# Finite-Time Attitude Control for Quadrotor with Input Constraints and Disturbances

Authors: Zijun Gao; Jin Wang; Yaping Tian
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

This paper investigates the adaptive output feedback attitude control of a quadrotor. First, a nonsingular terminal sliding-mode variable and auxiliary variable are introduced into a closed-loop structure. Meanwhile, a fuzzy logic system is incorporated into an adaptive algorithm to compensate for the adverse influence caused by lumped disturbances including system uncertainty and external disturbances on the attitude adjustment performance of a quadrotor. Then, a novel finite-time output feedback controller equipped with the saturation suppression algorithm is designed. Rigorous proof shows that the design control strategy ensures the closed-loop system stability and guarantees the attitude of the spacecraft to track desired command signals in finite time. Simulation results are presented to illustrate the performance of the proposed control scheme.


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## Body

## 1. Introduction

The challenges faced in the control of a quadrotor with excellent capability have been concerned and developed widely in the last several decades based on the demand for a reliable one for both commercial and military applications. During the quadrotor entry process, the attitude adjustment of the quadrotor involves attitude maneuvering through a wide range of physical limitation, lumped external disturbances, and system parameter uncertainties [1–3]. Many advanced flight control methods have been proposed for the quadrotor’s attitude control in order to improve the performance of the control system, such as adaptive backstepping method [4, 5], adaptive compensation control [6–8], sliding-mode control strategy [9, 10], and linearization control technique [11, 12].Although the high accuracy of attitude adjustment can be accomplished by using the previously discussed control strategy, these works require infinite time to achieve attitude maneuver. It is obvious that the convergence rate with infinite settling time is not ideal during critical phases of some kind of specialized real-time tasks. There can be no doubt that the finite-time control technology [13, 14] can provide higher tracking accuracy, faster dynamic response, and better robustness than the asymptotic control approach. Therefore, the finite-time control technique is one of the most effective control strategies that can handle the tracking control task and the attitude stabilization problem. In [15], a time-specified control strategy based on the nonsingular terminal sliding-mode technique was designed to solve the problem of trajectory tracking for robotic airships. In [16], a finite-time control method with time-delay estimation is proposed for shape memory alloy actuators. In [17], an adaptation supertwisting algorithm control scheme using the finite-time technique was designed for attitude tracking of the quadrotor. In [18], a smooth variable structure control scheme with multivariable form and disturbance observer was designed for quadrotor attitude adjustment. A nonsingular terminal sliding-mode control (NTSMC) method using the finite-time technique was thoroughly examined in [19] to achieve attitude tracking maneuver for the quadrotor with atmospheric disturbances. In a more recent work [20], an output feedback attitude controller based on the finite-time strategy was developed for the quadrotor with both matched and mismatched disturbances. Although many finite-time control strategies are focused on how to deal with attitude maneuver of the quadrotor, there is still a challenging problem for improving both the fast response ability and robustness of the closed-loop system with the factor of numerous disadvantages in different working environments, which is one motivation of this paper.Another problem to be noted is actuator saturation, which is generally encountered for nonsmooth nonlinear constraints of actuators in quadrotor applications. In fact, if an actuator falls into its physical saturation limitation boundary, any operation to enlarge the output of the actuator would not make any variation in the system output. It is likely to cause control performance degradation and even leading to task failure due to actuator saturation when the designed controller is not equipped with an effectiveness adjustment strategy to dump the saturated actuators [21–26]. Besides, taking quadrotor parameter uncertainties and large external disturbances into consideration makes the attitude tracking control problem of the quadrotor more difficult. How to cope with the simultaneous action of the abovementioned adverse factors has been turned into a challenging subject, and extensive attentions have been paid to solve the issue of saturation nonlinearity. To the best knowledge of the authors because of the complex characteristics of the quadrotor system, there are few results for the case where a finite-time control method is designed to achieve attitude adjustment of the quadrotor with control torque limitation and lumped disturbances.In order to respond to the above discussions, we will investigate the adaptive output feedback attitude control based on the finite-time control technique for quadrotor systems with lumped disturbance and input saturation in this paper. In particular, we are interested in developing an effective adaptation compensation algorithm capable of accommodating the lumped disturbances as well as input saturation suppression. The main contributions are summarized as follows:①
A control input auxiliary variable is designed into the design of the closed-loop system, and the design parameter update laws are able to make the control input boundedness by the saturation suppression algorithm only.②
A fuzzy logic system is incorporated into the parameter updating algorithm to estimate the complex nonlinear function including system uncertainty and external disturbances. Under the design NTSMC structure, a novel compensation control law equipped with the adaptive algorithm is designed to achieve both terminal surface variable stabilization and attitude tracking.③
A sliding-mode differentiator is utilized to relax the differential operation requirement of the virtual control law, which avoids the related singular problems. Moreover, the introduced differentiator can achieve virtual law approximation with lower computational complexity, facilitating the control law design.The paper is organized as follows. In Section2, the system model with adverse factors is formulated. The closed-loop system structure is established by the nonsingular terminal surface and the control input auxiliary variable, and the finite-time stability of the design control strategy with the parameters update law is analyzed in Section 3. Illustrative simulation of the design control strategy for the quadrotor and conclusion is given in Sections 4 and 5, respectively.


## 2. Problem Formulation

### 2.1. System Model

The quadrotor system model contains the rotational equations and the translational equations of motion. This paper will focus on the attitude controller design for the quadrotor, and therefore a control-oriented model can be expressed as [5–8](1)JΘ¨+NΘ˙=M+d,where Θ=α,β,σT∈ℝ3 is the attitude angle vector including the yaw angle α, pitch angle β, and roll angle σ; Θ˙ and Θ¨ are the attitude angular rate vector and the angular acceleration vector, respectively; d=d1,d2,d3T is the external disturbance; M=M1,M2,M3∈ℝ3 is control torque vector subject to nonlinear saturation described as(2)Mi=signui⋅uM,ifui≥uM,ui,ifui<uM,i=1,2,3,where uM is a known bound of Mi and ui is the design control command. In addition, J∈ℝ3×3 is a symmetric and positive definite matrix, and the matrix N∈ℝ3×3 is the Coriolis term of the quadrotor containing centrifugal and gyroscope terms (J and N will be given later).It is worth noting that the expression ofMi in (2) is a discontinuous function which will lead to the sharp corner. In order to improve the smoothness of the control input, the saturation nonlinear function Mi is approximated by a smooth function defined as(3)Γuui=uM⋅tanhuiuM,Γu=Γuu1,Γuu2,Γuu3T.In light of (3), (2) can be expressed as(4)Mi=Γuui+ΔMi,i=1,2,3,where ΔMi=Mi−Γuui≤uM1−tanh1 is a bounded approximation error. Furthermore, we have(5)M=Γu+ΔM,where ΔM=ΔM1,ΔM2,ΔM3T, satisfying ΔM=∑i=13ΔMi≤3⋅uM1−tanh1. Let Θ˙=ω, then (1) can be rewritten as(6)Θ˙=ω,Jω˙=−Nω+Γu+ΔM+d.In order to further develop our control scheme, the following lemmas and assumption are given.Assumption 1.
It is supposed that lumped disturbancesdii=1,2,3 is continuous, which satisfy maxd,d˙≤ρ where ρ is an unknown positive constant.Lemma 1 (see [27]).
A first-order sliding-mode differentiator is designed as(7)μ˙1=ς,ς=μ2−b1∗⋅μ1−Ft1/2⋅signμ1−Ft,μ˙2=−b2∗⋅signμ2−ς,where μ1 and μ2 are the states of the differentiator (7), ς is the output of the differentiator, b1∗ and b2∗ are the positive constants, and Ft is a known function. Then, ς can approximate the differential term F˙t to any arbitrary accuracy if the initial deviations μ1−Ft0 and ς−F˙t0 are bounded.Lemma 2 (see [27]).
Consider the systemx˙=fx,u. Suppose that there exist a continuous function Vx and scalars ψ1>0, 0<ψ2<1, and 0<ψ3<∞, such that(8)V˙x≤−ψ1⋅Vψ2x+ψ3.Then, this system’s trajectory is practical finite-time stable.


### 2.2. Description of a Fuzzy Logic System

From [28], any continuous functions on a compact set can be approximated by the fuzzy logic system, with arbitrary accuracy. The fuzzy logic system is composed of four principal sections: an inference engine, a fuzzy rule, a fuzzifier, and a defuzzifier. Indeed, a set of ‘If-Then’ linguistic rules can describe the fuzzy logic system, i.e.,(9)Rj:Ifx1isA1jand…andxnisAnj,thenzisBj,where j=1,2,…,N; x=x1,x2,…,xnT and z∈ℝ are the inputs and outputs of the fuzzy logic system, respectively; in addition, Aij,i=1,2,…,N and Bij are characterized by the fuzzy membership functions μAijxi and μBijz, respectively. Based on the singleton fuzzifier, product inference, center-average defuzzifier, and Gaussian membership function, the output of the fuzzy logic system can be written as(10)fx=∑j=1Mz¯j∏i=1nμAijxi∑j=1M∏i=1nμAijxi=Φ⋅ζx,where z¯j∈ℝ is the point at which μBijz achieves the maximum value, Φ=z¯1,z¯2,…,z¯M is an adjustable parameter vector, and ζ=ζ1x,ζ2x,…,ζnxT is the fuzzy basis function vector; each element of ζx is(11)ζix=∑j=1M∏i=1nμAijxi∑j=1M∏i=1nμAijxi.It should be pointed out that fuzzy logic system is a powerful function approximation tool, and it can effectively deal with uncertainty [29]. For a continuous nonlinear function D, it can be approximated by(12)D=fx=Φ⋅ζTx+ν,where Φ=Φ1,Φ2,…,ΦnT is an optimal weight vector and ν=ν1,ν2,…,νnT is the error vector of fuzzy approximation.Assumption 2.
The optimal weight and approximation error of FLS are bounded such thatΦ≤ΦM and ν≤νM, where ΦM and νM are the positive constants.
The aim of this paper is to construct a closed-loop system with the self-adaption compensation algorithm which guarantees that the commanded signalΘd is tracked by the system output Θ in finite time in the presence of the lumped disturbance and input constraints.


## 2.1. System Model

The quadrotor system model contains the rotational equations and the translational equations of motion. This paper will focus on the attitude controller design for the quadrotor, and therefore a control-oriented model can be expressed as [5–8](1)JΘ¨+NΘ˙=M+d,where Θ=α,β,σT∈ℝ3 is the attitude angle vector including the yaw angle α, pitch angle β, and roll angle σ; Θ˙ and Θ¨ are the attitude angular rate vector and the angular acceleration vector, respectively; d=d1,d2,d3T is the external disturbance; M=M1,M2,M3∈ℝ3 is control torque vector subject to nonlinear saturation described as(2)Mi=signui⋅uM,ifui≥uM,ui,ifui<uM,i=1,2,3,where uM is a known bound of Mi and ui is the design control command. In addition, J∈ℝ3×3 is a symmetric and positive definite matrix, and the matrix N∈ℝ3×3 is the Coriolis term of the quadrotor containing centrifugal and gyroscope terms (J and N will be given later).It is worth noting that the expression ofMi in (2) is a discontinuous function which will lead to the sharp corner. In order to improve the smoothness of the control input, the saturation nonlinear function Mi is approximated by a smooth function defined as(3)Γuui=uM⋅tanhuiuM,Γu=Γuu1,Γuu2,Γuu3T.In light of (3), (2) can be expressed as(4)Mi=Γuui+ΔMi,i=1,2,3,where ΔMi=Mi−Γuui≤uM1−tanh1 is a bounded approximation error. Furthermore, we have(5)M=Γu+ΔM,where ΔM=ΔM1,ΔM2,ΔM3T, satisfying ΔM=∑i=13ΔMi≤3⋅uM1−tanh1. Let Θ˙=ω, then (1) can be rewritten as(6)Θ˙=ω,Jω˙=−Nω+Γu+ΔM+d.In order to further develop our control scheme, the following lemmas and assumption are given.Assumption 1.
It is supposed that lumped disturbancesdii=1,2,3 is continuous, which satisfy maxd,d˙≤ρ where ρ is an unknown positive constant.Lemma 1 (see [27]).
A first-order sliding-mode differentiator is designed as(7)μ˙1=ς,ς=μ2−b1∗⋅μ1−Ft1/2⋅signμ1−Ft,μ˙2=−b2∗⋅signμ2−ς,where μ1 and μ2 are the states of the differentiator (7), ς is the output of the differentiator, b1∗ and b2∗ are the positive constants, and Ft is a known function. Then, ς can approximate the differential term F˙t to any arbitrary accuracy if the initial deviations μ1−Ft0 and ς−F˙t0 are bounded.Lemma 2 (see [27]).
Consider the systemx˙=fx,u. Suppose that there exist a continuous function Vx and scalars ψ1>0, 0<ψ2<1, and 0<ψ3<∞, such that(8)V˙x≤−ψ1⋅Vψ2x+ψ3.Then, this system’s trajectory is practical finite-time stable.


## 2.2. Description of a Fuzzy Logic System

From [28], any continuous functions on a compact set can be approximated by the fuzzy logic system, with arbitrary accuracy. The fuzzy logic system is composed of four principal sections: an inference engine, a fuzzy rule, a fuzzifier, and a defuzzifier. Indeed, a set of ‘If-Then’ linguistic rules can describe the fuzzy logic system, i.e.,(9)Rj:Ifx1isA1jand…andxnisAnj,thenzisBj,where j=1,2,…,N; x=x1,x2,…,xnT and z∈ℝ are the inputs and outputs of the fuzzy logic system, respectively; in addition, Aij,i=1,2,…,N and Bij are characterized by the fuzzy membership functions μAijxi and μBijz, respectively. Based on the singleton fuzzifier, product inference, center-average defuzzifier, and Gaussian membership function, the output of the fuzzy logic system can be written as(10)fx=∑j=1Mz¯j∏i=1nμAijxi∑j=1M∏i=1nμAijxi=Φ⋅ζx,where z¯j∈ℝ is the point at which μBijz achieves the maximum value, Φ=z¯1,z¯2,…,z¯M is an adjustable parameter vector, and ζ=ζ1x,ζ2x,…,ζnxT is the fuzzy basis function vector; each element of ζx is(11)ζix=∑j=1M∏i=1nμAijxi∑j=1M∏i=1nμAijxi.It should be pointed out that fuzzy logic system is a powerful function approximation tool, and it can effectively deal with uncertainty [29]. For a continuous nonlinear function D, it can be approximated by(12)D=fx=Φ⋅ζTx+ν,where Φ=Φ1,Φ2,…,ΦnT is an optimal weight vector and ν=ν1,ν2,…,νnT is the error vector of fuzzy approximation.Assumption 2.
The optimal weight and approximation error of FLS are bounded such thatΦ≤ΦM and ν≤νM, where ΦM and νM are the positive constants.
The aim of this paper is to construct a closed-loop system with the self-adaption compensation algorithm which guarantees that the commanded signalΘd is tracked by the system output Θ in finite time in the presence of the lumped disturbance and input constraints.


## 3. Control Strategy Design

In this section, an output feedback attitude control strategy is proposed under the framework of the NTSMC method, which guarantees the boundedness of all the signals in the control loop. To achieve high-accuracy tracking control, a control input updating algorithm is further designed to avoid control input constraints and to handle lumped disturbances in the system. The stability of the closed-loop system can be proved at the same time. Now, the detailed contents are stated as follows.First, we define the system tracking errore1∈ℝ3 and its derivative e2∈ℝ3 as(13)e1=e11,e12,e13T=Θ−Θd,(14)e2=e˙1=e21,e22,e23T=Θ˙−Θ˙d,where Θd=Θd1,Θd2,Θd3T represents the desired commanded signal. To ensure fast and accurate attitude tracking performance of system (1), an NTSM-type sliding surface [30] Sa∈ℝ3 and an auxiliary variable z∈ℝ3 are defined as(15)Sa=Sa1,Sa2,Sa3T=e1+c⋅e2κ⋅signe2,1<κ<2,c>0.5,(16)z=z1,z2,z3T=Γu−Γφ=Γuu1Γuu2Γuu3−Γφφ1Γφφ2Γφφ3=uM⋅tanhu1uMuM⋅tanhu2uMuM⋅tanhu3uM︸Γu−φM⋅tanhφ1φMφM⋅tanhφ2φMφM⋅tanhφ3φM︸Γφ,where e2κsigne2=e21κsigne21,e22κsigne22,e23κsigne23T, κ=p/q with odd integers p and q, and in addition, φM=uM⋅tanh1−b>0 (bis the design parameter satisfying 2b≪1) and φi (i=1,2,3) will be designed later. For analysis simplification, φ≜φ1,φ2,φ3T in which φ will be given later.Remark 1.
By choosing1<κ<2 and the expression form of Saii=1,2,3, the singularity problem for e2i=e˙1i=0 can be avoided. Moreover, function Sai=e1i+c⋅e2iκ⋅signe2i is continuous and differentiable although the signum operator and the absolute value are involved [31].
With the combination of (6), (13), (14), and (16), the derivative of the terminal sliding surface Sa is given as follows:(17)S˙a=S˙a1,S˙a2,S˙a3T=e2+κc⋅e2κ−1⋅e˙2=e2+κc⋅e2κ−1⋅Θ¨−Θ¨d=e2+κc⋅e2κ−1⋅−J−1⋅N⋅ω+J−1⋅z+J−1φ+J−1d−J−1φ−Γφ+J−1ΔM−Θ¨d=e2+κc⋅e2κ−1⋅φ+D,with e2κ−1=diage21κ−1,e22κ−1,e23κ−1≥0 and D=−J−1Nω+J−1d+J−1ΔM−Θ¨d+J−1z+J−1Γφ−φ. Note that the mixture term D is a nonlinear function including the external disturbances and parameter perturbations. To eliminate the adverse effect caused by the lumped disturbance D on the system, the fuzzy logic system represented by (10) is employed to learn the unknown nonlinear function D due to its universal approximation property, i.e.,(18)D=Φ⋅ζT+ν=Φ⋅Φ∗⋅ζT+ν,with Φ∗=Φ/Φ. With the help of Young’s inequality, the relationships (19) and (20) can be established:(19)SaTe2≤SaTe2222a12+a122,(20)κcSaTe2κ−1D=κcSaTe2κ−1⋅Φ⋅Φ∗⋅ζ+ν≤κ2c2SaTe2κ−122Φ22ζ222a22+a222+κ2c2SaTe2κ−122ν222a22+a222,where SaTe2κ−122=Sa1e21κ−12+Sa2e22κ−12+Sa3e23κ−12 and SaTe222=Sa1e212+Sa2e222+Sa3e232; moreover, since 0<κ−1=p−q/q<1, and based on the fact e2i2κ−1≤e2i2 when e2i≥1 and e2i2κ−1≤e2iκ−1 when e2i<1, we have(21)κ2c2SaTe2κ−122Φ22ζ222a22≤κ2c2SaTe222Φ22ζ222a22+κ2c2⋅S⌢aT⋅e⌢2κ−1⋅Φ22ζ222a22,κ2c2SaTe2κ−122ν222a22≤κ2c2SaTe222ν222a22+κ2c2⋅S⌢aT⋅e⌢2κ−1ν222a22,where S⌢a=Sa12,Sa22,Sa32T and e⌢2κ−1=e21κ−1,e22κ−1,e23κ−1T. From (21), (20) can be rewritten as(22)κcSaTe2κ−1D≤κ2c2SaTe222+S⌢aT⋅e⌢2κ−12a22⋅ηTψ+a22,where e2κ−1=diage21κ−1,e22κ−1,e23κ−1, η=Φ2,ν2T, and ψ=ζ2,1T. Next, a virtual control law φ is designed as(23)φ≜φ1,φ2,φ3T=−κc2a22Sa+e23−κSa⋅η^Tψ−e23−κSa,where e23−κ=diage213−κ,e223−κ,e233−κ, and the related adaptation law is choosen as(24)η^˙=γ⋅κ2c22a22⋅S⌢aT⋅e⌢2κ−1+SaTe222⋅ψ−a2⋅η^,η0=0,γ>0.
In this study,η^≜η^1,η^2T is employed to estimate η, and the estimate error η˜ denotes η˜=η−η^. It is worth noting that the time derivative of φ (in (23)) is required for the subsequent attitude controller and stability analysis. It is not difficult to find that the derivative of the virtual law φ can lead to the explosion problem of the complex term. In order to solve this problem, a first-order sliding-mode differentiator is introduced into the closed-loop system design to estimate each element of φ˙:(25)μ˙1i=ςi=μ2i−b1∗⋅μ1−φi1/2⋅signμ1−φi,μ˙2i=−b2∗⋅signμ2i−ςi,i=1,2,3,where μ1i and μ2i are the states of the differentiator, ςi is the output of the differentiator, and b1∗ and b2∗ are the positive design constants. According to (25) and Lemma 1, we obtain(26)φ˙=ς+ε,where ς=ς1,ς2,ς3T and ε=ε1,ε2,ε3T is the estimation error vector of the introduced differentiator. According to Lemma 1, we can get ε≤δ with a positive constant δ.
Noting thatSaTe2κ−1Sa=S⌢aTe⌢2κ−1 and SaTe2κ−1e23−κSa=SaTe222, we can get from (23) that(27a)κcSaTe2κ−1⋅φ≤−κ2c22a22⋅S⌢aT⋅e⌢2κ−1⋅η^Tψ−κ2c22a22SaTe222⋅η^Tψ−κcSaTe222.
Owing toη^Tψ+η˜Tψ=ηTψ, we have(27b)−κ2c22a22⋅SaTe222⋅η^Tψ−κ2c22a22SaTe222⋅η˜Tψ=−κ2c22a22SaTe222⋅ηTψ,−κ2c22a22⋅S⌢aTe⌢2κ−1⋅η^Tψ−κ2c22a22⋅S⌢aTe⌢2κ−1⋅η˜Tψ=−κ2c22a22⋅S⌢aTe⌢2κ−1⋅ηTψ.
By using (17), (19), (22), (24), (27a), and (27b), the following relationship can be established:(28)SaTS˙a−1γη˜Tη^˙≤12a12⋅SaTe222+a122−κ2c22a22⋅S⌢aTe⌢2κ−1⋅η^Tψ−κ2c22a22⋅SaTe222⋅η^Tψ−κc⋅SaTe222+a22+κ2c22a22SaTe222+S⌢aTe⌢2κ−1⋅ηTψ−κ2c22a22S⌢aTe⌢2κ−1+SaTe222⋅η˜Tψ+a2η˜Tη^≤−κc−2a12−1⋅SaTe222+0.5a12+a22+a2η˜Tη^.
To proceed, the dynamics of an auxiliary variablez in (16) can be calculated as(29)z˙=∂Γu∂u⋅u˙−∂Γφ∂φ⋅φ˙=Ξu⋅u˙−Ξφ⋅ς+ε,where u˙=u˙1,u˙2,u˙3T,(30)Ξu=diagΞu1,Ξu2,Ξu3=diag∂Γuu1∂u1,∂Γuu2∂u2,∂Γuu3∂u3=∂Γu∂u=4eu1/uM+e−u1/uM20004eu2/uM+e−u2/uM20004eu3/uM+e−u3/uM2,Ξφ=diagΞφ1,Ξφ2,Ξφ3=diag∂Γφφ1∂φ1,∂Γφφ2∂φ2,∂Γφφ3∂φ3=∂Γφ∂φ=4eφ1/φM+e−φ1/φM20004eφ2/φM+e−φ2/φM20004eφ2/φM+e−φ2/φM2=diag4eφ1/uM⋅tanh1−b+e−φ1/uM⋅tanh1−b2,4eφ2/uM⋅tanh1−b+e−φ2/uM⋅tanh1−bφ2/uM⋅tanh1−b2,4eφ3/uM⋅tanh1−b+e−φ3/uM⋅tanh1−bφ3/uM⋅tanh1−b2.
Now, the input update lawu˙ii=1,2,3 is designed as(31)u˙i=Ξui−1⋅σi−ui22uM2+b⋅1+σi2,ifui≥uM;Ξui−1⋅σi,if−uM<ui<uM;Ξui−1⋅σi+ui22uM2+b⋅1+σi2,ifui≤−uM;where ui0=0 and σ≜σ1,σ2,σ3T=−ς22z/2−czq/p−cz−czSap+q/p/z22, (ς22≜∑i=13ςi2, zq/p≜z1q/p,z2q/p,z3q/pT, Sap+q/p≜∑i=13Saip+q/p, and z22≜∑i=13zi2). To facilitate the stability analysis, we focus our attention on the term zTz˙ which satisfies the following relationship:①
Ifui≥uM, we obtain zi=Γuui−Γφφi>0; according to the fact Γuui≥ΓuuM=uM⋅tanh1 and Γφφi≤φM=uMtanh1−b, it yields ∑i=13ziui2/2uM2+b1+σi2≥0.Then, taking into account that(32)Ξφi=∂Γφφi∂φi=4eφi/uM⋅tanh1−b+e−φi/uM⋅tanh1−b2≤1,zTΞφς+ε≤0.5⋅z22ς22+Ξφ22+z22+δ⋅Ξφ22,this leads to(33)zTz˙=∑i=13Ξui⋅zi⋅u˙i−zTΞφ⋅ς+ε=zTσ−∑i=13zi⋅ui22uM2+b⋅1+σi2−zTΞφς+ε≤zT−czq/p−cz−ς22z2−cSap+q/p+z22ς222+3⋅1+δ2+z222=−czp+q/p−cSap+q/p−c−12z22+321+δ.②
Ifui<uM, we can prove that(34)zTz˙=∑i=13Ξuiziu˙i−zT⋅Ξφ⋅ς+ε=zTΞu−zTΞφς+ε≤zT−czq/p−cz−ς22z2−cSap+q/p+z22ς222+3⋅1+δ2+z222=−czp+q/p−cSap+q/p−c−12z22+31+δ2.③
Ifui≤−uM, it can be easily inferred that Γuui≤Γu−uM=−uM⋅tanh1. Owing to Γφφi≤φM=uM⋅tanh1−b, we further obtain zi=Γuui−Γφφi≤0, and this ensures that ∑i=13zi⋅ui2/2uM2+b⋅1+σi2≤0. Therefore,(35)zTz˙=∑i=13Ξuiziu˙i−zTΞφς+ε=zTσ+∑i=13zi⋅ui22uM2+b⋅1+σi2−zT⋅Ξφ⋅ς+ε≤zT−czq/p−cz−ς22z2−cSap+q/p+z22ς222+3⋅1+δ2+z222=−czp+q/p−cSap+q/p−c−12z22+31+δ2.
Noting that the design parameterc satisfies c>0.5 and combining with the above analysis, we thus obtain(36)zTz˙≤−c⋅zp+q/p−c⋅Sap+q/p+3⋅1+δ2.Remark 2.
It is noteworthy that the input update lawu˙i is necessary for the stability of the closed-loop system. Moreover, it can prevent ui from exceeding the working range. From (31), when ui=uM⋅1−2b (design parameter b satisfies 2b≪1), it means u˙i=Ξui−1σi−ui2/2uM2+b⋅1+σi2≤0; therefore, this prevents ui from getting greater than uM⋅1−2b. When ui=−uM1−2b, then u˙i=Ξui−1σi+ui2/2uM2+b⋅1+σi2≥0, and it implies that this prevents ui from getting smaller than ui=−1−2b⋅uM.
The following theorem established the criteria for selecting the controller parameters to warrant the bounded stability of the attitude tracking erroreii=1,2,3.Theorem 1.
Consider system (5) with Assumptions 1–2, and if the controller parameters a1,a2,c,b,andκ are chosen as follows:(37)0<a1,a2<1,c>0.5,κ⋅c>2a12−1,2b≪1,then the control structure composed of the virtual control law φ (in (23)) with parameter update (24) and input update law u˙i (in (31)) can insure that all the signals of the designed closed-loop system are bounded and e1ii=1,2,3 converges to a small neighborhood of the equilibrium point in finite time.Proof.
Consider the Lyapunov candidate function as(38)Vt=SaTSa2+zTz2+η˜Tη˜2γ.
Taking the time derivative ofVt and applying (28) and (36), the following inequality can be obtained based on η˜=η−η^:(39)V˙t=SaTS˙a+zTz˙−γ−1η˜Tη^˙≤−c⋅zp+q/p−c⋅Sap+q/p+a2η˜Tη^+a122+a22+321+δ.
By using Young’s inequality, we can obtain that(40)a2η˜Tη^=a2η˜Tη−η˜=−a2η˜Tη˜+a2η˜Tη≤a2−η˜Tη˜+12η˜Tη˜+12ηTη=−a2γ2γη˜Tη˜+a2ηTη2−a2γ2γη˜Tη˜p+q/2p+a2γ2γη˜Tη˜p+q/2p.
For Theorem1 proof, the two cases are required to be addressed in the analysis of (40).①
Ifa2γ⋅η˜Tη˜/2γ>1, the following inequality is true:(41)−a2γ⋅η˜Tη˜2γ+a2γ⋅η˜Tη˜2γp+q/2p≤0.②
Ifa2γ⋅η˜Tη˜/2γ≤1, the following relationship can be established:(42)−a2γ⋅η˜Tη˜2γ+a2γ⋅η˜Tη˜2γq+p/2p≤1.
By the combination of (41) and (42), we have(43)a2η˜Tη^≤−a2b⋅η˜Tη˜2γp+q/2p+0.5⋅a2⋅ηT22+1.
Using (43) results in(44)V˙t≤−c⋅2q+p/2p⋅zTz2q+p/2p−c⋅2q+p/2p⋅SaTSa2q+p/2p−a2γq+p/2p⋅a2γ⋅η˜Tη˜2γq+p/2p+Δ≤−ϖ⋅Vq+p/2p+Δ,where ϖ=minc⋅2q+p/2p,c⋅γq+p/2p and Δ=0.5⋅a12+a22+1.5⋅1+δ+0.5a22⋅ηT22+1. Using Lemma 2, the decrease in Vt can force the sliding-mode variable of the closed-loop system into Vq+p/2p≤Δ/1−β∗⋅ϖ with 0<β∗<1, which implies that the terminal sliding-mode variable is bounded in finite time as(45)limt⟶Tf1Sat∈Sa≤Δ1−β∗ϖ⋅2p/2p+qp/p+q.
In the expression of (45), the finite time Tf1 satisfies that Tf1≤Vx0p−q/p/ϖβ∗⋅p−q/p in which Vx0 is the initial value of V. To describe the boundedness of attitude tracking errors, we need to reconsider each element of the NTSS shown in (15):(46)e1i+c⋅e2iκ⋅signe2i=Sai,i=1,2,3,where Sai≤Δ/1−β∗ϖ⋅2p/2p+qp/p+q≜ξ. Then, (46) can be rearranged as(47)e1i+c¯⋅e2iκ⋅signe2i=0,where c¯≜c−Sai⋅e2i−κ⋅signe2i. From (45) and (47), we consider the following two cases:①
Ifc¯≤0, then it has been in the region ce2iκ≤ξ, and it is easy to obtain that e1i converges to the region e1i=Sai−c⋅e2iκ⋅signe2i≤2ξ in a finite time Tf1②
Ifc¯>0, (47) can be regarded as the sliding mode, and according to [31], it can be shown that the time required for e1i along the nonsingular terminal sliding surface to reach 0 is Tf2≤0.5⋅e1i2Tf11−κ¯/α∗1−κ¯ with κ¯=0.51+κ−1 and α∗=2κ¯⋅1/c¯1/κ; therefore, the convergence time of the tracking error e1i is Tf=Tf1+Tf2
According to the above analysis, the stability of the designed closed-loop system is ensured through the sense of Lyapunov and the finite-time reachability is obtained, and the total convergence time of the closed-loop system isTf≤Tf1+Tf2.Remark 3.
Note that the convergence accuracy of the tracking error depends on the size of parametersϖ and Δ. The smaller the desired e1ii=1,2,3, the larger the required parameters 1/a1, 1/a2, κ, c, and γ. In practical application, such control parameters can be determined by using the trial-and-error method under condition (37) until a good performance is obtained.Remark 4.
From the point of view of improving the robust capability of the closed-loop system, the information of lumped disturbance should be integrated into the control scheme design accordingly. In the design of the closed-loop system, the fuzzy logic system is employed to obtain the information of the nonlinear mixture termD. And, Φ is used in the fuzzy logic system estimation, as can be observed from η=Φ2,ν2T. So, the quantity of design update laws (24) can be completely independent from the amount of fuzzy logic rules.Remark 5.
In order to use the fuzzy approximation, the approximated function needs to be kept in a compact set during the operation process of the closed-loop system. For the proposed control strategy, such a condition is naturally fulfilled by the physical limitation of the quadrotor, and not only system states can be guaranteed to be bounded, but also all signals of the closed-loop system are confined in a compact set during the entire control process. The result is that the fuzzy logic system and fuzzy approximation can be reasonably used in the closed-loop system and perform its functions as approximation and learning for the lumped nonlinear function.Remark 6.
Compared with the input-saturation approximation approaches [32, 33], the proposed closed-loop control system utilizes the self-adjustment mechanism to prevent the design control command ui from exceeding uM, which is adaptively compensated for the whole control process.


## 4. Simulation Studies

### 4.1. Parameter Setting

In order to validate the theoretical result, a simulation study is performed in this section (based on MATLAB/Simulink environment). The inertial matrixJ and the Coriolis term N are given as follows:(48)J=J11J12J13J21J22J23J31J32J33,N=N11N12N13N21N22N23N31N32N33,with J11=I1sin2β+I2cos2βsin2α+I3cos2βcos2α, J12=cosβcosαsinαI2−I3, J13=−I1sinβ, J21=cosβcosαsinαI2−I3, J22=I2cos2α+I3sin2α, J23=0, J31=−I1sinβ, J32=0, J33=I1, N11=I1β˙sin2β+I2−β˙sin2βsin2σ+σ˙cos2βsin2σ−I3β˙sin2βcos2σ+σ˙cos2βsin2σ/2, N12=I1α˙sin2β/2−I2β˙sinβsin2σ/2−σ˙cosβcos2σ+α˙sin2βsin2σ/2 + I3−α˙sin2βcos2σ/2−σ˙cosβcos2σ+β˙sinβsin2σ/2, N13=−I1β˙cosβ+I2−I1α˙cos2βsin2σ/2, N21=−I1α˙sin2β/2+I2α˙sin2β⋅sin2σ/2+I3α˙sin2βcos2σ/2, N22=σ˙sin2σI2−I3/2, N23=I1α˙cosβ+I2−β˙sin2σ/2+α˙cosβcos2σ+I3β˙sin2σ/2−α˙cosβcos2σ, N31=α˙cos2βsin2σI3−I2/2, N32=−I1α˙cosβ+I2β˙sin2σ/2−α˙cosβcos2σ, and N33=0, where I1=I2=4.2×10−3N⋅m/rad/s2 and I3=8.5×10−3N⋅m/rad/s2.Meanwhile, the disturbanced is selected as(49)d=0.5⋅d∗,for0≤t<6s;2.5⋅d∗,for6s≤t<12s;where(50)d∗=0.01⋅sinπt125+sinπt250sinπt125+cosπt250cosπt125+sinπt250.The initial values of the quadrotor system areα0=−0.2rad, β0=0.5rad, σ0=1.2rad, and Θ˙=0.01rad/s. The control authority is assumed to be uM=0.4, and the attitude angle commands are set to be(51)Θd=αdβdσd=0.8sintsin0.4trad0.6cos0.5t1−cos0.7trad0.6cos0.7t1−cos0.45trad.The parameters for the controller are chosen as follows:p=5, q=3, a1=a2=0.05, c=2, b=0.001, and γ=5. The differentiator parameters are selected as b1∗=b2∗=5; moreover, the fuzzy membership functions are given as(52)μAi1xi=exp−xi+π/3π/182,μAi2xi=exp−xi+π/6π/182,μAi3xi=exp−xiπ/182,μAi4xi=exp−xi−π/6π/182,μAi5xi=exp−xi−π/3π/182.


### 4.2. Result Analysis

The attitude tracking response result is exhibited in Figures1–3. Obviously, we can see that the command signals can be tracked by our control strategy with consideration of external disturbances and control input constraints. As can be seen from Figures 4 and 5, the proposed method achieves the goal of attitude control with high accuracy and fast convergence speed for attitude tracking error trajectories e1ii=1,2,3. This is mainly because the proposed finite-time control scheme using the self-adaptation algorithm is capable of eliminating quickly the adverse influence caused by system uncertainty and external disturbances. During the increased disturbances t≥15s, the control performance based on the proposed control method is not obviously reduced because its self-adaption algorithm exerts accommodation process such that it can restore control performance as much as possible and further avoid the unstability of the closed-loop system with consideration of the increased external disturbances and control input constraints. The control input is as illustrated in Figure 6. Based on the above result and analysis, the effectiveness of the designed method can be validated.Figure 1
Attitude trackingα−αd.Figure 2
Attitude trackingβ−βd.Figure 3
Attitude trackingσ−σd.Figure 4
Attitude tracking error.Figure 5
Dynamics of the terminal sliding surface.Figure 6
Control input.


## 4.1. Parameter Setting

In order to validate the theoretical result, a simulation study is performed in this section (based on MATLAB/Simulink environment). The inertial matrixJ and the Coriolis term N are given as follows:(48)J=J11J12J13J21J22J23J31J32J33,N=N11N12N13N21N22N23N31N32N33,with J11=I1sin2β+I2cos2βsin2α+I3cos2βcos2α, J12=cosβcosαsinαI2−I3, J13=−I1sinβ, J21=cosβcosαsinαI2−I3, J22=I2cos2α+I3sin2α, J23=0, J31=−I1sinβ, J32=0, J33=I1, N11=I1β˙sin2β+I2−β˙sin2βsin2σ+σ˙cos2βsin2σ−I3β˙sin2βcos2σ+σ˙cos2βsin2σ/2, N12=I1α˙sin2β/2−I2β˙sinβsin2σ/2−σ˙cosβcos2σ+α˙sin2βsin2σ/2 + I3−α˙sin2βcos2σ/2−σ˙cosβcos2σ+β˙sinβsin2σ/2, N13=−I1β˙cosβ+I2−I1α˙cos2βsin2σ/2, N21=−I1α˙sin2β/2+I2α˙sin2β⋅sin2σ/2+I3α˙sin2βcos2σ/2, N22=σ˙sin2σI2−I3/2, N23=I1α˙cosβ+I2−β˙sin2σ/2+α˙cosβcos2σ+I3β˙sin2σ/2−α˙cosβcos2σ, N31=α˙cos2βsin2σI3−I2/2, N32=−I1α˙cosβ+I2β˙sin2σ/2−α˙cosβcos2σ, and N33=0, where I1=I2=4.2×10−3N⋅m/rad/s2 and I3=8.5×10−3N⋅m/rad/s2.Meanwhile, the disturbanced is selected as(49)d=0.5⋅d∗,for0≤t<6s;2.5⋅d∗,for6s≤t<12s;where(50)d∗=0.01⋅sinπt125+sinπt250sinπt125+cosπt250cosπt125+sinπt250.The initial values of the quadrotor system areα0=−0.2rad, β0=0.5rad, σ0=1.2rad, and Θ˙=0.01rad/s. The control authority is assumed to be uM=0.4, and the attitude angle commands are set to be(51)Θd=αdβdσd=0.8sintsin0.4trad0.6cos0.5t1−cos0.7trad0.6cos0.7t1−cos0.45trad.The parameters for the controller are chosen as follows:p=5, q=3, a1=a2=0.05, c=2, b=0.001, and γ=5. The differentiator parameters are selected as b1∗=b2∗=5; moreover, the fuzzy membership functions are given as(52)μAi1xi=exp−xi+π/3π/182,μAi2xi=exp−xi+π/6π/182,μAi3xi=exp−xiπ/182,μAi4xi=exp−xi−π/6π/182,μAi5xi=exp−xi−π/3π/182.


## 4.2. Result Analysis

The attitude tracking response result is exhibited in Figures1–3. Obviously, we can see that the command signals can be tracked by our control strategy with consideration of external disturbances and control input constraints. As can be seen from Figures 4 and 5, the proposed method achieves the goal of attitude control with high accuracy and fast convergence speed for attitude tracking error trajectories e1ii=1,2,3. This is mainly because the proposed finite-time control scheme using the self-adaptation algorithm is capable of eliminating quickly the adverse influence caused by system uncertainty and external disturbances. During the increased disturbances t≥15s, the control performance based on the proposed control method is not obviously reduced because its self-adaption algorithm exerts accommodation process such that it can restore control performance as much as possible and further avoid the unstability of the closed-loop system with consideration of the increased external disturbances and control input constraints. The control input is as illustrated in Figure 6. Based on the above result and analysis, the effectiveness of the designed method can be validated.Figure 1
Attitude trackingα−αd.Figure 2
Attitude trackingβ−βd.Figure 3
Attitude trackingσ−σd.Figure 4
Attitude tracking error.Figure 5
Dynamics of the terminal sliding surface.Figure 6
Control input.


## 5. Conclusion

In this study, a new model-free compensation control strategy using the finite-time technique is developed for attitude control of the quadrotor with adverse factors. The closed-loop system and the saturation suppression method are designed based on the NTSMC structure and the input parameter updating online strategy, respectively. A rigorous stability analysis and proof of the designed closed-loop system are combined with nonsingular terminal sliding-mode theory and Lyapunov theory. The numerical simulation on the attitude control of the quadrotor is performed to validate the developed method.



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