# Vibration Suppression of a Coupled Aircraft Wing with Finite-Time Convergence

Authors: Yiming Liu; Zhifeng Tan; Xiaofen Yang; Xiaowei Wang
Journal: Complexity
 (2022)

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## Abstract

A nonlinear coupled wing model subject to unknown external disturbances is proposed in this paper. Since the model is modeled by partial differential equations, the traditional control design scheme based on the ordinary differential equation model is not applicable, and the control law design becomes very complex. In this paper, a new antidisturbance boundary control scheme based on a finite time convergent disturbance observer is proposed. The control laws are designed based on the new disturbance observers to make the external disturbance errors converge to zero in a finite time and ensure the uniformly bounded stability of the controlled system. Finally, the effectiveness of the controllers and the finite-time convergence of disturbance errors are verified by the simulation and comparison.


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## Body

## 1. Introduction

Due to the characteristics of maneuverability, flexibility, and wide vision, aircrafts are widely used in various fields. Aircrafts include the fixed-wing, rotary-wing, and flapping-wing aircraft. Among them, the flapping-wing aircraft has unique advantages, including higher flight maneuverability and low flight costs. In addition, although the size is small and the flight resistance is large, the flight efficiency of the flapping-wing aircraft will not be reduced [1]. Therefore, more and more researchers are engaged in the research of flapping-wing aircraft [2–6]. Researchers combined the structural design with bionics to develop and research the flapping-wing aircraft. Some researchers used rigid materials to make the wings, which ignored the influence of fluid dynamic changes caused by the deformation of insect wings so that the fuselage may not be flexible enough and prone to failure during execution [7]. Because flexible materials have the advantages of improving the running speed of the mechanical system and further reducing the weight of the structure, they are widely used in the mechanical structure manufacturing in recent years [8, 9]. Therefore, we can use flexible materials to make flapping-wing aircraft wings [10]. Although the flexible wing can improve the flexibility of flapping-wing aircraft, the vibration and deformation of wings will affect the control effect and flight performance. Consequently, the problem of wing vibration needs to be solved urgently, which inspired our research. In this paper, we regard a single wing as a coupled distributed parameter system and use the combination of partial differential equation and ordinary differential equation (PDE-ODE) to describe the wing dynamic model. Due to the complexity of distributed parameter system control design, more and more attention has been paid in recent years [11–21].For the problem of vibration suppression of distributed parameter systems, several researchers have proposed various control methods including the modal reduction method [22] and boundary control [23–29]. The modal reduction method can effectively reduce the order of the infinite-dimensional system and treat the system as a finite dimension, but this method can easily cause spillover effects. The boundary control strategy can effectively solve this problem and improve the robustness of the system. In recent years, the research on vibration suppression of flexible structural systems based on the boundary control has made great progress. For example, in [30], with the method of barrier Lyapunov function, the controller could suppress the vibration of the system and cope with the input and output constraints at the same time. In [31], a boundary controller was constructed to suppress the vibration excursion of the hose using the backstepping control method, and a smooth hyperbolic tangent function was introduced to cope with the input amplitude and rate constraints. In [32], a boundary control was designed for the flexible string system with vibration and input backlash. In [33], with the help of a new disturbance observer, the external disturbance was effectively suppressed, and the purpose of vibration reduction was achieved. A robust adaptive controller was proposed to deal with parameter uncertainties and stabilize the system in [34]. In [35], the authors put forward an adaptive NN control strategy for flexible string systems with input constraints and actuator failures. Two iterative learning boundary control schemes were designed for a flexible microair vehicle under spatiotemporal variation disturbances to suppress structural vibration and make it track the target trajectory in [36]. As for the boundary control of the flexible aircraft wing system, many scholars have researched in recent years [37–39]. However, these studies did not consider that the system was affected by both distributed disturbances and boundary disturbances, and their existence may bring great side effects to the system. Therefore, disturbance suppression has become a key problem in the wing control design.In the past few years, one of the most commonly used antidisturbance methods is the disturbance observer technique. This method can be used for nonlinear systems with uncertainties and efficiently improve the robustness of the system [40–42]. The boundary disturbance observer-based control problem of a vibrating single-link flexible manipulator system with external disturbances was studied in [43]. In [44], considering the influence of extraneous disturbance acting on the wing, a new observer was proposed for controller design. A new disturbance observer was proposed to deal with the distributed disturbance and boundary disturbance of a flexible-link manipulator in [45]. However, it is noted that although the disturbance observer proposed in the above research could effectively track the change of unknown disturbance, it must be assumed that the disturbance changes slowly, and it could not ensure that the disturbance error converged in a finite time. For systems affected by the external disturbance with unknown varying frequency, it is very important to ensure that the disturbance error converges in a finite time [46]. So far, there has been no research on the finite-time convergence control of the coupled aircraft wing system, which motivates this research.In this article, the stability of nonlinear flexible coupled wing systems under external disturbances is studied. We briefly describe the contributions of this work: (i) Considering the unknown boundary disturbances of the flexible wing, the model is updated on this basis. (ii) The proposed observer can guarantee that the disturbance errors can converge to zero in a finite time. (iii) The vibration problem of the wing structure is well solved, and there is no spillover effect in the system control.The rest of this paper is organized as follows: the dynamics of the coupled aircraft wing system are presented in Section2. Section 3 presents the new finite-time control scheme and a detailed analysis of the stability of the closed-loop system. Simulation and analysis are carried out in Section 4. Finally, the conclusion is given in Section 5.


## 2. Problem Statement

### 2.1. System Model

A vibrating flexible flapping-wing aircraft subject to unknown disturbances is depicted in Figure1. In this paper, ℛ represents a collection of real numbers, xec is the length between the shear center and the mass center of the wing cross section, and xac is the distance from the aerodynamic center to the shear center of the wing. ρ is the mass per unit of the wing, Ip describes the polar moment of inertia, EIb denotes the bending rigidity, GJ represents torsion rigidity, ghz,t denotes the distributed disturbance, ξ represents the Kelvin–Voigt damping coefficient, and τ1t and τ2t represent the control inputs. Besides, g1t and g2t are the unknown disturbances at the wing tip z=s. For simplicity, some symbols are replaced with ⋅.=∂⋅/∂t, ⋅′=∂⋅/∂z, ⋅′.=∂2⋅/∂z∂t, ⋅″=∂2⋅/∂z2, and ⋅..=∂2⋅/∂t2.Figure 1
Flapping-wing robotic aircraft.The kinetic energyEkt of the flexible wing in this study is given directly as follows:(1)Ekt=12ρ∫0sq˙z,t2dz+12Ip∫0sθ.z,t2dz.Ept denotes the potential energy of the flexible wing:(2)Ept=12EIb∫0sq″z,t2dz+12GJ∫0sθ′z,t2dz.The virtual work of damping on the wing is expressed by(3)δHdt=−ξEIb∫0sq˙″z,tδq″z,tdz−ξGJ∫0sθ′.z,tδθ′z,tdz.The virtual work of the coupling of bending and torsion stiffness is presented as follows:(4)δHct=ρxec∫0sq¨z,tδθz,tdz+ρxec∫0sθ..z,tδqz,tdz.The virtual work of unknown disturbance is given as(5)δHft=∫0sghz,tδqz,t−xacghz,tδθz,tdz+g1tδqs,t+g2tδθs,t.The virtual workδHτt performed by the control inputs can be obtained as follows:(6)δHτt=τ1tδqs,t+τ2tδθs,t.Then, we add all the virtual works of the system:(7)δHt=δHct+Hdt+Hft+Hτt.Hamilton’s principle is formulated as follows [47, 48]:(8)∫t1t2δEkt−Ept+Htdt=0.According to (8), we can obtain the flexible wing dynamics as follows:(9)Ipθ¨z,t−GJθ″z,t−ρxecq¨z,t−ξGJθ˙″z,t=−xacghz,t,ρq¨z,t+EIbq‴′−ρxecθ¨z,t+ξEIbq˙‴′z,t=ghz,t.Moreover, we can get the boundary conditions of the system:(10)q0,t=q′0,t=q″s,t=θ0,t=0,EIbq‴s,t+ξEIbq˙‴s,t=−τ1t−g1t,GJθ′s,t+ξGJθ˙′s,t=τ2t+g2t.


### 2.2. Preliminaries

The lemmas and assumption are proposed here to facilitate the later controller design and stability analysis.Assumption 1.
Forg1t, g2t, and ghz,t, we suppose that there exist ζ1>0, ζ2>0, and ghmax>0 such that g1t≤ζ1, g2t≤ζ2, and ghz,t≤ghmax, z,t∈0,s×0,+∞. This assumption is reasonable because the energy of external disturbances is limited.Lemma 1 (see [49]).
If there existϖ1s,t,ϖ2s,t∈ℛ, p>0 with s,t∈0,f×0,+∞, we can obtain(11)ϖ1ϖ2≤1pϖ12+pϖ22.Lemma 2 (see [49]).
Ifϖs,t∈ℛ satisfies the condition ϖ0,t=0, then(12)ϖ2≤f∫0fϖ′2ds,where s,t∈0,f×0,+∞.Lemma 3.
The following inequality for the positive definite functionQt is used to derive our main results as follows:(13)Q˙t≤−ϕ1Qt−ϕ2Qϕ3t.Then, the functionQt will have an equilibrium point. It can converge to the point in a finite time as follows:(14)tf≤1ϕ11−ϕ3lnϕ1Q1−ϕ30+ϕ2ϕ2,with ϕ1>0, ϕ2>0, and 0<ϕ3<1 being undetermined constants.


## 2.1. System Model

A vibrating flexible flapping-wing aircraft subject to unknown disturbances is depicted in Figure1. In this paper, ℛ represents a collection of real numbers, xec is the length between the shear center and the mass center of the wing cross section, and xac is the distance from the aerodynamic center to the shear center of the wing. ρ is the mass per unit of the wing, Ip describes the polar moment of inertia, EIb denotes the bending rigidity, GJ represents torsion rigidity, ghz,t denotes the distributed disturbance, ξ represents the Kelvin–Voigt damping coefficient, and τ1t and τ2t represent the control inputs. Besides, g1t and g2t are the unknown disturbances at the wing tip z=s. For simplicity, some symbols are replaced with ⋅.=∂⋅/∂t, ⋅′=∂⋅/∂z, ⋅′.=∂2⋅/∂z∂t, ⋅″=∂2⋅/∂z2, and ⋅..=∂2⋅/∂t2.Figure 1
Flapping-wing robotic aircraft.The kinetic energyEkt of the flexible wing in this study is given directly as follows:(1)Ekt=12ρ∫0sq˙z,t2dz+12Ip∫0sθ.z,t2dz.Ept denotes the potential energy of the flexible wing:(2)Ept=12EIb∫0sq″z,t2dz+12GJ∫0sθ′z,t2dz.The virtual work of damping on the wing is expressed by(3)δHdt=−ξEIb∫0sq˙″z,tδq″z,tdz−ξGJ∫0sθ′.z,tδθ′z,tdz.The virtual work of the coupling of bending and torsion stiffness is presented as follows:(4)δHct=ρxec∫0sq¨z,tδθz,tdz+ρxec∫0sθ..z,tδqz,tdz.The virtual work of unknown disturbance is given as(5)δHft=∫0sghz,tδqz,t−xacghz,tδθz,tdz+g1tδqs,t+g2tδθs,t.The virtual workδHτt performed by the control inputs can be obtained as follows:(6)δHτt=τ1tδqs,t+τ2tδθs,t.Then, we add all the virtual works of the system:(7)δHt=δHct+Hdt+Hft+Hτt.Hamilton’s principle is formulated as follows [47, 48]:(8)∫t1t2δEkt−Ept+Htdt=0.According to (8), we can obtain the flexible wing dynamics as follows:(9)Ipθ¨z,t−GJθ″z,t−ρxecq¨z,t−ξGJθ˙″z,t=−xacghz,t,ρq¨z,t+EIbq‴′−ρxecθ¨z,t+ξEIbq˙‴′z,t=ghz,t.Moreover, we can get the boundary conditions of the system:(10)q0,t=q′0,t=q″s,t=θ0,t=0,EIbq‴s,t+ξEIbq˙‴s,t=−τ1t−g1t,GJθ′s,t+ξGJθ˙′s,t=τ2t+g2t.


## 2.2. Preliminaries

The lemmas and assumption are proposed here to facilitate the later controller design and stability analysis.Assumption 1.
Forg1t, g2t, and ghz,t, we suppose that there exist ζ1>0, ζ2>0, and ghmax>0 such that g1t≤ζ1, g2t≤ζ2, and ghz,t≤ghmax, z,t∈0,s×0,+∞. This assumption is reasonable because the energy of external disturbances is limited.Lemma 1 (see [49]).
If there existϖ1s,t,ϖ2s,t∈ℛ, p>0 with s,t∈0,f×0,+∞, we can obtain(11)ϖ1ϖ2≤1pϖ12+pϖ22.Lemma 2 (see [49]).
Ifϖs,t∈ℛ satisfies the condition ϖ0,t=0, then(12)ϖ2≤f∫0fϖ′2ds,where s,t∈0,f×0,+∞.Lemma 3.
The following inequality for the positive definite functionQt is used to derive our main results as follows:(13)Q˙t≤−ϕ1Qt−ϕ2Qϕ3t.Then, the functionQt will have an equilibrium point. It can converge to the point in a finite time as follows:(14)tf≤1ϕ11−ϕ3lnϕ1Q1−ϕ30+ϕ2ϕ2,with ϕ1>0, ϕ2>0, and 0<ϕ3<1 being undetermined constants.


## 3. Control Design

The emphasis of the research is how to construct boundary controllers to suppress the vibration of a coupled wing system. For the sake of the control goal, a new control scheme based on finite-time disturbance observer is adopted, which can ensure the stability of the closed-loop system. The control block diagram of the system is shown in Figure2.Figure 2
Finite-time control for the coupled aircraft wing.First, the boundary control laws are proposed as follows:(15)τ1t=−k1aqs,t+bq˙s,t−g^1t,τ2t=−k2aθs,t+bθ˙s,t−g^2t,where k1,k2>0.Step 1.
The auxiliary functions are defined as follows:(16)σ1t=−ξEIq‴s,t−ν1t,(17)ν˙1t=κ1σ1t+κ2signσ1t+κ3σ12a1−a2/a2t+τ1t+EIbq‴s,t,(18)σ2t=ξGJθ′s,t−ν2t,(19)ν˙2t=κ4σ2t+κ5signσ2t+κ6σ22b1−b2/b2t+τ2t−GJθ′s,t.
To arrive at the control objectives that deal with the unknown disturbancesg1t and g2t, we define the estimation form of disturbance terms as follows:(20)g^1t=κ1σ1t+κ2signσ1t+κ3σ12a1−a2/a2t,(21)g^2t=κ4σ2t+κ5signσ2t+κ6σ22b1−b2/b2t,where κ1,κ2,κ3,κ4,κ5, and κ6 are all positive numbers. Moreover, a1,a2,b1, and b2 are all odd numbers that satisfied a1<a2<2a1 and b1<b2<2b1.Step 2.
We choose Lyapunov candidate function as(22)Wt=Wat+Wot,where(23)Wat=b2ρ∫0sq˙z,t2dz+b2EIb∫0sq″z,t2dz+b2Ip∫0sθ˙z,t2dz+b2GJ∫0sθ′z,t2dz,Wot=aρ∫0sq˙z,tqz,tdz+aIp∫0sθ˙z,tθz,tdz−aρxec∫0sq˙z,tθz,t+qz,tθ˙z,tdz−bxec∫0sq˙z,tθ˙z,tdz,where a>0 and b>0.Remark 1.
In this paper, we use the Lyapunov direct method to design the controller given in the specific form of Lyapunov function, whereWat is derived from the system kinetic energy Ekt and potential energy Ept, which is called the energy term. Wot is derived from the coupling of various state quantities of the system and becomes a crossing term. Q1t and Q2t will be given later in this article, which represent the auxiliary items of the system to deal with disturbance errors. By adjusting the control laws (15)–(21) and Lyapunov candidate function, we ensure that the derivative of Lyapunov function Wt has an upper bound and let Q1tandQ2t satisfy Lemma 3 so as to prove the uniform boundedness of state variables and the finite-time convergence of disturbance errors.Remark 2.
The control signalsqs,t,q˙s,t,θs,t,θ˙s,t,q‴s,t,θ′s,t,q˙‴s,t, and θ˙′s,t in the control equations (15)–(21) can be obtained during execution, where qs,t and θs,t are obtained by the laser displacement sensors, and θ′s,t is obtained by the inclinometer. The remaining variables q˙s,t,θ˙s,t,q‴s,t,q˙‴s,t, and θ˙′s,t are further obtained by the backward difference algorithms.Theorem 1.
The Lyapunov function (22) has upper and lower bounds:(24)0≤λ2κt≤Wt≤λ1κt,where λ1 and λ2 are positive numbers.Proof.
A new function is defined as follows:(25)κt=∫0sq˙z,t2+q′′z,t2+θ˙z,t2+θ′z,t2dz.
Hence, we obtain(26)y2κt≤Wat≤y1κt,where y1 and y2 are two positive numbers, y1=b/2maxEIb,Ip,GJ,ρ, and y2=b/2minρ,EIb,Ip,GJ. For Wot, we can obtain(27)Wot≤aρ∫0sq˙z,t2dz+s4∫0sq″z,t2dz+aIp∫0sθ˙z,t2dz+s2∫0sθ′z,t2dz+aρxec∫0sq˙z,t2dz+∫0sθ˙z,t2dz+aρxecs4∫0sq″z,t2dz+s2∫0sθ′z,t2dz+bρxec∫0sq˙z,t2dz+∫0sθ˙z,t2dz+aIp+aρxec+bρxec∫0sθ˙z,t2dz+aIp+aρxecs2∫0sθ′z,t2dz≤y3κt,where y3=maxaρ+aρxec+bρxec, aρ+aρxecs4,aIp+aρxec+bρxec,aIp+aρxecs2 and b satisfies b>2y3/minρ,Ip,EIb,GJ.
Now, we can prove equation (24) as follows:(28)O≤λ2κt≤Wt≤λ1κt,where λ2=y2−y3 and λ1=y1+y3.Step 3.
The derivative ofWat gives(29)W˙at≤−bq˙s,tEIbq‴s,t+ξEIbq˙‴s,t+bθ˙s,tGJθ′s,t+ξGJθ˙′s,t+bρxec∫0sq˙z,tθ¨z,t+q¨z,tθ˙z,tdz−bξGJ2s2−η2bxac∫0sθ˙z,t2dz−bξEIb2s4−η1b∫0sq˙z,t2dz+bη1+bxacη2sghmax2−bξEIb2∫0sq˙″z,t2dz−bξGJ2∫0sθ˙″z,t2dz.
Similarly, differentiatingWot leads to(30)W˙ot=aρ∫0sq¨z,tqz,tdz+aIp∫0sθ¨z,tθz,tdz−aρxec∫0sq¨z,tθz,tdz−aρxec∫0sqz,tθ¨z,tdz+aρ∫0sq˙z,t2dz+aIp∫0sθ˙z,t2dz−bρxec∫0sq¨z,tθ˙z,tdz−bρxec∫0sq˙z,tθ¨z,tdz−2aρxec∫0sq˙z,tθ˙z,tdz.
Considering (10) and controllers (15)–(21), we obtain(31)W˙t≤−k2−1η9aθs,t+bθ˙s,t2+g˜22tη9−k1−1η8aqs,t+bq˙s,t2+g˜12tη8−aEIb−aξEIbη3−η6as4∫0sq′′z,tdz−aGJ−aξGJη4−η7s2xac∫0sθ′z,t2dz−bξEIb2s4−η1b−aρ−2aρxecη5∫0sq˙z,t2dz−bξGJ2s2−η2bxac−aIp−2aρxecη5∫0sθ˙z,t2dz−bξEIb2−aξEIbη3∫0sq˙′′z,t2dz−bξGJ2−aξGJη4∫0sθ˙'z,t2dz+bη1+bxacη2+aη6+axacη7sghmax2≤−λ3κt+ε,where ηi>0,i=1…9. In addition, because g^1t and g^2t are estimates of g1t and g2t, under the action of disturbance observers, the errors between them are also bounded. Thus, there exist positive numbers ζ3 and ζ4, satisfying g˜1t≤ζ3 and g˜2t≤ζ4.
Letμ1=k2−1/η9>0 and μ2=k1−1/η8>0. The selection of intermediate parameters is provided as follows:(32)μ3=bξEIb2s4−η1b−aρ−2aρxecη5>0,μ4=bξGJ2s2−η2bxac−aIp−2aρxecη5>0,μ5=aEIb−aξEIbη3−η6as4>0,μ6=aGJ−aξGJη4−η7as2xac>0,λ3=minμ3,μ4,μ5,μ6>0,ε=bη1+bxacη2+aη6+axacη7sghmax2+η9ζ42+η8ζ32<+∞.
Therefore, we can obtain(33)W˙t≤−λWt+ε,where λ=λ3/λ1.Theorem 2.
Through the dynamical system and the proposed control laws, the controlled system’s statesqz,t and θz,t are uniformly ultimately bounded.Proof.
Invoking (33) and multiplying eλt yields(34)W˙teλt≤−λWteλt+εeλt.
Integrating (34), we have(35)Wt≤W0−ελe−λt+ελ≤W0e−λt+ελ.
Furthermore, invoking (24) and Lemma 2, we have(36)1s3qz,t2≤1s2∫0sq′z,t2dz≤∫0sq″z,t2dz≤κt≤1λWt.
Finally, this results in(37)qz,t≤s3λ2W0e−λt+ελ,∀z,t∈0,s×0,+∞,θz,t≤sλ2W0e−λt+ελ,∀z,t∈0,s×0,+∞.Theorem 3.
Provided thatκ2≥ζ1 holds, the disturbance error g˜1t ensures convergence to zero before t≥tf1, where(38)tf1≤a22κ1a2−a1lnκ1Q1a2−a1/a20+2a1/a2−1κ32a1/a2−1κ3,with Q10 denoting an initial value of Q1t, and Q1t=σ12t/2Theorem 4.
Provided thatκ5≥ζ2 holds, the disturbance error g˜2t converges to zero before t≥tf2, where(39)tf2≤b22κ4b2−b1lnκ4Q2b2−b1/b20+2b1/b2−1κ62b1/b2−1κ6,with Q20 denoting an initial value of Q2t, and Q2t=σ22t/2.Proof.
DifferentiatingQ1t and invoking (16) and (17), we have(40)Q˙1t=σ1tσ˙1t=σ1tξEIbq˙‴s,t−ν˙1t=σ1tξEIbq˙‴s,t−κ1ϕ1t−κ2signϕ1t−κ3σ12a1−a2/a2t−τ1t−EIbqs,t=−κ1σ12t−κ2σ1t−κ3σ1t2a1−a2/a2t+σ1t+g1t≤−κ1σ12t−κ2−ζ1σ1t−κ3ϕ1tσ12a1−a2/a2t≤−κ1σ12t−κ3σ1tσ12a1−a2/a2t.
Therefore,Q1t will converge to an equilibrium point in a finite time tf1. We define p=a1/a2, and then we multiply (40) by Q1pt/1−p to derive the following:(41)dQ1≤−2κ11−pQ11−pt−2pκ31−pQ1pt1−pdt.
Lettingιt=Q11−pt, with the analysis made in (41), we derive(42)ι˙t≤−2κ11−pιt−2pκ31−p.
Multiplying (42) by e2κ11−pt results in(43)ι˙te2κ11−pt≤−2κ11−pe2κ11−ptιt−2pκ31−pe2κ11−pt.
Then, we obtain(44)dιte2κ11−ptdt≤−2pκ31−pe2κ11−pt.
We apply the integral of (44) as follows:(45)ιt≤−2p−1κ3κ1+ι0+2p−1κ3κ1e−2κ11−pt.
Equation (45) can be rewritten as(46)Q1−pt≤−2p−1κ3κ1+Q11−p0+2p−1κ3κ1e−2κ11−pt.
InvokingQ1t and (40) yields(47)Q1t≥0,Q˙1t≤0.
According to (46) and (47), we have(48)Q11−p0+2p−1κ3κ1e−2κ11−ptf1−2p−1κ3κ1≥0.
Whent=tf1 and Q1tf1=0, then we have Q1t≡0.
We can further obtain(49)e2κ11−ptf1≤κ1Q11−p0+2p−1κ32p−1κ3.
Taking the logarithm for it as follows:(50)tf1≤12κ11−plnκ1Q11−p0+2p−1κ32p−1κ3=a22κ1a2−a1lnκ1Q1a2−a1/a20+2a1/a2−1κ32a1/a2−1κ3.
In a word, consideringκ2≥ζ1, when t≥tf1, we derive(51)Q1t≡0.
Invoking (51) and Q1t, we obtain the following when t≥tf1,(52)σ1t≡0.
Moreover, whent≥tf1, we arrive at(53)σ˙1t≡0.
The disturbance estimation error is expressed as follows:(54)g˜1t=g1t−g^1t.
The combination of (10), (20), and (54) gives(55)g˜1t=−τ1t−ξEIbq˙‴s,t−EIbq‴s,t−g^1t=−ξEIbq˙‴s,t+−τ1t−EIbq‴s,t+κ1σ1t+κ2signσ1t+κ3σ12a1−a2/a2=σ˙1t.
Invoking (53) and (55), we can conclude that if κ2≥ζ1 holds, g˜1t converges to zero for ∀t≥tf1. Similarly, we can prove that if κ5≥ζ2 holds, g˜2t converges to zero for ∀t≥tf2.


## 4. Simulation

In this section, the finite difference method is used to approximate the dynamics of the system with system parameters ass=2.0m, xec=0.35m,EIb=0.2Nm2,GJ=0.5Nm2, ρ=10kg/m, xac=0.05N, Ip=1.5kgm, and ξ=0.6. The initial condition of the system is set as qz,0=z/s, and θz,0=πz/2s. Besides, we have q˙z,0=0 and θ˙z,0=0. The inevitable external disturbances are defined as follows:(56)ghz,t=1++3cos3πt+sinπtz/30g1t=0.02+0.06sin0.05tg2t=0.04+0.02sin0.1t.Figures3 and 4 show the displacement of the coupled wing without control, that is, τ1t=0 and τ2t=0. Figures 5 and 6 depict the three-dimensional representation of flexible wings with boundary control. The control parameters are chosen as k1=2, k2=1, κ1=0.8, κ2=1, κ3=4, κ4=0.8, κ5=1, κ6=4, a1=b1=7, and a2=b2=13. Figures 7 and 8 represent the response generated by the control inputs τ1t and τ2t. As shown in Figures 9 and 10, we can conclude that the control design can ensure that the flexible wing system has good performance by comparing with the freely vibrating situation. At last, we provide two pictures Figures 11 and 12 about the disturbance estimation errors. It can be concluded from the figures that the disturbance estimation errors can converge to zero in a finite time.Figure 3
Bending displacement of the wing without control.Figure 4
Twist displacement of the wing under without control.Figure 5
Bending displacement of the wing with the proposed control.Figure 6
Twist displacement of the wing with the proposed control.Figure 7
Designed control commandτ1t with the proposed control.Figure 8
Designed control commandτ2t with the proposed control.Figure 9
Boundary bending displacement of the wing with the proposed control.Figure 10
Boundary twist displacement of the wing with the proposed control.Figure 11
Disturbance estimation errorg˜1t with the proposed control.Figure 12
Disturbance estimation errorg˜2t with the proposed control.The simulation results Figures3–12 show that the designed controller can effectively suppress the vibration of flexible wing, and the bending and torsion deformation of a flexible wing qz,t and θz,t under the presented control are guaranteed to be stable. The last two figures show the finite-time convergence of the disturbance estimation errors g˜1t and g˜2t. In order to further highlight the advantages of the proposed control strategy, a boundary control strategy based on the nonlinear disturbance observers (DOBC) is expressed as follows:(57)τ1t=−k1aqs,t+bq˙s,t−g^1t,τ2t=−k2aθs,t+bθ˙s,t−g^2t,g^1t=−ξEIq‴s,t+σ1t,σ˙1t=−τ1t−EIbq‴s,t−g^1t,g^2t=ξGJθ′s,t+σ2t,σ˙2t=−τ2t+GJθ′s,t−g^2t,where parameter selections are consistent with the above design. Figures 13–20 show the control effect of the system using the DOBC strategy. It can be seen that the control effect is worse than that of the control strategy proposed in this paper. As can be seen from the figures, the inputs τ1t and τ2t and displacements qz,t, θz,t, qs,t, and θs,t cannot converge to zero quickly. Although the disturbance observers can track the change of external disturbances to a certain extent, they will still produce a certain disturbance errors. To sum up, we can get that the control effect of the derived finite-time convergence antidisturbance control strategy is better than that of the DOBC strategy.Figure 13
Bending displacement of the wing with DOBC.Figure 14
Twist displacement of the wing with DOBC.Figure 15
Designed control commandτ1t with DOBC.Figure 16
Designed control commandτ2t with DOBC.Figure 17
Boundary bending displacement of the wing with DOBC.Figure 18
Boundary twist displacement of the wing with DOBC.Figure 19
Disturbance estimation errorg˜1t with DOBC.Figure 20
Disturbance estimation errorg˜2t with DOBC.


## 5. Conclusion

In this paper, based on Lyapunov’s direct method and the new coupled wing model, we addressed the control problem for the flexible wing subject to external disturbances with a new finite-time convergence antidisturbance control strategy. Under the action of the controllers, the external disturbance errors converged to zero in a finite time. Hence, we concluded that the proposed control scheme could stabilize the flexible wing system with a better performance. Finally, we analyzed the stability of the closed-loop system and the effectiveness of the boundary controller through strict theoretical proof and simulation results. Future research directions may include the adaptive control [50–53], intelligent techniques [54–57], and the uncertainty and disturbance estimator-based control [58–60].



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