Stability of Infinite Dimensional Interconnected Systems with Impulsive and Stochastic Disturbances

and Applied Analysis 3 Definition 2 (see [19]). The zero solution of system (2) y i = 0 (i ∈ N) is string exponentially stable in the mean square sense if y i = 0 is string stable, and there exist constantsM > 0 and λ > 0 such that ‖y i (t)‖ 2 ∞ < M‖y i (0)‖ 2 e −λt holds. Definition 3 (see [13]). The zero solution of system (2) y i = 0 (i ∈ N) is string exponentially stable with mode constraint in the mean square sense if y i = 0 is string exponentially stable in themean square sense, and the inequality ‖y i (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 kf l , kq l (l = 1, 2) and df 1 , dq 1 such that f and q are globally Lipschitz in their arguments; that is, 󵄨 󵄨 󵄨 󵄨 f (y 1 , y 2 , y 3 ) − f (z 1 , z 2 , z 3 ) 󵄨 󵄨 󵄨 󵄨 ≤ k f 1 󵄨 󵄨 󵄨 󵄨 y 1 − z 1 󵄨 󵄨 󵄨 󵄨 + k f 2 󵄨 󵄨 󵄨 󵄨 y 2 − z 2 󵄨 󵄨 󵄨 󵄨 + d f 1 󵄨 󵄨 󵄨 󵄨 y 3 − z 3 󵄨 󵄨 󵄨 󵄨 , 󵄨 󵄨 󵄨 󵄨 q (y 1 , y 2 , y 3 ) − q (z 1 , z 2 , z 3 ) 󵄨 󵄨 󵄨 󵄨 ≤ k q 1 󵄨 󵄨 󵄨 󵄨 y 1 − z 1 󵄨 󵄨 󵄨 󵄨 + k q 2 󵄨 󵄨 󵄨 󵄨 y 2 − z 2 󵄨 󵄨 󵄨 󵄨 + d q 1 󵄨 󵄨 󵄨 󵄨 y 3 − z 3 󵄨 󵄨 󵄨 󵄨 . (4) Assumption A2. For every isolated subsystem (3), there exists positive definite function V i (t, y i (t)), y i ∈ R , which is continuously twice differentiable with respect to y i , and there exist positive constants α m (m = 1, 2, 3, 4, 5) such that (i) α 1 |y i | 2 ≤ V i (t, y i (t)) ≤ α 2 |y i | ; (ii) χ (2) V i (t, y i (t)) ≤ −α 3 |y i | ; (iii) |∂V i (t, y i (t))/∂y i,j | ≤ α 4 |x i |, |∂V i (t, y i (t))/∂y i,l y i,j | ≤ α 5 , where χ (2) (⋅) is an operator associated with (3) defined by


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][2][3][4][5][6][7][8][9][10][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][7][8][9][10][11].
The applied methods presented in [1][2][3][4][5][6][7][8][9][10][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][13][14].
The obtained stability results in [2][3][4][5][6][7][8][9][10][11][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 ẋ = (, ) (here  = col( 1 ,  2 , . . .,   )), the states are needed to be satisfied that sup ‖ 1 ()‖ ≥0 ≥ sup ‖ 2 ()‖ ≥0 ≥, . . ., ≥ sup ‖  ()‖ ≥0 or sup ‖ 1 ()‖ ≥0 ≥ max =2,3,... {sup ‖  ()‖ ≥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.

Mathematical Preliminaries
For convenience, some notations are introduced as follows: where | ⋅ | is the Euclidean norm and 0 <  < ∞,  ∈ N,  denotes the expectation of stochastic process, and N denotes the set of natural numbers.
The system (2) can be treated as an interconnection of isolated subsystems   given by (3)

Definitions and Assumptions.
Let   = 0 ( ∈ N) be the unique zero solution of system (2).
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 3 (see [13]).The zero solution of system (2)   = 0 ( ∈ N) is string exponentially stable with mode constraint in the mean square sense if   = 0 is string exponentially stable in the mean square sense, and the inequality Next, some assumptions will be given for the system (2) and the system (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  > 0 and  > 0 such that 2 ln   /(  −  −1 ) ≤  <  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 th 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  ̸ =   ,  ∈ N, calculating the operator  (1)   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  (1) [25], we can take the expectation of inequality (12) and rewrite it as Taking ( −,0 ) = 1,  = 0, 1, . . .,  − 1,  ∈ N. Substituting them into (13), we get −1 Since inequality (12) implies that the following inequality holds, it can be concluded that [ (1)   ] < 0,  ∈ N. Therefore, it follows from Lemma 4 that, for ∀ 0 >0, ∃ 0 > 0 such that Let  > 0, and satisfy Next, we will use the mathematical induction method to prove that sup hold, where When  = 1, it can be seen from ( 16) that ( 18) holds.Suppose that the following inequalities hold: From Assumption A3 and ( 19), we have Due to   > 1, we have We claim that (21) implies that If inequality (22) does not hold, there exist some  and (1)   ( * ) =  +   ( * ) ≥ 0, and Substituting (23) and inequality (24) into inequality (11), together with the properties of the operator , we get Since condition (12) implies that [ (1)   ( * )] < 0. This is a contradiction with  (1)   ( * ) ≥ 0.
By the mathematical induction method, it can be concluded that sup From condition 2 ln   /(  −  −1 ) ≤  < , we get that 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  ̸ =   ,  ∈ N, it is easy to obtain Abstract and Applied Analysis 7 Let  = ( 2 / 1 ) 1/2 ≥ 1, and where Define sets in state space: < 0. Therefore, for all   ∈   \ Π  and   ∈   \ Π  , we have This implies that, for   (0) ∈ Π  , we have   () ∈ Π  ,  > 0; that is, Note that condition (9) implies that the following inequality holds: Furthermore, inequality (33) can be rewritten as , ,  ∈ 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.
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  > 0 and  > 0 such that 2 ln   /(  −  −1 ) ≤  <  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.
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 lookahead 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.

Controller Design for Vehicle
Following System position, velocity, and acceleration of the th following vehicle are   , V  , 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 th following vehicle.
4.1.Dynamic Model.The rolling resistance friction is considered as the stochastic factor of the system.Let   =   + F  , where   is the certain part and F  is the stochastic part.It is assumed that F  /  is white noise process with mean value 0 and mean square error  2 .Let ( F  /  ) = ( ẋ  ).Therefore, the model of the longitudinal dynamics of a member vehicle with impulsive and stochastic disturbances is given by where, ,  ∈ N,   , ẋ  ,  ẋ  ,   ẋ 2  ,   ,   , and   denote the position, velocity, acceleration, effective aerodynamic drag, control effort, certain rolling resistance friction, and mass of the th following vehicle, respectively.Let V  (  ) be the velocity of the impulsive moment of the th following vehicle.Let the initial position of the th following vehicle be   (0).
It is well known that real number set  is a measurable set and  ∈ [0, ∞) ⊂  + .Consider that V  () is continuous on interval ( + −1 ,  −  ) and is a simple function in the set { −  ,  +  }, and so V  () is measurable on interval ( + −1 ,  +  ]; here,  ∈ N. V  () denote the integral of V  () in the set   , and let Mes(⋅) denote the measure; we have Due to the fact that Mes( that is, 4.2.Controller Design.Define an auxiliary error given by the following equation: where   () satisfies and Ṡ  is independent on ; that is to say,   is a finite variable [26].It is assumed that   ( +  ) =   ( −  ), ,  ∈ N. The expression of control law   () is chosen as follows: where Here,  > 0 with  ≥  > 0 is the control parameter and will be chosen later.

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   () = 0.5 2  (),  > 0,  ∈ N.

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 Section 3.
Substituting the expression of control law (52) into the system (44), we can obtain the following slide mode motion equation given by  vehicles in the platoon.It is assumed that when   = 8 s, V 1 ( +  ) = 1.05V 1 ( −  ); when   = 20 s, V 0 ( +  ) = 0.97V 0 ( −  ).The approach in [27] is used to generate Brown motion trajectory.
The simulation results are shown in Figures 2, 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.

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.

Figure 1 :
Figure 1: Spacing errors in a platoon.

Figure 4 :
Figure 4: Slide mode curves of vehicles.

Figure 5 :
Figure 5: Control input curves of vehicles.

Table 1 :
Initial states of vehicles.

Table 2 :
Parameters of vehicles.