Mathematical Problems in Engineering
Volume 2016 (2016), Article ID 5702182, 11 pages
http://dx.doi.org/10.1155/2016/5702182

Research Article

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

Meng-Meng Jiang and Xue-Jun Xie

Institute of Automation, Qufu Normal University, Shandong 273165, China

Received 1 May 2016; Accepted 31 July 2016

Academic Editor: Ricardo Aguilar-López

Copyright © 2016 Meng-Meng Jiang and Xue-Jun Xie. 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.

Abstract.

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.

1. Introduction

Since the concept of finite-time stability was introduced in [1], many efforts have been made to study the problem of finite-time stabilization because of faster convergence rates, higher accuracies, and better disturbance rejection properties. Based on the finite-time stability theorem in [2–4], some finite-time stabilization results have been achieved by combining finite-time stability with backstepping design method, for example, [5–9] and the references therein.

Recently, more attention of finite-time stability has been focused on a family of high-order nonlinear systems of the form

where is the control input, is the measured state, and denotes the unknown parameter vector. For , is an unknown and Lipschitz continuous function. : and are odd integers, . System (1) is called high-order system if there exists at least , .

For system (1), when is known, [10, 11] studied finite-time stability, where the order of state in (10) and (11) can be taken value in with . 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 [15–19] 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 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,

is the unmeasured state, referred to as dynamic uncertainty, is the unknown parameter vector, is an unknown and Lipschitz continuous function, is piecewise continuous with respect to and Lipschitz continuous with respect to , and ; 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.

2. 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 , denote . For , . A function is if it is continuous and is if it is continuously differential. denotes the set of all functions: that are continuous, strictly increasing, and vanishing at zero, and denotes the set of all functions that are of class and unbounded. For simplicity, we sometimes denote a function by or . Sign function is defined as if , if , and if .

Definition 1 (see [13]). Consider a system

where is continuous with respect to on an open neighborhood of the origin . The equilibrium of the system is (local) finite-time stable if it is Lyapunov stable and finite-time convergent in a neighborhood of the origin. By “finite-time convergence” one means the following: if, for any initial condition at any given initial time , there is a settling time , such that every solution of system (3) is defined with for and , for any . When , the origin is a globally finite-time stable equilibrium.

Definition 2 (see [21]). Consider a system

where is the input and is continuous with respect to . A continuous function is called FTISS-Lyapunov function for system (4) if there exist -functions , , and 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.

Lemma 3 (see [13]). Suppose that, for system (3), there is a positive-definite function (defined on , where is a neighborhood of the origin), and a real number and , such that (along the trajectory) on . Then is locally finite-time convergent or equivalently becomes locally in finite time, with its settling time for a given initial condition in a neighborhood of the origin in .

Lemma 4 (Young’s inequality). Let real numbers and satisfy ; then for any , and any given positive number , .

Lemma 5 (Jensen’s inequality). If , then for any , .

Lemma 6 (see [22]). If , then , for any , .

Lemma 7 (see [22]). If , then , for any , .

Lemma 8 (see [23]). For a continuous function with , , there exist smooth functions , , , and , such that , .

Lemma 9 (see [24]). If , , then for any , .

Lemma 10 (see [24]). is continuously differentiable, and , where , . Moreover, if , , then .

3. 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.

Assumption 1. The -subsystem has an FTISS-Lyapunov function that satisfies

where , are constants, and is a function. Moreover, and are functions such that

Assumption 2. For each , there is a constant satisfying and known nonnegative function and nonnegative function with and such that

where is an unknown constant, , , , , and

Assumption 3. We assume that and .

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.
(i) When and is known, Assumption 2 includes the growth condition in [11]

as its special case (i.e., in (8)), as well as the growth condition in [10]

as a special case (i.e., and is a constant).
From , , , it is easy to see that , which implies that the power in condition (8) defined by can take any value in an interval , while, for [10, 11], the powers only take values in .
(ii) When and is unknown, (8) is reduced to Assumption 1 in [14]:

(iii) When for all , system (2) becomes

which is studied by [20, 21].

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).

3.2. Design of Adaptive Finite-Time Controller

In what follows, we denote , which is unknown because is unknown, is the estimate of , and is the estimation error. Denote , , . For simplicity, denote , where , is sign function whose definition is in the notations explained in Section 2, , and is obviously if .

To give the design of controller, we first define the parameters recursively as

From (14), it follows that

Besides, from (15), it leads to ; then

Secondly, we introduce the following coordinate transformation:

where , , are positive functions to be specified later. For the sake of consistency, we let and .

Finally, to obtain the detailed expression of , we determine by induction.

Initial Step. Consider

Take ; then is positive-definite and . By (15), (17), Lemmas 4 and 6, Assumption 2, and ,

where is a nonnegative function and . By Lemma 8, we choose a positive function dominating the following function:

where is a function to be determined. Because of and Assumption 3, is a continuous function. After some manipulations, we have

where , , and will be determined later.

Inductive Step. We give this step by the following proposition.

Proposition 6. Suppose at Step , for system

there is a function which is positive-definite with respect to and positive function such that

where

is . Then, for system

the th function is and positive-definite with respect to , and one can find a positive function (see (A.14) in the Appendix) such that

where , (see (A.12) in the Appendix) is continuous, and

Proof. See the Appendix.

At Step , (27) holds with . Hence, by choosing , we can get

with an appropriate function and

such that

4. Finite-Time Convergence Analysis

We state the main result in this paper.

Theorem 7. The solutions of the closed-loop system (2), (29), and (30) are bounded, and the trajectory is finite-time convergent to the origin .

Proof. Define

From Assumption 3, is , , and, for , , , it follows that

The proof is divided into two parts.
Step  1. We show that the trajectories of the closed-loop system (2), (29), and (30) are bounded.
Take a Lyapunov function as

where is a continuous nondecreasing function with . It follows from (7) that when , ; when , . Then we observe from (30), (31), and (34) that

According to (33) and Lemma 8, we can find a desired function and a function such that

Substituting (36) into (35) leads to

which implies the boundedness of , , and .
Step  2. We show that the trajectories are finite-time convergent.
Consider , which is positive-definite with respect to . Then

At first, we consider local finite-time convergence in a small neighborhood around . It is easy to see that, locally around , , , and then , which implies that when is small enough, there is a constant such that . Thus, locally around , by (38), one has

Note that

where is continuous with because . Substituting (40) into (39), it is easy to obtain

where is a positive constant. By (15) and Lemma 9, then , which together with (24) and Lemma 5 mean that . Hence, (41) becomes

where is continuous with . Because is continuous and , it is obvious that there is a constant such that if . By Lemma 5 and , choosing , , we have . Thus, in a neighborhood , (42) becomes

which implies the local finite-time convergence in by Lemma 3.
We consider the global finite-time convergence. The finite-time convergence in has been achieved. We now study the situation outside . When the initial condition is outside , one has . It is easy to see that there is a constant such that . Let be the first time that intersects the boundary of . Using (37) and the above-mentioned argument, for any , one has

If do not enter in finite time, then . When , , while is a finite constant, then (44) leads to a contradiction. So will reach in finite time.

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 , where is a constant. We consider two cases.
Case  1 (). By (43), we have

Case  2 (). Before the state reaches , from (34) and (37), it follows that ; then . Therefore, by (44), will reach within :

After , it will stay in , and it will take

to reach the origin. Therefore, in this case, the settling time can be estimated as .

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.

5. A Simulation Example

Consider a simple system

where is an unknown parameter and is the unmeasurable dynamic uncertainty.

Choose ; then , . By (48), we have , , , , , , and . Choose ; then ; -subsystem is obviously finite-time input-to-state stable with . Hence, Assumptions 1–3 hold.

Take ; then . Following the design procedure in Section 3, one can obtain the adaptive finite-time controller

Choosing the initial conditions , , , and , we estimate the settling time. Since , , one has , which is in . According to Case in Remark 8, by , , one has .

With , Figure 1 shows the trajectories of , , , , , and one can see that , , reach the origin within .

Figure 1: Trajectories of , , , , .

6. 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 high-order 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, [25–36] 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.

Appendix

Proof of Proposition 6. We first prove that is . From (15) and being positive and and being , then is , so is .
Secondly, we prove that is positive-definite with respect to . When , by (15), (28), and Lemmas 7 and 9, one has

When , it can be shown that (A.1) also holds in a similar way.
From (A.1), the definition of , and the positive definiteness of with respect to , it is easy to see that and, for fixed , if and only if , which implies that is positive-definite with respect to .
Finally, we prove (27). From the definition of and (23), it follows that

We estimate the last three terms on the right-hand side of (A.2).
(i) By (15), (16), (17), (24), and Lemmas 4 and 9, one has

where is a constant dependent on , , , .
(ii) For , from (15), (17), (24), and Lemmas 4, 8, and 10, the following holds:

where is a general nonnegative function that may be implicitly changed in various places and is a nonnegative function. From (15), (17), (28), (A.4), and Lemma 9, it follows that

For , by Assumption 2, one has

where is a nonnegative function vanishing at the origin. For , by (15), (17), (24), (A.6), and Lemmas 6 and 9,

From (16), (A.5), (A.7), Lemma 4, and ,

where and are nonnegative functions with . It follows from (15), (16), (17), (24), (28), (A.6), and Lemmas 4 and 6 that

where and are nonnegative functions with . Combining (A.8) with (A.9) leads to

where ,.
(iii) For the last term of (A.2), from (17), (28), and Lemma 8, there is a nonnegative function such that . By (24), (25), and Lemma 4,

where is nonnegative function;

Substituting (A.3), (A.10), and (A.11) into (A.2), one has

Using Lemma 8, one can choose a positive function

such that

where . Substituting (A.15) into (A.13) leads to (27).

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (nos. 61273125 and 61673242).

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