1 Introduction

It is noteworthy that the properties of a number of practical engineering systems, such as electromagnetic waves, chemical physics, and fluid flow, cannot be adequately represented by integer-order dynamic systems but can be well embodied via employing fractional-order models. Due to their wide applications, fractional-order dynamic systems have attracted considerably attention from various fields, such as materials science [27], physics [8], pharmacokinetics [23], mechanics [10], supercapacitors [4], and neural networks [1], just to name a few. Please refer to the monograph [19] for more applications about fractional-order differential systems. Stability analysis [2] is one of the important and interesting topics for fractional-order dynamic systems. Many valuable methods on this issue have been reported, such as robust control [11], fractional-order controller design [28], adaptive control [9], sliding mode control [26], impulsive control [25], and so on.

Impulsive control only introduces transient control at certain discrete moments, which can achieve the control target through minimum control amount [20]. Impulsive control has gradually become a commonly used method in modern control because of its simple structure, lower control cost, and less information transmission [30]. It also has been widely used in coupled systems [6], neural network systems [37], chaotic secure communication [7] and system stabilization [33], etc. In recent years, with the rise of research on fractional-order control systems, many useful results have been proposed, for examples, impulsive synchronization of fractional-order complex networks [17], impulsive stabilization of fractional-order neural networks [31], impulsive control of fractional-order multi-agent systems [18], etc.

It should be noted that most of the results focused on fixed time-triggered impulsive control, that is, the impulse interval is preset. From the perspective of actual effects, impulsive control inputs at some moments are unnecessary, which will lead to the waste of system bandwidth resources [36]. Thanks to the proposal of event-triggered control theory, event-triggered mechanisms invoke data transmissions if predefined conditions on the data are satisfied. As a result, network and energy resources are consumed only when the data is necessary for control, which can achieve the control object with less information exchange. Thus, the design of certain event-triggered strategies have received increasing attention in recent years for integer-order dynamic systems [24, 34, 35, 37] and many references therein. Combining the advantages of impulsive control and eventtriggered control, event-triggered impulsive control (EIC) was proposed in recent years, where the impulsive instants are determined by some designed event-triggering functions and event-triggering conditions. Many scholars have carried out a series of fruitful researches in this field, such as applying the event-triggered impulse control method to synchronization analysis of discrete time-delay complex dynamical networks [12], nonlinear delay systems [15], input-to-state stability for heterogeneous multi-agent systems [13], discrete-time delayed systems and networks [16], Lyapunov stability problem for impulsive systems [14], consensus of multi-agent systems [3], neural networks [22], and so on. Compared with the event-triggered impulsive control for integer-order systems, there are few works on event-triggered impulsive control for fractional-order systems.

Based on the above discussion and inspired by the research in [16,32], this paper will study the exponential stabilization of general fractional-order continuous-time dynamic systems including the nonlinear and linear case via event-triggered impulsive control approach. The main contributions are as follows: (i) The event-triggered impulsive control method is applied to fraction-order continuous time dynamic systems. The impulsive instants are defined by some events depending on the state of the systems. Thus, the impulsive instants are not given in advance, which is different with the time-triggered impulsive control. (ii) The controller does not need to be updated continuously, and the Zeno-behavior of impulsive instants is excluded. Some unnecessary impulsive samples can be avoided by the control strategy and the online resources are saved. (iii) Based on the stability theory and inequality technique, some sufficient conditions on exponential stabilization of nonlinear and linear fractional-order dynamic systems are presented, respectively.

The rest of this paper is organized as follows. In Section 2, problem formulation is introduced. In Section 3, exponential stabilization is studied for nonlinear and linear fractional-order continuous-time dynamic systems via event-triggered impulsive control, respectively. The Zeno-behavior of impulsive instants is also excluded. In Section 4, simulation examples are presented to show the effectiveness of the theoretical results. Conclusions and future study are made in Section 5.

2 Problem formulation

2.1 Caputo fractional derivative

The Caputo and Riemann–Liouville (RL) fractional-order derivatives are the two broadly used to model the fractional-order dynamical systems. Since the initial conditions for fractional-order differential equations with Caputo fractional-order derivative take the same form as for the traditional integer-order differential equations, in this paper, we will adopt the Caputo fractional-order derivative to model the continuous time dynamic systems.

Definition 1. (See [21].) The Caputo fractional-order derivative of of order (0,1) is defined as follows:

where is the Gamma function, and denotes the derivative of .

For convenience, in the following, will be denoted as if no confusion arisen, where denotes the initial time.

Note that the differentiability of function is required in the definition of Caputo fractional-order derivative, but a number of function may not be differentiable. The right upper Dini order derivative of is introduced.

Definition 2. For (0,1), the upper right Dini fractional-order derivative is defined by

where is the right upper Dini derivative of .

Definition 3. (See [29].) For a Lebesgue-integrable function , the fractional-order integral of order (0,1) is defined as follows:

Lemma 1. (See [21].) If , then

It is obviously that Lemma 1 also holds for the upper right Dini fractional-order derivative.

Mittag-Leffler function defined as follows is often used to study the dynamic behavior of fractional-order dynamic systems [5]:

Especially, when Mittag-Leffler function with one parameter is

Lemma 2. (See [21].) Assume that and . Then

Lemma 3. (See [29].) Let Then is nonnegative, and the following statements are true:

• is monotonically nonincreasing andforwhen

• is monotonically nondecreasing and forwhen

Lemma 4. Letandwhere is a given continuous function. Then

Especially, when the following inequality holds:

Proof. Since , there exists a nonnegative function satisfying

and then

Let Then

Denote we have

Taking the Laplace transform on both sides, we can obtain that

where and are the Laplace transforms of and , respectively. Then

Thus, by the inverse Laplace transform we have

It follows that

The proof is completed.

2.2 System model and problem statement

Consider the following fractional-order continuous-time dynamic control systems with (0,1):

where and are the state and control input, respectively, : satisfies and there is a positive constant such that

In order to use the benefit of impulsive control and reduce the frequency of the controller update, the following event-triggered impulsive feedback controller (EIC) is used:

where denotes the left limit of function at satisfy and Lipschitz conditions, i.e., there are positive constants such thatk The impulsive instants t. are defined iteratively by

in which is called to be the triggering function defined as

where (0,1), and represents measurement error.

Definition 4. The fractional-order continuous-time dynamic system (1) with EIC (2) is said to be event-triggered impulsive exponential stabilization (EIES) if there exit such that

Definition 5. If there exists a constant such that inf where then the impulsive instants are called to no Zeno-behavior.

3 Exponential stabilization results

In this section, we first discuss that there is no Zeno-behavior for the impulsive instants determined by (3). Then we prove that system (1) is exponential stabilization under the EIC (2) with some given conditions. A corollary for linear case is also presented.

Theorem 1. There is no Zeno-behavior for the impulsive instants t. determined by (3).

Proof. For calculating the right upper Dini order derivative of we have

By Lemma 4 one can get

Then it follows from that

The next event will not be triggered until Thus,

If for some fixed , then one can get for any which implies the stability of system (1) is reached. Thus, without loss of generality, we assume that Then by (5) we have

Let Then

Therefore, there is a such that that is, there is no Zeno-behavior of impulsive instants.

Theorem 2. Assume that the impulsive instants are determined by (3) and Then system (1) with EIC (2) is EIES if the following inequality is satisfied:

where

Proof. For calculating the right upper Dini .-order derivative along the solution of (1), we have

By the definition of impulsive instants and triggering function (4) one can derive that

and then

In term of (6) and (7), we have

It follows from Lemma 4 that

and then

Notice that

Recalling (8), we have

Since and Then

Denote

and

we have

which implies that the fractional-order continuous-time dynamic system (1) is eventtriggered impulsive stabilization with the EIC (2).

According to Theorems 1 and 2, consider the following linear system:

where and, and the event-triggered impulsive feedback controller is designed as follows:

where are the control gain matrices determined later, impulsive instants t. are also determined by (3). Then we have the following corollary.

Corollary 1. Assume that the impulsive instants are determined by (3). Then system (9) with EIC (10) is EIES if the following inequality is satisfied:

where and .. Furthermore, there is no Zeno-behavior for the impulsive instants determined by (3).

4 Simulation examples

In this section, numerical examples for linear and nonlinear cases will be presented to illustrate the theoretical results.

Example 1. Consider the following linear fractional-order continuous-time dynamic system:

Assume that

and

If there is no impulsive control for system (11), that is, the controller (10) is replaced by Then we have

Let the initial condition for system (12) be The state of system (12) is depicted in Fig. 1, which shows system (11) with the continuous control is unstable.

Now, we apply the event-triggered impulsive control on system (11). Let

Assume by simple computation we can choose and Thus,

Therefore, by Corollary 1, system (11) is EIES.

The simulation results with the same initial condition as no impulsive control is presented in Fig. 2, which shows the fractional-order system (11) is stable with the event-triggered impulsive control (EIC).

The impulsive instants are depicted in Fig. 3, which implies the Zeno-behavior is excluded.

Example 2. Synchronization of fractional-order jerk chaotic systems.

where and

Construct the slave system as follows:

where is the control input and will be designed later.

Denote as the synchronization error. Then by master system (13) and slave system (14) we have

In order to avoid continuous update of the controller, the following event-triggered impulsive controller will be used:

where the impulsive instants tk are defined by (3). and in (4) are replaced by and respectively. The control gain matrix C is given as follows:

Then one can choose that By simple computation we have Thus,

Therefore, by Theorem 2 the zero solution of system (15) is exponential stability, which implies that the exponential synchronization between the master system (13) and slave system (14) can be achieved.

Let initial conditions be and The simulation results are depicted in Figs. 4–6, which show that not only the synchronization between the master and slave system with the event-triggered impulsive controller can be achieved, but also the Zeno-behavior of the impulsive instants is excluded.

5 Conclusions

Combining the advantages of impulsive control and event-triggered control, event-triggered impulsive stabilization of fractional-order continuous-time dynamic system is discussed. The impulsive instants depend on the states of the system and are not given in advance. Based on the stability theory and inequality technique, some sufficient conditions on exponential stabilization of nonlinear and linear fractional-order dynamic systems are presented, respectively. It proves that there is no Zeno-behavior for the impulsive instants determined by some designed events. As an application of the obtained theoretical results, the synchronization of fractional-order jerk chaotic systems is also presented in the simulation example. More general fractional-order dynamic systems and eventtriggered impulsive control problem of multi-agent systems will be considered in future study.

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