Finite-Time Stability Analysis: A Tutorial Survey

Xu, Honglei

doi:https://doi.org/10.1155/2020/1941636

Complexity

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Finite-time Control of Complex Systems and Their Applications

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Review Article | Open Access

Volume 2020 | Article ID 1941636 | https://doi.org/10.1155/2020/1941636

Finite-Time Stability Analysis: A Tutorial Survey

Honglei Xu¹

Academic Editor: Jianquan Lu

Received16 Jul 2020

Revised09 Sept 2020

Accepted23 Sept 2020

Published19 Oct 2020

Abstract

In the past decades, there has been a growing research interest in the field of finite-time stability and stabilization. This paper aims to provide a self-contained tutorial review in the field. After a brief introduction to notations and two distinct finite-time stability concepts, dynamical system models, particularly in the form of linear time-varying systems and impulsive linear systems, are studied. The finite-time stability analysis in a quantitative sense is reviewed, and a variety of stability results including state transition matrix conditions, the piecewise continuous Lyapunov-like function theory, and the converse Lyapunov-like theorem are investigated. Then, robustness and time delay issues are studied. Finally, fundamental finite-time stability results in a qualitative sense are briefly reviewed.

1. Introduction

Finite-time stability was first introduced in a Russian journal [1] and later appeared in the western literature [2–4]. The term short-time stability is another name for it [5]. In the current literature, there are two different concepts of finite-time stability. The first is the traditional finite-time stability concept which concerns the restrained system behavior during a specified interval of time. The initial and trajectory domains and the time interval need to be specified in advance, so the traditional concept is a quantitative one. We call it finite-time stability in a quantitative sense. The second one characterizes an asymptotically stable system whose state reaches zero in a finite time, called a settling time. Similar to the Lyapunov stability, it is a qualitative concept, and hence, we call the second concept finite-time stability in the qualitative sense. The analysis and synthesis results of both finite-time stability concepts can be applied to many practical applications such as ATM networks [6], car suspension systems [7], and robot manipulators [8].

Finite-time stability in a quantitative sense emphasizes the following characteristics: the system restrains its trajectory to a predefined time-varying domain over a finite time interval for a bounded initial condition. Even though it mimics the Lyapunov stability, it is quite different from the classical one due to its finite time interval and specified domains for initial conditions and system trajectories, i.e., a system is finite-time stable for some chosen initial and trajectory domains and time intervals but not finite-time stable for different ones. In the past few decades, many finite-time stability analysis and control design problems have been investigated and a variety of stability criteria have been obtained, see, for example, [3, 4, 9] and the references therein. Recently, computationally tractable finite-time stability criteria with less conservatism have been established under the help of new tools such as linear matrix inequalities [10], Lyapunov matrix equations [11], and differential linear matrix inequalities [12]. More recently, studies on the finite-time stability and stabilization have been extended from linear time-varying systems to complex dynamical systems such as switched systems [13–15] and stochastic systems [16].

On the other hand, finite-time stability in a qualitative sense has attracted much attention in recent years and become a growing interdisciplinary research area. It focuses on asymptotical stability analysis for dynamical systems whose trajectories reach an equilibrium point in a finite time. It is a stronger concept than asymptotical stability and has the settling-time characteristic. Relevant results on autonomous and nonautonomous nonlinear systems have been discussed in [17–20]. Later, switched versions and time-delay versions appeared in the literature, e.g., [21–23]. Recently, relevant issues of underactuated systems with disturbance have been considered in [24]. In the context of this paper, the readers should be not hard to distinguish whether the concept of finite-time stability is quantitative or qualitative, so we can use the term “finite-time stability” in most places without causing confusion.

In this paper, we will summarize the results of finite-time stability from both quantitative and qualitative aspects. Several excellent surveys on finite-time stability have been found, see, for example, the review papers [25–27], the books [7, 28, 29], and the references therein. These publications report and survey finite-time stability on one aspect or another. This paper aims to provide a unified self-contained tutorial review of finite-time stability to introduce the recent discoveries in the field.

The remainder of this paper is as follows. Section 2 gives some basic mathematical preliminaries including two finite-time stability concepts. Section 3 reviews finite-time stability results in a quantitative sense, mostly for linear time-varying systems. Results involving time-dependent and state-dependent impulses, time delays, and uncertainty are also investigated. In Section 4, we briefly overview some results on finite-time stability in a qualitative sense. Finally, a conclusion is drawn in Section 5.

2. Mathematical Preliminaries

2.1. Notations and Definitions

Let denote a set of nonnegative real numbers and the -dimensional Euclidean space, and consider the time interval . Let be the transpose of and be the identity matrix with an appropriate dimension. For a square matrix , we denote by , , and the set of eigenvalues, the maximum eigenvalues, and the minimum eigenvalues of , respectively. The symmetric components in a matrix are represented by . , called to be positive semidefinite (positive definite), means for all is equivalent to . Let denote the set of all vector-valued continuous functions on . For , it is represented by with the norm . Let and denote the open and closed sets of the allowable system states defined as and , respectively, where and is a symmetric positive definite real matrix. We denote and . For a set , the conical hull of is the set of all conical combinations, i.e., . The set of normalized extremal rays generating , denoted by with , , is the minimal set of unit vectors such that . Given a piecewise continuous matrix-valued (or vector-valued) function over and a positive real number , we denote and , i.e., and are the left and right limits, respectively. Let the set be an open set having the origin and a boundary .

We first look at the basic definition of finite-time stability for a dynamical systemwhere is the time variable, is the state variable, and is a -valued function. Suppose that system (1) has a unique solution. We first introduce the concepts of “finite-time stability” in a quantitative sense and in a qualitative sense, respectively.

Definition 1 (Finite-Time Stability in a Quantitative Sense). Given two sets and , , system (1) is said to be finite-time stable with respect to ifIn Figure 1, we can see a graphical explanation of Definition 1, where the initial set and the time-varying set can be in various forms. (1)When they are ellipsoids, which are the most common forms existing in the literature (see, e.g., [7, 30–32]), they can be formulated as and , where is a bounded and piecewise continuous matrix-valued function of time and . In many cases, and can also be expressed as and for [33].(2)When they are in the form of polytopes, they can be described by and , where and are the number of vertices of the polytopes and , and and are the -th vertex of the polytope and the -th vertex of the polytope [34].(3)They can be formulated as much generalized piecewise quadratic domains over conical partitions and . Then, and . It is obvious to see that the set of piecewise quadratic domains is a generalized set of ellipsoidal domains since an ellipsoidal domain is indeed a piecewise quadratic domain choosing for all . It can also represent the set of polytopic domains whose boundary is a polyhedral function’s level curve.

Definition 2 (Finite-Time Stability in a Qualitative Sense). System (1) is said to be finite-time stable if for any , and , there exist and such that implies , and for all . Here, is called the settling-time function of system (1), and the set is called the domain of attraction. Moreover, if , system (1) is globally finite-time stable.
Similarly, a schematic illustration of Definition 2 is given in Figure 2.

2.2. Mathematical Formulations

System (1) is a general model for both linear and nonlinear systems depending on the choice of the function . In the first part of this paper, we will focus on finite-time stability issues in a quantitative sense for continuous-time linear time-varying systems with and without finite jumps. In the second part of this paper, we will analyze finite-time stability in a qualitative sense for continuous-time nonlinear systems.

First, we introduce a linear time-varying system described asfor a given initial condition , which has been considered in many papers, see, e.g., [7, 30]. Here, is a continuous matrix-valued function.

In many practical scenarios, abrupt state changes and system jumping behaviors are commonly existing, and these finite jumps occur when the time points and/or the system states satisfy a certain triggering condition, say . When the impulses are triggered, the impulsive mappings can be described bywhere is a matrix-valued function, which describes the jumping behavior of system (3) with left continuity over the triggering set . We call a dynamical system modeled by (3) and (4) to be an impulsive linear system. According to the triggering set , impulsive linear systems expressed by (3) and (4) can be categorized into two main types: the time-dependent impulsive linear systems and the state-dependent impulsive linear systems, which can be described byrespectively [31, 32, 35]. For a time-dependent impulsive linear system, the impulses occur at the given time points, , so the triggering set can be written as , where . For a state-dependent impulsive linear system, the impulses happen when the system state reaches a preassigned set , and then, the triggering set is written as . It is worth pointing out that the well-posedness of the triggering times should be guaranteed and the Zeno phenomena need to be avoided in this paper.

3. Finite-Time Stability in a Quantitative Sense

In this section, we will provide some results on finite-time stability in a quantitative sense for linear time-varying system (3) and its variants with impulses (4).

3.1. State Transition Matrix

In linear system theory, we know that the solution of (3) can be described as , where the matrix-valued function has the following basic properties:

This matrix-valued function is called to be the state transition matrix, and an appropriate assumption on the nature of can ensure the existence and uniqueness of the state transition matrix. For system (3), it is stable in the sense of Lyapunov if there exists a positive constant such that for all . Moreover, system (3) is asymptotically stable if it is stable and . As for finite-time stability of the system (3), a necessary and sufficient stability condition will be provided in the following theorem [30].

Theorem 1. System (3) is finite-time stable with respect to , where and if and only if the state transition matrix of system (3) satisfies

The state transition matrix approach has been extended to impulsive linear system (5) in [31, 32, 36]. Letting , the solution of the impulsive linear system (5) will be , where , called the state transition matrix of (5), is a piecewise continuous matrix-valued function with discontinuous right-hand sides at the time instants . In detail, when , is the solution of the following matrix differential equation:

In the sequel intervals for , should satisfy

In the end, when , we have

Theorem 2. Impulsive linear system (5) is finite-time stable with respect to , where the sets and are the same with those in Theorem 1, if and only if for all , the following is satisfied:

The conditions in the form of state transition matrices (8) and (12) in Theorems 1 and 2 are valuable for theoretical analysis but hard to apply due to the high computational difficulty, particularly for the time-varying case.

To obtain computational conditions for finite-time stability, some Lyapunov-like functions are needed to establish conditions in the form of linear matrix equalities or Lyapunov matrix inequalities. In some early work such as [3], it concludes that system (1) is finite-time stable with respect to if and only if there exists a real-valued Lipschitz function , continuous on , and a real-valued integrable function such that, for , we have for all and for and . A piecewise continuous Lyapunov-like function is the most common one in the literature, and relevant results will be presented in the following subsection.

3.2. Piecewise Continuous Lyapunov-Like Functions

To obtain computationally tractable finite-time stability conditions, we choose a quadratic piecewise continuous Lyapunov-like function and establish the following conditions containing coupled differential Lyapunov matrix equations and differential linear matrix inequalities.

Theorem 3 (see [7, 30]). System (3) is finite-time stable with respect to , where the sets and are the same with those in Theorem 1, if and only if for all , there exists a symmetric piecewise differentiable matrix-valued function such that the following conditions involving the differential matrix equation with boundary conditions are satisfied:

We see that Theorem 3 provides a necessary and sufficient condition for finite-stability of system (3). However, it is not practicable to verify the differential matrix equation in (13) for every , and hence, Theorem 3 is not suitable for the computational purpose. Using Theorem 3, we can obtain a sufficient condition with computational tractability for finite-stability of system (3).

Theorem 4 (see [7, 30]). System (3) is finite-time stable with respect to , where the sets and are the same as those in Theorem 1, if and only if for all there exists a symmetric piecewise differentiable matrix-valued function such that the following conditions involving the differential matrix equation with boundary conditions are satisfied:

Nowadays, computational tools in the convex optimization framework such as linear matrix inequalities or differential linear matrix inequalities are very efficient, and we can obtain the following necessary and sufficient conditions including differential linear matrix inequalities, equivalent to (13):

The piecewise continuous Lyapunov-like function is also applied to the finite-time stability problem of impulsive linear systems. For linear time-varying systems with time-dependent impulses, we have the following finite-time stability results.

Theorem 5 (see [36]). System (5) is finite-time stable with respect to , where the sets and are the same with those in Theorem 1, if and only if for all there exists a piecewise differentiable positive definite matrix-valued function such that the following conditions involving differential linear matrix inequalities are satisfied:

Moreover, it is finite-time stable with respect to if and only if there exists a piecewise continuous positive definite matrix-valued solution such that the following conditions involving differential/difference Lyapunov equations are satisfied:where is a nonsingular matrix-valued function satisfying in .

As for a linear time-varying system with state-dependent impulses (6), we have the following theorem to provide a sufficient condition for its finite-time stability.

Theorem 6 (see [35]). System (6) is finite-time stable with respect to , where the sets and are the same with those in Theorem 1, if and only if for all there exists a piecewise differentiable positive definite matrix-valued function such that the following conditions involving linear differential/difference matrix inequalities are satisfied:

When the initial set and the time-varying set are two given polytopes and a quadratic function is chosen, we have following sufficient conditions of finite-time stability for a quadratic system:where .

Theorem 7 (see [34]). System (19) is finite-time stable with respect to , where the sets and are two given polytopes, if there exists a positive definite symmetric matrix such that

The initial set and the time-varying set can also be piecewise quadratic domains over conical partitions and , say and [37, 38]. In such cases, we choose a time-varying piecewise quadratic Lyapunov-like function defined over the abovementioned conical partition aswhere , are symmetric matrices. Then, a sufficient condition for the finite-time stability of linear-varying system (1) can be presented as follows.

Theorem 8 (see [37]). System (3) is finite-time stable with respect to , where the sets and are the given piecewise quadratic domains, if there exist positive definite symmetric matrices such thatFor and with .

The sufficient conditions in (22) are not applicable due to the infinite number of matrix inequalities. Applying -procedure arguments and considering the conical partition, a computationally tractable sufficient condition is also obtained in [37].

Theorem 9 (see [37]). System (3) is finite-time stable with respect to , where the sets and are the given piecewise quadratic domains, if there exist positive numbers , positive real-valued functions , and matrices , and , and positive definite symmetric matrices such that and there exist positive piecewise continuously differentiable matrix-valued functions , such that the following conditions involving differential linear matrix inequalities and linear matrix inequalities are satisfied:

3.3. Converse Lyapunov-Like Theorem

In [9], a converse Lyapunov-like theorem is established for finite-time uniformly stable continuous-time nonautonomous system (1). It provides the characterization of finite-time stability with regards to the existence of Lyapunov-like functions.

Theorem 10 (see [9]). If system (1) is uniformly stable with respect to , , then there exists a real-valued Lyapunov-like function satisfyingfor all , andfor and .

Moreover, two necessary and sufficient conditions for the existence of a Lyapunov-like function for finite-time stability are given as follows.

Theorem 11 (see [9]). System (1) is finite-time stable with respect to , , if and only if there exist a Lipschitz continuous real-valued Lyapunov-like function and a continuous real-valued function such that the following conditions are satisfied: (i) for all, (ii) for all , and (iii) .

Theorem 12 (see [39]). System (1) is finite-time stable with respect to , , if and only if there exist a continuous real-valued Lyapunov-like function such thatfor all , andfor all and with .

3.4. Miscellaneous Issues

Time delays are often encountered in many practical systems such as chemical processes, electric circuits, and networked systems, leading to unsatisfactory system behaviours and even instability. So, various stability problems for delayed systems have attracted much attention to lots of researchers. Among them, finite-time stability analysis has been of particular interest bringing forth many papers such as [40–42].

A linear time-invariant delayed system can be represented bywith an associated initial state function

To proceed, we need the following definitions.

Definition 3. System (28) associated with initial condition (29) is said to be finite-time stable with respect to if , which implies , where is a positive scalar-valued function satisfying for , and .

Definition 4. System (28) associated with initial condition (29) is said to be finite-time stable with respect to if implies .
Next, we introduce two sufficient conditions for finite-time stability of linear delayed system (28) in the following two theorems.

Theorem 13 (see [41]). System (28) associated with the initial condition (29) is finite-time stable with respect to if for all , we havewhere is the fundamental matrix of linear delayed system (28).

Theorem 14 (see [41]). System (28) associated with initial condition (29) is finite-time stable with respect to ifwherewith being the largest singular value of the corresponding matrix.

In [40], the authors construct a delay-dependent Lyapunov-like functionwhere is a differentiable matrix-valued function on such that the following differential matrix equation is satisfied:with the initial condition . Then, the following result based on Lyapunov function (33) can be given.

Theorem 15 (see [40]). System (28) associated with initial condition (29) is finite-time stable with respect to (α, β, h) if there exists a positive real number such that the following conditions are satisfied:where is a matrix measure of the given matrix and is the solution of the following transcendental matrix equation:

Next, consider a singular linear delayed systemwhere is a singular matrix with rank . There exist two nonsingular matrices and such that . Let

Then, a sufficient condition for the finite-time stability of singular linear delayed system (37) was established in [42].

Theorem 16 (see [42]). System (37) associated with initial condition (29) is finite-time stable with respect to if there exists a positive number , a symmetric positive definite matrix , a nonsingular matrix , and a matrix such that

More recently, the Lyapunov–Razumikhin approach is extended to finite-time stability for a nonlinear delayed systemin [43]. Sufficient conditions can be illustrated through the following theorem.

Theorem 17 (see [43]). System (40) associated with initial condition (29) is finite-time stable with respect to if there exists positive scalars with and , integrable real-valued function , class functions , , and a differentiable function such that (i) , (ii) , whenever for all , , whereand (iii)

Besides time delays, uncertainty is another important phenomenon commonly encountered in practical systems. The existence of uncertainty causes the poor performance and even instability. The finite-stability concept has been extended to uncertain linear systems [44–46]. The uncertainty can be expressed as norm-bounded uncertainty and structured uncertainty. Consider an uncertain linear systemwhere , is a norm-bounded uncertainty function such as and are known matrices of appropriate dimensions. The following theorem will present a necessary and sufficient condition for finite-time stability of system (43).

Theorem 18 (see [46]). System (43) is finite-time stable with respect to , where the sets and are the same as those in Theorem 1, if and only if for all , there exist a piecewise continuous function and a symmetric piecewise differentiable matrix-valued function such that the following conditions involving differential linear matrix inequalities are satisfied:

Similar to Theorem 3, necessary and sufficient conditions in Theorem 18 are not computationally tractable. For the computational purposes, we can see the following sufficient condition for finite-time stability of system (43).

Theorem 19 (see [46]). System (43) is finite-time stable with respect to , where the sets and are the same as those in Theorem 1, if and only if for all , there exist a piecewise continuous function and a symmetric piecewise differentiable matrix-valued function such that the following conditions involving differential linear matrix inequalities are satisfied:

Robustness analysis for a linear delayed system with structured uncertainty was conducted in [45], where the uncertain system is described bywhere are known matrices of appropriate dimension, and is the uncertain time-varying matrix with .

Theorem 20. (see [45]). System (46) associated with initial condition (29) is finite-time stable with respect to if there exists positive real numbers and symmetric positive definite matrices and such that the following conditions are satisfied:

If uncertain linear system (43) is affected by finite impulses (4), the system state will undergo abrupt changes at discrete time instants, which leads to more difficulty to analyze its stability performance. The following theorem from [46] gives a sufficient condition for finite-time stability of the linear time-varying system with both time-dependent impulses and uncertainty. More complex cases with a norm-bounded uncertainty on the impulsive matrix-valued function and state-dependent impulses were also provided in [46] as well.

Theorem 21 (see [44]). System (3) is finite-time stable with respect to , where the sets and are the same as those in Theorem 1, if and only if for all , there exists positive real number and a symmetric piecewise differentiable matrix-valued function such that the following conditions are satisfied:

All results mentioned above have illustrated the finite-time stability conditions in a quantitative sense, and we continue to introduce more results in a qualitative sense.

4. Finite-Time Stability in a Qualitative Sense

It is well known that a radially unbounded positive definite function with the property is a Lyapunov function. Lyapunov’s second method demonstrates that the existence of the Lyapunov function is also equivalent to the asymptotical stability of system (1), which provides the foundation for the following necessary and sufficient conditions for finite-time stability results.

Theorem 22 (see [27, 47]). Consider an autonomous nonlinear systemSystem (49) is finite-time stable if and only if there exists a smooth Lyapunov function (equivalently, all smooth Lyapunov functions) such that for all ,where the map is the inverse of .

Theorem 23 (see [27, 47]). System (49) is finite-time stable with a continuous settling-time function at the origin if and only if there exist a scalar , a positive scalar , and a smooth Lyapunov function such thatMoreover, the settling-time function should satisfy

A converse Lyapunov theorem was obtained for finite-time stability of nonlinear system (49) in [18]. As for the case that settling-time is continuous, we have the following converse theorem.

Theorem 24 (see [18]). If system (49) is finite-time stable with a continuous settling-time function at the origin and , then there exists a continuous function such that

The abovementioned Lyapunov-based methods for analysis of finite-time stability may not be suitable for constructive design. Recently, an implicit Lyapunov function method to solve an algebraic equation was derived in [48], which provides a design method for a robust controller for the closed-loop systems to handle exogenous disturbances. The implicit Lyapunov function theorem only verifies stability conditions in an implicit way and does not need to solve the equation.

Theorem 25 (see [48]). System (1) is finite-time stable with a settling-time function , where if there exists a continuously differentiable function such that for any , there exists a radially unbounded function such thatfor all , where and .

Recently, the notion of finite-time stability for nonlinear autonomous system (49) was extended to nonautonomous nonlinear system (1). These Lyapunov and converse Lyapunov results are derived and introduced in the following theorems.

Theorem 26. (see [17]). System (1) is finite-time stable if there exist a scalar , a positive function c(t), a class function , and a continuously differentiable function such that , , andMoreover, for the case , then system (1) is be globally finite-time stable. If there exists a class function such that , then system (1) is uniformly finite-time stable.

Theorem 27 (see [17]). Let and there exists a class function , where such that andIf system (1) is uniformly finite-time stable and the settling-time function is jointly continuous at , , then there exist a positive scalar c, a class function , and a continuously differentiable function such that , , and

5. Conclusions

This paper has overviewed the fundamental results of the finite-time stability analysis of dynamical systems. The concepts of finite-time stability are classified into those in the quantitative and qualitative senses. Finite-time stability in a quantitative sense is firstly investigated. Then, finite-time stability results in a qualitative sense are outlined. This review paper is far from complete due to our limitations and nonawareness. We hope that this paper can be a useful resource for practitioners, researchers, and graduate students working in this field.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported in part by the Australian Research Council under Grant DP160102819.

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Copyright © 2020 Honglei Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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