#### Abstract

This paper solves the tracking control problem of a class of stochastic pure-feedback nonlinear systems with external disturbances and unknown hysteresis. By using the mean-value theorem, the problem of pure-feedback nonlinear function is solved. The direction-unknown hysteresis problem is solved with the aid of the Nussbaum function. The external disturbance problems can be solved by defining new Lyapunov functions. Using the backstepping technique, a new adaptive fuzzy control scheme is proposed. The results show that the proposed control scheme ensures that all signals of the closed-loop system are semiglobally uniformly bounded and the tracking error converges to the small neighborhood of origin in the sense of mean quartic value. Simulation results illustrate the effectiveness of the proposed control scheme.

#### 1. Introduction

Hysteresis is widely found in mechanical equipment, which severely limits performance of the system and even leads to system instability. Therefore, the control problem of the system with hysteresis has been paid more and more attention. For the adaptive control system in [1–7], scholars study problems in different directions, such as hysteresis input in [1], dead zone input in [2], and time-delay input in [3]. There are several common hysteresis phenomena. The author solves the backlash-like hysteresis problem in [4]. The authors solve a class of traditional P-I hysteresis problems and propose an adaptive backstepping scheme in [5]. Scholars have studied a class of nonlinear systems with generalized P-I hysteresis inputs in [6, 7]. In addition, scholars have studied unmodeled dynamics deterministic systems in [8] and uncertain nonsmooth deterministic systems in [9]. The finite-time control problem of nonlinear deterministic systems is studied in [10]. However, the system studied above is a deterministic system and ignores the effects of stochastic disturbance.

Stochastic disturbance often occur in many systems, and the adaptive control problems of stochastic nonlinear systems are more difficult than those of deterministic nonlinear systems. Stochastic disturbance is added to the system, and differential operations on Lyapunov functions are more complicated. The research of stochastic nonlinear systems has been increasingly discussed in [11–21]. A finite time control method of switched stochastic systems is proposed in [11]. The control problem of nonlinear stochastic systems is discussed in [12–14]. Using the mean value theorem to solve the pure-feedback nonlinear function, the complexity of the system is increased. Further, more researchers have studied other types of stochastic nonlinear systems, such as from stochastic systems with unknown backlash-like hysteresis in [15] to pure-feedback stochastic nonlinear systems with unknown dead-zone input in [16], from SISO systems in [17] to pure-feedback MIMO systems in [18]. The stochastic systems with time-varying delays are proposed in [19, 20] and the stochastic systems with unknown direction hysteresis are proposed in [21]. The above studies have considered the effect of stochastic disturbance, without considering the external disturbances.

However, external disturbance often exists in practice. External disturbance cannot be ignored; it is also a source of system instability in practice. Scholars have extended the system without external disturbances in [22–25] to systems with external disturbances in [26–30], such as a determination system with external disturbances in [29] and a stochastic system with external disturbances in [30]. The system with external disturbances makes the design of the controller more difficult.

In this paper, the control problem of pure-feedback stochastic nonlinear system with external disturbance and unknown hysteresis is studied. For the determination system with external disturbance in [26], stochastic terms are not considered. The adaptive fuzzy control problem for stochastic nonlinear systems is studied in [14], without considering external disturbances. Therefore, a more general nonlinear system is processed in this paper. Furthermore, the difficulty is to deal with unknown direction hysteresis in [21]. The difficulty of this paper is how to solve the influence of external disturbance on the unknown direction hysteresis and ensure the stability of the nonlinear system. This problem can be solved by designing appropriate Lyapunov functions. The major contributions of this paper are described below:(1)The tracking control problem of the stochastic pure-feedback nonlinear systems with stochastic disturbances, direction-unknown hysteresis, and external disturbances is solved in this paper.(2)In the th step of the backstepping design, the Lyapunov function with the external disturbance term is defined, and the external disturbance problem is solved. A new adaptive control scheme is proposed.

The remainder of this article is as follows. The second part puts forward the preparation work and the problem formulation. The third part is the design process of the adaptive control method. The fourth part gives the simulation example. The fifth part summarizes the full text.

#### 2. Preparation and Problem Formulation

##### 2.1. Preliminary Knowledge

The stochastic nonlinear system is expressed as follows:where is the state variable, , are continuous functions. indicates that an independent -dimension standard Brownian motion, which is defined on the complete probability space.

*Definition 1 (see [31]). *For a quadratic continuous differentiable function , define a derivative operator expressed as follows:where is a trace of matrix.

*Remark 2. *The in correction term makes the design of the control scheme in the stochastic system more complicated than the design of the control scheme in the determined system.

Lemma 3 (see [32]). *For the stochastic system (1), let and are smooth functions defined on , , ; the function satisfes ( is a constant); is a Nussbaum-type function. The following inequality is satisfied: where is a nonnegative variable, is a real valued continuous local martingale. Then the functions , and are bounded on .*

Lemma 4 (see [33]). *For , the following inequality is established:where , , and .*

Lemma 5 (see [3]). *Consider the following dynamic system: where and are positive constants and is a positive function. Then for and any bounded initial condition , we have .*

##### 2.2. Problem Formulation

This paper considers the following stochastic pure-feedback nonlinear system:where and are the state vectors, is system output, and is defined as (1). and are unknown nonlinear functions. is a bounded external disturbance. is the system input and the output of an unknown Bouc-Wen hysteresis is defined as follows [1]:where and are unknown constants and have the same sign. is the input of the hysteresis. is the auxiliary variable, . In [1] we know that is bounded and can be expressed as: where , , are the unknown hysteresis parameters, and , .

*Remark 6. *This article has stochastic term , and the hysteresis output is different from [29]. If we ignore the external disturbance , the results of this paper are the same as [21]. Therefore, this paper considers a more general nonlinear system.

For the system (6), define with .

*Assumption 7. *For , there is an unknown constant such thatBy using the mean-value theorem, the pure-feedback nonlinear functions in (6) can be expressed aswhere , , and is point between and .

Substituting (11) into (6), the control system can be rewritten as

##### 2.3. Fuzzy Logic Systems

In order to approximate a continuous function with a fuzzy logic system, consider the following fuzzy rules: : If is and and is . Then is , ,

where is the input of system, is the output of the system, and are fuzzy sets in , and is the number of rules. The output form of the fuzzy logic system is as follows:whereLettingthe fuzzy system is written as

Lemma 8 (see [27]). *Let be a continuous function defined on a compact set . Then, for , there exists a fuzzy logic system such that**The goal of this paper is to design an adaptive controller, so that the system output converges to the reference signal and all signals of the closed-loop system are bounded.**Define a vector function as =, , where denotes the th order derivative of .*

*Assumption 9 (see [21]). *The reference signal and its order derivatives up to the th time are continuous and bounded.

#### 3. Adaptive Control Design

In this part, the adaptive fuzzy control is proposed by the backstepping technique, and the following coordinate transformation is defined to develop the backstepping technique: where is an intermediate function to be determined next.

In each step of the backstepping method, a fuzzy logic system will be used to approximate an unknown function . We define a constant , the estimation error is , as the estimation of , .

*Step 1*. For stochastic pure-feedback systems (12), according to , we know that dynamic error is satisfied We choose Lyapunov function as follows:where is a positive constant. By (2), (18), and (19), one has Applying Lemma 4 and Assumption 7, the following inequalities hold:where is a positive constant. Substituting (22) and (23) into (21), we can getDefining a new function , then the above inequality can be rewritten asBecause contains the unknown function and , it cannot be directly controlled in practice. Therefore, according to Lemma 8, for any given , there exists a fuzzy logic system such thatwhere . According to Lemma 4, it follows thatwhere is a positive parameter. We choose the following virtual control signal and adaptive law: where and are positive constants. Based on (28), Assumption 7, one has Substituting (27), (29), and (30) into (25), we haveIt is noted thatSubstituting (32) into (31), we havewhere

*Step 2*. Since and formula, one haswith Choose stochastic Lyapunov function as where is a positive design constant. Using the similar procedure as (21), it follows that It is noticed thatwhere is a positive parameter. Substituting (33), (38), and (39) into (37), we haveDefine a function as Furthermore, (40) can be rewritten as Since contains the unknown function , , and , it is not possible in practice. Thus, the fuzzy logic system is used to approximate , where Due to Lemma 8, can be written aswhere is any given positive constant. Repeating the method of (27), we havewhere is a positive parameter. We choose the following virtual control signal and adaptive law:where , are design constants. Similar to (30), the following inequality is obtained:Substituting (43), (45) and (46) into (41), we haveIt is noted that(47), can be rewritten in the formwhere ,

*Step *. According to and formula, one has with We consider the following Lyapunov function:where is a positive constant. Using the similar procedure as (21), it follows thatIt is noticed thatwhere is a positive constant. Using the similar procedure as (24), we haveDefine a function as Furthermore, (56) can be rewritten asSimilarly, The fuzzy logic system is used to approximate , where , with According to Lemma 8, can be expressed as where is any given positive constant. According to the method of (27), we can get where is a design constant. We choose the following virtual control signal and adaptive law:where and are positive parameters. Similar to (30), we have Similar to (31), we have It is noted that The above inequality can be rewritten as where ,

*Step *. Based on the coordinate transformation and formula, we can get with Consider stochastic Lyapunov function as follows: where and are positive constants. Denote as the estimation of , and the estimation error is . Similar to procedure (21), it follows that It is noticed that where is a positive constant. Based on (7), the the following inequality holds As [21], an even Nussbaum-type function is defined, and the following equality holds: where is the auxiliary virtual controller and is a positive parameter. Then, the following equality can be obtained: According to Lemma 4 and (8), it follows thatSubstituting (73)-(75) into (71), we have According to (65) with (), (70), and (76), (69) can be rewritten as follows: Define a function as . Furthermore, (77) can be rewritten as For any positive constant , the fuzzy logic system existed, such thatSimilarly, we can obtain where is a positive parameter. We choose the following virtual control signal and adaptive law: where , , and are positive constants. Substituting (80)-(83) into (78), we have Furthermore, The above inequality can be rewritten as where , , and

The control design of the adaptive fuzzy logic system has been completed by using the backstepping technique. The main theorem is described below.

Theorem 10. *Consider the stochastic pure-feedback nonlinear system (6) with Assumptions 7–9. For bounded initial conditions, combine with the virtual control signal and the adaptation law (60)-(61) and (81)-(83) guarantee that *(i)*all the signals of the closed-loop system are semi-globally uniformly bounded on , ;*(ii)*the steady-state tracking errors converge to in the sense of mean quartic value, which is defined as where is defined in (97).*

*Proof. *(i) Let Lyapunov function as , defining and . According to (86), it follows thatMultiplying by and based on formula, one has whereAccording to (88)-(89), the following inequality can be obtained: Integrating (91) on , one has Furthermore,Let ; according to Assumption 7 and the definition of , we have . Thus, according to the boundedness of and Lemma 3, we can get that and and are bounded on .

Next according to the definition of , we can get that and and are bounded on . Thereby, and are also bounded on . Due to and being bounded, we can get that is bounded. Based on is the function of and , thus is also bounded. Furthermore, we know that is bounded. Thus, all the signals of the closed-loop system are semiglobally uniformly bounded on , .

(ii) Define By taking expectation on (93) and applying (94), , we can get and therefore,Let and further, due to in the definition of (68), we have