Adaptive Fuzzy Control for Stochastic Pure-Feedback Nonlinear Systems with Unknown Hysteresis and External Disturbance

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 neighborhoodof origin in the sense ofmean quartic value. Simulation results illustrate the effectiveness of the proposed control scheme.


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][2][3][4][5][6][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][12][13][14][15][16][17][18][19][20][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][13][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 2 Complexity system instability in practice.Scholars have extended the system without external disturbances in [22][23][24][25] to systems with external disturbances in [26][27][28][29][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 purefeedback 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 Δ() 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.

Preparation and Problem Formulation
2.1.Preliminary Knowledge.The stochastic nonlinear system is expressed as follows: where  ∈   is the state variable,  :   ×  + →   , ℎ :   ×  + →  × 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  2 / 2 in ô correction term (1/ 2){ℎ  ( 2 / 2 )ℎ} makes the design of the control scheme in the stochastic system more complicated than the design of the control scheme in the determined system.
Remark 6.This article has stochastic term    (  ), and the hysteresis output  is different from [29].If we ignore the external disturbance Δ(), the results of this paper are the same as [21].Therefore, this paper considers a more general nonlinear system.

Fuzzy Logic Systems.
In order to approximate a continuous function () with a fuzzy logic system, consider the following fuzzy rules: where  = [ 1 ,  2 , . . .,   ]  ∈   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: where Letting the fuzzy system is written as Lemma 8 (see [27]).Let () be a continuous function defined on a compact set Ω.Then, for ∀ > 0, there exists a fuzzy logic system Φ  () 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.
Assumption 9 (see [21]).The reference signal   () and its order derivatives up to the th time are continuous and bounded.

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  −1 is an intermediate function to be determined next.
Step 1.For stochastic pure-feedback systems (12), according to  1 =  1 −   , we know that dynamic error is satisfied We choose Lyapunov function as follows: where  1 is a positive constant.By (2), (18), and ( 19), one has

Complexity
Applying Lemma 4 and Assumption 7, the following inequalities hold: where  1 is a positive constant.Substituting ( 22) and ( 23) into (21), we can get Defining a new function , then the above inequality can be rewritten as Because  1 contains the unknown function  1 and  1 , it cannot be directly controlled in practice.Therefore, according to Lemma 8, for any given  1 > 0, there exists a fuzzy logic system Φ  1  1 ( 1 ) such that where  1 = ( 1 ,   , ẏ  ).According to Lemma 4, it follows that where  1 is a positive parameter.We choose the following virtual control signal and adaptive law: where  1 and  1 are positive constants.Based on (28), Assumption 7, one has Substituting ( 27), (29), and ( 30) into (25), we have Substituting ( 32) into (31), we have where Step 2. Since  2 =  2 −  1 and ô formula, one has with Choose stochastic Lyapunov function as where  2 is a positive design constant.Using the similar procedure as (21), it follows that It is noticed that where  2 is a positive parameter.Substituting (33), (38), and (39) into (37), we have Define a function as Furthermore, (40) can be rewritten as Since  2 contains the unknown function  1 ,  1 , and  2 , it is not possible in practice.Thus, the fuzzy logic system Due to Lemma 8,  2 can be written as where  2 is any given positive constant.Repeating the method of ( 27), we have where  2 is a positive parameter.We choose the following virtual control signal and adaptive law: where  2 ,  2 are design constants.Similar to (30), the following inequality is obtained: Substituting (43), ( 45) and ( 46) into (41), we have It is noted that (47), can be rewritten in the form where Step  (3 ≤  ≤  − 1).According to   =   −  −1 and ô formula, one has with We consider the following Lyapunov function: where   is a positive constant.Using the similar procedure as (21), it follows that It is noticed that 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   =   − −1 and ô formula, we can get with Consider stochastic Lyapunov function as follows: where   and  are positive constants.Denote Δ as the estimation of Δ, and the estimation error is Δ = Δ− Δ.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 () =  2  is defined, and the following equality holds: where V 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 that Substituting ( 73)-( 75) into (71), we have According to (65) with ( =  − 1), (70), and ( 76), (69) can be rewritten as follows: Define a function as Furthermore, (77) can be rewritten as For any positive constant   > 0, the fuzzy logic system Φ     (  ) existed, such that Similarly, we can obtain where   is a positive parameter.We choose the following virtual control signal and adaptive law: (85) The above inequality can be rewritten as where   = (  /2  )  + (  /2)Δ 2 .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 semiglobally uniformly bounded on [0,   ), ∀  > 0; (ii) the steady-state tracking errors   converge to Ω  in the sense of mean quartic value, which is defined as where  is defined in (97). Proof Furthermore, Next according to the definition of (), we can get that   and θ and Δ are bounded on [0,   ).Thereby, θ and Δ are also bounded on [0,   ).Due to  1 =  1 −   and   being bounded, we can get that  1 is bounded.Based on  1 is the function of  1 and θ1 , thus  2 =  2 +  1 is also bounded.Furthermore, we know that   ,  = 3, 4, . . .,  is bounded.Thus, all the signals of the closed-loop system are semiglobally uniformly bounded on [0,   ), ∀  > 0.
and further, due to  =   in the definition of (68), we have Thus, the steady-state tracking errors   converge to Ω  in the sense of mean quartic value.

Simulation Example
In this section, a simulation example is given to prove the effectiveness of the proposed adaptive control scheme.

Conclusion
In this paper, we study a class of stochastic pure-feedback nonlinear systems with bounded external disturbance and   unknown hysteresis.This paper holds that stochastic disturbance and external disturbance exist simultaneously.By using the characteristics of the Nussbaum function, the unknown hysteresis problem is solved.Based on the approximation ability of fuzzy logic system, a new adaptive fuzzy control scheme is proposed.The scheme ensures that all signals of the closed-loop system are bounded and the tracking error converges to small domain of the origin.Finally, the simulation results show the effectiveness of the proposed scheme.

Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.

Figure 3 :
Figure 3: The true control input V.