Rapid exponential stabilization of nonlinear continuous systems via event - triggered impulsive control

: This article investigates the problem of rapid exponential stabilization for nonlinear continuous systems via event - triggered impulsive control ( ETIC ) . First, we propose a trigger mechanism that, when triggered by a prede ﬁ ned event, causes the closed - loop system exponentially stable. Then, the exponential stabilization is achieved by the designed ETIC with or without data dropout. The case where there are delays in the ETIC signals is also studied, and the exponential stabilization is proved. Finally, a numerical study is presented, along with numerical illustrations of the stability results.


Introduction
It has long been recognized that continuous control is incapable of solving many control problems of nonlinear systems [1]. Moreover, even in cases where continuous control can be used, many practical and theoretical obstacles arise when using this type of control, such as the lack of accurate and up-todate information at all time despite the availability of frequent state measurements, the lack of accurate state information at all times, or even the sensitivity of the closed-loop system behavior to state measurement errors [2]. Furthermore, the frequent use of the latest innovations in computer and communication technologies in control theory has added new difficulties of other kinds, such as the limited storage capacity of the large amount of data measured continuously or the network saturation in front of the large amount of data to be transported. All of these difficulties and challenges motivated scientists to develop new control methods. Initially, periodic control was applied, which was activated after a predetermined period of time, giving rise to time-triggered impulsive control (TTIC) [3][4][5]. To make the control system more effective against disturbances caused by impulsive control, the duration of the control period was shortened in some cases. However, in the case of system stability and the absence of disturbances, this control period becomes useless and represents a waste of system resources [2]. To further reduce the computational cost and waste of system resources, an alternative control strategy is proposed in which the impulsive control is activated only when certain predefined events occur [6,7]. In terms of this strategy, the event-triggered impulsive control (ETIC) is a control that is updated aperiodically only when certain predefined events occur [6,8,9].
In the last decade, various event-triggered control strategies have been proposed for many practical control systems, such as intelligent transportation systems [10], marine structures [11], active vehicle suspension systems [12,13], multi-agent systems [14,15], and for networked control systems [16][17][18][19]. In [20], the ETIC method is used to develop exponential stabilization criteria for continuous dynamical systems, with the stability criteria relying on the limiting interval between two events. In [21], Lyapunov stability problems for impulsive systems are studied using the ETIC strategy. It is shown that some conditions are sufficient for uniform stability and global asymptotic stability. The results in [21,20] do not include exogenous perturbations, which limits the application of the model. In [22], the input-to-state stability (ISS) property for continuous-time dynamical systems is obtained using the ETIC method involving three event levels, whereas in [23], a different approach is used to obtain the ISS exponential stabilization for a class of nonlinear systems in the presence of external disturbances. Recently, different event-triggered strategies for studying exponential stabilization of infinite dimensional systems were presented in [25,24]. In [25], an impulsive control was constructed to stabilize the damped wave equations derived from brain activity, whereas in [24], a different ETIC is used to achieve rapid exponential stabilization of the Lotka-McKendrick equation in population dynamics.
Continuing from [25], we study the robustness of the given ETIC in this article, taking into account various disturbances that can occur in the control process, such as the data dropout or the delay of signals carrying information about the event that triggers the control. These types of disturbances can affect the impact and effectiveness of the strategy that we have chosen.
The organization of this article is as follows. Section 2 describes the problem we are trying to solve as well as the corresponding control task. Section 3 describes our event-triggered control strategy and presents our main results. A numerical illustration of the main result is given in Section 4. Finally, Section 5 concludes with a summary of what has been achieved in this article and some possible research projects for the future.

Problem statement
Let us consider the following nonlinear system: where x t n ( ) ∈ is the system state, d t n ( ) ∈ is an external disturbance, t 0 is the initial time, and the function f t x , ( ) is piecewise continuous in t, locally Lipschitz in x with constant L, and satisfies f t, 0, 0 0 ( )= for all t t 0 ≥ . For systems (1) and (2), it is assumed that the state x t ( ) can be influenced at certain times t t t t 0 , with t k tending to +∞ as t toward +∞. This sequence will be chosen to achieve the exponential stability of systems (1) and (2) by an accurate choice of a certain controller u t k ( ). The controller u t k ( ) is chosen among q available values g x g x , , q , where for all j q 1 ≤ ≤ , g j is a function from n into n . More precisely, we are interested in constructing a sequence t u t , k k ( ( )) such that the solution of the following system is exponentially convergent: The following is a precise definition of the notion of rapid exponential stabilization of a solution. .

Main result
Throughout this section, we will set forth our main findings on how to design a control strategy that rapidly stabilizes our systems (1) and (2) in the sense of the aforementioned definition.

Event-triggered control design
Our strategy for constructing our impulsive control is dependent on two conditions: The first condition allows us to choose the intervention instants t k from a set of available choices, whereas the second condition imposes some constraints on the values u t k ( ) + that can be assigned to the control u to the right of the instants t k during the intervention. More specifically, we consider the following two assumptions.
Assumption 1 : The control instants satisfy: where p is a positive integer and τ i are positive constants for all i p 1, , .
Assumption 2 : There exists a positive real number μ 1 < such that Let β 1 < be a positive real number, which we will fix later. Now, before adopting a recurrence construction for our impulsive control t x t ,
Note that according to the definitions of events E 1 and E 2 , control is triggered only when event E 2 occurs. If N k is the number of occurrences of this event during the time interval [25] focuses on the exponential stabilization of infinitedimensional systems, the effectiveness of the presented method in the face of disturbances such as the loss of some information causing control triggering and delay in the transmission of the control signal has not been studied. Therefore, this study complements the aspects discussed in [25] by considering such perturbations that may affect the results of this strategy. [23] for the ISS study of nonlinear systems, with the difference that the convergence conditions are formulated in terms of the number of E 1 and E 2 events that can occur (Theorems 1 and 2 in [23]), which cannot be checked in practice before running the procedure. In addition, there is another difference between the results obtained in this study and those obtained in [23], which concerns the nature of the convergence in both studies. While this study demonstrated rapid exponential stabilization, the previous study only focused on ISS stabilization. 1 2 − strategy, systems (3)-(5) are controlled by the event-triggered control u defined as follows:

Remark 4. In light of our ETIC E E
The next execution time t k 1 + of the control u is determined by the event-triggering mechanism E E 1 2 − , which continuously checks whether the condition is satisfied or not. This condition contains information about the state variable x t k ( ) at the previous execution time t k , and t k 1 + can be written as Note here the difference from the classical ETIC, which can be formally written in our case as follows (see [6]): The difference between strategies (11) and (12) is that in (12), condition (10) must be satisfied every time t t k > , whereas in our proposed strategy (11), condition (10) must be satisfied only at points t τ i p , 1 k i + ≤ ≤ . This allows us to consider our ETIC strategy as a hybrid of the classical ETIC strategy and the TTIC strategy, but it complicates the checking process more difficult because the solution can take large values between the two time points t k and t k 1 + , contributing to a slow convergence of the solution to zero.

Rapidly exponential stabilization under ETIC
At this point, our first main result about the exponential stabilization of systems (1) and (2) can be established. Proof. We will proceed as in the proof of Theorem 1 in [25], noting first that, since we have for all it is obvious that Zeno behavior is avoided for systems (3)-(5) (see [4]). Let γ be any positive constant, and we seek, under assumptions 1 and 2 , a positive constant c such that the solutions of systems (3)-(5) verify the inequality in (6). is done according to the occurrence of two events E 1 and E 2 , resulting in the following two cases: (i) If t k 1 + results from the occurrence of the event E 1 , then from the definition of t k 1 + , we have x t x t β x t .
(ii) If t k 1 + results from the occurrence of event E 2 , then from the definition of x t k 1 ( ) + + , and Grnöwall's inequality applied to the solution x t ( ) of systems (3) and (4) on the interval t t , Taking into account (13) and (14), we obtain which gives In addition, by applying Gronwall's inequality for t t t , and assumption 1 states Therefore, combining (17), (18), and (19), it follows that where c e L γ τ 2 max ( ) = + . Hence, systems (3)-(5) are exponentially stable with decay rate γ 0 > and the proof of the theorem is finished. □ Now, we can set up the following criteria for rapid exponential stabilization of systems (3)-(5) with existing data dropouts in the ETIC.
Proof. From the proof of Theorem 1, it is easy to see that if there exist data dropouts of ETIC, then (15) should be changed to: Choosing β e γτ max = − and μ e 0, Using this, we obtain and by using (19), we obtain x t e e x .
Therefore, systems (3)-(5) are rapidly exponentially stable with convergence rate γ. This completes the proof. □ Remark 5. In comparison to the result proved in [23], we can observe that condition (11) considered in [23] to prove ISS stabilization with data dropouts in ETIC is expressed in terms of number of incidents E 1 and E 2 that lie in the future and thus have not yet occurred at the beginning of the process. However, in our research, convergence was not measured in terms of the number of these incidents. Moreover, our decay rate was randomly chosen to achieve exponential convergence under ETIC E E 1 2 ( ) ( ) − , which demonstrates our strategy's resilience in the face of data dropout during the transmission of the trigger signals.

Rapid exponential stabilization under time-delay in the ETIC
The results presented in Sections 3.2 and 3.3 were obtained without taking into account the communication delays that may exist in the communication network and affect the robustness of the ETIC scheme E 1 ( )-E 2 ( ). In this part, we will take into consideration the possibility of delays in the triggering of control. . In the delayed ETIC, at t k , the systems (3)-(5) will receive a previous ETIC signal u x t h k k k ( ( )) − , which will be used as the value of x t k ( ) + rather than u x t k k ( ( )). Thus, systems (3)-(5) take the following form: x t x .
Assumption : 3 Assume that there is no data dropout with regard to event E 2 .
The result concerning the stability of systems (25)- (27) in the presence of delays in the transmission of control signals is given in the following theorem: Theorem 3. Suppose that assumptions , 1 2 , and 3 are satisfied. Furthermore, it is assumed that there exists some m ∈ such that Proof. The proof of the avoidance of Zenon phenomena is the same as in Theorem 1. To prove the rapid exponential stability of systems (25)- (27), we distinguish two cases: Case 1: As only the delay occurring at the time of event E 2 can influence the control of the system, we can paraphrase this to state that if t k 1 + results from the event E 1 , then h Case 2: If t k 1 + results from the event E 2 , it follows that t t τ , where Id denotes the identity application of n 2 .
Now, it yields from (28) that It follows from (29) and (30) that As λ 0 1 < < , it follows from Theorem 4.2 in [26] that the discrete time-delay system (31) is exponentially stable. Then, according to the comparison principle for discrete systems (e.g., Proposition 1 in [27]), there exist two positive numbers α and M such that for all k ∈ , Combining (18), (19), and (32), we obtain does not confer any constraints and can be replaced by the standard assumption, which simply stipulates that the delays are finite from above. This is because if the delay exceeds τ max , then our strategy will act as if event E 1 had occurred.
(ii) Similarly, hypothesis (28) is not so restrictive, since we are concerned with studying the asymptotic behavior of the solutions of systems (25)- (27), we know that as soon as t t τ

Numerical application
To illustrate the effectiveness of the proposed event-triggered control strategy, we will apply this strategy to study the stability of trajectory tracking for a wheeled robot. The coordinates x y , ( ) of the vehicle center satisfy the following system [28][29][30][31]: where θ is the angle between the heading direction and the x-axis ( Figure 1). The problem of tracking a reference robot was originally introduced in [32] as follows: x t v t θ ṫ cos , The error coordinates will be defined similarly to that in [29][30][31][32][33] x t y t θ t θ t θ t θ t θ t x t x t y t y t θ t θ t   ( ) − are depicted in Figure 2, which shows their instability. To stabilize systems (40)-(42), we will apply Theorem 1, and for the simulation, we consider the following parameters: The impulsive control system takes the following form: x t x t t E B x t t E , , .

Conclusion
In this study, rapid exponential stabilization via event-triggered control has been investigated for nonlinear continuous dynamical systems. We have also studied the robustness of the ETIC for network systems where delays and data dropouts occur. In the presence of disturbances, the proposed ETIC is shown to be robust to dropouts and delays in data transmission. Nevertheless, larger delays may equate to a slower convergence speed in exponential stabilization. In addition, as an illustration of the theoretical results in this article, the stability of the trajectory tracking of a wheeled robot has been investigated. However, the complicated case of data dropout and delay in the transmission of ETIC simultaneously is still an open issue, and it will be on our agenda for future research. Other important future works of the ETIC method could involve the stabilization of some neutral fractional differential equations (see [34]) or some nonlinear fractional differential problems with boundary conditions (see [35]).