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Research Papers

# Lateral Motion Stability Control Via Sampled-Data Output Feedback of a High-Speed Electric Vehicle Driven by Four In-Wheel MotorsOPEN ACCESS

[+] Author and Article Information
Qinghua Meng

School of Mechanical Engineering,
Hangzhou Dianzi University,
Hangzhou 310018, Zhejiang, China
e-mail: mengqinghua@hdu.edu.cn

Chunjiang Qian

College of Engineering,
University of Texas at San Antonio,
San Antonio, TX 78249
e-mail: chunjiang.qian@utsa.edu

Pan Wang

School of Automation,
Southeast University,
Nanjing 210096, JiangSu, China
e-mail: panwangqf@126.com

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received April 2, 2016; final manuscript received May 30, 2017; published online August 29, 2017. Assoc. Editor: Azim Eskandarian.

J. Dyn. Sys., Meas., Control 140(1), 011002 (Aug 29, 2017) (8 pages) Paper No: DS-16-1170; doi: 10.1115/1.4037266 History: Received April 02, 2016; Revised May 30, 2017

## Abstract

This paper presents a lateral motion stability control method for an electric vehicle (EV) driven by four in-wheel motors subject to time-variable high speeds and uncertain disturbances caused by severe road conditions, siding wind forces, and different tire pressures. In order to tackle the uncertain disturbances, an almost disturbance decoupling method (ADD) using sampled-data output feedback control which is more suitable for computer implementation is proposed based on the domination approach. The proposed controller can attenuate the disturbances' effect on the output to an arbitrary degree of accuracy. Simulation results under different speeds by matlab show the effectiveness of the control method.

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## Introduction

Since electric vehicles (EVs), especially driven by four independent in-wheel motors, have shown superior potentials in energy efficiency, zero emission, performance benefits, and so on, more and more researchers and engineers are interested in the technologies such as vehicle motion control, energy optimization, and vehicle structural arrangement [14]. Unexpected disturbances, caused by perturbations such as severe road conditions, siding wind forces, and different tire pressures, may result in dangerous lateral motions when an EV runs under a high-speed. In order to drive an EV safely, drivers are required to react extremely quickly in such dangerous situations, which however is not an easy task. Therefore, improvement of the EV lateral motion dynamics by active vehicle control to avoid such catastrophic situations has been and is continuing to be a subject of active research [57]. However, most of these control algorithms are designed for the conventional vehicle architecture, not for the EV driven by four in-wheel motors. In the recent years, researchers proposed several lateral motion control methods. For example, a direct yaw-moment control system for an EV driven by four in-wheel motors was proposed in Ref. [8]. A linear approximation of vehicle dynamics model was used in the controller design. Four-wheel driven systems can provide vehicle stability by using the braking and acceleration abilities of independent in-wheel electric motors on each wheel in Refs. [9] and [10]. Two different approaches, one was a novel integrated lateral stability control system and the second was a regenerative braking-based lateral stability control system, were used in Ref. [11].

With the front wheel steering angle and four wheels' driving/braking torques as the control inputs, a vehicle driven by four in-wheel motors is a typical over-actuated system. Control allocation is a key process in the control of over-actuated system. There have been several different control-allocation methods, such as pseudo-inverse, daisy-chaining, linear programming, nonlinear programming, mixed-integer programming, and fixed-point methods [1214]. It is worthwhile to note, however, the numerical-optimization-based control-allocation methods usually have the drawback of high computational requirements, which may discourage their real-time implementations. Therefore, improving or replacing such methods with low-cost computing methods for the EV is more desirable.

As aforementioned, when an EV runs in cruise conditions, especially in high-speed cruise conditions, many factors will generate uncertain disturbances to the body and front steering wheels, which will influence the lateral stability of the EV seriously. The aforementioned control methods did not include uncertain disturbances or included disturbances as known to simplify the model and computation. In most conditions, some state parameters of the EV are not available, that means it is more difficult to control. Some controllers were designed using the state parameters acquired from equipped sensors which are expensive and sometime may fail to function well. In order to solve these problems, this paper studies the problem of disturbance attenuation with internal stability by designing a sampled-data output feedback controller, called almost disturbance decoupling (ADD) by sampled-data output, to control the lateral motion stability of the EV under high speeds despite of unknown system state parameters. The control objective of this paper is to use only output feedback to design a sampled-data controller which ensures almost disturbance decoupling with global asymptotic stability. The almost disturbance decoupling problem had been studied for both linear and nonlinear systems [1517]. Assuming that the nonlinearities satisfy a lower-triangular linear growth condition, the almost disturbance attenuation with internal stability via output feedback could be achieved using the feedback domination approaches in Ref. [18]. Disturbance observer-based control had been proven to be effective in compensating the effects of unknown external disturbances for nonlinear systems in Ref. [19] by designing a disturbance compensation gain vector or matrix. The method was extended to address the ADD problem for nonlinear systems with arbitrary disturbance relative degree [20].

However, the aforementioned achievements were achieved using continuous-time controllers, despite the fact that more and more systems are implemented digitally in practice. Sampled-data technologies were applied into control systems in the recent years [21,22]. It is necessary to solve the ADD problems by sampled-data controllers. A sampled-data observer-based output feedback fuzzy controller for nonlinear systems was designed using exact discrete-time design method in Ref. [23]. Based on the input delay approach, output feedback sampled-data polynomial controller for nonlinear systems was studied in Ref. [24]. A sampled-data output feedback controller was designed by using the domination approach with a tunable scaling gain and a tunable sampling period, which only based on the nominal linear system in Ref. [25]. However, these mentioned sampled-data output feedback control methods excluded disturbances in these systems.

This paper proposes an ADD controller based on sampled-data output feedback to keep the lateral motion stability of the EV driven by four in-wheel motors under high speeds. The main contributions of this paper lie in the following aspects:

1. (1)The time-variable speeds and uncertain disturbances from severe road, siding wind force, different tire pressures, and so on in the cruise conditions are included in the EV dynamic model.
2. (2)A linear state estimate discrete-time observer is designed to construct the sampled-data control law.
3. (3)A sampled-data output feedback controller achieving global asymptotically stable is designed without disturbances and dominating the disturbances' effect on the output using scaling gain.

The rest of this paper is organized as follows: The high-speed lateral motion dynamic model of an EV subject to tire forces, time-variable speeds, and uncertain disturbances is presented in Sec. 2. Section 3 presents the problem formulation. A sampled-data output feedback controller is designed in Sec. 4. Section 5 simulates the controller for lateral stability of an EV under different high speeds, which is followed by the conclusion in Sec. 6.

## High-Speed Lateral Motion Dynamic Model of an Electric Vehicle

The lateral motion model of an EV driven by four in-wheel motors under high speeds simplified to two wheels model is shown in Fig. 1 [26]. Different from the conventional car structures, each wheel is independently driven by an in-wheel motor, so an external yaw moment can be easily generated to regulate the car yaw and lateral motions because of the fast and precise torque response of motors. To design the controller, a set of widely used car lateral steering dynamics, ignoring the pitch and roll motions, can be expressed as follows:

Display Formula

(1)$β̇=Fyf+FyrmV−γγ̇=LfFyf−LrFyr+MzIz$

Mz is generated by longitudinal tire force difference between the left and right side wheels and can be controlled to compensate the car yaw rate.

Lateral forces of front and rear tires can be calculated according to the following linear tire model [27]: Display Formula

(2)$Fyf=Cfαf, Fyr=Crαr$

αf and αr can be calculated as Display Formula

(3)$αf=−(β+LfγV−δ(t)), αr=−(β−LrγV)$

Based on Eqs. (2) and (3), system (1) can be rewritten as Display Formula

(4)$β̇=−(Cf+Cr)βmV+(CrLr−CfLf)γmV2−γ+CfmVδ(t)γ̇=(CrLr−CfLf)βIz−(CfLf2+CrLr2)γIzV+MzIz+CfLfIzδ(t)$
In Eq. (4), the state vector includes the side slip angel β and the yaw rate γ, and the input is the external yaw moment Mz. δ(t) is uncertain disturbance which is generated by severe road conditions, siding wind forces, or different tire pressures and so on when an EV runs straightly under high speeds. Note that the front wheel steering angle δ(t) is usually very small under high speeds and tends to zero after some time, otherwise the EV will lose the lateral stabilization. The researchers generally regarded speed V as a constant in order to simply the analysis. However, it is known that the speed varies when the EV runs. In order to reflect the real conditions, this paper takes consideration of the case when the involved V is time-varying. In an analytical way, V can be formulated with uncertainties as the following form: Display Formula
(5)$ρ1(t)=1V, ρ2(t)=1V2$

Although V is time-varying, it has a limited highest speed. Therefore, we have Display Formula

(6)$0 ⩽ ρ1(t)≤θ1, 0 ⩽ ρ2(t)≤θ2$
Display Formula
(7)$|ρ̇2(t)|≤θ3$

for constants θ1, θ2, and θ3 ≥ 0.

Let $x(t)=(x1(t),x2(t))T∈ℝ2$ be system state, $u(t)∈ℝ, y(t)∈ℝ$, and $d(t)∈ℝ$ be the system input, output, and uncertain disturbance input, respectively. Under the following notations: Display Formula

(8)$x1(t)=β$
Display Formula
(9)$x2(t)=(CrLr−CfLf)ρ2(t)−mmγ$
Display Formula
(10)$u(t)=(CrLr−CfLf)ρ2(t)−mmIzMz$
Display Formula
(11)$d(t)=δ(t)$

system (4) can be rewritten as follows: Display Formula

(12)$ẋ1(t)=x2(t)+ϕ1(t,x(t))+g1d(t)ẋ2(t)=u(t)+ϕ2(t,x(t))+g2d(t)y(t)=x1(t)$

where Display Formula

(13)$g1(t)=Cfmρ1(t)$
Display Formula
(14)$g2(t)=(CrLr−CfLf)ρ2(t)−mmCfLfIz$
Display Formula
(15)$ϕ1(t,x(t))=−Cf+Crmρ1(t)x1(t)$
Display Formula
(16)$ϕ2(t,x(t))=[(CrLr−CfLf)ρ2(t)−m](CrLr−CfLf)mIzx1(t)+[−CrLr2+CfLf2Izρ1(t)+(CrLr−CfLf)ρ̇2(t)(CrLr−CfLf)ρ2(t)−m]x2(t)$

## Problem Formulation

The objective of this paper is to design a sampled-data output feedback controller for the EV lateral motion stability under high speeds such that the resulting closed-loop system is globally asymptotically stable at the origin when d(t) = 0 and also satisfies almost disturbance attenuation for every disturbance d(t) ∈ L2 to keep the lateral stability. Then, we can design an ADD controller via sampled-data output feedback. Given a real number γ > 0, a linear sampled-data feedback controller u(tk) is designed as Display Formula

(17)$ζ(tk+1)=Mζ(tk)+Ny(tk),u(tk)=−Kζ(tk),∀t∈[tk,tk+1), tk=kT, k=0,1,2,…,$

where the time instant tk and tk+1 are the sampling points, T is the sampling period, and $ζ∈ℝ2$ is the controller state, such that the following holds:

1. (1)When disturbance d(t) = 0, the closed-loop systems (12) and (17) are globally asymptotically stable at the equilibrium (x, ζ) = (0, 0).
2. (2)For every disturbance d(t) ∈ L2, the response of the closed-loop system (12) starting from the origin satisfies Display Formula
(18)$∫0∞|y(s)|2ds ⩽ γ2∫0∞‖d(s)‖2ds$

## Controller Design

In this section, a sampled-data output feedback controller to solve the problem of almost disturbance decoupling for system (1) will be designed. The main result is described as the following theorem.

Theorem 1.Under Eqs. (21) and (22), the problem of ADD for system (12) can be solved by a sampled-data output feedback controller in the form of Eq. (17).

Proof. The construction of the sampled-data output feedback controller can be divided into four steps. The first step is to change the coordinates for system (12). The second step is to design a linear discrete-time observer to estimate unmeasurable state. The third step is to construct a linear sampled-data control law using the estimated states. Finally, the scaling gain and sampling period are determined to globally stabilize the system (12) with the help of output feedback domination approach.

###### Pretreatment of System (12): Change of Coordinates.

First of all, a change of coordinates for system (12) is introduced. For a constant L ≥ 1 to be determined later, define the following change of coordinates: Display Formula

(19)$z1(t)=x1(t), z2(t)=x2(t)L, v(t)=u(t)L2$

Under this coordinates change, system (12) becomes Display Formula

(20)$ż1(t)=Lz2(t)+ϕ¯1(t,z(t))+g¯1(t)d(t)ż2(t)=Lv(t)+ϕ¯2(t,z(t))+g¯2(t)d(t)y(t)=z1(t)$

where $ϕ¯i(t,z(t))=ϕi(t,z(t))/Li−1, i=1,2, g¯i(t)=gi(t)/Li−1, i=1,2$.

From Eqs. (6), (15), and (16), it is obvious that a constant $c ⩾ 0$ always exists such that Display Formula

(21)$|ϕ1(t,x(t))|=|Cf+Crmρ1(t)x1(t)| ⩽ c|x1(t)||ϕ2(t,x(t))| ⩽ c(|x1(t)|+|x2(t)|)$

On the other hand, it is also obvious that there always exists a known constant $G0 ⩾ 0$ such that Display Formula

(22)$|gi| ⩽ G0, ∀t ⩾ 0, i=1,2$

From Eqs. (21) and (22), it can be verified that for i = 1, 2 Display Formula

(23)$|ϕ¯i(t,z(t))| ⩽ cLi−1(z1(t)+Li−1z2(t)) ⩽ c(z1(t)+z2(t))$
Display Formula
(24)$|g¯i(t)| ⩽ G0Li−1 ⩽ G0$

Define

$z(t)=[z1(t)z2(t)], A=[0100], B=[01], C=[10]Φ(⋅)=[ϕ¯1(⋅)ϕ¯2(⋅)], G(⋅)=[g¯1(⋅)g¯2(⋅)]$

under which system (20) can be rewritten as Display Formula

(25)$ż(t)=LAz(t)+LBv(t)+Φ(t,z(t))+G(t)d(t)y(t)=Cz(t)$

Since only the output y(t) = z1(t) is measurable at sampling points and the state z2(t) is not available, a linear observer should be designed to estimate the unmeasurable state of system (25).

###### Design of a Linear Discrete-Time Observer for System (25).

An observer is designed with continuous-time states over [tk, tk+1), discrete-time output z1(tk), and input v(tk) using the same method as in Refs. [28] and [29], without involving the $ϕ¯i(t,z(t))$ and $g¯i(t)$, i = 1, 2. More specially, we design the following observer: Display Formula

(26)$ẑ̇1(t)=Lẑ2(t)+La1(z1(tk)−ẑ1(t))ẑ̇2(t)=Lv(tk)+La2(z1(tk)−ẑ1(t)), ∀t∈[tk,tk+1)$

where a1 and a2 are the coefficients of the Hurwitz polynomial p1(s) = s2 + a2s + a1.

Defining

$ẑ(t)=[ẑ1(t) ẑ2(t)]T, H=[a1 a2]T, Â=A−HC$

observer (26) can be rewritten as Display Formula

(27)$ẑ̇(t)=LÂẑ(t)+LBv(tk)+LHy(tk), ∀t∈[tk,tk+1)$

###### Construction of a Sampled-Data Output Feedback Control Law.

Because the state z2 is not measurable, the sampled-data control law using the estimated $ẑ(tk)$ generated from the observer (27) is constructed as Display Formula

(28)$v(t)=v(tk)=−Kẑ(tk)=−k1ẑ1(tk)−k2ẑ2(tk),∀t∈[tk,tk+1), k=0,1,2,…,$
where k1 and k2 are the coefficients of Hurwitz polynomial p2(s) = s2 + k2s + k1 and both positive.

Substituting Eq. (28) into Eqs. (25) and (27), a closed-loop system is obtained for $t∈[tk,tk+1)$Display Formula

(29)$Ż(t)=[ż(t)ẑ̇(t)]=L[A−BKHCÂ−BK][z(t)ẑ(t)]−L[BB]K(ẑ(tk)−ẑ(t))+L[0HC](z(tk)−z(t))+[Φ(⋅)0]+[G(⋅)0]d(t)$

Denote

$A=[A−BKHCÂ−BK]=[A−BKHCA−HC−BK]$

As aforementioned, $Â=A−HC$ and A−BK are Hurwitz matrices. Thus, $A$ is a Hurwitz matrix as well from the following form:

$A=[I0−II]−1[A−BK−BK0Â][I0−II]$

Therefore, there is a positive definite matrix $P=PT∈ℝ4×4>0$ such that $ATP+PA=−I4×4$.

Construct a Lyapunov function V (Z) = ZTPZ with $Z=[z(t)ẑ(t)]T$. The derivative of V(Z) along system (29) is Display Formula

(30)$V̇(Z(t))=−L‖Z(t)‖2−2LZ(t)TP[BB]K(ẑ(tk)−ẑ(t))+2LZ(t)TP[0HC](z(tk)−z(t))+2Z(t)TP[Φ(⋅)0]+2Z(t)TP[G(⋅)0]d(t)=−L‖Z(t)‖2+2LZ(t)TP[0BK−HCBK][Z(t)−Z(tk)]+2Z(t)TP[Φ(⋅)0]+2Z(t)TP[G(⋅)0]d(t)$

###### Determination of Gain L and Computation of T.

To determine scaling gain L and sampling period T, it needs to find the estimates for some terms in Eq. (30). According to Eq. (23), the following holds: Display Formula

(31)$‖Φ(⋅)‖=ϕ¯12+ϕ¯22 ⩽ c1‖Z(t)‖$

where $c1=c1+2≥0$. With the help of Eq. (31), we have Display Formula

(32)$|2ZTP[Φ(⋅)0]|≤2c1λmax(P)‖Z(t)‖2$

According to Eq. (24), the following holds: Display Formula

(33)$|2ZTP[G(⋅)0]|d(t)≤4λmax(P)2G02‖Z(t)‖2+12‖d(t)‖2$
By Eqs. (29) and (31), it is straightforward to verify that Display Formula
(34)$‖Ż(t)‖≤c2‖Z(t)‖+c3‖Z(tk)‖+c4‖d(t)‖,∀t∈[tk,tk+1)$

for three constants c2, c3, and c4. With the help of Eq. (34), integrating Eq. (29) from tk to t yields Display Formula

(35)$‖Z(t)−Z(tk)‖≤∫tkt(c2‖Z(τ)‖+c3‖Z(tk)‖+c4‖d(τ)‖)dτ$

where t ∈ [tk, tk+1).

Define Display Formula

(36)$M(t)=∫tkt(c2‖Z(τ)‖+c3‖Z(tk)‖)dτ$

Taking derivative of $M(t)$ yields Display Formula

(37)$Ṁ(t)=c2‖Z(τ)‖+c3‖Z(tk)‖≤c2‖Z(τ)−Z(tk)‖+(c2+c3)‖Z(tk)‖≤c2M(t)+c2c4∫tkt‖d(τ)‖dτ+(c2+c3)‖Z(tk)‖$

By general solution of homogeneous linear equations, solving Eq. (37) for t ∈ [tk, tk+1) with $M(tk)=0$, we can get Display Formula

(38)$M(t)≤ec2(t−tk)∫tktec2(τ−tk)c2c4∫tkτ‖d(s)‖dsdτ+ec2(t−tk)∫tktec2(τ−tk)ec2(t−tk)(c2+c3)‖Z(tk)‖dτ=c2+c3c2‖Z(tk)‖[ec2(t−tk)−1]−c4∫tkt‖d(s)‖ds+c4ec2(t−tk)∫tkte−c2(τ−tk)‖d(τ)‖dτ$

Substituting Eq. (38) into Eq. (35), the following holds: Display Formula

(39)$‖Z(t)−Z(tk)‖≤c2+c3c2[ec2(t−tk)−1]‖Z(t)−Z(tk)‖+c2+c3c2[ec2(t−tk)−1]‖Z(t)‖)+c4ec2(t−tk)∫tkt‖d(τ)‖dτ$

If T is chosen small enough, denoting $Δ(t)=(c2+c3)/c2(ec2(t−tk)−1)$, there exists Display Formula

(40)$‖Z(t)−Z(tk)‖≤Δ(t−tk)1−Δ(t−tk)‖Z(t)‖+c4ec2(t−tk)1−Δ(t−tk)∫tkt‖d(τ)‖dτ$

Substituting Eqs. (32), (33), and (40) into Eq. (30) results in Display Formula

(41)$V̇(Z(t))≤−L‖Z(t)‖2+2LΓΔ(t−tk)1−Δ(t−tk)‖Z(t)‖2+2LΓ‖Z(t)‖c4ec2(t−tk)1−Δ(t−tk)∫tkt‖d(τ)‖dτ+2c1λmax(P)‖Z(t)‖2+4λmax(P)2G02‖Z(t)‖2+12‖d(t)‖2$

where Display Formula

(42)$Γ=λmax(P)‖0BK−HCBK‖$

According to Eq. (20), we have Display Formula

(43)$1γ2|y(t)|2=1γ2|[C0]Z(t)|2≤1γ2‖Z(t)‖2$

Combining Eqs. (41) and (43), we obtain Display Formula

(44)$V̇(Z(t))+1γ2|y(t)|2−‖d(t)‖2≤−L‖Z(t)‖2+2LΓΔ(t−tk)1−Δ(t−tk)‖Z(t)‖2+2LΓc4ec2(t−tk)1−Δ(t−tk)‖Z(t)‖∫tkt‖d(τ)‖dτ+2c1λmax(P)‖Z(t)‖2+4λmax(P)2G02‖Z(t)‖2+1γ2‖Z(t)‖2−12‖d(t)‖2$

where t ∈ [tk, tk+1).

Considering the stability of system (30), L can be fixed appropriately as Display Formula

(45)$L=2+2c1λmax(P)+4λmax(P)2G02+1γ2$

Since $t∈[tk,tk+1), Δ(t−tk)≤Δ(T)$. Substituting Eq. (45) into Eq. (44) and integrating both sides of Eq. (44) from tk to tk+1 result in Display Formula

(46)$∫tktk+1(V̇(Z(t))+1γ2|y(t)|2−‖d(t)‖2)dt≤−2∫tktk+1‖Z(t)‖2dt+2LΓΔ(T)1−Δ(T)∫tktk+1‖Z(t)‖2dt+2LΓc4ec2T1−Δ(T)∫tktk+1‖Z(t)‖∫tkt‖d(τ)‖dτdt+2Lλmax(P)∫tktk+1‖Z(t)‖2dt−12∫tktk+1‖d(t)‖2dt$

According to Cauchy–Schwarz inequality, we have Display Formula

(47)$∫tktk+1‖Z(t)‖∫tkt‖d(τ)‖dτdt≤12∫tktk+1‖Z(t)‖2dt∫tktk+11dt+12∫tkt‖d(t)‖2dt∫tktk+11dt=T2∫tktk+1‖Z(t)‖2dt+T2∫tkt‖d(t)‖2dt$

Substituting Eq. (47) into Eq. (46) results in Display Formula

(48)$∫tktk+1(V̇(Z(t))+1γ2|y(t)|2−‖d(t)‖2)dt≤[−2+2LΔ(T)+Tc4ec2T1−Δ(T)Γ]∫tktk+1‖Z(t)‖2dt+[TLΓc4ec2T1−Δ(T)−12]∫tktk+1‖d(t)‖2dt$
If T is chosen as a small enough constant such that Display Formula
(49)$−1+(2LΔ(T)1−Δ(T)+Tc4ec2T1−Δ(T))Γ<0$
Display Formula
(50)$TLΓc4ec2T1−Δ(T)−12<0$

The right side of Eq. (48) is negative definite. More specifically, we have Display Formula

(51)$∫tktk+1(V̇(Z(t))+1γ2|y(t)|2−‖d(t)‖2)dt≤−∫tktk+1‖Z(t)‖2dt$

from which it can be concluded that the equilibrium of the closed-loop system is uniformly globally asymptotically stable when d(t) = 0.

Moreover, the following can be obtained based on Eq. (51): Display Formula

(52)$V(Z(tn))−V(Z(t0))+∑k=0n−1∫tktk+11γ2|y(t)|2dt−∑k=0n−1∫tktk+1‖d(t)‖2dt≤0$

When Z(t0) = 0, based on the fact that $V(Z(tn))≥0$ and V(0) = 0, it can be concluded from Eq. (52) that Display Formula

(53)$∑k=0n−1∫tktk+11γ2|y(t)|2dt≤∑k=0n−1∫tktk+1‖d(t)‖2dt$

Because of t0 = 0 and letting $n→∞$, the following inequality holds: Display Formula

(54)$∫0∞|y(τ)|2dτ≤γ2∫0∞‖d(τ)‖2dτ$

Then, the almost disturbance decoupling in the L2 sense (18) is proved.

According to Eqs. (49) and (50), the following holds: Display Formula

(55)$Δ(T)≤min{1−Tc4ec2TΓ1+2LΓ,1−2TLc4ec2TΓ}$

Define

$W=min{1−Tc4ec2TΓ1+2LΓ,1−2TLc4ec2TΓ}$

Based on Eq. (55), the sampling period T can be calculated as Display Formula

(56)$T≤1c2ln(W+1)c2+c3c2+c3$

## Simulation Study

In this section, the simulation of the control system (25) is conducted by matlab. The EV parameters are listed in Table 1. Generally, the range of the high-speed V is about from 80 km/h to 120 km/h. When the EV runs under constant speed, the speed is allowed to vary about ±3%.

From Eq. (12), the simulated EV's lateral motion control system is Display Formula

(57)$ẋ1(t)=x2(t)+38.571ρ1(t)x1(t)+18.571ρ1(t)d(t)ẋ2(t)=u(t)+(0.0016ρ2(t)−1)x1(t)+21.6ρ1(t)x2(t)+8ρ̇2(t)8ρ2(t)−7x2(t)+(14.857ρ2(t)−13)d(t)$

where u(t) is the control torque. $d(t)=sin(t)/(1+t2)$ is used as the disturbance to simulate the disturbance generated by siding wind force. The disturbance is shown in Fig. 2. We also choose c ≥ 1.996 and G0 ≥ 19.96 to meet Eqs. (21) and (22). The sampled-data controller is

Display Formula

(58)$u(t)=u(tkc)=L2v(kT)=−17L2ẑ1(kT)−13L2ẑ2(kT)$

The observer is Display Formula

(59)$ẑ̇(t)=L[−171−130]ẑ(t)+[01]u(tk)L+L[170130]z(tk)$

In this paper, we just carried out electric vehicle stability control simulations under 120 km/h and 80 km/h. In the simulations, electric vehicle runs on the good city road. The initial values are set z(0) = [0, 0]T and $ẑ(0)=[0,0]T$ in the numerical simulation. The sampled-data controller input is shown in Fig. 3. It is demonstrated that the sampled-data controller responds with the disturbance immediately to stabilize the EV from Figs. 2 and 3. In the simulation, there is no disturbance applied to the EV from time 0 to 1st s when the EV runs straightly, and the controller is not active. The disturbance is applied to the EV from 1st s to 3rd s, and the controller reacts to control the EV's lateral motion stability. According to Eqs. (8) and (9), we map controller's states x1 and x2 into EV's parameters β and γ. The simulation results are shown from Figs. 49 under 120 km/h and 80 km/h. The simulation results show the effectiveness of the sampled-data output feedback controller. The controller can stabilize the EV's lateral motion under different high speeds, as shown from Figs. 49. The system state β and observer $β̂$ are stabilized to zero within 2 s as shown in Figs. 4 and 7, and the system state γ and observer $γ̂$ are stabilized to zero within 1.5 s as shown in Figs. 5 and 8. The overshoots of vehicle sideslip angle and yaw rate are small and attenuate rapidly. We can see that the output y(t) which also is the EV's sideslip angle β is stabilized to zero within 2 s as shown in Figs. 6 and 9, that means the controller can keep the EV's lateral motion stability.

## Conclusions

In this paper, for an EV driven by four in-wheels motors with uncertain disturbances under high speeds, the lateral motion stability control has been solved based on ADD via a sampled-data output feedback controller. The controller is not only more practical in practice but also more easy to design. First, the lateral motion dynamic model is built subject to the time-variable speeds and uncertain disturbances. Then, the sampled-data output feedback controller is constructed to solve the ADD problem for the EV control system which is solved usually by a continuous-time output feedback controller. The output feedback domination approach is also used to dominate the disturbances by using a scaling gain. With the help of these tools, the sampling period is calculated to guarantee global stability and disturbance attenuation. Finally, the simulations are conducted to prove the effectiveness of the sampled-data output feedback controller for the EV's lateral motion stability under 120 km/h and 80 km/h.

## Funding Data

• National Natural Science Foundation of China (Grant No. 51105124).

• Natural Science Foundation of Zhejiang Province (Grant No. LY16E050003).

• China Scholarship Council.

## Nomenclature

• Cf =

front tire cornering stiffness

• Cr =

rear tire cornering stiffness

• Fyf =

front tire lateral force

• Fyr =

rear tire lateral force

• Iz =

yaw inertia

• Lf =

distance between the center of gravity and the front wheel center

• Lr =

distance between the center of gravity and the rear wheel center

• m =

vehicle mass

• Mz =

external yaw moment

• u(t) =

controller input

• V =

vehicle speed

• x(t) =

system state

• y(t) =

system output

• $ẑ(t)$ =

observer state

• αf =

front wheel slip angle

• αr =

rear wheel slip angle

• β =

vehicle sideslip angle

• γ =

yaw rate

• δ =

front wheel steering angle

## References

Wang, J. , and Longoria, R. G. , 2009, “ Coordinated and Reconfigurable Vehicle Dynamics Control,” IEEE Trans. Control Syst. Technol., 17(3), pp. 723–732.
Shuai, Z. , Zhang, H. , Wang, J. , Li, J. , and Ouyang, M. , 2014, “ Combined AFS and DYC control of Four-Wheel-Independent-Drive Electric Vehicles Over CAN Network With Time-Varying Delays,” IEEE Trans. Veh. Technol., 63(2), pp. 591–602.
Chen, Y. , and Wang, J. , 2012, “ Design and Evaluation on Electric Differentials for Over-Actuated Electric Ground Vehicles With Four Independent In-Wheel Motors,” IEEE Trans. Veh. Technol., 61(4), pp. 1534–1542.
Meng, Q. , Xu, J. , and Dongfeng, W. , 2013, “ Power System of Electric Vehicle Driven by In-Wheel Motors,” Trans. Chin. Soc. Agric. Mach., 44(8), pp. 33–37.
Chen, B.-C. , and Kuo, C.-C. , 2014, “ Electronic Stability Control for Electric Vehicle With Four In-Wheel Motors,” Int. J. Automot. Technol., 15(4), pp. 573–580.
Hori, Y. , 2004, “ Future Vehicle Driven by Electricity and Control-Research on Four-Wheel-Motored UOT Electric March II,” IEEE Trans. Ind. Electron., 51(5), pp. 954–962.
Nam, K. , Fujimoto, H. , and Hori, Y. , 2012, “ Lateral Stability Control of In-Wheel-Motor-Driven Electric Vehicles Based on Sideslip Angle Estimation Using Lateral Tire Force Sensors,” IEEE Trans. Veh. Technol., 61(5), pp. 1972–1985.
Sakai, S.-I. , Sado, H. , and Hori, Y. , 1999, “ Motion Control in an Electric Vehicle With Four Independently Driven In-Wheel Motors,” IEEE/ASME Trans. Mechatron., 4(1), pp. 9–16.
Li, F. , Wang, J. , and Liu, Z. , 2009, “ Motor Torque Based Vehicle Stability Control for Four-Wheel-Drive Electric Vehicle,” IEEE Vehicle Power and Propulsion Conference (VPPC), Dearborn, MI, Sept. 7–10, pp. 1596–1601.
Sakai, S.-I. , Sado, H. , and Hori, Y. , 2002, “ Dynamic Driving/Braking Force Distribution in Electric Vehicles With Independently Driven Four Wheels,” Electr. Eng. Jpn., 138(1), pp. 79–89.
Em Irler, M. , Kahraman, K. , Şentürk, M. , Acar, O. , Güvenç, B. A. , Güvenç, L. , and Efend Ioğlu, B. , 2015, “ Lateral Stability Control of Fully Electric Vehicles,” Int. J. Automot. Technol., 16(2), pp. 317–328.
Johansen, T. A. , and Fossen, T. I. , 2013, “ Control Allocation—A Survey,” Automatica, 49(5), pp. 1087–1103.
Wang, J. , Solis, J. M. , and Longoria, R. G. , 2007, “ On the Control Allocation for Coordinated Ground Vehicle Dynamics Control Systems,” American Control Conference (ACC), New York, July 9–13, pp. 5724–5729.
Plumlee, J. H. , Bevly, D. M. , and Hodel, A. S. , 2004, “ Control of a Ground Vehicle Using Quadratic Programming Based Control Allocation Techniques,” American Control Conference (ACC), Boston, MA, June 30–July 2, Vol. 5, pp. 4704–4709.
Weiland, S. , and Willems, J. C. , 1989, “ Almost Disturbance Decoupling With Internal Stability,” IEEE Trans. Autom. Control, 34(3), pp. 277–286.
Lin, Z. , Bao, X. , and Chen, B. M. , 1999, “ Further Results on Almost Disturbance Decoupling With Global Asymptotic Stability for Nonlinear Systems,” Automatica, 35(4), pp. 709–717.
Marino, R. , Respondek, W. , and Van der Schaft, A. , 1989, “ Almost Disturbance Decoupling for Single-Input Single-Output Nonlinear Systems,” IEEE Trans. Autom. Control, 34(9), pp. 1013–1017.
Lin, W. , Qian, C. , and Huang, X. , 2003, “ Disturbance Attenuation of a Class of Non-Linear Systems Via Output Feedback,” Int. J. Robust Nonlinear Control, 13(13), pp. 1359–1369.
Yang, J. , Chen, W.-H. , and Li, S. , 2011, “ Non-Linear Disturbance Observer-Based Robust Control for Systems With Mismatched Disturbances/Uncertainties,” IET Control Theory Appl., 5(18), pp. 2053–2062.
Yang, J. , Chen, W. , Li, S. , and Chen, X. , 2013, “ Static Disturbance-to-Output Decoupling for Nonlinear Systems With Arbitrary Disturbance Relative Degree,” Int. J. Robust Nonlinear Control, 23(5), pp. 562–577.
Dharani, S. , Rakkiyappan, R. , and Cao, J. , 2015, “ Robust Stochastic Sampled-Data H Control for a Class of Mechanical Systems With Uncertainties,” ASME J. Dyn. Syst., Meas., Control, 137(10), p. 101008.
Lei, J. , 2013, “ Optimal Vibration Control for Uncertain Nonlinear Sampled-Data Systems With Actuator and Sensor Delays: Application to a Vehicle Suspension,” ASME J. Dyn. Syst., Meas., Control, 135(2), p. 021021.
Kim, D. W. , and Lee, H. J. , 2012, “ Sampled-Data Observer-Based Output-Feedback Fuzzy Stabilization of Nonlinear Systems: Exact Discrete-Time Design Approach,” Fuzzy Sets Syst., 201, pp. 20–39.
Lam, H. , 2011, “ Output-Feedback Sampled-Data Polynomial Controller for Nonlinear Systems,” Automatica, 47(11), pp. 2457–2461.
Qian, C. , and Du, H. , 2012, “ Global Output Feedback Stabilization of a Class of Nonlinear Systems Via Linear Sampled-Data Control,” IEEE Trans. Autom. Control, 57(11), pp. 2934–2939.
Zhao, H. , Gao, B. , Ren, B. , and Chen, H. , 2015, “ Integrated Control of In-Wheel Motor Electric Vehicles Using a Triple-Step Nonlinear Method,” J. Franklin Inst., 352(2), pp. 519–540.
Wang, R. , Zhang, H. , and Wang, J. , 2015, “ Robust Lateral Motion Control of Four-Wheel Independently Actuated Electric Vehicles With Tire Force Saturation Consideration,” J. Franklin Inst., 352(2), pp. 645–668.
Du, H. , Qian, C. , He, Y. , and Cheng, Y. , 2013, “ Global Sampled-Data Output Feedback Stabilisation of a Class of Upper-Triangular Systems With Input Delay,” IET Control Theory Appl., 7(10), pp. 1437–1446.
Chu, H. , Qian, C. , Yang, J. , Xu, S. , and Liu, Y. , 2016, “ Almost Disturbance Decoupling for a Class of Nonlinear Systems Via Sampled-Data Output Feedback Control,” Int. J. Robust Nonlinear Control, 26(10), pp. 2201–2215.
View article in PDF format.

## References

Wang, J. , and Longoria, R. G. , 2009, “ Coordinated and Reconfigurable Vehicle Dynamics Control,” IEEE Trans. Control Syst. Technol., 17(3), pp. 723–732.
Shuai, Z. , Zhang, H. , Wang, J. , Li, J. , and Ouyang, M. , 2014, “ Combined AFS and DYC control of Four-Wheel-Independent-Drive Electric Vehicles Over CAN Network With Time-Varying Delays,” IEEE Trans. Veh. Technol., 63(2), pp. 591–602.
Chen, Y. , and Wang, J. , 2012, “ Design and Evaluation on Electric Differentials for Over-Actuated Electric Ground Vehicles With Four Independent In-Wheel Motors,” IEEE Trans. Veh. Technol., 61(4), pp. 1534–1542.
Meng, Q. , Xu, J. , and Dongfeng, W. , 2013, “ Power System of Electric Vehicle Driven by In-Wheel Motors,” Trans. Chin. Soc. Agric. Mach., 44(8), pp. 33–37.
Chen, B.-C. , and Kuo, C.-C. , 2014, “ Electronic Stability Control for Electric Vehicle With Four In-Wheel Motors,” Int. J. Automot. Technol., 15(4), pp. 573–580.
Hori, Y. , 2004, “ Future Vehicle Driven by Electricity and Control-Research on Four-Wheel-Motored UOT Electric March II,” IEEE Trans. Ind. Electron., 51(5), pp. 954–962.
Nam, K. , Fujimoto, H. , and Hori, Y. , 2012, “ Lateral Stability Control of In-Wheel-Motor-Driven Electric Vehicles Based on Sideslip Angle Estimation Using Lateral Tire Force Sensors,” IEEE Trans. Veh. Technol., 61(5), pp. 1972–1985.
Sakai, S.-I. , Sado, H. , and Hori, Y. , 1999, “ Motion Control in an Electric Vehicle With Four Independently Driven In-Wheel Motors,” IEEE/ASME Trans. Mechatron., 4(1), pp. 9–16.
Li, F. , Wang, J. , and Liu, Z. , 2009, “ Motor Torque Based Vehicle Stability Control for Four-Wheel-Drive Electric Vehicle,” IEEE Vehicle Power and Propulsion Conference (VPPC), Dearborn, MI, Sept. 7–10, pp. 1596–1601.
Sakai, S.-I. , Sado, H. , and Hori, Y. , 2002, “ Dynamic Driving/Braking Force Distribution in Electric Vehicles With Independently Driven Four Wheels,” Electr. Eng. Jpn., 138(1), pp. 79–89.
Em Irler, M. , Kahraman, K. , Şentürk, M. , Acar, O. , Güvenç, B. A. , Güvenç, L. , and Efend Ioğlu, B. , 2015, “ Lateral Stability Control of Fully Electric Vehicles,” Int. J. Automot. Technol., 16(2), pp. 317–328.
Johansen, T. A. , and Fossen, T. I. , 2013, “ Control Allocation—A Survey,” Automatica, 49(5), pp. 1087–1103.
Wang, J. , Solis, J. M. , and Longoria, R. G. , 2007, “ On the Control Allocation for Coordinated Ground Vehicle Dynamics Control Systems,” American Control Conference (ACC), New York, July 9–13, pp. 5724–5729.
Plumlee, J. H. , Bevly, D. M. , and Hodel, A. S. , 2004, “ Control of a Ground Vehicle Using Quadratic Programming Based Control Allocation Techniques,” American Control Conference (ACC), Boston, MA, June 30–July 2, Vol. 5, pp. 4704–4709.
Weiland, S. , and Willems, J. C. , 1989, “ Almost Disturbance Decoupling With Internal Stability,” IEEE Trans. Autom. Control, 34(3), pp. 277–286.
Lin, Z. , Bao, X. , and Chen, B. M. , 1999, “ Further Results on Almost Disturbance Decoupling With Global Asymptotic Stability for Nonlinear Systems,” Automatica, 35(4), pp. 709–717.
Marino, R. , Respondek, W. , and Van der Schaft, A. , 1989, “ Almost Disturbance Decoupling for Single-Input Single-Output Nonlinear Systems,” IEEE Trans. Autom. Control, 34(9), pp. 1013–1017.
Lin, W. , Qian, C. , and Huang, X. , 2003, “ Disturbance Attenuation of a Class of Non-Linear Systems Via Output Feedback,” Int. J. Robust Nonlinear Control, 13(13), pp. 1359–1369.
Yang, J. , Chen, W.-H. , and Li, S. , 2011, “ Non-Linear Disturbance Observer-Based Robust Control for Systems With Mismatched Disturbances/Uncertainties,” IET Control Theory Appl., 5(18), pp. 2053–2062.
Yang, J. , Chen, W. , Li, S. , and Chen, X. , 2013, “ Static Disturbance-to-Output Decoupling for Nonlinear Systems With Arbitrary Disturbance Relative Degree,” Int. J. Robust Nonlinear Control, 23(5), pp. 562–577.
Dharani, S. , Rakkiyappan, R. , and Cao, J. , 2015, “ Robust Stochastic Sampled-Data H Control for a Class of Mechanical Systems With Uncertainties,” ASME J. Dyn. Syst., Meas., Control, 137(10), p. 101008.
Lei, J. , 2013, “ Optimal Vibration Control for Uncertain Nonlinear Sampled-Data Systems With Actuator and Sensor Delays: Application to a Vehicle Suspension,” ASME J. Dyn. Syst., Meas., Control, 135(2), p. 021021.
Kim, D. W. , and Lee, H. J. , 2012, “ Sampled-Data Observer-Based Output-Feedback Fuzzy Stabilization of Nonlinear Systems: Exact Discrete-Time Design Approach,” Fuzzy Sets Syst., 201, pp. 20–39.
Lam, H. , 2011, “ Output-Feedback Sampled-Data Polynomial Controller for Nonlinear Systems,” Automatica, 47(11), pp. 2457–2461.
Qian, C. , and Du, H. , 2012, “ Global Output Feedback Stabilization of a Class of Nonlinear Systems Via Linear Sampled-Data Control,” IEEE Trans. Autom. Control, 57(11), pp. 2934–2939.
Zhao, H. , Gao, B. , Ren, B. , and Chen, H. , 2015, “ Integrated Control of In-Wheel Motor Electric Vehicles Using a Triple-Step Nonlinear Method,” J. Franklin Inst., 352(2), pp. 519–540.
Wang, R. , Zhang, H. , and Wang, J. , 2015, “ Robust Lateral Motion Control of Four-Wheel Independently Actuated Electric Vehicles With Tire Force Saturation Consideration,” J. Franklin Inst., 352(2), pp. 645–668.
Du, H. , Qian, C. , He, Y. , and Cheng, Y. , 2013, “ Global Sampled-Data Output Feedback Stabilisation of a Class of Upper-Triangular Systems With Input Delay,” IET Control Theory Appl., 7(10), pp. 1437–1446.
Chu, H. , Qian, C. , Yang, J. , Xu, S. , and Liu, Y. , 2016, “ Almost Disturbance Decoupling for a Class of Nonlinear Systems Via Sampled-Data Output Feedback Control,” Int. J. Robust Nonlinear Control, 26(10), pp. 2201–2215.

## Figures

Fig. 1

The bike model of an EV driven by four in-wheel motors

Fig. 2

The disturbance used in the simulation

Fig. 3

The sampled-data output feedback controller input

Fig. 4

The system state β and observer state β̂ of the EV under 120 km/h (vehicle sideslip angle)

Fig. 5

The system state γ and observer state γ̂ of the EV under 120 km/h (yaw rate)

Fig. 6

The output of controller under 120 km/h

Fig. 7

The system state β and observer state β̂ of the EV under 80 km/h (vehicle sideslip angle)

Fig. 8

The system state γ and observer state γ̂ of the EV under 80 km/h (yaw rate)

Fig. 9

The output of controller under 80 km/h

## Tables

Table 1 The EV parameters used in the simulation

## Discussions

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