an enhanced fuzzy pd controller with two discrete nonlinear tracking differentiators

An enhanced fuzzy PD controller with two discretenonlinear tracking differentiators

Y.X. Su, C.H. Zheng, D. Sun and B.Y. Duan

Abstract: An enhanced fuzzy PD (EFPD) controller by incorporating two discrete nonlineartracking differentiators (TDs) into a conventional fuzzy PD controller is proposed. This work ismotivated by the fact that the performance of the controlled system is limited by how to select ahigh-quality differential signal in the presence of noise, based on the noncontinuous measuredposition signal only. The proposed TD is constructed based on the fact numerical integration canprovide more stable and accurate results than numerical differentiation in the presence of noise. Theproposed EFPD controller is easy to implement, and exhibits better control performance thanconventional fuzzy controllers, with especially high robustness against noise. Comparativesimulations and experiments with a linear observer performed on a permanent-magnet synchronousmotor system are presented to demonstrate the better performance of the controller.

1 Introduction

Since Mamdani and Assilian [1] first applied fuzzy logic tosteam engine control, fuzzy logic control has become anattractive and fruitful field of control theory [2, 3] and hasbeen successfully utilised in many industrial controlapplications. Regarded as one of the artificial intelligencemethodologies, fuzzy logic control provides a convenientmethod for constructing nonlinear controllers through theuse of heuristic information about how to control a plantrather than requiring an exact mathematical model. Thedevelopment of fuzzy controllers has progressed by leapsand bounds over the past two decades [2–10].

Fuzzy controllers for industrial applications assume thatthe system states are all available or that the velocity signalcan be directly obtained by numerical differentiation, inwhich the difference between successive positions isdivided by the time interval. However, in practice, anoptical encoder is still the most popular accurate positionsensor used in industrial fields because of its simpledetection circuit, high resolution, high accuracy and relativeease of adaptation to digital control systems. Due to theinherent digital characteristics of the incremental encoder,position error from quantisation always exists and this erroracts as a kind of measurement noise [11–21]. As a result the

numerical differentiation tends to magnify errors or noises.Because of this, measurement noise must be considered toaccurately detect the position and velocity information forhigh-precision control.

For a high-quality velocity signal based only on themeasured position some researchers employed a Kalmanfilter to estimate the velocity accurately [21]. Since thisapproach requires that the target velocity trajectory be sentto the filter, it cannot be applied in the case where anarbitrary velocity is measured [18]. With the developmentof control theory many approaches using observer theoryhave been extensively studied [11, 13–16]. For example,Bodson et al. [11] constructed a nonlinear observer by usingthe known dynamic model of the induction motor to obtainthe high-quality velocity signal. Arteaga [13] used a linearobserver to obtain velocity estimation with joint positionmeasurement only. Loria and Melhem [15] designed anobserver to solve the velocity estimation for fully actuatedEuler–Lagrange systems based on a new model thatconsists of a dynamic nonlinminimal realisation andconstraint equation. Harnefors and Nee [16] proposed alinear observer based on a phase-locked loop for high-quality velocity estimation with position measurement only.

An alternative to the use of an observer is to employfiltering techniques based on the measured position signal.Bodson et al. [11] solved for velocity estimation bycombining the backward difference of the position measure-ment with a low-pass filter. Queiroz et al. [12] used a high-pass filtering technique to develop a pseudo velocity signalthat serves as a surrogate for velocity measurements torealise the adaptive position=force control for a constrainedrobot manipulator. Other velocity estimation methods basedon position measurement only can be found in [17–20].

In this paper, based on the fact that numerical integrationcan provide more stable and accurate results than numericaldifferentiation in the presence of noise, a discrete nonlineartracking differentiator (TD) is developed to obtain a high-quality velocity signal based on position measurement onlywith simple calculation. Using the high-quality velocitysignal obtained, an enhanced fuzzy PD (EFPD) controller isproposed. The main advantages of the proposed method aretwofold. First, unlike other available Kalman filter or

q IEE, 2004

IEE Proceedings online no. 20041095

doi: 10.1049/ip-cta:20041095

Y.X. Su is with the Department of Manufacturing Engineering andEngineering Management, City University of Hong Kong, 83 Tat CheeAvenue, Kowloon, Hong Kong and is also with School of Electro-Mechanical Engineering, Xidian University, Xi’an 710071, P.R. China

C.H. Zheng is with the School of Electronic Engineering, XidianUniversity, Xi’an 710071, P.R. China

D. Sun is with the Department of Manufacturing Engineering andEngineering Management, City University of Hong Kong, 83 Tat CheeAvenue, Kowloon, Hong Kong

B.Y. Duan is with the School of Electro-Mechanical Engineering, XidianUniversity, Xi’an 710071, P.R. China

Paper first received 5th August 2003 and in revised form 3rd August 2004

IEE Proc.-Control Theory Appl., Vol. 151, No. 6, November 2004 675

observer-based approaches, the nonlinear tracking differ-entiator developed does not require the system model.Secondly, the developed EFPD controller is easy toimplement and exhibits better control performance. Moreimportantly, it appears to be more robust against noise.Extensive simulations and experimental results conductedon a permanent-magnet synchronous (PMAC) motor systemare presented to verify the effectiveness of the proposedEFPD controller.

2 Enhanced fuzzy PD controller

The proposed enhanced fuzzy PD controller is shown inFig. 1, which consists of two discrete nonlinear trackingdifferentiators and a conventional fuzzy PD controller. Eachblock in Fig. 1 is addressed in the following Sections.

2.1 Nonlinear tracking differentiator

The so-called nonlinear tracking differentiator [22–26] isconstructed based on the fact that numerical integration canprovide more stable and accurate results than numericaldifferentiation in the presence of noise, and refers to thefollowing system: given a reference signal r(t), the systemprovides two signals r1ðtÞ and r2ðtÞ; such that r1ðtÞ ! rðtÞand r2ðtÞ ! _rrðtÞ:

Theorem 1 [22, 26]: Suppose z(t) is a continuous functiondefined in ½0;1Þ; which satisfies limt!1 zðtÞ ¼ 0: If rðtÞ ¼zðRtÞðR >0Þ; the following expression holds for anarbitrarily given T >0:

limR!1

Z T

0jrðtÞjdt ¼ 0 ð1Þ

Proof: Based on the integral mean-value theorem there is at 2 ½0 T such thatZ T

0jrðtÞjdt ¼ TjrðtÞj ¼ TjzðRtÞj ð2Þ

Since limt!1 zðtÞ ¼ 0 it is straightforward that

limR!1

Z T

0jrðtÞjdt ¼ lim

R!1ðTjzðRtÞjÞ ¼ T lim

R!1ðjzðRtÞjÞ

¼ T 0 ¼ 0

Thus theorem 1 is established.

Theorem 2 [22, 26]: Consider the system

_zz1 ¼ z2

_zz2 ¼ f ðz1; z2Þ

�ð3Þ

If z1ðtÞ ! 0 and z2ðtÞ ! 0 as t ! 1; for an arbitrarilyconstant c and T >0; the solution r1ðtÞ to the system

_rr1 ¼ r2

_rr2 ¼ R2f r1 � c;r2

R

� �(ð4Þ

satisfies

limR!1

Z T

0jr1ðtÞ � cjdt ¼ 0 ð5Þ

Proof: First, changing the variable of integration,

t ¼ t

Rr1ðtÞ ¼ z1ðtÞ þ c

r2ðtÞ ¼ Rz2ðtÞ

8><>: ð6Þ

Differentiating r1ðtÞ and r2ðtÞ with respect to t yields

_rr1ðtÞ ¼ R_zz1ðtÞ_rr2ðtÞ ¼ R2_zz2ðtÞ

(ð7Þ

Then rewrite (3) as follows:

_rr1ðtÞ ¼ r2ðtÞ_rr2ðtÞ ¼ R2f r1ðtÞ � c;

r2ðtÞR

�8<: ð8Þ

and r1ðtÞ � c ¼ z1ðtÞ holds. Using the condition z1ðtÞ ! 0as t ! 1ðt ¼ RtÞ and theorem 1, it is straightforward that

limR!1

Z T

0jr1ðtÞ � cjdt ¼ 0

Therefore theorem 2 is justified. A

Theorem 3 [22, 26]: If the arbitrary solutions to system (3)satisfy z1ðtÞ ! 0 and z2ðtÞ ! 0 as t ! 1; then for anyarbitrarily bounded integrable function r(t) and givenconstant T >0; the solution r1ðtÞ to the system

_rr1 ¼ r2

_rr2 ¼ R2f r1 � r;r2

R

� �(ð9Þ

satisfies

limR!1

Z T

0jr1ðtÞ � rðtÞjdt ¼ 0 ð10Þ

Proof: The proof can be justified in the following two cases.Case 1: If r(t) is a constant function, theorem 3 holds fromtheorem 2.Case 2: If r(t) t 2 ½0; T is a bounded integral function, itis an element of the set of L1½0; T; where L1½0; T denotesthe set of all first-integrable function in the range of [0, T ].For an arbitrarily given e>0; there exists a simple seriesjnðtÞðn ¼ 1; 2; . . .Þ that uniformly converges to a continu-ous function cðtÞ 2 C½0; T such that [22]Z T

0jrðtÞ � jðtÞjdt <

e2

ð11Þ

Thus there exists an integer N0 such that jjðtÞ � jMðtÞj<e=4T for all M>N0: Consequently the following inequalityholds:Z T

0jrðtÞ � jMðtÞjdt

Z T

0jrðtÞ � cðtÞjdt

þZ T

0jcðtÞ � jMðtÞjdt<

e2

ð12Þ

Since jðtÞ is a continuous function, the simple series jMðtÞpartitions the range [0, T ] into some bounded intervalsdenoted by liði ¼ 1; 2; . . . ;mÞ: Select jMðtÞ as a

Fig. 1 Schematic diagram of enhanced hybrid fuzzy PDcontroller

IEE Proc.-Control Theory Appl., Vol. 151, No. 6, November 2004676

deterministic constant in each bounded interval, and basedon theorem 2, for all R>R0; there exists R0>0 such thatZ

li

jr1ðtÞ � jMðtÞjdt<e

2m; i ¼ 1; 2; . . . ;m ð13Þ

As a result Z T

0jr1ðtÞ � jMðtÞjdt <

e2

ð14Þ

Thereby for all R>R0; the following inequality holds:Z T

0jr1ðtÞ � rðtÞjdt <

Z T

0jr1ðtÞ � jMðtÞjdt

þZ T

0jjMðtÞ � rðtÞjdt < e ð15Þ

Finally we conclude that for an arbitrarily given e>0there exists R0>0 such that

R T0 jr1ðtÞ � rðtÞjdt< e

for all R > R0: Based on the definition of limit, we havelimR!1

R T0 jr1ðtÞ � rðtÞjdt ¼ 0:

The proof is justified. A

Theorem 3 shows that r1ðtÞ averagely converges to r(t). Ifthe bounded integrable function r(t) is viewed as ageneralised function, then r2ðtÞðr2ðtÞ ¼ _rr1ðtÞÞ weakly con-verges to the generalised derivative of r(t). Thereforesystem (9) can be used as a nonlinear tracking differentiatorto provide a smooth approach to the original generalisedfunction and its generalised derivative, in the sense of ave-rage convergence and weak convergence, respectively [22].

To reduce high-frequency chattering and make full use ofoptimal control theory a feasible discrete second-orderTD(I) with the Euler method can be expressed as [23, 24]

z1ðk þ 1Þ ¼ z1ðkÞ þ hz2ðkÞz2ðk þ 1Þ ¼ z2ðkÞ þ hfstðz1ðkÞ � rðkÞ; z2ðkÞ;R; h0Þ

�ð16Þ

where h is the sampling step, k denotes the kth samplinginstant, R is a velocity factor to determine the transitioncharacteristics, and h0 denotes the filtering factor for noisesuppression. The nonlinear function fstðx1; x2;R; h0Þ isdefined as

fstðx1; x2;R; h0Þ ¼ �RsgnðaÞ; jaj>d

Ra

d; jaj d

(ð17Þ

in which sgnðÞ denotes a standard sign function, and a and dcan be determined as follows:

a ¼x2 þ

ða0 � dÞ2

sgnðzÞ; jzj>d0

x2 þz

h0

; jzj d0

8><>: ð18Þ

with

d ¼ Rh0

d0 ¼ h0d

z ¼ x1 þ h0x2

a0 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffid2 þ 8Rjzj

p

8>>><>>>:

ð19Þ

Replacing r(k) by y(k) in (16) and using (17)–(19), z3 and z4

of ND (II) (recall Fig. 1) can be obtained, where z3ðkÞ !yðkÞ; z4ðkÞ ! _yyðkÞ; and y denotes the output of the system.

The discrete TD developed has high robustness to thevariation of the design parameters R and h0: In general,R can be chosen from the range of [5 5000] and large Rmay provide a fast convergence and better tracking, h0 canbe chosen in the range of [2 25] and large h0 is helpful tocancel out the noise. Usually a small sampling time requireslarge R to guarantee the fast convergence.

2.2 Fuzzy PD controller

Let the error e and the change of error c be two inputs to thefuzzy controller. Then the normalised inputs of thedeveloped EFPD controller can be defined by usingthe high-quality differential signal obtained by the devel-oped TDs, i.e.

eðtÞ ¼ Keðz1ðtÞ � z3ðtÞÞ ð20Þ

cðtÞ ¼ Kcðz2ðtÞ � z4ðtÞÞ ð21Þwhere Ke and Kc are scaling gains, respectively. The voltageto the servo drive is selected as the output of the fuzzycontroller. The voltage conversion factor for velocity is0:5 rad volt=s:

The fuzzy controller has 11 fuzzy sets with membershipfunctions uniformly distributed on each normalised inputuniverse of discourse. All membership functions used in thisfuzzy controller are triangular type with a base width of 0.4.Another 11 fuzzy sets with membership functions uniformlydistributed on normalised output universe of discourse areused for the output of the fuzzy controller. Zadeh’scompositional rule of inference and the standard centre of

Table 1: Rule base

Ci

uj;ki 21 20.8 20.6 20.4 20.2 0 0.2 0.4 0.6 0.8 1

Ej 21 1 1 1 1 1 1 0.8 0.6 0.4 0.2 0

20.8 1 1 1 1 1 0.8 0.6 0.4 0.2 0 20.2

20.6 1 1 1 1 0.8 0.6 0.4 0.2 0 20.2 20.4

20.4 1 1 1 0.8 0.6 0.4 0.2 0 20.2 20.4 20.6

20.2 1 1 0.8 0.6 0.4 0.2 0 20.2 20.4 20.6 20.8

0 1 0.8 0.6 0.4 0.2 0 20.2 20.4 20.6 20.8 21

0.2 0.8 0.6 0.4 0.2 0 20.2 20.4 20.6 20.8 21 21

0.4 0.6 0.4 0.2 20 20.2 20.4 20.6 20.8 21 21 21

0.6 0.4 0.2 0 20.2 20.4 20.6 20.8 21 21 21 21

0.8 0.2 0 20.2 20.4 20.6 20.8 21 21 21 21 21

1 0 20.2 20.4 20.6 20.8 21 21 21 21 21 21


gravity defuzzification technique are used to obtain a crispoutput. The normalised rule base for this application isshown in Table 1.

Since the convergence of the developed TD has beenjustified in the previous Section it is straightforward that thestability of the proposed EFPD controller by incorporatingtwo discrete TDs into a conventional FPD controller can beguaranteed according to the stability proofs of the fuzzycontroller given in [2, 5].

3 Simulations

3.1 Comparative simulations of TD and linearobserver

To validate the effectiveness of the developed discretenonlinear tracking differentiator, the comparison with alinear observer (LO) proposed by Harnefors and Nee in [16]was performed. The discrete form of the linear observerwith Euler’s method can be expressed as

yyðk þ 1Þ ¼ yyðkÞ þ hðooðkÞ þ 2rðrðkÞ � yyðkÞÞÞooðk þ 1Þ ¼ ooðkÞ þ hr2ðrðkÞ � yyðkÞÞ

�ð23Þ

where r is a user-defined design parameter, and yyðkÞ andooðkÞ are the estimated position and velocity of the referencer(k), respectively.

The simulations were programmed in Matlab with afourth-order Runge–Kutta method. The simulation periodwas determined as h ¼ 0:01s and all the initial values wereset to zero. The design parameters for the proposed TD andLO were as follows: h0 ¼ 5:0h and R ¼ 500 for the TD, andr ¼ 30 for the LO. Assume that the reference is perturbedby an additive white noise component with the maximum

amplitude of 0.01 rad. The first reference input is a unitsinusoidal position profile of rðtÞ ¼ sinðtÞrad: The esti-mated velocities and estimation errors by the developed TDand LO, respectively, are shown in Fig. 2. For this referenceit can be seen that the differential signal obtained by thedeveloped TD is almost as the same as that of the linearobserver. The second reference input is a low sinusoidalposition profile of rðtÞ ¼ sinð0:5tÞrad; the comparisonresults are shown in Fig. 3. It can be seen that the obtainedvelocity of the developed TD is better than that of the LO.

3.2 Comparative simulation of EFPD andFPD þ LO

The motion control of a permanent-magnet synchronous(PMAC) motor system is performed to validate the superiorperformance of the developed EFPD control. When rotorreference co-ordinates (d–q axes) are chosen as thereference co-ordinates, the dynamic model for a permanent-magnet synchronous motor can be described as [27]

dydt

¼ o

dodt

¼ KN

Jiq �

B

Jo

diddt

¼ �R

Lid þ Niqoþ 1

L�d

diq

dt¼ �R

Liq �

KN

Lo� Nidoþ 1

L�q

8>>>>>>>>>><>>>>>>>>>>:

ð24Þ

in which N denotes the number of pole pairs, R the statorresistance, L the stator inductance, K the torque constant,J the rotor moment inertia, o the angular velocity, id and iq

Fig. 2 Comparison of TD and LO from unit sinusoidal position profile

a Tracking differentiatorb Linear observer


are the currents on d–q reference frame, respectively, and�d and �q are the voltages in the same frame. In the motorvelocity mode for most industrial AC servo drives thedesired current component i�d in the d-axis is set to zero.As a result the desired perfect tracking error coverges in amanner such as

_eeo þ KP1eo þ KI1

Zeodt ¼ 0 ð25Þ

Then the desired current component i�q in the q-axis is

i�q ¼ J

KN

TL

Jþ o� þ KP1eo þ KI1

Zeodt

� �ð26Þ

where o� is the target volcity, eo ¼ o� � o the trackingerror, KP1 and KI1 are the proportional and integral gains ofthe velocity loop, respectively, and TL the external load.To reach the desired current component i� in the q-axisquickly the voltage control laws would be [27]

�d ¼ LR

Lid �Niqoþ _ii�d �KP2ðid � i�dÞ�KI2

Zðid � i�dÞdt

� �ð27Þ

�q ¼ L

�R

Liq þ

KN

LoþNidoþ _ii�q �KP2ðiq � i�qÞ

�KI2

Zðiq � i�qÞdt

�ð28Þ

where _ii�d and _ii�q are derivatives of i�d and i�q; respectively, andKP2 and KI2 denote the current-loop proportional andintegral gains, respectively. Since i�d ¼ 0 it is straightforwardthat _ii�d ¼ 0; and _ii�q can be obtained by differentiating (26)

with respect to time. When the servo drive works in thevelocity mode, (27) and (28) are the current-loop controllaws which enable the target current i�q to be reachedquickly. As a result, (26) is the voltage-loop control strategyfor velocity tracking. The target velocity o�ðtÞ for eachsampling interval is provided by the motion controller.

The PMAC motor used in this work is MSMA042A1GAC servomotor with the matched MSDA043A1A servodrive made by Panasonic Inc. The motor parameters are:rated power 0.4 kW, rated voltage 200 V, rated current2.5 A, rated speed 3000 rev=min, N ¼ 4; R ¼ 0:20O;L ¼ 0:0012 H; K ¼ 0:18 Nm=A and J ¼ 0:36 � 10�4 kgm2

: An incremental optical encoder was used to provide thefeedback position signal, and its resolution is4000 pulse=rev. The servo drive was set in the velocitymode and the gains were selected as KP1 ¼ 100; KI1 ¼ 5;KP2 ¼ 150 and KI2 ¼ 80:

The simulations were programmed in Matlab with afourth-order Runge–Kutta method. The sampling periodwas determined as T ¼ 0:01s: After several tests the sameparameters were determined for the EFPD and FPDþ LOcontrollers, i.e. Ke ¼ 20; Kc ¼ 1:0 and Ku ¼ 1:5: All theinitial values were set to zero.

The comparative simulations are shown in Fig. 4. Thereference is a command input rðtÞ ¼ sinð0:5tÞrad with theperturbation of an additive white noise component withthe maximum amplitude of 0.01 rad. It can be seen that theposition tracking performance of the proposed EFPDcontroller is better than that of the conventional FPDþLO controller. The improved performance results from thehigh-quality velocity signal selected by the developed TDs.Comparative simulations for the reference command with-out noise are also performed; the results are shown in Fig. 5.

Fig. 3 Comparison of TD and LO from low sinusoidal position profile

a Tracking differentiatorb Linear observer


It can be seen that the position tracking performance of theproposed EFPD controller is also better than that of theconventional FPDþ LO controller. The improved perform-ance is ascribed to the incorporation of the TDs whichfunction to arrange the transition of the controlled system.

4 Experiments

The experimental setup of the PMAC motor system isshown in Fig. 6. It consists of the aforementionedMSMA042A1G permanent-magnet synchronous motorand the matched MSDA043A1A servo drive, a six-channelD=A output card PCL-726, and a three-axis quadratureencoder card PCL-833. An incremental optical encoder withthe resolution of 4000 pulse=rev was used to provide thefeedback position signal. The control program were written

in Borland C and run on an industrial computer AdvantechPentium-233. The parameters for the proposed EFPDcontroller and the conventional FPDþ LO controller werethe same as in the simulation, namely Ke ¼ 20; Kc ¼ 1:0and Ku ¼ 1:5:

The first experiment is to test the position-trackingperformance for a unit step. Experimental results are shownin Fig. 7. It can be seen that the EFPD controller has a fasterresponse than the FPDþ LO controller. The improvedtransient performance is due mainly to the high-qualitydifferential signal selected by the proposed TDs with simplecalculation. In addition, the incorporation of the TDsarranges the transition of the controlled system such thatthe performance is further improved. The second experi-ment is to test the position-tracking performance for aperiodic sinusoidal command sinð0:5tÞrad: Experimental

Fig. 4 Comparative simulation of EFPD and FPDþ LO with noise

a Enhanced fuzzy PDb Fuzzy PDþ linear observer

Fig. 5 Position tracking error of EFPD and FPDþ LO without noise



Fig. 7 Experimental comparison of EFPD and FPDþ LO for step profile

Fig. 8 Experimental comparison of EFPD and FPDþ LO for low sinusoidal profile


Fig. 9 Experimental comparison of EFPD and FPDþ LO with additional noise


Fig. 6 Experimental setup


results are shown in Fig. 8. Obviously a better trackingperformance is obtained by using the proposed EFPDcontroller over that of the conventional FPDþ LO con-troller. The improved performance results from the high-quality differential signal selected by the proposed TDs andthe function of arranging the transition of the controlledsystem of the incorporated TDs.

To validate the high robustness against noise, an additivewhite-noise component with the maximum amplitude of0.01 rad is added to the feedback position signal obtained bythe optical encoder. The comparative results are shown inFig. 9. It can be seen from the comparison between Figs. 8with 9 that the maximum steady-state tracking error of theproposed EFPD controller has not changed much. For theFPDþ LO controller the maximum steady-state trackingerror increases from 0.020 rad without noise to 0.027 radwith noise, and the increased maximum tracking errormagnitude reaches 0.007 rad. Therefore it is clear that thedeveloped EFPD controller exhibits better robustnessagainst noise than the conventional FPDþ LO controller.

We conclude that the developed EFPD controller is betterthan the conventional FPDþ LO controller. The superiorperformance is mainly due to the high-quality differentialsignal selected by the proposed TDs with simple calcu-lation. In addition the incorporation of the TDs has thefunction of arranging the transition of the controlled systemsuch that performance is improved further.

5 Conclusions

This paper has presented a simple enhanced fuzzy PDcontroller by incorporating two nonlinear discrete trackingdifferentiators into a conventional fuzzy PD controller. Theincorporated differentiators are designed to a select high-quality differential signal from the position measurementonly, which is constructed based on the fact numericalintegration can provide more stable and accurate resultsthan numerical differentiation in the presence of noise. Themain improvement of the proposed enhanced fuzzy PDcontroller lies in its high robustness against noise and easeof implementation. The effectiveness of the proposedcontroller is verified in both simulations and experimentson a permanent-magnet synchronous AC motor controlsystem.

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an enhanced fuzzy pd controller with two discrete nonlinear tracking differentiators

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