engine torque simulated vibration control for drive train

IEEJ Journal of Industry ApplicationsVol.9 No.5 pp.618–628 DOI: 10.1541/ieejjia.9.618Translated from IEEJ Transactions on Industry Applications, Vol.140 No.1 pp.45–55

Paper(Translation ofIEEJ Trans. IA)

Engine Torque Simulated Vibration Control for Drive Train BenchSystem Using Generalized Periodic Disturbance Observer

Yugo Tadano∗ Senior Member, Takashi Yamaguchi∗∗ Member

Takao Akiyama∗∗ Senior Member, Masakatsu Nomura∗ Fellow

Drive train bench, which is a type of dynamometer, is used to test the power train components of a vehicle such astransmission, torque converter, etc. The test system has to simulate the vibration torque of the engine by motor control.Conventionally, a sine wave is used to simulate the vibration torque of an engine. However, the actual vibration torquewaveform contains harmonic distortion components inherent in the engine. Therefore, it is desirable to have a vibra-tion control technique capable of simulating the waveform of the engine torque more accurately. This paper proposesa novel engine torque simulated vibration control for the drive train bench system. The proposed method can achievea shaft torque detection value that follows the desired engine pulsation torque waveform by utilizing robust resonancesuppression control and a generalized periodic disturbance observer. The validity and usefulness of this technology areindicated by the theoretical explanation, simulation, and experimental results.

Keywords: generalized periodic disturbance observer, robust resonance suppression control, torque ripple suppression control,engine torque pulse simulation (ETPS)

1. Introduction

In order to shorten the development time and to improvethe environmental response and performance of vehicles, themotor control technology required by the dynamometer sys-tem (vehicle test equipment), is becoming more sophisti-cated (1)–(6). The virtual drive train evaluation system describedin this paper uses the components of the vehicle driving sys-tem such as various transmissions and torque converters ofautomobiles as specimens to measure and verify the powertransmission efficiency, endurance etc. The test specimen isconnected to a drive motor that replaces the engine, and anabsorb motor that absorbs the output. In this system, the ve-hicle components other than the test specimen can be con-figured as virtual components on a computer, and the sameload as that of driving on a real road can be applied to thetest specimen by motor control. In particular, the drive mo-tor that substitutes the engine is required to simulate the vi-bration torque of the engine. Conventionally, only the funda-mental component of the vibration torque is simply generatedby a sine-wave vibration control. However, the actual en-gine torque waveform includes the harmonic distortion com-ponents unique to combustion systems. Therefore, in order toconduct a more precise test, a vibration control technique thataccurately simulates the shape of the engine torque waveformincluding the harmonics is desired.

On the other hand, since the transmission system of drive

∗ R&D Group, MEIDENSHA CORPORATION515, Kaminakamizo, Higashimakado, Numazu, Shizuoka 410-8588, Japan

∗∗ Dynamometer Systems Business Unit, MEIDENSHA CORPO-RATION127, Nishishinmachi, Ota, Gunma 373-0847, Japan

train test equipment shows the characteristics of a multi-inertia resonance system comprising of a drive motor, a testspecimen and an absorb motor, and the test specimen, whichis a torque converter, has the characteristics of a non-linearspring, the resonance frequency, amplitude and phase char-acteristics change according to the torque. Although drivemotors often use low-inertia, high-power Interior PermanentMagnet Synchronous Motor (IPMSM) to achieve high re-sponse, due to specifications and structural restrictions, theyare susceptible to periodic disturbances such as torque rip-ples. Therefore, in order to simulate the desired engine vi-bration waveform in the shaft torque sensor installed on thecoupling shaft between the drive motor and the input of thespecimen, advanced vibration control technology that repro-duces the vibration frequency components while eliminatingthe influence of the above-mentioned complex mechanicalresonance characteristics and disturbances (aperiodic and pe-riodic disturbances) is desired. Further, to fulfil the functionsof the engine simulator in various operation modes, it is desir-able to have a configuration of the shaft torque feedback con-trol in which the waveform automatically follows the tran-sient changes in the vibration command, rotation speed andtorque.

Many prior studies (7)–(15) on resonance suppression controlof multi-inertia system, where angular velocity or angle isused as the control quantity have been reported, and are be-ing used in various industrial machinery. On the other hand,since there are only a few cases where the shaft torque re-quired for the vehicle test equipment is used as the controlledvariable, even though the shaft torque control methods forengine bench that are used in engine tests have been pro-posed (16)–(19), no consideration has been given to the vibrationcontrol. In the vibration control of drive train bench, cases

c© 2020 The Institute of Electrical Engineers of Japan. 618

Engine Torque Simulated Vibration Control for Drive Train Bench System（Yugo Tadano et al.）

where feed-forward vibration control is performed using theoutput waveform of the engine model (ETPS: Engine TorquePulse Simulation) as the command value of the driving mo-tor torque have been reported (3)–(6). However, as seen from thetest results of the literature, since it is affected by the torsionalresonance of the shaft, noise, and response delay of the high-frequency components due to the test equipment, reproduc-ing the complete engine torque waveform in the shaft torqueis considered difficult. Moreover, although there are manydescriptions on engine modeling, there has not been enoughstudy/consideration on the simulation of the engine torquewaveform with the shaft torque of the drive train test equip-ment. Therefore, vibration control by shaft torque feedbackis being studied (20). In the previous study, the shaft torquevibration control was performed by approximating the sys-tem characteristics of the vibration control frequency bandof the drive train test system to a three-inertia system, andseparating the low-frequency resonance points caused by thenon-linear spring characteristics of the test specimen from thehigh-frequency resonance points caused by the drive motor,shaft torque sensor, coupling stiffness, etc. of the test equip-ment. Specifically, vibration control by shaft torque feedbackis realized by applying I-PD control to the low-frequencysteady-state torque control and the μ-design method (21) to thehigh-frequency resonance suppression control, and combin-ing them. Further, for variable speed operation, it has a mech-anism to automatically adjust the amplitude of the vibrationwaveform. However, this function is limited to the amplitudeof the sinusoidal waveform, and automatic adjustment of theengine torque waveform is not considered.

In this paper, we follow the shaft torque feedback con-trol that combines the low-frequency steady state torque con-trol and the high-frequency resonance suppression control, asmentioned in the above prior study (20). On the other hand, in-stead of the automatic adjustment function of the amplitudeof the sinusoidal waveform (20), we propose the engine torquesimulated vibration control method (26), which is an improve-ment of the generalized periodic disturbance observer (22)–(25)

proposed by the authors. Since this method has a function toautomatically adjust multiple frequency components simul-taneously, in addition to realizing the engine torque simu-lation vibration control including harmonic components, itcan also suppress periodic disturbances such as torque rip-ples. Even if there are changes in the operating conditions(vibration waveform, rotation speed, torque, etc.), the desiredwaveform can be automatically followed. In other words, it isan arbitrary waveform tracking control method that can gen-erate periodic waveforms as specified by the command value,while comprehensively compensating for the system transfercharacteristics such as resonance, attenuation, periodic dis-turbance, aperiodic disturbance, phase lead/lag, etc. of thesystem being controlled.

The structure of this paper is as follows: First, we ex-plain the device configuration and the theory of the proposedmethod. Next, we show the validity and usefulness of thetheory from the simulation results for the three-inertia sys-tem configuration similar to the previous research (20). Lastly,we prove the feasibility based on experimental results.

2. System Configuration

2.1 Drive Train Test System Configuration Figure 1shows the configuration of the drive train test equipment dis-cussed in this paper. A driving motor and a power absorbingmotor are set as the input and output of the specimen torqueconverter respectively, and a shaft torque sensor is installedbetween the driving motor and the test specimen. The driv-ing motor performs vibration torque control by shaft torquefeedback such that the detected shaft torque value τdet be-comes the desired torque command value τ∗ (engine torquesimulated waveform). The power absorbing motor controlsthe rotation speed ωm such that it becomes the desired valueω∗m (corresponding to the engine rotation speed).

Figure 2 shows an example of the frequency transfer char-acteristics from the torque command τ∗inv of the driving mo-tor side inverter to the detected shaft torque value τdet. Ex-amples of physical parameters will be described in the nextsection. Since the specimen torque converter has non-linearspring characteristics in which the torsional stiffness of theshaft (K23) varies according to the torque, in the example inFig. 2, the resonance point fluctuation caused by the speci-men occurs in the band of about 6 [Hz] to 19 [Hz]. Further,since the high-frequency resonance point at about 356 [Hz]occurs due to the moment of inertia of the rotor, shaft torquesensor, couplings etc. of the driving motor and the torsionalstiffness of the coupling shaft, it is caused by the test equip-ment outside the test specimen. Thus, the drive train testequipment has the characteristics of a multi-inertia torsionalresonance system.

Fig. 1. System configuration of drive train bench

Fig. 2. Bode plot of drive train bench

619 IEEJ Journal IA, Vol.9, No.5, 2020


Since most commercial vehicles have four-cycle engineswhere each cylinder combusts the fuel once every two revolu-tions, large vibrations having a fundamental frequency com-ponent of the number of cylinders × 0.5 × speed of rotationare generated. When simulating an engine torque of 3 to 8cylinders for an engine speed of 600 to 6,000 [min−1], a vi-bration control band of 15 to 400 [Hz] will be required, asshown in Fig. 2. Further, to simulate the harmonic compo-nents of the vibration, it is desirable to be able to control ashigh a frequency band as possible. However, the frequencycontrol band includes the above-mentioned multi-inertia res-onance characteristics. Thus, this becomes an advanced tech-nical task where simultaneous tracking control of the ampli-tude and phase of multiple vibration frequency components isperformed, taking into consideration the attenuation charac-teristics such as resonance amplification and anti-resonancecaused by vibration and periodic disturbance.2.2 Three-inertia Model As described in the previ-

ous section, since a typical drive train testing equipment hastwo resonance points in the frequency control band, it can beapproximated to the transfer function model of three-inertiasystem (20) shown in Fig. 3. τ∗inv is the inverter torque com-mand value, τdet is the detected torque value, and θ is thedetected rotational phase value. Table 1 shows the mean-ing of each parameter and its numerical example. The non-linearity of the shaft torsional stiffness K23 of the specimenis set to 500 [N.m/rad], 2,500 [N.m/rad], and 5,000 [N.m/rad]respectively when the torque value is less than 50 [N.m], 50to 300 [N.m] and more than 300 [N.m]. Ginv(s) is the invertertorque current response, which is taken as a first-order lagsystem approximation of cut-off frequency 1 [kHz].

Gtm(s) is the detection delay of the shaft torque sensor.

Fig. 3. Block diagram of 3-inertial drive train bench model

Table 1. Parameters of 3-inertial model

From the phase characteristics of the Bode diagram shown inFig. 2, a second-order Pade approximation with a dead timeof 1.1 [ms] was used. It should be noted that the speed controlsystem on the power absorbing motor side in Fig. 1 controlsthe rotation speed with a response frequency of 1 [Hz], whichdoes not significantly affect the vibration control system.

From the characteristics shown in Fig. 2, the high-band res-onance frequency frH almost coincides with the two-inertialsystem resonance frequencies J1 - K12 - J2 as shown inEquation (1), and the low-band resonance frequency frL al-most coincides with the two-inertial system resonance fre-quencies [J1+J2] - K23 - J3 as shown in Equation (2). Thischaracteristic is used for the separation of high-frequencyband/low-frequency band in the control design describedlater.

frH =1

2π

√K12

(1J1+

1J2

)= 356 [Hz] · · · · · · · · · (1)

frL=1

2π

√K23

(1

J1+J2+

1J3

)=6～19 [Hz] · · · · · · (2)

3. Engine Torque Simulated Vibration Control

3.1 Overview of the Control Configuration In thispaper, we propose an arbitrary waveform tracking controlmethod that generates periodic waveforms according to thecommand value, while comprehensively compensating forthe system transfer characteristics (amplitude characteristicssuch as resonance and anti-resonance, phase lead/lag, anddisturbance). Figure 4 shows the overall control configura-tion diagram of the driving motor side torque control sys-tem. With the three-inertia model (Fig. 3) of the drive traintest equipment described in the previous chapter as the con-trol target, resonance suppression control (Fig. 6 describedlater) is used for the minor loop. Since the frequency trans-fer characteristics of the system to be controlled containingthe minor loop can be generalized by a complex vector ex-pression using the generalized periodic disturbance observerdescribed later, any method can be used for the resonancesuppression controller. Here, similar to Reference (20), the μdesign method (21), which is a type of robust control, is used.In the vibration torque controller of the major loop (Fig. 14,described later), an improved method of the generalized pe-riodic disturbance observer is applied to realize the periodicdisturbance suppression and engine torque waveform track-ing, that were difficult to achieve in Reference (20).

Considering the required performance of the drive traintest equipment, it is difficult to configure the shaft torque

Fig. 4. Overall control configuration diagram



Fig. 5. Generalized plant for high frequency resonancesuppression control

Fig. 6. Resonance suppression controller

feedback covering the entire frequency band of the vibra-tion control with a single controller (20). Therefore, as shownin Equations (1) and (2), focusing on the fact that the lowand high band resonance frequencies can be equivalently ex-pressed by their respective two-inertia models, we separatelydesigned the high-frequency resonance suppression control(Fig. 6 described later) and the low-frequency steady-statetorque control (Fig. 8 described later), and used a method tosuperimpose the output τ∗vib of the vibration torque controlleron the output τ∗st of the steady-state torque controller.

By inputting the superimposed value τ∗r to the high-frequency resonance suppression controller, resonance sup-pression control, steady-state torque control and vibrationtorque control are simultaneously achieved.3.2 High-frequency Resonance Suppression ControlThe high-frequency resonance point of about 356 [Hz]

shown in Fig. 2 almost coincides with the two-inertia sys-tem resonance frequencies frH of J1 - K12 - J2 as shown inEquation (1). In order to separate it from the low-frequencycontrol design using this characteristic, a generalized plantfor the equivalent model of the two-inertia system J1 - K12 -J2 is constructed as shown in Fig. 5, and the high-frequencyresonance suppression controller shown in Fig. 6 is designedusing the μ design method. Table 2 shows the meaning ofeach symbol in the generalized plant. Usually, the μ-designmethod that uses the structured singular value μ, takes intoaccount the structured perturbations explicitly to make a ro-bust design. However, when applying it to an actual dy-namometer system where various test specimens are replacedand tested, to simplify the operational control design process,rather than performing the parameter identification individ-ually taking the mechanical characteristics as the physicalmodel shown in Fig. 5, system identification is performed bya collective transfer function expression from input to output.

Hence, it is difficult to incorporate the fluctuations of thehigh-frequency resonance point into the generalized plant asstructured perturbations of K12 and J2. Therefore, in thispaper, only the inverter torque steady-state error ΔT is esti-mated as ±10[%], and the robust stability for the perturba-tion of mechanical characteristics is confirmed by numericalcalculation assuming resonance point fluctuation (maximum

Table 2. Parameters of generalized plant

±10 [Hz]) (20). The weight functions Wu(s) and We(s) weredesigned as shown in Equations (3) and (4). Considering thefact that a vibration frequency band of up to 400 [Hz] is re-quired, since Wu(s) is for high-frequency noise reduction ofthe inverter torque input, and We(s) is for avoiding interfer-ence with the low-frequency steady-state torque control, theresonance suppression control feedback gain is set such thatit decreases at low-frequencies.

Wu(s) =s + 2π · 400 · 0.010.01s + 2π · 400

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

We(s) =

(s + 2π · 400

s + 2π · 400 · 0.0001

)2

· · · · · · · · · · · · · · · · (4)

Next, the μ-design is implemented by calculating the gen-eralized plant configured above. Generally, in the μ-design,since it is not possible to directly find a compensator to makethe H∞ norm less than 1, the D-K iteration method, wherethe compensator K and the scaling matrix D are alternatelycalculated repeatedly, is used (21). By determining a local ap-proximate optimal solution from this, a compensator havingno practical problem is obtained. In this paper, the compen-sator obtained by three D-K iterations is balanced and trun-cated, and calculated as the 8th order resonance suppressioncontroller. Equations (5) and (6) show the transfer functionsdiscretized at a control period of 100 [μs] for the resonancesuppression controller calculated by μ-design (Fig. 6). Fur-ther, Fig. 7 shows their Bode characteristics. In the con-troller Cref (s) for the torque command input τ∗r of the reso-nance suppression controller, a notch characteristic of about8 [dB] is obtained near the high-range resonance frequencyof 356 [Hz]. Further, in the controller Ctm(s) for the de-tected shaft torque value τdet, since an effect of feedback gainreduction of 40 [dB/dec] is obtained in the low-frequencyrange due to the weight function We(s), the control spilloverwith the low-frequency steady-state torque control could beavoided.3.3 Low-frequency Steady-state Torque Control In

addition to being used in the engine torque simulated vibra-tion control dealt in this paper, the drive train test equipmentmust coexist with the existing control systems for various



Cref (z−1) =

0.1864 − 0.5496z−1 + 0.2928z−2 + 0.6410z−3 − 0.8614z−4 + 0.1231z−5 + 0.3452z−6 − 0.2171z−7 + 0.03971z−8

1 − 5.296z−1 + 12.09z−2 − 15.44z−3 + 11.90z−4 − 5.558z−5 + 1.472z−6 − 0.1799z−7 + 0.003703z−8

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (5)

Ctm(z−1) =0.06864 − 0.1938z−1+0.08408z−2+0.2415z−3 − 0.2842z−4+0.01998z−5+0.1199z−6 − 0.06770z−7+0.01153z−8

1 − 5.296z−1+12.09z−2 − 15.44z−3+11.90z−4 − 5.558z−5+1.472z−6 − 0.1799z−7+0.003703z−8

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

Fig. 7. Bode plot of resonance suppression controller

Fig. 8. Stationary torque controller with equivalent 2 in-ertia model in low frequency range

purposes such as rotational speed control by driving motor,electric inertia control where the moment of inertia of thedriving motor is made to look like some other moment ofinertia, etc. Further, in addition to the high-frequency res-onance suppression control described in the previous sec-tion, in the low-frequency range, it is necessary to control thesteady-state torque to a desired value, while considering theeffect of low-range resonance frequency fluctuation of about6 to 19 [Hz] shown in Fig. 2, is desired.

In the low-frequency characteristics shown in Fig. 2, al-though the value seems to be almost 0 [dB] even in uncon-trolled state, in reality, it does not become 0 [dB] at (J2+J3)/(J1+ J2+ J3). Hence, considering the inverter torqueerror ΔT and the disturbance of the power absorbing motor,steady-state torque control becomes necessary.

The low-frequency resonance point of about 6 to 19 [Hz]almost coincides with the two-inertia system resonance fre-quency frL of [J1+J2] - K23 - J3 according to Equation(2). In the control system design of low-frequency re-gion, since the influence of the shaft torque detection deadtime Gtm(s) and the inverter response Ginv(s) is small, thesteady-state torque control is designed excluding these byusing the two-inertia equivalent model. In this paper, simi-lar to Reference (20), a proportional differential leading typeI-PD controller as shown in Fig. 8 is constructed for thetwo-inertia system model whose resonance points fluctuategreatly. GSP(s) is a second-order low-pass filter used to avoid

the control spillover with the high-frequency resonance sup-pression controller described in the previous section. Sincethe highest resonance frequency in the low-frequency rangeis 19 [Hz], the cutoff frequency is set to 50 [Hz]. A first-orderlow-pass filter of cutoff frequency wPD [rad/s] is applied tothe PD controller unit. In the low-frequency I-PD control de-sign, since the effect of GSP(s) is negligible, the closed-looptransfer function from τ∗ → τdet can be approximated by thefourth-order system shown in Equation (7).

τdet

τ∗=

(K23 KI

J1 + J2s +

K23 KIwPD

J1 + J2

) /⎛⎜⎜⎜⎜⎜⎝s4 + wPDs3

+K23(J1 + J2 + J3 + J3 KDwPD)

(J1 + J2)J3s2

+K23((J1+J2)wPD+J3(KI+wPD+KPwPD))

(J1+J2)J3s

+K23 KIwPD

J1 + J2

⎞⎟⎟⎟⎟⎟⎠ · · · · · · · · · · · · · · · · · · · · · · · · · ·(7)

Next, the I-PD control parameters are designed such that theclosed-loop response of Equation (7) has the desired pole ar-rangement.

In this paper, we performed coefficient comparison so thatthe poles in Equation (7) are approximated to the pole ar-rangement of the fourth-order Butterworth standard formshown in Equation (8), and derived Equation (9).

τdet

τ∗=

w4c

s4 + a3wcs3 + a2w2c s2 + a1w3

c s + w4c

· · · · · · (8)

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

KP =J1 + J2 + J3

J3·⎛⎜⎜⎜⎜⎝k2(a1a3 − 1)

a23

− 1

⎞⎟⎟⎟⎟⎠KI =

J1 + J2 + J3J3

· wck2

a3

KD =J1 + J2 + J3

J3· a2k2 − 1

a3wcwPD = a3wc

· · · · · · · (9)

The coefficients of the fourth-order Butterworth standardform are a1 = 2.6131, a2 = 3.4142, and a3 = 2.6131. kis the coefficient of the resonance frequency f r, and wc is theresponse frequency of the standard form, and it is defined aswc = 2πk fr in terms of the resonance frequency f r. Here,taking k = 1, and the resonance frequency fr = 6 [Hz], thevalues detected were, wc = 37.699 [rad/s], KP = −0.2929, KI

= 1711.4, KD = 0.00082634, and wPD = 5843.1 [rad/s].3.4 Verification of the Control Characteristics by

Combining the High and Low frequency Regions Theresonance suppression control and steady-state torque controldesigned by separating high and low frequencies are com-bined as shown in Fig. 4 to realized control over the entire



Fig. 9. Bode plot of transfer characteristic (τ∗ → τdet)

Fig. 10. Bode plot of transfer characteristic (τ∗r (τ∗vib)→ τdet)

frequency band. Figure 9 shows the closed-loop transfercharacteristics (command value response) from the torquecommand value τ∗ to the detected shaft torque value τdet. Asthe command value response is about 3 [Hz], tracking con-trol of steady-state torque becomes possible without being af-fected by the resonance in the high-frequency range or the in-terference of the resonance suppression control. On the otherhand, for example, even if a vibration torque command (vi-bration frequency band of 15 to 400 [Hz]) that simulates theengine torque, is simply applied to the torque command τ∗, itcan be seen that the waveform cannot be reproduced, as thebandwidth is up to about 3 [Hz].

Figure 10 shows the closed-loop transfer characteristicsfrom the input torque τ∗r of the high-frequency resonance sup-pression controller in Fig. 4 (equivalent to vibration torquecontroller output τ∗vib, described later,) to the detected shafttorque value τdet. Due to the effect of high-frequency res-onance suppression control, the high-frequency resonancepoint of 356 [Hz] in Fig. 2 can be suppressed to less than0 [dB]. Further, since the effect of the τdet →I-PD controller→ τ∗r loop of the steady-state torque controller is also present,the non-linear resonance point in the low-frequency range of6 to 19 [Hz] also decreases. However, even if the desired vi-bration torque command is directly given to τ∗vib keeping thetransfer characteristics of Fig. 10 as it is, since the detectedshaft torque value τdet does not become the desired value, itis necessary to correct the amplitude and phase characteris-tics.

Figure 11 shows the closed-loop transfer characteristics(disturbance response) from the disturbance input d1 (Fig. 5)to the detected shaft torque value τdet. Due to the effect ofhigh-frequency resonance suppression control, although the

Fig. 11. Bode plot of transfer characteristic (d1 → τdet)

resonance in the high-frequency of 356 [Hz] is suppressed byabout 24 [dB] compared to the uncontrolled state in Fig. 2, again of about 5.6 [dB] remains. Thus, for example, if a peri-odic disturbance such as a torque ripple contained in the dis-turbance d1 is at a rotational speed that matches with this res-onance frequency, the effect of resonance becomes remark-able.

From the above characteristics, it is understood that the en-gine vibration torque cannot be simulated just by combiningthe low-frequency steady-state torque control and the high-frequency resonance suppression control, and the disturbancesuppression performance such as suppression of periodic dis-turbance etc., is also insufficient. Therefore, in this paper, inaddition to the resonance suppression control and the steady-state torque control, we propose a generalized periodic dis-turbance observer method that controls the vibration torque,while simultaneously suppressing the periodic disturbance,which is the cause for resonance. Thus, we realize a vibra-tion torque control that simulates the engine torque wave-form, while comprehensively suppressing resonance, aperi-odic disturbance, and periodic disturbance.3.5 Overview of the Generalized Periodic Distur-

bance Observer The generalized periodic disturbanceobserver is a method of extracting the arbitrary frequencycomponent to be controlled and configuring the disturbanceobserver in the rotational coordinate system synchronizedwith the frequency component (22)–(25). This section gives anoverview of the generalized periodic disturbance observer.First, as a method to extract the frequency components foran arbitrary time waveform y(t), it is converted to the d-axiscomponent Ydn, and qn-axis component Yqn, using the or-thogonal rotation coordinates dnqn synchronized with the nth

order frequency component defined by Equation (10). L isthe Laplace transform. In this paper, θ is the motor rotationphase, and the nth order frequency component is extractedwhile tracking it according to the change in rotation speed.[

Ydn

Yqn

]= 2

[GF(s)GF(s)

]· L

{[cos nθsin nθ

]· y(t)

}· · · · · · · · · · · (10)

GF(s) is a first-order low-pass filter for the extraction of thefrequency component shown in Equation (11). The cutofffrequency is set to wf = 2π × 3 [rad/s].

GF(s) =wf

s + wf· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (11)

Next, in the orthogonal rotation coordinates dnqn, a complex



Fig. 12. Definition of rotating coordinate system andcomplex vector

vector plane with the dn-axis component as the real part andqn-axis component as the imaginary part is defined as shownin Fig. 12. Pn is the nth-order frequency transfer character-istics vector of the system to be controlled, Un is the inputvector of the nth-order frequency component, and Yn is theoutput vector of the nth-order frequency component. Theyare expressed by Equations (12) to (14) respectively.

Pn = Pdn + jPqn · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (12)

Un = Udn + jUqn · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (13)

Yn = Ydn + jYqn · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (14)

Thus, by focusing only on a single frequency component ofthe orthogonal rotation coordinates dnqn, the frequency trans-fer characteristics of the system can be represented by a one-dimensional complex vector. Further, if complex vectors ofeach frequency component are prepared for the entire controlband, tracking during variable speed operation can also besupported. For example, preparing the real part Pd( fn) andthe imaginary part Pq( fn) of the system characteristic table,tabulated at intervals of 1 [Hz] as shown in Equation (15), oneset matching the nth-order frequency fn = nwm/2π of the mo-tor rotation speed wm is sequentially extracted from the table,and applied to the model Pn of the real system Pn.{

Pd( fn) : Pd(1), Pd(2), · · · Pd( fmax)Pq( fn) : Pq(1), Pq(2), · · · Pq( fmax)

· · · · · · · · · · · · · · (15)

As described above, even if the system to be controlled isa complex multi-inertia resonance system, it can always begeneralized as a simple one-dimensional complex vector onthe orthogonal rotation coordinates dnqn.

Figure 13 shows the basic configuration of the generalizedperiodic disturbance observer on the orthogonal rotation co-ordinates dnqn. The vector notation in the diagram denotesa complex vector. The actual system Pn of the nth-order fre-quency is given by Equation (12), and as described above, itis the nth-order frequency transfer characteristic from the in-put Un to the output Yn. The periodic disturbance Dn is takeninto consideration in the input section. The basic operationto negate the periodic disturbance Dn follows the method ofa conventional disturbance observer. However, the inversemodel Qn does not include the differential characteristics, andcan be derived simply from the reciprocal of Pn as shown inEquation (16). In Qn, similar to Pn, if the real part Qdn and theimaginary part Qqn are tabulated for each frequency, trackingcan be performed for variable speed operation.

Fig. 13. Basic system configuration of generalized peri-odic disturbance observer

Qn= Qdn+ jQqn=1

Pdn+ jPqn

=Pdn

P2dn+P2

qn

− jPqn

P2dn+P2

qn

∴ Qdn =Pdn

P2dn+P2

qn

, Qqn = − Pqn

P2dn+P2

qn

· · · · · · · · (16)

The estimated input value Un obtained from the detectedvalue Yn through Qn is given by Equation (17). By subtract-ing the input command value U∗n from Un via GF(s), the peri-odic disturbance estimated value Dn is obtained as shown inEquation (18).

Un = Qn · Yn · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · (17)

Dn = Un −GF(s) · U∗n · · · · · · · · · · · · · · · · · · · · · · · · · · (18)

D∗n is the periodic disturbance command value. However, itcan be suppressed by making the value 0. Further, by sub-tracting Dn in Equation (18), periodic disturbance Dn can becancelled. If vibration is required, the desired value can beentered in D∗n.

If this configuration is parallelized, multiple frequencycomponents can be controlled simultaneously, and an ar-bitrary periodic waveform can be generated by combiningthem.

Further, for details about the control performance and ro-bustness of the generalized periodic disturbance observer, seeReferences (22) to (25).3.6 Vibration Torque Control by Generalized Peri-

odic Disturbance Observer In this section, we describethe vibration torque control using the generalized periodicdisturbance observer.

As mentioned in Section 2.1, in a four-stroke engine, largevibrations are generated at the number of cylinders × 0.5 ×rotational frequency, and their harmonics. Taking a four-cylinder four-stroke engine as an example, periodic torquevibrations are generated at even orders like 2nd, 4th, 6th, 8th

and so on of the machine speed, and generally the lower or-der harmonics are larger. On the other hand, periodic distur-bances due to the dynamometer equipment such as IPMSMtorque ripple (6th order, 12th order, and so on of electricalfrequency), inverter current sensor error (1st order, 2nd order,and so on of electrical frequency), and dead-time error (6th

order etc. of electrical frequency) also occur in a complexmanner. In the 4-pole IPMSM used in this paper, periodicdisturbances of the 2nd, 4th, 12th and 24th-order of the ma-chine speed occur.

Based on the periodic characteristics described above, the



Fig. 14. Vibration torque controller with generalizedperiodic disturbance observer

purpose of controlling the vibration torque control unit is tofollow the desired engine torque for each frequency compo-nent, while eliminating the above periodic disturbances.

Figure 14 shows the configuration diagram where the vi-bration torque control unit of Fig. 4 is expanded. First, the de-viation Δτ between the torque command value τ∗ and the de-tected torque value τdet is taken, and the periodic disturbanceobserver is configured such that the nth-order frequency com-ponent ΔTn of the torque deviation included in Δτ is set to 0.Here, ΔTn ( = ΔTdn + jΔTqn) is extracted from Equation (19)using the dnqn orthogonal rotation coordinate transformation.[

ΔTdn

ΔTqn

]= 2

[GF(s)GF(s)

]· L

{[cos nθsin nθ

]· Δτ(t)

}· · · · · · · · (19)

In this paper, considering the 2nd, 4th, 6th and 8th order vibra-tions of a four-cylinder engine, and the 2nd, 4th, 12th, and the24th-order periodic disturbances, and by setting the orders ofN = [2, 4, 6, 8, 12, 24] for n, each frequency component ΔTn

is extracted in parallel.Next, the extracted ΔTn is multiplied by the inverse model

Qn. As shown in Equations (15) and (16) of the previous sec-tion, in Qn, the real part Qdn and the imaginary part Qqn aretabulated in 1 [Hz] increments, and the values synchronizingwith fn = nwm/2π are used for each order. Here, we describethe method of setting the table for the inverse model Qn. Thecontrol target system when viewed from the vibration torquecontrol unit τ∗vib of Fig. 4 has the closed-loop transfer char-acteristic τ∗vib→ τdet (Fig. 10). Figure 10 is the bode diagramof the minor loop that includes τdet →I-PD controller → τ∗rof the high-frequency resonance suppression control and thesteady-state torque controller. Converting this amplitude andphase characteristics (polar form) into a complex vector (realpart Pd( fn), and imaginary part Pq( fn)) in steps of 1 [Hz], andcalculating its inverse characteristics (real part Qd( fn), andimaginary part Qq( fn)) by Equation (16), we tabulate the in-verse model Qn as shown in Equation (20).{

Qd( fn) : Qd(1), Qd(2), · · · , Qd( fmax)Qq( fn) : Qq(1), Qq(2), · · · , Qq( fmax)

· · · · · · · · · (20)

The output through the inverse model Qn is given by Un.Since our aim is to set the frequency component ΔTn of thetorque deviation Δτ to 0, D∗n in Fig. 13 becomes 0. Whenthe control block Un → U∗n is transformed, it can be simpli-fied by the positive feedback method via GF(s) as shown byUn → T∗vibn in Fig. 14. By inversely transforming T∗vibn ( =T ∗vibdn + jT ∗vibqn) into the time domain by Equation (21), andcombining the waveform of each order set by N, the outputτ∗vib of the vibration torque control unit is obtained.

τ∗vib(t) =N(k)∑

n=N(1)

(T ∗vibdn cos nθ + T ∗vibqn sin nθ) · · · · · · · (21)

As described above, tracking of engine torque waveform thathas periodicity and suppression of periodic disturbance arerealized simultaneously for each frequency component.

4. Simulation Verification

Using the characteristics and parameters of the three-inertia system of Chapter 2, the validity and effectivenessof the proposed engine torque simulated vibration controlmethod are verified by numerical simulation. It should benoted that, here, the low-frequency steady-state torque con-trol (I-PD control), the high-frequency resonance suppressioncontrol, and the generalized periodic disturbance observer areabbreviated as I-PD, RSC, and GPDO respectively.

First, the engine torque simulation performance of the shafttorque is verified in a steady operation state. The vibrationtorque waveform of the 4-cylinder engine model is used forthe torque command value τ∗, and its average torque is setto 100 [N.m]. As a periodic disturbance, 1[%], 1[%], 3[%],and 3[%] respectively of the 2nd, 4th, 12th, and 24th-order pe-riodic disturbance described in Section 3.6 is applied, and asan aperiodic disturbance, Gaussian white noise of standardnormal distribution is applied to the inverter torque sectionand the shaft torque sensor detection section. Further, theeffect of the inverter torque error ΔT is also taken into con-sideration by assuming it to be −10[%]. The rotation speedis set to 1,780 [min−1] so that the 12th-order periodic distur-bance matches with the high-band resonance frequency frH =

356 [Hz], and the detected shaft torque value τdet is comparedand verified under strict conditions where the shaft torsionalresonance occurs.

Figure 15 shows the results when the vibration torque com-mand was directly applied to the test specimen having thecharacteristics of Fig. 2 in feed-forward (I-PD: OFF, RSC:OFF, GPDO: OFF). As there is no RSC/GPDO, a large tor-sional resonance occurs due to the 12th-order of the periodicdisturbance. Further, since there is no I-PD, average torqueis reduced to 90 [N.m] due to the effect of the inverter torqueerror. Figure 16 shows the results when only the RSC wasused to suppress resonance (I-PD: OFF, RSC: ON, GPDO:OFF). From the results of resonance suppression control, itcan be seen that although the resonance due to the 12th-orderperiodic disturbance is attenuated, the engine torque wave-form could not be followed. Further, the average torque alsodropped to 90 [N.m] due to the absence of I-PD. Figure 17

Fig. 15. Simulation result under the condition of I-PD:OFF, RSC: OFF, GPDO: OFF



Fig. 16. Simulation result under the condition of I-PD:OFF, RSC: ON, GPDO: OFF

Fig. 17. Simulation result under the condition of I-PD:ON, RSC: ON, GPDO: OFF

Fig. 18. Simulation result under the condition of I-PD:ON, RSC: ON, GPDO: ON (proposed method)

Fig. 19. Simulation result of torque step response withproposed method

shows the results when I-PD was added to RSC, and the vi-bration torque command was given to τ∗vib by feed-forward(I-PD:ON, RSC: ON, GPDO: OFF). By the effect of I-PD, al-though the average torque could be suppressed to 100 [N.m],the engine torque waveform could not be followed. Figure 18shows the results of the proposed method of using the gener-alized periodic disturbance observer (I-PD: ON, RSC: ON,GPDO: ON). The engine torque waveform could be success-fully simulated while comprehensively suppressing the peri-odic disturbances, aperiodic disturbances, and resonance).

Fig. 20. Simulation result of transient response duringrotation speed acceleration with proposed method

Next, we verify the transient control characteristics withrespect to the changes in torque and rotational speed, withthe proposed method. Figure 19 shows the results whenthe average torque is changed in steps from 0→100 [N.m],and the vibration torque amplitude is changed in steps from0→100 [N.m], at a rotational speed of 1,780 [min−1]. Theengine torque waveform could be tracked in approximately0.3 seconds after the step change, confirming that there is noproblem in the torque transient response.

Figure 20 shows the results when the rotational speed waschanged from 600→6,000 [min−1], while maintaining an en-gine torque waveform of average torque 100 [N.m] and vi-bration torque amplitude 100 [N.m]. Since the vibration fre-quency increases while simulating the engine torque wave-form, it is confirmed that it could follow the changes in therotational speed.

5. Experimental Verification

The feasibility of the proposed method is verified with anopposing-type motor test equipment. In this paper, for theconvenience of experimental system construction and post-ing of data, the experiment was performed in a two-inertiasystem with a resonance frequency of about 317 [Hz], with-out including the characteristics of the low-frequency drivetrain test specimen. Table 3 shows the main parameters ofthe experimental system. Although the low-frequency res-onance did not occur, similar to the simulation, the con-trol system testing was performed by combining the low-frequency steady-state torque control (I-PD control), high-frequency resonance suppression control (μ-design), and vi-bration torque control using the generalized periodic distur-bance observer.

Figure 21 the experimental results when a 4-cylinder en-gine torque waveform of average torque of 2 [N.m] and



Table 3. Experimental system parameters

Fig. 21. Experimental result under the condition of I-PD: ON, RSC: ON, GPDO: ON (N = [2, 4, 6, 8, 24,48])

Fig. 22. Experimental result under the condition of I-PD: ON, RSC: ON, GPDO: ON (N = [2, 4, 6, 8, 10, 12,14, 16, 18, 20, 22, 24, 26, 46, 48, 50])

vibration torque amplitude of 5 [N.m] was applied with arotation speed of 800 [min−1], where the 24th-order periodicdisturbance (6th-order electrical frequency × 4 pairs of poles)due to torque ripple almost coincided with the resonance fre-quency. Taking into consideration the lower-order compo-nents of the engine torque waveform such as 2nd, 4th, 6th, and8th-order components, and 24th and 48th-order periodic dis-turbances, the control target order of GPDO was set to N =[2, 4, 6, 8, 24, 48]. In general, although a shaft torque wave-form that follows the waveform of the command value is ob-tained, it can be seen that the other order components thatare not considered as control targets remain as disturbances.Since the actual IPMSMs and inverters have non-linearity, itis considered that sidebands and mutual interference due toperiodic disturbances and frequency components of the vi-bration torque occur in a complex manner. Therefore, theorder of the control target of the GPDO is expanded to N =[2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 46, 48, 50], andthe experimental results are shown in Fig. 22. By increas-ing the order of the target, the disturbance was sufficientlyeliminated, and a good shaft torque waveform similar to thesimulation result was obtained.

Fig. 23. Experimental result of torque step responsewith proposed method

Fig. 24. Experimental result of transient response dur-ing rotation speed acceleration with proposed method

Figure 23 shows the experimental results when the vibra-tion amplitude was changed in steps from 1→5 [N.m] whilesimulating the engine torque waveform, at a rotation speed of800 [min−1]. It was confirmed that the vibration torque am-plitude could follow in about 0.2 seconds. Figure 24 showsthe experimental results when the rotation speed was accel-erated from 200→1,400 [min−1] in about 3.5 seconds whilekeeping the vibration amplitude of the engine torque wave-form constant. The possibility of following the change in vi-bration frequency while maintaining the engine torque wave-form could be confirmed.

6. Conclusion

In this paper, we have proposed an engine torque simu-lated vibration control method using a generalized periodicdisturbance observer, and verified its validity and usefulness.From the simulation and experimental results, we have eluci-dated that it is possible to realize the tracking and simulationof engine torque waveform, while simultaneously suppress-ing the resonance, aperiodic disturbance and periodic distur-bance of the drive train test system, which was difficult inthe conventional method. Further, since the generalized pe-riodic disturbance observer is a simple algorithm and the re-producibility of the waveform can be improved by increasingthe number of orders parallelized, it is considered that it cansupport complex drive train tests such as engine misfire modeetc. We plan to continue examining the items required for thespecimen test.

In this paper, we have applied the robust resonance



suppression control by μ-design method to the minor loop ofthe generalized periodic disturbance observer for drive traintest system. However, other methods are also feasible. Fur-ther, in the feedback control, the shaft torque can also be re-placed with other physical quantities (such as current, volt-age or rotation speed). In other words, since it can be usedto suppress vibrations in various systems and track arbitraryrepetitive waveforms, we intend to continue expanding its ap-plications.

References

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Yugo Tadano (Senior Member) received the M.E. degree from MieUniversity, Japan, in 2002. In 2002, he joined MEI-DENSHA CORPORATION. He currently belongs tothe Basic & Core Technology Research Laboratories,R&D Group. He is engaged in research and develop-ment on power electronics and measurement controltechnology. In 2006, he received an IEE Japan BestPaper Presentation Award.

Takashi Yamaguchi (Member) received the M.E. degree from SeikeiUniversity, Japan, in 2006. In 2006, he joined MEI-DENSHA CORPORATION. He currently belongs tothe Dynamometer Systems Business Unit. He is en-gaged in research and development on automotivepower measurement systems and measurement con-trol technology. In 2015, he received Ph.D. degreefrom Tokyo University of Science.

Takao Akiyama (Senior Member) received the M.E. degree from Os-aka University, Japan, in 1994. In 1994, he joinedMEIDENSHA CORPORATION. He currently be-longs to the Dynamometer Systems Business Unit.He is engaged in research and development on au-tomotive power measurement systems and measure-ment control technology. In 2016, he receivedPh.D. degree from Hiroshima University. In 2017,he received an IEE Japan Electronics, Informationand Systems Society Distinguished Transaction Pa-

per Award.

Masakatsu Nomura (Fellow) received the M.E. degree from NagoyaUniversity, Japan, in 1977. In 1977, he joined MEI-DENSHA CORPORATION. He currently belongs tothe R&D Group. He is engaged in research and devel-opment on power electronics and measurement con-trol technology. In 2009, he received Ph.D. degreefrom Meiji University. In 1993, he received OhmTechnical Prize. In 2015, he received IEEJ IndustryApplications Society Technical Development Award.


engine torque simulated vibration control for drive train

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