speed control of torsional drive systems with backlash

Speed Control of Torsional Drive Systems with Backlash

S.Thomsen, F.W. FuchsInstitute of Power Electronics and Electrical Drives

Christian-Albrechts-University of Kiel, D-24143 Kiel, GermanyPhone: +49 (0) 431-8806107

Fax: +49 (0) 431-8806103Email: [email protected]

URL: http://www.tf.uni-kiel.de/etech/LEA

AcknowledgementsThis work was funded by the Deutsche Forschungsgemeinschaft (German Research Foundation).

Keywords<<Adjustable speed drive>>, <<Control of drive>>, <<Active damping>>, <<Variable speeddrive>>, <<Mechatronics>>.

AbstractThis paper presents the design, analysis and comparison of the conventional PI-control to two state spacecontrollers for speed control of drive systems with elastically coupled loads. A state space controllerof fourth order which considers only the mechanical system and a state space controller of fifth orderwhich takes an approximation of the electrical system into account are analyzed. Thereby, the effects ofbacklash in the drive are analyzed in each control.State space control yields a high performance, is able to damp torsional oscillations effectively and toreduce backlash effects. Measurement results confirm these statements.

IntroductionTorsional oscillations in electrical drive systems with elastic shafts are a well known problem [1]. Thenatural damping of such systems is very low and yields to a slow decay characteristic of torsional oscil-lations. Backlash is present in many mechanical systems [2]. If a motor is not directly connected to theload, backlash effects can disturb the system. They yield to high torque impulses which can excite tor-sional vibrations and reduce lifetime of the system significantly. Active damping of torsional vibrationsin drive systems with elastic shafts is a counter measure but yields to high requirements for the controller.Normally, control structures with proportional-integral (PI) controller are used for speed control of two-mass drive systems [3]. There are various design methods for tuning the PI control parameters whichlead to different control performances. For example the restricted pole placement with identical radius,with identical damping coefficient or with identical real part [4]. Another parameter tuning concept ofPI-controllers which yields to a maximum value for the phase margin in presented in [5]. However, PIcontrol method without additional feedback provides no free pole placement of the closed control loopand is not appropriate to damp torsional vibrations effectively [6].Better results can be achieved by feeding back additional system states. In [7] the derivative of the esti-mated shaft torque is used as additional feedback. But this approach is sensitive to measurement noise.A load torque observer and the results of this additional feedback are presented in [8]. A good tuningof a disturbance observer is shown in [1] and [9]. A systematic analysis of speed control with differentadditional feedbacks is done in [3]. All these approaches achieve an improved suppression of torsionaloscillations. Nevertheless, poles of the closed control loop and the resulting system dynamic cannot beset freely.The most promising approach for suppression of torsional oscillations can be obtained with the feedbackof all system states. The state space control method allows a free pole placement of the closed control

Speed Control of Torsional Drive Systems with Backlash THOMSEN Soenke

EPE 2009 - Barcelona ISBN: 9789075815009 P.1

loop. Therefore the performance of the controlled system can be chosen nearly freely. The problem ofstate space control is the design of the control parameters and the measurement of all system states orthe reconstruction from measurable signals. Control is especially difficult when not all system states aremeasurable [3]. Typically motor speed is the only measured variable of the mechanical system. Loadspeed and shaft torque are usually not measured in industrial applications [4].State space control is analyzed in [10], [11], [12], [13], [14] and [15]. Whereas [13], [14] and [15]present only simulation results. In [10] and [12] Kalman filters are used for estimation of the systemstates. Thus good results are received by feeding back the estimated states. Backlash and the resultingeffects are not analyzed. The attempt to reduce backlash effects is done in [11]. A gear torque observeris introduced which is used to compensate the backlash effects. However, by feeding back this estimatedgear torque, the overshoot of the load speed and the settling time increases. There has been a lot researchactivity in the area of controlling drive systems with elastic coupled loads. But the differences betweencontrol with and without backlash in the drive and the consequences for state space control have not beenclearly shown.The aim of this paper is to present three different control structures for speed control of drive systemswith elastically coupled loads. A conventional PI-controller, a state space controller of fourth order anda fifth order state space controller will be designed, analyzed and compared. Thereby, backlash effectswill be considered and investigated. Measurements will be shown to confirm theoretical results.

System DescriptionConventional speed controlled drive systems may include an inverter-fed motor which powers a loadvia gear and shaft, as shown in Fig. 1. The considered system can be divided into an electrical and amechanical subsystem. The electrical system consists of a current controller, a frequency converter andthe electromagnetic part of an induction motor. The mechanical system consists of the inertias of motor,gear, shaft and load and of the connecting elements between these parts. Multi-mass drive systems witha dominant resonant frequency can often be reduced to a two-mass system. The structure of a two-inertiasystem is presented in Fig. 2. The inertias of the motor ΘM and of the load ΘL are coupled by an elasticshaft which is modeled as a spring with torsional stiffness c and damping d. For this analysis the gear islocated close to the motor and is modeled as a dead zone with transmission ratio ΩM/ΩL = 1. The valueof the backlash gap is ±ϕL = 1. The dynamics of the electrical part of the drive are approximated asa first-order time-delay element with a time constant TE . Normally, this time constant is much smallerthan the time constant of the mechanical part. Due to this TE is neglected in most cases.

Figure 1: Overview of the drive system Figure 2: Block diagram of two-inertiasystem with backlash

Figure 3: Backlashas dead zone ele-ment

BacklashBacklash is one of the most important non-linearities in many applications that limit the performance ofspeed control [2]. Backlash can be modeled as a dead zone element as shown in Fig. 3. In dependenceof the value of the backlash gap ϕL and the angle ∆α = αM −αL (difference between motor positionand load position) a new output angle ϕ is generated which causes a rotation of the shaft. When thebacklash is open, the output angle ϕ is zero. Consequently the transmitted torque is equal to zero. Inthis case motor and load side are decoupled. The inertia which affects the motor is no longer the sumof all inertias of the drive system, but only the inertia of the motor and the transmission parts beforethe backlash. Hence the dynamic behavior varies. The movement of the load is autonomous while thebacklash gap is open and accordingly the load is neither observable nor controllable. When the backlashcloses, high torque impulses occur, which can excite torsional oscillations and reduce lifetime. Based onthis, the aim of the control has to be that the backlash opens as seldom as possible and closes softly. Theinfluence of backlash will be shown in the section with measurement results.



Two-Mass ModelThe internal damping d of the shaft is very low and can be neglected for control design analysis [16].The mathematical model is described in a per-unit system with time constants TM, TL, TC and TE , timeconstant of motor, load, spring and approximated electrical part, respectively. The state space model ofthe two-mass system is given by:

ddt

ωM

mS

ωL

=

0 − 1TM

01

TC0 − 1

TC

0 1TL

0

·ωM

mS

ωL

+

1TM

00

·mM +

00− 1

TL

·mL (1)

If the approximated time delay of the inner control loop is taken into account, the state space model isgiven by:

ddt

ωM

mS

ωL

mM

=

0 − 1

TM0 1

TM1

TC0 − 1

TC0

0 1TL

0 00 0 0 − 1

TE

·

ωM

mS

ωL

mM

+

0001

TE

·m∗M +

00− 1

TL

0

·mL (2)

Where ωM is the motor speed and ωL the load speed. mS, mM, mL and m∗M are the torsional shaft torque,

electromagnetic torque of the motor, load torque and the reference value of the torque.

ControlThis section shows the design of a conventional PI-controller and a state space controller for a two-inertia system. Two state space controllers are introduced. The first state space controller is designedby neglecting the inner control loop whereas the second one takes this control loop into account. Allpresented control methods are designed with neglected internal damping coefficient d of the shaft.

Conventional PI-ControlFig. 4 shows the block diagram of the conventional PI-controller with proportional gain kP and integralgain kI . The disadvantage of the conventional PI-speed controller is the limited pole placement. Theconventional control method contains no additional feedback from system states. Therefore it’s notpossible to locate all poles of the closed control loop independently. Consequently the dynamics of theclosed loop cannot be set freely and the torsional vibrations cannot be damped effectively.

ElectricalSystem

MechanicalSystem

Figure 4: Block diagram of conventional control with PI speed controller

The design of the control parameters kI and kP typically is done using standard optimization methodswith the aim to achieve a trade-off between reference reaction and disturbance reaction of the closedcontrol loop. Standard optimization methods such as the symmetrical optimum are not suitable for drivesystems with elastic shafts and yield to a high excitations of torsional oscillations. A restricted poleplacement is more appropriate to design the parameter of the PI-controller. Three different restrictedpole locations with either identical radius, identical damping coefficient or identical real part can bedistinguished [4]. An appropriate design for ratios from load inertia to motor inertia above two is a polelocation with identical damping coefficient, which will be presented and compared to state space control.

Design of Conventional PI-Control

The electrical system is neglected for design of the PI-controller. Then the transfer function of the closedcontrol loop from motor speed to motor reference speed is given by a fourth order function:

GW (s) =ωM(s)ω∗

M(s)=

(kI + s · kP)(

s2 · 1TM

+ 1TMTLTC

)s4 + s3 · kP

TM+ s2 · kITLTC+TM+TL

TMTLTC+ s · kP

TMTLTC+ kI

TMTLTC

(3)



The design of the control parameters takes place using pole placement. Therefore a forth order polyno-mial function P(s) with the desired poles of the closed control loop pz1 to pz4 is introduced:

P(s) = (s− pz1)(s− pz2)(s− pz3)(s− pz4) (4)

with:

pz1/z2 = ωz1

(−D1± j

√1−D2

1

); pz3/z4 = ωz2

(−D2± j

√1−D2

2

)(5)

The poles of the closed control loop are characterized by damping coefficients D1 and D2 and by theeigenfrequencies ωz1 and ωz2. Comparing the coefficients of the denominator of the closed loop transferfunction (3) and the forth order polynomial P(s) from equation (4) arises the following equations [4]:

kP = 2TM (ωz1D1 +ωz2D2) (6)

kI = TMTLTC(ω

2z1 ·ω2

z2)

(7)

ω2z1 +ω

2z2 +4ωz1ωz2D1D2−ω

2z1ω

2z2TLTC =

1TLTC

+1

TMTC(8)

ωz1D1

(ω

2z2−

1TLTC

)= ωz2D2

(1

TLTC−ω

2z1

)(9)

For the design with identical damping coefficient D = D1 = D2, the following control parameters arise:

kP = 2TMD(ωz1 +ωz2) (10)

kI = TMTLTC(ω

2z1 ·ω2

z2)

(11)

The eigenfrequencies result by system parameters and the selected damping coefficient D:

ωz1/z2 =

√TLTM−4D2 +4∓

√TLTM−4D2

2√

TLTC(12)

A good trade-off between reference reaction and disturbance reaction can be obtained with a dampingcoefficient D = 1/

√2 = 0.707 [17].

The two-mass mechanical system contains three poles. One is located in the origin and a conjugate-complex pole pair is located on the imaginary axis of the pole zero map. The resonance frequencyof the conjugate-complex pole pair of the mechanical system amounts to ω1 = 269 rad/s = 42.8 Hz.Designing the PI-controller with identical damping coefficient yields to the poles of the closed controlloop, as can be seen in Fig. 7. The eigenfrequencies ωz1 and ωz2 arise in dependence on the choice ofthis damping coefficient D. A damping coefficient D = 0.707 yields with the considered system to thefollowing frequencies of the closed control loop: ωz1 = 73.1 rad/s and ωz2 = 225 rad/s. The pole of theapproximated inner control loop is located on the real axis with eigenfrequency ω = 1500 rad/s and isoutside of the display area.

PI State Space ControlThe state space control method includes feedback of all system states. Fig. 5 shows the block diagramof the state space control if the electrical system is neglected and Fig. 6 shows the block diagram if theelectrical system is taken into account. A PI-controller is included to eleminate stationary control error.The reference value of the torque is calculated depending on the feedback of the system states, as can beseen in figures 5 and 6.The advantage of state space control is a theoretically free pole placement of the closed-loop control.Therefore high dynamic and high damping of torsional vibrations can be achieved. The disadvantage ofstate space control is the number of control parameters. The conventional control method includes twocontrol parameters kI and kP. The PI state space control includes five, respectively six control parameters,if the electrical system is considered. An appropriate method to design the control parameters for drivesystems with elastic shafts and backlash will be presented in this analysis.



Figure 5: Block diagram of PI state space controlof fourth order (electrical system is neglected)

ElectricalSystem

4

Figure 6: Block diagram of PI state space control of fifthorder (electrical system is taken into account)

Design of PI State Space Control

The closed-loop transfer function of the PI state space controlled system with neglected electrical systemis given by a fourth order function:

GW (s) =ωM(s)ω∗

M(s)=

(kI + s · kP)(

s2 · 1TM

+ 1TMTLTC

)s4 + s3 · kP−k1

TM+ s2 · kITLTC−k2TL+TM+TL

TMTLTC+ s · kP−k1−k3

TMTLTC+ kI

TMTLTC

(13)

The design of control parameters takes place using pole placement [18]. Comparing the denominator ofthe closed loop transfer function (13) with the polynomial P(s) from equation (4) and solving a systemof equations yields to following control parameters for the state space controller:

kI = ω2z1ω

2z2TMTLTC; kP =

kI

ωz1; k1 = kP−2TMωz1D1−2TMωz2D2 (14)

k2 = 1+TM

TL+TMTLTC

(TCω

2z1ω

2z2−ω

2z1−ω

2z2−4ωz1ωz2D1D2

)(15)

k3 = 2TMωz1D1 +2TMωz2D2−2TMωz1ω2z2D1TLTC−2TMω

2z1ωz2D2TLTC (16)

The dynamics of the closed-loop control can be chosen by the design parameters D1, D2, ωz1 and ωz2.In contrast to the conventional PI-controller, all considered poles of the closed control loop can be setfreely. If the electrical System is taken into account, the closed-loop transfer function increases to a fifthorder function:

GW (s) =ωM(s)ω∗

M(s)=

(kI + s · kP)(

s2 · 1TMTR

+ 1TMTLTCTR

)s5 + s4 · 1−k4

TR+ s3 · kPTC−k1TL+TM+TR

TMTCTR+ s2 · kITLTC−k2TL−k4(TM+TL)+TM+TL

TMTLTCTR

· · ·

· · ·+s · kP−k1−1

TMTLTCTR+ kI

TMTLTCTR

(17)

For the design of the control parameters, the order of the polynomial P(s) from equation (4) must beincreased to a fifth order function:

P(s) = (s− pz0)(s− pz1)(s− pz2)(s− pz3)(s− pz4) (18)

with one real pole pz0 = ωz0.Comparing the denominator of the closed loop transfer function (17) with the polynomial P(s) fromequation (18) and solving a system of equations yields to the control parameters of the state space con-troller. Due to limited space and long equations, the equations of control parameters are not shown hereexplicitly.The poles of the closed control loop of the state space controlled system can be seen in Fig. 7. Thepoles of the fourth order state space control are shown in the middle picture and the considered polesof the fifth order control are shown in the right picture. The damping of the poles are chosen as beforeD1 = D2 = 0.707. The eigenfrequencies are set to ωz1 = 30 rad/s and ωz2 = 300 rad/s. The fifth orderstate space control is able to influence the inner control loop. In order to reduce control input power, thepole of the inner control loop is shifted from ωz0 = 1500 rad/s to the eigenfrequency ωz0 = 300 rad/s.Damping coefficients D1 and D2 and eigenfrequencies ωz1 and ωz2 are chosen as with state space controlof fourth order.



Figure 7: Pole-zero maps: left: PI-controller tuned with identical damping coefficient D = 0.707; middle: fourthorder state space controller tuned with ωz1 = 30 rad/s, ωz2 = 300 rad/s, D1 = D2 = 0.707; right: fifth order statespace controller tuned with ωz0 = 300 rad/s, ωz1 = 30 rad/s, ωz2 = 300 rad/s, D1 = D2 = 0.707

Reconstruction of System States

In most applications it is not possible to measure all system states such as the load speed and the shafttorque. Therefore an observer is required which reconstructs these states. For an appropriate reconstruc-tion of all system states, a disturbance observer is used in this analysis [18]. Motor speed is the onlymeasured variable of the mechanical system which is used for the disturbance observer. Motor torqueMM is calculated by machine parameters and current. Load speed, shaft torque and disturbance torqueare estimated by the disturbance observer. To be fast enough and not too sensitive due to measurementnoise, the poles of the observer are three times further left of those of the closed control loop. Loadside is decoupled from the motor side, if the backlash is open. In this case, load variables are no longerobservable. This influence, relating to the state space control is analyzed in the next section.

Measurement Results and Backlash EffectsMeasurements were taken on the drive system shown in Fig. 8. A 5.5 kW induction motor is connectedvia backlash clutch, long shaft, torque sensor and flywheel to a 6.4 kW servo induction machine whichcan induce a disturbance torque. Incremental encoders with 5000 pulses/revolution on motor side and512 pulses/revolution on load side are used for speed measurement. The nominal speed of the inductionmotor amounts to 1455 1/min. The motor torque is limited to 36 Nm. A variable backlash gap can bechosen with an adjustable clutch on the motor side. A flywheel on the load side is used to increase theload inertia. The ratio of load inertia to motor inertia amounts to 3.4. The shaft torque is measuredby a torque sensor with a strain gauge. The long shaft consists of aluminum with a quill shaft and alength of 1500 mm. The control algorithm is implemented on a dSPACE DS1103 PPC controller boardand data are updated with a sampling frequency of 6 kHz. Previous simulation results are presented in[19]. Measurement results of the PI-controller designed with poles with identical damping coefficient

Figure 8: Laboratory setup

as described in the previous section are shown in Fig. 9. The upper picture on the left side shows the stepresponses of motor speed (blue line) and load speed (red line). At the time t = 0 s, the reference value(black dashed line) changes from zero to 20 % of the nominal motor speed and at the time t = 0.5 s from20 % to 0. It can be seen that torsional oscillations are excited during the changes of the reference speed.The resonance frequency amounts to 42.8 Hz. The overshoot is about 9 % and the settling time amountsto 0.3 seconds. The lower picture on the left side shows the shaft torque during the step responses.Tosional oscillations can be seen in the shaft torque to high amount. The maximum shaft torque amounts



to 1.65 times of the nominal value of motor torque. Results of disturbance steps can be found on theright side. The upper picture shows motor (blue line) and load (red line) speeds during a step changein the load. At the time t = 0 s, a load torque with 50 % of the nominal motor torque is induced andremoved at the time t = 1 s. The sag of motor and load speed amounts to 7.5 % and the overshoot to10.8 %. The corresponding shaft torque is shown in the lower picture on the right side. In contrast tostandard optimization methods like the symmetrical optimum, this design can avoid periodical torsionaloscillations. In Fig. 10, results of the PI-controlled drive system with backlash (ϕ = ±1) are shown.

Figure 9: PI-controller tuned with identical damping (D = 0.707) without backlash

It can be seen, that the amplitude of the oscillations of speeds and shaft torque are significantly higherthan without backlash. The difference between motor speed and load speed increases, consequently theshaft torque increases. The maximum value of the shaft torque exceeds 2.4 times the nominal value ofthe motor torque. This effect increases with increasing backlash gap. Furthermore, the backlash opensafter reaching the reference value and closes a short time later again, as can be seen in the enlarged view(upper picture, left side). This leads to a torque impulse (peak value amounts 0.3 times the nominalvalue of the motor torque) which excites oscillations in the shaft torque, as can be seen in the enlargedview in the lower picture on the left side. This effect is repeated during the reference step response from20 % to standstill at the time of 0.7 s. Overshoot and settling time are similar to the results withoutbacklash. The effects of backlash due to disturbance reaction can be seen on the right hand side. Thebehavior during a load step from 0 to 50 % of the nominal value of the motor torque is very similar tothe behavior without backlash. However, if the load torque jumps back to 0, backlash opens and closesagain which yields to torque impulses in the shaft torque, as can be seen in the lower picture at the timeof 1.15 s and 1.3 s. This effect increases with increasing backlash gap, as well. Results of the state

Figure 10: PI-controller tuned with identical damping (D = 0.707) with backlash

space controller of fourth order with backlash in the drive system can be found in Fig. 11. It can be seen,that the settling time of motor and load speed is similar to previous results but the overshoot is somewhat



smaller and amounts to 3.5 % (upper picture on the left side). Furthermore, the backlash does not openafter the reference step and therefore, no impulse in the shaft torque is induced. The measured shafttorque during the dynamic processes is similar to torque of the PI-controlled system. The maximumvalue of the shaft torque exceeds 2.4 times the nominal value of the motor torque and is comparable toprevious results. The disturbance reaction of the speeds and the shaft torque can be seen in the pictureson the right side. It can be found out that the impulses of the shaft torque during closing of the backlashare smaller than with PI-control. State space control achieves a better closing of the backlash duringdynamic processes. However, this state space control requires high control input power and cannot damptorsional oscillations during reference steps because of systems limitations. Measurement results of the

Figure 11: Fourth order state space controller with backlash

state space controller fifth order are shown in Fig. 12. The consideration of the electrical system providesto reduce the control input power and to damp torsional oscillations during dynamic processes effectively.It can be seen that oscillations during reference steps are reduced significantly. The maximum value ofthe shaft torque is much smaller than the PI-controlled system and amounts to 1.26 times the nominalvalue of the motor torque. High torque impulses in the shaft are eliminated. Overshoot amounts to 2.5 %and the settling time to 0.31 seconds. The disturbance step response shows good results, as well. Thereare no high torque impulses and almost no effects of backlash. In the previous results, measured system

Figure 12: Fifth order state space controller with backlash

states were fed back. If load speed and shaft torque is estimated by a disturbance observer, followingresults were achieved, see Fig. 13. Motor and load speeds looks almost the same as with measured states.The overshoot is slightly smaller (2 %) and the maximum value of the shaft torque is slightly bigger(1.42 times the nominal value of motor torque). Results of the disturbance step response show the sameperformances as with measured states.The estimated load speed and shaft torque during the reference steps and disturbance steps are shownin Fig. 14 and compared to the measured values. The red lines show the estimated values and the bluelines show the measured values. The estimated load speed and shaft torque show a good conformity with



measured values.Load side variables are not observable if the backlash opens. But the backlash opens only for a shorttime and yields just to small differences between measured and estimated values. This has almost noinfluence on the state space control. Analog results are achieved for other operating points.

Figure 13: Fifth order state space controller with backlash and observer

Figure 14: Comparison of measured and estimated states

ConclusionThree different control structures for speed control of drive systems with elastically coupled loads arepresented in this paper. A conventional PI-controller, a state space controller fourth order and a fifthorder state space controller are designed, analyzed and compared to each other. Thereby, the effects ofbacklash are considered.It has been shown that the conventional PI control method is not able to damp torsional vibrations and toreduce backlash effects effectively. Nevertheless, the design of the PI control parameters is of importancefor standard applications. A suitable design of the PI control parameters is given in this paper. The statespace controller of fourth order is appropriate to reduce torsional oscillations. But this approach requireshigh control input power and yields to oscillations during reference steps due to system limitations.Backlash effects are reduced but not eliminated. State space control of fifth order is able to damp torsionalvibrations effectively. Backlash effects are reduced significantly, as well. A suitable design of both statespace controls has been shown in this analysis. Load speed and shaft torque are reconstructed using adisturbance observer. It has been shown that the peak shaft torque of the PI-controlled system increasessignificantly if backlash is located in the drive system. Furthermore, the maximum value of the torqueimpulses increases with increasing backlash gap. To apply fifth order PI state space control gives bestresults concerning reference and disturbance reaction as well as in avoiding resonances and backlashinfluence. Measurement results confirm the statements.



Appendix

Table I: System parameters

MotorPower 5.5 kWTorque 36 NmSpeed 1455 min−1

LoadPower 6.4 kWTorque 39 kWSpeed 2490 min−1

MechanicsInertia of motor side 0.037 kgm2

Inertia of load side 0.1258 kgm2

Shaft stiffness 2070 Nm/rad

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speed control of torsional drive systems with backlash

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