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

Authors: Jie Wu; Dan Yang; Xinyi He; Xiaodi Li
Journal: Complexity
 (2020)

---

## Abstract

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 design methodology, 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.


---

## Body

## 1. Introduction

Over the past decades, many researchers have focused on Lyapunov asymptotic or exponential stability of dynamical systems, see [1–7]. 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. Thus, 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 time-delay 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 [8–13], 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 [18–20], 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 [22–26]. 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. Therefore, 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 Section2. Then, 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.


## 2. Preliminaries

Notations: letℝ denote the set of real numbers, ℝ+ the set of nonnegative numbers, ℝn the n-dimensional real spaces equipped with the Euclidean norm ⋅, and ℕ, D⊂ℝn the subspaces of ℝn. a∨b and a∧b are the maximum and minimum of a and b, respectively. The notation AT denotes the transpose of A.Consider the nonlinear system given by(1)x˙t=ft,xt,x0=x0,where t≥0, x∈ℝn denotes the state vector, f⋅:ℝ+×D⟶ℝn is a continuous function with ft,0=0 for all t≥0. We assume that ft,x satisfies suitable conditions so the solution xt=xt,0,x0 with initial state x0∈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:ℝn⟼ℝ+ and an open neighborhood ℕ⊆D such that the following statements hold:(i)
Finite-time convergence (FTC): for everyx0∈ℕ/0, xt∈ℕ/0 holds for all t∈0,Tx0 and xt⟶0 as t⟶Tx0−(ii)
Lyapunov stability (LS): for everyε ball Bε around the origin, there exists a δ ball Bδ around the origin such that, for every x0∈Bδ/0, xt∈Bε for all t∈0,Tx0Definition 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 x0∈O∩Bδ, xt∈Bε holds when t≥0; moreover, xt⟶0 as t⟶∞. Bδ and Bε are open balls around the origin of radius δ and ε, respectively.


## 3. Main Results

In this section, we present a generalization of the classicalFTS theorem [9]. Given the following definition(2)V˙t,xt≜∂V∂tt,x+∂V∂xt,xft,xt,for a continuously differentiable function Vt,x:ℝ+×ℝn⟶ℝ+. With a slight stealing concept, Vt,x:ℝ+×ℝn⟶ℝ+ is said to be positive definite and radially unbounded if there exists a positive definite and radially unbounded continuous function Wx:ℝn⟶ℝ+ such that Vt,x≥Wx, for t≥0,x∈ℝn.Theorem 1.
Consider system (1), if there exists a positive definite, continuously differentiable function V:ℝ+×ℝn⟶ℝ+, an integrable function ct:ℝ+⟼ℝ+, and two real numbers α∈0,1, σ>0, such that the derivative of V along the solution xt=xt,0,x0 of system (1) satisfies(3)V˙t,xt≤−ctVαt,xt,∀t≥0,where ct satisfies(4)∫0+∞csds≜β≥σ1−α1−α.Then, system (1) is locally FTS with respect to x0 satisfying V0<σ, where V0≜V0,x0. The settling-time function T:ℝn⟼ℝ+, depending on the initial state x0, is bounded by(5)Tx0≤inft>0:∫0tcsds=V01−α1−α.Moreover, whenβ=+∞ and V is radially unbounded, system (1) is globally FTS.Proof.
For a givenx0∈ℝn/0, let xt be the solution of system (1) through 0,x0. By transforming (3), we obtain(6)dVVα≤−ctdt,∀t≥0.
Integrating both sides of the abovementioned inequality, it gives(7)∫V0Vt,xt1VαdV≤∫0t−csds.
SinceV is a positive definite function, we can obtain(8)0≤V1−αt,xt1−α≤V01−α1−α−∫0tcsds.
Whenβ<+∞ and V0<σ, it follows from (4) that there exists a time point t∗>0 satisfying ∫0t∗csds<σ1−α/1−α, such that V01−α/1−α−∫0t∗csds=0. That is to say Vt∗,xt∗=0, from which the settling time Tx0 can be estimated, namely, (5). When t≥Tx0, V˙t,xt≤0, it follows from [27] that the solution of system (1) through Tx0,0 is unique in forward time. Therefore, Vt,xt≡0, for t≥Tx0, which means system (1) is locally FTC with respect to x0 satisfying V0<σ. Noting from (3), it is easy to obtain that system (1) is LS. Combining these two aspects, system (1) is locally FTS with respect to x0 satisfying V0<σ.
Whenβ=+∞ and V is radially unbounded, no matter what initial value x0 it is, there always exists a time point t∗∗>0, such that V01−α/1−α−∫0t∗∗csds=0, that is, Vt∗∗,xt∗∗=0. Through a similar procedure, we conclude that system (1) is globally FTS.Remark 1.
The 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, ct≡c>0. When infct:t≥0=0, the classical FTS theorem is unapplicable. However, in Theorem 1, we can still achieve the FTS of system (1), as long as (3) and (4) hold. This assertion can be verified in Section 5.


## 4. Applications

Consider a class of systems described by equations of the form(9)x˙=Gxu,where x∈ℝn, u∈ℝm, m<n, and Gx is a matrix of proper dimension. It has received considerable attention for the asymptotic stabilization of such systems during the past several decades. The 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]). The 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,(10)x˙1t=u1,x˙2t=u2,x˙3t=w1tx1u2−w2tx2u1,where t≥0,xit:ℝ+⟼ℝ,i=1,2,3, denotes the state component, ujt:ℝ+⟼ℝ,j=1,2, denotes the control input, wjt:ℝ+⟼ℝ,j=1,2, is continuous function denoting the disturbance term. For any integrable function ρt:ℝ+⟼ℝ, we introduce a set F as follows:(11)F=ρt∈ℝ:∫0tρsds⟶+∞ast⟶+∞.Assumption 1.
There exists a continuous functionηt:ℝ+⟼ℝ such that ηtw1t≥0, for t≥0, and the following inequality holds:(12)∫0+∞ηsw1sds=+∞.Assumption 2.
There exist two functionsρ1t,ρ2t∈F such that(13)w1tρ2t=w2tρ1t.Theorem 2.
Assume that Assumptions 1 and 2 hold, then the solution of system (10) converges to the origin and system (10) is RAS with respect to the set O≜x=x1,x2,x3T∈R3:x3=0,x1≠0 under the control input:(14)U=u1u2=−ρ1tx1−ηtx31/3x1−ρ2tx2.Proof.
We firstly show the attractivity of system (10). If x10≠0, then it follows from (14) that(15)x˙1t=−ρ1tx1,which implies(16)x1t=x10exp∫0t−ρ1sds.
Note thatρ1t∈F; then, it is easy to see that(17)x1t⟶0ast⟶+∞,which shows that x1t asymptotically tends to zero.
In addition, the third state component of system (10) becomes x˙3t=−ηtw1tx31/3 with control input u2, which gives that x3t is FTS combining (12) and Theorem 1. Moreover, the settling time T can be estimated by (5). For the second state component of system (10), when t>T, it holds x31/3/x1≡0; then, by (13), the derivative of x2 becomes −ρ2tx2. Similar to the above argument, we could conclude that the origin of x2t asymptotically tends to zero. For the case that x10=0, we can apply any open loop control to steer the system to a nonzero value of x1. This completes the proof of attractivity.
Secondly, we show the relative stability of system (10). When t>T, system (10) is transformed as(18)x˙1t=−ρ1tx1,x˙2t=−ρ2tx2,x3t=0,which implies that xt enters the set O from this moment, and x1t and x2t will converge to origin. Therefore, when x0∈O, x3t≡0x1t,x2t⟶0 as t⟶∞, then one can obtain that system (10) is RAS with respect to the set O. This completes the proof.Remark 2.
Theorem2 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 ηt,ρ1t,andρ2t, such that the FTS of x3t is achieved directly. In Section 5, we will show an example to illustrate our conclusion.Corollary 1.
Under conditions in Theorem1, when w1t=w2t, we just need ρ1t=ρ2t=p, where p is a positive constant. Then, the solution of system (10) converges to the origin and system (10) is RAS with respect to the set O≜x=x1,x2,x3T∈R3:x3=0,x1≠0 under the control input:(19)U=u1u2=−px1−ηtx31/3x1−px2.Remark 3.
It should be pointed out that, in Theorem2, the selection of set O and control input U is not unique. In Theorem 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 Brockett-like integrator, which is the result we will give in Theorem 3.
Consider the following Brockett-like integrator subject to time-varying disturbance:(20)x˙1t=u1,x˙2t=u2,x˙3t=w1tx1ku2−w2tx2u1,where t≥0,xit:ℝ+⟼ℝ,i=1,2,3, denotes the state component, ujt:ℝ+⟼ℝ,j=1,2, denotes the control input, wjt:ℝ+⟼ℝ,j=1,2, is continuous function denoting the disturbance term, k=a/b, a<b, and aandb are positive odd numbers.Assumption 3.
There exists a continuous functionμt:ℝ+⟼ℝ such that μtw1t≥0, for t≥0, and the following inequality holds:(21)∫0+∞μsw1sds=+∞.Assumption 4.
There exist two functionsφ1t,φ2t∈F such that(22)w1tφ2t=w2tφ1t.Assumption 5.
There exist two functionsTx1=inft>0,x1≠0:∫0tφ1sds=x11−k/1−k, Tx3=inft>0:∫0tμsw1sds=x32/3/2/3 such that the set K≜x=x1,x2,x3T∈R3:Tx3<Tx1 is not empty.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 Z≜x=x1,x2,x3T∈R3:K∪0,0,0T under the control input:(23)U=u1u2=−φ1tx1k−μtx31/3x1k−φ2tx2.Proof.
Ifx10≠0, it follows from (23) that(24)x˙1t=−φ1tx1k,x˙2t=−μtx31/3x1k−φ2tx2,x˙3t=−μtw1tx31/3.
About the first state component of system (20), we notice that φ1t∈F and k=a/b, a<b, and aandb are positive odd numbers, so obtained from Theorem 1, the first state component of system (20) is FTS. Meanwhile, combining (21), (22), and Theorem 1, the third state components of system (20) is FTS. Hence, for initial value x10and x30, the settling time can be established, that is, Tx10 and Tx30. Under the premise of Assumption 5, when t>Tx30, system (20) is transformed as(25)x˙1t=−φ1tx1k,x˙2t=−φ2tx2,x3t=0.
Similar to the proof process of Theorem2, we can obtain that system (20) is RAS with respect to the set Z. For the case that x10=0, we can apply any open loop control to steer the system to a nonzero value of x1. This completes the proof.


## 5. Numerical Simulations

In this section, we will demonstrate the effectiveness of the proposed results for the above applications through the following simulation examples.Example 1.
Consider the following 2D time-varying system:(26)x˙1t=sint−1x11/3+sintx22x1,x˙2t=cost−1x21/3−x12x2,where t≥0. Consider the Lyapunov function Vx=x12+x22, and the derivative along the trajectory of the system (26) is(27)V˙t=2x1sint−1x11/3+sintx22x1+2x2cost−1x21/3−x12x2=2sint−1x14/3+2cost−1x24/3+2sint−1x12x22≤2sint−1x14/3+2cost−1x24/3≤2sint−1∨cost−1x14/3+x24/3≤−ctV2/3t,where ct=2−2sint∨cost and ct≥0 on ℝ+. According to Lyapunov’s stability theory in [28], system (26) is LS. It can be calculated that the lower bound of ct is 0, so the classical FTS theorem in [9] is unapplicable. On the contrary, note that ct satisfies (4) with β=+∞, and it follows from Theorem 1 that system (26) is globally FTS and the settling time can be estimated as Tx0≤4.51. Figure 1 illustrates the state trajectories of system (26).Figure 1
Simulation results of states for system (26).Example 2.
Consider system (10) with w1t=t/20, w2t=1−cost/2, ηt=1, ρ1t=t/200, and ρ2t=1/10−cost/2/10. It is obvious that w1t, w2t∈F, ηt is a continuous function, and w1tρ2t=w2tρ1t. Hence, we can calculate that ∫0+∞ηsw1sds=+∞. It follows from Theorem 2 that system (10) is RAS with respect to the set O≜x=x1,x2,x3T∈ℝ3:x3=0,x1≠0, and the control input could be described by(28)U=u1u2=−t200x1−x31/3x1−110−cost/210x2.
It is worth pointing out that the classicFTS theorem in [9] is not applicable. This is due to the existence of the disturbance term, and the control input will reach infinity within a certain time, which is impractical. The numerical simulations for system (10) are follows.
In Figure2, it can be clearly seen that x3t firstly converges to zero, then x1tandx3t converge to zero. The control input u1andu2 change over time and turn into 0 when xt converges to zero. In Figure 3, the state vector xt enters the set O and finally reaches the origin in finite-time. All of these results are corresponding to the conclusion of Theorem 2, which illustrates the validity of the Theorem 2.Figure 2
Simulation results of states and control inputs for system (10).Figure 3
Simulation results of states for system (10).Example 3.
For convenience, the same parameters as in Example2 are used. Consider system (2) with w1t=t/20, w2t=1−cost/2, μt=1, φ1t=t/200, φ2t=1/10−cost/2/10, and k=1/3. It is not difficult to verify that the set K is not empty. Therefore, following from (21), (22), and Theorem 3, system (20) is RAS with respect to the set Z and the control input could be described by(29)U=u1u2=−t200x11/3−x31/3x11/3−110−cost/210x2.
It can be seen from Figure4 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 Theorem 3.Figure 4
Simulation results of states and control inputs for system (20).


## 6. Conclusion

The problem ofFTS 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 time-varying 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.



---

*Source: 8704505-2020-05-12.xml*