Paper

Backlash Vibration Suppression Control of Torsional System by

Novel Fractional Order PIDk Controller

Chengbin Ma∗Student Member

Yoichi Hori∗ Member

This paper proposes a novel fractional order PIDk controller for torsional system’s backlash vibrationsuppression control, in which the order k of D controller can not only be integer but also be any real number.Various methods have been proposed for two-inertia system’s speed control, but the control systems designedby these methods may not be able to suppress the vibration caused by gear backlash. In order to improvecontrol system’s robustness against the backlash, several methods have been proposed. However their designprocesses are very complicated. Clear and straightforward design concept is required in practical applications.As a novel approach, in this paper the D controller’s order is expanded to include any real number. RobustPIDk controller against backlash is designed by adjusting the order k directly. An approximation methodbased on Short Memory Principle is also introduced to realize the discrete Dk controller. Design process andexperimental results demonstrate straightforward robust control design through the novel FOC approach,the PIDk control system’s better robustness against backlash non-linearity and good approximation of therealization method.

Keywords: Fractional Order Control System, Torsional System, Backlash Vibration Control, Realization Method

1. Introduction

Gears are used widely in torsional system as an in-tegral part of power transmission and position devices.Any faults in gear transmission will considerably affectthe performance of such system. Especially, backlashnon-linearity between the teeth is one of the most impor-tant faults in torsional system’s speed control. It causesdelay, vibration, speed inaccuracy and degrades overallcontrol performance. In extreme cases, the severe back-lash vibration can make it intractable and cause possibledamage to the system.

Since in torsional system gear elasticity is much largerthan shaft elasticity, control system is commonly de-signed based on the simplified two-inertia model, inwhich driving servomotor and gears are treated as unityinertia. The backlash non-linearity between gears is notexpressed in the model. For two-inertia speed control,the history of control theory can be seen. Various designmethods such as classical PID control, time derivativefeedback, model following control, disturbance observer-based control, state feedback control and modern H∞control have been proposed (1)～(4). Among them, thePID control is the most widely used in real industrialapplications.

Common weak point of the above methods is that theexistence of backlash non-linearity is totally neglected.This weakness may make designed speed control systemsunstable and give rise to backlash vibration. In order tobe robust against backlash non-linearity, several meth-∗ Information & System Division, Electrical Control System En-

gineering, Institute of Industrial Science, University of Tokyo4-6-1 Komaba, Meguro-ku, Tokyo 153-8505

ods have been proposed, but their design processes arevery complicated. As an example, for PID control in-troducing a low-pass filter Kds/(Tds+1) and redesigningthe whole control system with three-inertia model canbe a solution (5). Due to the necessity of solving high or-der equations, the design is not easy to carry out. Clearand straightforward design concept is required in prac-tical applications.

In this paper a novel Fractional Order Control (FOC)approach of using fractional order PIDk controller isproposed for torsional system’s backlash vibration sup-pression control. In PIDk controller D’s order can beany real number, not necessarily be an integer. Tun-ing fractional order k can adjust control system’s fre-quency response directly; therefore a straightforward de-sign can be achieved for robust control against backlashnon-linearity.

The concept of FOC means controlled systems and/orcontrollers are described by fractional order differentialequations. Expanding derivatives and integrals to frac-tional orders had a firm and long standing theoreticalfoundation. Leibniz mentioned this concept in a letterto Hospital over three hundred years ago (1695) andthe earliest more or less systematic studies seem to havebeen made in the beginning and middle of the 19th cen-tury by Liouville (1832), Holmgren (1864) and Riemann(1953) (6). As its application in control engineering, FOCwas introduced by Tustin for the position control ofmassive objects half a century ago, where actuator sat-uration requires sufficient phase margin around and be-low the critical point (7). Some other pioneering workswere also carried out around 60’s (8). However the FOCconcept was not widely incorporated into control engi-

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Backlash Vibration Control by Fractional Order PIDk Controller

neering mainly due to the conceptually difficult idea oftaking fractional order, the existence of so few physicalapplications and the limited computational power avail-able at that time (9).

In the last few decades, researchers pointed out thatfractional order differential equations could model var-ious materials more adequately than integer order onesand provide an excellent tool for describing dynamic pro-cesses (6) (10) (11). The fractional order models need frac-tional order controllers for more effective control of thedynamic systems. This necessity motivated renewed in-terest in various applications of FOC (12) (13). With therapid development of computer performances, modelingand realization of the FOC systems also became possi-ble and much easier than before. But in motion controlthe FOC research is still in a primitive stage. This pa-per represents one of the first attempts towards applyingFOC to motion control.

The paper is organized as follows: in section 2, the ex-perimental setup and modeling of the torsional systemare introduced; in section 3, integer order PID con-trol’s robustness problem against backlash non-linearityis analyzed using Bode plot; in section 4, a novel PIDk

controller is proposed to improve the robustness of thesystem. Realization method for the discrete Dk con-troller is also introduced; in Section 5, experiments arecarried out to verify the effectiveness of the proposedPIDk controller for backlash vibration suppression con-trol. Finally, in section 6, conclusions are drawn.

2. Experimental Torsional System

2.1 Three-Inertia Model The experimentalsetup of torsional system is depicted in Fig. 1. A tor-sional shaft connects two flywheels while driving forceis transmitted from driving servomotor to the shaft bygears with gear ratio 1:2. Some system parameters arechangeable, such as gear inertia, load inertia, shaft’selastic coefficient and gears’ backlash angle. The en-coders and tacho-generators are used as position androtation speed sensors.

The simplest model of the torsional system with gearbacklash is the three-inertia model depicted in Fig. 2 andFig. 3, where Jm, Jg and Jl are driving motor, gear (driv-ing flywheel) and load’s inertias, Ks shaft elastic coef-ficient, ωm and ωl motor and load rotation speeds, Tm

input torque and Tl disturbance torque. In the model-ing, the gear backlash is simplified as a deadzone factorwith backlash angle band [−δ, +δ] and elastic coefficientKg. Frictions between components are neglected due totheir small values.

Parameters of the experimental torsional system are

Fig. 1. Experimental setup of torsional system

Fig. 2. Torsional system’s three-inertia model

Fig. 3. Block diagram of the three-inertia model

Table 1. Parameters of the three-inertia system

Jm Jg Jl Kg Ks δ

(Kgm2) (Kgm2) (Kgm2) (Nm/rad) (Nm/rad) (deg)

0.0007 0.0034 0.0029 3000 198.49 0.6

Fig. 4. Bode plot of the three-inertia model

shown in Table 1 with a backlash angle of ±0.6deg. Theopen-loop transfer function from Tm to ωm is

P3m(s) = {JgJls4 + [(Ks + Kg)Jl + KsJg]s2

+ KgKs}/{s{JmJgJls

4 + [Ks(Jg + Jl)Jm + Kg(Jm

+ Jg)Jl]s2 + (Jm + Jg + Jl)KsKg}}

=(s2 + ω2

h1)(s2 + ω2

h2)Jms(s2 + ω2

o1)(s2 + ω2o2)

· · · · · · · · · · (1)

where ωo1 and ωo2 are resonance frequencies while ωh1

and ωh2 are anti-resonance frequencies. ωo1 and ωh1 cor-respond to torsion vibration mode, while ωo2 and ωh2

correspond to gear backlash vibration mode (see Fig. 4).2.2 Simplified Two-Inertia Model Since the

gear elastic coefficient Kg is much larger than the shaftelastic coefficient Ks (Kg � Ks), for speed control de-sign the two-inertia model is commonly used in whichdriving motor inertia Jm and gear inertia Jg are simpli-fied to a single inertia Jmg(= Jm + Jg) (see Fig. 5). Theopen-loop transfer function for the two-inertia model is

P2m(s) =s2 + ω2

h

Jmgs(s2 + ω2o)

· · · · · · · · · · · · · · · · · · · · (2)

電学論 D，124巻 3 号，2004年 313

Fig. 5. Block diagram of the two-inertia model

Fig. 6. Set-point-I PID controller

where ωo is the resonance frequency and ωh is the anti-resonance frequency corresponding to the torsion vi-bration mode, while the existence of backlash vibra-tion mode is totally ignored in the simplified two-inertiamodel.

3. Design of PID Controller

In order to smooth the discontinuity of speed com-mand ωr by integral controller, a set-point-I PID con-troller is proposed for the torsional system’s speed con-trol (see Fig. 6).

Based on the simplified two-inertia model, the PIDcontroller’s parameters are designed by the standardform (γ1 = 2.5, γ2 = γ3 = 2) of Coefficient DiagramMethod (1) (15):

Kp =10√

211

√Jlks, Ki =

411

Ks, Kd =511

Jl − Jmg

· · · · · · · · · · · · · · · · · · · (3)

Equation (3) gives

Kp = 0.9754, Ki = 72.1782, Kd = −0.0028· · · · · · · · · · · · · · · · · · · (4)

where Kd < 0 means positive feedback of accelerationωm. Simulation results with the simplified two-inertiamodel show this integer order PID control system hasa superior performance for suppressing torsion vibration(see Fig. 7).

For three-inertia plant P3m(s), the close-loop transferfunction of integer order PID control system from ωr

to ωm is

Gclose(s) =CI(s)P3m(s)

1 + CI(s)P3m(s) + CPD(s)P3m(s)· · · · · · · · · · · · · · · · · · · (5)

where CI(s) is I controller and CPD(s) is the parallel ofP and D controllers in minor loop; therefore Gclose(s) isstable if and only if Gl = CI(s)P3m(s) + CPD(s)P3m(s)has positive gain margin and phase margin. But as de-picted in Fig. 8 the gain margin of Gl(s) is negative.With the existence of gear backlash the designed integer

Fig. 7. Time responses of the integer order PIDtwo-inertia system by simulation

Fig. 8. Bode plot of Gl(s) in PID control

Fig. 9. Bode plots of Gl(s) in PIDk control

order PID control system will easily be unstable andlead to backlash vibration.

4. Proposed PIDk Controller

4.1 Design of Fractional Order k In this pa-per, a novel fractional order PIDk controller is proposedto achieve a straightforward design of robust control sys-tem against gear backlash non-linearity. Instead of solv-ing high order equations, by changing the Dk controller’sfractional order k the frequency response of Gl(s) canbe directly adjusted (see Fig. 9). As depicted in Fig. 10,letting k be fractional order can improve PIDk con-trol system’s gain margin continuously. When k < 0.84the PIDk control system will be stable; therefore withproper selected fractional order k the backlash vibrationcan be suppressed.

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Backlash Vibration Control by Fractional Order PIDk Controller

Fig. 10. Gain margin versus fractional order k

Fig. 11. Gain plots of the PIDk control systemsand three-inertia plant

At the same time, for better backlash vibration sup-pression performance higher Dk controller’s order ismore preferable. As shown in open-loop gain plots of0.85, 0.8, 0.7 and 0.5 order PIDk control systems (seeFig. 11), higher the D controller’s order is taken lowerthe gain near gear backlash vibration mode is. Based onthe tradeoff between robustness and vibration suppres-sion performance, fractional order 0.7 is chosen as Dk

controller’s best order in this paper.4.2 Realization of Dk Controller The math-

ematical definition of fractional derivatives and integralshas been a subject of several different approaches (6) (10).One of the most frequently encountered definitions iscalled Grunwald-Letnikov definition, in which the frac-tional α order derivatives are defined as

t0Dαt = lim

h→0nh=t−t0

h−αn∑

r=0

(−1)r

(αr

)f(t − rh)

· · · · · · · · · · · · · · · · · · · (6)

where h is time increment and binomial coefficients are(α0

)= 1,

(αr

)=

α(α − 1) . . . (α − r + 1)r!

· · · · · · · · · · · · · · · · · · · (7)

It is clear that fractional order systems have an infi-nite dimension while integer order systems finite dimen-sional. Proper approximation by finite differential equa-tion is needed to realize the designed PIDk controller.In this paper, Short Memory Principle is adopted to re-alize the discrete fractional order Dk controller. Theprinciple is inspired by Grunwald-Letnikov definition.

Fig. 12. Binomial coefficients value versus term or-der j when approximating D0.5

Based on the approximation of the time increment hthrough the sampling time T , the discrete equivalent ofDk controller is given by

Z{Dk[f(t)]} ≈⎡⎣T−k

∞∑j=1

ckj z−(j−1)

⎤⎦ X(z) · · · (8)

where X(z) = Z{f(t)} and the binomial coefficients are

ck1 = 1, ck

j = (−1)(j−1)

(k

j − 1

)· · · · · · · · · · (9)

The semi-log chart of Fig. 12 shows binomial coeffi-cients value versus term order j in Equ. (8) when ap-proximating k = 0.5 derivative. The observation of thechart gives that the values of binomial coefficients near“starting point” t0 in Grunwald-Letnikov definition issmall enough to be neglected or “forgotten” for large t.

Therefore the finite dimension approximation of Dk

controller can be arrived by taking into account the be-havior of f(t) only in “recent past”, i.e. in the interval(t − L, t), where L is the length of “memory” and [L/T ]is the number of the recent sampled f(t) values remem-bered by discrete Dk controller:

t0Dkt [f(t)] ≈t−L Dk

t [f(t)], (t > t0 + L) · · · · · (10)

From Fig. 12, empirically memorizing 10 past valuesshould have good approximation. Clearly, in order tohave a better approximation, smaller sampling time andlonger memory length are preferable. A detailed discus-sion on fractional order controller’s realization methodswill be held in succeeding papers.

5. Experimental Results

As depicted in Fig. 13, the experimental torsional sys-tem is controlled by a PC with 1.6 GHz Pentium IVCPU and 512 M memory. Realtime operating systemRTLinuxTM distributed by Finite State Machine Labs,Inc. is used to guarantee the timing correctness of allhard realtime tasks. The control programs are writtenin RTLinux C threads which can be executed with stricttiming requirement of control sampling time. A 12-bitAD/DA multi-functional board is used whose conversiontime per channel is 10µsec.

Experiments on PIDk speed control are carried out

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Fig. 13. Digital control system of the experimentalsetup

Fig. 14. Time responses of the integer order PIDcontrol

with sampling time T=0.001 sec and memory length ofL/T = 10 (0.01 sec). Two encoders (8000 pulse/rev) areused as rotation speed sensors with coarse quantization±0.785 rad/sec.

Since the driving servomotor’s input torque commandTm has a limitation of maximum ±3.84 Nm, Ki is re-duced to 18.032 by trial-and-error to avoid large over-shoot caused by the saturation. Firstly, integer orderPID speed control experiment is carried out. As de-picted in Fig. 14 the PID control system can achieve sat-isfactory response when the backlash angle is adjustedto zero degree (δ = 0), while severe vibration occurs dueto the existence of backlash non-linearity (see δ = 0.6case). This experimental result is consistent with theanalysis in section 3.

Figure 15 depicts the experimental results of fractionalorder PIDk control with 0.7 and 0.5 order Dk con-trollers. Severe backlash vibration in the integer orderPID control case is effectively suppressed. The controlsystem’s stability and robustness against gear backlashnon-linearity can be greatly improved by the FOC ap-proach. PID0.7 control system has a good robustnessagainst backlash nonlinearity, while the error in the ex-perimental response curves is for encoders’ coarse quan-tization. The intermittent tiny vibrations in lower order0.5 case are due to its relatively high gain near gear back-lash vibration mode in open-loop frequency response.

It is interesting to find the vibration suppression per-formance of fractional order PIDk control system showssomewhat “interpolation” characteristic. As depictedin Fig. 16, PID1 control has the most severe backlashvibration, while PID0.85 is on the verge of instability.PID0.95 and PID0.9 have intermediate time responses.This experimental result is natural since these orders are

Fig. 15. Time responses of PIDk control

Fig. 16. Continuity of PIDk control’s vibrationsuppression performance

continuous. The “interpolation” characteristic is one ofmain points to understand the superiority of FOC asproviding more flexibility for designing robust controlsystems. At the same time, this experimental consis-tency with the logicality also verifies the good approxi-mation of the realization method based on Short Mem-ory Principle.

6. Conclusions

In this paper, an integer order PID controller is firstlydesigned by CDM’s standard form. Severe vibration oc-curs due to the gear backlash non-linearity. As men-tioned in introduction part, there are several methodsto suppress the backlash vibration of torsional systems.However fractional order PIDk control is an innovativeproposal. In this paper, PIDk controller is designedin a manner that all the PID parameters are kept un-changed and only D controller’s order is permitted to beadjusted fractionally. Experimental results show PIDk

control system’s improved robustness against backlashnon-linearity compared to the PID control design basedon CDM’s standard form. A good approximation of therealization method is also verified in experiments. Bychanging Dk controller’s order k, the control system’sfrequency response can be adjusted continuously. Thisflexibility can lead to straightforward design and bettertradeoff between stability margin loss and the strengthof backlash vibration suppression.

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Even having a little higher hardware demand, cheaperdesign cost and better robustness of FOC demonstratedin this paper still highlight its promising aspects. Rapiddevelopment of computational power also makes frac-tional order controller’s implementation not really prob-lematic. On the other hand, applying FOC concept tomotion control is still in a research stage. Future re-searches on FOC theory and its applications to morecomplex control problems are needed.

Finally the authors must point out the problem for in-teger order PID controller designed by using fixed gam-mas. Standard form was emphasized at CDM’s earlystage. In recent development, the selection of the gam-mas is recommended. Designers should choose propergammas that can guarantee not only stability but alsorobustness. As discussed in Ref. (15), what is reallyneeded for controller design of two-inertia resonant sys-tem is phase lag, not phase lead. For this reason, thevalue of D control becomes negative, which means phaselag. However positive feedback of D control will lead topoor robustness and should be avoided as much as possi-ble in control design. Namely, the big value of negativeKd causes problem for the integer order PID controldesign in section 3. In fact a PI controller, where theassignment is γ1 = 2.5, γ2 = 2 and Kd = 0, should givea satisfactory performance in both stability and robust-ness. That is, in order to avoid positive feedback of Dcontrol, γ3 can not be assigned and must be the valuecalculated under above assignment.

Therefore, care must be taken about this paper’s pur-pose. It is not to claim PIDk controller as a good con-troller for three-inertia system, but to contribute to be-ing a valuable experience for novel but still primitiveFOC research. Especially this paper shows the possibil-ity of better tradeoff between stability margin loss andthe strength of backlash vibration suppression by FOCapproach. This tradeoff is a common and natural prob-lem in oscillatory systems’ control (16). For such kindof systems, a fractional order controller in the followingform:

Gc(s) =Kps + Ki

(Tds + 1)ks· · · · · · · · · · · · · · · · · · · · · · · (11)

is expected to be a general solution, where better trade-off between stability margin and vibration suppressioncan be achieved by choosing proper fractional order k.Future research is required in this field.

(Manuscript received June 19, 2003,revised Oct. 21, 2003)

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Chengbin Ma (Student Member) was born in Chengchengcounty, Shannxi province, China on 21stNovember 1975. He received the master de-gree of engineering from Department of Elec-trical Engineering, the University of Tokyo inSeptember 2001 and is presently a Ph.D. stu-dent in the same department. His doctoral re-search fields are fractional order control theoryand applications to motion control.

Yoichi Hori (Member) received the B.S., M.S. and Ph.D.degrees in Electrical Engineering from the Uni-versity of Tokyo in 1978, 1980 and 1983, re-spectively. In 1983, he joined the University ofTokyo, the Department of Electrical Engineer-ing as a Research Associate. He later becamean Assistant Professor, an Associate Professor,and in 2000 a Professor. In 2002, he movedto the Institute of Industrial Science, the Uni-versity of Tokyo, as a Professor. His research

fields are control theory and its industrial application to motioncontrol, electric vehicle, and welfare system, etc. He is now a VicePresident of IEE-Japan IAS.

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