H ∞ -control of a rack-assisted electric power steering system

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H∞-control of a rack-assisted electricpower steering systemC. Dannöhl a , S. Müller b & H. Ulbrich ca BMW AG , Knorrstraße 147, D-80788 , München , Germanyb Lehrstuhl für Mechatronik , Technische UniversitätKaiserslautern , D-67663 , Kaiserslautern , Germanyc Lehrstuhl für Angewandte Mechanik , Technische UniversitätMünchen , D-85748 , Garching , GermanyPublished online: 19 Aug 2011.

To cite this article: C. Dannöhl , S. Müller & H. Ulbrich (2012) H∞-control of a rack-assisted electricpower steering system, Vehicle System Dynamics: International Journal of Vehicle Mechanics andMobility, 50:4, 527-544, DOI: 10.1080/00423114.2011.603051

To link to this article: http://dx.doi.org/10.1080/00423114.2011.603051

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Vehicle System DynamicsVol. 50, No. 4, April 2012, 527–544

H∞-control of a rack-assisted electric power steering system

C. Dannöhla*, S. Müllerb and H. Ulbrichc

aBMW AG, Knorrstraße 147, D-80788 München, Germany; bLehrstuhl für Mechatronik, TechnischeUniversität Kaiserslautern, D-67663 Kaiserslautern, Germany; cLehrstuhl für Angewandte Mechanik,

Technische Universität München, D-85748 Garching, Germany

(Received 19 February 2011; final version received 25 June 2011 )

Electric power steering (EPS) is more and more in use for passenger cars. Compared with hydraulicsteering systems there are many advantages, such as reduced CO2 emissions and the possibility touse the EPS motor torque for advanced driver assistance systems. One task of the steering system isto give the driver an adequate steering feel. This includes providing road feedback and the right levelof assistance torque. This article describes the steering torque control of a rack-assisted EPS. Thecontroller’s task is to follow a reference steering torque quickly and accurately. First, a mechanicalmodel of the EPS is shown. Then, an H∞-controller is designed, implemented and compared withother steering torque controllers. As steering torque discontinuities are a topic when looking at newcontrol algorithms, the phenomenon and its cause are analysed using a detailed mechanical model.The results of this analysis are considered in the controller design.

Keywords: electric power steering; steering torque control; EPS model; steering torquediscontinuities

1. Introduction

More and more passenger cars are equipped with electric power steering (EPS). This meansthat the assistance torque for the driver is given by an electric motor. Compared with hydraulicsteering systems, the EPS has many advantages: the CO2 emissions are reduced as energy isonly consumed when steering assistance is given, the system can be used for advanced driverassistance systems, for example, lane keeping, and there are more degrees of freedom fordesigning the steering feel. In the development process of EPS systems, there are many chal-lenges, such as providing enough electric power, small costs, and high quality. An importanttopic is also to choose a suitable assistance torque, which fits to the driving situation. Thereare different ways to control the supporting motor torque. One approach, used in many carstoday, is to choose the motor torque from a characteristic boost curve which depends on themeasured steering torque and on the vehicle speed, see [1]. Another approach is to calculatea reference steering torque depending on the driving situation, like in [2], and to use a torquecontroller which minimises the difference between the measured and the reference steering

*Corresponding author. Email: [email protected]

ISSN 0042-3114 print/ISSN 1744-5159 online© 2012 Taylor & Francishttp://dx.doi.org/10.1080/00423114.2011.603051http://www.tandfonline.com

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528 C. Dannöhl et al.

torque. In this paper, the second approach is considered for a rack-assisted EPS. The designprocess of the controller begins with describing the control task and with the specification ofthe requirements on the controller. These requirements are derived from a diagram which con-tains the drivers’ needs concerning the steering behaviour. The requirements on the controllerare specified by defining target values for a certain choice of control engineering parameters.Steering torque discontinuities are sometimes an issue. As the causes for this phenomenonhave not been thoroughly investigated yet, the search for the mechanical root cause is carriedout and the result is considered in the controller design. To develop a torque controller and toanalyse the steering system, a model of the EPS is needed. Here, the electrical subsystem is notmodelled in detail but in a simplified way as a PT1 element. Concerning the mechanical parts,a simple model is used for the controller design and a more detailed one for the system andcontroller analysis. For the examination of the root cause of steering torque discontinuities,an advanced model of the ball nut drive has been developed, which is included in the detailedmodel. Different controllers are derived and implemented. The controllers are compared byanalysing how well the specified requirements are reached. Numeric simulations and realvehicle measurements are performed.

2. Description and model of the EPS system

2.1. System description

The rack-assisted EPS considered in this work is shown in Figure 1. Its main componentsare: steering wheel (swh), steering column, torque sensor (tse), pinion (pin), rack (rac), andelectric motor. Between the wheels (whe) and the rack are the tie rods and the linkage (lin).The assistance is given by a brushless DC motor, which is positioned parallel to the rack. Themotion and the forces between the electric motor and the rack are translated by a belt (bel) anda ball screw drive (bsd). To obtain the requested torque from the electric motor, field-oriented

Figure 1. EPS system.

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Vehicle System Dynamics 529

Table 1. Nomenclature.

ϕ Angle x Position �q Vector of generalised coordinatesϕ Angular velocity x Velocity �h Vector of external forcesm Mass J Inertia b Damping coefficienti Transmission τ Time delay c Stiffness coefficientF Force T Torque or transfer function μ Friction coefficientFn Normal force gn Normal distance gt Tangential distancerm Mid-circle gn Normal velocity gt Tangential velocity

control is applied. An electrical control unit also belongs to the system. The nomenclatureused in this paper is given in Table 1.

2.2. Basic model of the EPS

2.2.1. Mechanical model of the EPS

The mechanical parts of the EPS are modelled as a four-mass oscillator:

• first mass: steering wheel and steering column,• second mass: rack, wheels, and pinion,• third mass: ball nut (bnu) of the ball screw drive,• fourth mass: rotor (rot) of the electric motor.

Between the four masses, spring–damper elements represent the torque sensor, the ball screwdrive, and the belt. The kinematical conditions are defined as follows:

ipin = ϕpin

xrac, ibsd = ϕbnu

xrac, ibel = ϕrot

ϕbnu. (1)

By applying the Lagrangian equations [3], the following differential equation system isobtained with �q = (ϕswh, xrac, ϕbnu, ϕmot)

T

M · �q + D · �q + K · �q = �h, �q ∈ �f , (M, D, K) ∈ �f ,f (2)

with

M =

⎛⎜⎜⎝

Jswh 0 0 00 mrac 0 00 0 Jbnu 00 0 0 Jrot

⎞⎟⎟⎠ , (3)

D =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

btse + bswh −btseipin 0 0

−btseipin brac + btsei2pin + bbsd −bbsd

ibsd0

0 −bbsd

ibsdbbnu + bbsd

i2bsd

+ bbel −bbel

ibel

0 0 −bbel

ibel

bbel

i2bel

+ brot

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

, (4)

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K =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

ctse −ctseipin 0 0

−ctseipin ctsei2pin + cbsd −cbsd

ibsd0

0 −cbsd

ibsd

cbsd

i2bsd

+ cbel −cbel

ibel

0 0 −cbel

ibel

cbel

i2bel

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

, (5)

�h =

⎛⎜⎜⎝

Tdriver − Tf 1

−Ff 2 − Fload

−Tf 3

Tmot − Tf 4

⎞⎟⎟⎠ . (6)

The stiffness parameters, the damping factors, and the transmissions are assumed to be linear.The nonlinear friction force is modelled as Coulomb friction by using a tanh function. Theequation is chosen as in [4]:

Ffri = tanh

(2 · vrel

v96

)· μG · Fn. (7)

The constant v96 is the velocity at which 96% of the maximum amplitude of the friction force isreached. The constant is chosen small to find a good compromise between a steady descriptionfor the simulation and between a realistic modelling.

2.2.2. Model of the electrical subsystem

For the complete system model, the electrical subsystem has to be considered in addition tothe mechanical parts. The dynamics of the electric motor are described in a simplified way byusing a PT1 element:

Tmot ref

Tmot actual= 1

1 + τ · s. (8)

As the behaviour of the electric motor is strongly influenced by its controller, the controlledelectric motor has to be described as a transfer function. The motor torque controller and theelectric motor make up the inner loop of a cascade control structure with the steering torquecontroller being in the outer loop. As the inner loop has much faster dynamics compared withthe outer one, the steering torque controller cannot react on high-frequency overshoots oroscillations in the inner loop. In the design process of the steering torque controller, a detailedmotor model only helps if the phenomena modelled more precisely compared with the PT1

behaviour can be influenced by the outer control loop. The simplification was chosen to keepthe whole investigation as simple as possible and also because other works, e.g. [5], showpromising results with comparable simplification for the controller design.

2.3. Further models

2.3.1. Reduced model

For a model-based controller design, the described system model is simplified by reducing theamount of masses and by linearising the nonlinear terms. The ball nut is not considered as aseparate component any more, but it is included in the second mass. A study of the eigenvaluesshows that reducing the masses from four to three means that the highest natural frequency

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at 81 Hz is not considered any more. For the studies described in this work, the consideredfrequency range is sufficient as the frequencies of the main amplitudes of excitation forces arebelow 30 Hz [4].

2.3.2. Detailed model

To analyse the reasons of steering torque discontinuities and for numeric simulations of thecontrolled system, a more detailed model is used. The root cause of steering torque dis-continuities has not been discovered yet. The following effects are considered as possiblecauses:

• the rotation of the rack because of its bearing (Figure 2(a)),• the translational movement of the ball nut because of its bearing (Figure 2(b)),• the nonlinear belt stiffness (Figure 2(c)),• the backlash in the ball screw drive (Figure 2(d)), and• the friction forces acting at the individual masses (Figure 2(e)).

For the analysis of the root cause of steering torque discontinuities, the system model isextended by the listed nonlinear effects. This leads to a model with rotational and translationaldegree of freedom both at the rack and at the ball nut. To combine the four independent degreesof freedom at the ball screw drive, a new model of this component has been developed. A forceelement is representing the numerous (more than 100) balls between the ball nut and the rack.The force can be calculated by

Fbsd = bbsd · gn + cbsd · gn, (9)

where gn and gn are the normal distance and velocity between the rack and the ball nutfollowing the two equations:

gn = cos γ · (xrac − xbnu) + rm · sin γ · (ϕrac − ϕbnu) (10)

gn = cos γ · (xrac − xbnu) + rm · sin γ · (ϕrac − ϕbnu). (11)

Figure 2. Possible causes for steering torque discontinuities. (a) Rack rotation, (b) rack displacement, (c) nonlinearbelt stiffness, (d) backlash, and (e) friction.

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An experimental approach is used to consider the friction in the ball screw drive. The frictionforce is described as a function of the tangential velocity gt between the rack and the ball nut:

gt = sin γ · (xrac − xbnu) + rm · cos γ · (ϕrac − ϕbnu). (12)

The friction force follows the equation

Ff = −μ(gt) · |Fbsd|. (13)

The detailed model can be described in terms of the generalised coordinates �qdet =(ϕswh, xrac, ϕrac, xbnu, ϕbnu, ϕmot)

T as follows:

Mdet · �qdet + Ddet · �qdet + Kdet · �qdet = �hdet (14)

with Mdet, Ddet, Kdet, and �hdet according to Equations (17)–(20). In Equation (18) and (19),the abbreviations s(γ ) = sin(γ ) and c(γ ) = cos(γ ) are used.

To model the rotation of the rack, a characteristic curve of the torque Ttorsion over the angleis used. Bench tests were carried out to obtain the data. The characteristic curve shows thatthere is much more torsion at low angles compared with the range of higher angles. To modelthis effect, a spring–damper element with a low stiffness value in the range of small anglesand a much higher stiffness in the region of higher angles was chosen. According to themeasurements, the rotational angle does not exceed around two degrees. The translationalmovement of the ball nut is modelled with the following equation for the force Fdispl resultingfrom the elastic bearing, where P stands for a polynomial of fifth order that describes thenonlinear behaviour of the bearing and the index ‘pre’ stands for pretension:

Fdispl(xbnu) ={

P(xpre − xbnu) − P(xpre + xbnu) for |xbnu| ≤ xpre,

−P(xpre + |xbnu|) · sgn(xbnu) for |xbnu| > xpre.(15)

The nonlinear stiffness of the belt was measured at the test bench. In the model, it is consideredby using a look-up table. There are different approaches to model backlash.A short overview isgiven in [6]. Here, the backlash in the ball screw drive is modelled with an elastic approach asthe numerous balls do not hit at the same time. The following equation describes the behaviour:

cbsdnl ={

cbsd for gn ≥ ubsd,

0 for gn < ubsd.(16)

The value of the backlash ubsd is taken from data sheets and test bench measurements. Thefriction is modelled as Coulomb friction as described in Equation (7). The friction values areobtained from special test bench measurements.

2.4. Parameterisation and model validation

The parameters of the models are obtained from data sheets and from numerous test benchmeasurements. The measurements have been performed with the whole assembly and alsowith single components. Additional measurements of the whole steering system were carriedout to do the validation of the different models by comparing the simulation results with the realmeasurements. The quality of all the three models is good. Figure 3 shows the comparison of

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20 21 22 23 24 25 26−6

−4

−2

0

2

4

6(a) (b)

Time (s)

Ste

erin

g to

rque

(N

m)

0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Frequency (Hz)

Am

plitu

de: s

teer

ing

torq

ue

MeasurementSimulation

MeasurementSimulation

Figure 3. Validation of the detailed model. (a) Time plot and (b) frequency plot.

the steering torque over time and over frequency for the detailed model. The system is excitedwith a motor torque with constant amplitude and increasing frequency:

Mdet =

⎛⎜⎜⎜⎜⎜⎜⎝

Jswh 0 0 0 0 00 xrac 0 0 0 00 0 Jrac 0 0 00 0 0 mbnu 0 00 0 0 0 Jbnu 00 0 0 0 0 Jrot

⎞⎟⎟⎟⎟⎟⎟⎠ , (17)

Ddet =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

bswh + btse −ipinbtse 0 0 0 0

−ipinbtse brac + btsei2pin bbsdc(γ )s(γ )rm −bbsdc(γ )2 −bbsdc(γ )s(γ )rm 0

+bbsdc(γ )2

0 bbsdc(γ )s(γ )rm bbsdr2ms(γ )2 −bbsdc(γ )s(γ )rm −bbsds(γ )2r2

m 0

0 −bbsdc(γ )2 −bbsdc(γ )s(γ )rm bbsdc(γ )2 bbsdc(γ )s(γ )rm 0

0 −bbsds(γ )c(γ )rm −bbsds(γ )2r2m bbsds(γ )c(γ )rm bbsds(γ )2r2

m −bbel1

ibel+bbnu + bbel

0 0 0 0 −bbel1

ibelbrot + bbel

1

i2bel

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

(18)

Kdet =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

ctse −ipinctse 0 0 0 0

−ipinctse ctsei2pin + cbsdc(γ )2 cbsdc(γ )s(γ )rm −cbsdc(γ )2 −cbsdc(γ )s(γ )rm 0

0 cbsdc(γ )s(γ )rm cbsdr2ms(γ )2 −cbsdc(γ )s(γ )rm −cbsds(γ )2r2

m 0

0 −cbsdc(γ )2 −cbsdc(γ )s(γ )rm cbsdc(γ )2 cbsdc(γ )s(γ )rm 0

0 −cbsds(γ )c(γ )rm −cbsds(γ )2r2m cbsds(γ )c(γ )rm cbsds(γ )2r2

m + cbel −cbel1

ibel

0 0 0 0 −cbel1

ibelcbel

1

i2bel

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

,

(19)

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10−1 100 1010

10

20

30

Frequency (Hz)

Am

plitu

de (

dB)

10−1 100 101−200

−100

0

100

Frequency (Hz)

Pha

se (º)

ModelMeasurement

Figure 4. Validation of the linear model.

�hdet =

⎛⎜⎜⎜⎜⎜⎜⎝

Tdriver − Tfri1

−Fload − Ffri2x + Ff sin(γ )

Ttorsion − Tfri2ϕ − Ff cos(γ )rm

−Fdispl − Ffri3x + Ff sin(γ )

−Tfri3ϕ − Ff cos(γ )rm

Tmot − Tfri4

⎞⎟⎟⎟⎟⎟⎟⎠ . (20)

Figure 4 shows the comparison of the steering torque in the frequency response plot for thelinear model with excitation at the electric motor. The peak of the frequency response plot ataround 8 Hz shows the natural frequency of the steering system. At this frequency, the steeringwheel and the steering column are moving against the remaining parts of the system.

3. System analysis

3.1. Linear analysis

Using the linear system model, the controllability, observability, and stability are proved. Thenatural frequencies of the undamped linearised four-mass oscillator can be found at around 8,54, and 81 Hz in addition to the uniform displacement of the three rigid bodies.

3.2. Steering torque discontinuities

Steering torque discontinuities are variations in the steering torque which are noticed by thedriver. It is an unwanted effect as it deteriorates the steering feel. The torque variations can befelt and measured in experiments in the car and at the test bench when the assistance torque

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13.2 13.4 13.6 13.8 14 14.2 14.4 14.6 14.8 15−5

−4

−3

−2

−1

0

1

2

3

4

5

Time (s)

Ste

erin

g to

rque

(N

m)

SimulationMeasurement

Figure 5. Comparison of test bench and simulation result.

is activated and also when it is deactivated. This means that the controller might amplify orattenuate the effect, but it is caused in the mechanical system. The root cause of steering torquediscontinuities is investigated using the detailed model described above. For the analysis,a measurement from the test bench with the electric motor as actuator is repeated in thesimulation and the steering torque signals from the simulation and from the measurement arecompared. During the numeric simulation, only one nonlinear effect of the described possiblecauses (see Section 2.3.2) is activated at a time. The analysis shows that the cause of steeringtorque discontinuities is mainly the dry friction in the steering system. The curve of the steeringtorque is qualitatively quite similar for the simulation with activated friction and the test benchmeasurement (Figure 5). The backlash in the ball screw drive also shows a small effect onsteering torque discontinuities. The effect gets bigger when the parameter for backlash isincreased in the model. Figure 6 shows the simulation result with variations of the backlashbetween 50% and 250% of the nominal value. The other examined effects do not show any

Figure 6. Results of the simulation with variations of the backlash parameter.

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difference in the course of the steering torque. The reference curve and the ones resultingfrom the simulation with nonlinear belt stiffness or rack rotation cannot be distinguished atall. The simulation result for the model with translational movement of the ball nut can onlybe distinguished from the reference curve at a very close look.

4. Steering torque control

4.1. Control task and specification of requirements

The task of the controller is to track a reference steering torque quickly and accurately. At thesame time disturbances have to be suppressed. The controller has to be robust against sensornoise, model inaccuracies, parameter changes during lifetime, and production tolerances.Disturbances are the changing load torque, friction forces, and the driver’s torque.

For the specification of the controller’s requirements, all important aspects have to beconsidered. For this reason, they are deduced from the driver’s requirements on the steeringbehaviour, which are road feedback (with disturbing or informative character), acoustics,turning circle, maximum steering angle velocity, yaw behaviour, steering force course, steeringforce level, steering return behaviour, and steering angle demand. These points are assignedto the steering torque controller, to the reference steering torque algorithm, and to steeringsystem design topics. For the steering torque controller, this leads to the requirement of agood tracking behaviour to reach an adequate steering force level, to follow a reference signalwhich contains informative road feedback and a certain steering force sequence. Additionally,the steering torque controller must have a good disturbance attenuation so that the driver isnot distracted by disturbing road feedback and to follow the reference steering torque evenif disturbances are present. Besides this, the torque controller has to be robust and stable.Objective parameters with target values for the subjective requirements have to be specifiedto enable the design of the appropriate controller. The correlation of subjective ratings toobjective parameters in the automotive field has been analysed in several works [7,8]. Theabove-listed requirements are translated to target values for characteristic parameters. Theresult of this transformation is listed in Table 2 for the tracking behaviour. The complete tablecan be found in [9]. These target values are partly based on experience.

4.2. Controller synthesis

The steering system model contains model inaccuracies as high-order dynamics are not consid-ered, lifetime and nonlinear effects (e.g. friction and backlash) are neglected, and production

Table 2. Target values for steering torque controller.

Reference steering torque Parameters Target value

Step response of 1 Nm Maximum overshoot Omax ≤0.1 NmRising time tris ≤0.1 sSettling time tset ≤0.2 s

Sine, constant frequency Maximum deviation between reference ≤0.2 NmAmplitude: 1 Nm, value and measurementFrequency: 0.2, 1, 3 Hz

Sine, varying frequency Frequency response 1 ± 0.2Amplitude: 1 Nm Amplitude of measured to reference Until 16 HzFrequency: 0.001–40 Hz steering torque

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tolerances are not considered. A controller is needed that works well, although its design isbased on the simplified model. Not only robustness because of model inaccuracies is needed,but also because of disturbing sensor noise. The idea of ‘robust control’ is to find a controllerthat works properly with a set of system models. This set of models contains the nominalmodel and further models with additional uncertainty compared with the nominal model, sothat the behaviour of one of the models in the set is similar to the real system behaviour.A robust control approach is applied here to consider that the model used for the controllerdesign does not contain information about the nonlinear effects (backlash, friction), the dis-turbances (e.g. rack load changes, driver’s torque), measurement noise, parameter changesbecause of production tolerances, operating conditions (e.g. different temperatures), and life-time effects. H∞-control, which is one approach of robust control, is chosen here because thefrequency domain specifications enable good transparency during the controller design andtuning. Another reason for selecting H∞-control is the fact that unstructured uncertainties canbe considered quite well in the approach and most of the model uncertainties described aboveare not parametric but unstructured. The controller for the defined control problem is calcu-lated by using a mathematical optimisation problem and the control objectives are describedin terms of closed-loop weighting functions.

4.2.1. Basic H infinity controller

An S/KS/T approach was chosen with the frequency-dependent weighting functions WS,WKS and WT for the sensitivity function (S), the control power function (KS), and thecomplementary sensitivity function (T). Details on the theory can be found in the techni-cal literature [10,11]. The control scheme is depicted in Figure 7. The weighting functionsare used as design parameters and the final tuning is shown in Figure 8. For WS, low-pass

Figure 7. Control scheme of the S/KS/T approach.

Figure 8. Weightings of the S/KS/T approach.

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behaviour is chosen to achieve good tracking behaviour which is particularly important in thelow-frequency range. WT has high-pass behaviour because sensor noises and model inaccura-cies increase with the frequency. To obtain the controller, the following optimisation problemhas to be solved with Fl being the abbreviation for a lower linear fractional transformation[10] and G representing the plant:

‖Fl(P, K)‖∞!= min (21)

with

⎡⎢⎢⎣

z1

z2

z3

yk

⎤⎥⎥⎦ =

⎡⎢⎢⎣

WS −WSG0 WTG0 WKS

1 −G

⎤⎥⎥⎦

︸︷︷︸P

·[

Ttse ref

u

], (22)

u = K · yk , e = Fl(P, K)w, (23)

�e = (z1, z2, z3)′, w = Ttse ref . (24)

As the resulting S/KS/T controller does not reach the requirements (see Section 4.3), severalextensions are chosen to improve it. This leads to a controller that is named ‘modified robustcontroller’ in this paper.

4.2.2. Modified robust controller

The signal-based approach, where the weights are used to introduce expected and knownfrequency content (see also [10]), is used. Compared with the basic S/KS/T controller, thefollowing changes are chosen:

• instead of using the deviation of reference and measured steering torque, the two signalsare used as separate inputs;

• not only the steering torque but also the rotor velocity ωrot is used as a measurement signal;• to consider nonlinear friction, an unknown external force is modelled at each of the three

masses of the system with the weight Wd . This approach is described in [12,13]. Theknowledge about where the nonlinear friction is acting is used here instead of working witha general uncertainty on the whole steering system transfer function;

• the measurement noises of the steering torque (n1) and of the rotor velocity (n2) are weightedwith Wn1 and Wn2 to give the known frequency content as input to the controller design.

Figure 9 shows the control scheme of the ‘modified robust controller’. The controller iscalculated by solving the following optimisation problem:

‖Fl(P, K)‖∞!= min (25)

with ⎡⎢⎢⎢⎢⎢⎢⎣

z1

z2

z3

ryk1

yk2

⎤⎥⎥⎥⎥⎥⎥⎦ =

⎡⎢⎢⎢⎢⎢⎢⎣

WS −WSG(1)Wd −WSWn1 0 −WSG(1)

0 WTG(1)Wd 0 0 WTG(1)

0 0 0 0 WKS

1 0 0 0 00 G(1)Wd Wn1 0 G(1)

0 G(2)Wd 0 Wn2 G(2)

⎤⎥⎥⎥⎥⎥⎥⎦

︸︷︷︸P

·

⎡⎢⎢⎢⎢⎣

Ttse ref

dn1

n2

u

⎤⎥⎥⎥⎥⎦ , (26)

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Figure 9. Control scheme of the modified robust controller.

u = (K1 K2 K3)︸︷︷︸K

·yk , e = Fl(P, K)w, (27)

�e =⎛⎝z1

z2

z3

⎞⎠ , �yk =

⎛⎝ r

yk1

yk2

⎞⎠ , �w =

⎛⎜⎜⎝

Ttse ref

dn1

n2

⎞⎟⎟⎠ , r = Ttse ref, �d =

⎛⎝d1

d2

d3

⎞⎠ . (28)

Further controllers are designed and implemented for a comparison of the described H∞-approach to other methods:

(1) A classical proportional-integral-derivative (PID) controller is implemented.(2) A cascade controller with a proportional-derivative (PD) controller in the inner loop to

control the rotor velocity and a PI controller to control the steering torque in the outerloop is designed.

(3) A state controller with a state observer, based on the linear quadratic Gaussian (LQG)approach, is used (see also [5,14]).

The controllers used for comparison were chosen because of the literature about steer-ing torque control (e.g. [5,15,16]). The classical controller and the cascade controller weredesigned using the frequency domain loop-shaping procedure. The only possibility to con-sider robustness in this approach is to check the phase margin after designing the controller.The phase margin and time domain behaviour can be changed iteratively to achieve the bestcompromise of the desired values. During the design of a linear quadratic regulator (LQR), itis not possible to consider explicitly robustness requirements. The assumptions of a linear sys-tem and white noise disturbances sometimes differ strongly from the real system conditions.LQR approaches have great robustness properties. However, if the linear quadratic controlleris combined with a Kalman filter, the robustness properties can be astonishing small [17]. Thebehaviour of an LQG compensator can be quite bad if the system model is different from thereal plant. With the loop transfer recovery (LTR) method, robustness can be recovered [10]. Forthe LQG controller analysed in this paper, the LTR procedure has been applied. To considerrobustness already in the controller design, a robust control method, for example, H∞-control,must be used. Another method of robust control is the parameter space approach [18] whichcan be applied to obtain a robust PID controller [19]. Regions in which the controller is stablefor a set of system parameters are in the focus of this approach.

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4.3. Controller validation

To compare and to validate the designed and implemented controllers, numeric simulationsare proceeded. It is checked whether the different controllers reach the specified requirements.To analyse how good a reference steering torque is followed, a step response and a sinusoidalreference torque are simulated. For the analysis of the disturbance behaviour, a disturbance stepbecause of a driver’s input of 20◦ steering angle and another one because of a changing rackforce of 500 N was chosen. To investigate the behaviour at different frequencies, the frequencyresponse was simulated and plotted for the reference and the disturbance behaviour. The usedsimulation model includes nonlinear friction and a driver model. The simulation results of thereference steering torque step response with an amplitude of 1 Nm are shown in Figure 10.The obtained values for rise time tris, maximum overshoot Omax, and settling time tset enablean objective comparison of the different controllers (Table 3). The results obtained with the‘modified robust controller’ and with the state controller are very good as the rise time andthe overshoot are rather small. The requirements are not met with the cascade controller andthe robust controller with the pure S/KS/T approach as they need too much time to arriveat 90% of the step input. Although the PID controller shows a fast response, its behaviour isnot desirable as it oscillates strongly. The step response was not analysed for larger valuesof the reference torque because sudden steps in the steering torque are usually not desired asthey have a negative effect on the steering feel. If larger amplitudes of step responses were

0.8 0.9 1 1.1 1.2 1.3−0.2

0

0.2

0.4

0.6

0.8

1

Time (s)

Ste

erin

g to

rque

(N

m)

Reference torquePID controllerCascade controllerState controllerRobust controller (S/KS/T)Robust controller (modified)

Figure 10. Step response comparison for the different controllers.

Table 3. Results of the 1 Nm reference step.

Requirement Omax (Nm) tris (s) tset (s)

Target value 0.100 0.100 0.200PID controller 0.039 0.016 0.164Cascade controller 0.058 0.202 0.255State controller – 0.044 0.072Robust controller (S/KS/T) 0.094 0.093 0.236Robust controller (modified) 0.032 0.033 0.050

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of interest, simulations with higher torque amplitude should be carried out in a future work.There is always the possibility to use different controller tunings depending on the value ofreference steering torque if necessary. The neglected nonlinearities can have an influence onthe step response behaviour. The controllers are all calculated based on a linear model. If therobustness behaviour is not good a controller can even become instable when simulated inthe model with the nonlinearities or when applied to the real system. Therefore, numeroussimulations were carried out in the system model described in Section 2.3.2 containing thenominal values for the nonlinear effects like backlash and dry friction. This was also donewith parameter variations for single nonlinear effects. All controllers show stable behaviourin this investigation. The nonlinearities, for example, backlash, which are in the transfer pathbetween the electric motor and the steering torque, can have an influence on how good the stepresponse behaviour is. Figure 11 shows the step response of the ‘modified robust controller’.The fastness and preciseness are quite good although applied in the extended model.

In the next step, driving manoeuvres are simulated with a model that includes the drivingdynamics of a vehicle. It is based on a nonlinear two-track model. The elasto-kinematics andthe kinematics of the vehicle are included in characteristic curves. The vehicle is modelledwith three rotational and three translational degrees of freedom. For the tyre, the PacejkaMagic Formula [20] is applied. One of the output signals of the vehicle model is the rackforce. The steering wheel angle is of sine shape with constant amplitude and frequency at aconstant vehicle speed. The reference steering torque is calculated in dependence of vehiclemeasurement values, for example, steering wheel angle, lateral acceleration. The manoeuvreis repeated for different steering angle frequencies. Figure 12 shows the simulation resultsof the different controllers with a steering angle frequency of 0.5 Hz and a vehicle speed of80 kph. The simulation results of all controllers show a bump at around 1.6 and 2.6 s. Thecause for this deviation is the friction in the system, which has a big impact on the resultwhen the motor is changing its position. The position of the bump can be found when the signchange occurs in the rotor velocity. The best results are obtained with the ‘modified robustcontroller’ and with the state controller as these two approaches consider the friction in thecontroller design.

0.8 0.9 1 1.1 1.2 1.3

0

0.2

0.4

0.6

0.8

1

1.2

Time (s)

Ste

erin

g to

rque

(N

m)

Reference torqueRobust controller (modified)

Figure 11. Step response using the nonlinear model.

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1 1.5 2 2.5 3

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Time (s)

Ste

erin

g to

rque

(N

m)

Reference torquePID controllerCascade controllerState controllerRobust controller (S/KS/T)Robust controller (modified)

Figure 12. Driving manoeuvre simulation with the different controllers.

Finally, the ‘modified robust controller’ is checked in a development vehicle. The trackingbehaviour is analysed by increasing the reference steering torque manually while the driveris holding the steering torque. The comparison of the reference and measured steering torqueis given in Figure 13. The disturbance behaviour is examined by moving the steering wheelwhen a reference steering torque of 0 Nm is chosen. A third test can give information aboutboth the tracking and the disturbance behaviour. At this test, a reference steering torque isgiven and the driver is moving the steering wheel. The controller has to follow the referencesteering torque and at the same time it has to attenuate the external disturbances. The controllerapproach is promising as the results are already good after a short time of fine-tuning.

5 10 15 20 25 30 35 40−4

−3

−2

−1

0

1

2

3

4

Time (s)

Ste

erin

g to

rque

(N

m)

MeasurementReference

Figure 13. Tracking behaviour in the vehicle.

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5. Summary and outlook

Today EPS is used more and more in passenger cars instead of hydraulic power steering. Thesteering assistance of the EPS has to be designed to satisfy the driver’s requirements. Here, acontroller that follows a reference steering torque is developed for a rack-assisted EPS. First,different models of the system with simplifying assumptions for the electrical subsystem arederived. Sometimes steering torque discontinuities, which are unwanted variations of the steer-ing torque, are a problem. The root cause of this phenomenon is analysed using a detailedmodel, which contains a new model of the ball screw drive. For the examined steering gear, thenonlinear friction in the steering system is identified as the root cause of the discontinuities. Inthe next step, the requirements on the steering torque controller are specified by defining targetvalues for characteristic control engineering parameters. An H∞-controller is developed andimplemented as a steering torque controller. The controller’s task is to follow an arbitrary refer-ence steering torque quickly and accurately even if disturbances are present, for example, loadforces from the road. Additionally, the controller shall compensate the friction to eliminate theeffect of steering torque discontinuities. When designing the controller, frequency-dependentweighting functions are used. This makes it possible to establish different priorities of conflict-ing requirements in different frequency ranges. Other controllers (a PID controller, a cascadecontroller, and a linear quadratic compensator) are designed and implemented. Numeric simu-lations are performed and the different controllers are compared referring to the defined targetvalues. The modified robust controller shows good behaviour in the numeric simulations, andit was also tested in a development vehicle with good results.

In the future, more fine-tuning of the H∞-controller should be done in the car to optimiseits behaviour. Besides this, a comparable analysis of all the controllers in the vehicle shouldbe carried out. The response of the controlled system to a J-turn manoeuvre should be testedin future work because if it works out well in this case then it should also be good in othermanoeuvres. For the specification of the controller’s requirements, more objective criteria forsubjective ratings are necessary. The μ-synthesis is a further development of the H∞-approachwhich makes it possible to consider structured model uncertainties, like parameter variations.It is worth knowing if the μ-synthesis gives additional improvement concerning the whole con-troller performance (including the friction compensation). One possible investigation would beto consider the friction in the steering system as parametric uncertainty of the linear dampingparameters when designing a controller according to the μ-synthesis approach. The ‘modifiedrobust controller’ is of high order. It works out fine in the testing vehicle with developmentequipment, but for usage in a series car it could be a problem. The computing time and theeffort when tuning the controller should be decreased by reducing the order of the controller.Therefore, it should be analysed if the controller performance still reaches the requirementsafter applying an order reduction method, see also [10].

References

[1] G. Karch and S. Grüner, Mechatronische Lenksysteme, Automatisierungstechnik 55(6) (2007), pp. 281–289.[2] B. Buschardt, Synthetische Lenkmomente, Ph.D. thesis, Technische Universität Berlin, 2003.[3] H. Ulbrich, Maschinendynamik, Teubner, Stuttgart, 1996.[4] M. von Groll, Modifizierung von Nutz- und Störinformationen am Lenkrad durch elektromechanische

Lenksysteme, Fortschritt-Berichte VDI, Reihe 12, Band 630, Ph.D. thesis, Universität Duisburg-Essen, 2006.[5] H. Henrichfreise and J. Jusseit, Optimale Regelung einer elektromechanischen Servolenkung, 5. VDI Mecha-

tronik Tagung, Fulda, 2003.[6] A.T. Wohnhaas, Simulation von Kraftfahrzeug-Lenkungen unter besonderer Berücksichtigung von Reibung und

Spiel, Ph.D. thesis, Universität Stuttgart, 1994.[7] M. Harrer, Characterisation of steering feel, Ph.D. thesis, University of Bath, 2007.

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[8] M. Decker, Zur Beurteilung der Querdynamik bei Personenkraftwagen, Ph.D. thesis, Technische UniversitätMünchen, 2009.

[9] C. Dannöhl, Entwicklung eines Algorithmus zur Servokraftregelung einer elektromechanischen Lenkung,Fortschritt-Berichte VDI, Reihe 12, Band 715, Ph.D. thesis, Technische Universität München, 2010.

[10] S. Skogestad and I. Postlethwaite, Multivariable Feedback Control, Wiley, Chichester, 2004.[11] J. Raisch, Mehrgrößenregelung im Frequenzbereich, Oldenbourg, München, 1994.[12] G.-W. van der Linden and P.F. Lambrechts, H∞ Control of an experimental inverted pendulum with dry friction,

IEEE Control Syst. Mag. 13 (1993), pp. 44–50.[13] J. Eich, B. Sattler, and A. Schönhoff, Robuste Regelung von mechanisch-elektrischen Antriebssystemen, VDI-

Tagung Antriebssysteme, Wiesloch, 1997.[14] H. Henrichfreise, Prototyping of a LQG compensator for a compliant positioning system with friction, 1.

Workshop TRANSMECHATRONIK – Entwicklung und Transfer von Entwicklungssystemen der Mechatronik,Heinz Nixdorf Institut, Universität Paderborn, Paderborn, 1997.

[15] J. S. Chen, Control of Electric Power Steering Systems, SAE Paper 981116, Steering and Suspension TechnologySymposium, Detroit, MI, 1998.

[16] J.-H. Kim and J.-B. Song, Control logic for an electric power steering system using assist motor, Mechatronics12 (2002), pp. 447–459.

[17] J.C. Doyle and G. Stein, Robustness with observers, IEEE Trans. Automat. Control 24(4) (1979), pp. 607–611.[18] J. Ackermann, Robust Control: The Parameter Space Approach, Springer Verlag, London, 2002.[19] J. Ackermann and D. Kaesbauer, Design of Robust PID Controllers, Proceedings of European Control

Conference, J.L. Martins de Carvalho, Porto, 2001.[20] H.B. Pacejka and I.J.M. Besselink, Magic formula tyre model with transient properties, Suppl. Veh. Syst. Dyn.

27 (1997), pp. 234–249.

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Documents

H ∞ -control of a rack-assisted electric power steering system