Adaptive Finite-Time Stabilization of High-Order Nonlinear Systems with Dynamic and Parametric Uncertainties

Under the weaker assumption on nonlinear functions, the adaptive finite-time stabilization of more general high-order nonlinear systems with dynamic and parametric uncertainties is solved in this paper. To solve this problem, finite-time input-to-state stability (FTISS) is used to characterize the unmeasured dynamic uncertainty. By skillfully combining Lyapunov function, sign function, backstepping, and finite-time input-to-state stability approaches, an adaptive state feedback controller is designed to guarantee high-order nonlinear systems are globally finite-time stable.


Introduction
Since the concept of finite-time stability was introduced in [1], many efforts have been made to study the problem of finitetime stabilization because of faster convergence rates, higher accuracies, and better disturbance rejection properties.Based on the finite-time stability theorem in [2][3][4], some finite-time stabilization results have been achieved by combining finitetime stability with backstepping design method, for example, [5][6][7][8][9] and the references therein.
For system (1), when  is known, [10,11] studied finitetime stability, where the order of state   ( = 1, . . ., ) in (10) and (11) can be taken value in (0, 1/(  . . . −1 )) with    −1 = 1.The restrictive condition was relaxed by [12], in which all the states in the bounding condition were allowed to be of both low order and high order.When  is unknown, it is well known that adaptive technique is an effective way to deal with control problem of nonlinear systems with parametric uncertainty.Reference [13] developed a continuous adaptive finite-time controller with the bounding condition of   being an order equal to 1.The latest paper [14] weakened the growth condition by allowing the order greater than 0. However, there is no dynamic uncertainty considered by these papers.
The analysis and control problem of nonlinear systems with dynamic uncertainty have been an active research topic because dynamic uncertainty often arises from many different control engineering applications; see [15][16][17][18][19] and the references therein.In view of the benefits of finite-time convergence, finite-time stabilization of nonlinear systems with dynamic uncertainty has been regarded as one of the important issues.By characterizing dynamic uncertainty with finite-time input-to-state stability (FTISS), [20] constructed a finite-time adaptive state feedback controller for one-order nonlinear systems with dynamic and parametric uncertainties.Reference [21] gave the explicit definition of FTISS and 2 Mathematical Problems in Engineering developed a framework for the finite-time control analysis and synthesis based on FTISS.However, for more general high-order nonlinear systems, to the best of the authors' knowledge, no result on finite-time stabilization has been achieved until now.
Based on the above discussion, an interesting problem is put forward spontaneously: for more general high-order systems with dynamic and parametric uncertainties, ∈   0 is the unmeasured state, referred to as dynamic uncertainty,  ∈   is the unknown parameter vector,   :   ×   0 ×  + →  ( = 1, . . ., ) is an unknown and Lipschitz continuous function,  :   0 ×  →  is piecewise continuous with respect to  and Lipschitz continuous with respect to  1 , and (0, 0) = 0; under the weaker assumption than Assumption 1 in [14] (see Remark 5 for a detailed discussion), can a finite-time stabilized continuous controller be designed?
In this paper, an affirmative solution to this problem is given.To solve this problem, finite-time input-to-state stability (FTISS) is used to characterize the unmeasured dynamic uncertainty.By skillfully combining Lyapunov function, sign function, backstepping, and FTISS approaches and overcoming some obstacles emerging in design and analysis owing to the relaxed condition on nonlinear functions, an adaptive state feedback controller is designed to guarantee high-order nonlinear system (2) is globally finite-time stable.An example demonstrates the theoretical result.
This paper is organized as follows.Section 2 gives preliminaries.Sections 3 and 4 provide the design and analysis of adaptive finite-time state feedback controller, following a simulation example in Section 5. Section 6 concludes the paper.The appendix proves Proposition 6 of Section 3.

Mathematical Preliminaries
Some notations and lemmas are to be used throughout this paper.
+ stands for the set of all the nonnegative real numbers.For any vector set of all functions:  + →  + that are continuous, strictly increasing, and vanishing at zero, and K ∞ denotes the set of all functions that are of class K and unbounded.For simplicity, we sometimes denote a function (()) by () or .Sign function sgn() is defined as sgn() = 1 if  > 0, sgn() = 0 if  = 0, and sgn() = −1 if  < 0.
Definition 2 (see [21]).Consider a system where V is the input and  is continuous with respect to (, V).A continuous function () is called FTISS-Lyapunov function for system (4) if there exist K ∞ -functions  1 ,  2 , and  3 and a positive constant  such that In the remainder of this section, we list several lemmas that serve as the basis for the design of state feedback controller for system (2).Lemma 3 is finite-time stability theorem.Lemmas 4-8 are used to enlarge inequalities.Lemmas 9 and 10 are used to deal with sign function.

Finite-Time Convergence Analysis
3.1.Problem Formulation and Assumptions.The purpose of this paper is to achieve a global finite-time control design for high-order nonlinear system (2) with dynamic and parametric uncertainties.
To achieve the purpose, we need the following assumptions.
Remark 4. Assumption 1 implies that -subsystem is characterized by finite-time input-to-state stability (FTISS).The inequalities in Assumption 3 are the small-gain conditions.
Remark 5.The following discussions in order demonstrate that Assumption 2 encompasses and generalizes the existing results.
By the discussions, it is highlighted that this paper substantially extends the results of these papers; namely, for more general high-order nonlinear systems (2) with dynamic and parametric uncertainties, the finite-time control problem is to be solved under weaker condition (8).

Inductive
Step.We give this step by the following proposition.
Proposition 6. Suppose at Step  − 1, for system where where (A.12) in the Appendix) is continuous, and Proof.See the Appendix.

Finite-Time Convergence Analysis
We state the main result in this paper.
Remark 8. We estimate the settling time.In practice, although we do not know the real value of  (or ), we always have its range.Without loss of generality, we assume that 0 <  < Σ, where Σ > 0 is a constant.We consider two cases.
Remark 9. Compared with [14], due to the appearance of dynamic and parametric uncertainties, and the weaker condition on nonlinear functions, the main difficulty in this paper is how to skillfully combine Lyapunov function, sign function, backstepping, and FTISS approaches to give the design and rigorous analysis of finite-time controller.

Conclusions
By characterizing the dynamic uncertainty with FTISS, under the weaker assumption on nonlinear functions, the problem of adaptive finite-time stabilization for more general highorder nonlinear systems with dynamic and parametric uncertainties is solved.Some interesting problems are still remaining: (1) For system (2) with possibly nonvanishing disturbances and more general dynamic uncertainty, can a finite-time convergent controller be given?(2) How can we construct output feedback to stabilize system (2) in finite time?(3) In recent years, some results on stochastic nonlinear systems with SISS/SiISS dynamic uncertainty have been obtained, for example, [25][26][27][28][29][30][31][32][33][34][35][36] and the references therein, but these papers only consider the global asymptotic stabilization.An important problem is whether finite-time stabilization can be obtained for stochastic nonlinear systems with dynamic and parametric uncertainties.