research article application of a new membership function...

Research ArticleApplication of a New Membership Function in Nonlinear FuzzyPID Controllers with Variable Gains

Xuda Zhang, Hong Bao, Jingli Du, and Congsi Wang

Key Laboratory of Electronic Equipment Structure Design of Ministry of Education, Xidian University, 2 Taibai Road,Xi’an 710071, China

Correspondence should be addressed to Hong Bao; [email protected]

Received 10 April 2014; Revised 29 July 2014; Accepted 7 August 2014; Published 28 August 2014

Academic Editor: Jiangang Zhang

Copyright © 2014 Xuda Zhang et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper proposes a nonlinear fuzzy PID control algorithm, whose membership function (MF) is adjustable, is universal, andhas a wide adjustable range. Appling this function to fuzzy control theory will increase system’s tunability. The continuity of thisfunction is proved.This method was employed in the simulation andHIL experiments. Effectiveness and feasibility of this functionare demonstrated in the results.

1. Introduction

Linear proportional-integral-derivative (PID) controller iscurrently the most widely used control method, and about90% of industrial processes employ this controller [1].Though PID controller is adequate for linear applications, itis found that this controller performs poorly for nonlinearsystems, time varying system, and system with time delay.Thus, nonlinear PID controllers are required to deal withthis problem. Various nonlinear PID controllers have beenproposed for complex plants [2–5], but these controllers areof less practical applicability due to the fact that the controlgains are fixed.Therefore, nonlinear controllers with variablegains become the popular research problem.

A variable gain controller is a controller in which at leastone gain varies with input variables. Fuzzy controller is akind of variable gain controllers. Fuzzy logic system theorywas first proposed by Zadeh in 1965, which has been widelyapplied in control field. In addition, fuzzy controllers havebeen proven to be an effective choice to solve many practicalproblems with less time [6]. It is proved, in the past 1980s,that some simple Mamdani fuzzy controllers are essentiallynonlinear PI or PD controllers with variable gains [7, 8].They all used linear fuzzy sets to fuzzify input variables. Thevariable gains enabled the fuzzy controllers to outperformtheir linear counterparts when controlling nonlinear or time-delay systems [9–12].However, since the relationship betweenadjustment of the consequents of the rules and properties

of controllers is not explicit, big trouble is caused to adjustcontroller parameters. Hence, T-S fuzzy controllers that uselinear or nonlinear input fuzzy sets are related to PID controlso that the adjustment of the consequents of fuzzy controllerrules can be implemented in a similar way of adjusting PIDcontroller parameters. Haj-Ali and Ying [13] proposed twotypes of constraint conditions of theMFs of this kind of fuzzycontrollers and the condition for controller structures thatcan be adjusted like PID controller. But, Haj-Ali and Ying[13] only presented the expressions of one kind of MFs, whilethose of other MFs were not presented. Moreover, the formatof these MFs is relatively fixed or of small adjustment range,making them only available to limited plants.

Aiming at solving the problems above, this paper presentsa new MF. Compared with the MF in Haj-Ali and Ying [13],this MF satisfies both two constraint conditions, which can,therefore, compensate the shortage of the second kind ofMF in [13]. The presented MF is of higher tenability andbigger tunable range.Therefore, it is more general.This paperdemonstrates effectiveness of the proposed method with anumerical example and a semiphysical experiment.

2. Configuration of Fuzzy Controllers

The error and error rate are selected as inputs for the fuzzycontroller as follows:

𝐸 (𝑛) = 𝑆 (𝑛) − 𝑦 (𝑛) , 𝐷 (𝑛) = 𝐸 (𝑛) − 𝐸 (𝑛 − 1) , (1)

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014, Article ID 482489, 7 pageshttp://dx.doi.org/10.1155/2014/482489

2 Journal of Applied Mathematics

where 𝑆(𝑛) is the reference output of the plant at samplingtime 𝑛 and 𝑦(𝑛) is the actual output of the plant. Here a two-dimensional fuzzy controller is considered with the inputs𝐸(𝑛) and𝐷(𝑛). The fuzzy rules are as follows.

IF 𝐸(𝑛) is 𝑃 AND𝐷(𝑛) is 𝑃 THEN Δ𝑢(𝑛) is 𝑃

IF 𝐸(𝑛) is 𝑃 AND𝐷(𝑛) is𝑁 THEN Δ𝑢(𝑛) is 𝑍

IF 𝐸(𝑛) is𝑁 AND𝐷(𝑛) is 𝑃 THEN Δ𝑢(𝑛) is 𝑍

IF 𝐸(𝑛) is𝑁 AND𝐷(𝑛) is𝑁 THEN Δ𝑢(𝑛) is𝑁,

where Δ𝑢(𝑛) denotes the change in controller output and “𝑃”stands for positive and “𝑁” for negative. Detailed explanationof these four rules can be found in [13].

A class-1 fuzzy set satisfies the following conditions:

𝑒𝑃 (𝑥) = 1 − 𝑒𝑁 (𝑥) , 𝑒

𝑃 (0) = 𝑑𝑁 (0) = 0.5,

𝑒𝑃 (𝑥) = 1 − 𝑑𝑃 (−𝑥) , 𝑒

𝑁 (𝑥) = 1 − 𝑑𝑁 (−𝑥) ,

(2)

and a class-2 fuzzy set satisfies

𝑒𝑃 (𝑥) = 𝑒𝑁 (−𝑥) , 𝑒

𝑃 (𝑥) = 𝑑𝑃 (𝑥) ,

𝑒𝑁 (𝑥) = 𝑑𝑁 (𝑥) .

(3)

For a fuzzy controller with the inputs 𝐸(𝑛) and 𝐷(𝑛), thesingleton output fuzzy sets, and the following MFs, consider

𝑒𝑃 (𝑥) =

1

1 + exp (−𝐾𝑃𝑥), 𝑒

𝑁 (𝑥) =1

1 + exp (𝐾𝑃𝑥),

(4)

𝑑𝑃 (𝑥) =

1

1 + exp (−𝐾𝑑𝑥), 𝑑

𝑁 (𝑥) =1

1 + exp (𝐾𝑑𝑥).

(5)

We can use the four rules above, product AND operator,and the centroid defuzzifier to structurally convert it intoa nonlinear PI or PD controller with variable gains [13] asfollows:

Δ𝑢 (𝑛) = (𝑒𝑃 (𝐸) 𝑑𝑃 (𝐷)𝐻 + 𝑒𝑁 (𝐸) 𝑑𝑁 (𝐷) (−𝐻))

× (𝑒𝑃 (𝐸) 𝑑𝑃 (𝐷) + 𝑒𝑃 (𝐸) 𝑑𝑁 (𝐷)

+ 𝑒𝑁 (𝐸) 𝑑𝑃 (𝐷) + 𝑒𝑁 (𝐸) 𝑑𝑁 (𝐷))

−1.

(6)

Theorem 1. A fuzzy controller that uses 𝐸(𝑛) and 𝐷(𝑛), thesingleton output fuzzy sets, product AND operator, the fourfuzzy rules, and the centroid defuzzifier structurally becomesa nonlinear PI or PD controller with variable gains in [−𝐿, 𝐿]×[−𝐿, 𝐿] if and only if 𝑒

𝑝(𝐸) = 1 − 𝑑

𝑝(−𝐷), 𝑒

𝑁(𝐸) = 1 −

𝑑𝑁(−𝐷), if the class-1 input fuzzy sets are used, or 𝑒

𝑝(𝐸) =

𝑑𝑃(𝑥), 𝑒𝑁(𝑥) = 𝑑

𝑁(𝑥), if the class-2 input fuzzy sets are

employed.

We will only prove the class-1 input fuzzy sets as follows.Consider

Δ𝑢 (𝑛) =𝑒𝑃 (𝐸) 𝑑𝑃 (𝐷)𝐻 + 𝑒𝑁 (𝐸) 𝑑𝑁 (𝐷) (−𝐻)

𝑒𝑃 (𝐸) 𝑑𝑃 (𝐷) + 𝑒𝑁 (𝐸) 𝑑𝑁 (𝐷)

.

NumDen

:

Num= 𝑒𝑃 (𝐸) 𝑑𝑃 (𝐷)𝐻 + 𝑒𝑁 (𝐸) 𝑑𝑁 (𝐷) (−𝐻)

= 𝑒𝑃 (𝐸) 𝑑𝑃 (𝐷) + (1 − 𝑒𝑃 (𝐸)) (1 − 𝑑𝑃 (𝐷))

× (−𝐻)

= 𝑒𝑃 (𝐸) 𝑑𝑃 (𝐷)𝐻 − 𝑒𝑃 (𝐸) 𝑑𝑃 (𝐷)𝐻

+ (𝑒𝑃 (𝐸) + 𝑑𝑃 (𝐷))𝐻 − 𝐻

+ (𝑒𝑃 (𝐸) + 𝑑𝑃 (𝐷))𝐻 − 𝐻

= (𝑒𝑃 (𝐸) + 𝑑𝑃 (𝐷) − 1)𝐻

Den = 𝑒𝑃 (𝐸) 𝑑𝑃 (𝐷) + 𝑒𝑃 (𝐸) 𝑑𝑁 (𝐷)

+ 𝑒𝑁 (𝐸) 𝑑𝑃 (𝐷) + 𝑒𝑁 (𝐸) 𝑑𝑁 (𝐷)

= 𝑒𝑃 (𝐸) 𝑑𝑃 (𝐷) + 𝑒𝑃 (𝐸) (1 − 𝑑𝑃 (𝐷))

+ (1 − 𝑒𝑃 (𝐸)) 𝑑𝑃 (𝐷) + (1 − 𝑒𝑃 (𝐸)) (1 − 𝑑𝑃 (𝐷))

= 1

(7)

Then,

Δ𝑢 = (𝑒𝑃 (𝐸) + 𝑑𝑃 (𝐷) − 1)𝐻. (8)

Taylor expansion of the fuzzy sets gives

𝑒𝑃 (𝐸) = 𝑎10 + 𝑎11𝐾𝑒𝐸 (𝑛) + 𝑎12(𝐾𝑒𝐸 (𝑛))

2

+ 𝑎13(𝐾𝑒𝐸 (𝑛))

3+ ⋅ ⋅ ⋅ ,

𝑑𝑃 (𝐷) = 𝑎20 + 𝑎21𝐾𝑑𝐷 (𝑛) + 𝑎22(𝐾𝑑𝐷 (𝑛))

2

+ 𝑎23(𝐾𝑑𝐷(𝑛))

3+ ⋅ ⋅ ⋅ .

(9)

Substituting them into (8) produces

Δ𝑢 = ( − 1 + 𝑎10+ 𝑎20+ 𝑎11𝐾𝑒𝐸 (𝑛) + 𝑎21𝐾𝑑𝐷 (𝑛)

+ 𝑎12(𝐾𝑒𝐸 (𝑛))

2+ 𝑎22(𝐾𝑑𝐷 (𝑛))

2

+ 𝑎13(𝐾𝑒𝐸 (𝑛))

3+ 𝑎23(𝐾𝑑𝐷 (𝑛))

3+ ⋅ ⋅ ⋅ )𝐻.

(10)

By noting that Δ𝑢 = 0, ∀(𝐾𝑒𝐸(𝑛) + 𝐾

𝑑𝐷(𝑛)) = 0 and

replacing 𝐾𝑑𝐷(𝑛) with −𝐾

𝑒𝐸(𝑛), we can rewrite (10) in a

polynomial of𝐾𝑒𝐸(𝑛) as follows:

Δ𝑢 = ( − 1 + 𝑎10+ 𝑎20+ (𝑎11− 𝑎21)𝐾𝑒𝐸 (𝑛) + (𝑎12 + 𝑎22)

× (𝐾𝑒𝐸 (𝑛))

2+ (𝑎13− 𝑎23) (𝐾𝑒𝐸 (𝑛))

3+ ⋅ ⋅ ⋅ )𝐻 ≡ 0.

(11)

Journal of Applied Mathematics 3

f(x)

G

Ar

0

1

Al

Figure 1: Examples of adjustable membership function.f(x)

G

Ar

Al0

1

K = 5

K = 10

K = 1

Figure 2: Membership functions with different values of 𝐾.

The condition the equation above holds is that all thecoefficients of𝐾

𝑒𝐸(𝑛)must be zero, leading to

𝑎10+ 𝑎20= 1,

𝑎1(2𝑘−1)

= 𝑎2(2𝑘−1)

≜ 𝑎2𝑘−1,

𝑎1(2𝑘)

= −𝑎2(2𝑘)

≜ 𝑎2𝑘

𝑘 = 1, 2, 3, . . . ;

(12)

namely, 𝑒𝑃(𝐸) = 1 − 𝑑

𝑃(−𝐷), 𝑒

𝑁(𝐸) = 1 − 𝑑

𝑁(−𝐷).

Here, the symbol “≜” stands for “defined as.” Substitutingall these equations back into (10) yields

Δ𝑢 = (𝑎1(𝐾𝑒𝐸 (𝑛) + 𝐾𝑑𝐷 (𝑛))

+ 𝑎2((𝐾𝑒𝐸(𝑛))2− (𝐾𝑑𝐷(𝑛))

2)

+ 𝑎3((𝐾𝑒𝐸(𝑛))3+ (𝐾𝑑𝐷(𝑛))

3) + ⋅ ⋅ ⋅ ) ⋅ 𝐻.

(13)

The proof is then completed by a direct application of thebinomial rules to this equation, which allows us to factor out𝐾𝑒𝐸(𝑛) + 𝐾

𝑑𝐷(𝑛) in the following equation:

Δ𝑢 = 𝜕 (𝐸,𝐷) (𝐾𝑒𝐸 (𝑛) + 𝐾𝑑𝐷 (𝑛)) ,

𝜕 (𝐸,𝐷)

= (𝑎1+ 𝑎2(𝐾𝑒𝐸 (𝑛) − 𝐾𝑑𝐷(𝑛))

+ 𝑎3((𝐾𝑒𝐸 (𝑛))

2− 𝐾𝑒𝐸 (𝑛)𝐾𝑑𝐷 (𝑛) + (𝐾𝑑𝐷 (𝑛))

2) + 𝜃)

⋅ 𝐻,

(14)

where 𝜃 represents the truncating error in Taylor expan-sion. The variable proportional gain and integral gain are𝜕(𝐸,𝐷)𝐾

𝑒and 𝜕(𝐸,𝐷)𝐾

𝑑, respectively.

In practical applications, 𝐻 is a design parameter and 𝐻could be 𝑃 or 0 for𝑍 and −𝐻 for𝑁. The determination of𝐾

𝑃

and𝐾𝑑in MFs is similar to that in PID controllers; therefore,

this method can be easily used in practical applications. But,we can see that the format of the MFs of the fuzzy controlleris fixed and as a consequence, it only applies to limited plants.

3. Adjustable Membership Function

3.1. Construction of NewMembership Function. To overcomethe drawbacks that the MF in Section 2 is of fixed format andpoor adjustability, we propose the following function:

𝑓 (𝑥)

=

{{{{{{{{{

{{{{{{{{{

{

0 𝑥 ≤ −𝑙

𝑥 + 𝑙

1 + 𝐾 (𝐴0− 𝑥 − 𝑙)

−𝑙 < 𝑥 ≤ 𝐴0− 𝑙

𝐴0+

(𝑥 + 𝑙 − 𝐴0) (1 + �̃�𝐴)

1 + �̃� (𝑥 + 𝑙 − 𝐴0)

𝐴0− 𝑙 < 𝑥 ≤ 𝐴

1− 𝑙

1 𝑥 > 𝐴1− 𝑙,

(15)

where 𝐴 = 𝐴1− 𝐴0and 𝐴

1− 𝐴0= 1. This function satisfies

membership function constraints in (3), and, in particular,the function will be moved left or right by changing 𝑙. But,the range of (15) will be changed with the change of the rangeof “𝑥,” that is, changing 𝐴

0and 𝐴

1. In order to satisfy the

condition that the range of MF must be limited to [0, 1],we multiply it by 1/𝐴

1. Then 𝐴

0and 𝐴

1can be set by

arbitrary values. So the final adjustable membership functionis determined as follows:

𝑓 (𝑥)

=

{{{{{{{{{{{{{{

{{{{{{{{{{{{{{

{

0 𝑥 ≤ −𝑙

1

𝐴1

(𝑥 + 𝑙

1 + 𝐾 (𝐴0− 𝑥 − 𝑙)

) −𝑙 < 𝑥 ≤ 𝐴0− 𝑙

1

𝐴1

[𝐴0+

(𝑥 + 𝑙 − 𝐴0) (1 + �̃�𝐴)

1 + �̃� (𝑥 + 𝑙 − 𝐴0)

] 𝐴0− 𝑙 < 𝑥

≤ 𝐴1− 𝑙

1 𝑥 > 𝐴1− 𝑙.

(16)

The shape of the adjustable function is shown in Figure 1.We defined 𝐺 = 𝐴

0− 𝑙 as the inflexion point, 𝐴

𝑙= −𝑙 as

the left endpoint, 𝐴𝑟= 𝐴1− 𝑙 as the right endpoint, and 𝐾

as function rate. And they can be adjusted by changing 𝐴𝑙,

𝐴𝑟, and 𝐺. Thereby, control performance is improved. This

membership function satisfies (2) and (3); furthermore, it haswide adjustment range.

TheMFs with different values of𝐾when𝐴𝑙= −0.5,𝐴

𝑟=

0.5, and 𝐺 = −0.1 are shown in Figure 2. It can be seen that


𝜃 m

l

mc

u

Figure 3: The inverted pendulum.

the curves of the functions vary significantly with differentvalues of 𝐾. Similarly, we can adjust other parameters, suchas 𝐴𝑙, 𝐴𝑟, and 𝐺, to improve control performance.

A mutation of the controller output may occur if themembership function is not smooth and continuous, thusthe control performance of the controller. The continuouslyderivable condition is given below.

3.2. Continuously Derivable Condition. The partial derivativeof the piecewise function (15) is

𝜕𝑓

𝜕𝑥=

{{{{{{{{{

{{{{{{{{{

{

0 𝑥 ≤ −𝑙

1 + 𝐾𝐴0

[1 + 𝐿 (𝐴0− 𝑥 − 𝑙)]

2−𝑙 < 𝑥 ≤ 𝐴

0− 𝑙

1 + �̃�𝐴

[1 + �̃� (𝑥 + 𝑙 − 𝐴0)]2𝐴0− 𝑙 < 𝑥 ≤ 𝐴

1− 𝑙

0 𝑥 > 𝐴1− 𝑙.

(17)

As long as (17) is continuous at the subsection points, (15)will be continuously differentiable. Hence, the continuousderivable condition is

𝐾𝐴0= �̃�𝐴. (18)

In addition, this condition will not reduce the regulationability of (16).

4. Simulation and Experiment

4.1. Simulation. Simulation on an inverted pendulum systemshown in Figure 3 is conducted.

The dynamics of the system is given by

�̇�1= 𝑥2,

�̇�2= [𝑔 ⋅ sin𝑥

1−𝑚 ⋅ 𝑙 ⋅ 𝑥

2

2⋅ cos𝑥

1⋅ sin𝑥

1

𝑚𝑐+ 𝑚

] +cos𝑥1

𝑚𝑐+ 𝑚

⋅𝑢

𝑙 (4/3 − 𝑚cos2𝑥1/ (𝑚𝑐+ 𝑚))

𝑦 = 𝑥1,

(19)

in which 𝑥1is the angular position of the pendulum, 𝑥

2is the

angular velocity, 𝑚𝑐is the mass of the cart, and 𝑚 is mass of

the pendulum.

0 2 31 4 5 6 7 8 9 10

0

0.2

0.4

0.6

−0.8

−0.6

−0.4

−0.2

Time (s)

Posit

ion

Tracking curveIdeal curve

(a)

0 2 4 6 8 10

0

0.2

0.4

0.6

−0.8

−0.6

−0.4

−0.2

Time (s)

Velo

city


31 5 7 9

(b)

Figure 4: Position and velocity tracking curves; (a) positiontracking curves and (b) velocity tracking curves.

A sinusoidal signal is chosen as input signal in thesimulation. Control parameters are as follows:𝐻 = 21,𝐾 = 1,𝐴1= 0.4, 𝐴

2= 1.8, and 𝑙 = 0.6. The simulation results are

shown in Figures 4, 5, and 6.From Figures 4 and 5, it can be found that the tracking

error at the initial stage is relatively large since the initial posi-tion error is somewhat big. Then tracking curve essentiallycoincides with the ideal curve after 1 s and is in a steady state.

Furthermore, a comparison with the control result usingthe membership function given in literature [13] is madeto illustrate the effectiveness of the method proposed. Acomparison of the tracking error under the same conditionsis shown in Figure 6.

We can conclude from Figure 6 that tracking errorsfluctuate up and down around zero. The simulation resultsshow that the position error of the controller with theadjustable MF is about 0.005 rad; on the other hand, thatusing the MF presented in literature [13] is about 0.01 rad, asshown in Figure 6(a). Control performance of the adjustable


0 21 3 4 5 6 7 8 9 10

0

2

4

6

8

10

12

14

16

−4

−2

Time (s)

u(N

)

Figure 5: System input 𝑢.

membership function can be demonstrated in this compari-son.

4.2. Experiment. A typical industrial mechatronic drives unit(IMDU) is depicted in Figure 7. This is a versatile systemthat can be used to illustrate fundamentals of servo controlas well as advanced topics, such as haptics and teleopera-tion, web winding control, backlash compensation, frictioncompensation, and high order coupling of complex industrialprocesses. The unit contains 4 shafts configured in a squarepattern. Two of the shafts are motor driven while the othertwo can freely rotate. All shafts are instrumented with opticalencoders. Each motor is driven by a linear current controlamplifier with the capability of 100watts.

In this paper, fuzzy controller and adjustablemembershipfunction are used to control one shaft of IMDU to trackstep signal and sinusoidal signal. Tracking errors of referencestep signal and sinusoidal signal are shown in Figures 8 and9, respectively. These figures are drawn based on the datacollected from real system.

We can conclude from Figures 8 and 9 that the con-trol performance of controller using adjustable membershipfunction is in the experiment’s permissible error range. Therise time is about 0.1 s and the steady-state error is around0 when tracking step signal. The system error is within fourdegrees when tracking sinusoidal signal. Although the resultdoes not reach our expectation, it still shows its practicality.

5. Conclusion

(1) This paper presents a novel membership function, that is,adjustable membership function. This membership functionis a supplement to the structural study of fuzzy controllerand PID controller. This function can be adjusted flexibly,including the adjustment of the left and right endpoints 𝐴

𝑙

and 𝐴𝑟, inflexion point 𝐺, and the function rate 𝐾 to meet

the control requirements.(2) When 𝐺 point is in the 𝑦-axis and 𝐴

𝑙= −𝐴

𝑟, this

function becomes class-1 MF in literature [13]. So class-1 MFcan also be seen as a special case of adjustable membershipfunction.

−0.02

−0.01

0

0 2 4 6 8 10 12 14 16 18 20

0.01

0.02

0.03

0.04

0.05

0.06

0.07

Time (s)

Posit

ion

erro

r (ra

d)

Literature MF tracking errorAdjustable MF tracking error

(a)

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 2 4 6 8 10 12 14 16 18 20

Time (s)

Velo

city

erro

r (ra

d/s)

Literature MF tracking errorAdjustable MF tracking error

(b)

Figure 6: Comparison of position and velocity tracking errors; (a)position tracking error and (b) velocity tracking error.

Figure 7: IMDU.


0 1 2 3 4 50

10

20

30

40

50

60

Time (s)

Ideal curveTracking curve

Posit

ion(∘)

(a)

Time (s)0 1 2 3 4 5

0

20

40

60

−20

Erro

r(∘ )

(b)

Figure 8: IMDU tracking step signal and tracking error; (a) IMDU tracking step signal and (b) IMDU tracking error.

Time (s)0 2 4 6 8 10

0

20

40

60


−20

−40

−60

Posit

ion(∘)

(a)

Time (s)0 2 4 6 8 10

0

2

4

−6

−4

−2

Erro

r(∘ )

(b)

Figure 9: IMDU tracking sinusoidal signal and tracking error; (a) IMDU tracking sinusoidal signal and (b) IMDU tracking error.

(3) Although this membership function can be adjustedflexibly, the structure of this fuzzy controller is too simple tosatisfy the strong nonlinear control tasks’ requirements, dueto the fact that nonlinear fuzzy PID are better in handlingnonlinear control problems and type-2 fuzzy controllers havebetter performance compared with type-1 fuzzy controllers.So our future work is extending the nonlinear fuzzy PIDcontroller to type-2 fuzzy PID controller and analyzing theinternal relations between the fuzzy controllers and nonlinearPID controller.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This project is supported by National Natural Science Foun-dation of China (no. 51105290 and 51035006), the special


fund of the Key Laboratory of Xinjiang Uygur AutonomousRegion, the special financial fund of astronomy of China, in2014, and the Shanghai Aerospace Science and TechnologyInnovation Fund. These supports are gratefully acknowl-edged. Many thanks are dedicated to the editor, associateeditor, and reviewers for their informative, valuable, andconstructive comments.

References

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[4] Y. Ni, L. Jiao, S. Chen, and B. Zhang, “Application of anonlinear PID controller on STATCOM with a differentialtracker,” inProceedings of the International Conference on EnergyManagement and Power Delivery, pp. 29–34, 1998.

[5] V. Parra-Vega and S. Arimoto, “Nonlinear PID control with slid-ing modes for tracking of robot manipulators,” in Proceedings ofthe IEEE International Conference on Control Applications (CCA’01), pp. 351–356, Mexico City, Mexico, September 2001.

[6] J. Yen, R. Langari, and D. P. Filev, Essentials of Fuzzy Modelingand Control, John Wiley & Sons, New York, NY, USA, 1994.

[7] C. Chen, S.Wang, C. Hsieh, and F. Chang, “Theoretical analysisof crisp-type fuzzy logic controllers using various t-normsum-gravity inference methods,” IEEE Transactions on FuzzySystems, vol. 6, no. 1, pp. 122–136, 1998.

[8] A. El Hajjaji and A. Rachid, “Explicit formulas for fuzzycontroller,” Fuzzy Sets and Systems, vol. 62, no. 2, pp. 135–141,1994.

[9] F. L. Lewis and K. Liu, “Towards a paradigm for fuzzy logiccontrol,” Automatica, vol. 32, no. 2, pp. 167–181, 1996.

[10] H. X. Li and H. B. Gatland, “New methodology for designing afuzzy logic controller,” IEEE Transactions on Systems, Man andCybernetics, vol. 25, no. 3, pp. 505–512, 1995.

[11] G. K. I. Mann, B.-G. Hu, and R. G. Gosine, “Analysis of directaction fuzzy PID controller structures,” IEEE Transactions onSystems,Man, andCybernetics B, vol. 29, no. 3, pp. 371–388, 1999.

[12] C.Wong, C. Chou, and D. Mon, “Studies on the output of fuzzycontroller with multiple inputs,” Fuzzy Sets and Systems, vol. 57,no. 2, pp. 149–158, 1993.

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