# HOSM Controller Using PI Sliding Manifold for an Integrated Active Control for Wheeled Vehicles

Authors: Antonio Navarrete Guzmán; Claudia Carolina Vaca García; Stefano Di Gennaro; Cuauhtémoc Acosta Lúa
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

This study considers the design of a modified high-order sliding mode (HOSM) controller using a PI sliding surface to the attitude control of a ground vehicle. A robust-modified HOSM controller is derived, so that the lateral velocity and yaw rate tracks the desired trajectory despite the environment actions acting on the ground vehicle and parameter variations. The stability is guaranteed with Lyapunov’s stability theorem function. The performance of the dynamic controllers is evaluated using the CarSim simulator considering a challenging double steer maneuver.


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## Body

## 1. Introduction

The integrated active control for wheeled vehicles is an important topic due to improving the drivability and safety of the vehicle in a critical situation. This active control employs electronic actuators in the vehicles. The new actuators offer great flexibility when a design control architecture allows the controller to be designed as a separate block. This decoupling helps to design the controller as an active front steering (AFS) or rear torque vectoring (RTV). The AFS controller imposes a steering angle to correct the action of the driver. The RTV controller imposes an active action in the yaw momentum of the vehicle. All these control actions render more safety to the ground vehicle and passengers.The control action is usually determined using approximation models because they capture the main aspect of the physics of the problem. But these mathematical models can be extended due to the presence of parameter uncertainties/variations and the presence of disturbances acting on the vehicle, among which are the change of parameters due to the road condition and the environmental disturbance acting on the vehicle (front and lateral wind). On this subject, the reader can find in [1–4]. However, many other mathematical models that include uncertainty in the parameters and external disturbances are available in the literature and could be considered, for instance, those in [5, 6].The article presents the combination of the modified HOSM with a PI sliding surface. The principal idea is to design a particular sliding surface on which the system dynamics are constrained to evolve by the modified HOSM. It is noted that the controller presented here maintains the inherent properties of the HOSM, such as robustness. For the above reasons, this article deals with designing a modified high-order sliding mode (HOSM) controller using a PI sliding surface to track the desired trajectory for lateral velocity and angular references despite variations in some parameters and external disturbances such as wind gusts. For this purpose, the active front steering (AFS) and rear torque vectoring (RTV) are considered as the integrated active control, and the aim is to improve the performance of the control system applied in automobiles. The proposed controller uses all the information from sensors that are currently available in modern vehicles. The closed-loop system stability has been proven in the sense of Lyapunov through the development of the candidate function. It was considered that the CarSim simulator to validate the controller is proposed. CarSim is a software that predicts dynamic vehicle behavior. In fact, CarSim delivers the most accurate, detailed, and efficient methods for simulating the performance of passenger vehicles and is supported by automotive enterprises, such as the Ford Motor Company and Chrysler.Earlier research on the integrated active controller is based on the nonlinear techniques as [7–9], and for more recent results, [10–14]. In [15], a nonlinear decoupling control approach is presented for a three-degree of freedom model. In [16], a yaw-stabilizing algorithm is presented, combining AFS with a low-level control of the longitudinal wheel slip, with adaptive yaw estimating the maximum tire-road friction parameter for each wheel. In [17], AFS and RTV are combined in an integrated controller to guarantee vehicle stability, making use of adaptive feedback.The HOSM was introduced in [18], and the main advantage of the high-order sliding mode is to overcome the chattering problem. A further notable advantage of HOSM is the possibility of considering sliding mode surfaces with a relative degree greater than one [19] and has great advantages compared to the sliding mode [20]. However, to add the integral action in discontinuous controllers as in the case of first-order sliding modes to improve the chattering phenomenon and smoothness of the controller action; the chattering is caused by reaching the sliding surface in finite time asymptotically fast, and this behavior generates the actuator commute quickly [21]. In the case of HOSM, adding a discontinuous integral action eliminates the effect of the disturbance; considering this action as an estimator of the disturbance [22], it has been shown that to continue adding integrators helps to decrease the chattering effect. Nonetheless, there are studies in which the integral action is added on the surface, improving the behavior of the controller in the transient time.Nevertheless, the reader can find works in which HOSM uses a PI controller as the sliding surface. In [23], the PID sliding mode control scheme is designed for a high-speed train (HST) subjected to actuator faults, asymmetric nonlinear actuator saturation. A combination of SMC with PID sliding surface is used to control a 2-degree of freedom (2-DOF) planar manipulator [24]. The study by Gu et al. [25] presents an SMC-PI sliding surface for robust tracking control of a nanopositioning stage composed of piezoceramic stack actuators (PSAs) and compliant flexure mechanisms. A second-order sliding mode using PI sliding surface applied in the control of DC motor drive is used in [26]. In [27], a sliding mode control with PID sliding surface is used to activate vibration damping of pneumatically actuated soft robots. In [28], the supertwisting algorithm combined with the proportional integral derivative sliding mode control modified is designed to solve trajectory tracking and stabilize the quadrotor attitude.The study is organized as follows. In Section2, the mathematical model of a vehicle is recalled, and the control problem is formulated. In Section 3, a modified HOSM with the PI sliding surface controller is presented. In Section 4, the resulting controller is tested with CarSim simulations. Some comments conclude the study.


## 2. Mathematical Model of the Vehicle Dynamics

In this article, to obtain the mathematical model of the ground vehicle, it is considered as a rigid body connected to the ground through the tires. This model is the so-called bicycle model or single-track model [29, 30] and is widely used by scientists to design nonlinear control due to preserving the essence of basic vehicle dynamics because it uses only longitudinal/lateral and yaw dynamics. The effectiveness of the bicycle model will be validated into a virtual automobile software (CarSim simulator) where it is testing the controller performance. In [31, 32], the active front steering (AFS) is considered as an input control; this actuator will impose an incremental steer angle δc. On the other hand, it also considered the yaw momentum as a control input Mz, which imposes negative longitudinal forces using the active brakes. Hence, the mathematical model of the ground vehicle is(1)mv˙x−vyωz=μxFx,f+Fx,r+Fd,x,mv˙y+vxωz=μyFy,f+Fy,r+Fd,y,Jzω˙z=μylfFy,f−lrFy,r+Mz+Md,z,where vx and vy are the longitudinal and lateral velocities of the vehicle center of mass, and αz and ωz are the yaw angle and yaw rate of the vehicle, m is the vehicle mass, Jz is the inertia momentum, lf is the distance between the center mass to the front wheels, and lr is the distance between the center mass to the rear wheels. Also, μxandμy are the longitudinal/lateral tire-road friction coefficients. Moreover, Fx,f and Fx,r are the front/rear longitudinal forces and Fy,f and Fy,r are the rear longitudinal/lateral forces, and Mz is the yaw moment. The external disturbance forces, due to environment actions as the wind blast, acting on the vehicle dynamics are Fd,x and Fd,y and Md,z is the torque generated by the external disturbance.The tire front lateral forceFy,f depends directly the front tire slip angle αf=δd+δc−vy+lfωz/vx, where δd is the road wheel angle imposed to the drive and the δc is the active front steering imposed by the designed control. Model (1) presents nonlinearities due to the characteristics of the tires. In this article, is used the compact Pacejka magic formula:(2)Fy,jαj=DjsinCjarctanBjαj,j=f,r,where Bj, Cj, and Dj are the values obtained experimentally [33].The tire slip angleαj, which depends all the forces acting in the vehicle dynamics, has some properties such as the angle has a minimum/maximum value, i.e., αj=±αj,max, and it is considered invertible in function to αj: αj∈−αj,max,αj,max.The tire front lateral force,Fy,fδd+δc,vy,ωz, is considered an invertible function with respect to the AFS control input δc; therefore, for a fixed value Fy,f,0, the solution of Fy,fδd+δc,vy,ωz=Fy,f,0 is unique and given by(3)δc=−δd+vy+lfωzvx+Fy,f−1Fy,f,0,ifαf∈−αf,max,αf,max,−δd+vy+lfωzvx±Fy,f,max,otherwise.The lateral forceFy,fδd+δc,vy,ωz is considered an invertible function; for this reason, it is possible to consider the following control input:(4)Δf=Fy,fδd+δc,vy,ωz−Fy,fδd,vy,ωz.SubstitutingΔf controller into the lateral velocity and the yaw rate velocity in the mathematical model of the ground vehicle (1), besides Fx,f+Fx,r=0, the vehicle dynamics is rewritten as(5)v˙x=vyωz+1mFd,x,v˙y=−vxωz+μymFy,fδd+Fy,r+μymΔf+1mFd,y,ω˙z=μyJzlfFy,fδd−lrFy,r+μylfJzΔf+1JzMz+1JzMd,z.The control aim is to design a modified HOSM controller using PI sliding surface, such as the lateral velocityvy that globally tracks a reference vy,ref, and ωz tends a reference yaw rate ωz,ref in finite time, despite the presence of the external disturbances and parameter uncertainties.In this article, it is considered that the longitudinal velocityvx, the lateral velocity vy and the yaw rate ωz are measurable variables. To solve the control problem, the following assumptions will be used.Assumption 1.
The steer angleδd is considered a smooth function, i.e., function C2.Assumption 2.
The variables of the referencevy,ref and ωz,ref and their derivatives v˙y,ref and ω˙z,ref are considered bounded (physically obvious).


## 3. Design a Modified HOSM Controller for Trajectory Tracking

In this section, a modified HOSM controller using PI sliding surface will be designed when the parameter uncertainties and external disturbances are known.Consider the tracking error as(6)ev,y=vy−vy,ref,eω,z=ωz−ωz,ref.and the dynamics of the error system is(7)e˙v,y=−vxωz+μymFy,fδd+Fy,r+μymΔf+1mFd,y−v˙y,ref,e˙ω,z=μyJzlfFy,fδd−lrFy,r+μylfJzΔf+1JzMz+1JzMd,z−ω˙z,ref.Using the tracking error (6), the PI sliding function is defined as(8)sy=kp,yev,yt+ki,y∫ev,ytdt,sz=kp,zeω,zt+ki,z∫eω,ztdt,where kp,y,ki,y,kp,z,ki,z>0. The gains in (8) provide flexibility for the construction of the PI sliding surface.Then, it is possible to obtain the derivative with respect to time of the PI sliding function (8) as(9)s˙y=kp,y−vxωz+μymFy,fδd+Fy,r+μymΔf+1mFd,y−v˙y,ref+ki,yev,y,s˙z=kp,zμyJzlfFy,fδd−lrFy,r+μylfJzΔf+1JzMz+1JzMd,z−ω˙z,ref+ki,zeωz.Hence, taking into account the PI surfaces˙y (9), the following input control Δf is proposed:(10)Δf=−mμy−vxωz+μymFy,fδd+Fy,r+1mFd,y−v˙y,ref−mμyki,ykp,yev,y+mμykp,y−λ1,ysy1/2signsy−λ2,ysy+∫0t−λ3,ysignsyτdτ+∫0t−λ4,ysyτdτ,where λ1,y,λ2,y,λ3,y,λ4,y>0.Similarly, the RTV controller takes into accounts˙z (9), and proposing the controller Mz as(11)Mz=−JzμyJzlfFy,fδd−lrFy,r+μylfJzΔf+1JzMd,z−ω˙z,ref−Jzki,zkp,zeωz+Jzkp,z−λ1,zsz1/2signsz−λ2,zsz+∫0t−λ3,zsignszτdτ+∫0t−λ4,zszτdτ,where λ1,z,λ2,z,λ3,z,λ4,z>0.Substituting the input control (10) into the PI sliding surface s˙y in (9), one obtains(12)s˙y=−λ1,ysy1/2signsy−λ2,ysy+χy,χ˙y=−λ3,ysignsy−λ4,ysy,and the input control Mz (11) into the PI sliding surface s˙z, one obtains(13)s˙z=−λ1,zsz1/2signsz−λ2,zsz+χz,χ˙z=−λ3,zsignsz−λ4,zsz.In fact, from (12) and (13), one gets the following differential inclusion:(14)s˙j=−λ1,jsj1/2−λ2,jsj+χj,χ˙j=−λ3,jsj0−λ4,jsj,where j=s,z, and sj1/2=sj1/2signsj and sj0=signsj.The proof of the stability of the origin to the PI sliding surfacesj can be considered as the following Lyapunov function:(15)V=12ξTPξ,λminPξ22≤Vξ≤λmaxPξ22,where V is continuous and differentiable when sj≠0, and(16)ξ=sj1/2sjχj,P=λ1,j2+4λ3,jλ1,jλ2,j−λ1,jλ1,jλ2,j2λ4,j+λ2,j2−λ2,j−λ1,j−λ2,j2.Letsj=sj1/2sj1/2 and sj0=1/2sj1/22sj1/2, the derivative of ξ with respect to time will be(17)ddtsj1/2=12sj1/2−λ1,jsj1/2−λ2,jsj+χj=−12sj1/2λ1,jsj1/2−χj−12λ2,jsj1/2,s˙j=−λ1,jsj1/2−λ2,jsj+χj,χ˙j=−λ3,jsj0−λ4,jsj=−12sj1/22λ3,jsj1/2−λ4,jsj.The derivative of (15) is(18)V.=ξTPξ˙,ξ˙=sj1/2.sj.χ˙j=−12s˙j1/2Λ1ξ+12Λ2ξ,with(19)Λ1=λ1,j0−10002λ3,j00,Λ2=λ2,j002λ1,j2λ2,j−202λ4,j0,so that(20)V.=−12sj1/2ξTPΛ1ξ+12ξTPΛ2ξ,PΛ1=λ1,jλ1,j2+2λ3,j0−λ1,j2λ2,jλ1,j2−2λ3,j0−λ1,jλ2,j−λ1,j20λ1,j,PΛ2=−−λ2,j−3λ1,j2+4λ3,j−2λ1,jλ2,j2−λ4,j2λ1,jλ2,j−λ1,j3λ2,j2+4λ4,j−2λ2,jλ2,j2+λ4,j2λ2,j2+4λ4,j3λ1,jλ2,j2λ2,j2−4λ4,j−2λ2,j.The first term of (20), i.e., −1/2sj1/2ξTPΛ1ξ can be rewritten as(21)−12|sj|1/2ξTPΛ1ξ=−12sj1/2λ1,j2λ3,j+λ1,j2sj1/22−2λ1,j2sj1/2xj+λ2,jλ1,j2−2λ3,jsj1/2sj−λ1,jλ2,jsjxj+λ1,jxj2,if −1/2sj1/2λ2,jλ1,j2−2λ3,jsj1/2sj=−1/2λ2,jλ1,j2−2λ3,jsj1/22, the first term in matrix form is written as(22)−12sj1/2ξTPΛ1ξ=−12ξTλ2,jλ1,j2−2λ3,j00000000ξ−12sj1/2ξTλ1,jλ1,j2+2λ3,j0−λ1,j200−12λ1,jλ2,j−λ1,j2−12λ1,jλ2,jλ1,jξ.Similarly, the second term of (20) is 1/2ξTPΛ2ξ, and it is written as(23)12ξTPΛ2ξ=12−λ2,j3λ1,j+4λ3,jsj1/22−2λ2,jλ2,j+λ4,jsj2−2λ2,jxj2−λ1,j5λ2,j2+2λ4,jsj1/2sj+5λ1,jλ2,jsj1/2xj+4λ2,j2sjxj,since sj1/2sj=1/2sj1/22sj2 and sj1/2xj=1/sj1/2sjxj, the second term in matrix form is(24)12ξTPΛ2ξ=−12λ2,jξT3λ1,j2+4λ3,j0002λ2,j2+λ4,j−2λ2,j0−2λ2,j2ξ−12sj1/2λ1,jξT00005λ2,j2+2λ4,j−52λ2,j0−52λ2,j0ξ.Finally, using (22) and (24), one rewrites (20) as(25)V.=−1sj1/2ξTQ1ξ−ξTQ2ξ,with(26)Q1=λ1,j2λ1,j2+2λ3,j0−λ1,j02λ4,j+5λ2,j2−3λ2,j−λ1,j−3λ2,j1,Q2=λ2,j22λ1,j2+λ3,j000λ2,j2+λ4,j−λ2,j0−λ2,j1,where V. is the negative definitive if and only if the matrix Q1,Q2>0, namely, λ1,j,λ2,j,λ3,j,λ4,j>0. Denoting by λminQ1,λminQ2 the minimum eigenvalues of the matrix Q1,Q2, one finally works out(27)V.≤1sj1/2λminQ1ξ22−λminQ2ξ22,and using the fact ev1/2≤ξ2≤2V/λminP, one obtains(28)V.≤−c1V1/2+c2V,where c1=2λminPλminQ1/λmaxP and c2=2λminQ2/λmaxP.


## 4. Sliding Modes Dynamics Stability

To determinate the stability of the sliding surface in (8), we focus when the sliding surface occurs, si=0;i=y,z, then(29)si=kp,iev,κt+ki,κ∫ev,κtdt,κ=y,z,ev,κt=−ki,κkp,κ∫ev,κtdt.For the analysis of stability of error (29), the Lyapunov candidate function is proposed as(30)Vv,κ=12ev,κt2.The dynamic of the Lyapunov function yields(31)V˙v,κ=−ki,κkp,κev,κ2tdt,Therefore, the errorev,κt is asymptotically stable when the gains ki,κ>0 and kp,κ>0.


## 5. Simulation Results

In this section, the simulation results taking into account the mathematical model of the vehicle (1) and the controllers (10) and (11) are presented using the CarSim simulator. CarSim delivers the most accurate, detailed, and efficient methods for simulating the performance of passenger vehicles. The test maneuver considered in this simulation is a so-called double-step steer (Figure 1). The maneuver consists of a rapid turn on the left of 100∘ at t=1s, followed by a turn on the right of −100∘ at t=3s, and finally, at t=5s, the steering wheel is set to zero. The parameters considered in (1) are given in Table 1.Figure 1
Steering wheel angleδd.Table 1
Nominal parameter values.
m0=1565kgBf,0=16Cf,0=1.41Jz,0=2075kgm2Br,0=16Cr,0=1.51Steer ratio: 16.01Df,0=8854NEf,0=0lf=1.38mDr,0=8394NEr,0=0lr=1.53m—The reference variablesvy,reft and ωz,reft will be considered as the behavior of an “ideal” or “reference” vehicle.(32)v˙y,r=−vxωz,ref+μy,rmrFy,f,rαf,r+Fy,r,rαr,r,ω˙z,r=μy,rJz,rFy,f,rαf,rlf,r−Fy,r,rαr,rlr,r,where αf,r=δd−vy,ref+lf,rωz,ref/vx, αr,r=−vy,ref−lr,rωz,ref/vx, and the tire-road friction coefficient reference μy,r, which is assumed as the road, is dry. The reference forces Fy,f,r and Fy,r,r are obtained using the compact Pacejka magic formula (2). The parameters considered in (32) are given in Table 2.Table 2
Constants of the reference system (32).
mr=1862kgBy,f,r=By,fDy,f,r=10000Nlf,r=lfBy,r,r=14.7Dy,r,r=10000Nlr,r=lrCy,f,r=Cy,fJz,r=2488kgm2μy,r=0.9Cy,r,r=1.2—Finally, a disturbance due to the wind is considered, which has been implemented in [2, 4]. The wind induces longitudinal/lateral forces Fd,x,Fd,y, and a yaw moment Md,z [34]. In terms of the front/lateral surfaces As,f, As,l of the vehicle, the air density ρ, and the (dimensionless) front/lateral aerodynamic coefficients ca,x, ca,y, the expression of the forces, and yaw torque due to the wind are Fd,x=−As,fρca,xvaw,x2/2, Fd,y=−As,lρca,yvaw,y2/2, Md,z=lcFd,y, where lc=lc0+0.025N is the distance between the center of mass and the center of pressure, with lc0 the nominal value and N a white noise. The parameters considered are given in Table 3.Table 3
Parameters of the external disturbance.
μyd=0.9As,f=2.59m2ca,x=0.3μyw=0.6As,l=5.1m2ca,y=0.6μx=μylc0=−0.20mρ=1.2kg/m3


### 5.1. CarSim Simulations

The simulations consider the performance of the controllers (10) and (11). The model of the vehicle considered in the CarSim simulator is a C-class hatchback automobile. The tires selected are 245/40–R17. The vehicle concerned has independent suspensions on both axles. The nominal values m0, Jz,0, Df,0, Dr,0, and lc0 have been used in the controller (Table 1), while the real automobile parameters are m=1.15m0, J=1.15Jz,0, Df=0.85Df,0, and Dr=0.85Dr,0. The variables considered measurables in the CarSim are the longitudinal/lateral velocities vx and vy and the yaw angle and rate αz and ωz. The inputs to the CarSim simulator are the steering angle δ=δd+δc, with δc calculated using (3), and the yaw moment Mz, and other inputs are the wind (wind-heading and wind-speed) and longitudinal/lateral tire-road.The initial conditions used to show the performance of the controllers (10) and (11) are vx0=100 km/h, vy0=0 km/h, ωz0=0 deg/s, and αz0=0 deg, and the simulation results are shown in Figures 1–7.Figure 2
(a) Longitudinal/lateral tire-road friction coefficientsμxandμy; and (b) sideslip angle β=vy/vx.
(a)(b)Figure 3
(a) Incremental steer angleδc; (b) Δf controller; and (c) yaw momentum Mz.
(a)(b)(c)Figure 4
(a) Lateral velocityvy; (b) lateral velocity reference vy,ref; and (c) ey=vy−vy,ref.
(a)(b)(c)Figure 5
(a) Yaw rateωz; (b) yaw rate reference ωz,ref; and (c) eω=ωz−ωz,ref.
(a)(b)(c)Figure 6
Longitudinal velocityvx.Figure 7
External disturbance: (a)Fd,x; (b) Fd,y; and (c) Md,z.
(a)(b)(c)The longitudinal/lateral tire-road friction coefficientsmuxand muy are given in Figure 2(a), and the side-slip angle behavior β=vy/vx is given in Figure 2(b). The control input Δf, the corresponding incremental steering angle (AFS) δc, and the yaw torque Mz are shown in Figure 3. In Figures 4 and 5 are given the controlled outputs vy and ωz and its references vy,ref and ωz,ref to track, and the tracking errors vy−vy,ref and ωz−ωz,ref. The longitudinal velocity vx is shown in Figure 6, and finally, the forces and moments Fd,x, Fd,y, Md,z are given in Figure 7.


## 5.1. CarSim Simulations

The simulations consider the performance of the controllers (10) and (11). The model of the vehicle considered in the CarSim simulator is a C-class hatchback automobile. The tires selected are 245/40–R17. The vehicle concerned has independent suspensions on both axles. The nominal values m0, Jz,0, Df,0, Dr,0, and lc0 have been used in the controller (Table 1), while the real automobile parameters are m=1.15m0, J=1.15Jz,0, Df=0.85Df,0, and Dr=0.85Dr,0. The variables considered measurables in the CarSim are the longitudinal/lateral velocities vx and vy and the yaw angle and rate αz and ωz. The inputs to the CarSim simulator are the steering angle δ=δd+δc, with δc calculated using (3), and the yaw moment Mz, and other inputs are the wind (wind-heading and wind-speed) and longitudinal/lateral tire-road.The initial conditions used to show the performance of the controllers (10) and (11) are vx0=100 km/h, vy0=0 km/h, ωz0=0 deg/s, and αz0=0 deg, and the simulation results are shown in Figures 1–7.Figure 2
(a) Longitudinal/lateral tire-road friction coefficientsμxandμy; and (b) sideslip angle β=vy/vx.
(a)(b)Figure 3
(a) Incremental steer angleδc; (b) Δf controller; and (c) yaw momentum Mz.
(a)(b)(c)Figure 4
(a) Lateral velocityvy; (b) lateral velocity reference vy,ref; and (c) ey=vy−vy,ref.
(a)(b)(c)Figure 5
(a) Yaw rateωz; (b) yaw rate reference ωz,ref; and (c) eω=ωz−ωz,ref.
(a)(b)(c)Figure 6
Longitudinal velocityvx.Figure 7
External disturbance: (a)Fd,x; (b) Fd,y; and (c) Md,z.
(a)(b)(c)The longitudinal/lateral tire-road friction coefficientsmuxand muy are given in Figure 2(a), and the side-slip angle behavior β=vy/vx is given in Figure 2(b). The control input Δf, the corresponding incremental steering angle (AFS) δc, and the yaw torque Mz are shown in Figure 3. In Figures 4 and 5 are given the controlled outputs vy and ωz and its references vy,ref and ωz,ref to track, and the tracking errors vy−vy,ref and ωz−ωz,ref. The longitudinal velocity vx is shown in Figure 6, and finally, the forces and moments Fd,x, Fd,y, Md,z are given in Figure 7.


## 6. Conclusions

In this study, an active front steering control and yaw moment was designed for the lateral and yaw dynamics of a vehicle. The active control of a robust nonlinear controller design uses the modified high-order sliding mode (HOSM) with the PI sliding surface. The stability performance of the PI sliding manifold was proved by the Lyapunov function. The simulation results highlight the performance of the proposed approach. The CarSim simulations show a good behavior of the HOSM-PI sliding surface even in the presence of external disturbances and parameter variations.



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