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.
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 [
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 [
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 [
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 [
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
The following notations, definition, and lemmas will be used throughout the whole paper.
Consider stochastic nonlinear time-delay system
The equilibrium
For system (
For any smooth function
For any real numbers
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.
In this paper, we consider the following stochastic nonlinear time-delay system:
The control objective is to design an adaptive state-feedback controller to render system (
For
In terms of Lemma
In addition, for
For stochastic nonlinear time-delay systems, previous works such as [
Before giving the detailed design procedure, introduce the state coordinate transformation as
In the sequel, we aim to give the adaptive design procedure by combining Lyapunov-Krasovskii functionals with backstepping. The detailed process is divided into
Consider Lyapunov function
Then, constructing the Lyapunov-Krasovskii functional
We give the inductive step through a proposition.
If at step
See Appendix.
By exactly following the design procedure at Step
We summarize the main result in the following theorem.
For system (
In view of (
On one hand, from (
Furthermore, considering
We emphasize two main points. (i) For system (
In this section, we give a simulation example to verify the proposed scheme in Section
Consider stochastic nonlinear time-delay system
Then, by exactly following the design procedure in Section
In simulation, choose
The state responses of the closed-loop system (
The control input responses of the closed-loop system (
Adaptive control law of the closed-loop system (
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 (
Firstly, in terms of (
According to (
Then, substituting (
The authors declare that they have no conflicts of interest.
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).