The Adaptive Neural Control for a Class of High-Order Uncertain Stochastic Nonlinear Systems

Qin, Xiaoyan

doi:https://doi.org/10.1155/2018/4620125

Mathematical Problems in Engineering

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Abstract Introduction Preliminaries Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Mathematical Theories and Applications for Nonlinear Control Systems

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Research Article | Open Access

Volume 2018 | Article ID 4620125 | https://doi.org/10.1155/2018/4620125

The Adaptive Neural Control for a Class of High-Order Uncertain Stochastic Nonlinear Systems

Xiaoyan Qin¹

Academic Editor: Weihai Zhang

Received23 May 2018

Accepted11 Jul 2018

Published09 Sept 2018

Abstract

This paper studies the problem of the adaptive neural control for a class of high-order uncertain stochastic nonlinear systems. By using some techniques such as the backstepping recursive technique, Young’s inequality, and approximation capability, a novel adaptive neural control scheme is constructed. The proposed control method can guarantee that the signals of the closed-loop system are bounded in probability, and only one parameter needs to be updated online. One example is given to show the effectiveness of the proposed control method.

1. Introduction

Ever since the stochastic stability theory was established by [1–3], the design and analysis of backstepping controller for stochastic nonlinear systems have achieved remarkable development in recent years; see [4–20] and the references therein. Based on the backstepping technique, Pan and Basar [8] firstly studied a class of stochastic nonlinear systems under a risk-sensitive cost criterion. Then, by combining backstepping technique with different nonlinear control methods, [9–13] obtained the state-feedback stabilization results of stochastic nonlinear systems in various structures. In the case of system states being unmeasurable, [14–19] further studied the problem of the output-feedback stabilization for stochastic nonlinear systems with the help of observer design. In addition, by applying the backstepping design and Lyapunov stability analysis, the finite-time control with fast convergence rate has been achieved for stochastic nonlinear systems in [10, 20, 21].

Note that when stochastic nonlinear system is of high-order, it may be nonsmooth and in general not stabilizable. How to deal with this problem is difficult. To handle this case, [9–13, 17, 20] have done remarkable work on stochastic high-order nonlinear systems and obtained different control results. Particularly, the homogeneous domination approach was extended to stochastic nonlinear system in [17], which provides an effective design methods for high-order stochastic nonlinear systems. However, the above-mentioned results were subjected to the nonlinear dynamics models which are known exactly or unknown parameters existing linearly. Thus, these results cannot be used for the stochastic systems with structured uncertainties. Naturally, one raises the problem of how to design the controller for the high-order stochastic nonlinear systems with structured uncertainties.

To handle the structured uncertainties for the stochastic nonlinear systems, the radial basis function neural network (RBF NN) or the fuzzy logic is used to approximate the uncertain functions, which ensures the growth assumptions can be weakened or removed. Based on these methods and some useful adaptive backstepping control approaches, fruitful results have been introduced and obtained in [22–30] and the references therein. Reference [22] studied the problem of fuzzy backstepping control for a class of stochastic nonlinear strict-feedback systems. Reference [23] considered the adaptive neural network output-feedback control for nonlinear systems with dynamical uncertainties. Reference [25] considered NN output-feedback control for stochastic nonlinear systems with unknown control coefficients. Reference [24] studied NN output-feedback control for stochastic time-delay nonlinear systems with unknown control coefficients. Furthermore, more problems of stochastic nonlinear systems with unmodeled dynamics were studied in [26–30].

Motivated by the aforementioned literatures, one raises the following meaningful problem: how to relax or remove the matching conditions on drift and diffusion terms by RBF NN? And how to design the adaptive NN state-feedback controller for a class of high-order stochastic nonlinear systems with unknown control directions?

In this paper, we will discuss the problem of adaptive neural control for a class of high-order stochastic nonlinear systems with structured uncertainties. By using backstepping recursive approach, Young’s inequality, etc., the restrictions on systems nonlinearities are removed and the procedure of the design is simpler. A novel adaptive neural controller is constructed, which assures that the closed-loop system is bounded in probability. In addition, in the design progress, there is only one parameter that needs to be updated online.

The rest of this paper is organized as follows. The notations and some preliminaries are provided in Section 2. In Section 3, we present the main results. In Section 4, we give the simulation example to illustrate the effectiveness of the proposed results, and the conclusion is drawn in Section 5.

2. Preliminaries and Problem Formulation

Notations. denotes the set of all real numbers, denotes the set of all nonnegative real numbers, and denotes the real -dimensional space. stands for the family of the functions with th continuous partial derivations. For a given vector or matrix , denotes its transpose, denotes its trance when is square, and is the Euclidean norm of a vector . For simplicity, the smooth function is sometimes denoted by .

Consider a class of high-order stochastic nonlinear systems as follows:where and are the system states and the control input, respectively. and . is an -dimensional standard Wiener process defined on the complete probability space with being a sample space, being a filtration, and being measure. and are unknown smooth functions, with , for . The disturbed virtual control coefficients () are unknown and continuous functions, respectively; is the Borel bounded measurable functions.

2.1. Preliminary Results

Next we introduce several technical lemmas which will play an important role in our later control design.

Consider the following stochastic nonlinear system:where and are the Borel measurable functions. and are assumed to be in their arguments.

Definition 1 (see [2]). Given for stochastic nonlinear system (2), the differential operator is defined as follows:where denotes all nonnegative functions on ; i.e., satisfies in and in . Simply, the smooth function is denoted by .

Definition 2 (see [15]). The solution of stochastic system (2) is said to be bounded in probability if it satisfies

Definition 3 (see [15]). Consider system (2) with and ; the equilibrium is globally stable in probability if, for any , there exists a class function such that

Lemma 4 (Young’s inequality [4]). For any , the following inequality holds:

Lemma 5 (see [15]). Consider the stochastic system (2). Assume that and are in their arguments, and and are bounded uniformly in if there exist functions , and constants such that Then the solution of (2) is bounded in probability.

The purpose of this paper is to construct a smooth adaptive neural state-feedback controller such that the solution process of system (1) is bounded in probability.

To design the controller for system (1), the following assumptions are needed: () are odd integers. () For any , there exists an ideal constant weight vector such that and () For , there are positive constants and , such that . () There exists constant such that .

Remark 6. If , system (1) becomes the strict-feedback form. The problem of the feedback control has been studied in [22–24, 26, 27]. However, they did not consider In this paper, we will consider the problem of the feedback control under the case

The following radial basis function neural network (RBFNN) will be considered and used to approximate unknown continuous functions:where is the input vector with being the neural networks input dimension. denotes the weight vector. is the neural network mode number. and are the basis function vectors. Here is the center of the receptive field, and is the width of the Gaussian function. Equation (8) can approximate any unknown continuous function over the compact set with arbitrary accuracy. Namely,The ideal constant weight vector is defined as , and is the approximation error.

From (9), we can easily getwhere ,

To design a state-feedback controller, we first introduce the following transformation:where , is the virtual control law and can be designed in the following form.

Using (11), we havewhere , , and .

3. Controller Design and Stability Analysis

3.1. Controller Design

In this section, by using the backstepping method and the BRFNN, we construct the adaptive neural controller and approximate the unknown nonlinear functions, respectively.

Step 1. Consider the Lyapunov function , where is the parameter error. By (3) and (12), we haveFrom Lemma 4 and assumptions , there exists constant , , such thatandSubstituting (14) and (15) into (13), we obtainwhere , . Obviously, is an unknown function since it has unknown function and . In practice, it cannot be used directly. Moreover, there exists a neural network , , such thatwhere . In the view of (16) and (17), we can getNow we choose the virtual control lawswhere is a constant to be chosen.
Substituting into (18), it can be rewritten asBy (11), Lemma 4, assumptions (), (), and , we obtainwhere , , and are constants.
Substituting (21) into (20), we can get

Step 2. We choose the Lyapunov function to design the control law . From (11), (12), and (22), we haveFrom the definition of and Lemma 4, there exist constants , , such thatandBy (23), (24), and (25), we can obtainFurther, adding and subtracting in the first bracket in (26) and using Lemma 4, there exists a constant , such thatThen, substituting (27) into (26), we getwhere , , and .
Similar to Step 1, obviously is an unknown function. Hence there exist the neural networks and such thatwhere .
We add and subtract in the formula (26), respectively, and, substituting (29) into (26), we can getNow we choose the virtual control law and , where is a constant to be chosen. Substitution into (30), (30) can be calculated asBy (11), Lemma 4, assumptions , and , we havewhere , , , .
Substituting (32) into (31), we get

Remark 7. Since contains , , the term cannot be used directly to design the virtual control law . And the function will be used to compensate for . As a result, the term will be considered in the later section.

Step i (). We choose the following Lyapunov function . From (3) and (12), we havewhere , andwhere Similar to the Steps 1 and 2, from the definition of , Lemma 4, and assumptions , there exist constants , , such thatandSubstituting , (37), and (38) into (34), we can getFurther, add and subtract in the first bracket in (39), and, using the Lemma 4, there exists a constant , such thatThen, substituting (40) into (39), we getwhereandSince the unknown functions and can be derived by , it cannot be used to design the control law directly. Thus there exist the neural networks and such thatwhere , .
We add and subtract in the formula (39), respectively, and, substituting (44) into (39), it hasChoose the virtual control law where is a constant to be chosen. Then, by substituting into (45) and using (11), Lemma 5, and , one yields where , , , .
In the view of (47), (45) can be calculated as

Remark 8. Since contains , the term cannot be used directly to design the virtual control law . And the function will be used to compensate for . As a result, the term will be considered in the later section.

Finally, when , is the actual control. Choose the actual controllerwhere is a design parameter to be chosen. It can be deduced thatwhere . We finish the controller design procedure.

3.2. Analysis of Stability

We have obtained the main result in the following theorem.

Theorem 9. Suppose that assumptions hold for the stochastic nonlinear system (1). Furthermore, suppose that the unknown functions can be approximated by the RBF neural networks. Given a control law with the virtual control signals , it is constructed in (49), and the adaptive law satisfieswhere the design parameter Then the signals of the closed-loop system are bounded in probability.

Proof. Choose the Lyapunov function such thatSubstituting the adaptive law (51) into the penultimate term in (52) results inIn the following, we will prove that the last term in (53) is negative. It is clear thatandSubstituting (55) into (54), one arrives atBy the definition of , which implies thatandsubstituting (57) and (58) into (53) yieldsFurthermore, let , , and , and it follows thatTherefore, according to Lemma 5, , and are bounded in probability. Since is a constant, is bounded in probability. It can be obtained that the control law , is also bounded in probability because is the function of and . So far we get that all the states of the closed-loop system (1) are bounded in probability.

4. Simulation Example

In this section, we will give an example to show the effectiveness of the proposed control method in this paper.

Example 1. Consider the following stochastic nonlinear system:

Obviously, the system satisfies and . Now according to Theorem 9, the virtual control function , the control law , and the adaptive law are chosen, respectively, aswhere , , , . In the simulation, neural network includes 7 nodes with centers spaced evenly in , neural network includes 81 nodes with centers spaced evenly in , and all the widths are chosen as 2. The design parameters are chosen as , , . , and . The initial condition . Figures 1 and 2 show the simulation results. From the figures, we can see that the proposed adaptive control method can guarantee that all the variables for the closed-loop system are bounded.

5. Conclusions

This paper has investigated the adaptive neural control for a class of high-order uncertain stochastic nonlinear systems. With the help of backstepping technique and separation technique, a smooth adaptive controller is constructed, and it ensures the closed-loop system is the global bounded in probability. Only one adaptive learning parameter needed to be updated online. One example has been given to show the effectiveness of the proposed analytical results. A further work is how to design the output-feedback tracking control for more high-order stochastic systems.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The author declares that they have no conflicts of interest.

Acknowledgments

This paper is supported by Shandong Province Natural Science Foundation of China (No. ZR2016AL05).

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Copyright

Copyright © 2018 Xiaoyan Qin. 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.

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