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Mathematical Problems in Engineering
Volume 2015, Article ID 529862, 13 pages
http://dx.doi.org/10.1155/2015/529862
Research Article

Adaptive Neural Output Feedback Control of Stochastic Nonlinear Systems with Unmodeled Dynamics

Department of Automation, College of Information Engineering, Yangzhou University, Yangzhou 225127, China

Received 9 April 2014; Revised 17 June 2014; Accepted 4 July 2014

Academic Editor: Ming Gao

Copyright © 2015 Xiaonan Xia and Tianping Zhang. 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.

Abstract

An adaptive neural output feedback control scheme is investigated for a class of stochastic nonlinear systems with unmodeled dynamics and unmeasured states. The unmeasured states are estimated by K-filters, and unmodeled dynamics is dealt with by introducing a novel description based on Lyapunov function. The neural networks weight vector used to approximate the black box function is adjusted online. The unknown nonlinear system functions are handled together with some functions resulting from theoretical deduction, and such method effectively reduces the number of adaptive tuning parameters. Using dynamic surface control (DSC) technique, Itô formula, and Chebyshev’s inequality, the designed controller can guarantee that all the signals in the closed-loop system are bounded in probability, and the error signals are semiglobally uniformly ultimately bounded in mean square or the sense of four-moment. Simulation results are provided to verify the effectiveness of the proposed approach.

1. Introduction

During the past decades, backstepping in [1] and dynamic surface control (DSC) in [2] have become two most popular methods for adaptive controller design. Many adaptive control schemes based on fuzzy/neural networks have been proposed for uncertain nonlinear systems using backstepping or dynamic surface control method in [313]. In the existing literature, three types of uncertainties were commonly considered, which included unknown system functions and parameter uncertainties and unmodeled dynamics. Unmodeled dynamics was dealt with by introducing an available dynamic signal in [3]. In addition, it was handled by a description method of Lyapunov function in [4]. In [4, 5], adaptive tracking control schemes were developed by backstepping and DSC for a class of strict-feedback uncertain nonlinear systems, respectively. In [710], adaptive control schemes were presented for a class of pure-feedback nonlinear systems. In [1113], the adaptive tracking approaches for single-input single-output (SISO) nonlinear systems were extended to uncertain large-scale nonlinear systems.

When system states are assumed to be unmeasurable, output feedback adaptive control based on filters or observers has attracted much attention. In [14], K-filters were firstly proposed, and adaptive output feedback control was developed using K-filters. Inspired by the work in [14], robust adaptive output feedback control schemes were studied for SISO uncertain nonlinear systems in [15, 16]. In [17], combining backstepping technique with small-gain approach, indirect adaptive output feedback fuzzy control was developed. In [18], decentralized adaptive output-feedback control was designed based on high-gain K-filters and dynamic surface control method for a class of uncertain interconnected nonlinear systems.

It is well known that due to the stochastic terms and the extra quadratic variation terms resulting from the Itô differentiation rule, both the structures and the controller design of stochastic systems are commonly more complicated than those of deterministic systems. In the past decade, much effort has focused on the study of adaptive control schemes for uncertain stochastic nonlinear systems and the proof of the control system stability in probability sense. In [1921], Deng et al. proposed the adaptive control scheme, based on backstepping for stochastic strict feedback or output-feedback nonlinear systems, and introduced a control Lyapunov function formula for stochastic disturbance attenuation earlier. In [22], by employing the stochastic Lyapunov-like theorem, adaptive backstepping state feedback control was developed for a class of stochastic nonlinear systems with unknown backlash-like hysteresis nonlinearities. In [23], the problem of decentralized adaptive output-feedback control was discussed for a class of stochastic nonlinear interconnected systems. In [24, 25], output feedback adaptive fuzzy control approaches were considered using backstepping method for a class of uncertain stochastic nonlinear systems. In [26], by combining stochastic small-gain theorem with backstepping design technique, an adaptive output feedback control scheme was presented for a class of stochastic nonlinear systems with unmodeled dynamics and uncertain nonlinear functions. In [27], a concept of stochastic integral input-to-state stability (SiISS) using Lyapunov function was first introduced, and output feedback control was developed for stochastic nonlinear systems with stochastic inverse dynamics. In [28], two linear output feedback control schemes were studied to make the closed-loop system noise-to-state stable or globally asymptotically stable in probability. In [29], by using the homogeneous domination technique and appropriate Lyapunov functions, an output-feedback stabilizing controller was designed to be globally asymptotically stable in probability. In [30], the small-gain control method was investigated for stochastic nonlinear systems with SiISS inverse dynamics. In [31], based on a reduced-order observer, small-gain type condition on SiISS and stochastic LaSalle theorem, an output feedback controller was developed for stochastic nonlinear systems. In [32], an adaptive output feedback control scheme was investigated by combining K-filters with DSC for a class of stochastic nonlinear systems with dynamic uncertainties and unmeasured states. In [33], adaptive control was developed using the backstepping method for a class of stochastic nonlinear systems with time-varying state delays and unmodeled dynamics.

Motivated by the above-mentioned results [4, 14, 32], in this paper, adaptive neural stochastic output feedback control is developed by combining K-filters with dynamic surface control to guarantee the stability of the closed-loop system. The main contributions of the paper lie in the following.(i)Adaptive neural output feedback control is developed using K-filters and dynamic surface control for a class of stochastic nonlinear systems with unmodeled dynamics and unmeasured states. The advantage of the design is that once the local system constructed by the filter signals is stabilized, all the signals in the closed-loop system are bounded in probability.(ii)Unmodeled dynamics is dealt with first by introducing a novel description based on Lyapunove function without using the dynamic signal to handle dynamic uncertainty in [32]. The novel description, which provides an effective method for dealing with unmodeled dynamics in output feedback adaptive controller design, is the development of original idea about handling unmodeled dynamics in [4].(iii)Utilizing the boundedness of continuous function, the unknown nonlinear system functions are handled together with some functions produced in stability analysis, rather than directly approximated before stability analysis in [6, 8, 9, 11, 12]. Therefore the design effectively reduces the order of filters and the number of adjustable parameters of the whole system, without estimating in [32].(iv)Using bounded input bounded output (BIBO) stability and the filter special structure, the stability of the closed-loop system is proved. Therefore, the difficulty, that the transfer function cannot be used in a stochastic system while it was widely used to analyze the boundedness of the K-filters signals in the deterministic systems in [4, 14, 1618], is solved by the proposed stability analysis approach in this paper.

The rest of the paper is organized as follows. The problem formulation and preliminaries are given in Section 2. The neural filters are designed, and adaptive stochastic output feedback control is developed based on dynamic surface control method. The stability in the closed-loop system in probability sense is analyzed in Section 3. Simulation results are presented to illustrate the effectiveness of the proposed scheme in Section 4. Section 5 contains the conclusions.

2. Problem Statement and Preliminaries

Consider the following uncertain stochastic nonlinear systems with unmodeled dynamics: where is the state; is the input, and is the output; is a known positive continuous function; is the unknown smooth function; is the unmodeled dynamics, and is the unknown smooth nonlinear dynamic disturbance; , is a Hurwitz polynomial; and are the unknown Lipschitz functions; is an -dimensional standard Brownian motion defined on the complete probability space with being a sample space, being a field, and being a probability measure. In this paper, it is assumed that only output is available for measurement.

The control objective is to design output feedback adaptive control for system 1 such that the output follows the specified desired trajectory , and all the signals of the closed-loop system are bounded in probability.

Assumption 1 (see [4]). The unknown nonlinear dynamic disturbances , , satisfy , and and are the unknown nonnegative smooth functions, and denotes the Euclidian norm of a vector.

Assumption 2. The system is globally exponentially stable when ; that is, there exists a Lyapunov function satisfying where ,, , are positive constants, and there exists such that , .

Assumption 3. There exists an unknown function , and , such that holds.

Assumption 4. The desired trajectory is known, where , and is a known constant.

Assumption 5. There exists a known constant such that the following inequality holds.

Remark 6. Assumption 2 is the extension of the description of unmodeled dynamics in [4], and it can effectively deal with unmodeled dynamics in output feedback adaptive controller design. To the best of authors’ knowledge, this assumption is first addressed.

Consider the following stochastic nonlinear system: where is the system state, is an -dimensional standard Brownian motion, , are locally Lipschitz and , are uniformly ultimately bounded. For any given , associated with the stochastic system 3, the infinitesimal generator is defined as follows: where is the trace of a matrix .

Definition 7 (see [34]). The stochastic process is said to be bounded in probability, if .

Definition 8. The solution of system 3 is said to be semiglobally uniformly ultimately bounded (SGUUB) in th moment (), if for some compact set and any initial state , there exists a constant and a time constant such that for all , especially, when , it is usually called SGUUB in mean square.

Lemma 9 (see [32]). For any stochastic process , if there exists a positive integer and a positive constant such that , , then is bounded in probability.

Lemma 10 (see [21]). Consider system 3 and suppose that there exists a function : , two constants , , class functions , such that for all and . Then, (i) for any initial state , there exists a unique strong solution for system 3; (ii) the solution of system 3 is bounded in probability; (iii)  , .

In order to design filters and observer, 1 can be rewritten as follows: where

3. Adaptive Robust Controller Design and Stability Analysis

3.1. Neural Filters and Controller Design

In order to estimate the state , we introduce the following filters: where , , is a Hurwitz matrix; that is where is a design constant.

Define the state estimate as follows: The observer error is defined as . Thus Denote the columns of as follows: Inspired by the work in [14], the filters are designed as follows: It is easy to show that where denotes dimensional vector with the th element being one and other elements being all zeros, .

Let be the th element of the vector and the th element of the vector , respectively. From [14], we know According to 11, we get where denotes the second row of the matrix , denotes the second element of the vector , and is the second element of .

Substituting 18 into 1, it yields where In view of 19 and 14, the system used to design adaptive output feedback DSC in next section is addressed as follows:

3.2. Stochastic Adaptive Dynamic Surface Controller Design

In this subsection, according to 21 and by using dynamic surface control method, we propose an output feedback stochastic adaptive tracking control scheme. Similar to backstepping, the whole design needs steps.

For convenience, some notations are presented below. , , where , will be given in the controller design later, , . , , is the output of a first-order filter with as the input, and is an intermediate control which will be developed for the corresponding th subsystem.

Define some Lyapunov functions as follows: where is given in Assumption 2.

Using Young’s inequality, the infinitesimal generator of satisfies According to Assumption 1 and by using Young’s inequality, we obtain According to Assumptions 2 and 3, using Young’s inequality, we get

Step  1. Let . Define the first dynamic surface as follows: Using the first equation of 21, we obtain

Choose the virtual control law as follows: where , are design constants, , , are the estimates of , , at time , respectively, and , , , and will be given later. Consider Therefore, we have

A first-order filter with as the input is designed as follows: Let ; thus, . Since , using Young’s inequality, it yields From Assumption 1, we obtain In view of 24, 25, 32, and 33 and by using Young’s inequality, we obtain where is a nonnegative continuous function, and where .

Let be a given compact set with being a design constant, and let be the approximation of the radial basis function neural networks on the compact set to . Then, we have , where denotes the approximation error and denotes the basis function vector with being chosen as the commonly used Gaussian functions, which have the form , and is the center of the receptive field and is the width of the Gaussian function; is an adjustable parameter vector.

According to 34 and by using Young’s inequality, it yields There exists a nonnegative continuous function satisfying Using Young’s inequality, we have where .

Step   (). Define the th dynamic surface , thus

Select the virtual control law as follows:

A first-order filter with the input is designed as follows: where is a design constant.

Let . Then . Noting , in view of 41 and 42, we obtain Step  . The control law will be determined in this step. Define the th dynamic surface as . The derivative of is

Choose the control law as follows: In view of 45 and 46, we have The parameters , , and are updated as follows: where , , , , , are the design constants.

3.3. Stability Analysis of Adaptive Control System

In this subsection, we will discuss the stability analysis of the closed-loop system. Firstly we define some Lyapunov functions and compact sets as follows: where , is a design constant, . It is easy to know that .

According to , we obtain From 4, we obtain There exist two nonnegative continuous functions ,,  and ,  such that From 53, we have From 52, 54, and 55, we obtain The infinitesimal generator of is There exist two nonnegative continuous functions , , ,  and , , ,  such that the following inequalities hold: From 59, we obtain From 58, 60, and 61, we obtain The continuous function on the compact set has a maximum , which depends on the constant , and on the compact set has a maximum , and on the compact set have the maximum and when are bounded.

Theorem 11. Consider the closed-loop system consisting of the plant 1 under Assumptions 15, the controller 46, and the adaptation laws 48 and 49. For any bounded initial conditions, there exist constants , , , , , , , , satisfying , such that all of the signals in the closed-loop system are bounded in probability, and , are SGUUB in four-moment, , , are SGUUB in mean square, and , , and satisfy where is a positive constant; will be given later in the proof of Theorem 11.

Proof. Choose the following Lyapunov function candidate: For any given positive constant, if , according to Lemma 9, we obtain that , , , , , are bounded in probability. , and, from Assumption 2, we obtain that ; that is, , so is bounded in probability.
Furthermore, 64 is rewritten as ; then , and choosing , we get .
From 14 and 49, we have that , , are also bounded in probability. It yields that are all bounded in probability. Noting , we obtain that is bounded in probability. From 14, we have that and . Thus we obtain that , are also bounded. Furthermore, from 42, we have that are bounded. According to 16 and 17, we obtain Since are bounded in probability, we have that are all bounded in probability. From 17, we get that is also bounded. In view of 40, 44, 4749, and 62, using Young’s inequality, we obtain Substituting 63 into 66, we obtain where . Since is a nonnegative continuous function, let , where .
If , and , then we have . Thus, if , then , ; that is, Similar to the discussion of Theorem 11 in [32], it is easy to know that the conclusion is true.

Remark 12. This paper differs from [32] in the following several aspects. (1) Unmodeled dynamics is dealt with by introducing a novel description based on Lyapunov function in this paper while the dynamic signal was handled with the help of a dynamic signal in [32]. (2) The unknown nonlinear system functions are handled together with some functions produced in stability analysis, but they were directly approximated before constructing the observer in [32]. Therefore, this brings out a good result that the filter order is reduced. (3) The neural networks weight vector used to approximate the black box function at the first design step is adjusted online in this paper such that much more information of weight vector can be used in adaptive law, whereas only the norm of weight vector acts as adaptive tuning parameter in [32]. (4) Utilizing bounded input bounded output stability and linear equations 65, the stability of the closed-loop system is proved in this paper, which avoids using the transfer function to make stability analysis in [32], which is questionable in probability sense.

Remark 13. The design parameters , and determined by 63 in Theorem 11 are only a sufficient condition. They provide a guideline for the designers. From 63, some suggestions are given for the choice of some key design parameters for any given positive constants and .(i)Increasing , , helps to increase , subsequently reduces .(ii)Decreasing , , helps to reduce and reduces .(iii)Increasing helps to increase and reduces .In practical applications, to obtain good tracking performance, some experiments need to be done before the valid parameters are given.

4. Simulation Results

To demonstrate the effectiveness of the proposed approach, two numerical examples are given.

Example 1. Consider the following third-order stochastic nonlinear system with unmodeled dynamics: where , , . The desired tracking trajectory is taken as . Select , , , , , ; then , , ; , , and it satisfies the conditions of Assumptions 2 and 3.
The filters are designed as follows: The adaptation laws are employed as follows: where .
The virtual control law is chosen as follows: The control law is employed as follows: where , , .
In the simulation, , , , , , , , , , , , , , , , , , . Simulation results are shown in Figures 1, 2, and 3. From Figure 1, it can be seen that fairly good tracking performance is obtained.

Figure 1: Output (solid line) and desired trajectory (dotted line).
Figure 2: Tracking error .
Figure 3: Control signal .

Remark 14. According to 69, we know that and . From the above selected design parameters and 63, it is easy to see that , . The constant is only used to analyze the stability in the closed-loop system. Therefore, 63 is true for the above selected design parameters , , , , , , , , .

Example 2. To compare the simulation results with [32], consider the following same stochastic nonlinear system with unmodeled dynamics in [32]: where , , . The desired tracking trajectory is taken as . The filters are designed as follows: The adaptation laws are employed as follows: where .
The virtual control law is chosen as follows: The control law is employed as follows: where , .
In the simulation, , , , , , , , , , , , , , , , , , . Simulation results are shown in Figures 46. If the proposed approach in [32] is utilized, and the design parameters of the adaptive controller are taken, the same values as in [32], the corresponding simulation results are as shown in Figures 79.
From Figures 4, 5, 7, and 8, it can be seen that better tracking performance can be obtained than [32]. However, 42 equations need to be solved online using the method in [32] while only 14 equations need to be solved online using the approach in this paper. Moreover, we know that increasing helps to improve the tracking precision.

Figure 4: Output (solid line) and desired trajectory (dotted line).
Figure 5: Tracking error .
Figure 6: Control signal .
Figure 7: Output (solid line) and desired trajectory (dotted line).
Figure 8: Tracking error .
Figure 9: Control signal .

5. Conclusions

Using K-filters and dynamic surface control, an adaptive output feedback neural control scheme has been proposed for a class of stochastic nonlinear systems with unmodeled dynamics. Unmodeled dynamics has been dealt with by introducing the novel description based on Lyapunov function. The unknown nonlinear system functions are handled together with some functions resulting from stability analysis, and the filter order is reduced. The neural network weight vector is adjusted online. Therefore, the more information included in radial basis function can be fully made use of. Using Chebyshev’s inequality and Itô formula, the designed controller can guarantee that all the signals in the closed-loop system are bounded in probability and the error signals are semiglobally uniformly ultimately bounded in the sense of four-moment or mean square. Simulation results illustrate the effectiveness of the proposed approach.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (61174046, 61473249, and 61473250).

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