torque control of electric power steering systems based on improved active … · 2020. 4. 29. ·...

Research ArticleTorque Control of Electric Power Steering Systems Based onImproved Active Disturbance Rejection Control

Shaodan Na,1 Zhipeng Li ,1 Feng Qiu,2 and Chao Zhang1

1School of Traffic and Transportation, Northeast Forestry University, Harbin 150040, China2College of Mechanical and Electrical Engineering, Northeast Forestry University, Harbin 150040, China

Correspondence should be addressed to Zhipeng Li; [email protected]

Received 15 January 2020; Accepted 25 March 2020; Published 29 April 2020

Academic Editor: Salvatore Strano

Copyright © 2020 Shaodan Na et al. *is is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In the electric power steering (EPS) system, low-frequency disturbances such as road resistance, irregular mechanical friction, andchanging motor parameters can cause steering wheel torque fluctuation and discontinuity. In order to improve the steering wheeltorque smoothness, an improved torque control method of an EPS motor is proposed in the paper. A target torque algorithm isestablished, which is related to steering process parameters such as steering wheel angle and angular speed. *en, a target torqueclosed-loop control strategy based on the improved ADRC is designed to estimate and compensate the internal and externaldisturbance of the system, so as to reduce the impact of the disturbance on the steering torque.*e simulation results show that theresponsiveness and anti-interference ability of the improved ADRC is better than that of the conventional ADRC and PI. *evehicle experiment shows that the proposed control method has good motor current stability, steering torque smoothness, andflexibility when there is low-frequency disturbance.

1. Introduction

An electric power steering (EPS) system has the advantagesof safety, energy saving, and comfortable steering [1], whichhas gradually replaced mechanical and hydraulic powersystems to achieve assist power steering function in thesteering system [2–4]. However, the EPS system also causessome issues. For example, the application of motor anddeceleration mechanism in EPS inevitably increases theinertia of the steering system and introduces unknownadditional friction. *e EPS controller sampling noise andimprecision of the control model will also cause disturbance.*ese issues can cause discontinuity and fluctuation forsteering torque, especially when the steering resistancetorque changes, greatly caused by unknown road bumps.*e frequency of the above disturbance is very low, evenclose to the range of input signal. It is difficult to filter out bymeans of a filter. *erefore, it is important to improve theresponsiveness and keep stability when unknown distur-bance occurs.

In recent years, many control strategies are proposed toimprove the steering feel and torque responsiveness for EPScontrol [5–7]. In [8], the PID strategy is used to establish atorque closed loop, in which the differential operationproduces high-frequency noise that drowns out the differ-ential signals.*erefore, designers tend to use the PI strategyin practical applications. However, when the unknown loaddisturbance varies dramatically, the PI strategy quickly in-creases the power gain to shorten the response time, whicheasily arouses the system fluctuation, resulting in unevensteering torque.

With the development of modern control theory tech-nology, many intelligent control methods have been appliedin EPS torque control. In [9–11], the genetic algorithm andparticle swarm optimization are used to find the optimalgain parameters. In [12], the ant colony optimization andparticle swarm optimization are proposed to find the op-timal PID parameters, which can improve the responsive-ness and stability. However, these algorithms all belong tothe random algorithm, which has some problems such as

HindawiMathematical Problems in EngineeringVolume 2020, Article ID 6509607, 13 pageshttps://doi.org/10.1155/2020/6509607

mailto:[email protected]://orcid.org/0000-0001-5964-1505https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2020/6509607

local optimal solution, premature convergence, and largecomputation.

Some scholars adopt the adaptive sliding mode observerto track the torque variation to ensure robustness in thesteering system, and the high-frequency noise of differentialsignal is eliminated by using a Kalman filter [13–15]. *esliding mode control is a bang-bang control, which willcause torque fluctuation and affect the smoothness. In ad-dition, the calculation of the Kalman filter is relatively large,so it is difficult to be applied in practice.

Many other scholars have applied the H-∞ theory totorque control. In [16–18], the H-∞ controller of thesteering system model is established. In [19], H-2 and H-∞are combined to design the observer to find the optimalsolution. *e key of H-∞ is to find out the frequency rangeof different errors in the system model and to determine theindex of the optimal solution. However, the index values arealways based on design experiences, and the control accu-racy is affected by the system model which varies by vehicletype.*erefore, the algorithm is not desirable in engineeringapplication.

*e ctive disturbance rejection control (ADRC) strategywas proposed in the late 1990s [20], which not only absorbedthe modern control theory of “system internal mechanismdescription” but is also based on the control strategy of“eliminating error by error.” *e core of ADRC is used tounify the internal and external disturbance of the system intoa total disturbance, which is estimated by an extended stateobserver (ESO) and compensated by closed-loop controlcontroller [21]. *en, the system can have good respon-siveness and anti-interference ability in the unknown dis-turbance environment.

Due to the nonlinearity and unpredictability of distur-bance, the nonlinear extended state observer (NESO) is usedto ensure the fast response. However, the calculation ofNESO is very large, which is not suitable for engineeringapplication. *e calculation of the linear extended stateobserver (LESO) is small, but the responsiveness of LESO ispoor. *e control system needs to consider both respon-siveness and calculation.

*e torque control accuracy of the DC permanentmagnet motor is studied under low-frequency disturbancein this paper.*emotor structure is simple, which contains asmall amount of the unknown parameters, resulting in highaccuracy for disturbance estimation. In addition, with theestimated frequency of disturbance becoming low, the es-timated burden of ESO becomes small. *erefore, theproposed ADRC strategy has good control effect on low-frequency disturbance.

*is paper proposes a torque control method based onimproved ADRC. *e system disturbance is estimated andcompensated by a parallel linear extended state observer (P-LESO) instead of the conventional ESO to ensure the goodresponsiveness and avoid excessive calculation.

*e rest of the paper is organized as follows. In Section 2,the mathematical models of the steering system and targetsteering torque algorithm are given. In Section 3, thetracking differentiators (TD), P-LESO, and feedback closed-loop control equation are applied to the steering torque

control, and simulation verification is performed. In Section4, the vehicle experiment verifies the effectiveness of thealgorithm is presented. Finally, Section 5 provides somegood conclusions.

2. Dynamics Analysis of the EPS System

2.1. Simplified Model of the EPS System. Figure 1 shows asimplified model of the EPS system, which includes two endsof the torque bar connected with the steering wheel and thesteering column, respectively. *e torque sensor installed onthe torque bar detects a shape variable to provide the signalfor the ECU controller, which can control the motor tosupply assist torque. *e motor assist torque is amplified bythe deceleration mechanism to overcome the internal fric-tion and road resistance torque [22].

Torque equation for the upper end of the torque bar canbe expressed as

Tsw − Ts � cswω + Jswasw, (1)

where Tsw is the steering wheel torque; Ts is the torque of thetorque bar, which is measured by the torque sensor; csw is thesteering wheel damping; ω is the angular speed of thesteering wheel; Jsw is the rotary inertia of the steering wheel;and asw is the steering wheel angular acceleration.

In fact, the moment of inertia Jsw and the steering wheeldamping csw are too small to be considered. *erefore,equation (1) can be simplified as

Ts ≈ Tsw. (2)

Torque equation for the lower end of the torque bar canbe expressed as

GTe + Ts � Jpap + cpωp + Tr, (3)

where G is the reduction ratio of the deceleration mecha-nism; Te is the motor electromagnetic torque; Jp is the rotaryinertia of the steering column; ap is the angular accelerationof the steering column; cp is the steering column damping;ωp is the angular speed of the steering column; and Tr is theroad resistance torque. In addition, the motor electromag-netic torque can be expressed as follows:

Te � Km · i, (4)

where Km is the motor torque constant and i is the motorcurrent.

As the motor is rigidly connected with the decelerationmechanism, the ωm is expressed as the angular speed of themotor and then ωp can be written as

ωm � Gωp. (5)

*erefore, by means of equations (2) to (5), the equationof steering wheel torque can be expressed as

Tsw � −GKmi + Tr + Jpap + cpωmG

. (6)

It can be seen that the steering wheel torque is greatlyaffected by the road disturbance and system damping. *eexternal disturbance can be expressed as a function a(t).

2 Mathematical Problems in Engineering

To ensure smooth and stable steering torque, the influence ofother interference terms needs to be compensated by con-trolling the motor current.

*e simplified mathematical model of the motor can beexpressed as

u � Ri + pLmi + Keωm, (7)

where u is the motor voltage; R is the equivalent resistance ofthe motor armature; p is the differential operator; Lm is themotor inductance; and Ke is the back-EMF coefficient of themotor.

*e main parameters in the above equation are shown inTable 1.

2.2. Target Steering Torque Algorithm. In order to avoid theinfluence of nonmajor factors in the algorithm, the targettorque control algorithm used in this study is establishedunder the following conditions: the vehicle steering systemstructure is rigidly connected without force deformation inthe steering mechanism of the vehicle; the tire side-slip angleduring steering is ignored.

Generally, the requirement of the vehicle to the steeringwheel torque varies with the driving condition. When thevehicle is driving at low speed or stationary, the steeringtorque should be flexible and light. As the vehicle speedincreases, the steering torque should be appropriately in-creased to ensure that the vehicle can resist the interferenceof lateral forces when driving in a straight line. In the case ofconstant vehicle speed, as the steering wheel angle increases,the steering torque should increase to ensure the steeringfeel; when the angular speed of the steering wheel is large, itmeans that the direction of the vehicle is changing rapidly,and the steering torque should be appropriately increased toensure the stability of the vehicle operation. *erefore, thetarget steering torque control algorithm can be expressed as

T∗sw � Kθ · Kv · v + 1( · θ − θd( , ω

*erefore, a variable α can be defined as a first-order inertiallink of θ, which can be expressed as

α(s) �θ(s)

Ts + 1. (10)

When T is small, it can be seen from the nature ofLaplace that α can approximate the input signal θ lagging Tin the time domain. *e approximation equation can beexpressed as

ωc(t) ≈1T

(θ(t) − θ(t − T)). (11)

However, due to the disturbance of sensor measurementnoise and controller sampling noise in practical applications,the angle signal θ is actually composed of the real anglesignal θa and the disturbance angle signal θn. *erefore,when the initial state of the system is 0, the convolutionproperty of Laplace shows that α can be expressed as

α(t) � ∞

0e

(1/T)(t− τ) θa(τ) + θn(τ)( dτ

� ∞

0e

(1/T)(t− τ)θa(τ)dτ + ∞

0e

(1/T)(t− τ)θn(τ)dτ.

(12)

In the second term of equation (12), the noise θn(τ) ishigh-frequency sampling noise, and the average value is 0.*us, α can be rewritten as

α(t) ≈ ∞

0e

(1/T)(t− τ)θa(τ)dτ ≈ θa(t − T). (13)

*e angular speed equation can be expressed as

ωc(t) �1T

θa(t) − θa(t − T) + θn(t)(

� _θa(t) +1Tθn(t).

(14)

As can be seen from equation (14), the smaller the timeconstant T, the larger the noise amplification. In practicalapplications, the system step size is usually less than 1ms,and the noise amplification is serious, which causes the realsignal to be drowned.

In this paper, in order to reduce the noise amplification,TD is used to calculate the angular speed as

ω(t) ≈θ t − τ1( − θ t − τ2(

τ2 − τ1, 0< τ1 < τ2, (15)

where ω is the approximate angular speed of the steeringwheel calculated by TD and τ1 and τ2 are two adjacentmoments. *e transfer function of equation (15) is

ω(s) �1

τ2 − τ1

θ(s)τ1s + 1

−θ(s)

τ2s + 1

�s

τ1τ2s2 + τ1 + τ2( s + 1θ(s).

(16)

Both τ1 and τ2 can be approximately recorded as τ whenthe time point is very close. *en, equation (16) can beexpressed as

ω(s) �r2s

s2 + 2rs + r2θ(s), (17)

where r is the reciprocal of τ. After a simple equation change,equation (17) can be rewritten as

θ1(s) �ω(s)

s,

θ1(s) �r2

s2 + 2rs + r2θ(s),

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(18)

where θ1 is the transition process of θ. It can be seen fromthe nature of Laplace, θ1 is the output of θ through thesecond-order linear system, and r is the damping parameterof the equation. When r> 1, θ1 follows θwithout overshoot.*e state space realization of equation (18) can be expressedas

_θ1(t) � ω(t),

_ω(t) � −r2 θ1(t) − θ(t)( − 2rω(t),

y � ω(t).

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(19)

TD

Torque angle sensor

1/b b

Motor

LESO

Target torque

algorithm

Steeringcolumn

Road resistance

– –++

+

–+

–

+

θ1i0 u0

z2

u

i

z1

ω

θ

T∗sw

Tsw

Tr

Tsw

eT eiKt

TeTa Km

Kpa (t)

a (t)

Figure 2: Block diagram of EPS motor control based on ADRC.


Equation (19) is the general form of TD in the timedomain, which has these characteristics:

(1) TD arranges θ1 as the transition process of θ to avoidthe step impact caused by abrupt θ changes.

(2) *e parameter r affects the tracking speed of thetransition process.*e parameter r is increasing withthe demand of the control accuracy of the systembecoming high.

(3) TD effectively reduces the influence of noise in thecalculation of differential signal and improves theaccuracy of differential signal.

Figure 3 shows the output of TD with different pa-rameters in step response.

From Figure 3, when the parameter r increases, θ1 cantrack θ faster and the approximate differentiation is moreaccurate. In this paper, r� 2500.

To simulate the practical applications, add a white noisewhose amplitude is 1% of the actual signal to the inputsignal. *e system step size is set to 1ms. Figure 4 shows theoutput of the TD in the step response and the sinusoidalresponse.

From Figure 4, although the noise is only accounts for1% of the signal, the differential error of conventional dif-ferentiators is too large to be applied. *e differential errorusing TD is 0.3% of that of the conventional differentialmethod. *erefore, a more accurate signal of steering wheelangular speed can be obtained by TD. Furthermore, a moreaccurate target torque algorithm can be obtained.

3.2. Improved ADRC Strategy Design and Stability Analysis.*e basis of torque closed loop is to complete the currentclosed loop. *e state equation of the motor is establishedas

pi � −R0

Lmi −

Rn

Lmi −

Ke

Lmωm +

1Lm

u,

y � i,

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(20)

where R0 is the static resistance of the motor; Rn is thechange in resistance when the motor is running; and u and iare the input and output of the equation of the state,respectively.

For the motor, the disturbance of the EPS system such assystem friction and road resistance is finally reflected in themotor speed, which is difficult to accurately judge. Mean-while, the motor parameters, such as motor resistance,change with the running state of the motor. *erefore, theinput current of the motor is not only related to the motorvoltage but also affected by internal and external distur-bance. It needs three steps to realize the core idea of ADRC:(1) the internal and external disturbances of the system areregarded as the total disturbance; (2) the total disturbance isestimated by ESO; and (3) the estimated result is com-pensated by the closed-loop control link. So the stateequation can be rewritten as

pi � a0i + f + bu,

y � i,

a0 � −R0

Lm,

b �1

Lm,

f � −Rn

Lmi −

Ke

Lmωm,

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(21)

where a0 and b are the system parameters, respectively, and fis the total disturbance. On the basis of equation (21), anESO is established as

ei � z1 − i,

_z1 � z2 − β1 · g ei( + a0i + bu,

_z2 � −β2 · g ei( ,

⎧⎪⎪⎨

⎪⎪⎩(22)

where z1 is the estimate of the current i; z2 is the estimate ofthe total disturbance; ei is the error between the true valueand the estimated value; β1 and β2 are the feedback gaincoefficients, respectively; and g(ei) is a feedback function ofthe error. When g(ei) is a linear function, the LESO isestablished. When g(ei) is a nonlinear function, the NESO isestablished. *e general nonlinear function is

fal ei, α, δ( �

ei

αsign ei( , ei

> δ,

ei

δ1−α, ei

≤ δ,

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(23)

where α and δ are the parameters of the function.When the ESO estimation effect is good, the propor-

tional control can achieve good current closed-loop control.*erefore, the total disturbance estimate is compensatedproportionally, and the current closed-loop control equationis established as

u0 � KP i0 − z1( ,

u � u0 −z2

b,

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(24)

where i0 is the motor target current obtained from the targettorque closed-loop control; u0 is the proportional errorfeedback between i0 and z1; Kp is the proportionality co-efficient; and u is the real output voltage after disturbancecompensation.

In fact, there is always an observation error between theestimate disturbance and the real disturbance:

f′ � f − z1, (25)

where f ′ is the error of disturbance observation. WhenNESO is adopted and appropriate parameters are matched, f ′ issmall and the system works well. However, the computation

Mathematical Problems in Engineering 5

of nonlinear function is very heavy in practical application. Onthe other hand, if LESO is adopted instead, the calculation issmall, but f ′ is large when the target gain is large or the in-terference frequency is high. *e responsiveness and anti-in-terference ability of ADRC is seriously affected.

Based on the principle of ADRC, the fundamentalpurpose of disturbance compensation is to transform theunknown system model into a first-order integration systemwhich is easy to solve. *erefore, the error of disturbanceobservation is the difference between the target current and

Ang

le (r

ad)

–0.2

0.2

0.6

1

0.47 0.49 0.51 0.53 0.550.45Time (s)

r = 750r = 1500

r = 2500Target step

(a)

The a

ngul

ar sp

eed

(rad

/s)

–0.2

0.2

0.6

1

0.47 0.49 0.51 0.53 0.550.45Time (s)

r = 750r = 1500

r = 2500Target step

(b)

Figure 3: Output of the tracking differentiator. (a) *e tracking process, θ1. (b) *e approximately differential, ω.

Ang

le (r

ad)

–0.2

0.2

0.6

1

1.4

0.2 0.4 0.6 0.8 10Time (s)

Real angleTransition of real angle

(a)

The a

ngul

ar sp

eed

(rad

/s)

–100

0

100

200

300

400

500

600

0.2 0.4 0.6 0.8 10Time (s)

Conventional differentiatorsTracking differentiators

(b)

Ang

le (r

ad)

0.5 1 1.5 20Time (s)

–1.5

–1

–0.5

0

0.5

1

1.5

Real angleTransition of real angle

(c)

The a

ngul

ar sp

eed

(rad

/s)

–30

–20

–10

0

10

20

30

40

0.5 1 1.5 20Time (s)

Conventional differentiatorsTracking differentiators

(d)

Figure 4: Output of TD. (a) Angle output under step response. (b) Angular speed output under step response. (c) Angle output undersinusoidal response. (d) Angular speed output under sinusoidal response.


the output current. *e target current is the first-orderintegral function of the target voltage:

f′(s) �bu0(s)

s + a0− i(s). (26)

As long as f ′ is compensated, the estimation accuracy ofdisturbance can be improved. f ′ is considered the output of asystem where the input u2 is zero. *en, a new LESO2 for thesystem to estimate f ′ is established. *e new LESO2 and theoriginal LESO1 form the parallel linear extended state observer(P-LESO). *e total disturbance of the system is observed andcompensated by LESO1, and the error of disturbance obser-vation in LESO1 is observed and compensated by LESO2.*erefore, P-LESO observationmethod can reduce the error ofdisturbance observation and improve the dynamic respon-siveness. *e P-LESO is constructed as follows:

ei � z11 − i,

_z11 � z12 − β1ei + a0i + bu,

_z12 � −β2ei,

⎧⎪⎪⎨

⎪⎪⎩(27)

ei′ � z21 − f′,_z21 � z22 − β1ei′,_z22 � −β2ei′,

⎧⎪⎪⎨

⎪⎪⎩(28)

where z11 is the estimate of the output current; z12 is theestimate of total system disturbance; z21 is the estimate of f ′;and β1, and β2 are the tunable parameters, and their valueswill directly affect the fastness and accuracy of the distur-bance estimation. Figure 5 shows the block diagram of EPSmotor control based on the improved ADRC strategy usingP-LESO.

*e LESO1 is expressed in the form of state space:_z1

_z2 �

−β1 1

−β2 0

z11

z12 +

β1 − a0β2

i +b

0 u. (29)

*e equation of the state is

Δ|sE − A| � s2 + β1 · s + β2, (30)

where A � −β1 1−β2 0

, which is the system matrix.

According to the Routh stability criterion, when β1> 0and β2> 0, the observer must be stable. Meanwhile, thesystem matrix of LESO2 is the same form as LESO1, andLESO2 is also stable.

In EPS systems, the observer is required to estimate lowfrequency (within 10Hz) disturbance quickly and accu-rately. *erefore, the observer bandwidth should be above100Hz. δ0 is the expected observer bandwidth. Whenβ1 � 2δ0 and β2 � δ

20, the expected observer characteristic

equation is

s + δ0( 2

� s2

+ 2δ0 · s + δ20. (31)

*e initial state of the system is assumed to 0, theP-LESO equation can be obtained by eliminating the in-termediate variables after Laplace transformation:

z12(s) + z21(s)

f(s)�

δ20s + δ0(

2. (32)

According to equation (32), the estimate disturbanceand the real disturbance can be abstracted as a second-order oscillation link. *erefore, the estimate disturbanceis obtained after performing the first-order low-pass fil-tering twice on the real disturbance. *e cut-off frequencyof each level of low-pass filter is δ0. However, whenβ1 � 2δ0 and β2 � δ

20, the damping ratio is 1, and the

equation has two equal negative real roots −δ0. In practicalapplications, the parameter is set to overdamping toprevent overshoot in step response. In this paper, β1 � 250and β2 �12,000.

Laplace transform is performed on equations (26), and(29) to obtain the transfer function between the inputvoltage and the output current of the motor:

i(s)

u(s)�

s

ks + β2+

kb2 − bβ1kbs + β2

. (33)

According to equation (33), the relationship betweenoutput current and input voltage in the improved ADRCstrategy is composed of a first-order inertia link and adifferential link. On the premise of ensuring the stability ofthe system, the output of the system can be corrected earlierto improve the responsiveness.

3.3. Simulation Analysis of Improved ADRC. According tothe working situation of the motor in the EPS system, thesignal frequency of torque and angle is within 3 Hz andthe typical frequency of mechanical disturbance is be-tween 10 Hz and 30 Hz, which is the low-frequencydisturbance.

Figure 6 shows the disturbance estimation of three kindsof ESO at different frequencies. When the disturbance is1Hz, all three observers agree well with the disturbance.When the disturbance is 10Hz, the disturbance estimationof LESO has a phase difference of 45 deg compared with thereal disturbance, but P-LESO and NESO still follow well.When the disturbance is 30Hz, the disturbance estimationof LESO has a phase difference of 90 deg compared with thereal disturbance, while the phase difference in P-LESO andNESO is 18 deg. *erefore, the designed P-LESO can stablyand accurately estimate the typical frequency of mechanicaldisturbance.

Figure 7 shows the step responses of different strategiesto typical frequency signals when there is disturbance noise.

From Figure 7, the ADRC based on different ESO isbetter than PI control in noise suppression and responsespeed, especially at higher signal frequencies, no matterwhich steps input signal frequency or disturbance frequency.When the signal frequency is low, the responsiveness of thethree ADRC strategies is similar. When the signal frequencyincreases, the responsiveness and anti-interference ability ofP-LESO is better than that of LESO, which is basically thesame of NESO.


Torque angle sensor

Motor

LESO1

Steeringcolumn

LESO2

b/(s + a0)

f′

Road resistance

TDTarget torque

algorithm – –+

+

+

+

–+

+

+

–

+

–

θ1

ω

θ

T∗sw

Tsw

Tsw

eT Kt

KmTeTaTr

i0

z11

ei Kpu0

u2 = 0

u i

a (t)

a (t)

1/b b

z12

z21

Figure 5: *e block diagram of EPS motor control based on improved ADRC.

–1.5

–1

–0.5

0

0.5

1

1.5

Curr

ent (

A)

0.6 1.2 1.8 2.4 30Time (s)

DisturbanceLESO

P-LESONESO

(a)

–1.5

–1

–0.5

0

0.5

1

1.5

Curr

ent (

A)

0.1 0.2 0.3 0.4 0.50Time (s)

DisturbanceLESO

P-LESONESO

(b)

–1.5

–1

–0.5

0

0.5

1

1.5

Curr

ent (

A)

0.04 0.08 0.12 0.16 0.20Time (s)

DisturbanceLESO

P-LESONESO

(c)

Figure 6: *e disturbance estimation results of three kinds of ESO. (a) 1Hz disturbance. (b) 10Hz disturbance. (c) 30Hz disturbance.


4. Control Strategy Experiment

On the basis of theory and simulation analysis, the controlperformances of steering torque and motor current in im-proved ADRC are verified by vehicle experiment. In order tosignificantly compare the various control strategies, an ex-periment vehicle with large and irregular friction in thesteering system is selected, and the experiment vehicle hasmultiple points of discontinuous resistance torque in theentire steering range. *e disturbance is especially obviouson the uneven road, and the vehicle keeps running at a lowspeed. *e rotation range of the steering wheel may be wide,but the steering wheel speed generally does not exceed 1 r/s.Considering the deceleration ratio, the maximum speed ofthe motor does not exceed 1000 r/min. *e steering systemparameters of the experiment vehicle are shown in Table 2.

*e motor control strategy of the improved ADRC isrealized by a cortex-M0 kernel motor controller. *e controlstep size of the system is 1ms, and the driving frequency ofPWM is 20 kHz. *e experiment vehicle is shown in Fig-ure 8, including the steering system, embedded hardwarecontroller, J-Link debugger, and host computer.

Curr

ent (

A)

–1.2–0.8–0.4

00.40.81.21.6

2

0.25 0.5 0.75 1 1.25 1.5 1.75 20Time (s)

iLESONESO

P-LESOPI

(a)

Curr

ent (

A)

–1.2–0.8–0.4

00.40.81.21.6

2

0.25 0.5 0.75 1 1.25 1.5 1.75 20Time (s)

iLESONESO

P-LESOPI

(b)

Curr

ent (

A)

0.25 0.5 0.75 10Time (s)

–1.2–0.8–0.4

00.40.81.21.6

2

iLESONESO

P-LESOPI

(c)

Curr

ent (

A)

–1.2–0.8–0.4

00.40.81.21.6

2

0.25 0.5 0.75 10Time (s)

iLESONESO

P-LESOPI

(d)

Figure 7: *e result of the system step response. (a) 10Hz noise and 1Hz signal. (b) 30Hz noise and 1Hz signal. (c) 10Hz noise and 3Hzsignal. (d) 30Hz noise and 3Hz signal.

Table 2: Main parameters of the experiment vehicle.

Parameters Unit ValueFront axle load kg 635Reduction ratio — 16.5*e motor rating W 270Maximum torque of motor N·m 2.4Steering wheel rotation range deg −720 to 720

Figure 8: Experiment vehicle.


Experiment 1. *e experiment vehicle is parked on the ir-regular rolling road, whose steering wheel is rotated from aresistance torque discontinuity point at a constant speed 1 r/s. *en, the relationship between target current and realcurrent in different control strategies is recorded in Figure 9.

In Figure 9(a), when the PI strategy is adopted withhigh target gain, the disturbance caused by irregularfriction can be amplified in the closed-loop process,resulting in the current oscillation amplitude reachingabout 4 A. In Figure 9(b), the torque oscillation using theADRC strategy can restore the stability within 0.13 s, andthe oscillation amplitude is reduced to 1.5 A. InFigure 9(c), the torque oscillation using improved ADRCcan restore stability within 0.05 s, and the oscillationamplitude does not exceed 1 A. Compared with the PI andADRC strategy, the motor current using the improvedADRC strategy is more stable.

Experiment 2. *e experiment vehicle is parked on the ir-regular rolling road. *e steering wheel is rotated in the fullrange of travel at different rotation speed.*en, the curves ofangle and torque of the steering wheel can be obtained andshown in Figure 10.

Figure 10(a) shows that when the steering wheel ro-tates at low speed, the steering torque fluctuation usingthe PI strategy increases to 2.5 N·m. *e steering torquefluctuation using the two ADRC strategies is less than1 N·m. *e responsiveness of two ADRC strategies isbetter than the PI strategy, which makes the steeringprocess smoother. In particular, when the steering wheelis rotated in the range of 400 to 500 deg, the steeringtorque using the PI strategy exceeds 4 N·m, and thesteering torque using the two ADRC strategies is 3 N·m.*e steering torque using the PI strategy is heavier thanusing the ADRC strategy, which affects the steeringflexibility.

Figure 10(b) shows that when the steering wheel isrotated at a high speed, the torque fluctuation using the PIstrategy and ADRC strategy increases to 3.5 N·m. Whenusing the improved ADRC strategy, the torque fluctuationis still less than 1 N·m. In the stage of large angle where theload torque fluctuates greatly, the steering torque usingthe PI strategy exceeds 5 N·m, and the steering torqueusing the ADRC strategy is 4 N·m. Compared to theADRC strategy, the torque fluctuation using the improvedADRC strategy is 3 N·m, which is decreased by 25%. *isindicates that when the load torque changes quickly and

Mot

or cu

rren

t (A

)

∆ = 3.9A

–8

–4

0

4

8

12

0.1 0.2 0.3 0.4 0.50Time (s)

Target currentReal current

(a)

Mot

or cu

rren

t (A

)

–2

0

2

4

6

8

10

0.1 0.2 0.3 0.4 0.50Time (s)


2.42

1.61.20.80.4

00.15 0.20.1

(b)

Mot

or cu

rren

t (A

)

–4

–2

0

2

4

6

8

10

0.1 0.2 0.3 0.4 0.50Time (s)


1.51

0.50

–0.50.125 0.150.1

(c)

Figure 9: Motor current response. (a) PI strategy. (b) ADRC strategy. (c) Improved ADRC strategy.


severely, the PI and ADRC strategies cannot providesufficient assisting torque in a timely manner. *e im-proved ADRC is still responsive to the requirements ofassist torque, which can keep the steering torque smoothand flexible.

Experiment 3. For the sake of safety, the vehicle speed isforced at 20 km/h.*e steering wheel is rotated left and rightrepeatedly at the speed of 0.5 r/s to keep the vehicle’s drivingroute approximately sinusoidal. *e curve of steering wheeltorque with time is shown in Figure 11.

Stee

ring

whe

el to

rque

(N·m

)

–6

–4

–2

0

2

4

6

–540 –360 –180 0 180 360 540 720–720Steering wheel angle (deg)

PIADRCImproved ADRC

(a)

Stee

ring

whe

el to

rque

(N·m

)

–8

–6

–4

–2

0

2

4

6

8

–540 –360 –180 0 180 360 540 720–720Steering wheel angle (deg)

PIADRCImproved ADRC

(b)

Figure 10: .Relationship between steering torque and steering angle. (a)*e steering wheel speed is 0.2 r/s. (b)*e steering wheel speed is 1 r/s.

Torq

ue (N

·m)

–4

–2

0

2

4

0.5 1 1.5 2 2.50Time (s)

Steering torqueTarget torque

(a)

Torq

ue (N

·m)

–4

–2

0

2

4

0.5 1 1.5 2 2.50Time (s)


(b)

Torq

ue (N

·m)

–4

–2

0

2

4

0.5 1 1.5 2 2.50Time (s)


(c)

Figure 11: Variation of steering wheel torque. (a) PI strategy. (b) ADRC strategy. (c) Improved ADRC strategy.


Figures 11(a) and 11(b) indicate the steering torquefluctuation violently due to irregular friction and unknownroad resistance. *e torque fluctuation using the PI strategyand ADRC strategy reaches above 1N·m and 0.5N·m, re-spectively. Figure 11(c) shows that the torque fluctuationusing improved ADRC is lower than 0.2N·m. Compared thethree control strategies, the improved ADRC strategy showsgood torque stability.

5. Conclusion

*is paper proposes a new method for torque feedbackcontrol of the EPSmotor, which has the advantages of stronganti-interference ability, smooth torque control, and smallcomputation. TD is used to calculate the angular speed of thesteering wheel to solve the problem of noise amplification indifferential calculation. *e improved ADRC strategy basedon the P-LESO method is designed to avoid the use ofnonlinear functions and reduce the computational burden ofthe system.

Simulation results show that the improved ADRCstrategy can effectively reduce the low-frequency distur-bance inside and outside the system.*e control process hasbetter responsiveness and anti-interference.

*e vehicle experiment shows that the steering torquecontrol method of the improved ADRC strategy reduces theoscillation of motor current and makes the steering wheeltorque more flexible, smooth, and stable when comparedwith the PI strategy and ADRC strategy.

In the case of road resistance, mechanical friction, andmotor parameter changes, the proposed method can stillkeep the steering torque of the vehicle in an appropriate andstable range.

Data Availability

*e data used to support the findings of this study are in-cluded within the article.

Conflicts of Interest

*e authors declare no conflicts of interest.

Acknowledgments

*is work was supported by grants from the National Sci-ence Foundation of China (grant number 51575097) and theFundamental Research Funds for the Central Universities(China) (no. 2572016AB72).

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