# RBFNN-Based Singularity-Free Terminal Sliding Mode Control for Uncertain Quadrotor UAVs

Authors: Meiling Tao; Xiongxiong He; Shuzong Xie; Qiang Chen
Journal: Computational Intelligence and Neuroscience
 (2021)

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## Abstract

In this article, a singularity-free terminal sliding mode (SFTSM) control scheme based on the radial basis function neural network (RBFNN) is proposed for the quadrotor unmanned aerial vehicles (QUAVs) under the presence of inertia uncertainties and external disturbances. Firstly, a singularity-free terminal sliding mode surface (SFTSMS) is constructed to achieve the finite-time convergence without any piecewise continuous function. Then, the adaptive finite-time control is designed with an auxiliary function to avoid the singularity in the error-related inverse matrix. Moreover, the RBFNN and extended state observer (ESO) are introduced to estimate the unknown disturbances, respectively, such that prior knowledge on system model uncertainties is not required for designing attitude controllers. Finally, the attitude and angular velocity errors are finite-time uniformly ultimately bounded (FTUUB), and numerical simulations illustrated the satisfactory performance of the designed control scheme.


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## Body

## 1. Introduction

Due to the advantages of a large range of applications of QUAVs, the attitude control of QUAVs has received extensive attention in many fields. Designing a suitable control law according to the type of flight mission is important for the study of QUAVs attitude control [1]. However, the control scheme proposed in [1] only achieves the asymptotic convergence of the attitude or tracking error, which means that the system states converge to zero within an infinite time. In practical applications, the controller design needs specific requirements for convergence speed and control accuracy. Thus, finite-time control (FTC) methods have been proposed to deal with the control problems of different nonlinear systems [2–4]. In [2], a parameter update law was constructed to compensate for parameter uncertainty, and an adaptive sliding mode control (SMC) scheme was constructed to ensure the finite-time convergence of the tracking error of QUAV. In [3], combining integral backstepping technology and terminal SMC, a FTC method was designed to guarantee the position and attitude tracking stability of QUAVs, such that the system states are semiglobal practical FTUUB. In [4], a FTC scheme was proposed to guarantee the finite-time convergence of position and attitude states in the QUAV tracking error system.To solve the dependence of the FTC on a nonlinear system model, multiple control methods have the capability to approximate nonlinear functions, which is used to implement the estimation task of the nonlinear system, i.e., adaptive control [5] and optimal control [6]. In the past years, a method called RBFNN has been widely introduced to approximate the dynamic parameters of nonlinear systems, such that no prior knowledge of model information is required in [7, 8]. In [7], a RBFNN method was devolved to estimate the unknown model uncertainties of robot manipulators. In [8], a RBFNN-based SMC scheme was presented to guarantee the asymptotic convergence of the system states under the model uncertainties. Compared with other estimation methods [5, 6], RBFNN has faster convergence speed and local approximation capability to avoid local minima problems. Thus, RBFNN is more suitable for real-time control, such as QUAV attitude control.Moreover, for complex coupled systems, extended state observer (ESO) as an alternative approach to solve the bounded disturbances is used to decouple the system by treating the coupling terms as a part of the lumped uncertainties. Due to its satisfactory disturbance estimation, ESO-based controllers are widely applied in different practical nonlinear systems, such as rigid spacecrafts [9] and robot manipulators [10]. In [9], a SMC based on backstepping and ESO methods was used to achieve a faster convergence in the rigid spacecraft attitude control system. In [10], an FTC with the model-assisted ESO was used to compensate the bounded uncertainties and guarantee the finite-time convergence of the system states. Although RBFNN and ESO have been successfully applied to a variety of uncertain nonlinear systems [7–10], it is less used in QUAVs.Inspired by the above discussions, an RBFNN-based finite-time adaptive attitude tracking controller is designed for the attitude tracking problem of QUAVs with inertial uncertainty and unknown external disturbances, and the main contributions are summarized in the following:(i)
Instead of employing any piecewise continuous functions, a SFTSMS is proposed to avoid the singularity directly in the differential of the sliding variable(ii)
An auxiliary function is designed to handle a potential singularity resulted from the use of the error-related inverse matrix in the attitude controller design(iii)
By employing RBFNN and ESO to estimate the unknown dynamics, prior knowledge on system model uncertainties is not required in the controller design, and the tracking errors are FTUUB by the proposed control lawThe structure of this article is given as follows. The attitude model and necessary preliminaries are given in Section2. Section 3 shows the detailed design process of SFTSMS. Controller design and rigorous theoretical proofs are depicted in Sections 4 and 5, respectively. Section 6 shows the effective simulations, and the conclusion is given in Section 7.


## 2. Model Description and Preliminaries

### 2.1. Quadrotor Attitude Dynamics

As depicted in Figure1, the dynamics of the QUAV is represented as follows [11]:(1)q˙0=−12qvTω,q˙v=12qv×+q0I3ω,where q0∈R and qv=q1,q2,q3T∈R3×1 denote the scalar and vector elements of the unit quaternion Q=q0,qvTT, respectively, and I3∈R3×3 is the identity matrix, and the skew-symmetric matrix A×∈R3×3 is given by(2)A×=0−A3A2A30−A1−A2A10.Figure 1
The schematic of quadrotor UAVs.The reference attitude vector is defined asQr=q0r,q1r,q2r,q3rT∈R4×1, and Q˙r and Q¨r are bounded. The attitude tracking error eq is(3)eq=EQr,Q=2q0rq0+∑i=13qirqi×−q0rq1+q1rq0+q2rq3−q3rq2−q0rq2−q1rq3+q2rq0+q3rq1−q0rq3+q1rq2−q2rq2+q3rq0,and the dynamics model is expressed as(4)Jω˙=−ω×Jω+u+d,where ω∈R3×1 denotes the angular velocity, J∈R3×3 is a positive definite inertia matrix, u=u1,u2,u3T∈R3×1 represents the generalized control torque produced by rotating propellers, and d=d1,d2,d3T∈R3×1 denotes the unknown but bounded continuous external disturbances.DefiningJ=J0+ΔJ, (4) is expressed as(5)J0ω˙=−ω×J0ω−ω×ΔJω−ΔJω˙+u+d,where J0 and ΔJ are denoted as the nominal and the unknown inertia matrix, respectively.The angular velocity tracking error is(6)eω=ω−ωr,where ωr is given by(7)ωr=2−q2rq˙v1r+q0rq˙v1r+qv3rq˙v2r−qv2rq˙v3r−qv2rq˙0r−qv3rq˙v1r+q0rq˙v2r+qv1rq˙v3r−qv3rq˙0r+qv2rq˙v1r−qv1rq˙v2r+q0rq˙v3r.From (5)–(7), the tracking error model is expressed as(8)e˙q=eω,e˙ω=−J0−1ω×J0ω−J0−1ω×ΔJω−J0−1ΔJω˙+J0−1u+J0−1d−ω˙r.Remark 1.
Unit quaternion is able to represent the attitude uniquely because the equilibrium states correspond to a unique physical equilibrium orientation [12] in the QUAV.


### 2.2. RBFNN

In practical systems, neural networks (NNs) are online estimation techniques for unknown nonlinear uncertainties. Due to the approximation characteristics and faster learning convergence, RBFNN is widely used in the estimation of nonlinear functions in the field of control. This section will introduce the structure of RBFNN.RBFNN consists of three parts: input layer, output layer, and hidden layer. As shown in Figure2, x∈x1,x2,…,xnT is the neural network input vector, W∈W1,W2,…,WmT is the weight of the mth network node, y represents the output vector, and ϕx=ϕ1,ϕ2,…,ϕmT is the basis function, which can approximate nonlinear uncertainties with high precision through the linear combination of Gaussian functions, which is given by the following [7]:(9)ϕkx=exp−x−μkTx−μkak2,k=1,2,…,m,where μk is the center of the RBF and ak means the scaling parameter of the network node m.Figure 2
The structure of RBFNN.Thus, the output vectory is expressed as(10)y=∑k=1mWkϕk.Considering that RBFNN has good nonlinear approximation ability [13], the approximate system model of the nonlinear function F is(11)Fx=WTϕx+ε,where ε represents the estimation error.


### 2.3. Useful Lemmas

Lemma 1 (see [14]).
ForΛ1>0, Λ2>0, and 0<ι<1, a Lyapunov condition of finite-time stability is expressed as V˙x+Λ1Vx+Λ2Vιx≤0, where the settling time satisfies T0≤1/Λ11−ιlnΛ1V1−ιx0+Λ2/Λ2, where Vx0 represents the initial state of Vx.Lemma 2 (see [15]).
Givena1,a2,…,an>0 and p>0, the following relationships hold:(12)∑i=1naip≥n1−p∑i=1naip,p>1,∑i=1naip≥∑i=1naip,0<p<1.Lemma 3 (see [16]).
Given a continuous functionfx=xι−ιx with 1≤ι≤2, for any x>0, there exists the following inequality:(13)xιι≥x+1−ιι.


## 2.1. Quadrotor Attitude Dynamics

As depicted in Figure1, the dynamics of the QUAV is represented as follows [11]:(1)q˙0=−12qvTω,q˙v=12qv×+q0I3ω,where q0∈R and qv=q1,q2,q3T∈R3×1 denote the scalar and vector elements of the unit quaternion Q=q0,qvTT, respectively, and I3∈R3×3 is the identity matrix, and the skew-symmetric matrix A×∈R3×3 is given by(2)A×=0−A3A2A30−A1−A2A10.Figure 1
The schematic of quadrotor UAVs.The reference attitude vector is defined asQr=q0r,q1r,q2r,q3rT∈R4×1, and Q˙r and Q¨r are bounded. The attitude tracking error eq is(3)eq=EQr,Q=2q0rq0+∑i=13qirqi×−q0rq1+q1rq0+q2rq3−q3rq2−q0rq2−q1rq3+q2rq0+q3rq1−q0rq3+q1rq2−q2rq2+q3rq0,and the dynamics model is expressed as(4)Jω˙=−ω×Jω+u+d,where ω∈R3×1 denotes the angular velocity, J∈R3×3 is a positive definite inertia matrix, u=u1,u2,u3T∈R3×1 represents the generalized control torque produced by rotating propellers, and d=d1,d2,d3T∈R3×1 denotes the unknown but bounded continuous external disturbances.DefiningJ=J0+ΔJ, (4) is expressed as(5)J0ω˙=−ω×J0ω−ω×ΔJω−ΔJω˙+u+d,where J0 and ΔJ are denoted as the nominal and the unknown inertia matrix, respectively.The angular velocity tracking error is(6)eω=ω−ωr,where ωr is given by(7)ωr=2−q2rq˙v1r+q0rq˙v1r+qv3rq˙v2r−qv2rq˙v3r−qv2rq˙0r−qv3rq˙v1r+q0rq˙v2r+qv1rq˙v3r−qv3rq˙0r+qv2rq˙v1r−qv1rq˙v2r+q0rq˙v3r.From (5)–(7), the tracking error model is expressed as(8)e˙q=eω,e˙ω=−J0−1ω×J0ω−J0−1ω×ΔJω−J0−1ΔJω˙+J0−1u+J0−1d−ω˙r.Remark 1.
Unit quaternion is able to represent the attitude uniquely because the equilibrium states correspond to a unique physical equilibrium orientation [12] in the QUAV.


## 2.2. RBFNN

In practical systems, neural networks (NNs) are online estimation techniques for unknown nonlinear uncertainties. Due to the approximation characteristics and faster learning convergence, RBFNN is widely used in the estimation of nonlinear functions in the field of control. This section will introduce the structure of RBFNN.RBFNN consists of three parts: input layer, output layer, and hidden layer. As shown in Figure2, x∈x1,x2,…,xnT is the neural network input vector, W∈W1,W2,…,WmT is the weight of the mth network node, y represents the output vector, and ϕx=ϕ1,ϕ2,…,ϕmT is the basis function, which can approximate nonlinear uncertainties with high precision through the linear combination of Gaussian functions, which is given by the following [7]:(9)ϕkx=exp−x−μkTx−μkak2,k=1,2,…,m,where μk is the center of the RBF and ak means the scaling parameter of the network node m.Figure 2
The structure of RBFNN.Thus, the output vectory is expressed as(10)y=∑k=1mWkϕk.Considering that RBFNN has good nonlinear approximation ability [13], the approximate system model of the nonlinear function F is(11)Fx=WTϕx+ε,where ε represents the estimation error.


## 2.3. Useful Lemmas

Lemma 1 (see [14]).
ForΛ1>0, Λ2>0, and 0<ι<1, a Lyapunov condition of finite-time stability is expressed as V˙x+Λ1Vx+Λ2Vιx≤0, where the settling time satisfies T0≤1/Λ11−ιlnΛ1V1−ιx0+Λ2/Λ2, where Vx0 represents the initial state of Vx.Lemma 2 (see [15]).
Givena1,a2,…,an>0 and p>0, the following relationships hold:(12)∑i=1naip≥n1−p∑i=1naip,p>1,∑i=1naip≥∑i=1naip,0<p<1.Lemma 3 (see [16]).
Given a continuous functionfx=xι−ιx with 1≤ι≤2, for any x>0, there exists the following inequality:(13)xιι≥x+1−ιι.


## 3. SFTSMS and Auxiliary Function Design

### 3.1. SFTSMS

A SFTSMS is constructed as(14)S=sigeq+Λ12−asig2−aeω+Λ2sigeq,where S=S1,S2,S3T∈R3×1, Λ1>0, Λ2>0, and 0<a=a1/a2<1, a1 and a2 are positive odd integers, and the term sigrx is given by(15)sigrx=x1rsgnx1,x2rsgnx2,x3rsgnx3T,where x=x1,x2,x3TR3×1 and r>0.The time derivative of (14) is expressed as(16)S˙=eω+Λ1eω+Λ2sigeq1−a⋅e˙ω+Λ2eω.Due to the facts that1−a>0, the singularity will not occur in (16).WhenS=0, (14) is rewritten as(17)sigeq+Λ12−asig2−aeω+Λ2sigeq=0.According to2−a>1, the following equation is obtained as(18)2−aΛ11/2−asig1/2−aeq=−eω+Λ2sigeq.From (18), the equivalent equation of (14) is obtained as follows [12]:(19)s=eω+Λ2sigeq+2−aΛ11/2−asig1/2−aeq=0.From (8) and (19), one has(20)e˙q=−Λ2sigeq−2−aΛ11/2−asig1/2−aeq.To illustrateeq is finite-time convergent, a Lyapunov function is chosen as(21)V1=12eqTeq.From (20), the time derivative of V1 is(22)V˙1=eqTe˙q=eqT−Λ2sigeq−2−aΛ11/2−asig1/2−aeq=−Λ2eq2−2−aΛ11/2−aeq1+1/2−a=−Λ¯1V1−Λ¯2V1ι0,where Λ¯1=2Λ2, Λ¯2=23−a/4−2a2−a/Λ11/2−a, and ι0=1/2+1/4−2a.According to Lemma1 and (8), eq and eω converge to the equilibrium within a finite-time Ts satisfying(23)Ts≤1Λ¯11−ι0lnΛ¯1V11−ι0eq0+Λ¯2Λ¯2,where eq0 represents the initial states of eqt.This completes the proof.


### 3.2. Auxiliary Function Design

Substituting (8) into (16) yields(24)S˙=eω+Λ1eω+Λ2sigeq1−a⋅−J0−1ω×J0ω−J0−1ω×ΔJω−J0−1ΔJω˙+J0−1Eu+Dt−ω˙r+Λ2eω=eω+YJ0−1u+D−J0−1ω×J0ω−J0−1ω×ΔJω−J0−1ΔJω˙−ω˙r+Λ2eω=eω+YF0+D+J0−1u,where Y=diagY1,Y2,Y3∈R3×3, Yi=Λ1Θi1−a with i=1,2,3, eq=eq1,eq2,eq3T, Θ=eω+Λsigeq, and F0=Y−1⋅eω+D−J0−1ω×J0ω−J0−1ω×ΔJω−J0−1ΔJω˙−ω˙r+Λ2eω.Due to the existence of the term ofF0 in the expression of Y−1, it may cause the potential singularity issue when Θ=0. Consequently, an auxiliary function φ is constructed to solve the singularity caused by Y−1 in the controller design.Thus, (24) is expressed as(25)S˙=YF0+J0−1u+eω+Λ2S−Λ2S=YJ0−1u+D−J0−1ω×J0ω−J0−1ω×ΔJω−J0−1ΔJω˙−ω˙r+Y⋅Λ2eω+eω+Λ2S−Λ2S.From the definition ofS and Θ, it has(26)eω+Λ2S=eω+Λ2sigeq+Λ12−asig2−aΘ=eω+Λ2sigeq+Λ2⋅Λ12−asig2−aΘ=Y⋅1Λ1sigaΘ+Λ22−aΘ.From (26), the auxiliary function φ is defined as(27)φ=1Λ1sigaΘ+Λ22−aΘ,and (25) can be rewritten as(28)S˙=ϒ⋅J0−1u+D−J0−1ω×J0ω−J0−1ω×ΔJω−J0−1ΔJω˙−ω˙r+φ+Λ2eω−Λ2S=ϒJ0−1u+F+D−Λ2S,where F denotes the lumped uncertainties presented by(29)F=−J0−1ω×J0ω−J0−1ω×ΔJω−J0−1ΔJω˙−ω˙r+φ+Λ2eω.


## 3.1. SFTSMS

A SFTSMS is constructed as(14)S=sigeq+Λ12−asig2−aeω+Λ2sigeq,where S=S1,S2,S3T∈R3×1, Λ1>0, Λ2>0, and 0<a=a1/a2<1, a1 and a2 are positive odd integers, and the term sigrx is given by(15)sigrx=x1rsgnx1,x2rsgnx2,x3rsgnx3T,where x=x1,x2,x3TR3×1 and r>0.The time derivative of (14) is expressed as(16)S˙=eω+Λ1eω+Λ2sigeq1−a⋅e˙ω+Λ2eω.Due to the facts that1−a>0, the singularity will not occur in (16).WhenS=0, (14) is rewritten as(17)sigeq+Λ12−asig2−aeω+Λ2sigeq=0.According to2−a>1, the following equation is obtained as(18)2−aΛ11/2−asig1/2−aeq=−eω+Λ2sigeq.From (18), the equivalent equation of (14) is obtained as follows [12]:(19)s=eω+Λ2sigeq+2−aΛ11/2−asig1/2−aeq=0.From (8) and (19), one has(20)e˙q=−Λ2sigeq−2−aΛ11/2−asig1/2−aeq.To illustrateeq is finite-time convergent, a Lyapunov function is chosen as(21)V1=12eqTeq.From (20), the time derivative of V1 is(22)V˙1=eqTe˙q=eqT−Λ2sigeq−2−aΛ11/2−asig1/2−aeq=−Λ2eq2−2−aΛ11/2−aeq1+1/2−a=−Λ¯1V1−Λ¯2V1ι0,where Λ¯1=2Λ2, Λ¯2=23−a/4−2a2−a/Λ11/2−a, and ι0=1/2+1/4−2a.According to Lemma1 and (8), eq and eω converge to the equilibrium within a finite-time Ts satisfying(23)Ts≤1Λ¯11−ι0lnΛ¯1V11−ι0eq0+Λ¯2Λ¯2,where eq0 represents the initial states of eqt.This completes the proof.


## 3.2. Auxiliary Function Design

Substituting (8) into (16) yields(24)S˙=eω+Λ1eω+Λ2sigeq1−a⋅−J0−1ω×J0ω−J0−1ω×ΔJω−J0−1ΔJω˙+J0−1Eu+Dt−ω˙r+Λ2eω=eω+YJ0−1u+D−J0−1ω×J0ω−J0−1ω×ΔJω−J0−1ΔJω˙−ω˙r+Λ2eω=eω+YF0+D+J0−1u,where Y=diagY1,Y2,Y3∈R3×3, Yi=Λ1Θi1−a with i=1,2,3, eq=eq1,eq2,eq3T, Θ=eω+Λsigeq, and F0=Y−1⋅eω+D−J0−1ω×J0ω−J0−1ω×ΔJω−J0−1ΔJω˙−ω˙r+Λ2eω.Due to the existence of the term ofF0 in the expression of Y−1, it may cause the potential singularity issue when Θ=0. Consequently, an auxiliary function φ is constructed to solve the singularity caused by Y−1 in the controller design.Thus, (24) is expressed as(25)S˙=YF0+J0−1u+eω+Λ2S−Λ2S=YJ0−1u+D−J0−1ω×J0ω−J0−1ω×ΔJω−J0−1ΔJω˙−ω˙r+Y⋅Λ2eω+eω+Λ2S−Λ2S.From the definition ofS and Θ, it has(26)eω+Λ2S=eω+Λ2sigeq+Λ12−asig2−aΘ=eω+Λ2sigeq+Λ2⋅Λ12−asig2−aΘ=Y⋅1Λ1sigaΘ+Λ22−aΘ.From (26), the auxiliary function φ is defined as(27)φ=1Λ1sigaΘ+Λ22−aΘ,and (25) can be rewritten as(28)S˙=ϒ⋅J0−1u+D−J0−1ω×J0ω−J0−1ω×ΔJω−J0−1ΔJω˙−ω˙r+φ+Λ2eω−Λ2S=ϒJ0−1u+F+D−Λ2S,where F denotes the lumped uncertainties presented by(29)F=−J0−1ω×J0ω−J0−1ω×ΔJω−J0−1ΔJω˙−ω˙r+φ+Λ2eω.


## 4. Controller Design

### 4.1. ESO

The bounded external disturbancesD in (28) are estimated by the ESO. Considering the disturbances D as an extended state, system (28) is rewritten as(30)S˙=YJ0−1u+D+F−Λ2S,D˙=ht,where ht represents the derivative of D.Then, the second-order ESO for (30) is constructed as(31)E1=Z1−S,Z˙1=YJ0−1u+Z2+F−Λ2S−ℏ1E1,Z˙2=−ℏ2falE1,where E1∈R3×1 means the ESO’s estimation error, Z1,Z2∈R3×1 denote the observer outputs, and ℏ1 and ℏ1 represent the observer gains. The function fal⋅ is given by the following [17]:(32)falE1i=E1iγE1i,E1i>δ,E1iδ1−γ,E1i≤δ,where i=1,2,3, 0<γ<1, and δ>0.Define the observer error(33)E2=D−Z2.According to the analysis in [18], the observer error satisfies E2i≤ζm,i=1,2,3, where ζm=maxi=1,2,3E2i>0.


### 4.2. Finite-Time Controller Design

Use (11) to approximate the nonlinear uncertainties (29).(34)Fixi=Wi∗Tϕ1xi+εi,where xi∈eqi,e˙qi,ωr,ω˙rT∈R4 is the NN input vector, W∗∈R4 represents the ideal weight vector, εi is the approximation error satisfying εi≤εN, εN>0, and ϕ1x∈R4 denotes the Gaussian function (9).An exponential reaching law is given by the following [19]:(35)S˙=−K1S+K2⋅sigγS,where K1>0, K2>0, and 0<γ<1.With the unknown nonlinear uncertaintiesF and the disturbances D estimated by the RBFNN and ESO, respectively, the finite-time control law u is given by(36)u=J0−K1S−K2sigγS−W^Tϕ1x−Z2−K3YS,where K3>0, W^=diagW^1,W^2,W^3, and W^i is used to estimate W^i∗ with i=1,2,3.The update law ofW^i is given by(37)W^˙i=δiYiSiϕ1i−ϖiW^i,where δi>0 and ϖi>0 with i=1,2,3.


## 4.1. ESO

The bounded external disturbancesD in (28) are estimated by the ESO. Considering the disturbances D as an extended state, system (28) is rewritten as(30)S˙=YJ0−1u+D+F−Λ2S,D˙=ht,where ht represents the derivative of D.Then, the second-order ESO for (30) is constructed as(31)E1=Z1−S,Z˙1=YJ0−1u+Z2+F−Λ2S−ℏ1E1,Z˙2=−ℏ2falE1,where E1∈R3×1 means the ESO’s estimation error, Z1,Z2∈R3×1 denote the observer outputs, and ℏ1 and ℏ1 represent the observer gains. The function fal⋅ is given by the following [17]:(32)falE1i=E1iγE1i,E1i>δ,E1iδ1−γ,E1i≤δ,where i=1,2,3, 0<γ<1, and δ>0.Define the observer error(33)E2=D−Z2.According to the analysis in [18], the observer error satisfies E2i≤ζm,i=1,2,3, where ζm=maxi=1,2,3E2i>0.


## 4.2. Finite-Time Controller Design

Use (11) to approximate the nonlinear uncertainties (29).(34)Fixi=Wi∗Tϕ1xi+εi,where xi∈eqi,e˙qi,ωr,ω˙rT∈R4 is the NN input vector, W∗∈R4 represents the ideal weight vector, εi is the approximation error satisfying εi≤εN, εN>0, and ϕ1x∈R4 denotes the Gaussian function (9).An exponential reaching law is given by the following [19]:(35)S˙=−K1S+K2⋅sigγS,where K1>0, K2>0, and 0<γ<1.With the unknown nonlinear uncertaintiesF and the disturbances D estimated by the RBFNN and ESO, respectively, the finite-time control law u is given by(36)u=J0−K1S−K2sigγS−W^Tϕ1x−Z2−K3YS,where K3>0, W^=diagW^1,W^2,W^3, and W^i is used to estimate W^i∗ with i=1,2,3.The update law ofW^i is given by(37)W^˙i=δiYiSiϕ1i−ϖiW^i,where δi>0 and ϖi>0 with i=1,2,3.


## 5. Stability Analysis

Theorem 1.
Considering the tracking error system (8) and the control schemes (36) and (37), all the signals of the closed-loop system are UUB, and the sliding variable S and the tracking errors eq and eω are FTUUB, respectively.Proof.
Design a Lyapunov functionV2(38)V2=12STS+∑i=1312δiW˜iTW˜i,where W˜i=Wi∗−W^i.
From (28), differentiating (38) leads to(39)V˙2=STS˙−∑i=131δiW˜iTW^˙i=STYJ0−1u+F+D−ST⋅Λ2S−∑i=131δiW˜iTW^˙i.
Substituting (34) and (36) into (39), one has(40)V˙2=STYJ0−1⋅J0−K1S−K2sigγS−W^Tϕ1−Z2−K3YS+W∗Tϕ1+ε+D−Λ2∑i=13Si2−∑i=131δiW˜iTW^˙i≤STY−K1S−K2sigγS+W˜1Tϕ1+ε+D−Z2−K3YS−∑i=131δiW˜iTW^˙i,where ε=ε1,ε2,ε3T.
From (33), (40) is modified as(41)V˙2=STY−K1S−K2sigγS+W˜1Tϕ1+ε+E2−K3YS−∑i=131δiW˜iTW^˙i.
Substituting (37) into (40) yields(42)V˙2≤−K1∑i=13YiSi2−K2∑i=13YiSi1+γ−K3∑i=13Yi2Si2+εN∑i=13YiSi+ζm∑i=13YiSi+∑i=13ϖiW˜iTW^i≤−K3′∑i=13Yi2Si2−∑i=13K3″YiSi−εN+ζm2K3″2+∑i=13ϖiW˜iTW^i+3εN24K3″+3ζm24K3″,where K3=K3′+K3″.
From Young’s inequality, the following relationships hold:(43)ϖiW˜iTW^≤ϖiW˜iTWi∗−W˜i≤−ϖi2W˜i2+ϖi2Wi∗2.
Substituting (43) into (42) yields(44)V˙2≤−K3′∑i=13Yi2Si2−ϖi2W˜i2+ϖi2Wi∗2+3εN24K3″≤−μV2+Φ1,where μ=min2K3′Yi2,δiϖi and Φ1=ϖi/2Wi∗2+3εN2/4K3″.
According to (38)–(44), one can conclude that S, W˜1, W˜2, and W˜3 are UUB. From (20) and the bounded values W1∗, W2∗, and W3∗, the uniform ultimate boundedness of eq, eω, W^1, W^2, and W^3 is guaranteed, and thus Yi is also UUB. Due to (8) and (36), e˙ω and u both are UUB. Since ϕ1x is bounded in (9), one can conclude STYW˜Tϕ1≤η1, STYε≤η2, and STYE2≤η3, where η1, η2, and η3 are positive constants.
Then, design a Lyapunov function(45)V3=12STS,and from (28), V˙3 is(46)V˙3=STS˙=STYJ0−1u+W∗Tϕ+ε+D−Λ2∑i=13Si2≤STYJ0−1u+W∗Tϕ+ε+D.
Substituting (36) into (46) yields(47)V˙3≤STY−K1S−K2sigγS+W˜Tϕ1+ε+E2−K3YS≤−K1∑i=13YiSi2−K2∑i=13YiSi1+γ−K3∑i=13Yi2Si2+STYW˜Tϕ1+STYε+STYE2≤−ρ1V3−ρ2V31+γ/2+Φ2,where ρ1=min2K1Yi,1, ρ2=min21+γ/2K2Yi,1, and Φ2=η1+η2+η3.
Thus, (47) can be transformed into the following form:(48)V˙3≤−ρ1−Φ2V3V3−ρ2V31+γ/2orV˙3≤−ρ1V3−ρ2−Φ2V31+γ/2V31+γ/2.
Due to Lemma2 and (48), S can converge into ΔS satisfying(49)ΔS=minΦ2ρ1,Φ2ρ22/1+γ,within a finite-time bounded by(50)Ts≤1ρ11−1+γ/2lnρ1V31−1+γ/2S0+ρ2ρ2,where S0 is the initial value of S.
From (14), one has(51)Sj=sigeqj+Λ12−asig2−aeωj+Λ2sigeqj=ηj,ηj≤ΔS,where j=1,2,3.
The termsΛ1/2−asig2−aeωj+Λ2sigeqj and sigeqj are both positive or negative. Considering only positive case and Lemma 3, the following relationship exists without loss of generality:(52)eωj+Λ2sigeqj2−a2−a≥eωj+Λ2sigeqj+a−12−a.
MultiplyingΛ1 and adding sigeqj on both sides of (52), it has(53)sigeqj+Λ1eωj+Λ2sigeqj2−a2−a≥sigeqj+Λ1eωj+Λ2sigeqj+Λ1a−12−a.
Substituting (53) into (51) yields(54)sigeqj+Λ1eωj+Λ1Λ2sigeqj≤ΔS¯,where ΔS¯=ΔS+Λ11−a/2−a.
From (54), the attitude tracking errors eq and angular velocity eω are finite-time stable. Thus, eq and eω converge into small regions:(55)eqj≤ΔS¯1+Λ1Λ2,eωj≤2ΔS¯Λ1,within a finite-time, respectively.
From the above discussion, the convergence timeT of closed-loop system states in (8) is bounded, which satisfies(56)T=Tr+Ts≤1Λ¯11−ι0lnΛ¯1V11−ι0eq0+Λ¯2Λ¯2+1ρ11−1+γ/2lnρ1V31−1+γ/2S0+ρ2ρ2.
Consequently, the sliding variableS and the tracking errors eq and eω are FTUUB.
This completes the proof.Remark 2.
The convergence time is determined by the parametersΛ¯1, Λ¯2, K1, K2, and ι0. From (56), we can find that when the parameters Λ¯1, Λ¯2, K1, and K2 are selected to be larger, the convergence speed will become faster while it will cause some chattering issues. Thus, the choice of Λ¯1, Λ¯2, K1, and K2 should be set appropriately to keep a balance between convergence speed and chattering.


## 6. Simulations Results

In this section, numerical simulations are displayed to show the attitude tracking performance. To indicate the superiority of the designed control scheme, two different control schemes are given for comparison, i.e., SFTSM controller [20] and adaptive linear sliding mode (ALSM) controller [21].To intuitively display the physical meaning of the reference attitude trajectories, the reference attitude quaternionq0r,qv1r,qv2r,qv3r is expressed as(57)q0rqv1rqv2rqv3r=cosψd2cosθd2cosϕd2+sinψd2sinθd2sinϕd2cosψd2cosθd2sinϕd2−sinψd2sinθd2cosϕd2cosψd2sinθd2cosϕd2+sinψd2cosθd2sinϕd2sinψd2cosθd2cosϕd2−cosψd2sinθd2sinϕd2,where ϕd,θd,ψd are the desired Euler angles.The parameters in (8) are J0=diag0.1,0.1,0.1kg/m2, ω0=0,0,0T, and ΔJ=diag1,1,2kg/m2, and the external disturbances are set as ϕ0=θ0=ψ0=0°, and dt=0.5sin0.1t,0.5sin0.1t,0.5costTN⋅m.In SFTSM, the parameters in (14) are set as Λ1=0.1, Λ2=0.4, and a=5/7. The parameters in (36) are chosen as K1=0.5, K2=1.1, K3=4, and γ=0.1. The number of RBFNN nodes is 10. The corresponding parameters in (9) and adaptive update law (37) are chosen as μi∈−2,2, ai=π, δi=0.1, ϖi=0.2, and W^0=0,0,0,0T.In ALSM, the linear sliding variableS is given by(58)S=eω+Λ0eq,where Λ0=7.5, and the controller is(59)u=J0Λ0eω+ω˙r+θ^0sgnS−K1S+K2sigγS,in which the updating law of θ^0 is given by(60)θ^˙0=c0S−ε0θ^0,where c0=1, and the other parameters in the ALSM are set the same as those of SFTSM.To illustrate the better tracking performance and transient convergence performance in SFTSM, the fixed initial Euler angles are selected withϕd=θd=ψd=10° as the reference trajectories. Figures 3 and 4 show the time response of control inputs in SFTSM and ALSM, respectively. It is concluded that the maximum amplitude of the control input is 10N⋅m. The comparative results of the Euler angle tracking performance are shown in Figures 5–7. From Figure 5, the convergence time of ϕ in SFTSM is 0.5 s, and the convergence time of ϕ in ALSM is 1.5 s. Due to the same analysis in Figures 6 and 7, SFTSM can achieve faster convergence rate and better transient performance of Euler angles than ALSM.Figure 3
Control input in SFTSM.Figure 4
Control input in ALSM.Figure 5
The tracking performance ofϕ.Figure 6
The tracking performance ofθ.Figure 7
The tracking performance ofψ.For the purpose of showing the superiority of SFTSM, the corresponding attitude quaternion tracking performance in ALSM is presented in Figures8–11. The convergence time of SFTSM is faster almost 1 s than ALSM, which means SFTSM can realize the better transient performance than ALSM. Based on the mentioned analysis, quaternion-based tracking performance in SFTSM is still outperformed than ALSM. According to Figures 3–11, it is illustrated that SFTSM can guarantee faster convergence speed and better tracking performance in the QUAV’s attitude tracking control.Figure 8
The tracking performance ofq0.Figure 9
The tracking performance ofqv1.Figure 10
The tracking performance ofqv2.Figure 11
The tracking performance ofqv3.


## 7. Conclusion

In this study, a finite-time convergent RBFNN-based adaptive controller has been constructed to resolve a tracking problem of quadrotor UAVs. Firstly, a SFTSMS is proposed to realize the finite-time convergence of the tracking errors, which can directly avoid the potential singularity problem without requiring any piecewise continuous functions. Besides, an auxiliary function is proposed to purposely prevent the hidden singularity issue caused by the error-related matrix in the controller design. Then, a finite-time attitude controller is designed to guarantee that the system states were FTUUB. With the presented control scheme by RBFNN and ESO, prior knowledge about the unknown nonlinear uncertainties and external disturbances is not required. Finally, comparative simulations have shown the effectiveness of the designed control scheme.



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*Source: 3576783-2021-08-19.xml*