Abstract
This paper addresses the problem of global finite-time stabilization by state feedback for a class of high-order nonlinear systems under weaker condition. By using the methods of adding a power integrator, a continuous state feedback controller is successfully constructed to guarantee the global finite-time stability of the resulting closed-loop system. A simulation example is provided to illustrate the effectiveness of the proposed approach.
1. Introduction
In this paper, we consider the following high-order nonlinear systems: where , are the system state and input, respectively; := and are positive odd integers, and are said to be the high orders of the system; , , are unknown continuous functions of all the states and the control input.
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 , 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 [2–10] 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 [12–21]. In particular, [22] solved the finite-time stabilization problem under the condition that satisfies with constants and . 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?
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, positive-definite, and proper Lyapunov function, we will focus on solving the above problem.
Notations. Throughout this paper, the following notations are adopted. denotes the set of all nonnegative real numbers and denotes the real -dimensional space. For any vector denote , , . denotes the set of all functions: , which are continuous, strictly increasing, and vanishing at zero; denotes the set of all functions which are of class and unbounded. A sign function is defined as follows: , if ; , if ; and , if . Besides, the arguments of the functions (or the functionals) will be omitted or simplified, whenever no confusion can arise from the context. For instance, we sometimes denote a function by simply , , or .
2. 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.
Assumption 1. For , there are smooth functions and constant such that 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. (i)In [14, 21], the system nonlinearities ’s are required to satisfy
where , , are functions. By (4), we get that . Furthermore, from , we have , . It means that
Letting be smooth functions and satisfying , we have
from this we can see that the main assumption in [14, 21] is a special case of Assumption 1 above. (ii)In [22], the system nonlinearities ’s are required to satisfy
with constants and . Obviously, when , inequality (4) degenerates to inequality (8). Moreover, the value range of in (4) is larger than that in (8). Thus, the main assumption in [22] is a special case of Assumption 1 above.
In the remainder of this section, we present the following lemmas which play an important role in the design process.
Lemma 3 (see [11]). Consider the nonlinear system
where is continuous with respect to on an open neighborhood of the origin . Suppose that there is a function defined in a neighborhood of the origin, real numbers and , such that (i) is positive definite on ; (ii), .
Then, the origin of (9) is finite-time stable with
for initial condition in some open neighborhood of the origin. If and is radially unbounded (i.e., as ), the origin of system (9) is globally finite-time stable.
Lemma 4 (see [6]). For , and being a constant, the following inequalities hold: If is odd, then
Lemma 5 (see [10]). If with , then for any
Lemma 6 (see [23]). Let be real variables; then for any positive real numbers , , and , one has where is any real number.
3. 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 for any and .
Step 1. Let and choose . Using (1) and Assumption 1, we have Obviously, the first virtual controller with being smooth results in
Step k . Suppose at step , there are a , proper and positive definite Lyapunov function , and a set of virtual controllers defined by with being smooth, such that
To complete the induction, at the step, we choose the following Lyapunov function: where
Noting that and using a similar method as in [10], can be shown to be , proper and positive definite. Moreover, we can obtain where .
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 right-hand 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 such that
Proposition 8. There exists a nonnegative smooth function such that
Proposition 9. There exists a nonnegative smooth function such that
Substituting (24)–(26) into (23) yields
Now, it easy to see that the virtual controller
where is a smooth function, renders
This completes the inductive step.
Using the inductive argument above, we can conclude that at the step, there exists a continuous state feedback controller of the form such that where
4. 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 ;(ii)the equilibrium of the closed-loop system is globally finite-time stable.
Proof. (i) Considering (18), system (1) can be equivalently transformed into a -system:
By the existence and continuation of the solutions, states are defined on , where the number may be infinite or not. The following analysis focuses on . Noting that is positive definite and radially unbounded, by (31), (32), and Lemma 4.3 in [24], it is not hard to obtain that there exist functions , , and such that
Since is a class function, then for any , one can always find a with such that , , which means that , . Suppose that is finite; then , which contradicts , . Hence, is well defined on , so is . (ii)Noticing that , by using (34), and Lyapunov stability theorem [24], we know that the equilibrium of the closed-loop -systems (30) and (33) is globally asymptotically stable. According to the definition of finite-time stability [11], if the global finite-time 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 . 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 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 .
5. Simulation Example
To illustrate the effectiveness of the proposed approach, we consider the following low-dimensional system: where and . 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 and high-order term . Choose ; then , , and . By Lemma 6, it is easy to get and . Clearly, Assumption 1 holds with and ; according to the design procedure proposed in Section 3, we can get where , , , and .
In the simulation, by choosing the initial values as and , Figure 1 is obtained to demonstrate the effectiveness of the control scheme.
(a) System states
(b) Control input
6. 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.
Appendix
Proof of Proposition 7. Noting that and , we have . Using (18), it follows from Lemma 5 that By (A.1) and Lemma 6, it can be obtained that where is a constant.
Proof of Proposition 8. According to (18), Assumption 1, and Lemma 4, it follows that
where and is a smooth function.
Using (A.3) and Lemmas 5 and 6, we obtain
where is a smooth function.
Proof of Proposition 9. Note that
where and .
Using (A.5), after simple calculations, it is not hard to obtain that for ,
By (22), (A.3), (A.6), and Lemmas 4 and 5, we get
where is a smooth function.
Noting that , by using Lemma 6, we have
where is a smooth function.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors thank the editor and the anonymous reviewers for their constructive comments and suggestions for improving the quality of the paper. This work has been supported in part by the National Nature Science Foundation of China under Grant 61073065 and the Key Program of Science Technology Research of Education Department of Henan Province under Grants 13A120016 and 14A520003.