# Stability of Infinite Dimensional Interconnected Systems with Impulsive and Stochastic Disturbances

Authors: Xiaohui Xu; Xiaofeng Yin; Jiye Zhang; Peng Wang
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

Some research on the stability with mode constraint for a class of infinite dimensional look-ahead interconnected systems with impulsive and stochastic disturbances is studied by using the vector Lyapunov function approach. Intuitively, the stability with mode constraint is the property of damping disturbance propagation. Firstly, we derive a set of sufficient conditions to assure the stability with mode constraint for a class of general infinite dimensional look-ahead interconnected systems with impulsive and stochastic disturbances. The obtained conditions are less conservative than the existing ones. Secondly, the controller for a class of look-ahead vehicle following systems with the above uncertainties is constructed by the sliding mode control method. Based on the obtained new stability conditions, the domain of the control parameters of the systems is proposed. Finally, a numerical example with simulations is given to show the effectiveness and correctness of the obtained results.


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## Body

## 1. Introduction

In the real industries, the control problem of many complex dynamic systems can be translated into the stability analysis. At present, there have been lots of research results about the stability analysis for the finite dimensional interconnected systems; see [1–11]. Nevertheless, considering that the connections or disconnections between the subsystems of the real interconnected systems are uncertain, which means that the dimension of the interconnected systems is uncertain, the interconnected systems can be described by infinite dimensional equations. On the other hand, there are some unavoidable disturbances in the real systems, such as stochastic disturbance and impulsive disturbance. Therefore some researchers have given stability analysis for some finite dimensional complex dynamic systems with impulsive and stochastic disturbances; see [6–11].The applied methods presented in [1–11] are based on the scalar Lyapunov function approach or the LMI tool. In fact, the LMI method is essentially a kind of the method using the scalar Lyapunov function method. It should be noted that until now there is no general constructive method for building the Lyapunov functions for nonlinear systems. In comparison with the vector Lyapunov function method, the scalar method or the LMI method needs to discuss the convergence of the scalar Lyapunov function when analyzing the stability of infinite dimensional systems. Hence the vector Lyapunov function method is more efficient. The research team led by Professor Zhang has studied the stability of some infinite dimensional nonlinear interconnected systems with stochastic disturbances based on the vector Lyapunov function approach and obtained some important stability results; see [12–14].The obtained stability results in [2–12, 14] are focused on the stability of the steady state of the systems without considering the size relationship of the state variables when the systems converge to steady-state process. For example, considering interconnected system x˙=f(t,x) (here  x=col(x1,x2,…,xn)), the states are needed to be satisfied that sup⁡∥x1(t)∥t≥0≥sup⁡∥x2(t)∥t≥0≥,…,≥sup⁡∥xn(t)∥t≥0 or sup⁡∥x1(t)∥t≥0≥maxi=2,3,…{sup⁡∥xi(t)∥t≥0}. The stability with the above constraint condition is named the stability with mode constraint. The Lyapunov stability in the general sense cannot describe the stability with mode constraint condition, and the existing Lyapunov function methods cannot be used to analyze the stability with mode constraint for the systems directly. The motivation for the stability problem with mode constraint comes from the analysis and the design of controllers for automated highway system [13], multirobot operation system [15], formation flying of unmanned aerial vehicles [16], and so on. In a formation one wants controllers to be designed so that any shock-wave arising from disturbance propagation should dampen as it travels away from the source. In other words the closed loop interconnected system for the formation needs to be stable with constraint condition.Automatic vehicle longitudinal following control is an important issue for coordinated control for a group of unmanned autonomous vehicles in automated highway system (for short, AHS). In AHS, vehicles are dynamically coupled by feedback control laws. The control objective is to dramatically improve the traffic flow capacity on a highway by enabling vehicles to travel together in tightly spaced platoons [15]. Therefore, the controller design for vehicle longitudinal following systems (for short, VLFS) is an interesting and challenging problem. Some significant research on the string stability analysis for VLFS has been done; see [17, 18]. Nevertheless, uncertain disturbance factors were not considered in [17, 18]. Uncertainties inevitably exist in the vehicle operating environment and vehicle systematic itself. In [19], some sufficient conditions, which assure the string stability of a class of stochastic VLFS with infinite dimensions, were obtained by using the vector Lyapunov function method. Since the Cauchy inequality technique was applied to deduce the stability conditions for the systems, the obtained criteria were relatively conservative. Besides, the controller design for VLFS was not studied in there. In [20], the problem of stabilization control system for a single vehicle in response to the exogenous impulsive disturbances was studied. The obtained results cannot be used to analyze the stability and controller design for the VFLS directly.To design the controller for the VFLS, there are various approaches, such as fuzzy control [21], sliding mode control [18], adaptive control [15], adaptive-sliding mode control [22], and fuzzy-sliding mode control [23]. However, the factor of uncertainties to the systems was not considered in the above references. On the other hand, the number of vehicles in VFLS is indeterminate as vehicles enter into or leave the platoon randomly. Therefore, the VLFS can only be described as infinite dimensional interconnected system.To sum up, this paper will present some sufficient conditions for assuring the stability with mode constraint for a class of infinite dimensional look-ahead interconnected systems with impulsive and stochastic disturbances by using the vector Lyapunov function approach. Furthermore, the controller for a class of look-ahead VLFS with the above uncertainties is constructed by the sliding mode control method. Based on the obtained new stability conditions, the domain of the control parameters of the systems is proposed.


## 2. Mathematical Preliminaries

For convenience, some notations are introduced as follows:(1)∥yi∥∞p=∥yi(·)∥∞p=sup⁡t≥0⁡E[|yi(t)|p],∥y(0)∥∞p=sup⁡i⁡E[|yi(0)|p],∥yi∥p=∥yi(·)∥p=(∫0∞E[|yi(t)|p]dt)1/p,∥y(0)∥p=(∑i=1∞E[|yi(0)|p])1/p,∥y(0)∥∞i-1=max⁡1≤j≤i-1⁡∥yj(0)∥,∥y(t)∥∞i-1=max⁡1≤j≤i-1⁡sup⁡t≥0⁡∥yj(t)∥,
where |·| is the Euclidean norm and 0<p<∞, i∈N, E denotes the expectation of stochastic process, and N denotes the set of natural numbers.


### 2.1. Model Description

Consider a class of general nonlinear infinite dimensional look-ahead interconnected systems with stochastic and impulsive disturbances described by(2)dyi=fi(yi,yi-1,y˙i-1)dt+qi(yi,yi-1,y˙i-1)dξi,t≠tkΔyi(tk)=yi(tk+)-yi(tk-),t=tk;
here, t∈[t0,+∞) and  yi∈Rn denotes the state of the ith subsystem, i∈N.   Δyi(tk) is the impulsive strength at discrete moment tk, and the discrete set {tk} is assumed to be satisfied that 0≤t0<t1<⋯<tk<⋯, and k→∞ as tk→∞, k∈N. It is assumed that yi(t) is right, continuous with yi(tk)=yi(tk+), and yi(tk-)=limt→tk-yi(t). Consider fi, qi: Rn×Rn×Rn→Rn;   let fi(0,0,0)=qi(0,0,0)=0. Assume that ξi is the one-dimensional independent standard Wiener processes defined on space (Ω,F,P); here Ω denotes sample space, F denotes σ algebra of subset of the sample space, and P denotes probability measures.The system (2) can be treated as an interconnection of isolated subsystems xi given by
(3)dyi=fi(yi,0,0)dt+qi(yi,0,0)dξi,t≠tk,Δyi(tk)=yi(tk+)-yi(tk-),t=tk,i,k∈N.


### 2.2. Definitions and Assumptions

Letyi=0  (i∈N) be the unique zero solution of system (2).Definition 1 (see [19]).
If, for∀ε>0,∃δ>0 such that ∥y(0)∥∞2<δ⇒supi∥yi(t)∥∞2<ε, then yi=0  (i∈N) is string stable in the mean square sense.
It should be noted that the string stability in this paper is defined for look-ahead interconnected system, which is a special class of interconnected systems. The string stability could guarantee that the state of every subsystem is uniformly bounded if the initial states of the subsystems are bounded.Definition 2 (see [19]).
The zero solution of system (2) yi=0  (i∈N) is string exponentially stable in the mean square sense if yi=0 is string stable, and there exist constants M>0 and λ>0 such that ∥yi(t)∥∞2<M∥yi(0)∥2e-λt holds.Definition 3 (see [13]).
The zero solution of system (2) yi=0  (i∈N) is string exponentially stable with mode constraint in the mean square sense if yi=0 is string exponentially stable in the mean square sense, and the inequality ∥yi(t)∥∞2≤(∥y(t)∥∞i-1)2 holds, t≥0,  i≥2.Next, some assumptions will be given for the system (2) and the system (3).Assumption A1. There exist positive constants klf,  klq (l=1,2) and d1f,  d1q such that f and q are globally Lipschitz in their arguments; that is,
(4)|f(y1,y2,y3)-f(z1,z2,z3)|≤k1f|y1-z1|+k2f|y2-z2|+d1f|y3-z3|,|q(y1,y2,y3)-q(z1,z2,z3)|≤k1q|y1-z1|+k2q|y2-z2|+d1q|y3-z3|.Assumption A2. For every isolated subsystem (3), there exists positive definite function Vi(t,yi(t)),  yi∈Rn, which is continuously twice differentiable with respect to yi, and there exist positive constants αm  (m=1,2,3,4,5) such that(i)
α1|yi|2≤Vi(t,yi(t))≤α2|yi|2;(ii)
χ(2)Vi(t,yi(t))≤-α3|yi|2;(iii)
|∂Vi(t,yi(t))/∂yi,j|≤α4|xi|, |∂2Vi(t,yi(t))/∂yi,lyi,j|≤α5,whereχ(2)(·) is an operator associated with (3) defined by
(5)χ(2)(·)=∂(·)∂t+∑j=1n∂(·)∂xi,jfj(yi,0,0)+0.5∑l=1n∑j=1n∂2(·)∂yi,j∂yi,lσlj(yi,0,0),
where σlj(yi,0,0)=ql(yi,0,0)qj(yi,0,0),l,j=1,2,…,n,i∈N.LetVi denote Vi(yi(t)) if there is no confusion, i∈N.Assumption A3. Let Δyi(tk)=Iik(yi(tk-)). The function Iik(·) with Iik(0)=0 denotes the strength of the intensity of impulsive disturbances, which is assumed to be continuous with t. Suppose that there exist positive constants γ_ik and γ-ik such that γ_ik|yi(tk-)|≤|Iik(yi(tk-))+yi(tk-)|≤γ-ik|yi(tk-)| holds; here, 0<γ_ik<1<γ-ik,  i,k∈N.Next a lemma established by us in [24] is given which will be used in the proof of the following theorem.Lemma 4.
Suppose thatWi(t)≥0, and E[Wi(t)]<∞,t≥0,i∈N. Consider the following inequalities:
(6)Eχ(1)(Wi)≤gi(E(Wi),E(Wi-1),…,E(W1),t)×{[∑j=1∞β′ijE(Wi-jmij)]2-βi0E(Wimii)+∑j=1∞βijE(Wi-jmij)+[E(Wimii)]-1[∑j=1∞βij′E(Wi-jmij)]2}
for Wi>0; it is assumed that gi(·)>0. Let βi0>0,βij≥0,βij′≥0, and βij=βij′=0  (j≥i),mii≤mij,   i,j∈N.  χ(1)(·) is an operator associated with (2) given by
(7)χ(1)(·)=∂(·)∂t+∑j=1n∂(·)∂yi,jfj(yi,yi-1,y˙i-1)+0.5∑l=1n∑j=1n∂2(·)∂yi,j∂yi,lσlj(yi,yi-1,y˙i-1),σlj(yi,yi-1,y˙i-1)=ql(yi,yi-1,y˙i-1)qj(yi,yi-1,y˙i-1),l,j=1,2,…,n,
where yi=[yi1,yi2,…,yin]∈Rn.If there existsW=(E(W10),E(W20),…) such that
(8)-βi0E(Wi,0mii)+∑j=1∞βijE(Wi-j,0mij)+[E(Wi,0mii)]-1[∑j=1∞βij′E(Wi-j,0mij)]2<0,
then, for ∀ε>0,  ∃δ>0 such that ∥W(0)∥∞<δ⇒sup⁡i∥Wi(t)∥∞<ε.


## 2.1. Model Description

Consider a class of general nonlinear infinite dimensional look-ahead interconnected systems with stochastic and impulsive disturbances described by(2)dyi=fi(yi,yi-1,y˙i-1)dt+qi(yi,yi-1,y˙i-1)dξi,t≠tkΔyi(tk)=yi(tk+)-yi(tk-),t=tk;
here, t∈[t0,+∞) and  yi∈Rn denotes the state of the ith subsystem, i∈N.   Δyi(tk) is the impulsive strength at discrete moment tk, and the discrete set {tk} is assumed to be satisfied that 0≤t0<t1<⋯<tk<⋯, and k→∞ as tk→∞, k∈N. It is assumed that yi(t) is right, continuous with yi(tk)=yi(tk+), and yi(tk-)=limt→tk-yi(t). Consider fi, qi: Rn×Rn×Rn→Rn;   let fi(0,0,0)=qi(0,0,0)=0. Assume that ξi is the one-dimensional independent standard Wiener processes defined on space (Ω,F,P); here Ω denotes sample space, F denotes σ algebra of subset of the sample space, and P denotes probability measures.The system (2) can be treated as an interconnection of isolated subsystems xi given by
(3)dyi=fi(yi,0,0)dt+qi(yi,0,0)dξi,t≠tk,Δyi(tk)=yi(tk+)-yi(tk-),t=tk,i,k∈N.


## 2.2. Definitions and Assumptions

Letyi=0  (i∈N) be the unique zero solution of system (2).Definition 1 (see [19]).
If, for∀ε>0,∃δ>0 such that ∥y(0)∥∞2<δ⇒supi∥yi(t)∥∞2<ε, then yi=0  (i∈N) is string stable in the mean square sense.
It should be noted that the string stability in this paper is defined for look-ahead interconnected system, which is a special class of interconnected systems. The string stability could guarantee that the state of every subsystem is uniformly bounded if the initial states of the subsystems are bounded.Definition 2 (see [19]).
The zero solution of system (2) yi=0  (i∈N) is string exponentially stable in the mean square sense if yi=0 is string stable, and there exist constants M>0 and λ>0 such that ∥yi(t)∥∞2<M∥yi(0)∥2e-λt holds.Definition 3 (see [13]).
The zero solution of system (2) yi=0  (i∈N) is string exponentially stable with mode constraint in the mean square sense if yi=0 is string exponentially stable in the mean square sense, and the inequality ∥yi(t)∥∞2≤(∥y(t)∥∞i-1)2 holds, t≥0,  i≥2.Next, some assumptions will be given for the system (2) and the system (3).Assumption A1. There exist positive constants klf,  klq (l=1,2) and d1f,  d1q such that f and q are globally Lipschitz in their arguments; that is,
(4)|f(y1,y2,y3)-f(z1,z2,z3)|≤k1f|y1-z1|+k2f|y2-z2|+d1f|y3-z3|,|q(y1,y2,y3)-q(z1,z2,z3)|≤k1q|y1-z1|+k2q|y2-z2|+d1q|y3-z3|.Assumption A2. For every isolated subsystem (3), there exists positive definite function Vi(t,yi(t)),  yi∈Rn, which is continuously twice differentiable with respect to yi, and there exist positive constants αm  (m=1,2,3,4,5) such that(i)
α1|yi|2≤Vi(t,yi(t))≤α2|yi|2;(ii)
χ(2)Vi(t,yi(t))≤-α3|yi|2;(iii)
|∂Vi(t,yi(t))/∂yi,j|≤α4|xi|, |∂2Vi(t,yi(t))/∂yi,lyi,j|≤α5,whereχ(2)(·) is an operator associated with (3) defined by
(5)χ(2)(·)=∂(·)∂t+∑j=1n∂(·)∂xi,jfj(yi,0,0)+0.5∑l=1n∑j=1n∂2(·)∂yi,j∂yi,lσlj(yi,0,0),
where σlj(yi,0,0)=ql(yi,0,0)qj(yi,0,0),l,j=1,2,…,n,i∈N.LetVi denote Vi(yi(t)) if there is no confusion, i∈N.Assumption A3. Let Δyi(tk)=Iik(yi(tk-)). The function Iik(·) with Iik(0)=0 denotes the strength of the intensity of impulsive disturbances, which is assumed to be continuous with t. Suppose that there exist positive constants γ_ik and γ-ik such that γ_ik|yi(tk-)|≤|Iik(yi(tk-))+yi(tk-)|≤γ-ik|yi(tk-)| holds; here, 0<γ_ik<1<γ-ik,  i,k∈N.Next a lemma established by us in [24] is given which will be used in the proof of the following theorem.Lemma 4.
Suppose thatWi(t)≥0, and E[Wi(t)]<∞,t≥0,i∈N. Consider the following inequalities:
(6)Eχ(1)(Wi)≤gi(E(Wi),E(Wi-1),…,E(W1),t)×{[∑j=1∞β′ijE(Wi-jmij)]2-βi0E(Wimii)+∑j=1∞βijE(Wi-jmij)+[E(Wimii)]-1[∑j=1∞βij′E(Wi-jmij)]2}
for Wi>0; it is assumed that gi(·)>0. Let βi0>0,βij≥0,βij′≥0, and βij=βij′=0  (j≥i),mii≤mij,   i,j∈N.  χ(1)(·) is an operator associated with (2) given by
(7)χ(1)(·)=∂(·)∂t+∑j=1n∂(·)∂yi,jfj(yi,yi-1,y˙i-1)+0.5∑l=1n∑j=1n∂2(·)∂yi,j∂yi,lσlj(yi,yi-1,y˙i-1),σlj(yi,yi-1,y˙i-1)=ql(yi,yi-1,y˙i-1)qj(yi,yi-1,y˙i-1),l,j=1,2,…,n,
where yi=[yi1,yi2,…,yin]∈Rn.If there existsW=(E(W10),E(W20),…) such that
(8)-βi0E(Wi,0mii)+∑j=1∞βijE(Wi-j,0mij)+[E(Wi,0mii)]-1[∑j=1∞βij′E(Wi-j,0mij)]2<0,
then, for ∀ε>0,  ∃δ>0 such that ∥W(0)∥∞<δ⇒sup⁡i∥Wi(t)∥∞<ε.


## 3. Stability Results

In this section, some sufficient conditions for judging the string exponential stability with mode constraint in the mean square for system (2) will be established.Theorem 5.
Suppose that Assumptions A1–A3 are satisfied. If there exist constantsη>0 and ξ>0 such that 2ln⁡η-k/(tk-tk-1)≤η<ξ and if the following inequality holds,
(9)(-α3+ξα2)(η_k)2α2-1/2+(η-k)2nα4α1-1/2(k2f+d1fk1f)×∑j=1i-1(d1f)j-1+12(η-k)2n2α5×[α1-1/2(k2q+d1qk1q)∑j=1i-1(d1q)j-1]×[2k1q+(α2α1)1/2(k2q+d1qk1q)∑j=1i-1(d1q)j-1]<0,
where η-k=sup⁡i{γ-ik},  η_k=inf⁡i{γ_ik},  k∈N, then the zero solution of system (2) is string exponentially stable with mode constraint in the mean square sense.Proof.
As mentioned in Assumption A2, the functionVi is the vector Lyapunov function of the ith isolated subsystem of the interconnected system (2). According to the vector Lyapunov function theory, in order to obtain the exponential string stability conditions with mode constraint for system (2), we choose the following vector Lyapunov function:
(10)Wi=eξtVi,i∈N.
Whent≠tk,  k∈N, calculating the operator χ(1)Wi along the zero solution of system (2) and applying Assumptions A1-A2, we get
(11)χ(1)Wi=ξeξtVi+eξt×[∑j=1n∂Vi∂yi,jfj(yi,yi-1,y˙i-1)+12∑l=1n∑j=1n∂2Vi∂yi,l∂yi,jσlj(yi,yi-1,y˙i-1)]≤ξeξtVi+eξt×{[(k2q+d1qk1q)∑j=1i-1(d1q)j-1|yi-j|]-α3|yi|2+nα4|yi|(k2f+d1fk1f)×∑j=1i-1(d1f)j-1|yi-j|+0.5n2α5×[2k1q|yi|+(k2q+d1qk1q)∑j=1i-1(d1q)j-1|yi-j|]×[(k2q+d1qk1q)∑j=1i-1(d1q)j-1|yi-j|]}≤(|yi|eξt/2)×{[(k2q+d1qk1q)∑j=1i-1(d1q)j-1eξt2|yi-j|]ξα2eξt/2|yi|-α3eξt/2|yi|+nα4(k2f+d1fk1f)×∑j=1i-1(d1f)j-1eξt/2|yi-j|+0.5n2α5[×∑j=1i-1(d1q)j-1|yi-j||yi|]2k1q+(k2q+d1qk1q)×∑j=1i-1(d1q)j-1|yi-j||yi|]×[(k2q+d1qk1q)∑j=1i-1(d1q)j-1eξt/2|yi-j|]}.
It follows from Assumption A2 and Wi=eξtVi that
(12)χ(1)Wi≤(|yi|eξt/2)×{[×∑j=1i-1(d1q)j-1(Wi-jWi)1/2(α2α1)1/2]2k1q+(k2q+d1qk1q)(-α3+ξα2)(Wiα2)1/2+nα4(k2f+d1fk1f)×∑j=1i-1(d1f)j-1(Wi-jα1)1/2+12n2α5×[×∑j=1i-1(d1q)j-1(Wi-jWi)1/2(α2α1)1/2]2k1q+(k2q+d1qk1q)×∑j=1i-1(d1q)j-1(Wi-jWi)1/2(α2α1)1/2]×[(k2q+d1qk1q)∑j=1i-1(d1q)j-1(Wi-jα1)1/2](-α3+ξα2)(Wiα2)1/2+nα4(k2f+d1fk1f)}.
From the properties of the operatorχ(1) [25], we can take the expectation of inequality (12) and rewrite it as
(13)Eχ(1)Wi≤E[(|yi|)]eξt/2×{[(k2q+d1qk1q)∑j=1i-1(d1q)j-1E[(Wi-jα1)1/2]](-α3+ξα2)E[(Wiα2)1/2]+nα4(k2f+d1fk1f)×∑j=1i-1(d1f)j-1E[(Wi-jα1)1/2]+12n2α5×[2k1q+(k2q+d1qk1q)∑j=1i-1(d1q)j-1×E[(Wi-jWi)1/2](α2α1)1/2[2k1q+(k2q+d1qk1q)∑j=1i-1(d1q)j-1]×[(k2q+d1qk1q)∑j=1i-1(d1q)j-1E[(Wi-jα1)1/2]]}.
TakingE(Wi-j,0)=1,  j=0,1,…,i-1,  i∈N. Substituting them into (13), we get
(14)Eχ(1)Wi≤E[(|yi|)]eξt/2×{[(k2q+d1qk1q)∑j=1i-1(d1q)j-1(α1)-1/2](-α3+ξα2)(α2)-1/2+nα4(k2f+d1fk1f)×∑j=1i-1(d1f)j-1(α1)-1/2+12n2α5×[×∑j=1i-1(d1q)j-1(α2α1)1/2]2k1q+(k2q+d1qk1q)×∑j=1i-1(d1q)j-1(α2α1)1/2]×[(k2q+d1qk1q)∑j=1i-1(d1q)j-1(α1)-1/2]}.
Since inequality (12) implies that the following inequality holds,
(15)(-α3+ξα2)α2-1/2+nα4α1-1/2(k2f+d1fk1f)×∑j=1i-1(d1f)j-1+12n2α5[α1-1/2(k2q+d1qk1q)∑j=1i-1(d1q)j-1]×[2k1q+(α2α1)1/2(k2q+d1qk1q)∑j=1i-1(d1q)j-1]<0,
it can be concluded that E[χ(1)Wi]<0,  i∈N. Therefore, it follows from Lemma 4 that, for ∀ε0>0,  ∃δ0>0 such that ∥Wi(0)∥∞<δ0⇒supi∥Wi(t)∥∞<ε0. Let ε>0, and satisfy ε0=α1ε2. Take δ0=α2δ2. When supi∥xi(0)∥<δ, we get Wi(0)=Vi(0)≤α2∥xi(0)∥2≤α2δ2=δ0. Furthermore, we have supi∥Wi(t)∥∞=supi∥Vi(t)∥∞eξt<ε0; namely,
(16)sup⁡i⁡∥yi(t)∥∞≤supi∥Vi(t)∥∞eξtα1e-ξt/2<ε0α1e-ξt/2=εe-ξt/2,0≤t<t1.
Next, we will use the mathematical induction method to prove that(17)sup⁡i⁡∥yi(t)∥∞<η-0η-1η-2⋯η-k-1εe-ξt/2,tk-1≤t<tk,i,k∈N,
hold, where η-0=1; namely,
(18)sup⁡i⁡∥yi(t)∥∞2<(η-0η-1η-2⋯η-k-1)2ε2e-ξt,tk-1≤t<tk,i,k∈N.
Whenk=1, it can be seen from (16) that (18) holds. Suppose that the following inequalities hold:
(19)sup⁡i⁡∥yi(t)∥∞2<(η-0η-1η-2⋯η-k-1)2ε2e-ξt,tp-1≤t<tp,p=1,2,…,k.
From Assumption A3 and (19), we have
(20)∥yi(tk+)∥∞2=∥Iik(yi(tk-))+yi(tk-)∥∞2≤γ-ik2∥yi(tk-)∥∞2≤η-k2∥yi(tk-)∥∞2,i,k∈N.
Due toη-k>1, we have
(21)∥yi(t)∥∞2<(η-0η-1η-2⋯η-k-1η-k)2ε2e-ξt,tk-1≤t≤tk,i,k∈N.
We claim that (21) implies that
(22)∥yi(t)∥∞2<(η-0η-1η-2⋯η-k-1η-k)2ε2e-ξt,tk≤t<tk+1,i,k∈N.
If inequality (22) does not hold, there exist some i and t*∈[tk,tk+1) such that
(23)∥yi(t*)∥∞2=(η-0η-1η-2⋯η-k-1η-k)2ε2e-ξt,χ(1)Wi(t*)=D+Wi(t*)≥0, and
(24)∥yj(t)∥∞2≤(η-0η-1η-2⋯η-k-1η-k)2ε2e-ξt,t∈[tk,t*],j=1,2,…,i-1.
Substituting (23) and inequality (24) into inequality (11), together with the properties of the operator χ, we get
(25)E[χ(1)Wi(t*)]≤∥yi∥{[(k2q+d1qk1q)∑j=1i-1(d1q)j-1]2}(-α3+ξα2)+nα4(k2f+d1fk1f)×∑j=1i-1(d1f)j-1+n2α5k1q(k2q+d1qk1q)×∑j=1i-1(d1q)j-1+12n2α5×[(k2q+d1qk1q)∑j=1i-1(d1q)j-1]2}×εη-0η-1η-2⋯η-k-1η-k.
Since condition (12) implies that
(26)(-α3+ξα2)+nα4(k2f+d1fk1f)∑j=1i-1(d1f)j-1+n2α5k1q(k2q+d1qk1q)∑j=1i-1(d1q)j-1+12n2α5[(k2q+d1qk1q)∑j=1i-1(d1q)j-1]2<0,E[χ(1)Wi(t*)]<0. This is a contradiction with χ(1)Wi(t*)≥0. By the mathematical induction method, it can be concluded that
(27)sup⁡i⁡∥yi(t)∥∞<η-0η-1η-2⋯η-k-1η-kεe-ξt/2,tk-1≤t<tk,k∈N.
From condition2ln⁡η-k/(tk-tk-1)≤η<ξ, we get that η-k≤e(1/2)η(tk-tk-1),k∈N. Furthermore,
(28)sup⁡i⁡∥yi(t)∥∞<e(1/2)η(t1-t0)e(1/2)η(t2-t1)⋯e(1/2)η(tk-1-tk-2)×εe-(1/2)ξt≤εe-(1/2)(ξ-η)t.
That is, supi∥yi(t)∥∞2<ε2e-(ξ-η)t,  t≥t0.
According to Definition2, the zero solution of system (2) is the string exponentially stable in the mean square sense with convergence rate ξ-η.
Next, we proceed to prove that the zero solution of system (2) is stable with the mode constraint. From the previous analysis, when t≠tk,  k∈N, it is easy to obtain
(29)χ(1)Vi≤-α3|yi|2+nα4|yi|(k2f+d1fk1f)|y|∞i-1∑j=1i-1(d1f)j-1+0.5n2α5[2k1q|yi|+(k2q+d1qk1q)|y|∞i-1∑j=1i-1(d1q)j-1]×[(k2q+d1qk1q)|y|∞i-1∑j=1i-1(d1q)j-1]≤-α3|yi|2+nα4(k2f+d1fk1f)×∑j=1i-1(d1f)j-1[|yi|∞2+(|y|∞i-1)2]2+n2α5k1q(k2q+d1qk1q)×∑j=1i-1(d1q)j-1[|yi|∞2+(|y|∞i-1)2]2+12n2α5(k2q+d1qk1q)2(|y|∞i-1)2[∑j=1i-1(d1q)j-1]2=[-α3+12nα4(k2f+d1fk1f)∑j=1i-1(d1f)j-1+12n2α5k1q(k2q+d1qk1q)∑j=1i-1(d1q)j-1](|yi|∞2)+{[∑j=1i-1(d1q)j-1]2}12nα4(k2f+d1fk1f)∑j=1i-1(d1f)j-1+12n2α5k1q(k2q+d1qk1q)∑j=1i-1(d1q)j-1×12n2α5(k2q+d1qk1q)2[∑j=1i-1(d1q)j-1]2}(|y|∞i-1)2.
Letc=(α2/α1)1/2≥1, and
(30)gm=θk(|y(t)|∞k-1)2×({α3-0.5nα4(k2f+d1fk1f)∑j=1k-1(d1f)j-1-12n2α5k1q(k2q+d1qk1q)∑j=1m-1(d1q)j-1})-1,t≠tk,k∈N,
where
(31)θm=12nα4(k2f+d1fk1f)∑j=1m-1(d1f)j-1+12n2α5k1q(k2q+d1qk1q)∑j=1m-1(d1q)j-1+12n2α5(k2q+d1qk1q)2[∑j=1m-1(d1q)j-1]2.
Define sets in state space:Π_i={xi:|yi(t)|2≤gi} and Π¯i={yi:|yi(t)|2≤cgi},i∈N. It is obvious that if yi∈Rn∖Π_i, then χ(1)Vi<0. Therefore, for all yi∈Rn∖Π-i and zi∈Rn∖Π-i, we have
(32)Vi(zi(t))≤α2|zi(t)|2<(c2)-1α2|zi(t)|2=α1|yi(t)|2≤Vi(yi(t)),t≠tk,k∈N.
This implies that, foryi(0)∈Π_i, we have yi(t)∈Π-i, t>0; that is,
(33)|yi(t)|2≤(α2α1)1/2θi(|y(t)|∞i-1)2×({α3-0.5nα4(k2f+d1fk1f)∑j=1i-1(d1f)j-1-0.5n2α5k1q(k2q+d1qk1q)∑j=1i-1(d1q)j-1})-1,t≠tk,i,k∈N.
Note that condition (9) implies that the following inequality holds:
(34)[-α3+12nα4(k2f+d1fk1f)∑j=1i-1(d1f)j-1+12n2α5k1q(k2q+d1qk1q)∑j=1i-1(d1q)j-1]α2-1/2+[[∑j=1i-1(d1q)j-1]212nα4(k2f+d1fk1f)∑j=1k-1(d1f)j-1+12n2α5k1q(k2q+d1qk1q)∑j=1k-1(d1q)j-1+12n2α5(k2q+d1qk1q)2[∑j=1i-1(d1q)j-1]2]α1-1/2<0;
namely,
(35)(α2α1)1/2θi×({α3-12nα4(k2f+d1fk1f)∑j=1i-1(d1f)j-1-12n2α5k1q(k2q+d1qk1q)∑j=1i-1(d1q)j-1})-1<1.
Let(36)r=(α2α1)1/2θi×({α3-12nα4(k2f+d1fk1f)∑j=1i-1(d1f)j-1-12n2α5k1q(k2q+d1qk1q)∑j=1i-1(d1q)j-1})-1.
Furthermore, inequality (33) can be rewritten as |yi(t)|2<r(|y(t)|∞i-1)2; that is,
(37)∥yi(t)∥∞2≤r(∥y(t)∥∞i-1)2<(∥y(t)∥∞i-1)2,tk-1≤t<tk,i,k∈N.
Whent=tk, k∈N, it follows from Assumption A3 that
(38)|yi(tk+)|∞=|Iik(yi(tk-))+yi(tk-)|∞≤η-k|yi(tk-)|∞<η-kr(|y(tk-)|∞i-1),i,k∈N.
Note that(39)|y(tk-)|∞i-1=max⁡1≤j≤i-1⁡{yj(tk-)}≤max⁡1≤j≤i-1⁡{(r_jk)-1|yj(tk+)|}≤(η_k)-1max⁡1≤j≤i-1⁡{|yj(tk+)|}.
Substituting (39) into (38), we have |yi(tk+)|∞<η-k(η_k)-1rmax1≤j≤i-1{|yj(tk+)|}, i,k∈N. According to condition (9), we have η-k(η_k)-1r<1, so
(40)|yi(tk+)|∞<max⁡1≤j≤i-1⁡{|yj(tk+)|};
namely, ∥yi(tk+)∥∞2<(∥y(tk+)∥∞i-1)2, i,k∈N. This along with (39) means that the zero solution of system (2) is stable with mode constraint.
Combining (28), (37), and (40), it follows from Definition 3 that the zero solution of system (2) is string exponentially stable with mode constraint in the mean square.Whend1f and d1q in Assumption A3 satisfy 0<d1f<1, 0<d1q<1, the stability of system (2) can be judged by the following corollary.Corollary 6.
Consider the system (2). Suppose that Assumptions A1–A3 are satisfied. If there exist constants η>0 and ξ>0 such that 2ln⁡η-k/(tk-tk-1)≤η<ξ and if the following inequality holds,
(41)(-α3+ξα2)η_kα2-1/2+nη-kα4α1-1/2(k2f+d1fk1f)×(1-d1f)-1+12η-kn2α5×[α1-1/2(k2q+d1qk1q)(1-d1q)-1]×[2k1q+(α2α1)1/2(k2q+d1qk1q)(1-d1q)-1]<0,
where η-k=sup⁡i{γ-ik}, η_k=inf⁡i{γ_ik}, k∈N, 0<d1f<1, and 0<d1q<1, then the zero solution of system (2) is string exponentially stable with mode constraint in the mean square sense.The proof of Corollary6 can be done by induction as the proof of Theorem 5, and so we omit it here.Remark 7.
By using the vector Lyapunov function method, the stability of a class of infinite dimensional look-ahead interconnected systems is studied in this paper. It should be noted that a comparison system aggregated by Lyapunov functions is usually a linear system. So when applying the stability condition of such a linear system to the original nonlinear system, “super-sufficient” stability conditions are obtained in general, as analyzed in [17, 19]. That is to say, the obtained stability conditions are relatively conservative. It can be seen from inequality (14) that the comparison systems aggregated by the Lyapunov functions in this paper are still nonlinear systems, which means that our obtained results are less conservative than the existing ones.Remark 8.
The dynamic behavior of some stochastic look-ahead interconnected systems without considering impulsive disturbance has been analyzed in [19, 24], and some sufficient conditions ensuring the string stability of the system have been obtained by the vector Lyapunov function methods. The obtained conditions in [19, 24] cannot be used to judge the stability with mode constraint for the systems. On the other hand, the models studied in [19, 24] are derived from the context of the controller design problem of look-ahead vehicle longitudinal following systems. However, the authors did not further study how to find the suitable parameters domain of the controller for the systems based on their established stability conditions.Remark 9.
Some research has been studied by us in [13] on the string stability with mode constraint for the system (2) without impulsive disturbance. It is easy to see that the results in [13] are contained in the obtained results in Theorem 5 in this paper.The research studied in [13, 19, 24] did not pay attention to how to choose parameters domains of the controller for the look-ahead VLFS by using their obtained stability conditions. In the next section, we will design the controller for a class of look-ahead VLFS with stochastic and impulsive disturbances. Based on the obtained stability results in this paper, we will give the domains of the control parameters chosen for the system.


## 4. Controller Design for Vehicle Following System

Consider a platoon of vehicles using a longitudinal control system for vehicle following, as shown in Figure1. The position, velocity, and acceleration of the ith following vehicle are xi, vi, and ai, respectively. The constant spacing policy in [18] is employed by all automated vehicles in the platoon. Let Li be the desired intervehicular distance of the ith following vehicle.Figure 1
Spacing errors in a platoon.


### 4.1. Dynamic Model

The rolling resistance friction is considered as the stochastic factor of the system. LetFi=F-i+F˘i, where F-i is the certain part and F˘i is the stochastic part. It is assumed that F˘i/Mi is white noise process with mean value 0 and mean square error σ2. Let d(F˘i/Mi)=g(x˙i)dξ. Therefore, the model of the longitudinal dynamics of a member vehicle with impulsive and stochastic disturbances is given by
(42)dx˙i(t)=-cix˙i2(t)+ui(t)-F-iMi+g(x˙i)dξ,t≠tkΔvi(tk)=vi(tk+)-vi(tk-)=Iik(vi(tk-)),t=tk,
where, i,k∈N, xi, x˙i, dx˙i, cix˙i2, ui, F-i, and Mi denote the position, velocity, acceleration, effective aerodynamic drag, control effort, certain rolling resistance friction, and mass of the ith following vehicle, respectively. Let vi(tk) be the velocity of the impulsive moment of the ith following vehicle. Let the initial position of the ith following vehicle be xi(0).In order to avoid the collision among vehicles in the platoon in the presence of the impulsive disturbances, it is assumed thatγ_ikvi(tk-)≤vi(tk+)≤γ-ikvi(tk-) and 0<γ_ik<1<γ-ik; here, γ_ik and γ-ik are positive constants, i,k∈N.Letεi(t) be the spacing error of the ith vehicle, which is given by
(43)εi(t)=xi(t)-xi-1(t)+Li,i∈N.Obviously, we haveε˙i(t)=x˙i(t)-x˙i-1(t) and  ε¨i(t)=x¨i(t)-x¨i-1(t). Furthermore, (42) can be rewritten as
(44)dε˙i(t)=ui(t)-ci[ε˙i(t)+x˙i-1(t)]2-F-iMidt+g(x˙i)dξ-dx˙i-1(t),Δε˙i(tk)=ε˙i(tk+)-ε˙i(tk-),k∈N,i∈N.Whent=tk, k∈N, due to γ_ikvi(tk-)≤vi(tk+)≤γ-ikvi(tk-), there exists η_k≥supi{γ_ik} and η-k≥supi{γ-ik} such that
(45)ε˙i(tk+)=vi(tk+)-vi-1(tk+)=Iik(vi(tk-))+vi(tk-)-I(i-1)k×(vi-1(tk-))-vi-1(tk-)≤γ-ik[vi(tk-)-(γ_(i-1)kγ-ik)vi-1(tk-)]≤η-k[vi(tk-)-vi-1(tk-)],ε˙i(tk+)=vi(tk+)-vi-1(tk+)=Iik(vi(tk-))+vi(tk-)-I(i-1)k(vi-1(tk-))-vi-1(tk-)≥γ_(i-1)k[(γ-ikγ_(i-1)k)vi(tk-)-vi-1(tk-)]≥η_k[vi(tk-)-vi-1(tk-)];
namely,
(46)η_kε˙i(tk-)≤ε˙i(tk+)≤η-kε˙i(tk-),i,k∈N.Assume that there exists a constantη>0 such that 2ln⁡η-k/(tk-tk-1)≤η.It is well known that real number setR is a measurable set and t∈[0,∞)⊂R+. Consider that vi(t) is continuous on interval (tk-1+,tk-) and is a simple function in the set {tk-,tk+}, and so vi(t) is measurable on interval (tk-1+,tk+]; here, k∈N. Furthermore, vi(t) is Lebesgue integral on (tk-1+,tk+], k∈N. Let Ek=(tk-1+,tk+]=Ek1⋃Ek2⋃Ek3; here, Ek1=(tk-1+,tk-), Ek2={tk-}, and Ek3={tk+}. Obviously, Ek1⋂Ek2⋂Ek3=Φ. Let (L)∫Ekvi(t)dt denote the integral of vi(t) in the set Ek, and let Mes(·) denote the measure; we have
(47)xi(tk+)=xi(tk-1-)+(L)∫Ekvi(t)dt=xi(tk-1-)+(L)∫Ek1vi(t)dt+(L)∫Ek2vi(t)dt+(L)∫Ek3vi(t)dt,xi(tk-)=xi(tk-1-)+(L)∫Ek1vi(t)dt+(L)∫Ek2vi(t)dt.Due to the fact thatMes(Ek2)=Mes(Ek3)=0, xi(tk+)=xi(tk-). Furthermore,
(48)εi(tk+)=xi(tk+)-xi-1(tk+)+Li=xi(tk-)-xi-1(tk-)+Li<η-kεi(tk-)εi(tk+)=xi(tk+)-xi-1(tk+)+Li=xi(tk-)-xi-1(tk-)+Li>η_kεi(tk-);
that is,
(49)η_kεi(tk-)<εi(tk+)<η-kεi(tk-),i,k∈N.


### 4.2. Controller Design

Define an auxiliary error given by the following equation:(50)Si(t)=q1[x˙i(t)-x˙i-1(t)]+q2[xi(t)-xi-1(t)+Li]+q3[x˙i(t)-v0(t)]+q4[xi(t)-x0(t)+∑j=1iLj]+σi(t),
where σi(t) satisfies
(51)dσi(t)=q1dx˙i-1+q3dv0-(q1+q3)g(x˙i)dξ
and S˙i is independent on ξ; that is to say, Si is a finite variable [26]. It is assumed that σi(tk+)=σi(tk-),  i,k∈N.The expression of control lawui(t) is chosen as follows:
(52)ui(t)=uiequ(t)+uiN(t),i∈N,
where
(53)uiequ(t)=cix˙i2(t)+F-i-Mi(q1+q3)×[q2(x˙i(t)-x˙i-1(t))+q4(x˙i(t)-v0(t))],uiN(t)=-Miq1+q3λSi(t).
Here, λ>0 with λ≥η>0 is the control parameter and will be chosen later.


### 4.3. Stability Analysis

#### 4.3.1. Reachability of Slide Mode

In this section, we will analyze the fact that the slide mode is asymptotically reachable.Choose the control vector Lyapunov functionVi(t)=0.5Si2(t),    t>0,   i∈N.(i)
Whent≠tk, k∈N, calculating the right upper derivative of Vi(t) along (44), we get
(54)D+Vi(t)=SiSi˙=Si[(q1+q3)-cix˙i2+ui-F-iMi+(q2+q4)x˙i-q2x˙i-1-q4v0[(q1+q3)-cix˙i2+ui-F-iMi],00i∈N.Substituting (52) into the above equation, we obtain D+Vi(t)=-λSi2(t), i∈N. It is obvious that when Si(t)≠0, D+Vi(t)<0. Therefore, Vi(t) is strictly monotone decreasing, t>0,    t≠tk, i,k∈N.(ii)
Whent=tk, k∈N, according to (50) and σi(tk+)=σi(tk-), we obtain
(55)Si(tk+)<q1ηk[vi(tk-)-vi-1(tk-)]+q2[xi(tk-)-xi-1(tk-)+Li]+q3ηk[vi(tk-)-v0(tk-)]+q4×[xi(tk-)-x0(tk-)+∑j=1iLj]+ηkσi(tk+)<ηkSi(tk-),i,k∈N.Due to the fact thatS˙i(t)+λSi(t)=0, Si(tk+1+)≤η-k+1Si(tk+)exp⁡[-λ(tk+1+-tk+)]. From the condition 2ln⁡⁡η-k/(tk-tk-1)≤η, λ≥η>0, we get ηk+1exp⁡[-λ(tk+1+-tk+)]≤exp⁡[-(λ-0.5η)(tk+1+-tk+)]<1, and so
(56)Si(tk+1+)<Si(tk+),i,k∈N.From (56), the impulsive sequence {Si(tk+)} is strictly monotone decreasing, i,k∈N. On the other hand, we know from the preceding analysis that when t≠tk, Si(t) is strictly monotone decreasing. To sum up, it can be concluded that Si(t)→0 as t→∞. Namely, the slide mode Si(t) is asymptotically reachable, i∈N.


#### 4.3.2. Stability of Slide Mode Motion

In this section, the stability domain of the control parameters will be proposed based on the new stability conditions established in Section3.Substituting the expression of control law (52) into the system (44), we can obtain the following slide mode motion equation given by
(57)dx˙i(t)=(q1+q3)-1×[-q4(x˙i(t)-v0(t))-λSi(t)]-q4(x˙i(t)-v0(t))-λSi(t)]q1dx˙i-1(t)+q3dv0(t)-q2(x˙i(t)-x˙i-1(t))-q4(x˙i(t)-v0(t))-λSi(t)]dt+g(x˙i(t))dξ,t≠tk,Δvi(tk)=vi(tk+)-vi(tk-),t=tk,i,k∈N.By the same way, we get(58)dx˙i-1(t)=(q1+q3)-1×[-q4(x˙i-1(t)-v0(t))-λSi-1(t)]q1dx˙i-2(t)+q3dv0(t)-q2(x˙i-1(t)-x˙i-2(t))-q4(x˙i-1(t)-v0(t))-λSi-1(t)]dt+g(x˙i-1(t))dξ,t≠tk,Δvi-1(tk)=vi-1(tk+)-vi-1(tk-),t=tk,i,k∈N.In the surface of the slide mode, due to the fact thatSi(t)=Si-1(t)=0, d(Si(t)-Si-1(t))=0,  i∈N. Furthermore, according to (44) and the assumption conditions, we have
(59)dε˙i(t)=[-q2+q4q1+q3ε˙i(t)+q2q1+q3ε˙i-1(t)]dt+[g(x˙i(t))-g(x˙i-1(t))]dξ,t≠tk,Δε˙i(tk)=ε˙i(tk+)-ε˙i(tk-),t=tk,i,k∈N.Letyi(t)=ε˙i(t)=x˙i(t)-x˙i-1(t),  ε0(t)=0,  ε˙0(t)=0, and ε¨0(t)=0. Set l1=(q2+q4)/(q1+q3) and l2=q2/(q1+q3). From (59), we have
(60)dy1(t)=-l1y1(t)dt+[g(x˙1(t))-g(x˙0(t))]dξ,t≠tk,dyi(t)=[-l1yi(t)+l2yi-1(t)]dt+[g(x˙i(t))-g(x˙i-1(t))]dξ,t≠tk,Δyi(tk)=yi(tk+)-yi(tk-),t=tk,i,k∈N.
Consider that yi(t)=ε˙i(t)=x˙i(t)-x˙i-1(t); (60) can be rewritten as
(61)dy1(t)=-l1y1(t)dt+g(y1(t))dξ,t≠tkdyi(t)=[-l1yi(t)+l2yi-1(t)]dt+[g(yi(t)+yi-1(t)+⋯+y1(t))-g(yi-1(t)+yi-2(t)+⋯+y1(t))]dξ,t≠tk,Δyi(tk)=yi(tk+)-yi(tk-),t=tk,i,k∈N,
where η_k|yi(tk-)|≤|yi(tk+)|≤η-k|yi(tk-)|.The isolate subsystems of (61) are of the following forms:
(62)y˙i(t)=-l1yi(t)+g(yi(t))dξ,t≠tk,Δyi(tk)=yi(tk+)-yi(tk-),t=tk,i,k∈N.Obviously, (61) with isolate subsystems (62) is a special case of (2) with isolate subsystems (3).Taking the vector function LyapunovVi(t)=yi2(t),   i∈N. It is easy to get α1=1,  α2=1,  and  α4=α5=2. From Theorem 5, for judging the stability for (2), some sufficient conditions for assuring the string exponential stability with mode constraint in the mean square sense for (61) can be established as follows:(i)
it is assumed thatg satisfies the global Lipschitz condition; that is, there exists constant θ>0 such that |g(y)-g(z)|≤θ|y-z|;(ii)
α3≥2l1-θ2;(iii)
there exists constantλ>0 such that 2ln⁡η-k/(tk-tk-1)≤η<λ, k∈N;(iv)
ηk_(λ-2l1+θ2)+2η-kl2<0,    k∈N.By the same analysis with the proof of Theorem5, it is easy to obtain that the zero solution yi(t)=0 of (61) is string exponentially stable with mode constraint in the mean square. From the relations among yi(t), εi(t), and Si(t), it is easy to conclude that the vehicle following errors ε˙i(t) and εi(t) converge to zero, i∈N.Remark 10.
The stability with mode constraint for the vehicle following error system (59) could guarantee that the error of every following vehicle is not only uniformly bounded, but also less than the error of the leading vehicle.


## 4.1. Dynamic Model

The rolling resistance friction is considered as the stochastic factor of the system. LetFi=F-i+F˘i, where F-i is the certain part and F˘i is the stochastic part. It is assumed that F˘i/Mi is white noise process with mean value 0 and mean square error σ2. Let d(F˘i/Mi)=g(x˙i)dξ. Therefore, the model of the longitudinal dynamics of a member vehicle with impulsive and stochastic disturbances is given by
(42)dx˙i(t)=-cix˙i2(t)+ui(t)-F-iMi+g(x˙i)dξ,t≠tkΔvi(tk)=vi(tk+)-vi(tk-)=Iik(vi(tk-)),t=tk,
where, i,k∈N, xi, x˙i, dx˙i, cix˙i2, ui, F-i, and Mi denote the position, velocity, acceleration, effective aerodynamic drag, control effort, certain rolling resistance friction, and mass of the ith following vehicle, respectively. Let vi(tk) be the velocity of the impulsive moment of the ith following vehicle. Let the initial position of the ith following vehicle be xi(0).In order to avoid the collision among vehicles in the platoon in the presence of the impulsive disturbances, it is assumed thatγ_ikvi(tk-)≤vi(tk+)≤γ-ikvi(tk-) and 0<γ_ik<1<γ-ik; here, γ_ik and γ-ik are positive constants, i,k∈N.Letεi(t) be the spacing error of the ith vehicle, which is given by
(43)εi(t)=xi(t)-xi-1(t)+Li,i∈N.Obviously, we haveε˙i(t)=x˙i(t)-x˙i-1(t) and  ε¨i(t)=x¨i(t)-x¨i-1(t). Furthermore, (42) can be rewritten as
(44)dε˙i(t)=ui(t)-ci[ε˙i(t)+x˙i-1(t)]2-F-iMidt+g(x˙i)dξ-dx˙i-1(t),Δε˙i(tk)=ε˙i(tk+)-ε˙i(tk-),k∈N,i∈N.Whent=tk, k∈N, due to γ_ikvi(tk-)≤vi(tk+)≤γ-ikvi(tk-), there exists η_k≥supi{γ_ik} and η-k≥supi{γ-ik} such that
(45)ε˙i(tk+)=vi(tk+)-vi-1(tk+)=Iik(vi(tk-))+vi(tk-)-I(i-1)k×(vi-1(tk-))-vi-1(tk-)≤γ-ik[vi(tk-)-(γ_(i-1)kγ-ik)vi-1(tk-)]≤η-k[vi(tk-)-vi-1(tk-)],ε˙i(tk+)=vi(tk+)-vi-1(tk+)=Iik(vi(tk-))+vi(tk-)-I(i-1)k(vi-1(tk-))-vi-1(tk-)≥γ_(i-1)k[(γ-ikγ_(i-1)k)vi(tk-)-vi-1(tk-)]≥η_k[vi(tk-)-vi-1(tk-)];
namely,
(46)η_kε˙i(tk-)≤ε˙i(tk+)≤η-kε˙i(tk-),i,k∈N.Assume that there exists a constantη>0 such that 2ln⁡η-k/(tk-tk-1)≤η.It is well known that real number setR is a measurable set and t∈[0,∞)⊂R+. Consider that vi(t) is continuous on interval (tk-1+,tk-) and is a simple function in the set {tk-,tk+}, and so vi(t) is measurable on interval (tk-1+,tk+]; here, k∈N. Furthermore, vi(t) is Lebesgue integral on (tk-1+,tk+], k∈N. Let Ek=(tk-1+,tk+]=Ek1⋃Ek2⋃Ek3; here, Ek1=(tk-1+,tk-), Ek2={tk-}, and Ek3={tk+}. Obviously, Ek1⋂Ek2⋂Ek3=Φ. Let (L)∫Ekvi(t)dt denote the integral of vi(t) in the set Ek, and let Mes(·) denote the measure; we have
(47)xi(tk+)=xi(tk-1-)+(L)∫Ekvi(t)dt=xi(tk-1-)+(L)∫Ek1vi(t)dt+(L)∫Ek2vi(t)dt+(L)∫Ek3vi(t)dt,xi(tk-)=xi(tk-1-)+(L)∫Ek1vi(t)dt+(L)∫Ek2vi(t)dt.Due to the fact thatMes(Ek2)=Mes(Ek3)=0, xi(tk+)=xi(tk-). Furthermore,
(48)εi(tk+)=xi(tk+)-xi-1(tk+)+Li=xi(tk-)-xi-1(tk-)+Li<η-kεi(tk-)εi(tk+)=xi(tk+)-xi-1(tk+)+Li=xi(tk-)-xi-1(tk-)+Li>η_kεi(tk-);
that is,
(49)η_kεi(tk-)<εi(tk+)<η-kεi(tk-),i,k∈N.


## 4.2. Controller Design

Define an auxiliary error given by the following equation:(50)Si(t)=q1[x˙i(t)-x˙i-1(t)]+q2[xi(t)-xi-1(t)+Li]+q3[x˙i(t)-v0(t)]+q4[xi(t)-x0(t)+∑j=1iLj]+σi(t),
where σi(t) satisfies
(51)dσi(t)=q1dx˙i-1+q3dv0-(q1+q3)g(x˙i)dξ
and S˙i is independent on ξ; that is to say, Si is a finite variable [26]. It is assumed that σi(tk+)=σi(tk-),  i,k∈N.The expression of control lawui(t) is chosen as follows:
(52)ui(t)=uiequ(t)+uiN(t),i∈N,
where
(53)uiequ(t)=cix˙i2(t)+F-i-Mi(q1+q3)×[q2(x˙i(t)-x˙i-1(t))+q4(x˙i(t)-v0(t))],uiN(t)=-Miq1+q3λSi(t).
Here, λ>0 with λ≥η>0 is the control parameter and will be chosen later.


## 4.3. Stability Analysis

### 4.3.1. Reachability of Slide Mode

In this section, we will analyze the fact that the slide mode is asymptotically reachable.Choose the control vector Lyapunov functionVi(t)=0.5Si2(t),    t>0,   i∈N.(i)
Whent≠tk, k∈N, calculating the right upper derivative of Vi(t) along (44), we get
(54)D+Vi(t)=SiSi˙=Si[(q1+q3)-cix˙i2+ui-F-iMi+(q2+q4)x˙i-q2x˙i-1-q4v0[(q1+q3)-cix˙i2+ui-F-iMi],00i∈N.Substituting (52) into the above equation, we obtain D+Vi(t)=-λSi2(t), i∈N. It is obvious that when Si(t)≠0, D+Vi(t)<0. Therefore, Vi(t) is strictly monotone decreasing, t>0,    t≠tk, i,k∈N.(ii)
Whent=tk, k∈N, according to (50) and σi(tk+)=σi(tk-), we obtain
(55)Si(tk+)<q1ηk[vi(tk-)-vi-1(tk-)]+q2[xi(tk-)-xi-1(tk-)+Li]+q3ηk[vi(tk-)-v0(tk-)]+q4×[xi(tk-)-x0(tk-)+∑j=1iLj]+ηkσi(tk+)<ηkSi(tk-),i,k∈N.Due to the fact thatS˙i(t)+λSi(t)=0, Si(tk+1+)≤η-k+1Si(tk+)exp⁡[-λ(tk+1+-tk+)]. From the condition 2ln⁡⁡η-k/(tk-tk-1)≤η, λ≥η>0, we get ηk+1exp⁡[-λ(tk+1+-tk+)]≤exp⁡[-(λ-0.5η)(tk+1+-tk+)]<1, and so
(56)Si(tk+1+)<Si(tk+),i,k∈N.From (56), the impulsive sequence {Si(tk+)} is strictly monotone decreasing, i,k∈N. On the other hand, we know from the preceding analysis that when t≠tk, Si(t) is strictly monotone decreasing. To sum up, it can be concluded that Si(t)→0 as t→∞. Namely, the slide mode Si(t) is asymptotically reachable, i∈N.


### 4.3.2. Stability of Slide Mode Motion

In this section, the stability domain of the control parameters will be proposed based on the new stability conditions established in Section3.Substituting the expression of control law (52) into the system (44), we can obtain the following slide mode motion equation given by
(57)dx˙i(t)=(q1+q3)-1×[-q4(x˙i(t)-v0(t))-λSi(t)]-q4(x˙i(t)-v0(t))-λSi(t)]q1dx˙i-1(t)+q3dv0(t)-q2(x˙i(t)-x˙i-1(t))-q4(x˙i(t)-v0(t))-λSi(t)]dt+g(x˙i(t))dξ,t≠tk,Δvi(tk)=vi(tk+)-vi(tk-),t=tk,i,k∈N.By the same way, we get(58)dx˙i-1(t)=(q1+q3)-1×[-q4(x˙i-1(t)-v0(t))-λSi-1(t)]q1dx˙i-2(t)+q3dv0(t)-q2(x˙i-1(t)-x˙i-2(t))-q4(x˙i-1(t)-v0(t))-λSi-1(t)]dt+g(x˙i-1(t))dξ,t≠tk,Δvi-1(tk)=vi-1(tk+)-vi-1(tk-),t=tk,i,k∈N.In the surface of the slide mode, due to the fact thatSi(t)=Si-1(t)=0, d(Si(t)-Si-1(t))=0,  i∈N. Furthermore, according to (44) and the assumption conditions, we have
(59)dε˙i(t)=[-q2+q4q1+q3ε˙i(t)+q2q1+q3ε˙i-1(t)]dt+[g(x˙i(t))-g(x˙i-1(t))]dξ,t≠tk,Δε˙i(tk)=ε˙i(tk+)-ε˙i(tk-),t=tk,i,k∈N.Letyi(t)=ε˙i(t)=x˙i(t)-x˙i-1(t),  ε0(t)=0,  ε˙0(t)=0, and ε¨0(t)=0. Set l1=(q2+q4)/(q1+q3) and l2=q2/(q1+q3). From (59), we have
(60)dy1(t)=-l1y1(t)dt+[g(x˙1(t))-g(x˙0(t))]dξ,t≠tk,dyi(t)=[-l1yi(t)+l2yi-1(t)]dt+[g(x˙i(t))-g(x˙i-1(t))]dξ,t≠tk,Δyi(tk)=yi(tk+)-yi(tk-),t=tk,i,k∈N.
Consider that yi(t)=ε˙i(t)=x˙i(t)-x˙i-1(t); (60) can be rewritten as
(61)dy1(t)=-l1y1(t)dt+g(y1(t))dξ,t≠tkdyi(t)=[-l1yi(t)+l2yi-1(t)]dt+[g(yi(t)+yi-1(t)+⋯+y1(t))-g(yi-1(t)+yi-2(t)+⋯+y1(t))]dξ,t≠tk,Δyi(tk)=yi(tk+)-yi(tk-),t=tk,i,k∈N,
where η_k|yi(tk-)|≤|yi(tk+)|≤η-k|yi(tk-)|.The isolate subsystems of (61) are of the following forms:
(62)y˙i(t)=-l1yi(t)+g(yi(t))dξ,t≠tk,Δyi(tk)=yi(tk+)-yi(tk-),t=tk,i,k∈N.Obviously, (61) with isolate subsystems (62) is a special case of (2) with isolate subsystems (3).Taking the vector function LyapunovVi(t)=yi2(t),   i∈N. It is easy to get α1=1,  α2=1,  and  α4=α5=2. From Theorem 5, for judging the stability for (2), some sufficient conditions for assuring the string exponential stability with mode constraint in the mean square sense for (61) can be established as follows:(i)
it is assumed thatg satisfies the global Lipschitz condition; that is, there exists constant θ>0 such that |g(y)-g(z)|≤θ|y-z|;(ii)
α3≥2l1-θ2;(iii)
there exists constantλ>0 such that 2ln⁡η-k/(tk-tk-1)≤η<λ, k∈N;(iv)
ηk_(λ-2l1+θ2)+2η-kl2<0,    k∈N.By the same analysis with the proof of Theorem5, it is easy to obtain that the zero solution yi(t)=0 of (61) is string exponentially stable with mode constraint in the mean square. From the relations among yi(t), εi(t), and Si(t), it is easy to conclude that the vehicle following errors ε˙i(t) and εi(t) converge to zero, i∈N.Remark 10.
The stability with mode constraint for the vehicle following error system (59) could guarantee that the error of every following vehicle is not only uniformly bounded, but also less than the error of the leading vehicle.


## 4.3.1. Reachability of Slide Mode

In this section, we will analyze the fact that the slide mode is asymptotically reachable.Choose the control vector Lyapunov functionVi(t)=0.5Si2(t),    t>0,   i∈N.(i)
Whent≠tk, k∈N, calculating the right upper derivative of Vi(t) along (44), we get
(54)D+Vi(t)=SiSi˙=Si[(q1+q3)-cix˙i2+ui-F-iMi+(q2+q4)x˙i-q2x˙i-1-q4v0[(q1+q3)-cix˙i2+ui-F-iMi],00i∈N.Substituting (52) into the above equation, we obtain D+Vi(t)=-λSi2(t), i∈N. It is obvious that when Si(t)≠0, D+Vi(t)<0. Therefore, Vi(t) is strictly monotone decreasing, t>0,    t≠tk, i,k∈N.(ii)
Whent=tk, k∈N, according to (50) and σi(tk+)=σi(tk-), we obtain
(55)Si(tk+)<q1ηk[vi(tk-)-vi-1(tk-)]+q2[xi(tk-)-xi-1(tk-)+Li]+q3ηk[vi(tk-)-v0(tk-)]+q4×[xi(tk-)-x0(tk-)+∑j=1iLj]+ηkσi(tk+)<ηkSi(tk-),i,k∈N.Due to the fact thatS˙i(t)+λSi(t)=0, Si(tk+1+)≤η-k+1Si(tk+)exp⁡[-λ(tk+1+-tk+)]. From the condition 2ln⁡⁡η-k/(tk-tk-1)≤η, λ≥η>0, we get ηk+1exp⁡[-λ(tk+1+-tk+)]≤exp⁡[-(λ-0.5η)(tk+1+-tk+)]<1, and so
(56)Si(tk+1+)<Si(tk+),i,k∈N.From (56), the impulsive sequence {Si(tk+)} is strictly monotone decreasing, i,k∈N. On the other hand, we know from the preceding analysis that when t≠tk, Si(t) is strictly monotone decreasing. To sum up, it can be concluded that Si(t)→0 as t→∞. Namely, the slide mode Si(t) is asymptotically reachable, i∈N.


## 4.3.2. Stability of Slide Mode Motion

In this section, the stability domain of the control parameters will be proposed based on the new stability conditions established in Section3.Substituting the expression of control law (52) into the system (44), we can obtain the following slide mode motion equation given by
(57)dx˙i(t)=(q1+q3)-1×[-q4(x˙i(t)-v0(t))-λSi(t)]-q4(x˙i(t)-v0(t))-λSi(t)]q1dx˙i-1(t)+q3dv0(t)-q2(x˙i(t)-x˙i-1(t))-q4(x˙i(t)-v0(t))-λSi(t)]dt+g(x˙i(t))dξ,t≠tk,Δvi(tk)=vi(tk+)-vi(tk-),t=tk,i,k∈N.By the same way, we get(58)dx˙i-1(t)=(q1+q3)-1×[-q4(x˙i-1(t)-v0(t))-λSi-1(t)]q1dx˙i-2(t)+q3dv0(t)-q2(x˙i-1(t)-x˙i-2(t))-q4(x˙i-1(t)-v0(t))-λSi-1(t)]dt+g(x˙i-1(t))dξ,t≠tk,Δvi-1(tk)=vi-1(tk+)-vi-1(tk-),t=tk,i,k∈N.In the surface of the slide mode, due to the fact thatSi(t)=Si-1(t)=0, d(Si(t)-Si-1(t))=0,  i∈N. Furthermore, according to (44) and the assumption conditions, we have
(59)dε˙i(t)=[-q2+q4q1+q3ε˙i(t)+q2q1+q3ε˙i-1(t)]dt+[g(x˙i(t))-g(x˙i-1(t))]dξ,t≠tk,Δε˙i(tk)=ε˙i(tk+)-ε˙i(tk-),t=tk,i,k∈N.Letyi(t)=ε˙i(t)=x˙i(t)-x˙i-1(t),  ε0(t)=0,  ε˙0(t)=0, and ε¨0(t)=0. Set l1=(q2+q4)/(q1+q3) and l2=q2/(q1+q3). From (59), we have
(60)dy1(t)=-l1y1(t)dt+[g(x˙1(t))-g(x˙0(t))]dξ,t≠tk,dyi(t)=[-l1yi(t)+l2yi-1(t)]dt+[g(x˙i(t))-g(x˙i-1(t))]dξ,t≠tk,Δyi(tk)=yi(tk+)-yi(tk-),t=tk,i,k∈N.
Consider that yi(t)=ε˙i(t)=x˙i(t)-x˙i-1(t); (60) can be rewritten as
(61)dy1(t)=-l1y1(t)dt+g(y1(t))dξ,t≠tkdyi(t)=[-l1yi(t)+l2yi-1(t)]dt+[g(yi(t)+yi-1(t)+⋯+y1(t))-g(yi-1(t)+yi-2(t)+⋯+y1(t))]dξ,t≠tk,Δyi(tk)=yi(tk+)-yi(tk-),t=tk,i,k∈N,
where η_k|yi(tk-)|≤|yi(tk+)|≤η-k|yi(tk-)|.The isolate subsystems of (61) are of the following forms:
(62)y˙i(t)=-l1yi(t)+g(yi(t))dξ,t≠tk,Δyi(tk)=yi(tk+)-yi(tk-),t=tk,i,k∈N.Obviously, (61) with isolate subsystems (62) is a special case of (2) with isolate subsystems (3).Taking the vector function LyapunovVi(t)=yi2(t),   i∈N. It is easy to get α1=1,  α2=1,  and  α4=α5=2. From Theorem 5, for judging the stability for (2), some sufficient conditions for assuring the string exponential stability with mode constraint in the mean square sense for (61) can be established as follows:(i)
it is assumed thatg satisfies the global Lipschitz condition; that is, there exists constant θ>0 such that |g(y)-g(z)|≤θ|y-z|;(ii)
α3≥2l1-θ2;(iii)
there exists constantλ>0 such that 2ln⁡η-k/(tk-tk-1)≤η<λ, k∈N;(iv)
ηk_(λ-2l1+θ2)+2η-kl2<0,    k∈N.By the same analysis with the proof of Theorem5, it is easy to obtain that the zero solution yi(t)=0 of (61) is string exponentially stable with mode constraint in the mean square. From the relations among yi(t), εi(t), and Si(t), it is easy to conclude that the vehicle following errors ε˙i(t) and εi(t) converge to zero, i∈N.Remark 10.
The stability with mode constraint for the vehicle following error system (59) could guarantee that the error of every following vehicle is not only uniformly bounded, but also less than the error of the leading vehicle.


## 5. Numerical Simulations

In this section, some numerical simulations are performed for a four-vehicle platoon with one leading vehicle and three following vehicles. The reference model of VLFS is referred to in Section4 in this paper (Figure 1).Letg(x˙i)=x˙i,  i=1,2,3. The control laws for every following vehicle are of the forms in (56). According to the conditions in Section 4.3.2, the control parameters are designed as q1=1, q2=2, q3=3, q4=4, and λ=1.4. The leading vehicle in the platoon makes the following acceleration maneuver [18]: when 0s≤t<4s, a0=0m/s2; when 4s≤t<7s, a0=-0.25(t-2)m/s2; when 7s≤t<10s, a0=-0.75m/s2; when 10s≤t<16s, a0=0.25(t-10)-0.75m/s2; when 16s≤t<19s, a0=0.75m/s2; when 19s≤t<22s, a0=0.25(19-t)+0.75m/s2; when 22s≤t≤30s, a0=0m/s2.It is assumed that the desired intervehicular distance of the three following vehicles isLi=8m, i=1,2,3. Table 1 shows the initial states of vehicles in the platoon. From (46), it can be obtained that e1(0)=0.5m, e2(0)=0.4m, and e3(0)=0.3m. Table 2 shows the parameters of the following vehicles in the platoon. It is assumed that when tk=8s, v1(tk+)=1.05v1(tk-); when tk=20s, v0(tk+)=0.97v0(tk-). The approach in [27] is used to generate Brown motion trajectory.Table 1
Initial states of vehicles.
Vehicle numberi
a
i
(
0
) (m/s2)
v
i
(
0
) (m/s)
x
i
(
0
) (m)
0
0
25.0
50.0
1
0
25.5
42.5
2
0
26.0
34.9
3
0
26.5
27.2Table 2
Parameters of vehicles.
Vehicle numberi
M
i (kg)
c
i (Ns2/m2)
F
i (N)
1
1000
0.50
200
2
900
0.60
180
3
1100
0.55
220The simulation results are shown in Figures2, 3, 4, and 5. From the simulation results, it can be seen that the errors of following vehicles not only converge to zero but also satisfy the mode constraint condition. That is to say, the stability results obtained in this paper are correct and practical. Since the domains of control parameters are large, the controllers are easy to be designed in practice.Figure 2
Velocity curves of vehicles.Figure 3
Spacing error curves of vehicles.Figure 4
Slide mode curves of vehicles.Figure 5
Control input curves of vehicles.


## 6. Conclusions

In this paper, the problem of string exponential stability with mode constraint of infinite dimensional nonlinear interconnected systems with stochastic and impulsive disturbances has been studied by using the vector Lyapunov function method. Sufficient conditions of string exponential stability with mode constraint have been derived for a class of general infinite dimensional look-ahead interconnected systems with impulsive and stochastic disturbances. Moreover, the controller for a class of look-ahead vehicle longitudinal following systems with the above uncertainties has been proposed by the sliding mode control method. Based on the obtained new stability conditions, the domain of the control parameters of the systems has been obtained, and the domain of the control parameters of the systems is enlarged. A numerical example with simulations has been given to show the effectiveness and correctness of the obtained results.



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