# Adaptive Robust Dynamic Surface Integral Sliding Mode Control for Quadrotor UAVs under Parametric Uncertainties and External Disturbances

Authors: Ye Zhang; Ning Xu; Guoqiang Zhu; Lingfang Sun; Shengxian Cao; Xiuyu Zhang
Journal: Complexity
 (2020)

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## Abstract

A robust adaptive fuzzy nonlinear controller based on dynamic surface and integral sliding mode control strategy (ADSISMC) is proposed to realize trajectory tracking for a class of quadrotor UAVs. In this study, the composite factors including parametric uncertainties and external disturbances are added to controller design, which make it more realistic. The quadrotor model is divided into two subsystems of attitude and position that make the control design become feasible. The main contributions of the proposed ADSISMC strategy are as follows: (1) The combination of dynamic surface and integral sliding mode makes the system always in sliding stage by finding the appropriate initial position compared with the common sliding mode, and the complexity of explosion in backstepping method is eliminated. (2) By introducing the fuzzy system, the unknown functions and uncertainties can be approximated which significantly improves the robustness and the tracking performance. (3) The switching control strategy is utilized to compensate for the errors between estimated and ideal inputs; the tracking performance of the whole system has been significantly improved. The simulation results show the effectiveness of the proposed control method.


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## Body

## 1. Introduction

As a newborn member of the small unmanned aerial vehicle (UVA) family, quadrotor has attracted much research interest due to its extensive utility in several important applicants, such as commercial photography, military surveillance, rescue mission, and agricultural investigation [1–4]. Compared with traditional unmanned fixed-wing flight vehicles and manned airplanes, the main advantages of the quadrotor lie in small size, low cost, stable hovering, vertical take-off and landing (VTOL), convenient portability, and versatile features [5]. However, trajectory tracking control of the quadrotor is a thorny problem because of its nonlinear, underactuated dynamics, and strong coupling [6, 7]. Moreover, the quadrotor system is susceptible to external disturbances such as wind and nonlinear frictions. What is more, taking robustness of the trajectory tracking controller into consideration poses a bigger challenge [8].In early quadrotor research stage, many studies used conventional linear control methods such as proportional-integral-derivative (PID) [9, 10] and linear quadratic regulator (LQR) [11] to design the quadrotor controller in order to improve the simpleness and practicability. The linear control technology was developed to stabilize the quadrotor by neglecting the unimportant factors and linearizing the dynamic model. Therefore, it is poor and even not acceptable for the tracking accuracy and the robustness of the quadrotor. To overcome the drawbacks of the aforementioned linear control approaches, a large number of nonlinear control strategies, including backstepping control [12–15], sliding mode control (SMC) [16–20], and feedback linearization control, are utilized to improve the tracking performance of the flight control system.Backstepping control method has received extensive attention, not only in the quadrotor control, but also in some mechanical systems. The main idea of backstepping technology is to select the appropriate state variable function as the recursive virtual control input. Once the final control input is received, the stability of the whole system is guaranteed. In [21–24], a backstepping controller has been designed to stabilize the attitude system of quadrotor. In [24], attitude control using hybrid backstepping methodology based on Frenet–Serret theory is studied in detail. The results show that the controller has good robustness under wind disturbance. To solve the problem of trajectory tracking, in [25], an adaptive controller combining parameter adaptive and backstepping control is designed. However, the obvious limitation of conventional backstepping design is the problem of “complexity of explosion” caused by the repeated differentiation of some nonlinear functions and the lack of robustness against uncertainties. To overcome this limitation of traditional backstepping control, dynamic surface control (DSC) is proposed as an effective alternative method [26–30]. In [27], a dynamic surface control method based on RBF neural network approximation is proposed for a class of nonlinear time-delay systems with state variables all measurable, which greatly simplifies the design process of the controller.Moreover, to enhance the attitude performance robustness, disturbance observer (DOB), parameter estimation [31], and the approximation-based adaptive control are generally combined with DSC to handle external disturbances and parameterized uncertainties. For instance, a class of adaptive control methods using fuzzy logic systems or neural networks to approximate unknown functions in nonlinear systems have been proposed in [32–38]. In [33], a dynamic surface control-based adaptive fuzzy control method is proposed to overcome the “explosion of complexity” problem of classical backstepping. In [35], a robust dynamic surface controller based on extended state observer is presented for a quadrotor UAV subject to external disturbances and parametric uncertainties. In [36], both indirect and direct global neural controllers with the dynamic surface design are developed for the strict-feedback systems. The simulation results are presented to demonstrate the feasibility of the proposed global neural DSC design. A robot control scheme based on dynamic surface considering output error constraints, unknown dynamics, and bounded disturbance has been proposed in [39]; by introducing an improved virtual variable, the robustness of the control system was improved. However, the performance properties and robustness are not taken into account in these papers. As a commonly used nonlinear control method, the SMC is utilized as an effective method to design robust controllers for a specific class of nonlinear tracking problems in the presence of uncertain conditions [40–45]. Traditional SMC features the low sensitivity to the disturbances and parameter variations of the system [46–51]. In [50], a method based on second-order sliding mode control is used to avoid the chattering phenomenon for quadrotor UAVs. In [51], a robust backstepping sliding mode nonlinear controller for quadrotor UAVs is proposed to improve the robustness of the controller against model uncertainty and external disturbances. Compared with traditional sliding mode control method, the integral sliding mode (ISM) can guarantee that the system always meets the desired dynamic performance index during the whole arrival period which significantly improves the robustness of the control system.Motivated by the aforementioned observations, a new control methodology combined with dynamic surface and ISMC is proposed for the quadrotor trajectory tracking problem under parametric uncertainties and external disturbances. The main contributions of this paper are summarized as follows: (1) By fusing the technique of DSC and the integral SMC, a new integral sliding mode robust dynamic surface trajectory tracking controller is designed, which eliminates the “explosion of complexity” in the backstepping and improves the robustness of the whole system. (2) The FLSs are introduced to approach the ideal control law. And the estimations of the weight vector norm are utilized in the FLSs to significantly reduce the number of online estimation parameters. Therefore, the amount of calculation is obviously reduced, and the structure of the proposed controller is simplified. (3) The adaptive switching control is introduced to compensate the error between the real control law and the ideal control law, and the tracking performance of the whole system has been significantly improved.The rest of this paper is organized as follows. The modeling of a quadrotor and some preliminaries are introduced in Section2. The control algorithms are introduced in Section 3. Section 4 gives the stability analysis of the control system. Extensive simulations under different operating scenarios are given in Section 5. This paper ends with the conclusions in Section 6.


## 2. Problem Formulation and Preliminaries

### 2.1. The Mathematical Model of UAV

The quadrotor UAV is an underactuated system because it has six degrees of freedom, but only four actual inputs [1, 2]. In this paper, the quadrotor UAV with four rotors is shown in Figure 1. The equations of the dynamic quadrotor UAV are basically a rotating rigid body with six degrees of freedom [4, 5] which are usually derived by Newton–Euler formulas [8–10].Figure 1
Schematic of the quadrotor UAV.Defineξ=ϕ,θ,φT and ωb=p,q,rT, with φ, θ, and ϕ being the angle of roll, pitch, and yaw with respect to the inertia frame. p, q, and r are the angular velocity of roll, pitch, and yaw with respect to the body-fixed frame. The rotation matrix from the rigid frame to the inertia frame can be expressed as(1)Rt=CφCθCφSθSϕ−SφCϕCφSθCϕ+SφSϕSφCθSφSθSϕ+CφCϕSφSθCϕ−CφSϕ−SθCθSϕCθCϕ,where S⋅ and C⋅ denote sin⋅ and cos⋅, respectively. According to the rotation matrix Rt, the relationship between ξ˙ and ωb can be described as(2)ωb=Rrξ˙=10−Sθ0CϕCθSϕ0−SϕCθCϕϕ˙θ˙φ˙.DefineMb as the torque provided by the rotors with respect to the body-fixed frame, and it is presented as follows:(3)Mb=MbxMbyMbz=lkΩ42−Ω22lkΩ32−Ω12lΩ42+Ω22−Ω12−Ω32,where Ωi denotes the rotary speed of the front, right, rear, and left rotors, respectively; lk is the distance between a rotor and the center of mass of the quadrotor; k is the drag force coefficient; and l is the reverse moment coefficient. Using the Newton–Euler equation, the rotational dynamic equation of the quadrotor is obtained as follows:(4)Mb=Jbω˙b+ωb×Jbωb+Mg+Md,where Jb=diagJx,Jy,Jz is a symmetric positive definite constant matrix with Jx, Jy, and Jz being the rotary inertia with respect to the ObXb, ObYb, and ObZb axes, respectively; the notation × denotes cross multiplication; Mg and Md are the resultant torques due to the resultant of aerodynamic frictions torque and the gyroscopic effects. They are given as(5)Mg=∑i=14ωb×Jr0,0,−1i+1ΩiT,Md=diagdϕ,dθ,dφξ˙,where Jr denotes the moment of inertia of each rotor; dϕ, dθ, and dφ are the corresponding aerodynamic drag coefficients. According to (4), the following equation can be obtained:(6)ω˙b=Jb−1Mb−Mg−Md−ωb×Jbωb.Furthermore, with the help of approximation of Euler angles at equilibrium point, the following dynamic equations can be obtained:(7)ϕ¨=θ˙φ˙Jy−Jz−Jrθ˙ϖ−dϕϕ˙+MbxJx,θ¨=ϕ˙φ˙Jz−Jx−Jrϕ˙ϖ−dθθ˙+MbyJyφ¨=ϕ˙θ˙Jx−Jy−dφφ˙+MbzJz,,where ϖ=Ω4+Ω3−Ω2−Ω1 can be got easily online. It should be noted that to make the roll and pitch angles physically meaningful, they are both limited to (−π/2, π/2). In particular, the yaw angle is also limited to (−π/2, π/2) in this study, while P=x,y,zT∈R3 is the position with respect to the inertial frame. The translational dynamic equations of the quadrotor are given as(8)mP¨=Rt⋅F+00−mg−dxx˙dyy˙dzz˙,where dx, dy, and dz are the air drag coefficients which are added in (8) to model the drag force caused by translational motions; F is the lift force generated by rotors with respect to the body-fixed frame.(9)F=00kΩ12+Ω22+Ω32+Ω42.By combining (7) and (8), a compact affine nonlinear equation of the quadrotor UAV is given as(10)X˙=fX+gXU,whereX=x,x˙,y,y˙,z,z˙,ϕ,ϕ˙,θ,θ˙,φ,φ˙T∈R12 is the state variable; fX and gX are smooth functions on X. Equation (10) is expended as follows:(11)x˙1=x2,x˙2=Cx7Sx9Cx11+Sx7Sx11U1−a1x2+d1,x˙3=x4,x˙4=Cx7Sx9Sx11−Sx7Sx11U1−a2x4+d2,x˙5=x6,x˙6=Cx7Cx9U1−g−a3x6+d3,x˙7=x8,x˙8=a4x10x12+a5ϖx10−a6x8+U2+d4,x˙9=x10,x˙10=a7x8x12+a8ϖx8−a9x10+U3+d5,x˙11=x12,x˙12=a10x8x10−a11x12+U4+d6,where Ui,i=1,2,3,4 represents the control inputs defined as follows:(12)U1=lkΩ12+Ω22+Ω32+Ω42m,U2=lkΩ42−Ω22Jx,U3=lkΩ32−Ω12Jy,U4=lkΩ42+Ω22−Ω32−Ω12Jz,and ai,i=1,2,…,11, are the normalized parameters defined as follows:(13)a1=dxm,a2=dym,a3=dzm,a4=Jy−JzJx,a5=JrJx,a6=dϕJx,a7=Jz−JxJy,a8=JrJy,a9=dθJy,a10=Jx−JyJz,a11=dφJz.


### 2.2. Fuzzy Logic Systems (FLSs)

In this study, the Fuzzy Logic Systems (FLSs) are introduced to approximate the continuous unknown functions on a given compact set. The FLSs consist of three main parts: fuzzy rule base, fuzzification, and defuzzification operators. The form of the fuzzy rules of the fuzzy controller isRulel: If x1 is F1l and x2 is F2l and …and xn is Fnl.Theny is Gl,l=1,2,…,N.where xt=x1,x2,…,xnT and y are the input and output of the whole fuzzy system, respectively. And N is the number of the rules. The fuzzy basis functions forms are defined as(14)yx=∑l=1Ny¯l∏i=1nμFilxi∑l=1N∏i=1nμFilxi,where y¯l=maxy∈RμGly.(15)ξlx=∏i=1nμFilxi∑l=1N∏i=1nμFilxi.DenotingαT=y¯1,y¯2,…,y¯N=α1,α2,…,αN and ξx=ξ1x,ξ2x,…,ξNxT, then equation of the fuzzy system can be rewritten as(16)yx=αTξx.Lemma 1.
For a continuous nonlinear functionfx in a compact set Ωx, it can be effectively approximated by FLSs with any small approximated error ε>0. fx can be expressed as follows:(17)supx∈Ωxfx−αTξx<ε,if x∈Ωx, then the smooth nonlinear function fx can be expressed as(18)fx=α∗Tξx+εx,where α∗ is the optimal fuzzy parameter vector, εx is the approximation error satisfies εx≤ε¯ε¯>0.


## 2.1. The Mathematical Model of UAV

The quadrotor UAV is an underactuated system because it has six degrees of freedom, but only four actual inputs [1, 2]. In this paper, the quadrotor UAV with four rotors is shown in Figure 1. The equations of the dynamic quadrotor UAV are basically a rotating rigid body with six degrees of freedom [4, 5] which are usually derived by Newton–Euler formulas [8–10].Figure 1
Schematic of the quadrotor UAV.Defineξ=ϕ,θ,φT and ωb=p,q,rT, with φ, θ, and ϕ being the angle of roll, pitch, and yaw with respect to the inertia frame. p, q, and r are the angular velocity of roll, pitch, and yaw with respect to the body-fixed frame. The rotation matrix from the rigid frame to the inertia frame can be expressed as(1)Rt=CφCθCφSθSϕ−SφCϕCφSθCϕ+SφSϕSφCθSφSθSϕ+CφCϕSφSθCϕ−CφSϕ−SθCθSϕCθCϕ,where S⋅ and C⋅ denote sin⋅ and cos⋅, respectively. According to the rotation matrix Rt, the relationship between ξ˙ and ωb can be described as(2)ωb=Rrξ˙=10−Sθ0CϕCθSϕ0−SϕCθCϕϕ˙θ˙φ˙.DefineMb as the torque provided by the rotors with respect to the body-fixed frame, and it is presented as follows:(3)Mb=MbxMbyMbz=lkΩ42−Ω22lkΩ32−Ω12lΩ42+Ω22−Ω12−Ω32,where Ωi denotes the rotary speed of the front, right, rear, and left rotors, respectively; lk is the distance between a rotor and the center of mass of the quadrotor; k is the drag force coefficient; and l is the reverse moment coefficient. Using the Newton–Euler equation, the rotational dynamic equation of the quadrotor is obtained as follows:(4)Mb=Jbω˙b+ωb×Jbωb+Mg+Md,where Jb=diagJx,Jy,Jz is a symmetric positive definite constant matrix with Jx, Jy, and Jz being the rotary inertia with respect to the ObXb, ObYb, and ObZb axes, respectively; the notation × denotes cross multiplication; Mg and Md are the resultant torques due to the resultant of aerodynamic frictions torque and the gyroscopic effects. They are given as(5)Mg=∑i=14ωb×Jr0,0,−1i+1ΩiT,Md=diagdϕ,dθ,dφξ˙,where Jr denotes the moment of inertia of each rotor; dϕ, dθ, and dφ are the corresponding aerodynamic drag coefficients. According to (4), the following equation can be obtained:(6)ω˙b=Jb−1Mb−Mg−Md−ωb×Jbωb.Furthermore, with the help of approximation of Euler angles at equilibrium point, the following dynamic equations can be obtained:(7)ϕ¨=θ˙φ˙Jy−Jz−Jrθ˙ϖ−dϕϕ˙+MbxJx,θ¨=ϕ˙φ˙Jz−Jx−Jrϕ˙ϖ−dθθ˙+MbyJyφ¨=ϕ˙θ˙Jx−Jy−dφφ˙+MbzJz,,where ϖ=Ω4+Ω3−Ω2−Ω1 can be got easily online. It should be noted that to make the roll and pitch angles physically meaningful, they are both limited to (−π/2, π/2). In particular, the yaw angle is also limited to (−π/2, π/2) in this study, while P=x,y,zT∈R3 is the position with respect to the inertial frame. The translational dynamic equations of the quadrotor are given as(8)mP¨=Rt⋅F+00−mg−dxx˙dyy˙dzz˙,where dx, dy, and dz are the air drag coefficients which are added in (8) to model the drag force caused by translational motions; F is the lift force generated by rotors with respect to the body-fixed frame.(9)F=00kΩ12+Ω22+Ω32+Ω42.By combining (7) and (8), a compact affine nonlinear equation of the quadrotor UAV is given as(10)X˙=fX+gXU,whereX=x,x˙,y,y˙,z,z˙,ϕ,ϕ˙,θ,θ˙,φ,φ˙T∈R12 is the state variable; fX and gX are smooth functions on X. Equation (10) is expended as follows:(11)x˙1=x2,x˙2=Cx7Sx9Cx11+Sx7Sx11U1−a1x2+d1,x˙3=x4,x˙4=Cx7Sx9Sx11−Sx7Sx11U1−a2x4+d2,x˙5=x6,x˙6=Cx7Cx9U1−g−a3x6+d3,x˙7=x8,x˙8=a4x10x12+a5ϖx10−a6x8+U2+d4,x˙9=x10,x˙10=a7x8x12+a8ϖx8−a9x10+U3+d5,x˙11=x12,x˙12=a10x8x10−a11x12+U4+d6,where Ui,i=1,2,3,4 represents the control inputs defined as follows:(12)U1=lkΩ12+Ω22+Ω32+Ω42m,U2=lkΩ42−Ω22Jx,U3=lkΩ32−Ω12Jy,U4=lkΩ42+Ω22−Ω32−Ω12Jz,and ai,i=1,2,…,11, are the normalized parameters defined as follows:(13)a1=dxm,a2=dym,a3=dzm,a4=Jy−JzJx,a5=JrJx,a6=dϕJx,a7=Jz−JxJy,a8=JrJy,a9=dθJy,a10=Jx−JyJz,a11=dφJz.


## 2.2. Fuzzy Logic Systems (FLSs)

In this study, the Fuzzy Logic Systems (FLSs) are introduced to approximate the continuous unknown functions on a given compact set. The FLSs consist of three main parts: fuzzy rule base, fuzzification, and defuzzification operators. The form of the fuzzy rules of the fuzzy controller isRulel: If x1 is F1l and x2 is F2l and …and xn is Fnl.Theny is Gl,l=1,2,…,N.where xt=x1,x2,…,xnT and y are the input and output of the whole fuzzy system, respectively. And N is the number of the rules. The fuzzy basis functions forms are defined as(14)yx=∑l=1Ny¯l∏i=1nμFilxi∑l=1N∏i=1nμFilxi,where y¯l=maxy∈RμGly.(15)ξlx=∏i=1nμFilxi∑l=1N∏i=1nμFilxi.DenotingαT=y¯1,y¯2,…,y¯N=α1,α2,…,αN and ξx=ξ1x,ξ2x,…,ξNxT, then equation of the fuzzy system can be rewritten as(16)yx=αTξx.Lemma 1.
For a continuous nonlinear functionfx in a compact set Ωx, it can be effectively approximated by FLSs with any small approximated error ε>0. fx can be expressed as follows:(17)supx∈Ωxfx−αTξx<ε,if x∈Ωx, then the smooth nonlinear function fx can be expressed as(18)fx=α∗Tξx+εx,where α∗ is the optimal fuzzy parameter vector, εx is the approximation error satisfies εx≤ε¯ε¯>0.


## 3. DSISMC Controller Design Procedure

In this section, the design process of dynamic surface integral sliding mode controller is proposed and the control system block diagram is shown in Figure2. The controller design process is divided into two parts: the position tracking controller design and attitude tracking controller design, as shown in Tables 1 and 2, respectively. To make the presentation clear, the specific controller design process is given in Appendix A.Figure 2
Schematic diagram of the proposed control scheme.Table 1
Position control algorithm of UAVs.
Step 1e1=x1−x1d,(T1.1)x¯2=x˙1d−c1e1,(T1.2)τ1x˙2d+x2d=x¯2,x2d0=x¯20,(T1.3)S1=x2−∫0tx˙2d−k1e˙1−k2e1dt,(T1.4)v1fs=α1Tξ1,α^˙1=−η1S1ξ1,(T1.5)v1vs=−E^1sgnS1,E^˙1=ρ1S1,v1=v1fs+v1vs.(T1.6)Step 2e3=x3−x3d,(T1.7)x¯4=x˙3d−c2e3,(T1.8)τ2x˙4d+x4d=x¯4,x4d0=x¯40,(T1.9)S2=x4−∫otx˙4d−k3e˙3−k4e3dt,(T1.10)v2fs=α2Tξ2,α^˙2=−η2S2ξ2,(T1.11)v2vs=−E^2sgnS2,v2vs=−E^2sgnS2,v2=v2fs+v2vs.(T1.12)Step 3e5=x5−x5d,(T1.13)x¯6=x˙5d−c3e5,(T1.14)τ3x˙6d+x6d=x¯6,x6d0=x¯60,(T1.15)S3=x6−∫0tx˙6d−k5e˙5−k6e5dt,(T1.16)v3fs=α3Tξ3,α^˙3=−η3S3ξ3,(T1.17)v3vs=−E^3sgnS3,v3vs=−E^3sgnS3,v3=v3fs+v3vs.(T1.18)Table 2
Attitude control algorithm of UAVs.
Step 4e7=x7−x7d,(T2.1)x¯8=x˙7d−c4e7,(T2.2)τ4x˙8d+x8d=x¯8,x8d0=x¯80,(T2.3)S4=x8−∫0tx˙8d−k7e˙7−k8e8dt,(T2.4)U2fs=α4Tξ4,α^˙4=−η4S4ξ4,(T2.5)U2vs=−E^4sgnS4,E^˙4=ρ4S4,U2=U2fs+U2vs.(T2.6)Step 5e9=x9−x9d,(T2.7)x¯10=x˙9d−c5e9,(T2.8)τ5x˙10d+x10d=x¯10,x10d0=x¯100,(T2.9)S5=x10−∫0tx˙10d−k9e˙9−k10e11dt,(T2.10)U3fs=α5Tξ5,α^˙5=−η5S5ξ5,(T2.11)U3vs=−E^5sgnS5,E^˙5=ρ5S5,U3=U3fs+U3vs.(T2.12)Step 6e11=x11−x11d,(T2.13)x¯12=x˙11d−c6e11,(T2.14)τ6x˙12d+x12d=x¯12,x12d0=x¯120,(T2.15)S6=x12−∫0tx˙12d−k11e˙11−k12e12dt,(T2.16)U4fs=α6Tξ6,α^˙6=−η6S6ξ6,(T2.17)U4vs=−E^6sgnS6,E^˙6=ρ6S6,U4=U4fs+U4vs.(T2.18)In Table1, ei,i=1,2,3 are the tracking error, and x¯i,i=2,4,6 are the virtual laws. The low-pass first-order filters (T1.3), (T1.9), and (T1.15) are presented in each step to get a new variable xid,i=2,4,6 with the time constants τi,i=1,2,3. The integral sliding mode surfaces Si,i=1,2,3 are selected in each step which improves the robustness of the system against disturbances and parameters uncertainties. Due to the existence of unknown functions and parameters, the fuzzy system vifs,i=1,2,3 are utilized to approximate vi,i=1,2,3 with αi,i=1,2,3 and ξi,i=1,2,3 being the adjustable parameters and fuzzy basis vectors, respectively. Then, the switching control law vivs,i=1,2,3 are introduced to compensate the error of vi,i=1,2,3 and the ideal input. Ei,i=1,2,3 are the switching gain, ci, ηi, ρi, i=1,2,3, and ki,i=1,2,…,6 are positive constant. It is worth noting that v=v1,v2,v3T is a group of virtual control laws and they can be given by v1=Cx7Sx9Cx11+Sx7Sx11U1, v2=Cx7Sx9Sx11−Sx7Sx11U1, v3=Cx7Cx9U1.In Table2, ei,i=4,5,6 are the tracking error, andx¯i,i=8,10,12 are the virtual laws. The low-pass first-order filters (T2.3), (T2.9), and (T2.15) are presented in each step to get a new variable xid,i=8,10,12 with the time constants τi,i=4,5,6. The integral sliding mode surfaces Si,i=4,5,6 are selected in each step to improve the robustness of the system. The fuzzy system Uifs,i=2,3,4 are utilized to approximate Ui,i=2,3,4 with αi,i=4,5,6 and ξi,i=4,5,6 being adjustable parameters and fuzzy basis vectors, respectively. Then, the switching control law Uivs,i=2,3,4 are introduced to compensate for Ui,i=2,3,4 and the ideal input. Ei,i=4,5,6 are the switching gain. ci, ηi, ρi, i=4,5,6, and ki,i=7,8,…,12 are positive constant.Remark 1.
v1, v2, and v3 are combinations of available terms that can be given directly or measured in Tables 1 and 2. Therefore, the control input U1 can be solved by U1=v3/CX7CX9.


## 4. Stability Analysis of the Closed-Loop System

In this section, stability analysis of the proposed control system is established to confirm that all signals in the closed loop are ultimately bounded. The errors of the first-order filter are presented as follows:(19)yi=xid−x¯i,i=2,4,…,12.From (T1.3), (T1.9), (T1.15), (T2.3), (T2.9), and (T2.15), one can obtain(20)x˙id=−yiτi/2,i=2,4,…,12.The derivative ofyii=2 in time can be obtained as(21)y˙2=x˙2d−x¯2=−y2τ1−x¨2d+c1e˙1,then, one can obtain(22)y˙2=−y2τ1+B2e1,e2,y2,x¨2d,where B2e1,e2,y2,x¨2d=−x¨2d+c1e˙1 is a continuous function. The following formula can also be obtained:(23)y˙i=−yiτi/2+Bi⋅,i=2,4,6,8,10,12.Then, the following inequalities hold:(24)yiy˙i≤−yi2τi/2+Biyi,i=2,4,6,8,10,12.Consider the Lyapunov function candidate(25)V=V1+V2,with V1=1/2∑i=16e2i−12+y2i2 and V2=1/2∑i=16Si2+1/ηiα˜iTα˜i+1/ρiE˜i2. Then, the following theorem can be obtained.Theorem 1.
Consider the closed-loop system which consists of the position and attitude system, with the virtual control signals (T1.2), (T1.8), (T1.14), (T2.2), (T2.8), and (T2.14), the adaptive laws (T1.5), (T1.11), (T1.17), (T2.5), (T2.11), and (T2.17), and the switching control laws (T1.6), (T1.12), (T1.18), (T2.6), (T2.12), and (T2.18). If all the parametersci, ηi, ρi,i=1,2,…,6, ki,i=1,2,…,12 and the time constant of first-order filter τi,i=1,2,…,6 are designed properly to satisfy V0≤p, p>0, all the closed-loop signals are uniformly bounded and the tracking error can be kept arbitrarily small.Proof.
The specific proof process is presented in Appendix B.


## 5. Simulations

In this section, the following simulations are given to validate the effectiveness and the performance of the proposed adaptive dynamic surface integral sliding mode control. The parameters for the quadrotor UAV adopted in this paper are presented in Table3. In the following simulation, the desired trajectory of the position and yaw angle xt,yt,zt,φtare chosen as sint,cost,0.5t,sin0.5t. The controller parameters chosen for simulation are ci=0.01, τi=0.001, ηi=200, ρi=0.1,i=1,2,…,6, ki=9,i=1,3,5,7,9,11, and ki=20,i=2,4,6,8,10,12. The fuzzy membership functions are chosen as follows: ulsi=exp−si+π/6−l−1×π/12/π/242,i=1,2,3;l=1,2,3,4,5,6. The disturbances are chosen as follows: d1=cost, d2=sint, d3=sintcost, d4=0.5sin0.5t, d5=0.5cos0.5t, d6=0.25sin0.5tcos0.5t.Table 3
Quadrotor parameters.
SymbolCase1Case 2Case 3Case 4Unitsm2222kgl0.20.20.20.2mκ2.982.982.982.9810−6N⋅s2⋅rad−2τ1.141.141.141.1410−7N⋅s2⋅rad−2dϕ1.21.21.21.210−2N⋅s⋅rad−1dθdϕ1.21.21.21.210−2N⋅s⋅rad−1Jx1.251.251.251.25N⋅s2⋅rad−1Jy1.251.251.251.25N⋅s2⋅rad−1Jz2.502.883.253.75N⋅s2⋅rad−1Case 1. In this case, the parameters of the quadrotor are assumed normal. The simulations for this case are presented.Cases 2–4. Uncertainty (15%, 30%, and 50% added) in the rotary inertia as well as the presence of external disturbance. In these cases, we give a consideration to both external disturbance and uncertainty in rotary inertia. The uncertainty in yaw axis is 15%, 30%, and 50%, respectively.The simulation results of Case 1 are presented from Figures3 to 7. Figure 3 shows the 3D tracking trajectory by using the proposed controller. The position and yaw angle trajectories are shown in Figure 4, and the tracking errors are given in Figure 5. Figure 6 shows the control signals. Figure 7 shows the change of roll and pitch angles. From the tracking performance, it can be concluded that the proposed control scheme can guarantee that all the variables are bounded and that the control system has strong robustness. Figures 8 to 10 illustrate the tracking performance under the different uncertainty cases. The proposed scheme has the better robustness against external disturbances and uncertainty parameters. Figures 11 to 13 show the tracking error comparisons between the ADSISMC, the ADSC, and the ASMC methods. Meanwhile, the maximum values (MVTE) and the root mean square values (RMSVTE) of tracking error in steady of the proposed scheme and the other two schemes are given in Table 4. The simulation results show that the proposed scheme has better tracking performance and robustness compared with the ADSC and the ASMC methods.Figure 3
Space diagram of position in normal case.Figure 4
Reference signal and the actual outputs.Figure 5
The actual tracking error.Figure 6
Control inputs.Figure 7
Change of roll and pitch.Figure 8
Space diagram of position of different uncertainty cases added in yaw axis.Figure 9
Reference singnal and the actual outputs of different uncertainty cases added in yaw axis.Figure 10
The actual tracking error of different uncertainty cases addes in yaw axis.Figure 11
Space diagram of position by using DSISMC, SMC, and DSC.Figure 12
Reference signal and the actual outputs by using DSISMC, SMC and DSC.Figure 13
The actual tracking error by using DSISMC, SMC, DSC.Table 4
The MVTE and RMSVTE of different schemes.
Kind of errorsProposed schemeADSC schemeASMC schemeMVTE ofx (m)7.66e−44.90e−29.76e−3MVTE ofy (m)6.58e−45.03e−27.15e−3MVTE ofz (m)5.70e−41.50e−23.15e−3MVTE ofφ (rad)4.83e−41.53e−27.84e−4RMSVTE ofx (m)3.33e−43.81e−27.02e−3RMSVTE ofy (m)2.92e−44.03e−25.16e−3RMSVTE ofz (m)2.16e−49.82e−32.73e−3RMSVTE ofφ(rad)1.34e−41.16e−25.04e−4


## 6. Conclusion

This paper proposed a dynamic surface integral sliding mode control scheme for a quadrotor UAV under the conditions of parameter uncertainty and external disturbances. Virtual control inputs are introduced in the robust controller design to guarantee the trajectory tracking performance, and the problem of “explosion of complexity” in the backstepping design has been greatly simplified. The fuzzy systems are utilized to approximate the ideal control inputs and the switch control is introduced to compensate for errors between estimated and ideal inputs which improves the control performance and robustness of the whole system. In addition, the stability analysis of the overall system through Lyapunov stability theory is presented, and all signals of the closed loop are ultimately bounded. Finally, the simulation results show that robustness and improved tracking performance can be achieved with the proposed control scheme.



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