Abstract
This paper investigates the problem of state-feedback stabilization for a class of stochastic high-order nonlinear systems with time-varying delays. Under the weaker conditions on the power order and the nonlinear growth, by using the method of adding a power integrator, a state-feedback controller is successfully designed, and the global asymptotic stability in the probability of the resulting closed-loop system is proven with the help of an appropriate Lyapunov-Krasovskii functional. A simulation example is given to demonstrate the effectiveness of the proposed design procedure.
1. Introduction
It is well known that stochastic modeling has come to play an important role in the field of engineering where stochastic differential equations have been applied for the analysis and control of stochastic systems. As a consequence, the study of stochastic nonlinear systems has drawn an increasing attention in the past decades [1–9]. Especially, by applying the so-called method of adding a power integrator [10], which can be viewed as the latest achievement of the traditional backstepping method, a series of research results for stochastic high-order nonlinear systems have been achieved [11–18].
On the other hand, time delay extensively exists in various engineering systems, such as electrical networks, microwave oscillator, and hydraulic systems. It is well known that the existence of time delay often deteriorates the control performance of systems and even causes the instability of closed-loop systems [19]. Therefore, the control design and stability analysis of time-delay systems has been an active research area within the automation and control community. In recent years, by employing the Lyapunov-Krasovskii method or Lyapunov-Razumikhin method to deal with the time delay, control theory and techniques for time-delay linear systems were developed and many advanced methods such as time-delay partitioning method and input-output method have been established; see, for instance, [20–25] and reference therein. However, for the time-delay nonlinear systems, there exist many open problems which are so important and interesting at least from the theoretical point of view and have been paid careful attention; see, for example, the lastly published papers [26–29].
In this paper, we consider a class of stochastic high-order systems with time-varying delays described by where is system state vector and is control input, respectively; , ; is the time-varying delay satisfying for a known constant ; and are positive odd integers, and are said to be the high orders of the system; , , are disturbed virtual control coefficients; is an -dimensional independent standard Wiener process defined on a complete probability space with being a sample space, being a -field, being a filtration, and being a probability measure. The functions and are assumed to be locally Lipschitz with and .
It is worth pointing out that, when and , system (1) degenerates to the stochastic time-delay system in strict-feedback form, whose study on feedback control problem has achieved great development in recent years; see, for example, [30–33] and reference therein. However, when , the control design for system (1) is challenging because on one hand, the presence of the fractional power makes system (1) not only nonsmooth but also having the uncontrollable unstable linearization, and on the other hand, the existence of time-delay effect makes the common assumption on the high-order systems nonlinearities infeasible and which conditions should be placed to the nonlinearities remains unanswered. To the best of the authors’ knowledge, up to now only [34, 35] two papers considered the state-feedback stabilization problem of system (1) under the strict power order restriction () and some rigorous growth condition. However, from both practical and theoretical points of view, it is somewhat restrictive to require system (1) to satisfy such restrictions. Immediately, the following interesting questions are proposed. Is it possible to relax the power order restriction and the nonlinear growth condition? Under these weaker conditions, how to design a state-feedback controller such that the closed-loop system is globally asymptotically stable in probability?
Inspired by the recent works [15, 29], we will solve the aforementioned problems here. The contribution of this paper is highlighted as follows.(i)By comparison with the existing results in [34, 35], the power order restriction is completely removed, the nonlinear growth condition is largely relaxed, and a much weaker sufficient condition is given.(ii)Different from the result in the deterministic case [29], in which the controller need not satisfy the locally Lipschitz condition, for stochastic system, to guarantee the existence and uniqueness of the solution, the designed controller must guarantee that the closed-loop system satisfies the locally Lipschitz condition. Hence, how to construct the controller constitutes one of the main contributions.(iii)Due to the appearance of high-order, time-varying delay and nonlinear assumption, how to construct an appropriate Lyapunov-Krasovskii functional for system (1), especially under the assumption that and are not bounded linearly, is a nontrivial work.
The remainder of this paper is organized as follows. Section 2 offers notations and some preliminary results. Section 3 presents the control design procedure and the main result, while Section 4 gives a simulation example to illustrate the theoretical finding of this paper. Finally, concluding remarks are proposed in Section 5.
2. Notations and Preliminary Results
Throughout this paper, the following notations are adopted. denotes the set of all nonnegative real numbers and denotes the real -dimensional space. and positive integers}, and are positive odd integers, and . For a given vector or matrix , denotes its transpose, denotes its trace when is square, and is the Euclidean norm of a vector . denotes the space of continuous -value functions on endowed with the norm defined by for . denotes the family of all -measurable bounded -valued random variables . denotes the set of all functions with continuous th partial derivatives; denotes the family of all nonnegative functions on which are in and in ; denotes the family of all functions which are in the first argument and in the second argument.
Consider the following stochastic time-delay system: with initial data , where is a Borel measurable function; is an -dimensional independent standard Wiener process defined on a complete probability space . The functions and are locally Lipschitz with and .
Definition 1 (see [32]). For any given associated with system (2), the differential operator is defined as
Lemma 2 (see [32]). For system (2), if there is a function and two class functions and , such that
where is a nonnegative continuous function, then(i)there exists a unique solution on ;(ii)the equilibrium is globally stable in probability and the solution satisfies
Especially, if is a positive definite function, then the equilibrium is globally asymptotically stable (GAS) in probability.
In the remainder of this section, we list several lemmas that serve as the basis for the development of a state-feedback controller for system (1). They are the key tools for adding a power integrator technique.
Lemma 3 (see [36]). For , , and being a constant, the following inequalities hold: If is odd, then
Lemma 4 (see [37]). For and positive real number , the following inequality holds: where for and for .
Lemma 5 (see [38]). Let and be real variables; then for any positive real numbers , , and , one has where is any real number.
3. Controller Design and Main Result
The objective of this paper is to design a state-feedback controller of the form such that system (1) is GAS in probability at the origin.
To this end, the following assumptions regarding system (1) are imposed.
Assumption 6. For , there are positive constants and such that
Assumption 7. For , there are nonnegative smooth functions , and constants such that where and . Let and , . Meanwhile, one of the following conditions should be satisfied:(i) is required if or , ;(ii) is required if condition (i) does not hold.
Remark 8. Assumption 7 is a generalization of the homogeneous growth condition introduced in [15], where . The assumption plays an essential role in ensuring the locally Lipschitz condition of the closed-loop system, which guarantees the existence and uniqueness of the solution. Moreover, it is worth pointing out that Assumption 7 encompasses the assumption in existing results [34, 35]. Specifically, when , , and , Assumption 7 can be seen to reduce to the condition used in [35]. Moreover, when and by choosing , Assumption 7 reduces to the following condition: which is equivalent to that in [34].
Remark 9. It should be emphasized that the available state-feedback results for high-order stochastic time-delay system (1) in [34, 35] are all based on the strict power order restriction (). In this paper, for , we only need a much more general condition: , , which means that the power order restriction is completely removed in this paper.
Without loss of generality, we assume that with being any even integer and being any odd integer, under which and with the definition of in Assumption 7, we know that . For general , a similar technique in [39] can be used.
Choose to satisfy and , . With Assumption 7, is chosen in the following manner.(a)Choose if condition (i) of Assumption 7 is satisfied.(b) can be chosen as any satisfying if condition (ii) of Assumption 7 holds.
Now, we are in the position to give the controller design procedure by applying the method of adding a power integrator. To simplify the deduction procedure, we sometimes denote by .
Step 1. Define and choose the Lyapunov functional , where . With the help of Assumptions 6 and 7 and Lemma 5, we have where × × and × × are nonnegative smooth functions.
Obviously, the first virtual controller with being smooth results in
Remark 10. It is necessary to mention that in the first design step, the functions and have been provided with explicit expression in order to deduce the completely explicit virtual controller. However, in the later design steps, sometimes for the sake of briefness, we will not explicitly write out the functions which are easily defined.
Inductive Step. Suppose at step that 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 th step, we choose the following Lyapunov functional candidate where
Noting that , using a similar method as in [40], can be shown to be , proper and positive definite. Moreover, we can obtain where , .
Using (19), (20), and (21), it follows that
In order to proceed further, an appropriate bounding estimate should be given for each term on the right-hand side of (22). This is accomplished in the following propositions whose technical proofs are given in the Appendix.
Proposition 11. There exists a positive constant such that
Proposition 12. There exists a nonnegative smooth function such that
Proposition 13. There exists a nonnegative smooth function such that
Proposition 14. There exists a nonnegative smooth function such that
Proposition 15. There exists a nonnegative smooth function such that
Proposition 16. There exists a nonnegative smooth function such that
Proposition 17. There exists a nonnegative smooth function such that
Substituting (23)–(29) into (22) yields Now, it easy to see that the virtual controller renders
Finally, when , is the actual control. By choosing the actual control law, we get where
We are now ready to state the main theorem of this paper.
Theorem 18. If Assumptions 6 and 7 hold for stochastic high-order nonlinear time-delay system (1), under the state-feedback controller (33), then the closed-loop system has a unique solution on and the equilibrium at the origin of the closed-loop system is GAS in probability.
Proof . We prove Theorem 18 in two steps.
Step 1. We first prove that in (1) satisfies the locally Lipschitz condition.
From (17), (33), and , one has
where . Then, for ,
where
By the definition of , we know that , which implies that
from which, (37), (38), and ’s being smooth, we obtain that is continuous. Therefore, is .
Since and are assumed to be locally Lipschitz, so the system consisting of (1), (17), and (33) satisfies the locally Lipschitz condition.
Step 2. Then we prove that the origin of closed-loop system is GAS in probability.
From the definitions of ’s, we easily see that , as the function of , is positive, definite, and radially unbounded. Then, by (34), (35), and Lemma 4.3 in [41], there exist functions , , and such that
With the help of Lemma 2, we conclude that the closed-loop system has a unique solution on , and the equilibrium is GAS in probability. This together with the definitions of ’s directly concludes that the origin of system (1) is also GAS in probability. Thus, the proof is completed.
Remark 19. In the deterministic work [29], strong stability theory [42] guarantees the existence of a solution; hence, the locally Lipschitz condition need not to be satisfied. While for stochastic system, according to the existing stochastic stability theory, to guarantee the existence and uniqueness of the solution, the designed controller must guarantee that the closed-loop system satisfies the locally Lipschitz condition. What should be emphasized is that how to guarantee the constructed Lyapunov functional being and the closed-loop systems satisfying the locally Lipschitz condition simultaneously is one of the difficulties in this paper.
Remark 20. Since Itô stochastic differentiation involves not only the gradient but also the higher order Hessian term in the Lyapunov design procedure, many tedious nonlinear terms will arise in the design process, especially in the case of the appearance of high order, time-varying delay and nonlinear assumption. How to deal with these terms is another difficulty of this paper.
Remark 21. From the above design procedure, we can see that the upper bound of the change rate of time delays has important impact on the control effort. To keep the control effort within the certain range, the upper bound of the change rate of time delays cannot be arbitrarily close to 1, which should be considered in practical engineering design.
4. Simulation Example
To illustrate the effectiveness of the proposed controller, we consider the following low-dimensional system where .
It is evident that, even though system (41) is simple, it could not be globally asymptotically stabilized using the design method presented in [34, 35] because of the unsatisfiability of the power order restriction. However, if we choose which together with and implies that , it is easy to get , , and . Clearly, Assumption 7 is satisfied. Moreover, noting that and , the controller proposed in this paper is applicable.
By choosing and , following the design procedure given in Section 3, we can get where , , , × × , × , and .
In the simulation, we choose the initial values and . Figure 1 gives the responses of (41) and (42). From the figure, we can see that under the constructed controller, the solution process of the closed-loop system asymptotically converges to zero almost surely. We can also see that a little larger control effort is needed at the beginning. As mentioned in [32], when there exist stochastic disturbances and time delays, the effort of a controller designed based on the backstepping method is bigger than the common case, to which attention should be paid in practical use.
(a) System states
(b) Control input
5. Conclusion
This paper deals with the state-feedback stabilization problem for a class of stochastic high-order nonlinear time-delay systems under weaker conditions. The designed state-feedback controller ensures that the origin of the closed-loop system is GAS in probability. 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 stabilization controller for the systems studied in the paper if only partial state vector is being measurable, which is now under our further investigation.
Appendix
Proof of Proposition 11. There are two different cases for the proof.
If , using (17), it follows from Lemma 3 that
Noting that , by (A.1), from Assumptions 6 and 7 and Lemma 5, it can be obtained that
where is a positive constant.
If , from (17) and Lemma 4, we can get
where and are positive constants.
By applying Lemma 5, we have
where is a positive constant.
By choosing , with (A.2) and (A.4), we can obtain that
Proof of Proposition 12. According to (17), Assumption 7, and Lemma 3, it follows that
where and is a nonnegative smooth function.
By using (A.6) and Lemma 5, we have
where is a nonnegative smooth function.
Proof of Proposition 13. Note that
where
Then, for , we have
By (17), (21), (A.6), (A.10), and Assumptions 6 and 7, we get
where , , and are nonnegative smooth functions.
It is noted that satisfies and and by Lemma 5, we have
where is a nonnegative smooth function.
Proof of Proposition 14. From (17), Assumption 7, and Lemma 3, it follows that
where and is a nonnegative smooth function.
According to (17) and (21), we have
where , , and are nonnegative smooth functions.
Noting that and , by applying Lemma 5 to the above inequality, we have
where is a nonnegative smooth function.
In a similar way, Propositions 15–17 can be proved and they are omitted here.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
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. The authors thank the editor and the anonymous reviewers for their constructive comments and suggestions for improving the quality of the paper.