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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 471481, 14 pages

http://dx.doi.org/10.1155/2014/471481

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

^{1}School of Transportation and Automotive Engineering, Xihua University, Chengdu 610039, China^{2}State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu 610031, China

Received 29 March 2014; Revised 8 May 2014; Accepted 8 May 2014; Published 15 June 2014

Academic Editor: Hongli Dong

Copyright © 2014 Xiaohui Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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.

#### 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 (here ), the states are needed to be satisfied that or . 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: where is the Euclidean norm and , , denotes the expectation of stochastic process, and 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 here, and denotes the state of the subsystem, . is the impulsive strength at discrete moment , and the discrete set is assumed to be satisfied that , and as , . It is assumed that is right, continuous with , and . Consider , : ; let . Assume that is the one-dimensional independent standard Wiener processes defined on space ; here denotes sample space, denotes algebra of subset of the sample space, and denotes probability measures.

The system (2) can be treated as an interconnection of isolated subsystems given by

##### 2.2. Definitions and Assumptions

Let () be the unique zero solution of system (2).

*Definition 1 (see [19]). *If, for such that , then () 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) () is string exponentially stable in the mean square sense if is string stable, and there exist constants and such that holds.

*Definition 3 (see [13]). *The zero solution of system (2) () is string exponentially stable with mode constraint in the mean square sense if is string exponentially stable in the mean square sense, and the inequality holds, , .

Next, some assumptions will be given for the system (2) and the system (3).

*Assumption A1*. There exist positive constants , () and , such that and are globally Lipschitz in their arguments; that is,

*Assumption A2.* For every isolated subsystem (3), there exists positive definite function , , which is continuously twice differentiable with respect to , and there exist positive constants such that(i);(ii);(iii), ,

where is an operator associated with (3) defined by where .

Let denote if there is no confusion, .

*Assumption A3*. Let . The function with denotes the strength of the intensity of impulsive disturbances, which is assumed to be continuous with . Suppose that there exist positive constants and such that holds; here, , .

Next a lemma established by us in [24] is given which will be used in the proof of the following theorem.

Lemma 4. *Suppose that , and . Consider the following inequalities:
**
for ; it is assumed that . Let , and , . is an operator associated with (2) given by
**
where .*

If there exists such that then, for , such that .

#### 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 constants and such that and if the following inequality holds,
**
where , , , 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 function is the vector Lyapunov function of the 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:

When , , calculating the operator along the zero solution of system (2) and applying Assumptions A1-A2, we get
It follows from Assumption A2 and that

From the properties of the operator [25], we can take the expectation of inequality (12) and rewrite it as

Taking , , . Substituting them into (13), we get

Since inequality (12) implies that the following inequality holds,
it can be concluded that , . Therefore, it follows from Lemma 4 that, for , such that . Let , and satisfy . Take . When , we get . Furthermore, we have ; namely,

Next, we will use the mathematical induction method to prove that
hold, where ; namely,

When , it can be seen from (16) that (18) holds. Suppose that the following inequalities hold:

From Assumption A3 and (19), we have

Due to , we have

We claim that (21) implies that

If inequality (22) does not hold, there exist some and such that
, and

Substituting (23) and inequality (24) into inequality (11), together with the properties of the operator , we get
Since condition (12) implies that
. This is a contradiction with . By the mathematical induction method, it can be concluded that

From condition , we get that . Furthermore,
That is, , .

According to Definition 2, the zero solution of system (2) is the string exponentially stable in the mean square sense with convergence rate .

Next, we proceed to prove that the zero solution of system (2) is stable with the mode constraint. From the previous analysis, when , , it is easy to obtain

Let , and
where

Define sets in state space: and . It is obvious that if , then . Therefore, for all and , we have

This implies that, for , we have , ; that is,

Note that condition (9) implies that the following inequality holds:
namely,

Let

Furthermore, inequality (33) can be rewritten as ; that is,

When , , it follows from Assumption A3 that

Note that
Substituting (39) into (38), we have , . According to condition (9), we have , so
namely, , . 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.

When and in Assumption A3 satisfy , , 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 and such that and if the following inequality holds,
**
where , , , , and , then the zero solution of system (2) is string exponentially stable with mode constraint in the mean square sense.*

The proof of Corollary 6 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 Figure 1. The position, velocity, and acceleration of the following vehicle are , , and , respectively. The constant spacing policy in [18] is employed by all automated vehicles in the platoon. Let be the desired intervehicular distance of the following vehicle.

##### 4.1. Dynamic Model

The rolling resistance friction is considered as the stochastic factor of the system. Let , where is the certain part and is the stochastic part. It is assumed that is white noise process with mean value 0 and mean square error . Let . Therefore, the model of the longitudinal dynamics of a member vehicle with impulsive and stochastic disturbances is given by where, , , , , , , , and denote the position, velocity, acceleration, effective aerodynamic drag, control effort, certain rolling resistance friction, and mass of the following vehicle, respectively. Let be the velocity of the impulsive moment of the following vehicle. Let the initial position of the following vehicle be .

In order to avoid the collision among vehicles in the platoon in the presence of the impulsive disturbances, it is assumed that and ; here, and are positive constants, .

Let be the spacing error of the vehicle, which is given by

Obviously, we have and . Furthermore, (42) can be rewritten as

When , , due to , there exists and such that namely,

Assume that there exists a constant such that .

It is well known that real number set is a measurable set and . Consider that is continuous on interval and is a simple function in the set , and so is measurable on interval ; here, . Furthermore, is Lebesgue integral on , . Let ; here, , , and . Obviously, . Let denote the integral of in the set , and let denote the measure; we have

Due to the fact that , . Furthermore, that is,

##### 4.2. Controller Design

Define an auxiliary error given by the following equation: where satisfies and is independent on ; that is to say, is a finite variable [26]. It is assumed that , .

The expression of control law is chosen as follows: where Here, with 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 function , , .(i)When , , calculating the right upper derivative of along (44), we get

Substituting (52) into the above equation, we obtain , . It is obvious that when , . Therefore, is strictly monotone decreasing, , , .(ii)When , , according to (50) and , we obtain

Due to the fact that , . From the condition , , we get , and so

From (56), the impulsive sequence is strictly monotone decreasing, . On the other hand, we know from the preceding analysis that when , is strictly monotone decreasing. To sum up, it can be concluded that