Int. J. Electric and Hybrid Vehicles, Vol. 5, No. 2, 2013 143

Optimisation of energy in electric power-assistedsteering systems

Kumeresan A. Danapalasingam

Faculty of Electrical Engineering,Department of Control and Mechatronics Engineering,Universiti Teknologi Malaysia,UTM Skudai, 81310 Johor, MalaysiaE-mail: kumeresan@fke.utm.my

Abstract: Electric Power-assisted Steering (EPS) is a control system wherean electric motor is used to provide assistance in vehicle steering. In thiswork controllers are designed for column-type EPS systems to enable energyoptimisation. Two EPS systems are considered, namely brushed DC motorequippedEPSandbrushlessDC(BLDC)motorequippedEPS.Usingmathematicalmodels of EPS controllers are developed based on nonlinear adaptive regulationmethod to generate driver torque. Proportional-Integral-Derivative (PID) control isthen applied to produce assistance torque in accordance to desired energy saving.Simulation results using Matlab show the trade-off between driver’s comfort andenergy consumption. The control paradigm introduced here fits appropriately inelectric vehicles (EVs) where electrical energy is scarce.

Keywords: EPS; electric power-assisted steering; brushless DC motor; energyoptimisation; electric vehicle.

Reference to this paper should be made as follows: Danapalasingam, K.A.(2013) ‘Optimisation of energy in electric power-assisted steering systems’,Int. J. Electric and Hybrid Vehicles, Vol. 5, No. 2, pp.143–154.

Biographical notes: Kumeresan A. Danapalasingam received his BE (Electrical– Mechatronics) and ME (Electrical – Mechatronics and Automatic Control)Degrees from Universiti Teknologi Malaysia (UTM) in 2003 and 2006respectively. In 2010, he was awarded a PhD (Electrical and Electronic)by Aalborg University, Denmark. At present, he is a Senior Lecturerat the Department of Control and Mechatronics Engineering, Faculty ofElectrical Engineering, UTM. His research interests include control engineering,autonomous helicopter, artificial intelligence and hybrid electric vehicle.

1 Introduction

In a vehicle EPS assists a human driver by measuring torque exerted by the driver andproviding additional torque by means of an electric motor. Conventionally an EPS systemrelies on vehicle velocity and driver torque measurements to determine target motor currentlevels from assist characteristic (Zhang et al., 2009) or boost (Chitu et al., 2011) curve.

Copyright © 2013 Inderscience Enterprises Ltd.

144 K.A. Danapalasingam

The assist characteristic curve is a set of graphs of motor current vs. driver torque andvehicle velocity that is generated experimentally to produce desired assistance torque andthus predetermined steering comfort (Mehrabi et al., 2011). While in internal combustionengine vehicles provision of maximum steering comfort to drivers is possible, carefulconsiderations are needed in EVs. Given a limited power supply in an EV a higher torqueassistance would result in a faster battery drain. Therefore the use of the assist characteristiccurve to obtain reference assistance torque levels is not applicable in EVs and a betterapproach enabling energy optimisation is justified.

While the use of a traditional brushed dc motor in an EPS requires a low cost and a simplecontrol some disadvantages include high maintenance of the brushes, low overall powerdensity and Electromagnetic Interference (EMI) problems associated with commutatorarcing (Torrey and Kokernak, 2002). These drawbacks motivate some manufacturers toapply a BLDC in an EPS system. In this paper both of the EPS systems, namely the onethat is equipped with a brushed DC motor and the one with a BLDC motor, are consideredfor electrical energy optimisation.

Some previous works on EPS will now be reviewed. In Mehrabi et al. (2011) a modifiedLinear Quadratic Gaussian controller is shown to be able to track the characteristic curveand attenuate external disturbances for a column-type EPS system. From the simulationresults the controller performs well despite the inclusion of nonlinear rotational frictionterms even though explicit expressions of the friction are not provided. Chitu et al. (2011)applied a Linear Quadratic Regulator to derive an optimal controller for an EPS system. Bothsimulation and dSPACE ControlDesk real-time application utilising the assist characteristiccurve show stability in frequency, robustness and closed-loop stable step responses duringparameter variation. In Zang and Chen (2011) a Fuzzy PID control strategy is simulatedfor assist motor current tracking. The simulation model that includes a simple road surfacedisturbance demonstrates the effectiveness of the proposed controller. While brushed DCmotors are considered in the above works an EPS with a BLDC actuator is studied in Zhuand Patankar (2004). Zhu et al. propose an adaptive control method to achieve torque controlby approximating motor electrical dynamics with back EMF compensation error correction.In another related work, Hu et al. (2004) constructed a test bed and developed a controllerfor a BLDC EPS system and reported an acceptable system performance.

Out of the four control objectives of an EPS system (cf. Liu et al., 2007) thebasic function of an EPS i.e., assistance torque control is considered here. To achieveenergy optimisation the author develops mathematical models of two column-type EPSsystems with a brushed DC and a BLDC assist motors. Nonlinear rotational friction andLuGre dynamic tyre friction are included in the mathematical models for accurate EPSrepresentations. The models are then used to design controllers based on nonlinear adaptiveregulation for driver torque generation to track a reference wheel angle trajectory. Toenable an option to select torque assist level and consequently amount of energy savingdesired eco factor E is introduced in later sections of this paper. Target assistance torqueis then achieved using PID control of motor current with Pulse Width Modulation (PWM)implementation.

The paper is organised as follows; in Section 2 the EPS mathematical model is discussed.A brief explanation on controller designs are provided in Section 3 after which simulationresults and conclusion are given in Sections 4 and 5 respectively.

Optimisation of energy in electric power-assisted steering systems 145

2 Mathematical model

The following are the mathematical models of two column-type EPS systems (see Figure 1).Since a brushed DC motor equipped EPS system and a BLDC motor equipped EPS systemare considered, both motor models are provided below.Torque sensor model:

Tc = Ks

(θs −

xr

Rs

). (1)

Steering column model:

Jsθs = Td − Tc −Bsθs − T sf . (2)

DC assist motor model:

Tm = Kaia(Jm +

JGG2

)θm = Tm +

Tc

G−(Bm +

BG

G2

)θm − Tm

f

Lbia = V −Rbia −Kbθm.

BLDC assist motor model:

Tm =np

θm(Eaia + Ebib + Ecic)

Vn =1

3(Va + Vb + Vc)−

1

3(Ea + Eb + Ec)

Lbl ia = Va − Vn −Rblia − Ea

Lbl ib = Vb − Vn −Rblib − Eb

Lbl ic = Vc − Vn −Rblic − Ec(Jm +

JGG2

)θm = Tm +

Tc

G−(Bm +

BG

G2

)θm − Tm

f .

Rack and pinion model:

Ta = GTm

Mrxr =Ta + Tc

Rs− FTR −Brxr −Krxr, (3)

where

θs: Steering wheel angleθm: Assist motor armature shaft angleTd: Driver torqueTc: Steering column torque sensor measurementTm: Assist motor torqueTa: Assistance torque

146 K.A. Danapalasingam

T sf : Steering column friction

Tmf : Assist motor and reduction gear friction

Bs: Steering column viscous damping coefficientBm: Assist motor viscous damping coefficientBG: Reduction gear viscous damping coefficientBr: Pinion and rack viscous damping coefficientJs: Steering column moment of inertiaJm: Assist motor moment of inertiaJG: Reduction gear moment of inertiaKs: Steering column stiffnessKa: Assist motor torque coefficientKb: Assist motor back emf coefficientKr: Rack equivalent spring constantV : DC motor voltageia: DC motor currentRb: DC motor resistanceLb: DC motor inductanceVa, Vb, Vc: BLDC motor three-phase voltageia, ib, ic: BLDC motor three-phase currentEa, Eb, Ec: BLDC motor three-phase trapezoidal back EMFVn: BLDC motor neutral voltagenp: BLDC motor number of polesRbl: BLDC motor resistanceLbl: BLDC motor inductanceG: Reduction gear ratioxr: Horizontal rack displacementRs: Pinion radiusMr: Pinion, rack and wheel equivalent massFTR: Dynamic tyre friction.

The nonlinear rotational friction is given by

T if = (αi

0 + αi1e

−αi2|θi|)sgn1(θi)

+(αi3 + αi

4e−αi

5|θi|)sgn2(θi), i = s,m,

where

sgn1(θi) =

1 θi ≥ 0

0 θi < 0,

sgn2(θi) =

0 θi ≥ 0

−1 θi < 0, i = s,m

and αij ∈ R, αi

j > 0, j = 0, . . . , 5 and αi0 = αi

3, αi1 = αi

4, αi2 = αi

5, i = s,m in general(Kara and Eker, 2004).

The LuGre dynamic tyre friction can be expressed as

τa = Ms +Msa,

FTR =τar,

Optimisation of energy in electric power-assisted steering systems 147

where r is the steering arm length and Ms and Msa are sticking and self-alignment torquesrespectively. For complete expressions of Ms and Msa please refer to Tordesillas et al.(2011) and Velenis et al. (2002).

Figure 1 A column-type EPS

3 Controller design

The main objective of this work is to demonstrate the option made available to choose adesired steering comfort. Since a better steering feel is provided by a higher torque assista higher energy consumption would result and the reverse is also true. The amount ofassistance torque Ta to be produced is determined by the eco factor E and driver torqueTd. Therefore in this section the controller design is done in two stages. In the first part acontroller based on nonlinear adaptive regulation is developed to generate driver torque Td

to track a reference wheel angle trajectory. Next a reference assistance torque T ∗a is obtained

using the generated Td and E. In the second stage DC and BLDC motor controls are carriedout using PID and PWM application to track T ∗

a .

3.1 Driver torque generation

Consider equation (3) with Ta = 0,

Mrxr =Tc

Rs− FTR −Brxr −Krxr (4)

and a reference wheel angle trajectory θ∗w given by

x∗r =

N∑i=1

Ai cos(Ωit+ φi), (5)

θ∗w =x∗r

r,

148 K.A. Danapalasingam

with a fixed N , unknown amplitudes Ai, phases φi and frequencies Ωi. Then fromequations (4) and (5) a steady state steering column torque T ∗

c needed to track θ∗w is given by

T ∗c = (FTR +Brxr +Krxr +Mrx

∗r)Rs. (6)

From equations (1) and (6) a reference steering wheel angle is obtained as follows.

θ∗s =T ∗c

Ks+

xr

Rs. (7)

By solving for a steady state driver torque T ∗d from equation (2) using equation (7) we have

T ∗d = Tc +Bsθs + T s

f + Jsθ∗s . (8)

Note that in steady state T ∗d will produce θ∗s , T ∗

c and consequently x∗r .

Using the nonlinear adaptive regulation method expressions (7) and (8) yield controllersof the forms

e1 = x∗r − xr,

u1st = k12(e1 + k11e1),

g1st = Gu1st,

ξ1 = (F +GΨ1)ξ1 + g1st,

Tuc = Ψ1ξ1 + u1

st, k11, k12 > 0 (9)

and

θus =Tuc

Ks+

xr

Rs,

e2 = θus − θs,

u2st = k22(e2 + k21e2),

g2st = Gu2st,

ξ2 = (F +GΨ2)ξ2 + g2st,

Td = Ψ2ξ2 + u2st, k21, k

22 > 0 (10)

whose performances are subject to the tuning of k11 , k12 , k21 and k22 . For definitions of thecontrol parameters and detailed explanation of control designs (9) and (10) readers areadvised to refer to Isidori et al. (2003a, 2003b). The above controllers can be shown to beable to track x∗

r in a globally asymptotically and locally exponentially stable manner.

3.2 Assist motor control

The controller designed in the previous section will generate driver torque Td for referencewheel angle θ∗w tracking. From the steering column torque measurement Tc desired assistmotor torque T ∗

m is obtained as follows.

T ∗a = ETc,

T ∗m =

T ∗a

G. (11)

Optimisation of energy in electric power-assisted steering systems 149

Given the value in equation (11) reference assist motor current i∗a = T ∗m/Ka is then

computed for DC and BLDC motor controls as depicted in Figure 2. A PID controller istuned to achieve desired accuracy in i∗a tracking of the assist motor that is powered by PWM.

Note that E = 1 would mean that equivalent steering effort has to be put by both ahuman driver and the assist motor. Even though that results in a comfortable steering itleads to a higher battery usage as compared to setting E = 0.1.

Figure 2 Assist motor control

4 Simulation results

The mathematical model of a column-type EPS system given in Section 2 together with thecontrol designs in Section 3 are simulated using Matlab. Parameters of the EPS model isadopted from Zang and Chen (2011), Tordesillas et al. (2011) and Velenis et al. (2002).

Recall that in Section 3.1 a controller is designed to generate driver torque Td forreference wheel angle θ∗w tracking. From Figure 3 it could be seen that controllers (9) and(10) performs well since a very close tracking is achieved.

Figure 3 Reference wheel angle θ∗w and actual wheel angle θw (see online version for colours)

In Figures 4 and 5 steering column torque Tc and assistance torque Ta are plotted forEPS systems with brushed and brushless DC assist motors respectively, with E = 0.6.As the values of Ta are always approximately 60% of Tc as desired the effectiveness of thePID controller with PWM from Section 3.2 is demonstrated.

150 K.A. Danapalasingam

Figure 4 Steering column torque Tc and assistance torque Ta of EPS with brushed DC motor(E = 0.6) (see online version for colours)

Figure 5 Steering column torque Tc and assistance torque Ta of EPS with BLDC motor (E = 0.6)(see online version for colours)

Figures 6 and 7 show steering column torque Tc of EPS systems with brushed and brushlessDC assist motors respectively, for different values of E. Figures 8 and 9 show assist motortorque Tm of EPS systems with brushed and brushless DC assist motors respectively, fordifferent values of E. Note the inverse relationship between Tc and Tm. As a higher Tm isdesired (by setting a higher value of E) a lower Tc is required for steering.

Optimisation of energy in electric power-assisted steering systems 151

Figure 6 Steering column torque Tc of EPS with brushed DC motor (see online version for colours)

Figure 7 Steering column torque Tc of EPS with BLDC motor (see online version for colours)

As expected brushed and brushless DC assist motor currents ia is proportional to Tm inFigures 10 and 11. From Figures 6, 10 and 7, 11 it is clearly shown that a lighter steeringfeel (a higher E) requires a higher current draw from a car battery. However if users aregiven the option to choose E accordingly a better management between steering comfortand power usage could be achieved for energy optimisation.

152 K.A. Danapalasingam

Figure 8 Assist motor torque Tm of EPS with brushed DC motor (see online version for colours)

Figure 9 Assist motor torque Tm of EPS with BLDC motor (see online version for colours)

5 Conclusion

Energy optimisation of EPS in EVs has been the primary focus of this paper. In this workthe conventional method of determining target assist motor current using a lookup table iscompletely eliminated to give way to an approach enabling battery energy saving. Ratherthat using fixed values of reference assist motor current to generate required assistancetorque here the author enabled an option to set the level of steering comfort as desired.

Optimisation of energy in electric power-assisted steering systems 153

This flexibility is made available by means of the eco factor E introduced in this paper.As a consequence drivers could choose to save battery energy given the inverse relationshipbetween steering comfort and power consumed by the assist motor. Simulation results verifythe feasibility of the proposed control methods in achieving energy optimisation in two EPSsystems, namely brushed DC motor equipped EPS and BLDC motor equipped EPS.

Figure 10 Assist motor current ia of EPS with brushed DC motor (see online version for colours)

Figure 11 Assist motor current ia of EPS with BLDC motor (see online version for colours)

154 K.A. Danapalasingam

Acknowledgements

This work was done while the author was undertaking the Local Expert Attachment Program(LEAP) in the Research Department, Perusahaan Otomobil Nasional Bhd. (PROTON),Selangor, Malaysia.

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