# EPS System of Tractor Automatic Driving: An Improved LMI with Mixed Sensitivity Control Method

Authors: Huanxiao Pang; Guangfei Xu; Zhicai Song; Dianbao Zhao; Wenjun Wang; Peisong Diao
Journal: Journal of Electrical and Computer Engineering
 (2021)

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## Abstract

Electric power steering (EPS) is widely used in tractor automatic driving because of its good operation stability. However, there is a lack of research about solving robust problems and response ability simultaneously when the tractor encounters emergency steering in harsh fields. The traditional robust controller has poor tracking performance and antidisturbance ability when encountering emergency steering. This paper proposes to add the corresponding mixed sensitivity operator to the corresponding performance index in the controller. By adjusting the amplitude of the mixed sensitivity operator, the tracking performance and the speed of disturbance attenuation can be both adjusted for the tractor EPS system. Simulation and hardware in the loop experiments verify the antidisturbance ability of the controller and the torque tracking performance. The results show that the control method has strong robustness and robust stability, which can meet the practical requirements. Also, the power steering characteristic of theH∞ controller with hybrid sensitivity design method is better than that of an unoptimized one, and its robustness is better, and under external pavement interference, the following ability is stronger for the ideal hand torque and the steering is more stable.


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## Body

## 1. Introduction

A tractor is the power source in field operation, its working environment is complex, and often it is used in the rugged field work; therefore, power steering system is essential for tractors [1], and scholars have done many kinds of research on it.In the research process, the vehicle stability [2], the model uncertainty, and the interference problems are mainly concerned and various advanced control methods and research schemes are adopted to improve the overall performance of electric power steering (EPS) under different working conditions [3].The special working environment of tractors determines that the EPS system of tractors needs to consider the motor control strategy under special working conditions. Bei et al. [4] designed the motor control strategy including the compensation of return-to-normal process, the compensation of emergency avoidance process, the compensation of full-load large-angle turning, and other special steering conditions. However, the influence of random interference and measurement noise on EPS system is not considered. In the research process, the influence of random interference and sensor noise on EPS control cannot be ignored. Therefore, robust control methods have been applied to the research of EPS. Zhao et al. [5] designed a hybrid H2/H∞ robust controller considering tire nonlinearity and external interference. A displacement and force coupling control design for active front steering (AFS) system of vehicle is proposed in their paper. The robustness is controlled by H∞, and the performance is monitored by H2. He et al. [6] put forward a rolling time-domain control method. A robust H∞ coordination control scheme is developed based on linear matrix inequality (LMI) [7, 8]. It is proposed to simultaneously suppress lateral path-tracking deviation while maintaining autonomous vehicle stability under dynamic driving situations at handling limits.However, many control systems including EPS need to deal with multiobjective constraint problem, which is difficult for the aforementioned algorithms. As LMI can take all control targets into consideration by establishing Lyapunov function [9, 10], a series of control method based on LMI is proposed. Menhour et al. [11] proposed a two-PID control method based on LMI theory. The controllers are based on LMI theory stabilizing optimal and on linear time-invariant models of the vehicle, obtained around its nominal operating point. The different gains of both PID controllers are managed by a switch depending on the velocity. It can get good control effectiveness, but the author lacks the influence of uncertainty on the controller. Therefore, this paper proposes an LMI control method based on mixed sensitivity operator and applies it to the EPS system. It can be used for tractor power steering under special conditions and solving the influence of random interference and sensor noise on EPS system so that the EPS control strategy of tractors has better operation performance and safety performance.This paper is arranged as follows. In Section2, the space model of EPS is established; in Section 3, the controller is designed based on the LMI method; then in Section 4, the simulation analysis is carried out, and the problems of the controller based on the LMI method are found; in Section 5, the existing problems are solved, and the LMI method with mixed sensitivity operator is proposed, and the robust performance is verified; and in Section 6, the algorithm is tested in practice. Section 7 is the conclusion.


## 2. Establishment of EPS Model for Tractor

EPS dynamic modeling is the key of this paper; therefore, we need to carry out mathematical modeling of the main components of the steering power transmission system [5, 12]. Figure 1 shows the tractor steering system. This paper mainly studies the electric power steering system. The following is the mathematical modeling of the main components.Figure 1
Tractor steering system.


### 2.1. Torsion Bar Equation

According to the vehicle dynamics, the torsion bar can be regarded as a scale model in the range of stiffness [11, 13]:(1)δ=xrrp,Tc=Ksα−δ,where δ is the steering gear angle, rad; xr is the steering gear rack displacement, m; rp is the radius of steering gear, m; Ks is the torsional rigidity; Tc is the torsion bar torque, N; and α is the steering wheel angle, rad.


### 2.2. Electrical Machinery Model

The system uses a brushed DC motor. The differential equation can be obtained by Kirchhoff laws of voltage:(2)u=RIa+LdIadt+Kbθ˙,θ˙=Nδ˙,Tm=KaIa,where u is the motor terminal voltage, V; R is the armature resistance, Ω; θ is the angle of motor, rad; N is the drive ratio which is the steering shaft to the motor; Ka is the torque coefficient of motor; Kb is the coefficient of back electromotive force of motor; L is the inductance coefficient of motor, H; Ia is the motor current, A; and Tm is the electromagnetic torque of motor, N m.


### 2.3. Tire Model

It is not easy to use the model to accurately build the dynamic characteristics of the tire when the tractor is running, but for the EPS model of the tractor in this paper, it is enough to establish the equation of tire angle and resistance moment. Fujiwara and other experts proposed that the relationship between the steering angle and the tire resistance moment is approximately linear when the vehicle is stationary or at low speed, and the tractor is always at low speed during operation [14], which fully conforms to the aforementioned theory. So the resistance moment of the steering wheel is(3)Ttire=θtire1Ktire,where Ttire is the steering resisting torque, N m; θtire1 is the front wheel actual angle, rad; and Ktire is the proportion coefficient of electric power torque-rotation angle.It is known from the vehicle system dynamics [15] that there is a difference between the theoretical angle θtire and the actual angle θtire1, which directly affects the return torque of the steering system.(1)
Theoretical angular equation:(4)θtire=θshaftN2,whereθtire is the theoretical angle, rad; θshaft is the intermediate shaft angle, rad; and N2 is the transmission ratio of steering gear.(2)
Actual angle equation: in the actual angle, when the stiffness coefficient of the intermediate shaft is larger, the deformation angle of the front wheel will be smaller, so(5)θtire1=θtire−θδ=θtire−θtireKrigid≈θtire.(3)
Resistance moment equation of intermediate shaft [16]:(6)T1=TtireN2,whereT1 is the resistance moment equation of intermediate shaft, N m.Therefore, it can be deduced from the aforementioned formula:(7)T1=KtireN22θshaft.If(8)T1=KtireN22θshaft≈δ,then the steering load torque can be expressed as(9)T1=δK1.


### 2.4. Controller Model

The controller plays an important role in EPS, which is the basis of the road feel control strategy and assist control strategy. Assist characteristic curve is an important factor of EPS assist characteristic. Figure2 shows a typical assist characteristic curve. It can be seen from Figure 1 that(10)i=0,0<Tc≤1,kvTc,1<Tc≤7,7kv,Tc>7,where i is the motor current and k(v) is the slope at different speeds.Figure 2
Assist characteristic curve.It is also known from the electrical characteristics of DC motor that(11)i=u−kbθ˙R.We can get(12)kvTc=u−kbθ˙R.Then,(13)u=kvTcR+kbNα˙−kbNT˙cks.Among them,(14)kd=−kdNks,kp=kvR,kN=kbN,where kd is the differential coefficient; kp is the proportional coefficient; and kN is the integral coefficient.The deformation can be(15)u=kpTc+kdT˙c−kNα˙.It can be seen that the armature voltage of the motor is determined by the torque and the steering wheel rotating speed, so the electric power steering system mainly regulates the relationship among the steering resistanceTl, the motor power Tm, and the driver's hand torque Th. Therefore, the controller model can control the motor current or voltage to achieve the coordination of the three.In conclusion, the nonlinear dynamic model of EPS controlled system is obtained as follows:(i)
Steering column:(16)Ihα¨+Chα¨+Tc=Th+fhα,α˙.(ii)
Output shaft:(17)Ieδ¨+Ceδ˙=Tc+NTα−Tl+feδ,δ˙.(iii)
Assisted motor:(18)Imθ¨+Ceθ˙=Tm−Tα+fmθ,θ˙.(iv)
Pinion-and-rack mechanism:(19)Mrx¨r+Crx˙r+krxr+TR=Tlrp+frx˙r,xr.Among them,fhα,α˙, feδ,δ˙, fmθ,θ˙, and frx˙r,xr are time-varying and nonlinear uncertain models.The deformation can be(20)a¨=1Ih−Ksα−Ksδ+Chα˙+Th+fhα,α˙,θ¨=1Im−Kmθ−Cm+KaKbRθ˙+KmNδ+KαRu+fmθ,θ˙,x¨r=1MRKsrpα+Nkmrpθ−Kr+Ksrp2+N2Kmrp2xr−CRx˙r+feδ,δ˙,TR−TR,δ¨=1MRrpKsrpα+Nkmrpθ−Krrp+Ksrp+N2Kmrpxr−CRrpx˙r+feδ,δ˙,TR−TR.


## 2.1. Torsion Bar Equation

According to the vehicle dynamics, the torsion bar can be regarded as a scale model in the range of stiffness [11, 13]:(1)δ=xrrp,Tc=Ksα−δ,where δ is the steering gear angle, rad; xr is the steering gear rack displacement, m; rp is the radius of steering gear, m; Ks is the torsional rigidity; Tc is the torsion bar torque, N; and α is the steering wheel angle, rad.


## 2.2. Electrical Machinery Model

The system uses a brushed DC motor. The differential equation can be obtained by Kirchhoff laws of voltage:(2)u=RIa+LdIadt+Kbθ˙,θ˙=Nδ˙,Tm=KaIa,where u is the motor terminal voltage, V; R is the armature resistance, Ω; θ is the angle of motor, rad; N is the drive ratio which is the steering shaft to the motor; Ka is the torque coefficient of motor; Kb is the coefficient of back electromotive force of motor; L is the inductance coefficient of motor, H; Ia is the motor current, A; and Tm is the electromagnetic torque of motor, N m.


## 2.3. Tire Model

It is not easy to use the model to accurately build the dynamic characteristics of the tire when the tractor is running, but for the EPS model of the tractor in this paper, it is enough to establish the equation of tire angle and resistance moment. Fujiwara and other experts proposed that the relationship between the steering angle and the tire resistance moment is approximately linear when the vehicle is stationary or at low speed, and the tractor is always at low speed during operation [14], which fully conforms to the aforementioned theory. So the resistance moment of the steering wheel is(3)Ttire=θtire1Ktire,where Ttire is the steering resisting torque, N m; θtire1 is the front wheel actual angle, rad; and Ktire is the proportion coefficient of electric power torque-rotation angle.It is known from the vehicle system dynamics [15] that there is a difference between the theoretical angle θtire and the actual angle θtire1, which directly affects the return torque of the steering system.(1)
Theoretical angular equation:(4)θtire=θshaftN2,whereθtire is the theoretical angle, rad; θshaft is the intermediate shaft angle, rad; and N2 is the transmission ratio of steering gear.(2)
Actual angle equation: in the actual angle, when the stiffness coefficient of the intermediate shaft is larger, the deformation angle of the front wheel will be smaller, so(5)θtire1=θtire−θδ=θtire−θtireKrigid≈θtire.(3)
Resistance moment equation of intermediate shaft [16]:(6)T1=TtireN2,whereT1 is the resistance moment equation of intermediate shaft, N m.Therefore, it can be deduced from the aforementioned formula:(7)T1=KtireN22θshaft.If(8)T1=KtireN22θshaft≈δ,then the steering load torque can be expressed as(9)T1=δK1.


## 2.4. Controller Model

The controller plays an important role in EPS, which is the basis of the road feel control strategy and assist control strategy. Assist characteristic curve is an important factor of EPS assist characteristic. Figure2 shows a typical assist characteristic curve. It can be seen from Figure 1 that(10)i=0,0<Tc≤1,kvTc,1<Tc≤7,7kv,Tc>7,where i is the motor current and k(v) is the slope at different speeds.Figure 2
Assist characteristic curve.It is also known from the electrical characteristics of DC motor that(11)i=u−kbθ˙R.We can get(12)kvTc=u−kbθ˙R.Then,(13)u=kvTcR+kbNα˙−kbNT˙cks.Among them,(14)kd=−kdNks,kp=kvR,kN=kbN,where kd is the differential coefficient; kp is the proportional coefficient; and kN is the integral coefficient.The deformation can be(15)u=kpTc+kdT˙c−kNα˙.It can be seen that the armature voltage of the motor is determined by the torque and the steering wheel rotating speed, so the electric power steering system mainly regulates the relationship among the steering resistanceTl, the motor power Tm, and the driver's hand torque Th. Therefore, the controller model can control the motor current or voltage to achieve the coordination of the three.In conclusion, the nonlinear dynamic model of EPS controlled system is obtained as follows:(i)
Steering column:(16)Ihα¨+Chα¨+Tc=Th+fhα,α˙.(ii)
Output shaft:(17)Ieδ¨+Ceδ˙=Tc+NTα−Tl+feδ,δ˙.(iii)
Assisted motor:(18)Imθ¨+Ceθ˙=Tm−Tα+fmθ,θ˙.(iv)
Pinion-and-rack mechanism:(19)Mrx¨r+Crx˙r+krxr+TR=Tlrp+frx˙r,xr.Among them,fhα,α˙, feδ,δ˙, fmθ,θ˙, and frx˙r,xr are time-varying and nonlinear uncertain models.The deformation can be(20)a¨=1Ih−Ksα−Ksδ+Chα˙+Th+fhα,α˙,θ¨=1Im−Kmθ−Cm+KaKbRθ˙+KmNδ+KαRu+fmθ,θ˙,x¨r=1MRKsrpα+Nkmrpθ−Kr+Ksrp2+N2Kmrp2xr−CRx˙r+feδ,δ˙,TR−TR,δ¨=1MRrpKsrpα+Nkmrpθ−Krrp+Ksrp+N2Kmrpxr−CRrpx˙r+feδ,δ˙,TR−TR.


## 3. Controller Design Based on LMI Method

The controllerK(s) is designed through the EPS mathematical model, and the H∞ robust control theory is combined to realize the H∞ robust control standardization of the EPS controlled closed-loop system model [17–19].The state equation of the system can be obtained as follows:(21)x˙=Ax+B1w+B2u,z=C1x+D11w+D12u,y=C2x+D21w+D22u,u=ky.Take state variables asx = (α, δ, θ, α˙, δ˙, θ˙). The control input is u = [u]∈Rn. The road signal input is w=dδThTR∈Rq. dδ is sensor noise. TR is external disturbance. The control output is z=eaefuT∈Rr.y=Tc+dTc∈Rp is the measured value.Among them, the coefficient matrices are variable.(22)A=000100000010000001−KsIhKsIh0−ChIh00KsMRrp2MMNKmMRrp0−CRMR00kmN−KsIm00HH.Among them,(23)MM=−1MRrpKrrp+Ksrp+N2Kmrp,HH=−CmR+KaKbImR,B1=00000000001Ih000−1MRrp000,B2=00000kaImRT,D11=00000−kfrp000,D12=001T,C1=kaksksks0Nkm−kaksks−ks+kskrrp20−km000000−kscrrp20−cm00T,C2=ks−ks0000,D21=100,D22=0.Formula (23) is the uncertain space state equation based on H∞ robust control, as shown in Figure 3.Figure 3
H∞ standardized control model.(24)zy=Pwu=p11p12p21p22wu.The output and input ofH∞ standardized control system can be transformed into formula (24).Using the conversion algorithm of frequency transfer function representation and state space representation, the following can be obtained: the expression of the transfer relationship from the external disturbancew = [dδ ThTR]T to the evaluation index z = [eaef−u]T of the controlled system is Tzw. Then, the following control objective based on the standard H∞ control model is to minimize Tzw on the basis of solving the K of the controller, that is, the influence of external interference on the evaluation index of the controlled system is reduced to the lowest point.Let the desired controllerK be [20](25)x˙k=Akxk+Bkyu=Ckxk+Dky,where the control space of the controller is xk ∈ Rnxk and Ak, Bk, Ck, and Dk are the parameter space matrices of the controlled system controller to be determined.From the model of the controlled system (formula (21)) and its controller (formula (25)), the closed-loop system expression of the controlled object can be obtained as follows:(26)x˙c=Ac1xc+Bc1wu=Cc1xc+Dc1w,where xc = [xTxkT]T is the space state of the controlled object. It is converted to a matrix format of(27)Ac1Bc1Cc1Dc1=A0B0C0D11+B¯D¯12KC¯D¯21,where K=AkBkCkDk.Therefore,(28)A0B0B¯C0D11D¯12C¯D¯21KT=A0B1B200000IC0D11D120C0D21DkTDkT0I0CkTAkT.The design requirement ofH∞ robust controller is to find K, so that the closed-loop expression of the controlled plant (formula (25)) can satisfy the controlled system model (formula (21)) [21]. Assuming that the model of the controlled system and its controller have been determined, the sufficient and necessary condition for the controller of the controlled system to be an H∞ controller is that there is a symmetric positive definite matrix Xc1 such that the following formula holds:(29)Ac1TXc1+Ac1Xc1Xc1Bc1Cc1TBc1TXc1−IDc1TCc1Dc1−I<0.From the controlled system model (formula (21)) and its controller (formula (24)), we can see that formula (28) is the LMI problem. According to the matrix theory, BMI problem can be transformed into LMI problem.Let(30)Hxc1=A0TXc1+Ac1X0Xc1B0C0TB0TXc1−ID11TC0D11−I,Pxc1=B¯TXc10D11T,Q=C¯D¯210.Then formula (28) can be transformed into(31)Hxc1+Pxc1TKQ+QTKTPxc1<0.Set(32)Xc1,Txc1=A0Xc1−1+Ac1X0B0Xc1−1C0TB0T−ID11TC0Xc1−1D11−I.Then the matrix model of the model parameters of the controlled system can be obtained as follows:(33)P=B¯T0D¯12T.IfXc1 > 0, then(34)NPxc1THxc1NPxc1<0,NQTTxc1NP<0.From the aforementioned study, we can draw the following conclusions: if there is a symmetric positive definite matrixXc1 and(35)NPTTxc1NP<0,NQTHxc1NQ<0,then there is an nk-order H∞ controller for the controlled system model (formula (21)).Ifnk and n are the order of H∞ controller and system model, respectively, Xc1 is a real symmetric matrix of (n + nk) × (n + nk) dimension. If X and Y are n × n-dimensional symmetric matrices, the matrix variables Xc1 and Xc1−1 can be expressed as(36)Xc1=XX2X2TX3,Xc1−1=YY2Y2TY3.From this,NPTTxc1NP<0 and NQTHxc1NQ<0 is equivalent to(37)Nc00IAY+YATYC1TB1C1Y−ID11B1TC1D11−INc00I<0,(38)N000IATX+XAXBC1TB1TX−ID11C1D11−IN000I<0,where N0 and Nc are matrices with any set of base vectors in kernel space ker([C2D21]) and ker([C2D21]) as column vectors, respectively. Feasp, the solver of LMI software package in MATLAB engineering, is used to obtain the feasible solutions of X and Y in the above equation. If nk is a positive integer, X and Y are symmetric positive definite matrices given in Rnxn when(39)XIIY>0,rankXIIY≤n+nk.Formula (39) makes the matrix X2, Y2∈Rnxnk and the positive definite symmetric matrix X3, Y3∈Rnkxnk satisfy(40)XX2X2TX3>0,XX2X2TX3−1=YY2Y2TY3.Therefore, formulas (37) and (38) have solutions, and their solutions X and Y satisfy equation (39). Then Xc1 composed of X and Y satisfies equation (36). If the order of H∞ controller has nk ≥ n (as rankXIIY≤2n), then its rank can satisfy equation (39). The feasible solution of Xc1 is obtained. By substituting this solution into formula (29), the LMI containing only K can be obtained. The Feasp solver in the toolbox is used to obtain the feasible solution K of linear matrix inequality, and finally, the controller is obtained.


## 4. Problems in Simulation Based on LMI

Based on robust control theory, the control objective of EPS uncertain controlled system is designed to evaluate whether it can meet the designed performance to ensure the stability of the controlled system. Table1 shows the choice of simulation experiment parameters.Table 1
Engineering parameters of EPS system components.
ProjectParameterSymbolValuesMotor parameterMoment of inertia of motor (kg m2)Im0.00018Viscous damping coefficient of motor (N m s/rad)Cm0.003399Resistance (Ω)R1/6Back EMF constant (V/rad−1)Kd0.057Electromagnetic torque constant (N m/A)Ka0.04Rigidity coefficient of motor (N M/rad−1)Km125Steering wheel and column parametersMoment of inertia (kg/m2)Ih0.0012Viscosity damping coefficient (N m s/rad)Ch0.261Torsional stiffness of torsion bar (N m/rad)Ks84.33Output shaft parametersMoment of inertia (kg/m2)Ie0.001Viscosity damping coefficient (N m s/rad)Ce0.031Reduction ratio of worm gear reducerN16.5Rack and pinion parametersMass (kg)Mr0.89Viscosity damping coefficient (N s)Cr200Steering pinion radius (m)rp0.007The assist torque deviation is the difference between the ideal assist torqueTa∗ and the actual assist torque Ta [22]. The main reason is that there is a difference between the actual torque and the measured torque due to the hysteresis effect caused by nonlinear mechanical force transmission, component damping, and inertia in the steering system [23]. Torque tracking performance is an important performance of EPS. Therefore, the controller can reduce the torque deviation as much as possible to improve the tracking performance of the whole steering control system. In other words, the designed controller must be able to control the torque deviation ea = Ta–Ta∗ as small as possible under the uncertainty of road and human disturbance [24]. The following is a simulation study of ea response to road interference force TR and driver input hand torque Th.Figure4 shows the response curve of ea to driver input hand torque Th unit step. It can be seen from Figure 5 that when the driver inputs unit Th, the EPS assist torque deviation ea will appear a transient process of following. About 0.05 seconds after Th is applied, ea will reach a maximum following deviation with a peak value of 9.5% and then decrease sharply. After 2 to 3 seconds of transient time, the steady assist torque deviation is about 1.8%. From the ea to Th unit step response curve, the peak value of assist torque deviation is small, and the stable assist torque deviation is also small. The response effect of EPS assist torque deviation to Th can meet the requirements of EPS controller stability control objectives. From the time effect of stability, the transient process from response to stability takes a long time, and the normal transient time should be less than 2 seconds. Therefore, this obviously cannot meet the sensitivity requirements of EPS in fast steering.Figure 4
Unit step response curve ofea to Th.Figure 5
Sine response curve ofea to Th. (a) 2 Hz response of input hand torque and (b) 20 Hz response of input hand torque.
(a)(b)As shown in Figure5, the driver's input Th is a sine wave at two different frequencies. In Figure 5(a), the frequency of Th is a low-frequency sine wave of 2 Hz, and the assist torque deviation ea keeps the synchronous frequency change with the driver's input Th, and the phase is the same. In Figure 5(b), the frequency of Th is a 20 Hz high-frequency sine wave, the power torque deviation ea and the driver input Th cannot keep the synchronous frequency change, and the phase lags behind the phase of Th. Therefore, the controller cannot meet the sensitivity requirements of fast steering.


## 5. LMI Control Method with Mixed Sensitivity

From the aforementioned simulation analysis, it can be seen that theea following sensitivity cannot meet the designed control requirements when the controller inputs high frequency. In order to further improve the control quality of the designed controller, LMI algorithm with mixed sensitivity operator is adopted. The integrated control structure is shown in Figure 6.Figure 6
The control structure.


### 5.1. Simulation and Evaluation of EPS Assist Torque Deviation Control

Since the system sensitivityS0(s) is essentially the closed-loop system transfer function matrix between the output y of the controlled system and the disturbance signal d, the designed controller requires that the singular value of S0(s) is very small, so the ability of the controller to suppress the disturbance will be robust. Similarly, for the EPS closed-loop system, S0(s) is also the transfer function matrix between the torque error e and the ideal input r. Therefore, S0(s) will determine the tracking performance of the assist torque error e. The smaller the singular value of S0(s), the stronger the robustness of the system controller and the better the tracking performance. To adjust a certain performance index of the controlled system, it is very important to determine the corresponding weighting function. The mixed sensitivity operator W(s) is introduced to adjust the sensitivity of the assist following deviation ea. The amplitude of W(s) can be used to adjust the tracking performance of ea and the attenuation speed of disturbance. Therefore, the control quality of ea can be improved by adjusting the weighting function Ws1(s) of ea.As shown in Figure7, the unit step response curve of ea to Th when the weighting function is Ws1s=40/s+0.5. It can be seen from the figure that the EPS assist torque deviation will have a following transient process. About 0.05 seconds after Th is applied, ea will reach a maximum follow-up deviation of 3.8% and then decrease sharply. After 0.1-second transient time, the deviation of stable assist torque is about 0.026%. Compared with Figure 5, the control quality of ea by the controller has been significantly improved in both response speed and phase synchronization. Therefore, the control performance of the controller model based on H∞ robust control theory has been significantly improved.Figure 7
Response curve ofea to Th. (a) Unit step and (b) sine wave (20 Hz).
(a)(b)


### 5.2. Simulation and Evaluation of EPS Road Feel Control

EPS road feel control steering system control performance index is another important performance index, which is related to the driver's comfort in the process of driving. In the actual driving state, steering resisting torqueT1 and road interference TR will change with the change of road conditions. It is significant to study this performance index for safe and stable driving.The control effect of EPS road feel is evaluated by road feel erroref. Figure 8 shows the unit step response curve of ef to Th. In Figure 8(a), the mixed sensitivity operator is Ws1s=1/s+0.5. The peak response of ef to Th unit step is about 8.7. There is a transient process of 0.8 seconds after Th is applied, and then it goes into steady state, which can meet the stability requirements of the steering system. However, if the response peak value of ef to Th unit step is too large, it will make the driver feel “overshoot” for a very short time, which may cause the oscillation of the steering system. Therefore, it is necessary to adjust the weighting function of ef. Adjust the original Ws1s=1/s+0.5 to Ws1s=100/s+0.5, and the simulation effect is shown in Figure 8(b). It can be seen from the figure that the peak value of ef response to Th unit step is about 2.2, there is a transient process of 0.6 seconds after Th is applied, and then it enters the steady state. At this time, it can not only meet the stability of EPS steering system but also prevent “overshoot” feeling. By adjusting the parameters of Ws1(s) weighting function, the control performance of EPS can be improved continuously, so as to realize the smoothness of EPS working state.Figure 8
Response curve ofef to Th. (a) Ws1s=1/s+0.5 and (b) Ws1s=100/s+0.5.
(a)(b)


### 5.3. Control Simulation and Evaluation of EPS Robustness

By adjusting the corresponding performance weighting functionWS(s), the performance of the controlled system can be improved to a certain extent without modifying the parameters of the mechanical parts of EPS.EPS inevitably faces the interference of structural uncertainty. The main structural disturbance factors include steering column vibration caused by uneven road surface, torque disturbance of steering system caused by electromagnetic control system, disturbance of modeling parameters (such as change of system working environment, change of gap between components, wear between components, aging of components, and zero drift of electronic components). The simulation study is a comprehensive evaluation of the stability of the controller model under structural uncertainty disturbance based onH∞ robust control theory. In order to better compare the adjustment of single index with the comprehensive adjustment of robustness, the simulation condition is that except for the selected parameters to be disturbed, the weighted function matrix W and other parameters of the controlled system remain unchanged. By formula,(41)a=ksIh.That is, the variation range of torsional stiffnessKs = 84.33 N m/rad in actual operation is −27.4% to 27.4%; the moment of inertia Ih = 0.0012 kg/m2 of steering column and steering wheel in actual operation is −31.8% to 31.8%. Figure 9 shows that the unit step response of ea to Th is a curve cluster when the disturbance interval of a is 90275 to 50275. It can be seen from the performance of curve cluster in Figure 9 that ea reaches the peak value in about 0.1 seconds when Th is applied, and the fluctuation range is 0.06% to 0.094%, and the jitter range relative to its central curve is −0.02% to –0.01%. After the peak value, the curve in the curve cluster drops sharply and tends to be stable in about 1.6 seconds. Therefore, as a closed-loop system, EPS is robust. The research on different parameter weights shows that the robust performance of the whole system is better than before after the H∞ robust controller is applied to the control system.Figure 9
Unit step response curve cluster I ofea to Th.In the process of simulation, it is very important to adjust the parameters of weighting function. For example, the variation range of parametera in Figures 9 and 10 is 90275 to 50275, which reflects the tolerance effect of Th and external interference on ea. But the weighting function selected in Figure 9 is WT1s=0.01s+3/s+6000, while the weighting function selected in Figure 10 is WT1s=50s+30/s+6000. It can be seen from the two figures that the stability accuracy of the controlled system in Figure 10 is worse than that in Figure 9, and the ea response curve in Figure 10 is relatively divergent than that in Figure 9. In general, the amplitude and follow-up of the response curves in Figures 9 and 10 reflect the robust stability of the controlled system.Figure 10
Unit step response curve cluster II ofea to Th.EPS controller may produce a certain range of disturbance in the process of its use. The causes may be as follows: first, from the hardware point of view, it may be due to the time shift of physical devices in the process of loading the program; second, from the perspective of software, the transfer function in our design ofH∞ controller may be high-order, but in practical application, as hardware operation speed and accuracy cannot keep up with the ideal controller, in this case, we can only reduce the order of the controller algorithm, and then, this also leads to a certain degree of error in the actual controller. So the disturbance of the controller also has the problem of stability. Using the same method, if other conditions are the same, only the uncertainty of the controller itself is changed to |Δ|, so that |Δ|≤ 10%, then the generalized controlled system model is as follows:(42)Gps=1+ΔG0s.At this time,ea response curve cluster to Th is shown in Figure 11.Figure 11
Step response fromTh to ea with controller uncertainty.It can be seen that the response curve cluster has good stability and robust stability in terms of the convergence of amplitude and final curve and the divergence of curve cluster.


## 5.1. Simulation and Evaluation of EPS Assist Torque Deviation Control

Since the system sensitivityS0(s) is essentially the closed-loop system transfer function matrix between the output y of the controlled system and the disturbance signal d, the designed controller requires that the singular value of S0(s) is very small, so the ability of the controller to suppress the disturbance will be robust. Similarly, for the EPS closed-loop system, S0(s) is also the transfer function matrix between the torque error e and the ideal input r. Therefore, S0(s) will determine the tracking performance of the assist torque error e. The smaller the singular value of S0(s), the stronger the robustness of the system controller and the better the tracking performance. To adjust a certain performance index of the controlled system, it is very important to determine the corresponding weighting function. The mixed sensitivity operator W(s) is introduced to adjust the sensitivity of the assist following deviation ea. The amplitude of W(s) can be used to adjust the tracking performance of ea and the attenuation speed of disturbance. Therefore, the control quality of ea can be improved by adjusting the weighting function Ws1(s) of ea.As shown in Figure7, the unit step response curve of ea to Th when the weighting function is Ws1s=40/s+0.5. It can be seen from the figure that the EPS assist torque deviation will have a following transient process. About 0.05 seconds after Th is applied, ea will reach a maximum follow-up deviation of 3.8% and then decrease sharply. After 0.1-second transient time, the deviation of stable assist torque is about 0.026%. Compared with Figure 5, the control quality of ea by the controller has been significantly improved in both response speed and phase synchronization. Therefore, the control performance of the controller model based on H∞ robust control theory has been significantly improved.Figure 7
Response curve ofea to Th. (a) Unit step and (b) sine wave (20 Hz).
(a)(b)


## 5.2. Simulation and Evaluation of EPS Road Feel Control

EPS road feel control steering system control performance index is another important performance index, which is related to the driver's comfort in the process of driving. In the actual driving state, steering resisting torqueT1 and road interference TR will change with the change of road conditions. It is significant to study this performance index for safe and stable driving.The control effect of EPS road feel is evaluated by road feel erroref. Figure 8 shows the unit step response curve of ef to Th. In Figure 8(a), the mixed sensitivity operator is Ws1s=1/s+0.5. The peak response of ef to Th unit step is about 8.7. There is a transient process of 0.8 seconds after Th is applied, and then it goes into steady state, which can meet the stability requirements of the steering system. However, if the response peak value of ef to Th unit step is too large, it will make the driver feel “overshoot” for a very short time, which may cause the oscillation of the steering system. Therefore, it is necessary to adjust the weighting function of ef. Adjust the original Ws1s=1/s+0.5 to Ws1s=100/s+0.5, and the simulation effect is shown in Figure 8(b). It can be seen from the figure that the peak value of ef response to Th unit step is about 2.2, there is a transient process of 0.6 seconds after Th is applied, and then it enters the steady state. At this time, it can not only meet the stability of EPS steering system but also prevent “overshoot” feeling. By adjusting the parameters of Ws1(s) weighting function, the control performance of EPS can be improved continuously, so as to realize the smoothness of EPS working state.Figure 8
Response curve ofef to Th. (a) Ws1s=1/s+0.5 and (b) Ws1s=100/s+0.5.
(a)(b)


## 5.3. Control Simulation and Evaluation of EPS Robustness

By adjusting the corresponding performance weighting functionWS(s), the performance of the controlled system can be improved to a certain extent without modifying the parameters of the mechanical parts of EPS.EPS inevitably faces the interference of structural uncertainty. The main structural disturbance factors include steering column vibration caused by uneven road surface, torque disturbance of steering system caused by electromagnetic control system, disturbance of modeling parameters (such as change of system working environment, change of gap between components, wear between components, aging of components, and zero drift of electronic components). The simulation study is a comprehensive evaluation of the stability of the controller model under structural uncertainty disturbance based onH∞ robust control theory. In order to better compare the adjustment of single index with the comprehensive adjustment of robustness, the simulation condition is that except for the selected parameters to be disturbed, the weighted function matrix W and other parameters of the controlled system remain unchanged. By formula,(41)a=ksIh.That is, the variation range of torsional stiffnessKs = 84.33 N m/rad in actual operation is −27.4% to 27.4%; the moment of inertia Ih = 0.0012 kg/m2 of steering column and steering wheel in actual operation is −31.8% to 31.8%. Figure 9 shows that the unit step response of ea to Th is a curve cluster when the disturbance interval of a is 90275 to 50275. It can be seen from the performance of curve cluster in Figure 9 that ea reaches the peak value in about 0.1 seconds when Th is applied, and the fluctuation range is 0.06% to 0.094%, and the jitter range relative to its central curve is −0.02% to –0.01%. After the peak value, the curve in the curve cluster drops sharply and tends to be stable in about 1.6 seconds. Therefore, as a closed-loop system, EPS is robust. The research on different parameter weights shows that the robust performance of the whole system is better than before after the H∞ robust controller is applied to the control system.Figure 9
Unit step response curve cluster I ofea to Th.In the process of simulation, it is very important to adjust the parameters of weighting function. For example, the variation range of parametera in Figures 9 and 10 is 90275 to 50275, which reflects the tolerance effect of Th and external interference on ea. But the weighting function selected in Figure 9 is WT1s=0.01s+3/s+6000, while the weighting function selected in Figure 10 is WT1s=50s+30/s+6000. It can be seen from the two figures that the stability accuracy of the controlled system in Figure 10 is worse than that in Figure 9, and the ea response curve in Figure 10 is relatively divergent than that in Figure 9. In general, the amplitude and follow-up of the response curves in Figures 9 and 10 reflect the robust stability of the controlled system.Figure 10
Unit step response curve cluster II ofea to Th.EPS controller may produce a certain range of disturbance in the process of its use. The causes may be as follows: first, from the hardware point of view, it may be due to the time shift of physical devices in the process of loading the program; second, from the perspective of software, the transfer function in our design ofH∞ controller may be high-order, but in practical application, as hardware operation speed and accuracy cannot keep up with the ideal controller, in this case, we can only reduce the order of the controller algorithm, and then, this also leads to a certain degree of error in the actual controller. So the disturbance of the controller also has the problem of stability. Using the same method, if other conditions are the same, only the uncertainty of the controller itself is changed to |Δ|, so that |Δ|≤ 10%, then the generalized controlled system model is as follows:(42)Gps=1+ΔG0s.At this time,ea response curve cluster to Th is shown in Figure 11.Figure 11
Step response fromTh to ea with controller uncertainty.It can be seen that the response curve cluster has good stability and robust stability in terms of the convergence of amplitude and final curve and the divergence of curve cluster.


## 6. Hardware in the Loop Test

In the part of simulation test, the robustness of EPS is simulated and evaluated by MATLAB. In this section, the designed controller is embedded into the built-in EPS experimental platform, and the tracking effect between the actual value and the target value is mainly observed, so as to verify the performance of the controller. The EPS hardware in the loop test bed [25] is shown in Figure 12.Figure 12
EPS hardware in the loop test bed.Under the condition of in situ simulation, the hand torque is applied on the steering wheel, and the dSPACE software console records the actual current to the target current data under different conditions in real time. The tracking effect between the actual torque and the target torque and between the actual current and the target current is analyzed [26], and then the performance of the controller is verified. The hardware in the loop test results are shown in Figures 13 and 14.Figure 13
Response curve under step signal. (a) Angle contrast curve. (b) Torque contrast curve. (c) Current contrast curve.
(a)(b)(c)Figure 14
Response curve under sinusoidal signal. (a) Angle contrast curve. (b) Torque contrast curve. (c) Current contrast curve.
(a)(b)(c)Figures13 and 14 are the response curves under two different input paths, respectively. From the angle comparison in Figure 13(a), it can be seen that the real angle and the target angle follow well, and the real angle fluctuates about 0.8%. After 50 ms, the real angle reaches a stable state and can always be stable near the expected value. To sum up, when the input changes, although the rotation angle will fluctuate slightly, the real rotation curve will follow the target curve smooth and stable.Comparing Figure13(b) with Figure 13(c) and Figure 14(b) with Figure 14(c), we can see that the torque changes with the change of current, both of them reach the peak value in 1400 and 200 ms, respectively, then there is a small amplitude oscillation, and finally reach the stable value immediately. The algorithm ensures the fast response of the current and the small fluctuation in the target range. The results show that the optimization of the power steering system is good, which can make the driver get less delay road feel and smooth feel, and can make the power motor reach the predetermined torque in a short time.From the overall test results, the hysteresis between the power steering torque and the target torque is very small, which indicates that the sensitivity of the system is very good. In the comparison diagram of actual angle and target angle, the curve following performance is better, and there is no large fluctuation and overshoot phenomenon, which indicates that the power steering system will not appear steering jitter phenomenon in the actual field work.


## 7. Conclusion

In this article, the EPS controller based on LMI control method with mixed sensitivity operator is designed. First, nonlinear dynamic model of EPS controlled system is derived. Then, LMI controller with mixed sensitivity operator is derived, and the controller stability is verified. Simulation is carried out to validate the control effectiveness. The main work of this article mainly has the following aspects:(a)
When the EPS controller is designed with LMI algorithm simply, if the hand torque input frequency becomes larger, the steering wheel angle and steering gear angle cannot keep up with the change of hand torque in amplitude and frequency. It can be concluded that the single LMI controller cannot meet the sensitivity requirements of EPS fast steering.(b)
The LMI controller with mixed sensitivity operator is designed. The antidisturbance ability of the controller and the control effect of the power steering system are verified by simulation and hardware in the loop experiment. Compared with the simple LMI controller and the LMI controller with mixed sensitivity, the tracking performance of the gear angle to the steering wheel angle is robust. That is to say, in the process of vehicle driving, the tracking performance target of the controlled system can be guaranteed under the changes of road interference, frequency, and amplitude of steering wheel input hand torque.



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*Source: 5513378-2021-08-14.xml*