Finite-time stabilization of time-varying nonlinear systems based on a novel differential inequality approach
Introduction
In recent years, much attention has been attracted to the Lyapunov stability over a finite time interval, i.e., the finite-time stability (FTS), rather than the classical Lyapunov asymptotic stability [1], [2], [3], [4], [5], [6]. This is mainly due to the superiority of FTS that gives rise to the better generalization performance, the faster convergence rate, and the better robustness against perturbations. These superiorities render that the method of finite-time stabilization becomes one of the most appealing tools in engineering applications, such as the attitude stabilization of rigid spacecraft [7], [8] and the tracking control of wheeled mobile industrial robot [9], [10]. On the flip side, since the Lyapunov theory of FTS was developed in [11], [12], [13], [14], the problem of finite-time stability and stabilization of various nonlinear systems has been extensively studied, see [15], [16], [17], [18], [19], [20] and references therein. Generally, the control strategies for finite-time stabilization can be divided into three classes: non-smooth finite-time control, nonchattering continuous control, and impulsive control. For instance, in [15], a continuous control law for high-order nonlinear system was proposed in light of backstepping-like procedure and adaptive idea. Further, by improving conventional nonlinear systems subject to high-order or low-order nonlinear growth rates, a higher precision and faster convergence controller was designed in [16] and its practical application in finite-time adaptive control of uncertain systems was presented in [17]. It should be pointed out that these control strategies belong to the non-smooth finite-time control, which includes the sign function in the control signal. Nevertheless, bringing the sign function into system dynamics always results in a chattering effect to the system state, which is unfavorable to the stabilization of system. Thereby, in [21], [22], [23], the nonchattering continuous control was presented via the adding a power integrator approach, which does not include the sign function. In particular, if we further consider that the control signal is allowed to occur only at certain discrete instants [24], the impulsive control strategy is generated. Based on impulsive control theory and resetting events analysis [25], some essential FTS Lyapunov theorems for nonlinear impulsive systems were established in [26] and two kinds of impulse including stabilizing or destabilizing effects were studied. Furthermore, the possible bound of the settling time was estimated in [26], which depended on both the initial value of system and the impulse effect. One may observe that these control strategies ([15], [16], [17], [18], [21], [22], [23], [25], [26]) are all based on the classical finite-time differential inequalities in [12], which is feasible for the finite-time stabilization of time-invariant systems. If the problem of finite-time control is concentrated on time-varying systems, they may no longer be applicable. “Can we give the finite-time differential inequality in time-varying form and apply it to the corresponding finite-time control strategy?” This is an interesting and critical problem to be addressed in our concerning.
On the other hand, the synchronization control problem of complex networks invites much attention from theoretical analysis and practical applications. In fact, synchronization is a considerable collective dynamic behavior of complex networks, which possesses a bright application possibilities in recent years, see [27], [28], [29], [30], [31]. The finite-time synchronization control means that synchronization target can be arrived over a finite time interval, which is investigated and popularized in many kinds of fields, see [21], [22], [32], [33]. For instance, the problem of finite-time synchronization control was addressed for delayed complex networks with impulses in [32]. Considering that the initial value of system cannot be obtained accurately, the control strategy of fixed-time synchronization for the complex networks with impulsive effects was investigated in [21]. Furthermore, [22] extends such synchronization strategy to stochastic systems. However, one may find that the classical finite-time differential inequalities in [12] are incorporated into most existing synchronization strategies, such as those in [21], [22], [32], [33]. Thereby, they are only applicable to complex networks with time-invariant parametric matrices. Hence, if we further consider the complex networks with time-varying parametric matrices, one will naturally ask whether we can establish the control strategies of finite-time synchronization for the afore-mentioned complex networks.
In this paper, inspired by the above consideration, the main purpose is to solve the finite-time control problem of time-varying systems. By the extended finite-time differential inequality, we establish some FTS lemmas for time-varying systems. Meanwhile, two kinds of control schemes including continuous control and hybrid control are proposed. Compared with most existing results [15], [16], [17], [18], [21], [22], [23], [25], [26], the main advantages and novelties of this paper include: (1) In this paper, the classical finite-time differential inequality is extended to time-varying systems and two novel lemmas are proposed for global FTS and local FTS, respectively. And for a class of time-varying nonlinear systems, two kinds of finite-time control schemes are presented. (2) Our proposed control schemes are applicable to the finite-time stabilization of complex networks with time-varying parametric matrices. The placement of this paper is designed as follows. In Section 2, we introduce the time-varying system and give some preliminary definitions and essential lemmas. The main theoretical results and the synchronization of complex networks with time-varying parametric matrices are derived in Section 3 and Section 4, respectively. Two illustrated examples are presented in Section 5. Finally, in Section 6, the conclusion is provided.
Notations. Let and denote the set of all real numbers and the set of all nonnegative real numbers, and the -dimensional and -dimensional real spaces equipped with the Euclidean norm , respectively, the set of positive integers, the open set in around the origin. Let or represent the positive or negative definite symmetric matrix . (or ) denotes the maximum (or minimum) eigenvalue of matrix , the -dimensional identity matrix, and the Kronecker product. For and , let represent an open ball domain with center and radius .
Section snippets
Preliminaries
Consider the following nonlinear systemwhere represents the system state vector, denotes the initial value, and the continuous function is the nonlinear term with for all . In particular, for all initial value , we always assume that meets certain conditions such that system (1) possesses the unique forward solution , see [34]. Some essential definitions are given in the following. Definition 1 ([12])
Main results
Consider a specific class of system (1):where represents the system state vector, is the control input to be designed. and denote the continuous, matrix-valued functions, is a nonlinear function with the diagonal Lipschitz matrix . In addition, when there is no control input, we assume that system (5) has a trivial solution . Theorem 1 If there exist continuous functions ,
Application
In this section, the finite-time synchronization of complex networks with time-varying parametric matrices will be investigated. First of all, consider the complex networks with N identical nodes in the following formwhere denotes the system state vector with the initial value , . and are continuous matrix-valued functions. is the
Examples
In this part, some numerical examples are presented to demonstrate the distinctiveness of the proposed control schemes. Example 1 Consider the 2-D nonlinear system (5) with time-varying parametric matricesGiven the diagonal matrix-valued functions and in the formBy simple computation, the Lipschitz matrix , constants , and can be determined by , , and ,
Conclusion
In this paper, the finite-time stability and stabilization of time-varying nonlinear systems is investigated. In terms of the extended finite-time differential inequality (i), two control schemes have been studied for the corresponding closed-loop systems. The control parameters are given by solving the differential inequalities (i), which is different from the design of LMI-based control parameters. Then we applied the propose results to complex networks with time-varying parametric matrices
Acknowledgments
This work was supported by National Natural Science Foundation of China (62173215), Major Basic Research Program of the Natural Science Foundation of Shandong Province in China (ZR2021ZD04, ZR2020ZD24), and the Support Plan for Outstanding Youth Innovation Team in Shandong Higher Education Institutions (2019KJI008). The paper has not been presented at any conference.
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