# A Novel Nonlinear Quadrotor Attitude Regulation with Disturbances and Preassigned Convergences

Authors: Yanping Gao; Shiyu Zhen; Lihong Zhang; Zuojun Liu
Journal: International Journal of Aerospace Engineering
 (2022)

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## Abstract

This article develops a novel unknown system dynamic estimator-based funnel control scheme for nonlinear quadrotor attitude regulation with preassigned convergence subject to parametric uncertainties and external perturbations. An invariant manifold equipped with first-order filtering is established. To online identify the lump disturbances, an unknown system dynamic estimator is employed with a simple formula, which need a lower computation burden. Based on aforementioned estimator, a novel funnel control via utilizing funnel variable is investigated, where an exponentially decaying funnel function is preset with a prior preassigned convergence for regulation angle error. The angle tracking errors are proved to be ultimately uniformly bounded, and angle regulation error can evolve within the preset funnel boundary. Simulation results demonstrate the effectiveness of the developed control scheme.


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## Body

## 1. Introduction

In recent years, quadrotors have been one of obbligato aircrafts in monitoring, disaster rescue, forest fire detection, power inspection, express transportation [1–5], and other fields due to their simple structure, low-budget manufacturability, and flexible maneuverability. However, it is not trivial to design a controller for attitude regulation featured with multiple input and multiple output and having nonlinear systems. Besides, parametric uncertainties and unknown external environmental disturbances are ubiquitous and unavoidable in practical scenarios, which the performance of quadrotors may be affected by aforementioned lumped disturbances. Therefore, effective and robust attitude regulation control for quadrotors in the presence of uncertainty and external disturbance is a meaning and challenging problem.With the requirement of strong disturbance rejection as well as precise control performance, some attempts have been made by adding feedforward compensation based on the controller. To guarantee the improvement of feedforward performance, the disturbance observer and corresponding theory have been investigated for quadrotors in traditional techniques, such as sliding mode observers (SMO) [6–9], function approximators by neural networks (NN) [10–13], fuzzy logistic system (FLS) [14, 15], and extended state observers (ESO) [16, 17]. In [8], a first-order SMO equipped with high gain observer is designed to estimate unknown disturbance. Based on the estimation of high-order SMO, the composite control is structured in position and attitude loop with different controller in [9], respectively. In [11], a robust and adaptive controller-based NN is conducted for quadrotor by introducing the generalized regression neural network. To realize the finite-time convergence, an adaptive integral sliding mode control is proposed with a novel fully connected recurrent neural networks with finite time learning process in [13]. In [14], an adaptive backstepping control is proposed with the command filtering technique for quadrotor trajectory tracking, where FLS is employed to estimate the uncertainty dynamics in the quadrotor model. By incorporating FLS into control scheme, the adaptive quantized control is considered for trajectory tracking in position loop and attitude loop in [15]. By constructing ESO to estimate the uncertainties, a robust state feedback controller is designed in [16] for linear quadrotor model derived by the flatness theory. In [17], a robust backstepping sliding mode control is exploited in position loop, and ESO is applied in the attitude loop for quadrotor to derive desired trajectory. Although the above literature achieves satisfactory performance with prominent disturbance estimation, the chattering phenomenon may arise owing to design of SMO. The heavy computational burden that is caused by function approximation with adjusting parameters repeatedly in NN and FLS is not practical in practical engineering applications. In addition, many piratical problems in engineer usually focus not only on steady-state error but also on overshoot and convergence time, which is not taken into account in above-mentioned controller design.Another significant issue related to controlling performance is the preassigned convergence of the tracking error. To tackle this issue, a funnel boundary is first established to regulate both transient and steady-state behavior in [18], which depict the convergence by predefine funnel function. By utilizing funnel controller (FC), the tracking error is converted to a funnel variable such that it can remain with the proposed funnel boundary. Compared to prescribed performance control (PPC) proposed in [19], it is flexible to design the corresponding controller based on funnel control without encountering nonsingularity problem, which will exist during presenting inverse transformation function in PPC. Thus, there has been some research focusing on funnel controller. In [20], a low-complexity FC tracking controller is developed for a class of system with more inputs than outputs. For arbitrary relative degree nonlinear system, the tracking controller is considered such that the evolvement of tracking error is within the range of predefined funnel [21]. By introducing high-gain observer to derive virtual output, a FC is proposed in [22] for nonlinear systems via transforming arbitrary relative degree system into relative degree one system. However, the FC in [21, 22] are only developed for certain system. For nonlinear system with disturbance, the FC-based backstepping control in [23] is explicated for quadrotors with improved funnel function, where SMO is employed to estimate the disturbance. In [24], by incorporating dynamic surface control with NN, an adaptive tracking controller is developed for hydraulic manipulators to ensure that tracking error evolves within the new funnel function. Although the tracking performance can be guaranteed with the funnel boundary by the robust control in [23–25], the high estimation of disturbance may affect the tracking performance.The unknown system dynamic estimation (USDE) proposed by [26] is a novel estimator with simple formula and few parameter adjustment to handle the uncertainties and disturbance for nonlinear system, avoiding utilizing the function approximation by NN or FLS with inducing computational burden. Different from the ESO and other estimators, there is only one parameter tuning without intermediate variable during the USDE designer. Based on improved unknown disturbance estimator (MUDE), a motion control is considered in [27] for robotic systems to settle the lumped disturbance. In [28], sliding mode control is developed by employing USDE to compensate for disturbance, achieving fast convergence and strong robustness. However, due to quadrotor with coupling and multivariables, it is challenging to design USDE-based controller for quadrotor attitude regulation, unlike the above literature USDE aimed at system in a chain of integrators. In addition, the above controller cannot guarantee the preassigned convergence of tracking error, which may cause disaster practical with obstacle. Hence, it is imperative and urgent to investigate USDE-based funnel control to eliminate tracking errors of quadrotor attitude.Inspired by the foregoing statements, a novel USDE-based tracking control with funnel control is investigated for quadrotor attitude regulation with total disturbances. First, the USDE is adopted for quadrotor for precise estimation of the uncertainty and external disturbance with less computational burden. Second, a novel funnel control involving the approximation by USDE is designed to such that the attitude regulation error remains the preset funnel function. The novelties of this work are twofold.(i)
Unlike the above stating SMO-based controller [6–9] and controller with function approximators [10–15], where chattering and heavy computational burden with adjusting parameters always arise, we adopt the unknown system dynamic estimator (USDE) with an invariant manifold for quadrotor attitude regulation to confirm the uncertainty in quadrotor model. In comparison with the ESO-based controller [16, 17], due to state in quadrotor calculated with simple filter, a perspicuous framework consisting of USDE with low computation is conducted(ii)
A novel USDE-based funnel control is proposed for quadrotor attitude regulation to enable preassigned convergence with existence of estimation error. Different from the conventional controllers [13, 29], where the tracking error of quadrotors is only guaranteed to be ultimately uniformly bounded (UUB), a FC is evolved with the estimation of USDE with novel funnel variable to guarantee preassigned convergences defined by funnel function, which is regarded as the prior criterion for transient-state and stead-state regulation errors


## 2. Problem Statements

Figure1 describes the quadrotor attitude dynamics by establishing the body-fixed coordinate B and the inertial-fixed coordinate E, and Fi denote the thrusts generated by the rotor actuations. Based on the previous work [30, 31], a model can be established when faced with external disturbances.
(1)Θ̇=Γω,Jω̇=−ω×Jω+F+d,where Θ=ϕ,θ,ψT is the Euler angle in frame E, consisting of roll angle ϕ, pitch angle θ, and yaw angle ψ. ω=ϕ̇,θ̇,ψ̇T, where ϕ̇,θ̇,ψ̇ denotes the angular rate. In addition, matrix Γ=1,tanθsinϕ,tanθcosϕ;0,cosϕ,−sinϕ;0,secθsinϕ,secθcosϕ reveals the relationship between angle Θ and angular velocity ω in the attitude loop. J=diagJϕ+Jϕ∗,Jθ+Jθ∗,Jψ+Jψ∗ stands for the positive diagonal matrix, where Jϕ,Jθ,Jψ are inertia nominal moments and Jϕ∗,Jθ∗,Jψ∗ are inertia moment uncertainties. The control input vector is defined as F=Fϕ,Fθ,FψT. And d=dϕ,dθ,dψT could be used to represent the external disturbances faced by the attitude dynamics of the quadrotors.Figure 1
Elementary diagram of quadrotor.Furtherly, it can be obtained along (1) with JΘ=diagJϕ,Jθ,Jψ and JΘ∗=diagJϕ∗,Jθ∗,Jψ∗ representing uncertainty parameters as
(2)Θ¨=−ΓJΘ−1ω×Jω+ΓJΘ−1d−JΘ∗ω̇+Γ̇ω+JΘ−1ΓF.Taking into accountΔ=−ΓJΘ−1ω×Jω+ΓJΘ−1d−JΘ∗ω̇+Γ̇ω and u=JΘ−1ΓF and introducing new vector variable with Ω1=Θ=Ω1,ϕ,Ω1,θ,Ω1,ψT and Ω2=Γω=Ω2,ϕ,Ω2,θ,Ω2,ψT, it can be rewritten as a chain of integrators:
(3)Ω̇1=Ω2,Ω̇2=u+Δ.The aim of this paper is to design attitude regulation control for quadrotor based on unknown system dynamic estimator to deal with total disturbances in model, which can accurately bring the angles to converge preset attitude with preassigned convergence.


## 3. Main Results

### 3.1. Filter-Based Unknown System Dynamic Estimator Design

Noting thatΔ is unknown due to the lump disturbances, which is mainly based on the second equation of (3), USDE is designed to estimate Δ. In line with [26, 32], filtering operations are introduced firstly for the available states Ω2 and u.
(4)kΩ̇2f+Ω2f=Ω2,Ω2f0=0∈ℝ3,ku̇f+uf=u,uf0=0∈ℝ3,where control inputs u=uϕ,uθ,uψT, Ω2f, and uf represent the corresponding auxiliary filtering vector for Ω2 and u and k is the adjustable filtering parameters.Lemma 1 [33].
Given the positive filtering parameterk∈0,+∞, the auxiliary vector η=k−1Ω2−Ω2f−uf+Δ is bounded and the following condition holds:
(5)limk⟶0limt⟶∞1kΩ2−Ω2f−uf+Δ=0.Notably, the auxiliary vectorη denotes the invariant manifold that indicates a mapping between filtered vectors Ω2f and uf and lumped disturbances Δ. Therefore, the USDE can be developed based on filtered vectors as
(6)Δ^=1kΩ2−Ω2f−uf,where Δ^=Δ∧ϕ,Δ∧θ,Δ∧ψT is conducted to approximate lumped disturbances. To characterize the estimation, the following assumption is necessary.Assumption 2 [34–36].
The time derivatives of unknown lump disturbances can be bounded with an unknown positive constantδ, i.e., Δ̇≤δ.Theorem 3.
For angular velocity dynamics and estimation (6) with filtered vector in (4), the disturbance estimation errors Δ~=Δ−Δ^ can converge to the following neighborhood of origin:
(7)Δ~≤Δ~02e−t/k+k2δ2.Proof.
By adding the first-order filter1/ks+1 on the second equation of (3), it can be given with Laplace operator s in the form of
(8)1ks+1Ω̇2=1ks+1u+1ks+1Δ.
From the quadrotor model (1), its transformation (3), and the filtering manipulation (4), we can get
(9)Ω̇2f=1kΩ2−Ω2f=uf+Δf,with Δf=Δ/ks+1, so that Δf=Δ^ can be deduced from (6).
Thus, the estimation errors can be expressed by(10)Δ~=Δ−Δf=ksks+1Δ,and its derivative with respect to time will then be given along (1), (3), and (4) as
(11)Δ~̇=Δ̇−Δ̇f=Δ̇−Ω̇2−Ω̇2f+ku̇fk=Δ̇−Ω̇2−Ω2−Ω2f/k+ku̇fk=Δ̇−u+Δ−Ω2−Ω2f/k+ku̇fk=Δ̇−Δ−Δfk.
Now, the estimation errorΔ~ is investigated by considering the following Lyapunov function:
(12)V1=12Δ~TΔ~.
Based on (11), the time derivative of V1 can be expressed as
(13)V̇1=Δ~TΔ~̇=Δ~TΔ̇−Δ−Δfk=Δ~TΔ̇−1kΔ~TΔ−Δf=Δ~TΔ̇−1kΔ~TΔ~.
It can be easily derived from Young’s inequalityab≤a2β/2+b2/2β for positive constant β that
(14)V̇1≤12kΔ~2+kδ22−1kΔ~TΔ~=−1kV1+kδ22.
One will have(15)V1t≤V10e−t/k+k2δ22.
Thus, we obtain the following conclusion about estimation errors:(16)Δ~=2V1t≤Δ~02e−t/k+k2δ2.
It can be shown that, for the given filtering parameterk⟶0, the estimation error Δ~ is guaranteed to converge exponentially to the origin when time approximates to be infinite.


### 3.2. Controller Design

During the controller design,r=ϕd,θd,ψdT represents the given reference signal. Then, e=Ω1−r, where e=eϕ,eθ,eψT is defined as the angle tracking error. In order to drive the tracking error within the preset funnel function −ϑit<eit<ϑit, the funnel boundary is described by the following function as [26]
(17)ϑit=αe−lt+γ,fori=ϕ,θ,ψ,where parameters α,γ,l are selected to meet the conditions that α≥γ>0 and ei0<ϑi0=α+γ.To fulfill preassigned convergence, we consider a novel funnel variable:(18)ζit=eitϑit−eit,fori=ϕ,θ,ψ,with derivative of (18) yields
(19)ζ̇i=ėiϑi−ei−1/2eiϑ̇i−ėi−1/2ϑi−eiϑi−ei,fori=ϕ,θ,ψ.Furthermore, it could be redescribed as(20)ζ̇i=1ϑi−eiėi−1/2ϑ̇i−ėieiϑi−ei=χiėi+μi,where χi=1/ϑi−ei and μi=−ϑ̇i−ėiei/2ϑi−ei.Thus, the virtual control inputτi can be established by stabilizing the funnel variable ζit:
(21)τi=−kΘ,iχi−1ζi+ṙi−μi,fori=ϕ,θ,ψ,where the subsystem gain kΘ,i>0 can be adjusted to stabilize system. Naturally, we can get angular rate errors:
(22)z=Ω2−τ,where z=zϕ,zθ,zψT and τ=τϕ,τθ,τψT.Based on the dynamical model (3), calculating the differential of angular rate errors yields
(23)ż=u+Δ−τ̇.Noting thatΔ is unavailable, the control input u can be constructed via substituting the disturbance estimation (6) as follows:
(24)ui=−kω,izi+τ̇i−Δ^ifori=ϕ,θ,ψ,where the controller gain kω,i>0 can be regulated.


## 3.1. Filter-Based Unknown System Dynamic Estimator Design

Noting thatΔ is unknown due to the lump disturbances, which is mainly based on the second equation of (3), USDE is designed to estimate Δ. In line with [26, 32], filtering operations are introduced firstly for the available states Ω2 and u.
(4)kΩ̇2f+Ω2f=Ω2,Ω2f0=0∈ℝ3,ku̇f+uf=u,uf0=0∈ℝ3,where control inputs u=uϕ,uθ,uψT, Ω2f, and uf represent the corresponding auxiliary filtering vector for Ω2 and u and k is the adjustable filtering parameters.Lemma 1 [33].
Given the positive filtering parameterk∈0,+∞, the auxiliary vector η=k−1Ω2−Ω2f−uf+Δ is bounded and the following condition holds:
(5)limk⟶0limt⟶∞1kΩ2−Ω2f−uf+Δ=0.Notably, the auxiliary vectorη denotes the invariant manifold that indicates a mapping between filtered vectors Ω2f and uf and lumped disturbances Δ. Therefore, the USDE can be developed based on filtered vectors as
(6)Δ^=1kΩ2−Ω2f−uf,where Δ^=Δ∧ϕ,Δ∧θ,Δ∧ψT is conducted to approximate lumped disturbances. To characterize the estimation, the following assumption is necessary.Assumption 2 [34–36].
The time derivatives of unknown lump disturbances can be bounded with an unknown positive constantδ, i.e., Δ̇≤δ.Theorem 3.
For angular velocity dynamics and estimation (6) with filtered vector in (4), the disturbance estimation errors Δ~=Δ−Δ^ can converge to the following neighborhood of origin:
(7)Δ~≤Δ~02e−t/k+k2δ2.Proof.
By adding the first-order filter1/ks+1 on the second equation of (3), it can be given with Laplace operator s in the form of
(8)1ks+1Ω̇2=1ks+1u+1ks+1Δ.
From the quadrotor model (1), its transformation (3), and the filtering manipulation (4), we can get
(9)Ω̇2f=1kΩ2−Ω2f=uf+Δf,with Δf=Δ/ks+1, so that Δf=Δ^ can be deduced from (6).
Thus, the estimation errors can be expressed by(10)Δ~=Δ−Δf=ksks+1Δ,and its derivative with respect to time will then be given along (1), (3), and (4) as
(11)Δ~̇=Δ̇−Δ̇f=Δ̇−Ω̇2−Ω̇2f+ku̇fk=Δ̇−Ω̇2−Ω2−Ω2f/k+ku̇fk=Δ̇−u+Δ−Ω2−Ω2f/k+ku̇fk=Δ̇−Δ−Δfk.
Now, the estimation errorΔ~ is investigated by considering the following Lyapunov function:
(12)V1=12Δ~TΔ~.
Based on (11), the time derivative of V1 can be expressed as
(13)V̇1=Δ~TΔ~̇=Δ~TΔ̇−Δ−Δfk=Δ~TΔ̇−1kΔ~TΔ−Δf=Δ~TΔ̇−1kΔ~TΔ~.
It can be easily derived from Young’s inequalityab≤a2β/2+b2/2β for positive constant β that
(14)V̇1≤12kΔ~2+kδ22−1kΔ~TΔ~=−1kV1+kδ22.
One will have(15)V1t≤V10e−t/k+k2δ22.
Thus, we obtain the following conclusion about estimation errors:(16)Δ~=2V1t≤Δ~02e−t/k+k2δ2.
It can be shown that, for the given filtering parameterk⟶0, the estimation error Δ~ is guaranteed to converge exponentially to the origin when time approximates to be infinite.


## 3.2. Controller Design

During the controller design,r=ϕd,θd,ψdT represents the given reference signal. Then, e=Ω1−r, where e=eϕ,eθ,eψT is defined as the angle tracking error. In order to drive the tracking error within the preset funnel function −ϑit<eit<ϑit, the funnel boundary is described by the following function as [26]
(17)ϑit=αe−lt+γ,fori=ϕ,θ,ψ,where parameters α,γ,l are selected to meet the conditions that α≥γ>0 and ei0<ϑi0=α+γ.To fulfill preassigned convergence, we consider a novel funnel variable:(18)ζit=eitϑit−eit,fori=ϕ,θ,ψ,with derivative of (18) yields
(19)ζ̇i=ėiϑi−ei−1/2eiϑ̇i−ėi−1/2ϑi−eiϑi−ei,fori=ϕ,θ,ψ.Furthermore, it could be redescribed as(20)ζ̇i=1ϑi−eiėi−1/2ϑ̇i−ėieiϑi−ei=χiėi+μi,where χi=1/ϑi−ei and μi=−ϑ̇i−ėiei/2ϑi−ei.Thus, the virtual control inputτi can be established by stabilizing the funnel variable ζit:
(21)τi=−kΘ,iχi−1ζi+ṙi−μi,fori=ϕ,θ,ψ,where the subsystem gain kΘ,i>0 can be adjusted to stabilize system. Naturally, we can get angular rate errors:
(22)z=Ω2−τ,where z=zϕ,zθ,zψT and τ=τϕ,τθ,τψT.Based on the dynamical model (3), calculating the differential of angular rate errors yields
(23)ż=u+Δ−τ̇.Noting thatΔ is unavailable, the control input u can be constructed via substituting the disturbance estimation (6) as follows:
(24)ui=−kω,izi+τ̇i−Δ^ifori=ϕ,θ,ψ,where the controller gain kω,i>0 can be regulated.


## 4. Stability Analysis

To depict the controller property, we first give the statements on funnel-based angle errors and angle rate error. The derivate of funnel vector can be rewritten with (21) as
(25)ζ̇i=−kΘ,iζi+χizi,fori=ϕ,θ,ψ.Likewise, one can take derivation of angular rate errors along (24) that
(26)żi=−kω,izi+Δ~i,fori=ϕ,θ,ψ.Theorem 4.
For the system (3), the proposed novel control laws (21) and (24) for angle and angle rate with estimation from USDE (6) can drive all the regulation error into the neighborhood of origin. Moreover, the angle errors eit can be regulated with preassigned convergence via preset funnel function provided that the premier angle errors exist in interval ei0∈−ϑi0,ϑi0 with positive variable ϑi0i=ϕ,θ,ψ and corresponding suitable controller gains kΘ and kω.Proof.
Taking into account the whole attitude kinetics, we can construct the Lyapunov function:(27)V=12∑i=ϕ,θ,ψζi2+zi2.
Based on error dynamics (25) and (26), the derivation of (27) can be taken as
(28)V̇=∑i=ϕ,θ,ψζiζ̇i+ziżi=∑i=ϕ,θ,ψζi−kΘ,iζi+χizi+zi−kω,izi+Δ~i=∑i=ϕ,θ,ψ−kΘ,iζi2−kω,izi2+ζiχizi+ziΔ~i.
Then, the second term and last term can be relaxed by Young’s inequality:(29)∑i=ϕ,θ,ψζiχizi≤12λmaxχζ2+z2,∑i=ϕ,θ,ψziΔ~i≤12z2+12Δ~2,where ζ=ζϕ,ζθ,ζψT, χ=diagχϕ,χθ,χψ, and its maximum eigenvalue is expressed as λmax⋅.
Therefore, we can restate (28) with simple operation as
(30)V̇≤−λminkΘ−12λmaxχζ2−λminkω−12λmaxχ−12z2+12Δ~2,where kΘ=diagkΘ,ϕ,kΘ,θ,kΘ,ψ, kω=diagkω,ϕ,kω,θ,kω,ψ, and λmin⋅ is the minimum value of the corresponding matrix.
Based on Theorem3 results, one has
(31)limt⟶∞Δ~≤kδ.
We can expressV̇ as
(32)V̇≤−−12λmaxχ+λminkΘζ2−−12λmaxχ+λminkω−12z2+12k2δ2,so that we can integrate (32) and derive
(33)V̇≤−κV+σ,where σ=k2δ2/2 and κ=min2κ1,2κ2,>0 with suitable controller gains satisfying
(34)κ1=−12λmaxχ+λminkΘ>0,κ2=−12λmaxχ+λminkω−12>0.
Furthermore, by solving the inequality (33), it can be obtained as
(35)0≤Vt≤σκ1−e−κt+V0e−κt.
The boundedness of tracking errorsζ and z can be guaranteed from (35). When the time tends to infinity, the upper boundedness can be indicated as
(36)ζ≤2σ/κ,z≤2σ/κ.
Then, for any positive constantε, it can be further expressed as
(37)ζi2≤2σκ<ε,fori=ϕ,θ,ψ.
In particularly, we can obtain along (18) with ζit as
(38)ei2ϑit+eitϑit−eit≤ei2ϑit−eitϑit−eit=ei2ϑit−eit<ε.
By simple arithmetic, angle error satisfies(39)ei2≤εϑi2−ei2<εϑi2−ei2+ei2ϑi2ε.
It follows that(40)ei2+ε2ei2<ei2+ε2ϑi2.
Thus, for arbitrarilyt≥0, one has
(41)−ϑit<eit<ϑitfori=ϕ,θ,ψ.


## 5. Simulation Results

As is shown in Section2, we set the parameters of quadrotor dynamics as follows: the known moment of inertia is given as diagJϕ,Jθ,Jψ=diag0.16,0.16,0.32. To verify the robustness of controller, the uncertainties in inertia moment and external disturbance are set as diagJϕ∗,Jθ∗,Jψ∗=diag0.012,0.012,0.012, Δϕ=2sint+sin1.5t, Δθ=2cost+sin0.5tcos0.8t, and Δψ=2sint. The initial angles are set as Θ0=−2,3,1, and the angle reference is ϕd=15sin0.25t,5cos0.25t,5T.To display the effectiveness of our proposed controller scheme, the simulation results are listed in Figures2–5 with controller parameters in Table 1. In Figure 2, it is shown that the attitude is governed by the proposed controller to regulate the reference attitude with total disturbances. Meanwhile, it shows that USDE can promptly observe and capture the unmeasurable perturbations with an improved accuracy in Figure 3. Figure 4 reveals that the tracking error evolves the funnel boundary defined funnel function and control inputs are displayed in Figure 5. To present the superiority of proposed controller, conventional control scheme [37] is employed to the nonlinear quadrotor attitude regulation for comparison. Although the attitude successfully tracks the reference attitude, a slower convergence rate and a worse tracking accuracy with control consumptions are derived in Figure 6. Furthermore, we compare the time for converging to steady state, tracking precision with index of standard derivation and control inputs in Table 2.Figure 2
Response of attitude regulation.Figure 3
Estimation performances for total disturbances based on USDE.Figure 4
The angle tracking errors under the FC.Figure 5
Control inputs.Table 1
Parameters of controllers.
SectionsValuesUSDEk=0.01Funnel controlα=5,l=0.6,γ=0.3Control gainskΘ,ϕ=1,kΘ,θ=1,kΘ,ψ=1,kω,ϕ=4,kω,θ=4,kω,ψ=4Figure 6
The angle tracking errors under conventional controller [37].Table 2
Contrastive outcomes between suggested method and conventional controller [37].
IndicesNotationProposed methodConventional controllers [37]Convergence timeeϕ4.53610.240eθ3.02110.238eψ2.98011.786Tracking precisioneϕ0.00220.1087eθ0.02610.1117eψ0.02630.0781Integrated absolute of control inputsuϕ35.829.86uθ38.9529.73uψ108.3106.4


## 6. Conclusion

This article investigated a novel funnel control for quadrotor attitude regulation with uncertainty parameters and external disturbance by incorporating unknown system dynamic estimator to ensure tracking preassigned convergences. The USDE is designed to online estimate the lumped disturbances by establishing connection between environmental perturbations and filtered dynamics. Furthermore, the USDE-based novel funnel control is proposed to drive the angle regulation error within preset funnel boundary, where funnel function and corresponding variable are introduced to adjust the angle error. Eventually, the tracking error of angle and angle rate are UUB with the preassigned convergence. For quadrotor attitude regulation, the validity and availability of proposed control method are verified by simulations. Note that time-triggered and infinite convergence are investigated in this work and more event-trigger and appointed-time techniques based on funnel control can be considered as in [38, 39].



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