Abstract
This paper considers a more general stochastic nonlinear time-delay system driven by unknown covariance noise and investigates its adaptive state-feedback control problem. As a remarkable feature, the growth assumptions imposed on delay-dependent nonlinear terms are removed. Then, with the help of Lyapunov-Krasovskii functionals and adaptive backstepping technique, an adaptive state-feedback controller is constructed by overcoming the negative effects brought by unknown time delay and covariance noise. Based on the designed controller, the closed-loop system can be guaranteed to be globally asymptotically stable (GAS) in probability. Finally, a simulation example demonstrates the effectiveness of the proposed scheme.
1. Introduction
In control fields, stochastic noises (white noise, Levy noise, etc.) extensively occur in real plants including parameter perturbations, stochastic errors, and external environment variations. Therefore, the investigation of stochastic nonlinear systems is meaningful both theoretically and practically. During the past decades, the backstepping technique presented by [1] for stochastic nonlinear systems has been proven to be an effective design tool. Based on the backstepping technique, Lyapunov function method, and stochastic stability theory, recent years have witnessed considerable results on stochastic nonlinear systems; see [2–17] and the references therein. Particularly, adaptive backstepping technique, a recursive design procedure, has been extended to stochastic nonlinear systems with various uncertainties and many significant developments have been achieved in [9–17].
As is well-known time delays frequently exist in practical systems such as electrical networks, microwave oscillator, and chemical reactor systems. The existence of time delays may deteriorate system performance and cause instability. Therefore, the control and design for stochastic nonlinear time-delay systems has been one of the active research topics and already obtained fruitful results [18–33]. In [18–22], by using Lyapunov-Krasovskii functional method, the output-feedback stabilization problems were solved for stochastic nonlinear systems with time delays only presenting in system output. References [23, 24] considered the control problems of high-order stochastic nonlinear time-delay systems by introducing the adding a power integrator technique. However, the growth conditions assumed on system nonlinearities somewhat restrict the extension of the proposed control schemes in [18–24].
In recent years, how to weaken or remove the traditional nonlinear growth assumptions has been the main focus and difficulty in stochastic nonlinear time-delay systems control. In [25–27], the homogeneous domination approach was extended to stochastic nonlinear time-delay systems and the assumptions on nonlinearities in drift and diffusion vector fields were relaxed. In addition, the adaptive control technique has been applied with neural network approximation approach to weaken the assumptions on system time-delay nonlinearities in [28–32] and the related references. Recently, the traditional pure growth assumptions were further relaxed for stochastic nonlinear time-delay systems in [33] with the help of parameter-based controller design method. Despite the remarkable efforts obtained on relaxing the growth assumptions, the existing results including [25–33] all failed to remove these nonlinear growth assumptions.
On the other hand, it is well-known that the noise of unknown covariance is also a source of uncertainties, which may bring some negative effects on systems. In the past decades, the control problems for stochastic nonlinear systems driven by noise of unknown covariance have been studied in [34–36] by using Lyapunov functions and stochastic stability theorem. However, to the best of the authors’ knowledge, for stochastic nonlinear time-delay systems driven by unknown covariance noise, there are few related results. Motivated by the aforementioned discussions, a natural problem arises:
How to remove the growth assumptions on system nonlinearities and further stabilize stochastic nonlinear time-delay systems driven by unknown covariance noise?
This paper will focus on handling the above problem. The main contributions are listed as follows: (i) this paper considers a more general class of stochastic nonlinear systems disturbed by both unknown time delay and covariance noise. A distinctive novelty is that the growth assumptions imposed on time-delay nonlinearities in existing results are proven to be unnecessary and can be removed. (ii) By utilizing adaptive control technique and Lyapunov-Krasovskii functional method, the adverse effects brought by unknown covariance noise and time delay are compensated and an adaptive state-feedback controller is designed. It is proven that the designed controller can render the closed-loop system globally asymptotically stable (GAS) in probability.
The remainder of this paper is organized as follows. Section 2 gives the mathematical preliminaries. The design process and analysis procedure are given in Sections 3 and 4, respectively. In Section 5, a simulation example is presented. Section 6 concludes this paper. Some necessary proof is provided in Appendix.
2. Mathematical Preliminaries
The following notations, definition, and lemmas will be used throughout the whole paper.
Notations. denotes the set of all the nonnegative real numbers; denotes the -dimensional Euclidean space; denotes the family of all the functions with continuous th partial derivations; denotes the Euclidean norm of a vector or a square matrix; denotes the transpose of a given vector or matrix and denotes its trace when is square; denotes the space of continuous -valued functions on endowed with the norm defined by for ; stands for the family of all -measurable bounded -valued random variables ; 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; is the set of all functions , which are of for each fixed and decrease to zero as for each fixed .
Consider stochastic nonlinear time-delay systemwhere the initial data is for ; is a constant delay; is an -dimensional standard wiener process defined on a complete probability space , where is a sample space, is a -field, and is the probability measure with a natural filtration (i.e., ); the drift term and the diffusion term are locally Lipschitz functions with and . Obviously, system (1) admits a trivial solution . For any given , the differential operator along system (1) is defined as
Definition 1 (see [22]). The equilibrium of system (1) is said to be globally asymptotically stable (GAS) in probability if for any , there exists a function such that for any and , where .
Lemma 2 (see [22]). For system (1), if there exist functions and , such thatwhen , then there exists a unique solution on for system (1) and the equilibrium is GAS in probability with .
Lemma 3 (see [9]). For any smooth function , , there exists a smooth function such that .
Lemma 4 (see [37]). For any real numbers , , and continuous function , holds with being a real design constant.
3. State-Feedback Controller Design
In this section, we first present the problem to be investigated and a key lemma used in the design procedure. Then, based on adaptive backstepping technique, the recursive design procedure is given to construct an adaptive state-feedback controller.
3.1. Problem Formulation
In this paper, we consider the following stochastic nonlinear time-delay system:where and are system control input and measurable states, respectively; , , and ; is a constant time delay; is defined as in (1); is an unknown bounded nonnegative definite Borel measurable matrix function and denotes the infinitesimal covariance function of the driving noise ; for , the drift terms are functions with and the diffusion terms are smooth functions with .
The control objective is to design an adaptive state-feedback controller to render system (5) to be GAS in probability by removing the growth assumptions on delay-dependent . To make this feasible, we first show a lemma, which plays a key role in designing the ideal controller.
Lemma 5. For functions , , with , there exist nonnegative smooth functions such that
Proof. In terms of Lemma in [38], it holds thatwhere and are nonnegative smooth functions. Then, there must exist sufficiently large nonnegative smooth functions such that . Considering this with (7), then (6) can hold by choosing .
In addition, for , since and are smooth, using Lemma 3, one getswhere () are smooth functions.
Remark 6. For stochastic nonlinear time-delay systems, previous works such as [18–33] all gave the growth conditions similar to (6) through imposing assumptions on system nonlinearities. In this paper, Lemma 5 proves that these assumptions are unnecessary and can be removed, which is a main distinctive feature of this paper. In addition, we give an example to show Lemma 5 can hold. Considering , one gets , , and , . Then, it can be verified that and . Thus, Lemma 5 is satisfied with and .
3.2. Design of State-Feedback Controller
Before giving the detailed design procedure, introduce the state coordinate transformation aswhere are virtual control laws to be determined and is the estimate of with the formAccording to (5), (9), and Itô’s differentiation formula, it is easy to getwhere , , and denote and , respectively; and .
In the sequel, we aim to give the adaptive design procedure by combining Lyapunov-Krasovskii functionals with backstepping. The detailed process is divided into steps.
Step 1. Consider Lyapunov function for system (5), where is the estimate error of and is the adaptive gain constant. Then, by means of (2), (8), (10)-(11), Lemmas 4 and 5, and Itô’s rule, it can be verified thatwhere with being a design constant; and .
Then, constructing the Lyapunov-Krasovskii functionaland the first virtual control lawtogether with (2) and (12), one yieldswhere is a positive design constant.
Step i (). We give the inductive step through a proposition.
Proposition 7. If at step there exist a series of virtual control laws making the Lyapunov-Krasovskii functional satisfywhere is a positive design constant and , then, there exists a virtual control lawsuch that the th Lyapunov-Krasovskii functionalsatisfieswhereand , , and are positive design constants; , , and , are nonnegative continuous functions and .
Proof. See Appendix.
Step n. By exactly following the design procedure at Step , one can obtain the adaptive state-feedback controllerwhich renders the Lyapunov-Krasovskii functionalto satisfywhere and are known nonnegative continuous functions; , , , , , , , and are positive design constants with
4. Stability Analysis
We summarize the main result in the following theorem.
Theorem 8. For system (5), there exists an adaptive control law (21) such that (i) the closed-loop system consisting of (5), (9), (14), (17), and (21) is GAS in probability; (ii) and .
Proof. In view of (23), it is obvious that is on and . In addition, the inequality (4) in Lemma 2 is satisfied with , which is a -class function with . In the sequel, we focus on verifying inequality (3) in Lemma 2.
On one hand, from (22) and , one hasLet ; obviously, and hold. On the other hand, by the mean value theorem, one can achievewhere and . Define ; then . Hence, inequality (3) in Lemma 2 is satisfied. Thus, one concludes from Lemma 2 that (i) holds with .
Furthermore, considering (), , and (9), one further gets . In addition, from (22)-(23), it holds that converges a.s. to a finite limit as ; that is, . Hence, conclusion (ii) is proved, which completes the proof of Theorem 8.
Remark 9. We emphasize two main points. (i) For system (5), this paper completely removes the growth assumptions imposed on system time-delay nonlinearities. (ii) The construction of adaptive controller (21) is difficult and the proof of Theorem 8 is not a trivial work.
5. A Simulation Example
In this section, we give a simulation example to verify the proposed scheme in Section 3.
Example 1. Consider stochastic nonlinear time-delay systemwhere is defined as in (5) and is a time delay. It can be verified that , , , , and in Lemma 5 and (8).
Then, by exactly following the design procedure in Section 3, one can get the adaptive controller with the formwhere , , , , , , , , , , , and are positive design constants; ; ; ; with , , , , , ; with ; with ; ; and are all nonnegative continuous functions.
In simulation, choose , , , , , , , , , , , , , and . The initial values are given by , , and . Figures 1–3 demonstrate the effectiveness of the control scheme.
6. Conclusions
This note solves the adaptive state-feedback control for stochastic nonlinear time-delay systems driven by unknown covariance noise. The traditional assumptions imposed on system nonlinearities are removed and the negative effects generated by unknown covariance noise are eliminated by using Lyapunov-functionals and adaptive backstepping technique. In addition, an adaptive state-feedback controller is designed to enable the closed-loop system to be GAS in probability. One more problem under investigation is how to solve the output-feedback control problem for system (5).
Appendix
Proof of Proposition 7. Firstly, in terms of (2), (11), (16), and (18), one arrives atTo proceed further, we try to estimate the third-eighth terms in the right-hand side of (A.1).
According to (9), one haswhere . Using (A.2) and Lemmas 4 and 5, it is easy to verify thatwherewith . Now, we turn to give the estimate procedure. By applying (9)-(10), (A.2)-(A.3), and Lemma 4, it can be verified that where , , , , , , and are positive design constants; ; , ; for , for and for ; ; ; ; and .
Then, substituting (A.5) into (A.1) yieldswhere . Hence, by choosing as (17), as (20), and , one can finally get (19). This completes the proof.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work is supported by National Natural Science Foundation of China (nos. 61573172, 61503166), 333 High-Level Talents Training Program in Jiangsu Province (no. BRA2015352), Program for Fundamental Research of Natural Sciences in Universities of Jiangsu Province (no. 15KJB510011), and Shandong Province Natural Science Foundation of China (no. ZR2016AL05).