# Study on the Nonsingular Problem of Fractional-Order Terminal Sliding Mode Control

Authors: Kening Li; Jianyong Cao; Fan Yu
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

An improved type of control strategy combining the fractional calculus with nonsingular terminal sliding mode control named non-singular fractional terminal sliding mode control (NFOTSM) is proposed for the nonlinear tire-road friction control system of vehicle in this paper. A fractional-order switching manifold is proposed, and the corresponding control law is formulated based on the Lyapunov stability theory to guarantee the sliding condition. The proposed controller ensures the finite time stability of the closed-loop system. Then, a terminal attractor is introduced to solve the singularity problem of fractional terminal sliding mode control (FOTSM). Finally, the performance of the NFOTSM is fully investigated compared with other related algorithms to find the effectiveness for the tire-road friction system. The results show that the NFOTSM has better performance than other related algorithms.


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## Body

## 1. Introduction

Fractional calculus is an old mathematical with a 300 years old history [1, 2]. For many years, this branch of science was considered as a sole mathematical and theoretical subject with nearly no applications [3]. But, in recent decades, fractional calculus has become an interesting topic among researchers, and different applications have been proposed for this field of science [4–6]. Designing fractional-order controllers is one of these applications.The idea of designing a fractional-order (FO) controller was firstly proposed by Oustaloup in 1988 [7]. He introduced a robust fractional-order control scheme which is called Commande Robuste d’Ordre Non-Entier (CRONE) [8, 9]. A frequency domain approach has been introduced in [10]. Variable order fractional controllers are introduced by Valério et al. [11]. Tuning FO-PID controller is introduced by Padula et al. [12]. Model Predictive Control of Fractional-Order Nonlinear Discrete-Time Systems is proposed by Domek [13]. The adaptive FO controllers [14] and the effect of fractional order in variable structure control are investigated and verified by Tenreiro Machado [15].Sliding mode control (SMC) is a well-known efficient control scheme which has been widely applied for both linear and nonlinear systems [16]. This control is also considered as an effective approach for control of the dynamical systems with uncertainties. In general, an arbitrary linear manifold is considered as a sliding surface, which can guarantee the asymptotic stability and desired performance of the closed-loop control system. However, the main disadvantage of SMC scheme is that the system states cannot reach the equilibrium point in finite time. But, in recent years, a new control scheme called terminal sliding mode control (TSMC) is proposed to overcome this drawback utilizing nonlinear sliding surface [17, 18]. Nonlinear switching hyperplanes in TSMC can improve the transient performance substantially. Besides, compared with the conventional SMC with linear sliding manifold, TSMC offers some superior properties such as faster, finite time convergence and higher control precision. And a nonsingular terminal sliding mode (NTSM) method is proposed to avoid the singularity in TSM control systems [19, 20].However, TSMC designs are restricted to integer-order controllers, and there is only few papers in which utilizing the means of FO calculus in the design procedure of a fractional-order TSMC (FOTSMC) is considered, especially for the nonsingular of the controllers [20]. Therefore, designing an NFOTSMC for dynamical systems is still an open problem.This paper is organized as follows: Section2 presents nonsingular fractional terminal sliding mode control strategy. The adopted control structure of tire-road friction Ext-ABS is introduced and realized in Section 3. At last, numerical simulation results are shown in Section 4.


## 2. Nonsingular Fractional-Order Terminal Sliding Mode Control

In this section, nonsingular fractional terminal sliding mode control strategy is introduced and realized for the tire-road friction servocontrol system.


### 2.1. Basic Definition and Preliminaries of Fractional-Order Calculus

Fractional-order integration and differentiation are the generalization of the integer-order ones. Efforts to extend the specific definitions of the traditional integer-order to the more general arbitrary order context led to different definitions for fractional derivatives [1, 2, 19]. Two of the most commonly used definitions are Riemann-Liouville and Caputo definitions.Definition 1.
Theαth-order Riemann-Liouville fractional derivative of function f(t) with respect to t and the terminal value t0 is given by
(1)Dαf(t)=dαf(t)dtα=1Γ(m-α)dmdtm∫t0tf(τ)(t-τ)α-m+1dτ
and the Riemann-Liouville definition of the αth-order fractional integration is given by
(2)t0Itαf(t)=Iαf(t)=1Γ(α)dmdtm∫t0tf(τ)(t-τ)1-αdτ,
where m is the first integer larger than α and Γ is the Gamma function.Definition 2.
The Caputo fractional derivative of orderα of a continuous function f:R+→R is defined as follows:
(3)Dαf(t)={1Γ(m-α)∫t0tf(m)(τ)(t-τ)α-m+1dτ,m-1<α<m,dmdtmf(t),α=m,
where m is the first integer larger than α.Property 1.
If the fractional derivativeαDtαy(t)(k-1≤α<k) of a function y(t) is integrable, then
(4)αItα(αDtαy(t))=y(t)-∑j=1k[αDtα-jy(t)]t=α(t-α)α-jΓ(α-j+1).Lemma 3.
The fractional integration operatorαItα with [α]>0 is bounded
(5)∥Itαy∥p≤K∥y∥p,1≤p≤∞.Theorem 4.
Letx=0 be an equilibrium point for the nonautonomous fractional-order system
(6)Dαx(t)=f(x,t),
where f(x,t) satisfies the Lipschitz condition with Lipschitz constant l>0. Assume that there exists a Lyapunov candidate V(t,x(t)) satisfying
(7)α1∥x∥α≤V(t,x)≤α2∥x∥,V˙(t,x)≤-α3∥x∥,
where α>0, α1>0, α2>0, and α3>0. Then the equilibrium point of system (6) is Mittag-Leffler stable, and the stability implies asymptotic stability.Theorem 5.
Letx=0 be an equilibrium point for the nonautonomous fractional-order system (6). Assume that there exist a Lyapunov function V(t,x(t)) and class-K functions αi(i=1,2,3) satisfying
(8)α1∥x∥≤V(t,x)≤α2∥x∥,DβV(t,x)≤-α3∥x∥,
where β∈(0,1). Then the equilibrium point of system (6) is asymptotic stability.


### 2.2. Review of Integer-Order Nonsingular Terminal Sliding Mode Control

In this paper, thecontrol problem is to get the system to track a desired vector Xd (Xd=[xd,x˙d]T). So, the fractional-order system can be defined as
(9)x˙1=x2,x˙2=f(x1,x2,t)+u,
where u is the control effort, x=[x1,x2]T.And the tracking error can be defined as(10)e(t)=x(t)-xd(t).The control goal is to design a robust nonsingular fractional-order terminal sliding mode controller to stabilize state variables and track the desired vectorXd in finite time.Here, definition and some properties of NTSM are briefly reviewed for essential preparation of the fractional-order NTSMC design.Definition 6.
Terminal sliding manifold can be defined as the following nonlinear differential equation:(11)s=e˙(t)+ξe(t)p/q,
where ξ>0 is a constant, p, q are the given positive odd integers, and p<q<2p.Considering the constraints˙=0, the control input can be defined as
(12)u(t)=x¨d-f(x1,x2,t)-ξpqe˙(t)e(t)(p/q-1)+Ksign(s).According to finite time stability theory, the equilibrium pointe=0 of error system is globally finite time stable, and the time is
(13)Ttsm=qξ(q-p)e(0)(1-p/q).


### 2.3. Nonsingular Fractional Terminal Sliding Mode Controller

In this section we will develop nonsingular fractional-order terminal sliding mode control for trajectory tracing of a class of dynamical systems, for example, the servosystem. For this purpose, the forms introduced in the previous section are used. As a result, the high precision tracking can be acquired with faster convergence speed compared with the integer-order terminal sliding mode control.Definition 7.
Fractional terminal sliding manifold can be defined as(14)s=Dαe¨(t)+ξe(t)p/q,α∈(0,1).
If the terminal sliding mode control law is designed as (15), then the tracking error converges to zero in finite time [20]. Consider
(15)u(t)=x¨d-f(x1,x2,t)-D-1-α[ξpqe˙(t)e(t)(p/q-1)]+D-1-α[Ks+Ksign(s)+δ].
But this system has a singularity problem because of(p/q-1)<0. Consequently, to avoid this phenomenon, a new “terminal attractor” is proposed to develop a singularity free TSM control.Definition 8.
Nonsingular fractional terminal sliding manifold can be defined as follows:(16)s=Dαe¨(t)+ξ1e˙(t)p/q+ξ2e(t)g/h,α∈(0,1)
where g, h are the given positive odd integers and 1<g/h<2.And the “terminal attractor” can be defined as(17)s˙=(-ϕs-γsζ1/ζ2)e˙p/q-1,
where ϕ∈R+; γ∈R+; ζ1 and ζ2 are the given positive odd integers and 0<ζ1/ζ2<1.From (9), (10), (16), and (17), the NFOTSM control law can be obtained as follows:
(18)u(t)=x¨d-f(x1,x2,t)+qξ1p[ξ2gh(-ϕs-γsζ1/ζ2)-Dα+1e¨e˙1-p/q-ξ2gheg/h-1e˙2-p/q].Proof.
Firstly, forg/h-1>0 and 2-p/q>1-p/q>0, we can get that the proposed NFOTSM control law is a singularity free TSM control.
Secondly, the Lyapunov function is defined in the form(19)V=12s2.
Then, we have(20)V˙=ss˙=s(Dα+1e¨+ξ1pqe˙p/q-1e¨+ξ2gheg/h-1e˙)=s(Dα+1e¨+ξ1pqe˙p/q-1(f(x1,x2,t)+u-x¨d)+ξ2gheg/h-1e˙).
Substituting (18) into (20) results in
(21)V˙=s(-ϕs-γsζ1/ζ2)=-ϕs2-γs(ζ1+ζ2)/ζ2≤-ϕs2.
So, the sliding condition is satisfied.
According to (16), we get that the tracking error converges to zero. To show that this happening (exact tracking, e=0) occurs in finite time, we can define a stopping time as follows:
(22)τs=inf{t≥tr:e(t)=0},
where tr is the reaching time, the time for the system states to reach the sliding surface. So, all we need is to prove that there exists a tr≤ts<∞, such that τs≤ts.
Taking the concept of fractional integral and derivative operators into account and using (16), assume the reaching time is tr; one obtains
(23)D-α(Dαe¨(t))=-D-α(ξ1e˙(t)p/q+ξ2e(t)g/h).
So, one can conclude that(24)e¨(t)-[trDα-1e¨(t)]t=tr(t-tr)α-1Γ(α)=-D-α(ξ1e˙(t)p/q+ξ2e(t)g/h),
where
(25)[trDα-1e¨(t)]t=tr(t-tr)α-1Γ(α)=0.
So, it can be concluded that(26)e(t)-[trDte(t)]t=tr(t-tr)2-[e(t)]t=tr=-D-α-2(ξ1e˙(t)p/q+ξ2e(t)g/h).
According to Lemma3, we have
(27)trD-α-2(e˙(t)p/q+e(t)g/h)=trItα+2(e˙(t)p/q+e(t)g/h)≤∥trItα+2(e˙(t)p/q+e(t)g/h)∥≤N∥e˙(t)p/q+e(t)g/h∥.
Substituting (27) into (26) results in
(28)e(t)-[tr1Dte(t)]t=tr1(t-tr1)2-[e(t)]t=tr1≤N∥ξ1e˙(t)p/q+ξ2e(t)g/h∥.
So, we get(29)∥e(t)-[tr1Dte(t)]t=tr1(t-tr1)2∥-∥[e(t)]t=tr1∥≤N∥ξ1e˙(t)p/q+ξ2e(t)g/h∥.
Noting thate(t)=e˙(t)=0 at t=ts, it yields
(30)∥[trDte(t)]t=tr2∥(t-tr)≤∥[e(t)]t=tr∥.
Consequently, if[trDte(t)]t=tr, then ts=tr. Else, we have
(31)ts≤2∥[e(t)]t=tr∥∥[tr1Dte(t)]t=tr/2∥+tr1=2∥e(tr)∥∥e˙(tr)∥+tr.
Therefore, it can be concluded that the system trajectories attain to reference trajectory in finite time.
The proof is complete.


## 2.1. Basic Definition and Preliminaries of Fractional-Order Calculus

Fractional-order integration and differentiation are the generalization of the integer-order ones. Efforts to extend the specific definitions of the traditional integer-order to the more general arbitrary order context led to different definitions for fractional derivatives [1, 2, 19]. Two of the most commonly used definitions are Riemann-Liouville and Caputo definitions.Definition 1.
Theαth-order Riemann-Liouville fractional derivative of function f(t) with respect to t and the terminal value t0 is given by
(1)Dαf(t)=dαf(t)dtα=1Γ(m-α)dmdtm∫t0tf(τ)(t-τ)α-m+1dτ
and the Riemann-Liouville definition of the αth-order fractional integration is given by
(2)t0Itαf(t)=Iαf(t)=1Γ(α)dmdtm∫t0tf(τ)(t-τ)1-αdτ,
where m is the first integer larger than α and Γ is the Gamma function.Definition 2.
The Caputo fractional derivative of orderα of a continuous function f:R+→R is defined as follows:
(3)Dαf(t)={1Γ(m-α)∫t0tf(m)(τ)(t-τ)α-m+1dτ,m-1<α<m,dmdtmf(t),α=m,
where m is the first integer larger than α.Property 1.
If the fractional derivativeαDtαy(t)(k-1≤α<k) of a function y(t) is integrable, then
(4)αItα(αDtαy(t))=y(t)-∑j=1k[αDtα-jy(t)]t=α(t-α)α-jΓ(α-j+1).Lemma 3.
The fractional integration operatorαItα with [α]>0 is bounded
(5)∥Itαy∥p≤K∥y∥p,1≤p≤∞.Theorem 4.
Letx=0 be an equilibrium point for the nonautonomous fractional-order system
(6)Dαx(t)=f(x,t),
where f(x,t) satisfies the Lipschitz condition with Lipschitz constant l>0. Assume that there exists a Lyapunov candidate V(t,x(t)) satisfying
(7)α1∥x∥α≤V(t,x)≤α2∥x∥,V˙(t,x)≤-α3∥x∥,
where α>0, α1>0, α2>0, and α3>0. Then the equilibrium point of system (6) is Mittag-Leffler stable, and the stability implies asymptotic stability.Theorem 5.
Letx=0 be an equilibrium point for the nonautonomous fractional-order system (6). Assume that there exist a Lyapunov function V(t,x(t)) and class-K functions αi(i=1,2,3) satisfying
(8)α1∥x∥≤V(t,x)≤α2∥x∥,DβV(t,x)≤-α3∥x∥,
where β∈(0,1). Then the equilibrium point of system (6) is asymptotic stability.


## 2.2. Review of Integer-Order Nonsingular Terminal Sliding Mode Control

In this paper, thecontrol problem is to get the system to track a desired vector Xd (Xd=[xd,x˙d]T). So, the fractional-order system can be defined as
(9)x˙1=x2,x˙2=f(x1,x2,t)+u,
where u is the control effort, x=[x1,x2]T.And the tracking error can be defined as(10)e(t)=x(t)-xd(t).The control goal is to design a robust nonsingular fractional-order terminal sliding mode controller to stabilize state variables and track the desired vectorXd in finite time.Here, definition and some properties of NTSM are briefly reviewed for essential preparation of the fractional-order NTSMC design.Definition 6.
Terminal sliding manifold can be defined as the following nonlinear differential equation:(11)s=e˙(t)+ξe(t)p/q,
where ξ>0 is a constant, p, q are the given positive odd integers, and p<q<2p.Considering the constraints˙=0, the control input can be defined as
(12)u(t)=x¨d-f(x1,x2,t)-ξpqe˙(t)e(t)(p/q-1)+Ksign(s).According to finite time stability theory, the equilibrium pointe=0 of error system is globally finite time stable, and the time is
(13)Ttsm=qξ(q-p)e(0)(1-p/q).


## 2.3. Nonsingular Fractional Terminal Sliding Mode Controller

In this section we will develop nonsingular fractional-order terminal sliding mode control for trajectory tracing of a class of dynamical systems, for example, the servosystem. For this purpose, the forms introduced in the previous section are used. As a result, the high precision tracking can be acquired with faster convergence speed compared with the integer-order terminal sliding mode control.Definition 7.
Fractional terminal sliding manifold can be defined as(14)s=Dαe¨(t)+ξe(t)p/q,α∈(0,1).
If the terminal sliding mode control law is designed as (15), then the tracking error converges to zero in finite time [20]. Consider
(15)u(t)=x¨d-f(x1,x2,t)-D-1-α[ξpqe˙(t)e(t)(p/q-1)]+D-1-α[Ks+Ksign(s)+δ].
But this system has a singularity problem because of(p/q-1)<0. Consequently, to avoid this phenomenon, a new “terminal attractor” is proposed to develop a singularity free TSM control.Definition 8.
Nonsingular fractional terminal sliding manifold can be defined as follows:(16)s=Dαe¨(t)+ξ1e˙(t)p/q+ξ2e(t)g/h,α∈(0,1)
where g, h are the given positive odd integers and 1<g/h<2.And the “terminal attractor” can be defined as(17)s˙=(-ϕs-γsζ1/ζ2)e˙p/q-1,
where ϕ∈R+; γ∈R+; ζ1 and ζ2 are the given positive odd integers and 0<ζ1/ζ2<1.From (9), (10), (16), and (17), the NFOTSM control law can be obtained as follows:
(18)u(t)=x¨d-f(x1,x2,t)+qξ1p[ξ2gh(-ϕs-γsζ1/ζ2)-Dα+1e¨e˙1-p/q-ξ2gheg/h-1e˙2-p/q].Proof.
Firstly, forg/h-1>0 and 2-p/q>1-p/q>0, we can get that the proposed NFOTSM control law is a singularity free TSM control.
Secondly, the Lyapunov function is defined in the form(19)V=12s2.
Then, we have(20)V˙=ss˙=s(Dα+1e¨+ξ1pqe˙p/q-1e¨+ξ2gheg/h-1e˙)=s(Dα+1e¨+ξ1pqe˙p/q-1(f(x1,x2,t)+u-x¨d)+ξ2gheg/h-1e˙).
Substituting (18) into (20) results in
(21)V˙=s(-ϕs-γsζ1/ζ2)=-ϕs2-γs(ζ1+ζ2)/ζ2≤-ϕs2.
So, the sliding condition is satisfied.
According to (16), we get that the tracking error converges to zero. To show that this happening (exact tracking, e=0) occurs in finite time, we can define a stopping time as follows:
(22)τs=inf{t≥tr:e(t)=0},
where tr is the reaching time, the time for the system states to reach the sliding surface. So, all we need is to prove that there exists a tr≤ts<∞, such that τs≤ts.
Taking the concept of fractional integral and derivative operators into account and using (16), assume the reaching time is tr; one obtains
(23)D-α(Dαe¨(t))=-D-α(ξ1e˙(t)p/q+ξ2e(t)g/h).
So, one can conclude that(24)e¨(t)-[trDα-1e¨(t)]t=tr(t-tr)α-1Γ(α)=-D-α(ξ1e˙(t)p/q+ξ2e(t)g/h),
where
(25)[trDα-1e¨(t)]t=tr(t-tr)α-1Γ(α)=0.
So, it can be concluded that(26)e(t)-[trDte(t)]t=tr(t-tr)2-[e(t)]t=tr=-D-α-2(ξ1e˙(t)p/q+ξ2e(t)g/h).
According to Lemma3, we have
(27)trD-α-2(e˙(t)p/q+e(t)g/h)=trItα+2(e˙(t)p/q+e(t)g/h)≤∥trItα+2(e˙(t)p/q+e(t)g/h)∥≤N∥e˙(t)p/q+e(t)g/h∥.
Substituting (27) into (26) results in
(28)e(t)-[tr1Dte(t)]t=tr1(t-tr1)2-[e(t)]t=tr1≤N∥ξ1e˙(t)p/q+ξ2e(t)g/h∥.
So, we get(29)∥e(t)-[tr1Dte(t)]t=tr1(t-tr1)2∥-∥[e(t)]t=tr1∥≤N∥ξ1e˙(t)p/q+ξ2e(t)g/h∥.
Noting thate(t)=e˙(t)=0 at t=ts, it yields
(30)∥[trDte(t)]t=tr2∥(t-tr)≤∥[e(t)]t=tr∥.
Consequently, if[trDte(t)]t=tr, then ts=tr. Else, we have
(31)ts≤2∥[e(t)]t=tr∥∥[tr1Dte(t)]t=tr/2∥+tr1=2∥e(tr)∥∥e˙(tr)∥+tr.
Therefore, it can be concluded that the system trajectories attain to reference trajectory in finite time.
The proof is complete.


## 3. Control Structure of Ext-ABS

In this section, Ext-ABS, which is designed to track any slip ratio based on the exiting antilock system (ABS), is chosen as illustrative example to verify the effectiveness of the proposed controllers. Here, we only consider the design of its control law using the proposed NFOTSM.


### 3.1. Quarter-Car Model

Tire friction estimation is a prerequisite for longitudinal friction control. For an electric vehicle with in-wheel motors, the brake and drive torque are measured by transducers mounted in motors. And there has been an attempt to measure the brake torque of conventional vehicles by force transducer mounted on the brake caliper support. In this paper, assume that the brake torque can be measured, and then the tire friction can be estimated using the method discussed below.A quarter-car model is shown in Figure1. The dynamic equations of motion are as follows:
(32)mdudt=-Fxb(33)Jdωdt=rbFxb-Tb(34)FZ=mg,
where m is the mass of the quarter car, u is the vehicle horizontal speed, Fxb is the tire friction, J is the wheel inertia, ω is the wheel angular speed, rb is the wheel radius, Tb is the brake torque, FZ is the vertical force, and g is the acceleration of gravity.Figure 1
Quarter-car model.Then the tire friction can be estimated by the following equation:(35)F^xb=Jrbdωdt+Tbrb.And the longitudinal slip ratioλ^xb with respect to F^xb is estimated by
(36)λ^xb=u-ωrbu.The following equation is to describe the nonlinear relationship of the tire frictionFxb and the corresponding λxb(37)Fxb=FZ·μxb(λxb),
where μxb is friction coefficient.The desired longitudinal slip ratioλRef, which is corresponding to the desired tire friction FRef, can be calculated by using the numerical method of nonlinear equations as follows:
(38)FRef-FZ·μRef*(λRef)=0.


### 3.2. Tire Model

The magic formula is applied in this paper, which is widely used to calculate steady-state tire force and moment in vehicle dynamics studies. For a constant vertical load, the longitudinal friction coefficient is given by(39)μL=μ(λ,BL,CL,DL,EL)=DLsin[CLarctan{BLλ-EL(BLλ-arctan(BLλ))}],
where μL is the longitudinal friction coefficient,  λ is the longitudinal slip, BL is the stiffness factor, CL is the shape factor, DL is the peak value, and EL is the curvature factor.


### 3.3. NFOTSM Control Law for Ext-ABS

The control architecture of the Ext-ABS is shown in Figure2. Firstly, let us define
(40)e=λ-λRef.
By (32), (33), and (36), we have
(41)λ˙=d((u-ωrb)/u)dt=-rb2Fxb/J+(1-λ)u˙u+TbrbuJ.Figure 2
Control architecture of the Ext-ABS.So, we get(42)e˙=-rb2Fxb/J+(1-λ)u˙u+TbrbuJ-λ˙Ref.As it was previously mentioned, we assume that the order of derivatives in the applied controller isα=0.9. So, the NFOTSM control input can be obtained as
(43)u(t)=T^b=uJrb{λ˙Ref--rb2Fxb/J+(1-λ)u˙u-qξ2p[ξ1ghDα+1e¨·e1-p/q+ξ1ghe˙g/h-1·e¨·e1-p/q-ϕs-γsζ1/ζ2]}.


## 3.1. Quarter-Car Model

Tire friction estimation is a prerequisite for longitudinal friction control. For an electric vehicle with in-wheel motors, the brake and drive torque are measured by transducers mounted in motors. And there has been an attempt to measure the brake torque of conventional vehicles by force transducer mounted on the brake caliper support. In this paper, assume that the brake torque can be measured, and then the tire friction can be estimated using the method discussed below.A quarter-car model is shown in Figure1. The dynamic equations of motion are as follows:
(32)mdudt=-Fxb(33)Jdωdt=rbFxb-Tb(34)FZ=mg,
where m is the mass of the quarter car, u is the vehicle horizontal speed, Fxb is the tire friction, J is the wheel inertia, ω is the wheel angular speed, rb is the wheel radius, Tb is the brake torque, FZ is the vertical force, and g is the acceleration of gravity.Figure 1
Quarter-car model.Then the tire friction can be estimated by the following equation:(35)F^xb=Jrbdωdt+Tbrb.And the longitudinal slip ratioλ^xb with respect to F^xb is estimated by
(36)λ^xb=u-ωrbu.The following equation is to describe the nonlinear relationship of the tire frictionFxb and the corresponding λxb(37)Fxb=FZ·μxb(λxb),
where μxb is friction coefficient.The desired longitudinal slip ratioλRef, which is corresponding to the desired tire friction FRef, can be calculated by using the numerical method of nonlinear equations as follows:
(38)FRef-FZ·μRef*(λRef)=0.


## 3.2. Tire Model

The magic formula is applied in this paper, which is widely used to calculate steady-state tire force and moment in vehicle dynamics studies. For a constant vertical load, the longitudinal friction coefficient is given by(39)μL=μ(λ,BL,CL,DL,EL)=DLsin[CLarctan{BLλ-EL(BLλ-arctan(BLλ))}],
where μL is the longitudinal friction coefficient,  λ is the longitudinal slip, BL is the stiffness factor, CL is the shape factor, DL is the peak value, and EL is the curvature factor.


## 3.3. NFOTSM Control Law for Ext-ABS

The control architecture of the Ext-ABS is shown in Figure2. Firstly, let us define
(40)e=λ-λRef.
By (32), (33), and (36), we have
(41)λ˙=d((u-ωrb)/u)dt=-rb2Fxb/J+(1-λ)u˙u+TbrbuJ.Figure 2
Control architecture of the Ext-ABS.So, we get(42)e˙=-rb2Fxb/J+(1-λ)u˙u+TbrbuJ-λ˙Ref.As it was previously mentioned, we assume that the order of derivatives in the applied controller isα=0.9. So, the NFOTSM control input can be obtained as
(43)u(t)=T^b=uJrb{λ˙Ref--rb2Fxb/J+(1-λ)u˙u-qξ2p[ξ1ghDα+1e¨·e1-p/q+ξ1ghe˙g/h-1·e¨·e1-p/q-ϕs-γsζ1/ζ2]}.


## 4. Simulation Results and Analysis

In this section, simulations are carried out to verify the effectiveness of the proposed controllers. The tracking ability of Ext-ABS is tested, and the superiority of the NFOTSM is compared with nonsingular terminal sliding mode control (NTSM). For the details, the whole performance of the friction servocontrol system (Ext-ABS) is shown underμ-jump road surface and sine input signal.


### 4.1. Simulation Results underμ-Jump Road Surface

In the simulation ofμ-jump road surface, the desired tire friction FRef input is 2500 N after 0.5 s. And at 1.5 s, the vehicle drives from asphalt, dry road to asphalt, wet road. Simulation results are shown in Figures 3–5.Figure 3
Tracking performance of tire friction.Figure 4
Tracking error of tire friction.Figure 5
Vehicle acceleration.As is depicted in Figures3 and 4, with respect to tire friction, since NFOTSM can use the nonlinear fractional-order term Dαe¨(t) to obtain a corrective control force, the values of NFOTSM track the reference value in faster response than that of NTSM after 0.5s and 1.5s. Consequently, the response of vehicle acceleration of NFOTSM is also more sensitive than that of NTSM, as is shown in Figure 5. This proves the proposed control strategy to be effective in tracking desired tire friction and in adaptation to the variation in road surface.


### 4.2. Simulation Results under Sine Input Signal

In the simulation of sine input signal, the desired tire frictionFRef input is a sine input. Simulation results are shown in Figures 6–8, under the proposed NFOTSM control law (41) and NTSM control law. Figure 6 represents, respectively, the state time response for tracking a periodic orbit sine input signal. It can be seen that the dynamics of system states are stabilized to a periodic motion. Time responses of the corresponding tracking error of tire friction are depicted in Figure 7. Figure 8 shows the state time responses of the vehicle acceleration. From the above corresponding responses, it can be concluded that the obtained theoretic results are feasible and efficient for controlling dynamical systems, and the proposed controller guarantees the finite time stability of the closed-loop system. Furthermore, it is obvious that the new control law makes the system stable and alleviates the chattering problem.Figure 6
Tracking performance of tire friction.Figure 7
Tracking error of tire friction.Figure 8
Vehicle acceleration.


## 4.1. Simulation Results underμ-Jump Road Surface

In the simulation ofμ-jump road surface, the desired tire friction FRef input is 2500 N after 0.5 s. And at 1.5 s, the vehicle drives from asphalt, dry road to asphalt, wet road. Simulation results are shown in Figures 3–5.Figure 3
Tracking performance of tire friction.Figure 4
Tracking error of tire friction.Figure 5
Vehicle acceleration.As is depicted in Figures3 and 4, with respect to tire friction, since NFOTSM can use the nonlinear fractional-order term Dαe¨(t) to obtain a corrective control force, the values of NFOTSM track the reference value in faster response than that of NTSM after 0.5s and 1.5s. Consequently, the response of vehicle acceleration of NFOTSM is also more sensitive than that of NTSM, as is shown in Figure 5. This proves the proposed control strategy to be effective in tracking desired tire friction and in adaptation to the variation in road surface.


## 4.2. Simulation Results under Sine Input Signal

In the simulation of sine input signal, the desired tire frictionFRef input is a sine input. Simulation results are shown in Figures 6–8, under the proposed NFOTSM control law (41) and NTSM control law. Figure 6 represents, respectively, the state time response for tracking a periodic orbit sine input signal. It can be seen that the dynamics of system states are stabilized to a periodic motion. Time responses of the corresponding tracking error of tire friction are depicted in Figure 7. Figure 8 shows the state time responses of the vehicle acceleration. From the above corresponding responses, it can be concluded that the obtained theoretic results are feasible and efficient for controlling dynamical systems, and the proposed controller guarantees the finite time stability of the closed-loop system. Furthermore, it is obvious that the new control law makes the system stable and alleviates the chattering problem.Figure 6
Tracking performance of tire friction.Figure 7
Tracking error of tire friction.Figure 8
Vehicle acceleration.


## 5. Conclusions

In this paper, the problem of designing NFOTSMC for a class of dynamical systems is investigated. Based on the Lyapunov stability criteria, the nonsingular FO terminal sliding mode control law is designed, and the finite time stability of the closed-loop system is guaranteed under the proposed controller. To further verify the control effect, the Ext-ABS using NFOTSM is designed to realize self-tuning tire friction control system adaptive to variation of road condition. In spite of a higher computation load compared with other control methods, a drastic optimization of the algorithms allows the controller to provide a fast adaptation to the change in road surface condition, and its stability is proven thanks to a sufficient number of simulation scenarios, such as a sudden change of road condition and a sudden change of reference force condition.



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*Source: 523251-2013-08-13.xml*