Finite-time stabilization of time-varying nonlinear systems based on a novel differential inequality approach

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Highlights

  • The problem of finite-time control is investigated for a class of time-varying nonlinear systems in this paper.

  • Two kinds of control schemes including continuous control and hybrid control are presented.

  • The above theoretical results are applied to the finite-time synchronization of complex networks with time-varying structure.

  • Our proposed results improve and extend some recent publications.

Abstract

In this paper, the finite-time stabilization problem is investigated for a class of time-varying nonlinear systems. The classical finite-time differential inequality is extended to time-varying systems and some new lemmas are derived for global finite-time stability (FTS) and local FTS of the corresponding closed-loop systems. Then based on the hybrid control theory and the extended time-varying differential inequality, we present two control schemes including continuous control and hybrid control. It is shown that the continuous control is formulated to make the system converge in finite time, while the impulsive part involved in hybrid control accelerates the stabilization. In addition, the theoretical results are applied to the finite-time synchronization of complex networks with time-varying parametric matrices. Ultimately, two numerical examples are presented to demonstrate the distinctiveness and the effectiveness of our proposed results.

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 R and R+ denote the set of all real numbers and the set of all nonnegative real numbers, Rn and Rn×m the n-dimensional and m×n-dimensional real spaces equipped with the Euclidean norm ||, respectively, Z+ the set of positive integers, N the open set in Rn around the origin. Let Λ>0 or Λ<0 represent the positive or negative definite symmetric matrix Λ. λmax() (or λmin()) denotes the maximum (or minimum) eigenvalue of matrix , In the n-dimensional identity matrix, and the Kronecker product. For θRn and ϵ>0, let Bε(θ) represent an open ball domain with center θ and radius ε.

Section snippets

Preliminaries

Consider the following nonlinear systemx˙(t)=g(t,x(t)),where x(t)=[x1(t),,xn(t)]TRn represents the system state vector, x(0)x0 denotes the initial value, and the continuous function g(t,x):R+×RnRn is the nonlinear term with g(t,0)=0 for all t0. In particular, for all initial value x0Rn/{0}, we always assume that g(t,x) meets certain conditions such that system (1) possesses the unique forward solution x(t)=x(t,0,x0), see [34]. Some essential definitions are given in the following.

Definition 1

([12])

Main results

Consider a specific class of system (1):x˙(t)=Ac(t)x(t)+Ad(t)f(x(t))+u(t),where x(t)Rn represents the system state vector, u(t)Rn is the control input to be designed. Ac(t):R+Rn×n and Ad(t):R+Rn×n denote the continuous, matrix-valued functions, f(x)=[f1(x),,fn(x)]TRn is a nonlinear function with the diagonal Lipschitz matrix LRn×n. In addition, when there is no control input, we assume that system (5) has a trivial solution x(t)0.

Theorem 1

If there exist continuous functions l(t),n(t):R+R+,

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 formxi˙(t)=Ao(t)xi(t)+Bo(t)f(xi(t))+c(t)j=1NhijΓxj(t),where xi(t)=[xi1(t),,xin(t)]TRn denotes the system state vector with the initial value xi(0)xi0, i=1,,N. Ao(t):R+Rn×n and Bo(t):R+Rn×n are continuous matrix-valued functions. f(xi(t))=[f1(xi1(t)),,fn(xin(t))]T 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 matricesAc(t)=[cost2+sint112+sint0.5],Ad(t)=[1.42+sint002+sint0.4cost2+sint],f(x)=[1cosx1(t)sinx2(t)].Given the diagonal matrix-valued functions P(t) and Q(t) in the formP(t)=Q(t)=[1002+sint].By simple computation, the Lipschitz matrix L, constants ω1, and ω2 can be determined by L=I2, ω1=1, and ω2=3,

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.

References (42)

  • S.G. Nersesov et al.

    Finite-time stabilization of nonlinear impulsive dynamical systems

    Nonlinear Anal. Hybrid Syst

    (2008)
  • X. Li et al.

    Finite-time stability and settling-time estimation of nonlinear impulsive systems

    Automatica

    (2019)
  • J. Lu et al.

    A unified synchronization criterion for impulsive dynamical networks

    Automatica

    (2010)
  • S. Yu et al.

    Continuous finite-time control for robotic manipulators with terminal sliding mode

    Automatica

    (2005)
  • Y. Shen et al.

    Semi-global finite-time observers for nonlinear systems

    Automatica

    (2008)
  • B. Zhou

    On asymptotic stability of linear time-varying systems

    Automatica

    (2016)
  • V. Lakshmikantham et al.

    Theory of impulsive differential equations

    (1989)
  • W.M. Haddad et al.

    Impulsive and hybrid dynamcial systems: Stability, dissipativity and control

    (2006)
  • X. Liao et al.

    Stability of dynamical systems

    (2007)
  • J. Wang et al.

    Extended dissipative control for singularly perturbed PDT switched systems and its application

    IEEE Trans. Circuits Syst. I Regul. Pap.

    (2020)
  • X. Liu et al.

    Interval type-2 fuzzy passive filtering for nonlinear singularly perturbed PDT-switched systems and its application

    Journal of Systems Science and Complexity

    (2021)
  • Cited by (4)

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