAR LEVEL QUANTITATIVE FACTOR TUNING

The role of quantitative factor is to quantify e, Δ e to the seted area in figure 3, so that kp, ki, kd will be switched as pre-seted.

For the control object P(s), given as 30, emax=30. The overshoot should be less than 1, e [-1 1] that before quantified should be seted in the callback area e [-0.25 0], take the quantitative factor Gc=0.1, the e membership function's center is 10*[-1 -0.5 0 0.5 1] corresponding to e before quantified.

The steady-state gain of ( )P s� is 0.02, the limit of controller input ranges in [-25, 25], the sampling period is 0.2s, thus we can getΔ emax=25*0.02*0.2 =0.1, set the quantitative factor Gc=10, the e� membership function's center is 0.1*[-1 -0.5 0 0.5 1] corresponding to Δ e before quantified. If the control object is unknown,Δ emax can also be obtained by experimental methods.

V. THE SIMULATION RESULTS

the range of kp, ki, kd 's values determined by simulation are respectively shown as follows: kpmin=0, kpmax=5, kimin=0, kimax=0.005, kdmin=0, kdmax=500.

The step response of P(s) is shown in figure 8. Fuzzy PID has the integral term, if the actor exists dead zone and other non-linear aspects, it can quickly eliminate the static error. P(s) has a pole of 0, but the existence of integral term definitely did not cause significant overshoot.

By the simulation results we can see, compared with control

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effects of constant-coefficient PID controller (kp=3, ki=0, kd=500), the fuzzy PID has a small overshoot, and short setting time.

Fuzzy PIDPID

Figure 8: step response of y

Fuzzy PIDPID

Figure 9: controller's output u

VI. CONCLUSION

This paper analyses the method of designing fuzzy PID rule base on the phase plane in detail, gives a general design procedure, and it overcomes randomness and blindness of the fuzzy PID rule base design by making phase plane as a tool. The rule base can be modified flexibly based on the objects' dynamic characteristics on the phase plane, thus to describe the objects' nonlinear characteristics correctly. This design method is intuitive and common, and it has a strong leading significance and engineering value for the tuning of fuzzy PID controller.

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[2] D. Liu, C. Xia, M. Zhang, and Y. Wang, “ Control of brustless DC motor using fuzzy set based immune feedback PID controller,” in Proc. ISIE2007, pp. 1045-1049.

[3] S. Wang, S. Yu, and Z. Feng , “ A method for controlling a loading system based on a fuzzy PID controller,” Mechanical Science and Technology for Aerospace Engineering, vol. 30, no. 1, pp. 166-172, 2011.

[4] Q. Zhang , X. Zhou, and Q. Guo, “ Study on control methodology for parallel-connected induction motors based on adaptive fuzzy PID control,” Electric Drive Automation, vol. 33, no. 1, pp. 5-18, 2011 .

[5] J. Jian, J. Jiang, X. Wang, Y. Wang, and J. Xu, “ A Model Helicopter Self-return System Based on Fuzzy PID Control,” Aircraft Design, vol. 31, no. 1, pp. 61-65, 2011.

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