# Guaranteeing Prescribed Performance Control for Gyrostabilized Platform with Unknown Control Direction Preceded by Hysteresis

Authors: Yexing Wang; Humin Lei; Jikun Ye; Xiangwei Bu; Yali Xue
Journal: International Journal of Aerospace Engineering
 (2019)

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## Abstract

This paper investigates the problem of precise and quick tracking for gyrostabilized platform (GSP) with unknown hysteresis, unknown control directions, and unknown compound disturbance. Firstly, the dynamic model of GSP is transformed into a strict feedback formulation by designed FD to facilitate the backstepping control system. Secondly, performance functions are constructed at each step of backstepping design to force tracking errors to fall within the prescribed boundaries. Besides, through ingenious transformation, radial basis function neural network (RBFNN) is applied to estimate the unknown control gains preceded by hysteresis. Hence, the problem of prescribed performance control with unknown compound disturbances, unknown hysteresis, and unknown control directions is creatively solved. Furthermore, the exploited controllers are accurate model independent, which guarantees satisfactory robustness of control laws against unknown uncertainties. Finally, the stability of the closed-loop control system is confirmed via Lyapunov stability theory, and numerical simulations are given for a GSP to validate the effectiveness of the proposed controller.


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## Body

## 1. Introduction

Gyrostabilized platform (GSP) is a kind of precise servo tracking system, which is usually mounted on a mobile carrier for stable tracking of moving targets. Varieties of detectors are mounted on GSP to isolate the motion of carrier as well as to get high-performance information of targets. GSP is originally used in weapons including nonstrapdown seekers [1], aerial shooting, and airborne remote sensing system [2]. Recently, it is also widely used in robotics [3], deep space exploration, and other high-precision tracking systems. Therefore, it has broad application prospects, and it is worth to investigate further.GSP is a complex time-varying nonlinear system with compound disturbances [4, 5]. The internal disturbances of system include hysteresis nonlinearity of motor and the perturbation of model, while the external disturbances of system include the motion of basement and friction torque between the shafts. Faced with such problems, researches have tried varieties of approaches to realize better dynamic response performance and stronger robustness of GSP. For most actual GSP system, proportional-integral-derivative (PID) control or modified PID has been widely used because of its reliability and simple control structure [6].Recently, to solve the problem of compound disturbances, many control strategies have been designed [7]. Active disturbance rejection control technique is commonly used to compensate for disturbance because of its strong robustness against various disturbances, X. Y. Zhou and et al. combine the feedforward control with friction observer to compensate for friction disturbance, and the friction is eliminated in large scale because of the precise estimation. Besides, they designed a backstepping integral adaptive compensator to compensate for disturbance [8, 9]. While the shortcoming is that the parameters of friction are estimated offline. Moreover, RBFNN is commonly used to estimate the disturbance for its approximation ability [10, 11]. In reference [11], an adaptive RBFNN was proposed to generate the feedback control parameters online, while the extended state observer is used to compensate for composite disturbances. The control strategy eliminates disturbances in large scale, while the overshoot of response is large. In reference [12], an adaptive neural network is applied to estimate the uncertain disturbances as well as eliminating “chattering phenomenon,” the strategy is simple and efficient.Though satisfying robustness and high precision can be achieved through the abovementioned methods, there are still some shortcomings to these methods. A fatal one is that most researchers focus on the characteristic of controlled object, while the nonlinearity of actuator is ignored. The servo motor plays a role of actuator, its precision is mainly restrained by trigger deadzone and hysteresis, while the deadzone can be seen as external disturbance to eliminate. Many researchers have drawn a common conclusion that that hysteresis of motor will deteriorate its response characteristic and tracking precision [13, 14]. Therefore, to promote the performance of GSP, hysteresis problem has to be solved. Besides, the existing research focuses on improving steady performance of GSP. The robustness and precision of GSP are enhanced, but the dynamic performance of GSP cannot be guaranteed.The study of hysteresis nonlinearities has been drawing much attention in the control community for a long time [15, 16]. For several classes of deterministic nonlinear system with unknown backlash-like hysteresis, adaptive control was proposed by some researchers, while the method has received little attention [17]. Nevertheless, few researchers investigate the problem with both unknown backlash-like hysteresis and unknown control directions [18]. In reference [18], Yu et al. adopt backstepping control to control the mentioned system. By designing a state observer, the states of transformed system are estimated, besides, a RBFNN is adopted to acquire estimated unknown functions, while the control strategy brings in two additional states, which makes controller design complex.Motivated by previous investigations, this paper will concentrate on solving the problem of compound disturbances and unknown control directions as well as guaranteeing prescribed performance for tracking errors. The compound disturbances include two parts. One part is friction torque, which is a strong nonlinear disturbance that affects the tracking performance at angular velocity “crossing zero” point sharply [2]. And the other is the movement of basement. The disturbances are all considered unknown and estimated online by RBFNN. RBFNN plays a crucial part in disturbance compensation and enhancing the robustness. The key point of application of RBFNN is the model transformation. It is model transformation that normalizes the disturbances. Meanwhile, the designed FD enables transformation by estimating newly defined states. Furthermore, because of the high dynamic performance of GSP, prescribed performance control is adopted. Performance functions are defined under the structure of backstepping control. At the last step of backstepping control, there exists unknown control gain; therefore, a Nussbaum gain function (NGF) is adopted to solve the problem. Thus, prescribed performance control under compound disturbances and unknown control directions is creatively achieved.Special contributions of this paper are summarized as follows:(1)
This paper focuses on the problem of compound disturbances for GSP. Different from existing researches on GSP, in this paper, hysteresis problem is creatively taken into account. Different from most investigations into hysteresis, the hysteresis is completely unknown. Especially, compared with reference [18], input-driven observer is not needed
(2)
Prescribed output quality is guaranteed for GSP through prescribed performance control; the control strategy is creatively proposed and applied to GSP
(3)
The presented control approach is independent of accurate models. Thus, its disturbance rejection ability is fine and the computational cost is relatively lowThe paper is organised as follows: in Section2, the model of GSP is built and its working preliminaries are presented. In Section 3, necessary control techniques are introduced, and the controller is designed along with the stability analysis. In Section 4, simulations are carried out to demonstrate the effectiveness.


## 2. System Modeling and Problem Formulation

### 2.1. Constitution and Operating Principle of Two-Axis GSP

Figure1 shows the schematic diagram of two-axis GSP. We can see that stabilized platform consists of two gimbals, which are pitch gimbal and yaw gimbal, respectively. The system is driven by two servo motors; the detective sensor is placed in the inner frame.Figure 1
Schematic diagram of the two-axis GSP.From Figure1, we can see the relationships between two gimbals: gyroscopes measuring the angular rate of pitch and yaw gimbals, angle sensors measuring the angle of pitch and yaw gimbals, and current sensors measuring the current of pitch and yaw motors.GSP is fixed at the projectile body to track the target [19].Because of the low coupling and similar characteristic of pitch and yaw channels [20], we choose pitch channel to analyze.


### 2.2. Dynamic Model of the GSP

Figure2 shows the pitch channel block diagram of GSP. The block within the red imaginary line stands for the servo motor, while the block within the blue imaginary line stands for the friction disturbance; θd represents the angle conference signal of the system; kPWM is the power amplifier coefficient; θ̇ is the angular rate of stabilized platform in inertial space; ϑ̇ is the disturbance of basement movement; Tturb is the disturbance moment; kg is the simplified transfer function of rate gyroscope; Tc is the moment output of servo motor; i is the electric current of the servo motor; Cm is the moment coefficient of motor; Ce is the coefficient of counter electromotive force.Figure 2
Block diagram of GSP tracking loop.Combining with the dynamic equation of stable platform and the dynamic equation of motor, mathematical model of GSP is acquired as follows [21]:
(1)ẋ1=x2,ẋ2=Cmx3JL−TturbJL,ẋ3=−CeLax2−RaLax3+kPWMLauv−Ceϑ̇La,where x1,x2, and x3 are state variables, which represent θl,θ̇l, and i, respectively. uv is the system input and the output of the following backlash-like hysteresis described as [22]
(2)u̇v=αv̇cv−uv+Bv̇,where v is the input of the backlash-like hysteresis, α and B are constants, and c>0 is the slope of lines satisfying c>B. This paper assumes that the parameters of (2) are unknown.In the light of the analysis in [18], (2) can be solved as
(3)ut=cvt+dv,where
(4)dv=u0−cv0e−αv−v0sgnv̇+e−αvsgnv̇∫v0vB−ceαςsgnv̇dς,where u0 and v0 are the initial values of u and v, respectively. As is shown in [21], dv is bounded by an unknown constant value dmax.The disturbance torque mainly results from friction between gimbals [23]. Stribeck friction model is chosen as the torque disturbance model in this paper. The mathematical formulation is given as follows:
(5)Tturb=Fc+Fs−Fce−v/vs2sgnv+Bvv,where Fc and Fs are Coulomb friction and static friction, respectively.Remark 1.
The controller output isv in Figure 2, while there exists unknown hysteresis, which can be expressed as u̇v=αv̇cv−uv+Bv̇, is regarded as a completely unknown function. Thus, u in Figure 2 is regarded as the newly defined output of controller that drives the motor directly. The hysteresis makes the control directions unknown.Remark 2.
In this paper, “unknown hysteresis” means that in (3), the coefficient c and function dv are unknown. “Unknown control directions” result from unknown hysteresis, as is shown in (1). In (1), the designed controller output is v, while the control item is kPWM/Lauv. The map from v to uv is unknown, so the problem is called “problem with unknown control directions.” “Unknown compound disturbances” mean that Tturb,ϑ̇ are completely unknown.Remark 3.
To enhance the control precision, some researchers identify the parameters of friction model [2, 24]. While the process of identification is complex owing to its complex dynamic behavior, besides, identification has to be carried out repeatedly for different systems. In this paper, friction torque is regarded unknown, which simplifies the process of controller design, and the algorithm is more universal to apply.


### 2.3. Control Problems for GSP

There are some troublesome characteristics in the GSP:(1)
When there exist unknown disturbances, high-performance angle tracking is hard to guarantee
(2)
The unknown hysteresis results in unknown control direction problem for GSP, which increases the difficulty of controller design


## 2.1. Constitution and Operating Principle of Two-Axis GSP

Figure1 shows the schematic diagram of two-axis GSP. We can see that stabilized platform consists of two gimbals, which are pitch gimbal and yaw gimbal, respectively. The system is driven by two servo motors; the detective sensor is placed in the inner frame.Figure 1
Schematic diagram of the two-axis GSP.From Figure1, we can see the relationships between two gimbals: gyroscopes measuring the angular rate of pitch and yaw gimbals, angle sensors measuring the angle of pitch and yaw gimbals, and current sensors measuring the current of pitch and yaw motors.GSP is fixed at the projectile body to track the target [19].Because of the low coupling and similar characteristic of pitch and yaw channels [20], we choose pitch channel to analyze.


## 2.2. Dynamic Model of the GSP

Figure2 shows the pitch channel block diagram of GSP. The block within the red imaginary line stands for the servo motor, while the block within the blue imaginary line stands for the friction disturbance; θd represents the angle conference signal of the system; kPWM is the power amplifier coefficient; θ̇ is the angular rate of stabilized platform in inertial space; ϑ̇ is the disturbance of basement movement; Tturb is the disturbance moment; kg is the simplified transfer function of rate gyroscope; Tc is the moment output of servo motor; i is the electric current of the servo motor; Cm is the moment coefficient of motor; Ce is the coefficient of counter electromotive force.Figure 2
Block diagram of GSP tracking loop.Combining with the dynamic equation of stable platform and the dynamic equation of motor, mathematical model of GSP is acquired as follows [21]:
(1)ẋ1=x2,ẋ2=Cmx3JL−TturbJL,ẋ3=−CeLax2−RaLax3+kPWMLauv−Ceϑ̇La,where x1,x2, and x3 are state variables, which represent θl,θ̇l, and i, respectively. uv is the system input and the output of the following backlash-like hysteresis described as [22]
(2)u̇v=αv̇cv−uv+Bv̇,where v is the input of the backlash-like hysteresis, α and B are constants, and c>0 is the slope of lines satisfying c>B. This paper assumes that the parameters of (2) are unknown.In the light of the analysis in [18], (2) can be solved as
(3)ut=cvt+dv,where
(4)dv=u0−cv0e−αv−v0sgnv̇+e−αvsgnv̇∫v0vB−ceαςsgnv̇dς,where u0 and v0 are the initial values of u and v, respectively. As is shown in [21], dv is bounded by an unknown constant value dmax.The disturbance torque mainly results from friction between gimbals [23]. Stribeck friction model is chosen as the torque disturbance model in this paper. The mathematical formulation is given as follows:
(5)Tturb=Fc+Fs−Fce−v/vs2sgnv+Bvv,where Fc and Fs are Coulomb friction and static friction, respectively.Remark 1.
The controller output isv in Figure 2, while there exists unknown hysteresis, which can be expressed as u̇v=αv̇cv−uv+Bv̇, is regarded as a completely unknown function. Thus, u in Figure 2 is regarded as the newly defined output of controller that drives the motor directly. The hysteresis makes the control directions unknown.Remark 2.
In this paper, “unknown hysteresis” means that in (3), the coefficient c and function dv are unknown. “Unknown control directions” result from unknown hysteresis, as is shown in (1). In (1), the designed controller output is v, while the control item is kPWM/Lauv. The map from v to uv is unknown, so the problem is called “problem with unknown control directions.” “Unknown compound disturbances” mean that Tturb,ϑ̇ are completely unknown.Remark 3.
To enhance the control precision, some researchers identify the parameters of friction model [2, 24]. While the process of identification is complex owing to its complex dynamic behavior, besides, identification has to be carried out repeatedly for different systems. In this paper, friction torque is regarded unknown, which simplifies the process of controller design, and the algorithm is more universal to apply.


## 2.3. Control Problems for GSP

There are some troublesome characteristics in the GSP:(1)
When there exist unknown disturbances, high-performance angle tracking is hard to guarantee
(2)
The unknown hysteresis results in unknown control direction problem for GSP, which increases the difficulty of controller design


## 3. Controller Design

### 3.1. Finite-Time-Convergent Tracking Differentiator Design

To estimate newly defined states, a finite-time tracking differentiator (FD) is designed.Assumption 1 (see [25]).
The input signalxt is continuous and piecewise n-order derivable with the following characteristics. The derivatives of xt up to order n−2 exist on the whole time domain and xt is not n−1 order at some instants tj,j=1,2,…k may hold.Based on the assumption, to estimate newly defined states, a high-order tracking differentiator (FD) is designed. The FD is formulated as follows:(6)χ̇1=χ2χ̇2=χ3⋮χ̇i=Rn−a1tanhχ1−xt−a2tanhχ2R−⋯−aitanhχiRn−1.Based on reference [26], there exist ϕ>0 and ιϕ>n such that χi−xi−1t=O1/Rιϕ−i+1,i=1,2,…,n where R>0,ai>0,i=1,2,…n are positive constants to be designed; O1/Rιϕ−i+1 means the approximation of 1/Rιϕ−i+1 order between χi and xi−1t; ϕ=1−ϖ/ϖ and xt; ϖ∈0,minι/ι+n1/2,n≥2 xt is the input signal; χi,i=1,2,…,n donate the states of FD and they are xi−1t,i=1,2,…,n, respectively. The corresponding estimation errors are defined as follows:
(7)s1=χ1−xt,s2=χ2−ẋt,…,sn=χn−xn−1t.The designed FD is proven finite-time-convergence and stable in reference [20], and the estimation error is bounded.


### 3.2. Neural Network

To guarantee the controller’s robustness, an adaptive RBFNN is introduced to approximate the uncertain functions. The adaptive RBFNN is defined as the mapping relationship between input vectorX=x1,x2,…,xnT∈Rn and the output y∈R [26, 27].
(8)y=WThX,where W=w1,w2,…,wmT∈Rm stands for weight vector; m and n represent the node number and input number, respectively; hX=h1X,h2X,…hmXT∈Rm with hjX is defined as follows:
(9)hjX=exp−X−cσj2∈Rm,j=1,2,…,m,where c=c1,c2,…,cnT∈Rm and σ=σj1,σj2,…,σjnT∈Rn mean a center and a width vector of hjX, respectively [28, 29].For an arbitrary continuous unknown functionFX, it has to be proven that there exists an ideal weight vector W=w1,w2,…,wmT∈Rm such that
(10)FX=W∗hX+ε,where ε is approximate error, which satisfies that ε≤εmax. It should be noted that W∗ is unknown, its elements w1∗,w2∗,…,wm∗ are required to be adjusted adaptively.Define the error between the ideal weight vectorW∗ and the estimated weight vector W^ as
(11)W~=W^‐W∗.The adaptation laws ofW^ is designed in next section.


### 3.3. Prescribed Performance

By prescribed performance, we mean that the tracking errore evolves strictly within the prescribed decaying bounds as follows:
(12)‐δ¯ℏt<et<δ¯ℏt,where the performance function ℏt=ℏ0−ℏ∝e−lt+ℏ∝ is bounded and strictly decreasing with ℏ∝≤ℏt≤ℏ0. And ℏ0>ℏ∝>0, l>0, 0<δ¯<1, and 0<δ¯<1 are design parameters [26].Ife remains within the prescribed bound of (12), the maximum overshoot of e is restrained less than max δ¯ℏ0,δ¯ℏ0, and the steady value is no more than max δ¯ℏ∝,δ¯ℏ∝. Thus, the transient performance and steady performance of e are guaranteed by choosing appropriate parameters for (12).Considering that it is unable to devise controller directly based on (12), an error transfer function μεt=δ¯eεt−δ¯e−εt/eεt+e−εt is applied to transfer (12) into the following formulation [28].
(13)et=μεtℏt,where εt is transfer error.Appropriately, (13) is equivalent illustrated by (12). Furthermore, μεt is bounded and strictly increasing.From (13), we have
(14)εt=μ−1ε=12lne/ℏ+δ¯δ¯−e/ℏ.Furthermore,ε̇t is acquired as
(15)ε̇t=rė−eℏ̇tℏt.With(16)ℏ̇t=−lℏ0−ℏ∝e−lt∈−lℏ0−ℏ∝,0,r=12ℏt1e/ℏt+δ¯−1e/ℏt−δ¯=12ℏt1μεt+δ¯−1μεt−δ¯.It is obvious thatr is bounded.


### 3.4. Backstepping Prescribed Performance Controller Design for GSP

#### 3.4.1. Model Transformation

The first step of controller design is to transfer the mathematical model in (1). The mathematical model will be transferred into pure-feedback formation.Definey1=x1, y2=x2, and y3=Cmx3/JL−Tturb/JL, then we have
(17)ẏ1=y2,ẏ2=y3,ẏ3=f1y2,y3+f2v+fd,where f1y2,y3=−Ce/Lay2−RaJL/LaCmy3Cm/JL, and f2v=CmkPWM/JLLacvt+dv is control item, while fd=Cm/JL−Ce/Laϑ̇−Ṫturb/Cm−Ra/CmLaTturb represents transferred disturbance.Then, separating the linear part of control item from nonlinear part, the following equation is acquired.(18)ẏ1=y2,ẏ2=y3,ẏ3=f1y2,y3+Acvt+f2dv+fd,where the constant A=CmkPWM/JLLa, c is unknown gain, and f2d=CmkPWM/JLLadv is unknown function regarded as disturbance to be estimated.Remark 4.
Aiming at the problem of unknown complex disturbances, a RBFNN will be adopted. Because of unknown control directions, RBFNN cannot be applied directly. Through equivalent transformation, unknown gainc becomes a part of unknown functions to be estimated.Thus, the model of GSP in (1) is transformed into pure-feedback system in (18).


#### 3.4.2. Control Law Design

Assumption 2.
We assume that the angle reference signalθd is limited, while its derivative and its second-order derivative are also limited. Furthermore, the disturbance of basement ϑ̇ is also limited.Step 1.
Define the angle tracking errore1.
(19)e1=y1−y1d.Define a performance functionℏ1t=ℏ01−ℏ1∝e−l1t+ℏ1∝ to restrain e1.
(20)‐δ¯1ℏ1t<e1t<δ¯1ℏ1t,where ℏ01>0, ℏ1∝>0, l1>0, 0<δ1¯<1, and 0<δ1¯<1 are design parameters and satisfying ℏ01>ℏ1∝, ‐δ¯1ℏ1t<e10<δ¯1ℏ1t, and ℏ1∝<ℏ1t<ℏ10.We convert (20) into the following formulation:
(21)e1=μ1ε1tℏ1t,where μ1ε1t=δ¯1eε1t−δ¯1e−ε1t/eε1t+e−ε1t∈−δ1¯,δ1¯ is a transformed function and ε1t is transformed error. From (14), we get the formulation of ε1t
(22)ε1t=μ1−1ε1t=12lne1/ℏ1t+δ1¯δ1¯−e1/ℏ1t.Furthermore,ε̇1t is derived as
(23)ε̇1t=r1ė1−e1ℏ̇1tℏ1t=r1ẏ1−ẏ1d−e1ℏ̇1tℏ1t=r1y2−ẏ1d−e1ℏ̇1tℏ1t,where r1=1/2ℏ1t1/δ¯1e1/ℏ1t+δ¯1−1/δ¯1e1/ℏ1t−δ¯1≥1/2δ¯1ℏ1t+1/2δ¯1ℏ1t>0.The virtual control law is designed as(24)z2d=−k1ε1t+ẏ1d+e1ℏ̇1tℏ1t,where k1 is a design parameter, ℏ̇1t=−l1ℏ10−ℏ1∝e−l1t.Step 2.
Define angular velocity tracking errore2.
(25)e2=y2−y2d.Construct a performance functionℏ2t=ℏ02−ℏ2∝e−l2t+ℏ2∝>0 to restrain e2.
(26)‐δ¯2ℏ2t<e2t<δ¯2ℏ2t.where ℏ02>0, ℏ2∝>0, l2>0, 0<δ2¯<1, and 0<δ2¯<1 are design parameters and satisfying ℏ02>ℏ2∝, ‐δ2¯ℏ2t<e20<δ2¯ℏ2t, and ℏ2∝<ℏ2t<ℏ20.Equation (26) is further converted into the following formulation:
(27)e2=μ2ε2tℏ2t.where μ2ε2t=δ¯2eε2t−δ2¯e−ε2t/eε2t+e−ε2t∈−δ2¯,δ2¯ is a transformed function and ε2t is transformed error. From (27), we can get
(28)ε2t=μ2−1ε2t=12lne2/ℏ2t+δ2¯δ2¯−e2/ℏ2t.The time derivative ofε2t is
(29)ε̇2t=r2ė2−e2ℏ̇2tℏ2t=r2ẏ2−ẏ2d−e2ℏ̇2tℏ2t=r2y3−ẏ2d−e2ℏ̇2tℏ2t,where r2=1/2ℏ2t1/δ2¯e2/ℏ2t+δ2¯−1/δ2¯e2/ℏ2t−δ2¯≥1/2δ2¯ℏ2t+1/2δ2¯ℏ2t>0.The virtual control law is devised as(30)y3d=−k2ε2t+ẏ2d+e2ℏ̇2tℏ2t,where k2 is a design parameter, ℏ̇2t=−l2ℏ20−ℏ2∝e−l2t.Step 3.
Define current tracking errore3.
(31)e3=y3−y3d.Construct a performance functionℏ3t=ℏ03−ℏ3∝e−l3t+ℏ3∝>0 to restrain e3.
(32)‐δ3¯ℏ3t<e3t<δ3¯ℏ3t,where ℏ03>0, ℏ3∝>0, l3>0, 0<δ3¯<1, and 0<δ3¯<1 are design parameters and satisfying ℏ03>ℏ3∝, ‐δ3¯ℏ3t<e30<δ3¯ℏ3t, and ℏ3∝<ℏ3t<ℏ30.Equation (28) is further converted into the following formulation:
(33)e3=μ3ε3tℏ3t,where μ3ε3t=δ¯3eε3t−δ3¯e−ε3t/eε3t+e−ε3t∈−δ3¯,δ3¯ is a transformed function and ε3t is transformed error. From (33), we can get
(34)ε3t=μ3−1ε3t=12lne3/ℏ3t+δ3¯δ3¯−e3/ℏ3t.The time derivative ofε3t is
(35)ε̇3t=r3ė3−e3ℏ̇3tℏ3t=r3ẏ3−ẏ3d−e3ℏ̇3tℏ3t=r3f1y2,y3+Acvt+f2dv+fd−ẏ3d−e3ℏ̇3tℏ3t,where r3=1/2ℏ3t1/δ3¯e3/ℏ3t+δ3¯−1/δ3¯e3/ℏ3t−δ3¯≥1/2δ3¯ℏ3t+1/2δ3¯ℏ3t>0.Define Lyapunov function(36)V=ε122+ε222+ε322+1γ1W~TW~̇.Employing (23), (29), and (35), V̇ is acquired as follows:
(37)V̇=ε1ε̇1+ε2ε̇2+ε3ε̇3+1γ1W~TW~̇=ε1r1y2−ẏ1d−e1ℏ̇1ℏ1+ε2r2y3−ẏ2d−e2ℏ̇2ℏ2+ε3r3f1y2,y3+Acvt+f2dv+fd−ẏ3d−μ3ℏ̇3t+1γ1W~TW~̇=ε1r1y2−ẏ1d−e1ℏ̇1ℏ1+ε2r2y3−ẏ2d−e2ℏ̇2ℏ2+ε3r3Acvt+ε3r3f1y2,y3−ẏ3d−μ3ℏ̇3t+ε3r3f2dv+fd+1γ1W~TW~̇.Here, a RBFNN is applied to offset the disturbancefd as well as the unknown input function f2dv. Considering that the input of the system contains an unknown gain c, RBFNN cannot be applied directly. Through an equivalent transformation, the following equation is acquired.Remark 5.
The unknown compound disturbance isD=fd+f2dv/Ac, where fd=Cm/JL−Ce/Laϑ̇−Ṫturb/Cm−Ra/CmLaTturb, f2d=CmkPWM/JLLadv, Tturb is friction disturbance, and it is one part of unknown compound disturbance. Besides, the unknown function of unknown hysteresis dv is estimated too. The compound disturbance will be estimated in whole.
(38)V̇=ε1r1y2−ẏ1d−e1ℏ̇1ℏ1+ε2r2y3−ẏ2d−e2ℏ̇2ℏ2+ε3r3Acv1t−W^hx+ε3r3f1y2,y3−ẏ3d−μ3ℏ̇3t+ε3r3f2dv+fd+1γ1W~TW~̇=ε1r1y2−ẏ1d−e1ℏ̇1ℏ1+ε2r2y3−ẏ2d−e2ℏ̇2ℏ2+ε3r3f1y2,y3−ẏ3d−μ3ℏ̇3t+ε3r3Acv1t+ε3r3Ac−W^hx+f2dv+fdAc+1γ1W~TW~̇.Substituting (19), (25), and (31) into (38) leads to
(39)V̇=ε1tr1e2−k1ε1+ε2tr2e3−k2ε2t+ε3tr3Acv1t−W^hx+ε3tr3f1y2,y3−ẏ3d−μ3ℏ̇3t+ε3tr3f2dv+fd+1γ1W~TW~̇=r1μ2ε2ℏ2tε1t−k1ε12t+r2μ3ε3ℏ3tε2t−k2ε22t+ε3r3Acv1t−W^hx+ε3r3f1y2,y3−ẏ3d−μ3ℏ̇3t+ε3r3f2dv+fd+1γ1W~TW~̇.Define a total disturbanceD=fd+f2dv/Ac, we mean that D=W^hx+ε, where x=x1,x2,x3. That is to say, when ε is small enough, W^hx will converge to D in high precision [26]. Thus, vt=v1t−W^hx.To analyze disturbance item individually, define a new function as follows.(40)V̇d=ε3r3Ac−W^hx+f2dv+fdAc+1γ1W~TW~̇≤ε3r3AcW~hx+εmax+1γ1W~TW~̇.Define a positive constantc¯ satisfying c≤c¯. Applying Young’s inequality [24], we can acquire that
(41)V̇d≤W~hx+εmax22+ε3r3Ac¯22+1γ1W~TW~̇=εmax22+ε3r3Ac¯22+W~TW~hxThx2+εmaxhx+W~̇γ1.SinceW∗ is a constant, it should be noted that W~̇=W^̇.According to (41), adaption law W^̇ is designed as follows:
(42)W~̇=‐γ1W~hxThx2+εmaxhx.Then, (41) becomes
(43)V̇d≤εmax22+ε3r3Ac¯22.Combining with (39), the following inequality is acquired.
(44)V̇≤r1μ2ε2ℏ2tε1t−k1ε12t+r2μ3ε3ℏ3tε2t−k2ε22t+ε3r3f1y2,y3−ẏ3d−μ3ℏ̇3t+ε3r3Acv1t+εmax22+ε3r3Ac¯22.It is not hard to conclude that there exists a positive constant∑0 such that f1y2,y3−ẏ3d−μ3ℏ̇3t≤∑0. Furthermore, we know that ℏ2∝<ℏ2t<ℏ20,ℏ3∝<ℏ3t<ℏ30, r1≥1/2ℏ1t1/δ1+1/δ1>0,r2≥1/2ℏ2t1/δ2+1/δ2¯>0, μ2ε2t=δ¯2eε2t−δ2¯e−ε2t/eε2t+e−ε2t∈−δ2¯,δ2¯, μ3ε3t=δ¯3eε3t−δ3¯e−ε3t/eε3t+e−ε3t∈−δ3¯,δ3¯.Then (44) becomes
(45)V̇≤r1ε1tδ¯2ℏ2t−k1ε1t+r2ε2tδ¯3ℏ3t−k2ε2t+ε3r3∑0+ε3r3Acv1t+εmax22+ε3r3Ac¯22.Finally, the actual control law is chosen as(46)v1t=N3ξ3κ31ε3t+κ32r3ε3t2‐W^hx,ξ̇3=κ31r3ε32t+κ32r32ε32t2,where N3ξ3=eξ32cosπξ3/2 is a Nussbaum function [30]; κ31 and κ32 are design parameters.Invoking (41), (44) becomes
(47)V̇≤r1ε1tδ¯2ℏ2t−k1ε1t+r2ε2tδ¯3ℏ3t−k2ε2t+ε3r3∑0+ε3r3AcN3ξ3κ31ε3t+κ32r3ε3t2+εmax22+ε3r3Ac¯22.Based on Young’s inequality, it is easy to getε3tr3∑0≤κ32r32ε32t/2+∑02/2κ32.Then (47) becomes
(48)V̇≤r1ε1tδ¯2ℏ2t−k1ε1t+r2ε2tδ¯3ℏ3t−k2ε2t+κ32r32ε32t2+∑022κ32+ε3r3AcN3ξ3κ31ε3t+κ32r3ε3t2+εmax22+ε3r3Ac¯22=r1ε1tδ¯2ℏ2t−k1ε1t+r2ε2tδ¯3ℏ3t−k2ε2t++κ32r32ε32t2+∑022κ32+AcN3ξ3ξ̇3+εmax22+ε3r3Ac¯22=r1ε1tδ¯2ℏ2t−k1ε1t+r2ε2tδ¯3ℏ3t−k2ε2t+−κ31r32ε32t+κ31r32ε32t+κ32r32ε32t2+∑022κ32+AcN3ξ3ξ̇3+εmax22+ε3r3Ac¯22=r1ε1tδ¯2ℏ2t−k1ε1t+r2ε2tδ¯3ℏ3t−k2ε2t+AcN3ξ3ξ̇3+ξ̇3−κ31r32ε32t+∑022κ32+εmax22+ε3r3Ac¯22.It is obvious to observe from (48) that ε1t and ε2t are bounded if ε1t≤δ¯2ℏ20/k1 and ε2t≤δ¯3ℏ30/k2. Else if ε1t>δ¯2ℏ20/k1 and ε2t>δ¯3ℏ30/k2, we obtain r1ε1tδ¯2ℏ20−k1ε1t<0 and r2ε2tδ¯3ℏ30−k2ε2t<0. Moreover, there exist adequately small constants 0<k3<r1k1 and 0<k4<r2k2 such that r1ε1tδ¯2ℏ20−k1ε1t+k3ε12t<0 andr2ε2tδ¯3ℏ30−k2ε2t+k4ε22t<0.Thus, (48) becomes
(49)V̇≤r1ε1tδ¯2ℏ2t−k1ε1t+k3ε12t−k3ε12t+r2ε2tδ¯3ℏ3t−k2ε2t+k4ε22t−k4ε22t+AcN3ξ3ξ̇3+ξ̇3−κ31r32ε32t+∑022κ32+εmax22+ε3r3Ac¯22≤−k3ε12t−k4ε22t−κ31−Ac¯22r32ε32t+AcN3ξ3ξ̇3+ξ̇3+∑022κ32+εmax22≤−ιhV+AcN3ξ3ξ̇3+ξ̇3+∑022κ32+εmax22.Withι1=min2k3,2k4,κ31r3−r3Ac¯/22, κ31>r3Ac¯/22.Multiplyingeι1t on both sides of (49) leads to
(50)ddxVeι1t≤AcN3ξ3ξ̇3eι1t+ξ̇3eι1t+∑022κ32+εmax22.Integrating (45) over 0,t, we obtain
(51)0≤V≤V0+e−ι1t∫0tAcN3ξ3ξ̇3eι1τdτ+e−ι1t∫0tξ̇3eι1τdτ+∫0t∑022κ32+εmax22e−ι1t−τdτ.Noticing∫0t∑02/2κ32+δmax/22e−ι1t−τdτ=∑02/2κ32+εmax/221/ι11−e−ι1t∈0,∑02/2κ32+εmax/22, we know that ∫0t∑02/2κ32+εmax/22e−ι1t−τdτ is bounded. Furthermore, (46) becomes
(52)0≤V≤V0+∑022κ32+εmax22+e−ι1t∫0tAcN3ξ3ξ̇3eι1τdτ+e−ι1t∫0tξ̇3eι1τdτ.Invoking lemmas 1 and 2 presented in [30, 31], we know that V, e−ι1t∫0tAcN3ξ3ξ̇3eι1τdτ, and e−ι1t∫0tξ̇3eι1τdτ are all bounded. Thus, all the signals involved are bounded. From the boundedness of ε1,ε2, and ε3, we know that there exist positive constants ε¯1,ε¯2,ε¯3 such that ε1≤ε¯1,ε2≤ε¯2, and ε3≤ε¯3. Furthermore, ‐δ¯iℏit<−δ¯ieεit+δi¯e−εit/eεit+e−εitℏit≤eit≤δ¯ieεit−δi¯e−εit/eεit+e−εitℏit<δi¯ℏit, i=1,2,3. It is obvious that the prescribed performance for e1,e2, and e3 is guaranteed.The design procedure of GSP is completed.Remark 6.
It should be noted that the controller design is not relying on model of GSP, which guarantees the robustness against model uncertain and unknown disturbance.Remark 7.
Though similar control framework is designed as reference [31], compared with [31], the robustness against compound disturbances and unknown control directions is enhanced in large scale. Especially, the stability of system under compound disturbances is proven. However, in [31], the stability analysis as well as controller design did not consider any disturbances, which was not rigorous.


## 3.1. Finite-Time-Convergent Tracking Differentiator Design

To estimate newly defined states, a finite-time tracking differentiator (FD) is designed.Assumption 1 (see [25]).
The input signalxt is continuous and piecewise n-order derivable with the following characteristics. The derivatives of xt up to order n−2 exist on the whole time domain and xt is not n−1 order at some instants tj,j=1,2,…k may hold.Based on the assumption, to estimate newly defined states, a high-order tracking differentiator (FD) is designed. The FD is formulated as follows:(6)χ̇1=χ2χ̇2=χ3⋮χ̇i=Rn−a1tanhχ1−xt−a2tanhχ2R−⋯−aitanhχiRn−1.Based on reference [26], there exist ϕ>0 and ιϕ>n such that χi−xi−1t=O1/Rιϕ−i+1,i=1,2,…,n where R>0,ai>0,i=1,2,…n are positive constants to be designed; O1/Rιϕ−i+1 means the approximation of 1/Rιϕ−i+1 order between χi and xi−1t; ϕ=1−ϖ/ϖ and xt; ϖ∈0,minι/ι+n1/2,n≥2 xt is the input signal; χi,i=1,2,…,n donate the states of FD and they are xi−1t,i=1,2,…,n, respectively. The corresponding estimation errors are defined as follows:
(7)s1=χ1−xt,s2=χ2−ẋt,…,sn=χn−xn−1t.The designed FD is proven finite-time-convergence and stable in reference [20], and the estimation error is bounded.


## 3.2. Neural Network

To guarantee the controller’s robustness, an adaptive RBFNN is introduced to approximate the uncertain functions. The adaptive RBFNN is defined as the mapping relationship between input vectorX=x1,x2,…,xnT∈Rn and the output y∈R [26, 27].
(8)y=WThX,where W=w1,w2,…,wmT∈Rm stands for weight vector; m and n represent the node number and input number, respectively; hX=h1X,h2X,…hmXT∈Rm with hjX is defined as follows:
(9)hjX=exp−X−cσj2∈Rm,j=1,2,…,m,where c=c1,c2,…,cnT∈Rm and σ=σj1,σj2,…,σjnT∈Rn mean a center and a width vector of hjX, respectively [28, 29].For an arbitrary continuous unknown functionFX, it has to be proven that there exists an ideal weight vector W=w1,w2,…,wmT∈Rm such that
(10)FX=W∗hX+ε,where ε is approximate error, which satisfies that ε≤εmax. It should be noted that W∗ is unknown, its elements w1∗,w2∗,…,wm∗ are required to be adjusted adaptively.Define the error between the ideal weight vectorW∗ and the estimated weight vector W^ as
(11)W~=W^‐W∗.The adaptation laws ofW^ is designed in next section.


## 3.3. Prescribed Performance

By prescribed performance, we mean that the tracking errore evolves strictly within the prescribed decaying bounds as follows:
(12)‐δ¯ℏt<et<δ¯ℏt,where the performance function ℏt=ℏ0−ℏ∝e−lt+ℏ∝ is bounded and strictly decreasing with ℏ∝≤ℏt≤ℏ0. And ℏ0>ℏ∝>0, l>0, 0<δ¯<1, and 0<δ¯<1 are design parameters [26].Ife remains within the prescribed bound of (12), the maximum overshoot of e is restrained less than max δ¯ℏ0,δ¯ℏ0, and the steady value is no more than max δ¯ℏ∝,δ¯ℏ∝. Thus, the transient performance and steady performance of e are guaranteed by choosing appropriate parameters for (12).Considering that it is unable to devise controller directly based on (12), an error transfer function μεt=δ¯eεt−δ¯e−εt/eεt+e−εt is applied to transfer (12) into the following formulation [28].
(13)et=μεtℏt,where εt is transfer error.Appropriately, (13) is equivalent illustrated by (12). Furthermore, μεt is bounded and strictly increasing.From (13), we have
(14)εt=μ−1ε=12lne/ℏ+δ¯δ¯−e/ℏ.Furthermore,ε̇t is acquired as
(15)ε̇t=rė−eℏ̇tℏt.With(16)ℏ̇t=−lℏ0−ℏ∝e−lt∈−lℏ0−ℏ∝,0,r=12ℏt1e/ℏt+δ¯−1e/ℏt−δ¯=12ℏt1μεt+δ¯−1μεt−δ¯.It is obvious thatr is bounded.


## 3.4. Backstepping Prescribed Performance Controller Design for GSP

### 3.4.1. Model Transformation

The first step of controller design is to transfer the mathematical model in (1). The mathematical model will be transferred into pure-feedback formation.Definey1=x1, y2=x2, and y3=Cmx3/JL−Tturb/JL, then we have
(17)ẏ1=y2,ẏ2=y3,ẏ3=f1y2,y3+f2v+fd,where f1y2,y3=−Ce/Lay2−RaJL/LaCmy3Cm/JL, and f2v=CmkPWM/JLLacvt+dv is control item, while fd=Cm/JL−Ce/Laϑ̇−Ṫturb/Cm−Ra/CmLaTturb represents transferred disturbance.Then, separating the linear part of control item from nonlinear part, the following equation is acquired.(18)ẏ1=y2,ẏ2=y3,ẏ3=f1y2,y3+Acvt+f2dv+fd,where the constant A=CmkPWM/JLLa, c is unknown gain, and f2d=CmkPWM/JLLadv is unknown function regarded as disturbance to be estimated.Remark 4.
Aiming at the problem of unknown complex disturbances, a RBFNN will be adopted. Because of unknown control directions, RBFNN cannot be applied directly. Through equivalent transformation, unknown gainc becomes a part of unknown functions to be estimated.Thus, the model of GSP in (1) is transformed into pure-feedback system in (18).


### 3.4.2. Control Law Design

Assumption 2.
We assume that the angle reference signalθd is limited, while its derivative and its second-order derivative are also limited. Furthermore, the disturbance of basement ϑ̇ is also limited.Step 1.
Define the angle tracking errore1.
(19)e1=y1−y1d.Define a performance functionℏ1t=ℏ01−ℏ1∝e−l1t+ℏ1∝ to restrain e1.
(20)‐δ¯1ℏ1t<e1t<δ¯1ℏ1t,where ℏ01>0, ℏ1∝>0, l1>0, 0<δ1¯<1, and 0<δ1¯<1 are design parameters and satisfying ℏ01>ℏ1∝, ‐δ¯1ℏ1t<e10<δ¯1ℏ1t, and ℏ1∝<ℏ1t<ℏ10.We convert (20) into the following formulation:
(21)e1=μ1ε1tℏ1t,where μ1ε1t=δ¯1eε1t−δ¯1e−ε1t/eε1t+e−ε1t∈−δ1¯,δ1¯ is a transformed function and ε1t is transformed error. From (14), we get the formulation of ε1t
(22)ε1t=μ1−1ε1t=12lne1/ℏ1t+δ1¯δ1¯−e1/ℏ1t.Furthermore,ε̇1t is derived as
(23)ε̇1t=r1ė1−e1ℏ̇1tℏ1t=r1ẏ1−ẏ1d−e1ℏ̇1tℏ1t=r1y2−ẏ1d−e1ℏ̇1tℏ1t,where r1=1/2ℏ1t1/δ¯1e1/ℏ1t+δ¯1−1/δ¯1e1/ℏ1t−δ¯1≥1/2δ¯1ℏ1t+1/2δ¯1ℏ1t>0.The virtual control law is designed as(24)z2d=−k1ε1t+ẏ1d+e1ℏ̇1tℏ1t,where k1 is a design parameter, ℏ̇1t=−l1ℏ10−ℏ1∝e−l1t.Step 2.
Define angular velocity tracking errore2.
(25)e2=y2−y2d.Construct a performance functionℏ2t=ℏ02−ℏ2∝e−l2t+ℏ2∝>0 to restrain e2.
(26)‐δ¯2ℏ2t<e2t<δ¯2ℏ2t.where ℏ02>0, ℏ2∝>0, l2>0, 0<δ2¯<1, and 0<δ2¯<1 are design parameters and satisfying ℏ02>ℏ2∝, ‐δ2¯ℏ2t<e20<δ2¯ℏ2t, and ℏ2∝<ℏ2t<ℏ20.Equation (26) is further converted into the following formulation:
(27)e2=μ2ε2tℏ2t.where μ2ε2t=δ¯2eε2t−δ2¯e−ε2t/eε2t+e−ε2t∈−δ2¯,δ2¯ is a transformed function and ε2t is transformed error. From (27), we can get
(28)ε2t=μ2−1ε2t=12lne2/ℏ2t+δ2¯δ2¯−e2/ℏ2t.The time derivative ofε2t is
(29)ε̇2t=r2ė2−e2ℏ̇2tℏ2t=r2ẏ2−ẏ2d−e2ℏ̇2tℏ2t=r2y3−ẏ2d−e2ℏ̇2tℏ2t,where r2=1/2ℏ2t1/δ2¯e2/ℏ2t+δ2¯−1/δ2¯e2/ℏ2t−δ2¯≥1/2δ2¯ℏ2t+1/2δ2¯ℏ2t>0.The virtual control law is devised as(30)y3d=−k2ε2t+ẏ2d+e2ℏ̇2tℏ2t,where k2 is a design parameter, ℏ̇2t=−l2ℏ20−ℏ2∝e−l2t.Step 3.
Define current tracking errore3.
(31)e3=y3−y3d.Construct a performance functionℏ3t=ℏ03−ℏ3∝e−l3t+ℏ3∝>0 to restrain e3.
(32)‐δ3¯ℏ3t<e3t<δ3¯ℏ3t,where ℏ03>0, ℏ3∝>0, l3>0, 0<δ3¯<1, and 0<δ3¯<1 are design parameters and satisfying ℏ03>ℏ3∝, ‐δ3¯ℏ3t<e30<δ3¯ℏ3t, and ℏ3∝<ℏ3t<ℏ30.Equation (28) is further converted into the following formulation:
(33)e3=μ3ε3tℏ3t,where μ3ε3t=δ¯3eε3t−δ3¯e−ε3t/eε3t+e−ε3t∈−δ3¯,δ3¯ is a transformed function and ε3t is transformed error. From (33), we can get
(34)ε3t=μ3−1ε3t=12lne3/ℏ3t+δ3¯δ3¯−e3/ℏ3t.The time derivative ofε3t is
(35)ε̇3t=r3ė3−e3ℏ̇3tℏ3t=r3ẏ3−ẏ3d−e3ℏ̇3tℏ3t=r3f1y2,y3+Acvt+f2dv+fd−ẏ3d−e3ℏ̇3tℏ3t,where r3=1/2ℏ3t1/δ3¯e3/ℏ3t+δ3¯−1/δ3¯e3/ℏ3t−δ3¯≥1/2δ3¯ℏ3t+1/2δ3¯ℏ3t>0.Define Lyapunov function(36)V=ε122+ε222+ε322+1γ1W~TW~̇.Employing (23), (29), and (35), V̇ is acquired as follows:
(37)V̇=ε1ε̇1+ε2ε̇2+ε3ε̇3+1γ1W~TW~̇=ε1r1y2−ẏ1d−e1ℏ̇1ℏ1+ε2r2y3−ẏ2d−e2ℏ̇2ℏ2+ε3r3f1y2,y3+Acvt+f2dv+fd−ẏ3d−μ3ℏ̇3t+1γ1W~TW~̇=ε1r1y2−ẏ1d−e1ℏ̇1ℏ1+ε2r2y3−ẏ2d−e2ℏ̇2ℏ2+ε3r3Acvt+ε3r3f1y2,y3−ẏ3d−μ3ℏ̇3t+ε3r3f2dv+fd+1γ1W~TW~̇.Here, a RBFNN is applied to offset the disturbancefd as well as the unknown input function f2dv. Considering that the input of the system contains an unknown gain c, RBFNN cannot be applied directly. Through an equivalent transformation, the following equation is acquired.Remark 5.
The unknown compound disturbance isD=fd+f2dv/Ac, where fd=Cm/JL−Ce/Laϑ̇−Ṫturb/Cm−Ra/CmLaTturb, f2d=CmkPWM/JLLadv, Tturb is friction disturbance, and it is one part of unknown compound disturbance. Besides, the unknown function of unknown hysteresis dv is estimated too. The compound disturbance will be estimated in whole.
(38)V̇=ε1r1y2−ẏ1d−e1ℏ̇1ℏ1+ε2r2y3−ẏ2d−e2ℏ̇2ℏ2+ε3r3Acv1t−W^hx+ε3r3f1y2,y3−ẏ3d−μ3ℏ̇3t+ε3r3f2dv+fd+1γ1W~TW~̇=ε1r1y2−ẏ1d−e1ℏ̇1ℏ1+ε2r2y3−ẏ2d−e2ℏ̇2ℏ2+ε3r3f1y2,y3−ẏ3d−μ3ℏ̇3t+ε3r3Acv1t+ε3r3Ac−W^hx+f2dv+fdAc+1γ1W~TW~̇.Substituting (19), (25), and (31) into (38) leads to
(39)V̇=ε1tr1e2−k1ε1+ε2tr2e3−k2ε2t+ε3tr3Acv1t−W^hx+ε3tr3f1y2,y3−ẏ3d−μ3ℏ̇3t+ε3tr3f2dv+fd+1γ1W~TW~̇=r1μ2ε2ℏ2tε1t−k1ε12t+r2μ3ε3ℏ3tε2t−k2ε22t+ε3r3Acv1t−W^hx+ε3r3f1y2,y3−ẏ3d−μ3ℏ̇3t+ε3r3f2dv+fd+1γ1W~TW~̇.Define a total disturbanceD=fd+f2dv/Ac, we mean that D=W^hx+ε, where x=x1,x2,x3. That is to say, when ε is small enough, W^hx will converge to D in high precision [26]. Thus, vt=v1t−W^hx.To analyze disturbance item individually, define a new function as follows.(40)V̇d=ε3r3Ac−W^hx+f2dv+fdAc+1γ1W~TW~̇≤ε3r3AcW~hx+εmax+1γ1W~TW~̇.Define a positive constantc¯ satisfying c≤c¯. Applying Young’s inequality [24], we can acquire that
(41)V̇d≤W~hx+εmax22+ε3r3Ac¯22+1γ1W~TW~̇=εmax22+ε3r3Ac¯22+W~TW~hxThx2+εmaxhx+W~̇γ1.SinceW∗ is a constant, it should be noted that W~̇=W^̇.According to (41), adaption law W^̇ is designed as follows:
(42)W~̇=‐γ1W~hxThx2+εmaxhx.Then, (41) becomes
(43)V̇d≤εmax22+ε3r3Ac¯22.Combining with (39), the following inequality is acquired.
(44)V̇≤r1μ2ε2ℏ2tε1t−k1ε12t+r2μ3ε3ℏ3tε2t−k2ε22t+ε3r3f1y2,y3−ẏ3d−μ3ℏ̇3t+ε3r3Acv1t+εmax22+ε3r3Ac¯22.It is not hard to conclude that there exists a positive constant∑0 such that f1y2,y3−ẏ3d−μ3ℏ̇3t≤∑0. Furthermore, we know that ℏ2∝<ℏ2t<ℏ20,ℏ3∝<ℏ3t<ℏ30, r1≥1/2ℏ1t1/δ1+1/δ1>0,r2≥1/2ℏ2t1/δ2+1/δ2¯>0, μ2ε2t=δ¯2eε2t−δ2¯e−ε2t/eε2t+e−ε2t∈−δ2¯,δ2¯, μ3ε3t=δ¯3eε3t−δ3¯e−ε3t/eε3t+e−ε3t∈−δ3¯,δ3¯.Then (44) becomes
(45)V̇≤r1ε1tδ¯2ℏ2t−k1ε1t+r2ε2tδ¯3ℏ3t−k2ε2t+ε3r3∑0+ε3r3Acv1t+εmax22+ε3r3Ac¯22.Finally, the actual control law is chosen as(46)v1t=N3ξ3κ31ε3t+κ32r3ε3t2‐W^hx,ξ̇3=κ31r3ε32t+κ32r32ε32t2,where N3ξ3=eξ32cosπξ3/2 is a Nussbaum function [30]; κ31 and κ32 are design parameters.Invoking (41), (44) becomes
(47)V̇≤r1ε1tδ¯2ℏ2t−k1ε1t+r2ε2tδ¯3ℏ3t−k2ε2t+ε3r3∑0+ε3r3AcN3ξ3κ31ε3t+κ32r3ε3t2+εmax22+ε3r3Ac¯22.Based on Young’s inequality, it is easy to getε3tr3∑0≤κ32r32ε32t/2+∑02/2κ32.Then (47) becomes
(48)V̇≤r1ε1tδ¯2ℏ2t−k1ε1t+r2ε2tδ¯3ℏ3t−k2ε2t+κ32r32ε32t2+∑022κ32+ε3r3AcN3ξ3κ31ε3t+κ32r3ε3t2+εmax22+ε3r3Ac¯22=r1ε1tδ¯2ℏ2t−k1ε1t+r2ε2tδ¯3ℏ3t−k2ε2t++κ32r32ε32t2+∑022κ32+AcN3ξ3ξ̇3+εmax22+ε3r3Ac¯22=r1ε1tδ¯2ℏ2t−k1ε1t+r2ε2tδ¯3ℏ3t−k2ε2t+−κ31r32ε32t+κ31r32ε32t+κ32r32ε32t2+∑022κ32+AcN3ξ3ξ̇3+εmax22+ε3r3Ac¯22=r1ε1tδ¯2ℏ2t−k1ε1t+r2ε2tδ¯3ℏ3t−k2ε2t+AcN3ξ3ξ̇3+ξ̇3−κ31r32ε32t+∑022κ32+εmax22+ε3r3Ac¯22.It is obvious to observe from (48) that ε1t and ε2t are bounded if ε1t≤δ¯2ℏ20/k1 and ε2t≤δ¯3ℏ30/k2. Else if ε1t>δ¯2ℏ20/k1 and ε2t>δ¯3ℏ30/k2, we obtain r1ε1tδ¯2ℏ20−k1ε1t<0 and r2ε2tδ¯3ℏ30−k2ε2t<0. Moreover, there exist adequately small constants 0<k3<r1k1 and 0<k4<r2k2 such that r1ε1tδ¯2ℏ20−k1ε1t+k3ε12t<0 andr2ε2tδ¯3ℏ30−k2ε2t+k4ε22t<0.Thus, (48) becomes
(49)V̇≤r1ε1tδ¯2ℏ2t−k1ε1t+k3ε12t−k3ε12t+r2ε2tδ¯3ℏ3t−k2ε2t+k4ε22t−k4ε22t+AcN3ξ3ξ̇3+ξ̇3−κ31r32ε32t+∑022κ32+εmax22+ε3r3Ac¯22≤−k3ε12t−k4ε22t−κ31−Ac¯22r32ε32t+AcN3ξ3ξ̇3+ξ̇3+∑022κ32+εmax22≤−ιhV+AcN3ξ3ξ̇3+ξ̇3+∑022κ32+εmax22.Withι1=min2k3,2k4,κ31r3−r3Ac¯/22, κ31>r3Ac¯/22.Multiplyingeι1t on both sides of (49) leads to
(50)ddxVeι1t≤AcN3ξ3ξ̇3eι1t+ξ̇3eι1t+∑022κ32+εmax22.Integrating (45) over 0,t, we obtain
(51)0≤V≤V0+e−ι1t∫0tAcN3ξ3ξ̇3eι1τdτ+e−ι1t∫0tξ̇3eι1τdτ+∫0t∑022κ32+εmax22e−ι1t−τdτ.Noticing∫0t∑02/2κ32+δmax/22e−ι1t−τdτ=∑02/2κ32+εmax/221/ι11−e−ι1t∈0,∑02/2κ32+εmax/22, we know that ∫0t∑02/2κ32+εmax/22e−ι1t−τdτ is bounded. Furthermore, (46) becomes
(52)0≤V≤V0+∑022κ32+εmax22+e−ι1t∫0tAcN3ξ3ξ̇3eι1τdτ+e−ι1t∫0tξ̇3eι1τdτ.Invoking lemmas 1 and 2 presented in [30, 31], we know that V, e−ι1t∫0tAcN3ξ3ξ̇3eι1τdτ, and e−ι1t∫0tξ̇3eι1τdτ are all bounded. Thus, all the signals involved are bounded. From the boundedness of ε1,ε2, and ε3, we know that there exist positive constants ε¯1,ε¯2,ε¯3 such that ε1≤ε¯1,ε2≤ε¯2, and ε3≤ε¯3. Furthermore, ‐δ¯iℏit<−δ¯ieεit+δi¯e−εit/eεit+e−εitℏit≤eit≤δ¯ieεit−δi¯e−εit/eεit+e−εitℏit<δi¯ℏit, i=1,2,3. It is obvious that the prescribed performance for e1,e2, and e3 is guaranteed.The design procedure of GSP is completed.Remark 6.
It should be noted that the controller design is not relying on model of GSP, which guarantees the robustness against model uncertain and unknown disturbance.Remark 7.
Though similar control framework is designed as reference [31], compared with [31], the robustness against compound disturbances and unknown control directions is enhanced in large scale. Especially, the stability of system under compound disturbances is proven. However, in [31], the stability analysis as well as controller design did not consider any disturbances, which was not rigorous.


## 3.4.1. Model Transformation

The first step of controller design is to transfer the mathematical model in (1). The mathematical model will be transferred into pure-feedback formation.Definey1=x1, y2=x2, and y3=Cmx3/JL−Tturb/JL, then we have
(17)ẏ1=y2,ẏ2=y3,ẏ3=f1y2,y3+f2v+fd,where f1y2,y3=−Ce/Lay2−RaJL/LaCmy3Cm/JL, and f2v=CmkPWM/JLLacvt+dv is control item, while fd=Cm/JL−Ce/Laϑ̇−Ṫturb/Cm−Ra/CmLaTturb represents transferred disturbance.Then, separating the linear part of control item from nonlinear part, the following equation is acquired.(18)ẏ1=y2,ẏ2=y3,ẏ3=f1y2,y3+Acvt+f2dv+fd,where the constant A=CmkPWM/JLLa, c is unknown gain, and f2d=CmkPWM/JLLadv is unknown function regarded as disturbance to be estimated.Remark 4.
Aiming at the problem of unknown complex disturbances, a RBFNN will be adopted. Because of unknown control directions, RBFNN cannot be applied directly. Through equivalent transformation, unknown gainc becomes a part of unknown functions to be estimated.Thus, the model of GSP in (1) is transformed into pure-feedback system in (18).


## 3.4.2. Control Law Design

Assumption 2.
We assume that the angle reference signalθd is limited, while its derivative and its second-order derivative are also limited. Furthermore, the disturbance of basement ϑ̇ is also limited.Step 1.
Define the angle tracking errore1.
(19)e1=y1−y1d.Define a performance functionℏ1t=ℏ01−ℏ1∝e−l1t+ℏ1∝ to restrain e1.
(20)‐δ¯1ℏ1t<e1t<δ¯1ℏ1t,where ℏ01>0, ℏ1∝>0, l1>0, 0<δ1¯<1, and 0<δ1¯<1 are design parameters and satisfying ℏ01>ℏ1∝, ‐δ¯1ℏ1t<e10<δ¯1ℏ1t, and ℏ1∝<ℏ1t<ℏ10.We convert (20) into the following formulation:
(21)e1=μ1ε1tℏ1t,where μ1ε1t=δ¯1eε1t−δ¯1e−ε1t/eε1t+e−ε1t∈−δ1¯,δ1¯ is a transformed function and ε1t is transformed error. From (14), we get the formulation of ε1t
(22)ε1t=μ1−1ε1t=12lne1/ℏ1t+δ1¯δ1¯−e1/ℏ1t.Furthermore,ε̇1t is derived as
(23)ε̇1t=r1ė1−e1ℏ̇1tℏ1t=r1ẏ1−ẏ1d−e1ℏ̇1tℏ1t=r1y2−ẏ1d−e1ℏ̇1tℏ1t,where r1=1/2ℏ1t1/δ¯1e1/ℏ1t+δ¯1−1/δ¯1e1/ℏ1t−δ¯1≥1/2δ¯1ℏ1t+1/2δ¯1ℏ1t>0.The virtual control law is designed as(24)z2d=−k1ε1t+ẏ1d+e1ℏ̇1tℏ1t,where k1 is a design parameter, ℏ̇1t=−l1ℏ10−ℏ1∝e−l1t.Step 2.
Define angular velocity tracking errore2.
(25)e2=y2−y2d.Construct a performance functionℏ2t=ℏ02−ℏ2∝e−l2t+ℏ2∝>0 to restrain e2.
(26)‐δ¯2ℏ2t<e2t<δ¯2ℏ2t.where ℏ02>0, ℏ2∝>0, l2>0, 0<δ2¯<1, and 0<δ2¯<1 are design parameters and satisfying ℏ02>ℏ2∝, ‐δ2¯ℏ2t<e20<δ2¯ℏ2t, and ℏ2∝<ℏ2t<ℏ20.Equation (26) is further converted into the following formulation:
(27)e2=μ2ε2tℏ2t.where μ2ε2t=δ¯2eε2t−δ2¯e−ε2t/eε2t+e−ε2t∈−δ2¯,δ2¯ is a transformed function and ε2t is transformed error. From (27), we can get
(28)ε2t=μ2−1ε2t=12lne2/ℏ2t+δ2¯δ2¯−e2/ℏ2t.The time derivative ofε2t is
(29)ε̇2t=r2ė2−e2ℏ̇2tℏ2t=r2ẏ2−ẏ2d−e2ℏ̇2tℏ2t=r2y3−ẏ2d−e2ℏ̇2tℏ2t,where r2=1/2ℏ2t1/δ2¯e2/ℏ2t+δ2¯−1/δ2¯e2/ℏ2t−δ2¯≥1/2δ2¯ℏ2t+1/2δ2¯ℏ2t>0.The virtual control law is devised as(30)y3d=−k2ε2t+ẏ2d+e2ℏ̇2tℏ2t,where k2 is a design parameter, ℏ̇2t=−l2ℏ20−ℏ2∝e−l2t.Step 3.
Define current tracking errore3.
(31)e3=y3−y3d.Construct a performance functionℏ3t=ℏ03−ℏ3∝e−l3t+ℏ3∝>0 to restrain e3.
(32)‐δ3¯ℏ3t<e3t<δ3¯ℏ3t,where ℏ03>0, ℏ3∝>0, l3>0, 0<δ3¯<1, and 0<δ3¯<1 are design parameters and satisfying ℏ03>ℏ3∝, ‐δ3¯ℏ3t<e30<δ3¯ℏ3t, and ℏ3∝<ℏ3t<ℏ30.Equation (28) is further converted into the following formulation:
(33)e3=μ3ε3tℏ3t,where μ3ε3t=δ¯3eε3t−δ3¯e−ε3t/eε3t+e−ε3t∈−δ3¯,δ3¯ is a transformed function and ε3t is transformed error. From (33), we can get
(34)ε3t=μ3−1ε3t=12lne3/ℏ3t+δ3¯δ3¯−e3/ℏ3t.The time derivative ofε3t is
(35)ε̇3t=r3ė3−e3ℏ̇3tℏ3t=r3ẏ3−ẏ3d−e3ℏ̇3tℏ3t=r3f1y2,y3+Acvt+f2dv+fd−ẏ3d−e3ℏ̇3tℏ3t,where r3=1/2ℏ3t1/δ3¯e3/ℏ3t+δ3¯−1/δ3¯e3/ℏ3t−δ3¯≥1/2δ3¯ℏ3t+1/2δ3¯ℏ3t>0.Define Lyapunov function(36)V=ε122+ε222+ε322+1γ1W~TW~̇.Employing (23), (29), and (35), V̇ is acquired as follows:
(37)V̇=ε1ε̇1+ε2ε̇2+ε3ε̇3+1γ1W~TW~̇=ε1r1y2−ẏ1d−e1ℏ̇1ℏ1+ε2r2y3−ẏ2d−e2ℏ̇2ℏ2+ε3r3f1y2,y3+Acvt+f2dv+fd−ẏ3d−μ3ℏ̇3t+1γ1W~TW~̇=ε1r1y2−ẏ1d−e1ℏ̇1ℏ1+ε2r2y3−ẏ2d−e2ℏ̇2ℏ2+ε3r3Acvt+ε3r3f1y2,y3−ẏ3d−μ3ℏ̇3t+ε3r3f2dv+fd+1γ1W~TW~̇.Here, a RBFNN is applied to offset the disturbancefd as well as the unknown input function f2dv. Considering that the input of the system contains an unknown gain c, RBFNN cannot be applied directly. Through an equivalent transformation, the following equation is acquired.Remark 5.
The unknown compound disturbance isD=fd+f2dv/Ac, where fd=Cm/JL−Ce/Laϑ̇−Ṫturb/Cm−Ra/CmLaTturb, f2d=CmkPWM/JLLadv, Tturb is friction disturbance, and it is one part of unknown compound disturbance. Besides, the unknown function of unknown hysteresis dv is estimated too. The compound disturbance will be estimated in whole.
(38)V̇=ε1r1y2−ẏ1d−e1ℏ̇1ℏ1+ε2r2y3−ẏ2d−e2ℏ̇2ℏ2+ε3r3Acv1t−W^hx+ε3r3f1y2,y3−ẏ3d−μ3ℏ̇3t+ε3r3f2dv+fd+1γ1W~TW~̇=ε1r1y2−ẏ1d−e1ℏ̇1ℏ1+ε2r2y3−ẏ2d−e2ℏ̇2ℏ2+ε3r3f1y2,y3−ẏ3d−μ3ℏ̇3t+ε3r3Acv1t+ε3r3Ac−W^hx+f2dv+fdAc+1γ1W~TW~̇.Substituting (19), (25), and (31) into (38) leads to
(39)V̇=ε1tr1e2−k1ε1+ε2tr2e3−k2ε2t+ε3tr3Acv1t−W^hx+ε3tr3f1y2,y3−ẏ3d−μ3ℏ̇3t+ε3tr3f2dv+fd+1γ1W~TW~̇=r1μ2ε2ℏ2tε1t−k1ε12t+r2μ3ε3ℏ3tε2t−k2ε22t+ε3r3Acv1t−W^hx+ε3r3f1y2,y3−ẏ3d−μ3ℏ̇3t+ε3r3f2dv+fd+1γ1W~TW~̇.Define a total disturbanceD=fd+f2dv/Ac, we mean that D=W^hx+ε, where x=x1,x2,x3. That is to say, when ε is small enough, W^hx will converge to D in high precision [26]. Thus, vt=v1t−W^hx.To analyze disturbance item individually, define a new function as follows.(40)V̇d=ε3r3Ac−W^hx+f2dv+fdAc+1γ1W~TW~̇≤ε3r3AcW~hx+εmax+1γ1W~TW~̇.Define a positive constantc¯ satisfying c≤c¯. Applying Young’s inequality [24], we can acquire that
(41)V̇d≤W~hx+εmax22+ε3r3Ac¯22+1γ1W~TW~̇=εmax22+ε3r3Ac¯22+W~TW~hxThx2+εmaxhx+W~̇γ1.SinceW∗ is a constant, it should be noted that W~̇=W^̇.According to (41), adaption law W^̇ is designed as follows:
(42)W~̇=‐γ1W~hxThx2+εmaxhx.Then, (41) becomes
(43)V̇d≤εmax22+ε3r3Ac¯22.Combining with (39), the following inequality is acquired.
(44)V̇≤r1μ2ε2ℏ2tε1t−k1ε12t+r2μ3ε3ℏ3tε2t−k2ε22t+ε3r3f1y2,y3−ẏ3d−μ3ℏ̇3t+ε3r3Acv1t+εmax22+ε3r3Ac¯22.It is not hard to conclude that there exists a positive constant∑0 such that f1y2,y3−ẏ3d−μ3ℏ̇3t≤∑0. Furthermore, we know that ℏ2∝<ℏ2t<ℏ20,ℏ3∝<ℏ3t<ℏ30, r1≥1/2ℏ1t1/δ1+1/δ1>0,r2≥1/2ℏ2t1/δ2+1/δ2¯>0, μ2ε2t=δ¯2eε2t−δ2¯e−ε2t/eε2t+e−ε2t∈−δ2¯,δ2¯, μ3ε3t=δ¯3eε3t−δ3¯e−ε3t/eε3t+e−ε3t∈−δ3¯,δ3¯.Then (44) becomes
(45)V̇≤r1ε1tδ¯2ℏ2t−k1ε1t+r2ε2tδ¯3ℏ3t−k2ε2t+ε3r3∑0+ε3r3Acv1t+εmax22+ε3r3Ac¯22.Finally, the actual control law is chosen as(46)v1t=N3ξ3κ31ε3t+κ32r3ε3t2‐W^hx,ξ̇3=κ31r3ε32t+κ32r32ε32t2,where N3ξ3=eξ32cosπξ3/2 is a Nussbaum function [30]; κ31 and κ32 are design parameters.Invoking (41), (44) becomes
(47)V̇≤r1ε1tδ¯2ℏ2t−k1ε1t+r2ε2tδ¯3ℏ3t−k2ε2t+ε3r3∑0+ε3r3AcN3ξ3κ31ε3t+κ32r3ε3t2+εmax22+ε3r3Ac¯22.Based on Young’s inequality, it is easy to getε3tr3∑0≤κ32r32ε32t/2+∑02/2κ32.Then (47) becomes
(48)V̇≤r1ε1tδ¯2ℏ2t−k1ε1t+r2ε2tδ¯3ℏ3t−k2ε2t+κ32r32ε32t2+∑022κ32+ε3r3AcN3ξ3κ31ε3t+κ32r3ε3t2+εmax22+ε3r3Ac¯22=r1ε1tδ¯2ℏ2t−k1ε1t+r2ε2tδ¯3ℏ3t−k2ε2t++κ32r32ε32t2+∑022κ32+AcN3ξ3ξ̇3+εmax22+ε3r3Ac¯22=r1ε1tδ¯2ℏ2t−k1ε1t+r2ε2tδ¯3ℏ3t−k2ε2t+−κ31r32ε32t+κ31r32ε32t+κ32r32ε32t2+∑022κ32+AcN3ξ3ξ̇3+εmax22+ε3r3Ac¯22=r1ε1tδ¯2ℏ2t−k1ε1t+r2ε2tδ¯3ℏ3t−k2ε2t+AcN3ξ3ξ̇3+ξ̇3−κ31r32ε32t+∑022κ32+εmax22+ε3r3Ac¯22.It is obvious to observe from (48) that ε1t and ε2t are bounded if ε1t≤δ¯2ℏ20/k1 and ε2t≤δ¯3ℏ30/k2. Else if ε1t>δ¯2ℏ20/k1 and ε2t>δ¯3ℏ30/k2, we obtain r1ε1tδ¯2ℏ20−k1ε1t<0 and r2ε2tδ¯3ℏ30−k2ε2t<0. Moreover, there exist adequately small constants 0<k3<r1k1 and 0<k4<r2k2 such that r1ε1tδ¯2ℏ20−k1ε1t+k3ε12t<0 andr2ε2tδ¯3ℏ30−k2ε2t+k4ε22t<0.Thus, (48) becomes
(49)V̇≤r1ε1tδ¯2ℏ2t−k1ε1t+k3ε12t−k3ε12t+r2ε2tδ¯3ℏ3t−k2ε2t+k4ε22t−k4ε22t+AcN3ξ3ξ̇3+ξ̇3−κ31r32ε32t+∑022κ32+εmax22+ε3r3Ac¯22≤−k3ε12t−k4ε22t−κ31−Ac¯22r32ε32t+AcN3ξ3ξ̇3+ξ̇3+∑022κ32+εmax22≤−ιhV+AcN3ξ3ξ̇3+ξ̇3+∑022κ32+εmax22.Withι1=min2k3,2k4,κ31r3−r3Ac¯/22, κ31>r3Ac¯/22.Multiplyingeι1t on both sides of (49) leads to
(50)ddxVeι1t≤AcN3ξ3ξ̇3eι1t+ξ̇3eι1t+∑022κ32+εmax22.Integrating (45) over 0,t, we obtain
(51)0≤V≤V0+e−ι1t∫0tAcN3ξ3ξ̇3eι1τdτ+e−ι1t∫0tξ̇3eι1τdτ+∫0t∑022κ32+εmax22e−ι1t−τdτ.Noticing∫0t∑02/2κ32+δmax/22e−ι1t−τdτ=∑02/2κ32+εmax/221/ι11−e−ι1t∈0,∑02/2κ32+εmax/22, we know that ∫0t∑02/2κ32+εmax/22e−ι1t−τdτ is bounded. Furthermore, (46) becomes
(52)0≤V≤V0+∑022κ32+εmax22+e−ι1t∫0tAcN3ξ3ξ̇3eι1τdτ+e−ι1t∫0tξ̇3eι1τdτ.Invoking lemmas 1 and 2 presented in [30, 31], we know that V, e−ι1t∫0tAcN3ξ3ξ̇3eι1τdτ, and e−ι1t∫0tξ̇3eι1τdτ are all bounded. Thus, all the signals involved are bounded. From the boundedness of ε1,ε2, and ε3, we know that there exist positive constants ε¯1,ε¯2,ε¯3 such that ε1≤ε¯1,ε2≤ε¯2, and ε3≤ε¯3. Furthermore, ‐δ¯iℏit<−δ¯ieεit+δi¯e−εit/eεit+e−εitℏit≤eit≤δ¯ieεit−δi¯e−εit/eεit+e−εitℏit<δi¯ℏit, i=1,2,3. It is obvious that the prescribed performance for e1,e2, and e3 is guaranteed.The design procedure of GSP is completed.Remark 6.
It should be noted that the controller design is not relying on model of GSP, which guarantees the robustness against model uncertain and unknown disturbance.Remark 7.
Though similar control framework is designed as reference [31], compared with [31], the robustness against compound disturbances and unknown control directions is enhanced in large scale. Especially, the stability of system under compound disturbances is proven. However, in [31], the stability analysis as well as controller design did not consider any disturbances, which was not rigorous.


## 4. Simulation and Analysis

To verify the effectiveness of the proposed control scheme, a seeker servo system is chosen as simulation case.The parameters of the control plant are shown in Table1.Table 1
Model Parameters.
Parameter
Value
Parameter
Value
JL
1.2×10−3kg⋅m2
Ce
0.75
Cm
0.625N⋅m/A
vs
0.2
La
0.0062H
FC
0.03
Ra
5.1Ω
FS
0.05
kPWM
3.75
Bv
0.017The control parameters of controller are chosen as follows:k1=20,k2=15,κ31=−6,κ32=3, δ1¯=1,δ1¯=1,ℏ01=5,ℏ1∝=0.1,l1=0.6,δ2¯=1,δ2¯=1,ℏ02=15,ℏ2∝=0.15,l2=0.5,δ3¯=1,δ3¯=1,ℏ03=30,ℏ3∝=5, and l3=0.3. Moreover, all the model coefficients in (1) are assumed to be uncertain by defining C=C01+0.3cos2t, where C is the uncertain coefficient and C0 is the nominal value of C. While the parameters of RBFNN are chosen as γ1=100,εmax=0.001, the number of node is 100, and the center of RBFNN is evenly spaced in c∈−5,4.9, the width is chosen as b=10.We choosesint as angular reference signal, while the initial disalignment angle is 0.5rad. Because the GSP possesses great isolation towards high-frequency signals [32], the basement disturbance is ϑ̇=0.5cos2trad/s. The tracking performance of the presented control approach is depicted in Figures 3–6.Figure 3
Angle tracking performance.Figure 4
Angle tracking error.Figure 5
Angular velocity tracking error.Figure 6
Current tracking error.Figures4–6 show that all tracking errors are forced within the prescribed bounds in the presence of compound disturbances. Thus, the control object is achieved. Furthermore, we can observer from Figures 4 and 5 that the system output of the proposed method converges faster and the precision is higher. Results show that the steady error of angle and angular velocity are no more than 0.02rad and 0.12rad/s, respectively. It indicates that better transient and steady performance can be achieved through designed controller. Besides, the controller output is shown in Figure 7; because of nonlinear friction disturbance, the controller output is nonsmooth at velocity zero point. Especially, the designed controller is independent of precise model, and the method is able to achieve off-line estimation of unknown certain disturbance and the method enables great robustness against uncertain disturbance.Figure 7
Controller output.


## 5. Conclusion

In this paper, a prescribed performance controller based on RBFNN is exploited for GSP with the backstepping framework. The problem of unknown hysteresis, unknown friction, and unknown compound disturbance is solved through a model transformation and RBFNN estimation of compound disturbances. Furthermore, Nussbaum function is adopted in this paper, which overcomes the problem of GSP with “unknown control directions”. By constructing performance functions, angle and angular velocity tracking errors are restrained within the prescribed boundaries. The stability of the proposed method is proved via Lyapunov stability synthesis. Finally, the tracking performance and superiority of the designed method are proved through simulations. Especially, the research of GSP can be improved by considering interactions between gimbals, which will be more meaningful in practical systems.



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