Finite-Time Stability for a Class of Underactuated Systems Subject to Time-Varying Disturbance

Based on the classical finite-time stability theory, the problem of finite-time stability (FTS) for time-varying nonlinear systems is investigated in this paper. Several FTS theorems involving global form and local form are presented, and an estimate of the settling-time of such systems is obtained. As an application, we consider the problem of asymptotic stabilization of the Brockett integrator subject to time-varying disturbance. By the switched finite-time controller designmethodology, we establish a sufficient condition to guarantee the relative asymptotic stability. For Brockett-like integrator subject to time-varying disturbance, we achieve better convergence performance. Examples and their simulations are given to demonstrate the applicability of the proposed results.


Introduction
Over the past decades, many researchers have focused on Lyapunov asymptotic or exponential stability of dynamical systems, see [1][2][3][4][5][6][7]. One of the typical features of the asymptotic or exponential stability is that the solution will tend to an equilibrium state as time tends to infinity. us, it is difficult to achieve fast transient and high-precision performances. Finite-time stability (FTS) served as a special case of asymptotic stability means that the system reaches an equilibrium state in finite time, which presents an efficient tool for many engineering problems. Lots of interesting results on FTS have been raised from theoretical and practical points of view. Haimo [8] introduced a definition of continuous finite-time differential equations as fast accurate controllers for dynamical systems. Bhat and Bernstein [9] proposed Lyapunov theorem on FTS of continuous autonomous systems. Bhat and Bernstein [10] achieved globally FTS of the double integrator. Yang et al. [11] concerned the stabilization of switched dynamical networks with logarithmic quantization couplings in finite time. Moulay and Perruquetti [12] studied the FTS for a class of continuous systems using Lyapunov function. Furthermore, Moulay et al. [13] established some FTS theorems for timedelay systems based on Lyapunov functionals and the extension of Artsteins transformation. Polyakov and Hu et al. [14,15] introduced the concept of fixed-time stability, which means that the system is globally finite-time stable and the settling-time is bounded by some positive constant for any initial values. However, most of existing results, such as [8][9][10][11][12][13], are based on the framework of time invariant. Recently, Haddad et al. [16] obtained a sufficient condition of FTS for nonlinear time-varying systems. However, it can only be applied to some special cases due to the strict restriction that the derivative of Lyapunov function is less than zero almost everywhere. Moulay and Perruquetti [17] presented the FTS conditions for nonautonomous continuous systems. However, it was simplified to the time-invariant form through the inequality of the Lyapunov function.
Hence, more general methods should be established for nonlinear time-varying systems. To avoid confusion, it should be pointed out that the FTS considered in this paper is different from another FTS concept adopted in [18][19][20], which dealt with the finite-time boundedness.
As stated in Brockett's result [21], Brockrtt integrator (Nonholonomic integrator), covered in underactuated systems, was the first example of locally controllable nonlinear system which is not smoothly stabilizable and requires additional constraints. In order to overcome the abovementioned limitations, various methods have been proposed for the problem of asymptotic stabilization of the Brockett integrator [22][23][24][25][26]. Astolfi [22] addressed the problem of almost exponential stabilization with bounded control. Banavar and Sankaranarayanan [23] studied a switched finite-time controller design methodology. Rehman [24] dealt with steering control of nonholonomic systems with drift for the extended nonholonomic double integrator. Rehman and Ahmed [25] presented piecewise constant and state-dependent feedback control for the nonholonomic integrator. Chihchen [26] investigated the time-varying control for globally exponential stabilization of the Brockett integrator. However, the abovementioned works did not take disturbance into account and converged in infinite time domain. erefore, based on the FTS theorem for time-varying systems, we will explore stability issues of the Brockett integrator subject to time-varying disturbance.
In this paper, we firstly address some essential stability definitions in Section 2. en, in Section 3, based on the classical finite-time stability theorem [9], the FTS theorem for time-varying nonlinear systems is considered. We apply it to solve the problem of asymptotic stabilization of the Brockett integrator subject to time-varying disturbance, as shown in Section 4. By some examples, we verify the main results in Section 5. In Section 6, conclusions are be presented.

Preliminaries
Notations: let R denote the set of real numbers, R + the set of nonnegative numbers, R n the n-dimensional real spaces equipped with the Euclidean norm | · |, and N, D ⊂ R n the subspaces of R n . a ∨ b and a∧b are the maximum and minimum of a and b, respectively. e notation A T denotes the transpose of A. Consider the nonlinear system given by _ where t ≥ 0, x ∈ R n denotes the state vector, f(·): R + × D ⟶ R n is a continuous function with f(t, 0) � 0 for all t ≥ 0. We assume that f(t, x) satisfies suitable conditions so the solution x(t) � x(t, 0, x 0 ) with initial state x 0 ∈ D uniquely exists in forward time for all initial conditions except possibly the origin (see [9]).
Definition 1 (see [9]). System (1) is said to be FTS if there exist a function T: R n ⟼R + and an open neighborhood N ⊆ D such that the following statements hold: ) and x(t) ⟶ 0 as t ⟶ T(x 0 ) − (ii) Lyapunov stability (LS): for every ε ball B ε around the origin, there exists a δ ball B δ around the origin such that, for every Definition 2 (see [23]). For a given set O that contains the origin, system (1) is said to be relatively asymptotically stable (RAS) with respect to the set O, if for any ε > 0, there exists δ > 0 such that, for all x 0 ∈ O∩B δ , x(t) ∈ B ε holds when t ≥ 0; moreover, x(t) ⟶ 0 as t ⟶ ∞. B δ and B ε are open balls around the origin of radius δ and ε, respectively.

Main Results
In this section, we present a generalization of the classical FTS theorem [9]. Given the following definition

Theorem 1. Consider system (1), if there exists a positive definite, continuously differentiable function
where (4) e settling-time function T: R n ⟼R + , depending on the initial state x 0 , is bounded by Moreover, when β � +∞ and V is radially unbounded, system (1) is globally FTS.
Proof. For a given Integrating both sides of the abovementioned inequality, it gives Since V is a positive definite function, we can obtain 2 Complexity When β < +∞ and V 0 < σ, it follows from (4) that there exists a time point t * > 0 satisfying , it is easy to obtain that system (1) is LS. Combining these two aspects, system (1) is locally FTS with respect to x 0 satisfying V 0 < σ.
When β � +∞ and V is radially unbounded, no matter what initial value x 0 it is, there always exists a time point e classical FTS theorem mentioned in [9] has been widely used in many fields, such as finite-time synchronizion of complex networks, finite-time attitude stabilization for spacecraft, and terminal sliding mode method of nonlinear systems, see [10,23]. However, these theoretical results are applied under the framework of time invariant, that is, { } � 0, the classical FTS theorem is unapplicable. However, in eorem 1, we can still achieve the FTS of system (1), as long as (3) and (4) hold. is assertion can be verified in Section 5.

Applications
Consider a class of systems described by equations of the form where x ∈ R n , u ∈ R m , m < n, and G(x) is a matrix of proper dimension. It has received considerable attention for the asymptotic stabilization of such systems during the past several decades. e reason for such an interest lies in the fact that system (9) cannot be asymptotically stabilized by any continuous differentiable and state feedback control laws (see [22]). e Brockett integrator, as a special case of system (9), has plenty of theoretical results with time-invariant form. However, there are few theoretical results considering the form of time-varying systems. In this paper, we study the Brockett integrator subject to time-varying disturbance, that is, where t ≥ 0, x i (t): R + ⟼R, i � 1, 2, 3, denotes the state component, u j (t): R + ⟼R, j � 1, 2, denotes the control input, w j (t): R + ⟼R, j � 1, 2, is continuous function denoting the disturbance term. For any integrable function ρ(t): R + ⟼R, we introduce a set F as follows: Assumption 1. ere exists a continuous function η(t): R + ⟼R such that η(t)w 1 (t) ≥ 0, for t ≥ 0, and the following inequality holds: Theorem 2. Assume that Assumptions 1 and 2 hold, then the solution of system (10) converges to the origin and system (10) is Proof. We firstly show the attractivity of system (10). If x 1 (0) ≠ 0, then it follows from (14) that which implies Note that ρ 1 (t) ∈ F; then, it is easy to see that which shows that x 1 (t) asymptotically tends to zero. In addition, the third state component of system (10) becomes _ x 3 (t) � −η(t)w 1 (t)x 3 1/3 with control input u 2 , which gives that x 3 (t) is FTS combining (12) and eorem 1. Moreover, the settling time T can be estimated by (5). For the second state component of system (10), when t > T, it holds x 3 1/3 /x 1 ≡ 0; then, by (13), the derivative of x 2 becomes −ρ 2 (t)x 2 . Similar to the above argument, we could conclude that the origin of x 2 (t) asymptotically tends to zero. For the case that x 1 (0) � 0, we can apply any open loop control to steer the system to a nonzero value of x 1 .
is completes the proof of attractivity.
Secondly, we show the relative stability of system (10). When t > T, system (10) is transformed as which implies that x(t) enters the set O from this moment, and x 1 (t) and x 2 (t) will converge to origin. erefore, when x(0) ∈ O, x 3 (t) ≡ 0 x 1 (t), x 2 (t) ⟶ 0 as t ⟶ ∞, then one can obtain that system (10) is RAS with respect to the set O.
is completes the proof.

Remark 2.
eorem 2 provides a method for the problem of asymptotic stabilization of the Brockett integrator subject to time-varying disturbance. A relatively asymptotical stabilization is achieved by weakening the stability conditions. Besides, in the process of designing controller U, we apply a switched finite-time controller design methodology. In fact, it is also a discontinuous control. Worth mentioning that it is of vital importance to design functions η(t), ρ 1 (t), and ρ 2 (t), such that the FTS of x 3 (t) is achieved directly. In Section 5, we will show an example to illustrate our conclusion.

en, the solution of system (10) converges to the origin and system (10) is RAS with respect to the set
Remark 3. It should be pointed out that, in eorem 2, the selection of set O and control input U is not unique. In eorem 2, we only get the FTS of one state component. In order to achieve better performance, we can further consider multiple components to achieve the FTS for the Brockettlike integrator, which is the result we will give in eorem 3.
Consider the following Brockett-like integrator subject to time-varying disturbance: where t ≥ 0, x i (t): R + ⟼R, i � 1, 2, 3, denotes the state component, u j (t): R + ⟼R, j � 1, 2, denotes the control input, w j (t): R + ⟼R, j � 1, 2, is continuous function denoting the disturbance term, k � a/b, a < b, and a and b are positive odd numbers.

Assumption 3.
ere exists a continuous function μ(t): R + ⟼R such that μ(t)w 1 (t) ≥ 0, for t ≥ 0, and the following inequality holds: +∞ 0 μ(s)w 1 (s)ds � +∞. (21) Assumption 4. ere exist two functions φ 1 (t), φ 2 (t) ∈ F such that Theorem 3. Assume that Assumptions 3, 4, and 5 hold; then, the solution of system (20) converges to the origin, and system (20) is RAS with respect to the set Proof. If x 1 (0) ≠ 0, it follows from (23) that About the first state component of system (20), we notice that φ 1 (t) ∈ F and k � a/b, a < b, and a and b are positive odd numbers, so obtained from eorem 1, the first state component of system (20) is FTS. Meanwhile, combining (21), (22), and eorem 1, the third state components of system (20) is FTS. Hence, for initial value x 1 (0)and x 3 (0), the settling time can be established, that is, T(x 1 (0)) and T(x 3 (0)). Under the premise of Assumption 5, when t > T(x 3 (0)), system (20) is transformed as Similar to the proof process of eorem 2, we can obtain that system (20) is RAS with respect to the set Z. For the case that x 1 (0) � 0, we can apply any open loop control to steer the system to a nonzero value of x 1 . is completes the proof.

Numerical Simulations
In this section, we will demonstrate the effectiveness of the proposed results for the above applications through the following simulation examples. 4 Complexity Example 1. Consider the following 2D time-varying system: where t ≥ 0. Consider the Lyapunov function V(x) � x 1 2 + x 2 2 , and the derivative along the trajectory of the system (26) is where c(t) � 2 − 2(sin t ∨ cos t) and c(t) ≥ 0 on R + . According to Lyapunov's stability theory in [28], system (26) is LS. It can be calculated that the lower bound of c(t) is 0, so the classical FTS theorem in [9] is unapplicable. On the contrary, note that c(t) satisfies (4) with β � +∞, and it follows from eorem 1 that system (26) is globally FTS and the settling time can be estimated as T(x 0 ) ≤ 4.51. Figure 1 illustrates the state trajectories of system (26).
Hence, we can calculate that +∞ 0 η(s)w 1 (s)ds � +∞. It follows from eorem 2 that system (10) is RAS with respect to the set O ≜ x � (x 1 , x 2 , x 3 ) T ∈ R 3 : x 3 � 0, x 1 ≠ 0 , and the control input could be described by It is worth pointing out that the classic FTS theorem in [9] is not applicable.
is is due to the existence of the disturbance term, and the control input will reach infinity within a certain time, which is impractical. e numerical simulations for system (10) are follows.
In Figure 2, it can be clearly seen that x 3 (t) firstly converges to zero, then x 1 (t) and x 3 (t) converge to zero. e control input u 1 and u 2 change over time and turn into 0 when x(t) converges to zero. In Figure 3, the state vector x(t) enters the set O and finally reaches the origin in finitetime. All of these results are corresponding to the conclusion of eorem 2, which illustrates the validity of the eorem 2.
Example 3. For convenience, the same parameters as in Example 2 are used. Consider system (2) with w 1 (t) � t/20,   Complexity φ 2 (t) � 1/10, − cos(t/2)/10, and k � 1/3. It is not difficult to verify that the set K is not empty. erefore, following from (21), (22), and eorem 3, system (20) is RAS with respect to the set Z and the control input could be described by It can be seen from Figure 4 that the first and third state components of the system (20) is FTS, and the second state component of system (20) asymptotically tends to zero, which shows the validity of eorem 3.

Conclusion
e problem of FTS for time-varying nonlinear systems is investigated in this paper, where the results of global FTS and local FTS are proposed, respectively. As an application, the stabilization problem of the Brockett integrator subjecting to time-varying disturbance is studied. By the switched finite-time controller design methodology, we achieve the RAS for the Brockett integrator subject to timevarying disturbance. Further research topics would be considered to extend the main results of this paper to other more complex problems, such as the finite-time stabilization for wheeled mobile robot subject to time-varying disturbance and the FTS theorem for time-varying nonlinear systems in the sense of Lyapunov.

Data Availability
No data were used to support this study.