Global Finite-Time Stabilization for a Class of Uncertain High-Order Nonlinear Systems

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The importance for studying such system is exemplified in [1], where state feedback controller was used to stabilize the underactuated, weakly coupled, unstable mechanical system.Since Jacobian linearization of system (1) at the origin is neither controllable nor feedback linearizable for the case of  > 1, the traditional design tools including feedback linearization or backstepping are hardly applicable to the system (1).Mainly thanks to the adding a power integrator method, a series of stabilizing results have been achieved over the last decades; for example, one can see [2][3][4][5][6][7][8][9][10] and the references therein.However, it should be mentioned that the aforementioned works only consider the feedback stabilizer that makes the trajectories of the systems converge to the equilibrium as the time goes to infinity.
Compared to the asymptotic stabilization, the finite-time stabilization, which renders the trajectories of the closed-loop systems convergent to the origin in a finite time, has many advantages such as fast response, high tracking precision, and disturbance-rejection properties [11].Hence it is more meaningful to investigate the finite-time stabilization problem than the classical asymptotical stabilization.In recent years, the finite-time stabilization of system (1) has been studied fairly extensively with various restrictions on the integrator powers and the system nonlinearities [12][13][14][15][16][17][18][19][20][21].In particular, [22] with constants  > 0 and  ∈ (−2/(2 + 1) 1 ⋅ ⋅ ⋅  −1 , 0).However, from both practical and theoretical points of view, it is somewhat restrictive to require system (1) to satisfy such restriction.Therefore, the following interesting problem is proposed: is it possible to relax the nonlinear growth condition?Under the weaker condition, can a finite-time stabilizing controller be designed? 2

Abstract and Applied Analysis
In this paper, by necessarily modifying the method of adding a power integrator and by successfully overcoming some essential difficulties such as the weaker assumption on the system growth, the appearance of the sign function, and the construction of a continuously differentiable, positivedefinite, and proper Lyapunov function, we will focus on solving the above problem.

Problem Statement and Preliminaries
The objective of this paper is to develop a recursive design method for globally finite-time stabilizing system (1) via state feedback under the following assumption.
where   's are defined as Remark 2. It is worth pointing out that Assumption 1, which gives the nonlinear growth condition on the system drift terms, encompasses the assumptions in the closely related works [14,21,22].To clearly show this, we would like to make the following comparisons to reveal the relationship between Assumption 1 and the counterparts in [14,21,22]; that is, Assumption 1 includes those as special cases.
In the remainder of this section, we present the following lemmas which play an important role in the design process.

Finite-Time Control Design
In this section, we will construct a continuous state feedback controller by applying the method of adding a power integrator.For simplicity, we denote sgn()||  ≜ []  for any  ∈  + and  ∈ .
with  1 > 0, . . .,  −1 > 0 being smooth, such that To complete the induction, at the th step, we choose the following Lyapunov function: where Noting that 2 −  +1   ≥ 1 and using a similar method as in [10],   can be shown to be  1 , proper and positive definite.Moreover, we can obtain where  = 1, . . .,  − 1.
Using ( 20)- (22), it follows that In order to proceed further, an appropriate bounding estimate should be given for the last three terms on the righthand side of inequality (23).This is accomplished in the following propositions whose technical proofs are given in the appendix.Proposition 7.There exists a positive constant  1 such that ( Now, it easy to see that the virtual controller where   ≥ ( −  + 1 +  1 +  2 +  3 ) 1/( +1   ) is a smooth function, renders This completes the inductive step.
Using the inductive argument above, we can conclude that at the th step, there exists a continuous state feedback controller of the form such that where

Stability Analysis
We state the main result in this paper.
Theorem 10.If Assumption 1 holds for system (1), under the continuous state feedback controller (30), then the following holds: (i) all the solutions of the closed-loop system are well defined on [0, +∞); (ii) the equilibrium  = 0 of the closed-loop system is globally finite-time stable.
(ii) Noticing that   = +∞, by using (34), and Lyapunov stability theorem [24], we know that the equilibrium  = 0 of the closed-loop -systems (30) and ( 33) is globally asymptotically stable.According to the definition of finite-time stability [11], if the global finitetime attractivity of the closed-loop system can be guaranteed, then the global finite-time stabilization result will be obtained.To this end, let us prove the global finite-time attractivity.First of all, by using Lemma 5, it is easy to see that So we have the following estimate: Let  = 2/(2 − ).With (36) and (31) in mind, by Lemma 4, it is not difficult to obtain that Therefore, by Lemma 3, we obtain that the equilibrium  = 0 of the closed-loop -systems (30) and (33) is globally finite-time stable.This together with the definitions of  *  's directly concludes that the globally finite-time stability of the closed-loop systems (1), (18), and (30) at the equilibrium  = 0.

Simulation Example
To illustrate the effectiveness of the proposed approach, we consider the following low-dimensional system: where  1 = 5/3 and  2 = 1.It is worth pointing out that although system (38) is simple, it cannot be globally finite-time stabilized using the design method presented in [14,21,22] because of the presence of both low-order term   ).
In the simulation, by choosing the initial values as  1 (0) = 1 and  2 (0) = −1, Figure 1 is obtained to demonstrate the effectiveness of the control scheme.

Conclusion
In this paper, a continuous state feedback stabilizing controller is presented for a class of high-order nonlinear systems under weaker condition.The controller designed preserves the equilibrium at the origin and guarantees the global finite-time stability of the systems.It should be noted that the proposed controller can only work well when the whole state vector is measurable.Therefore, a natural and more interesting problem is how to design output feedback stabilizing controller for the systems studied in the paper if only partial state vector being measurable, which is now under our further investigation.