#### Abstract

The problem of fuzzy-based direct adaptive tracking control is considered for a class of pure-feedback stochastic nonlinear systems. During the controller design, fuzzy logic systems are used to approximate the packaged unknown nonlinearities, and then a novel direct adaptive controller is constructed via backstepping technique. It is shown that the proposed controller guarantees that all the signals in the closed-loop system are bounded in probability and the tracking error eventually converges to a small neighborhood around the origin in the sense of mean quartic value. The main advantages lie in that the proposed controller structure is simpler and only one adaptive parameter needs to be updated online. Simulation results are used to illustrate the effectiveness of the proposed approach.

#### 1. Introduction

During the past decades, many control methods have been developed to control design of nonlinear systems, such as adaptive control [1–3], backstepping control [4], and fault tolerant control [5–9]. Particularly, backstepping-based adaptive control has been an effective tool to deal with the control problem of nonlinear strict-feedback systems without satisfying matching condition. So far, many interesting results have been obtained for deterministic nonlinear systems in [4, 10–21] and for stochastic cases in [22–35]. In the aforementioned papers, however, the existing results required that the nonlinear functions be in the affine forms; that is, systems are characterized by input appearing linearly in the system state equation.

Pure-feedback nonlinear systems, which have no affine appearance of the state variables that can be used as virtual control and the actual control input, stand for a more representative form than strict-feedback systems. Many practical systems are in nonaffine structure, such as biochemical process [4] and mechanical systems [36]. Therefore, the study on stability analysis and controller synthesis for pure-feedback nonlinear system is important both in theory and in practice applications [37–42]. By combining adaptive neural control and backstepping, in [37, 38], a class of pure-feedback systems was investigated, which contained only partial nonaffine functions and at least a control variable or virtual control signal existed in affine form. In [39], an “ISS-modular” approach combined with the small-gain theorem was presented for adaptive neural control of completely nonaffine pure-feedback systems. Recently, in [41], an adaptive neural tracking control scheme was presented for a class of nonaffine pure-feedback systems with multiple unknown state time-varying delays. On the other hand, it is well known that stochastic disturbance often exists in practical systems and is usually a source of instability of control systems. So, the consideration of the control design for pure-feedback nonlinear systems with stochastic disturbance is a meaningful issue and has attracted increasing attention in the control community in recent years [43–45]. In [43], the problem of adaptive fuzzy control for pure-feedback stochastic nonlinear systems has been reported. Then, Yu et al. [44] presented an adaptive neural controller to guarantee the four-moment boundedness for a class of uncertain nonaffine stochastic nonlinear system with time-varying delays. However, the considered systems in [43–45] are of a special kind of pure-feedback stochastic nonlinear systems in which only the last equation was viewed as nonaffine equation.

Motivated by the above observations, we will develop a novel fuzzy-based direct adaptive tracking control scheme for a class of pure-feedback stochastic nonlinear systems. The presented controller guarantees that all the signals in the closed-loop system remain bounded in probability and the tracking error converges to a small neighborhood around the origin in the sense of mean quartic value. The main contributions of this paper lie in that the structure of the proposed controller is simpler and only one adaptive parameter needs to be updated online. As a result, the computational burden is significantly alleviated and the control scheme may be more implemented in practice.

The remainder of this paper is organized as follows. The problem formulation and preliminaries are given in Section 2. An adaptive fuzzy tracking control scheme is presented in Section 3. The simulation example is given in Section 4, and it is followed by Section 5 which concludes the work.

#### 2. Problem Formulation and Preliminaries

To introduce some useful conceptions and lemmas, consider the following stochastic system: where is state vector, is a -dimensional independent standard Brownian motion defined on the complete probability space with being a sample space, being a -field, being a filtration, and being a probability measure, and , are locally Lipschitz functions in and satisfy , .

*Definition 1 (see [25]). *For any given , associated with the stochastic differential equation (1), define the differential operator as follows:
where is the trace of .

*Remark 2. *The term is called Itô correction term, in which the second-order differential makes the controller design much more difficult than that of the deterministic system.

*Definition 3 (see [46]). *The trajectory , of stochastic system (1) is said to be semiglobally uniformly ultimately bounded in th moment, if, for some compact set and any initial state , there exist a constant and a time constant such that , for all . Particularly, when , it is usually called semiglobally uniformly ultimately bounded in mean square.

Lemma 4 (see [46]). *Suppose that there exist a function , two constants and , and class functions and such that
**
for all and . Then, there is a unique strong solution of system (1) for each and it satisfies
*

In this paper, we consider a class of pure-feedback stochastic nonlinear systems described by where and are defined in (1), and represent the control input and the system output, and respectively, , , () are unknown smooth nonlinear functions.

The control objective is to design a fuzzy-based adaptive tracking control law for system (5) such that the system output follows a desired reference signal in the sense of mean quartic value, and all signals in the closed-loop system remain bounded in probability. To this end, define , , where denotes the th time derivative of .

For the system (5), define where .

*Assumption 5 (see [42]). *The signs of , , are known, and there exist unknown constants and such that for ,

*Assumption 6 (see [17]). *The reference signal and its th order derivatives are continuous and bounded.

In this note, fuzzy logic system will be used to approximate a continuous function defined on some compact sets. Adopt the singleton fuzzifier, the product inference, and the center-average defuzzifier to deduce the following fuzzy rules:: IF is and is and and is . Then is (),

where and are the input and output of the fuzzy system, respectively. and are fuzzy sets in . Since the strategy of singleton fuzzification, center-average defuzzification, and product inference is used, the output of the fuzzy system can be formulated as where is the point at which fuzzy membership function achieves its maximum value, which is assumed to be 1. Let , and . Then, the fuzzy logic system can be rewritten as If all memberships are chosen as Gaussian functions, the lemma below holds.

Lemma 7 (see [47]). *Let be a continuous function defined on a compact set . Then, for any given constant , there exists a fuzzy logic system (10) such that
*

The following lemma will be used in this note.

Lemma 8 (Young’s inequality [23]). *For , the following inequality holds:
**
where , , , and .*

#### 3. Adaptive Fuzzy Control Design

In this section, a fuzzy-based adaptive tracking control scheme is proposed for the system (5). The backstepping design procedure contains steps and is developed based on the coordinate transformation , , where . The virtual control signal and the adaption law will be constructed in the following forms: where (), , and are positive design parameters and , is the estimation of unknown constant which is specified as Specially, is the actual control input .

For simplicity, in the following, the time variable and the state vector are omitted from the corresponding functions, and let .

*Step 1. *Since , the error dynamic satisfies
Consider a Lyapunov function candidate as
where is the parameter error. By (2) and the completion of squares, one has
where is a constant. Define a new function
then (18) can be rewritten as
Based on Assumption 5 and the fact of , one has
According to Lemma 1 in [37], for each value of and , there exists a smooth ideal control input such that
Applying mean value theorem [48], there exists () which makes
where , .

Apparently, Assumption 5 on is still valid for . Substituting (23) into (20) and applying the result (22) and results in
Since contains the unknown function , cannot be directly implemented in practice. According to Lemma 7, there exists a fuzzy logic system which can model the unknown function over a compact set , such that, for any given ,
where is approximation error. Then, based on Young’s inequality, it follows that
where the unknown constant is defined in (15) and the property of is used in (26). By choosing virtual control signal in (13) with , and using the property of and Assumption 5, the following result holds:
Substituting (26) and (27) into (20) and using Young’s inequality to the term produces
with .

*Step 2. *Based on the coordinate transformation and Itô formula, one has
with
Take the following Lyapunov function:
Then, by using a similar procedure as Step 1, it follows that
It is noticed that
where is a constant.

Furthermore, it can be verified by substituting (28) and (33) into (32) that

From the definition of in (14), we have By combining (30), (34), and (35), the following result holds: where It can be seen from (37) that . Then, from Assumption 5, we have According to Lemma 1 in [37], there exists a smooth ideal control input such that By applying mean value theorem [48], there exists () such that where ,. Assumption 5 on is still valid for .

Substituting (39) and (40) into (36) produces where .

Further, fuzzy logic system is used to approximate the desired unknown virtual signal over a compact set such that, for any given positive constant , can be shown as with being the approximation error.

Then, constructing the virtual control signal in (13) and repeating the same methods used in (26) and (27), one has By taking (43) into account and using Young’s inequality to the term , (41) can be rewritten as where ,.

*Remark 9. *The adaptive law defined in (14) is a function of all the error variables. So, unlike the conventional approximation-based adaptive control schemes, in (30) cannot be approximated directly by fuzzy logic system . To solve this problem, in (35), is divided into two parts. The first term in the right hand side of (35) is contained in to be modeled by , and the last term in (35) which is a function of the latter error variables, namely, , , will be dealt with in the later design steps. This design idea will be repeated at the following steps.*Step i .* From and Itô formula, we have
where
Choose a Lyapunov function as
Then, it follows that
where the term in (48) can be obtained in the following form by straightforward derivations as those in former steps:
where .

According to the definition of in (14), one has Then, substituting (49) into (48) and taking (50) into account, (48) can be rewritten as where Noting , it follows that Based on Lemma 1 in [37], there exists a smooth ideal control input which makes In addition, applying mean value theorem [48], there exists () such that where ,. Assumption 5 on is still valid for . Combining (51)–(55) yields where . Subsequently, using fuzzy logic system to model over a compact set such that for any given , can be expressed as with being the approximation error. Furthermore, constructing virtual control signal in (13) and following the same line as the procedures used from (26) to (27), one has By substituting (58) into (56) and using Young’s inequality, we have where .

*Step n.* In this step, the actual control input signal will be obtained. By and Itô formula, we have
where is defined in (46) with . Choosing the Lyapunov function
then, the following inequality can be easily verified by using (2), the completion of squares, and taking (59) with into account:
where
From the definition of , Assumption 5 and Lemma 1 in [37], for every value and , there exists a smooth ideal control input such that . By using mean value theorem [48], there exists () such that
where ,.

Furthermore, (62) can be rewritten as Next, using fuzzy logic system to approximate the unknown function on the compact set , constructing the actual control in (13) with , and repeating the same procedures as (58)-(59), one has Further, choosing the adaptive law in (14) and using the inequality result in where ,.

Now, the actual control signal is constructed. The main result will be summarized by the following theorem.

Theorem 10. *Consider the pure-feedback stochastic nonlinear system (5), the controller (13), and adaptive law (14) under Assumptions 5 and 6. Assume that there exists sufficiently large compacts ,, such that , for all ; then, for bounded initial conditions (where is an appropriately chosen compact set),*(i)*all the signals in the closed-loop system are bounded in probability;*(ii)*there exists a finite time such that the quartic mean square tracking error enters inside the area for all ,
* *where the time will be given later.*

*Proof. *(i) For the stability analysis of the closed-loop system, choose the Lyapunov function as . From (67), it follows that
where , and . Therefore, based on Lemma 1 in [37], and remain bounded in probability. Because is a constant, is bounded in probability. It can be further seen that is a function of and . So, is also bounded in probability. Hence, we conclude that all the signals in the closed-loop system (5) remain bounded in the sense of probability.

(ii) From (69), the following inequality can be obtained directly by [24, Theorem 4.1]
Let , then (70) satisfies
Then, it can be easily verified that there exists a time such that

#### 4. Simulation Results

In this section, to illustrate the effectiveness of the proposed control scheme, we consider the following second-order pure-feedback stochastic nonlinear system: The control objective is to design an adaptive controller for the system such that all the signals in the closed-loop system remain bounded and the system output tracks the given reference signal . According to Theorem 10, choose the virtual control law in (13), actual control input in (13) with , and adaptive law in (14). The design parameters are taken as follows: , , and . The initial conditions are given by and .

The simulation results are shown in Figures 1–4. Figure 1 shows the system output and the reference signal . Figure 2 and Figure 3 show that the state variable and adaptive parameter are bounded. Figure 4 displays the control input signal .

#### 5. Conclusions

In this paper, a novel fuzzy-based adaptive control scheme has been presented for pure-feedback stochastic nonlinear systems. The proposed controller guarantees that all the signals in the closed-loop systems are bounded in probability and tracking error eventually converges to a small neighborhood around the origin in the sense of mean quartic value. The main advantages of this control scheme are that the controller is simpler than the existing ones and only one adaptive parameter needs to be estimated online for an -order system. Numerical results have been provided to show the effectiveness of the suggested approach. Our future research will mainly focus on the multi-input and multioutput (MIMO) pure-feedback stochastic nonlinear systems based on the result in this paper.

#### Conflict of Interests

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

#### Acknowledgments

This work is partially supported by the Natural Science Foundation of China (61304002, 61304003, and 11371071), the Program for New Century Excellent Tallents in University (NECT-13-0696), the Program for Liaoning Innovative Research Team in University under Grant (LT2013023), the Program for Liaoning Excellent Talents in University under Grant (LR2013053), and the Education Department of Liaoning Province under the general project research under Grant (no. L2013424).