- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 673869, 11 pages
State-Feedback Stabilization for Stochastic High-Order Nonlinear Systems with Time-Varying Delays
1School of Mathematics and Statistics, Anyang Normal University, Anyang 455000, China
2School of Civil and Architecture Engineering, Anyang Institute of Technology, Anyang 455000, China
Received 15 June 2013; Revised 3 September 2013; Accepted 12 December 2013; Published 30 January 2014
Academic Editor: Shawn X. Wang
Copyright © 2014 Fangzheng Gao et al. 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.
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.
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 , 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 . 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 , 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 .
Lemma 2 (see ). 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 ). For , , and being a constant, the following inequalities hold: If is odd, then
Lemma 4 (see ). For and positive real number , the following inequality holds: where for and for .
Lemma 5 (see ). 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 , 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 . Moreover, when and by choosing , Assumption 7 reduces to the following condition: which is equivalent to that in .
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  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 .
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 , can be shown to be , proper and positive definite. Moreover, we can obtain where , .
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
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 , 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 , strong stability theory  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 , 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.
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.
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
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.
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.
- H. J. Kushner, Stochastic Stability and Control, vol. 33 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1967.
- M. Krstić and H. Deng, Stabilization of Nonlinear Uncertain Systems, Communications and Control Engineering Series, Springer, New York, NY, USA, 1998.
- H. Deng, M. Krstić, and R. J. Williams, “Stabilization of stochastic nonlinear systems driven by noise of unknown covariance,” IEEE Transactions on Automatic Control, vol. 46, no. 8, pp. 1237–1253, 2001.
- Y. Liu, Z. Pan, and S. Shi, “Output feedback control design for strict-feedback stochastic nonlinear systems under a risk-sensitive cost,” IEEE Transactions on Automatic Control, vol. 48, no. 3, pp. 509–513, 2003.
- Y.-G. Liu and J.-F. Zhang, “Reduced-order observer-based control design for nonlinear stochastic systems,” Systems & Control Letters, vol. 52, no. 2, pp. 123–135, 2004.
- Y.-G. Liu and J.-F. Zhang, “Practical output-feedback risk-sensitive control for stochastic nonlinear systems with stable zero-dynamics,” SIAM Journal on Control and Optimization, vol. 45, no. 3, pp. 885–926, 2006.
- S.-J. Liu, Z.-P. Jiang, and J.-F. Zhang, “Global output-feedback stabilization for a class of stochastic non-minimum-phase nonlinear systems,” Automatica, vol. 44, no. 8, pp. 1944–1957, 2008.
- J. H. Zhang, M. F. Ren, Y. Tian, et al., “Constrained stochastic distribution control for nonlinear stochastic systems with non-gaussian noises,” International Journal of Innovative Computing, Information and Control, vol. 9, no. 4, pp. 1759–1767, 2013.
- H. Q. Wang, B. Chen, and C. Lin, “Adaptive neural tracking control for a class of stochastic nonlinear systems with unknown dead-zone,” International Journal of Innovative Computing, Information and Control, vol. 9, no. 8, pp. 3257–3269, 2013.
- W. Lin and C. Qian, “Adding one power integrator: a tool for global stabilization of high-order lower-triangular systems,” Systems & Control Letters, vol. 39, no. 5, pp. 339–351, 2000.
- X.-J. Xie and J. Tian, “State-feedback stabilization for high-order stochastic nonlinear systems with stochastic inverse dynamics,” International Journal of Robust and Nonlinear Control, vol. 17, no. 14, pp. 1343–1362, 2007.
- J. Tian and X. J. Xie, “Adaptive state-feedback stabilization for more general high-order stochastic nonlinear systems,” Acta Automatica Sinica, vol. 34, no. 9, pp. 1188–1191, 2008.
- X.-J. Xie and J. Tian, “Adaptive state-feedback stabilization of high-order stochastic systems with nonlinear parameterization,” Automatica, vol. 45, no. 1, pp. 126–133, 2009.
- X.-J. Xie and N. Duan, “Output tracking of high-order stochastic nonlinear systems with application to benchmark mechanical system,” IEEE Transactions on Automatic Control, vol. 55, no. 5, pp. 1197–1202, 2010.
- W. Li, Y. Jing, and S. Zhang, “Adaptive state-feedback stabilization for a large class of high-order stochastic nonlinear systems,” Automatica, vol. 47, no. 4, pp. 819–828, 2011.
- W. Li, X.-J. Xie, and S. Zhang, “Output-feedback stabilization of stochastic high-order nonlinear systems under weaker conditions,” SIAM Journal on Control and Optimization, vol. 49, no. 3, pp. 1262–1282, 2011.
- X.-J. Xie, N. Duan, and X. Yu, “State-feedback control of high-order stochastic nonlinear systems with SiISS inverse dynamics,” IEEE Transactions on Automatic Control, vol. 56, no. 8, pp. 1921–1926, 2011.
- W. Li, X. Liu, and S. Zhang, “Further results on adaptive state-feedback stabilization for stochastic high-order nonlinear systems,” Automatica, vol. 48, no. 8, pp. 1667–1675, 2012.
- K. Gu, V. L. Kharitonov, and J. Chen, Stability of Time-Delay Systems, Control Engineering, Birkhäauser, Berlin, Germany, 2003.
- Q. C. Zhong, Robust Control of Time-Delay Systems, Springer, London, UK, 2006.
- L. Wu, X. Su, P. Shi, and J. Qiu, “A new approach to stability analysis and stabilization of discrete-time T-S fuzzy time-varying delay systems,” IEEE Transactions on Systems, Man, and Cybernetics B, vol. 41, no. 1, pp. 273–286, 2011.
- X. Su, L. Wu, and P. Shi, “Senor networks with random link failures: distributed filtering for T-S fuzzy systems,” IEEE Transactions on Industrial Informatics, vol. 9, no. 3, pp. 1739–1750, 2013.
- X. Su, P. Shi, L. Wu, and S. Nguang, “Induced l2 filtering of fuzzy stochastic systems with time-varying delays,” IEEE Transactions on Systems, Man, and Cybernetics, vol. 43, no. 4, pp. 1251–1264, 2013.
- X. Su, P. Shi, L. Wu, and Y. Song, “A novel control design on discrete-time Takagi-Sugeno fuzzy systems with time-varying delays,” IEEE Transactions on Fuzzy Systems, vol. 21, no. 4, pp. 655–671, 2013.
- F. Li and X. Zhang, “Delay-range-dependent robust H-filtering or singular LPV systems with time variant delay,” International Journal of Innovative Computing Information and Control, vol. 9, no. 1, pp. 339–353, 2013.
- Z. Sun, Y. Liu, and X. Xie, “Global stabilization for a class of high-order time-delay nonlinear systems,” International Journal of Innovative Computing, Information and Control, vol. 7, no. 12, pp. 7119–7130, 2011.
- Z.-Y. Sun, X.-J. Xie, and Z.-G. Liu, “Global stabilisation of high-order nonlinear systems with multiple time delays,” International Journal of Control, vol. 86, no. 5, pp. 768–778, 2013.
- Z. Sun, X. Zhang, and X. Xie, “Continuous global stabilisation of high-order time-delay nonlinear systems,” International Journal of Control, vol. 86, no. 6, pp. 994–1007, 2013.
- F. Gao, F. Yuan, and Y. Wu, “Global stabilisation of high-order non-linear systems with time-varying delays,” IET Control Theory & Applications, vol. 7, no. 13, pp. 1737–1744, 2013.
- Y. Fu, Z. Tian, and S. Shi, “State feedback stabilization for a class of stochastic time-delay nonlinear systems,” IEEE Transactions on Automatic Control, vol. 48, no. 2, pp. 282–286, 2003.
- Y. Fu, Z. Tian, and S. Shi, “Output feedback stabilization for a class of stochastic time-delay nonlinear systems,” IEEE Transactions on Automatic Control, vol. 50, no. 6, pp. 847–851, 2005.
- S.-J. Liu, S. S. Ge, and J.-F. Zhang, “Adaptive output-feedback control for a class of uncertain stochastic non-linear systems with time delays,” International Journal of Control, vol. 81, no. 8, pp. 1210–1220, 2008.
- W. Chen, L. Jiao, J. Li, and R. Li, “Adaptive NN backstepping output-feedback control for stochastic nonlinear strict-feedback systems with time-varying delays,” IEEE Transactions on Systems, Man, and Cybernetics B, vol. 40, no. 3, pp. 939–950, 2010.
- W. Chen, J. Wu, and L. C. Jiao, “State-feedback stabilization for a class of stochastic time-delay nonlinear systems,” International Journal of Robust and Nonlinear Control, vol. 22, no. 17, pp. 1921–1937, 2012.
- X.-J. Xie and L. Liu, “A homogeneous domination approach to state feedback of stochastic high-order nonlinear systems with time-varying delay,” IEEE Transactions on Automatic Control, vol. 58, no. 2, pp. 494–499, 2013.
- J. Polendo and C. Qian, “A generalized homogeneous domination approach for global stabilization of inherently nonlinear systems via output feedback,” International Journal of Robust and Nonlinear Control, vol. 17, no. 7, pp. 605–629, 2007.
- J. Polendo and C. Qian, “A generalized framework for global output feedback stabilization of genuinely nonlinear systems,” in Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference (CDC-ECC '05), pp. 2646–2651, Sevilla, Spain, December 2005.
- B. Yang and W. Lin, “Nonsmooth output feedback design with a dynamic gain for uncertain systems with strong nonlinearity,” in Proceedings of the 46th IEEE Conference on Decision and Control (CDC '07), pp. 3495–3500, New Orieans, La, USA, December 2007.
- C. Qian, “A homogeneous domination approach for global output feedback stabilization of a class of nonlinear systems,” in Proceedings of the American Control Conference (ACC '05), vol. 7, pp. 4708–4715, June 2005.
- C. Qian and W. Lin, “Non-Lipschitz continuous stabilizers for nonlinear systems with uncontrollable unstable linearization,” Systems & Control Letters, vol. 42, no. 3, pp. 185–200, 2001.
- H. K. Khalil, Nonlinear Systems, Prentice-Hall, Upper Saddle River, NJ, USA, 1996.
- J. Kurzweil, “On the inversion of Lyapunov's second theorem on the stability of motion,” Transactions of the American Mathematical Society, vol. 2, pp. 19–77, 1956.