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Research Papers

# Damage-Mitigating Predictive Control of Airfoil Flutter for a General Hypersonic Flight VehiclePUBLIC ACCESS

[+] Author and Article Information
Xiaohui Zhang

College of Automation Engineering,
Nanjing University of Aeronautics and Astronautics,
Nanjing 211106, China
e-mail: zhangxiaoh@nuaa.edu.cn

Yuhui Wang

College of Automation Engineering,
Nanjing University of Aeronautics and Astronautics,
Nanjing 211106, China
e-mail: wangyh@nuaa.edu.cn

Xingkai Feng

College of Automation Engineering,
Nanjing University of Aeronautics and Astronautics,
Nanjing 211106, China
e-mail: fengstar@nuaa.edu.cn

Siyuan Hou

College of Automation Engineering,
Nanjing University of Aeronautics and Astronautics,
Nanjing 211106, China
e-mail: housiyuan@nuaa.edu.cn

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the Journal of Vibration and Acoustics. Manuscript received August 1, 2018; final manuscript received April 10, 2019; published online June 5, 2019. Assoc. Editor: Alper Erturk.

J. Vib. Acoust 141(5), 051007 (Jun 05, 2019) (9 pages) Paper No: VIB-18-1323; doi: 10.1115/1.4043511 History: Received August 01, 2018; Accepted April 10, 2019

## Abstract

This paper aims to investigate the airfoil flutter damage-mitigating problem in hypersonic flow. A new adaptive robust nonlinear predictive control law is designed in this paper to mitigate the damage during airfoil flutter of a generic hypersonic flight vehicle. A three-degrees-of-freedom airfoil dynamic motion model is established, in which the third piston theory is employed to derive the unsteady aerodynamics. Then, the complicated responses of the hypersonic airfoil flutter model are analyzed. In order to mitigate the damage of the airfoil, a predictive controller is designed by introducing an adaptive predictive period, and asymptotical stability analysis of the robust nonlinear predictive controller is performed. Subsequently, based on the nonlinear aerodynamics of the airfoil and damage accumulation model, the damage of the airfoil is observed online. Simulation results illustrate the effectiveness of the proposed method.

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## Introduction

The flutter of an aeroelastic system is a nonlinear dynamic unstable phenomenon. It is very dangerous to aircraft structures, because the vibration amplitude and damage accumulation of the airfoil may increase dramatically. Therefore, to restrict the negative effects of airfoil flutter and mitigate the fatigue damage of the airfoil in hypersonic flow, various strategies have been explored to suppress flutter [110].

In the last few decades, aeroelastic analysis and active control have received a great deal of attention in the hypersonic airflow region [510]. Most of them focused on the calculation of the unsteady aerodynamics and active controller design. In Refs. [11,12], the first-order and the third-order approximation piston theories were employed to calculate the unsteady aerodynamics of the airfoil with concentrated nonlinearities. In Refs. [4,5,8,9], the recurrent neural networks [4] and control [9] were successfully used to design controllers for suppressing airfoil flutter.

To achieve a better control effect and mitigate airfoil flutter damage, a novel damage-mitigation predictive control is presented in this paper. Predictive control based on the optimization of a cost function of tracking error is one of the most promising control methods in engineering. In Ref. [13], long-range generalized predictive control (GPC) was applied in linear systems successfully. With further study, great efforts have been taken to extend GPC to nonlinear systems. In Ref. [14], a nonlinear predictive control (NPC) for a class of multivariable nonlinear system was proposed, and the importance of predictive period was verified. In Ref. [15], a finite-horizon nonlinear predictive controller using the Taylor approximation was presented and applied to the planning motion problem of the mobile robot and the tracking problem of the rigid link robot manipulator. In Ref. [16], the NPC law based on an uncertain nonlinear system was raised to optimize the receding-horizon criterion of the attitude tracing error for a hypersonic flight vehicle (HFV). In Ref. [17], a robust nonlinear predictive control (RNPC) with an integral term was proposed to enhance the robustness of the systems. Undoubtedly, these methods based on NPC have contributed a lot to the control problems with complicated models. The predictive periods of the NPC in Refs. [1417] were set as constant values; in most cases, it is reasonable to do so, but due to the complex nonlinearities of the airfoil flutter dynamics of an HFV, a fixed predictive period may cause an additional damage accumulation.

Due to the destructiveness of the airfoil flutter, any minor structural damage may evolve into a disaster. Therefore, high-precision modeling for the damage dynamics of the vehicles is crucial to obtain the damage evolution process and achieve the satisfied damage-mitigating control effect. In Refs. [1820], a fatigue crack growth model was considered to analyze the damage evolution under variable-amplitude loading. In Refs. [21,22], a hierarchical life extending control was applied to a boiler system, and two continuum fatigue damage models were described. Similarly, for the airfoil flutter system of HFVs, it is expected that the vibration can be restricted by effective control methods to slow down the evolution of fatigue damage.

As discussed above, a novel adaptive robust nonlinear predictive (RNPC) scheme is designed to mitigate the damage caused by the airfoil flutter of a generic HFV. Based on the third-order piston theory, the nonlinear aerodynamics of the airfoil is calculated, and a three degrees-of-freedom (3DOF) airfoil motion model is established. Based on the stress–fatigue damage model [1822], a logarithmic damage model with a consideration of the influence of the current linear damage is constructed to estimate the nonlinear damage of the airfoil online. Subsequently, a damage-mitigating predictive controller with the adaptive predictive period is designed to control the airfoil flutter of the HFV. Numerical simulations are carried out to demonstrate the effectiveness and feasibility of the proposed control scheme. The main contribution of this work is that a novel RNPC scheme is proposed to mitigate the damage caused by airfoil flutter of a generic HFV, in which the predictive period θ is regulated adaptively by the error e.

## Nonlinear Airfoil Motion Model

Considering a typical 3DOF hypersonic airfoil model (as shown in Fig. 1) with cubic structure nonlinearities and aerodynamic nonlinearities, the equations of the airfoil motion with a control surface based on the energy method can be presented as follows [6,11,12]: Display Formula

(1)
$mh¨+Sαα¨+Sββ¨+chh˙+Khh=−Qα$
Display Formula
(2)
$Sαh¨+Iαα¨+[(d−a)bSβ+Iβ]β¨+cαα˙+Kαα+δαK^αα3=Mα$
Display Formula
(3)
$Sβh¨+[(d−a)bSβ+Iβ]α¨+Iββ¨+cββ˙+Kββ=Mβ+u$
where h, α, and β denote the plunging (upward considered positive), pitching (upward deflection considered positive), and flapping displacement (downward deflection considered positive), respectively; a and d are the length percentage of the half chord b from the leading edges to the elastic axis and the length percentage of b from the leading edges to the hinge, respectively; V denotes the airspeed; m denotes the structural mass per unit span; Sα and $Sβ$ are the static unbalance of the airfoil about the elastic axis and the static unbalance of the flag about the hinge, respectively; Iα and $Iβ$ denote the inertial mass moment of the airfoil about the elastic axis and the inertial mass moment of the flag about the hinge, respectively; ch, cα, and $cβ$ are the damping coefficients in the plunging, pitching, and flapping, respectively; and Kh, Kα, and $Kβ$ denote the linear structural stiffness coefficients in the plunging, pitching, and flapping, respectively; $δα$ that denotes the tracing quantity of nonlinear pitching stiffness can take the value 1 or 0 depending on whether the nonlinear is accounted for (here, $δα$ = 1); $K^α$ denotes the nonlinear pitching stiffness coefficients; Qh, Mα, and $Mβ$ are the unsteady aerodynamic force and moments, respectively; and u denotes the control input.

Based on the third-order piston theory method [9,11,12], the aerodynamic differential pressure $△p$ can be described as Display Formula

(4)
$△p=4qdλMA[(∂z(x,t)∂t1V+∂z(x,t)∂x)+(k+1)λ2MA212×(∂z(x,t)∂t1V+∂z(x,t)∂x)3]$
where qd is the dynamic pressure, MA = V/a is the Mach number, where a is the speed of sound, and $λ=MA2/Ma2+1$ is the correction factor. k is the ratio of specific heats (k = 1.4), x and z(x, t) are the lateral and longitudinal displacements of any point of the airfoil as shown in Fig. 1, respectively. And Display Formula
(5)
$z(x,t)={h+(x−ab)α,x≤dbh+(x−ab)α+(x−db)β,db

Subsequently, the unsteady aerodynamic lift Qh and moments Mα and $Mβ$ are obtained from the integration of the aerodynamic differential pressure $△p$ of the airfoil, as follows: Display Formula

(6)
$Qα=∫0db△Pdx+∫db2b△Pdx$
Display Formula
(7)
$Mα=−∫0db△P(x−ab)dx−∫db2b△P(x−ab)dx$
Display Formula
(8)
$Mβ=−∫db2b△P(x−db)dx$

Based on Eqs. (6), (7), and (9), the equations of airfoil motion (Eqs. (1)(3)) can be rewritten as follows: Display Formula

(9)
$mh¨+Sαα¨+Sββ¨+(ch+8qdλbMAV)h˙+8qdλ(1−a)b2MAVα˙+2qdλ(2−d)2b2MAVβ˙+Khh+8qdλbMAα+4qdλ(2−d)bMAβ=−2qdλ3(k+1)MAb3α3$
Display Formula
(10)
$Sαh¨+Iαα¨+[(d−a)bSβ+Iβ]β¨+8qdλ(1−a)b2MAVh˙+[cα+4qdλb3MAV(83−4a+2a2)]α˙+2qdλ(d−3a+4)(2−d)2b33MAVβ˙+[Kα+8qdλ(1−a)b2MA]α+2qdλ(2−d)(d−2a+2)b2MAβ=−2qdλ3(k+1)MA(1−a)b23α3−BKαα3$
Display Formula
(11)
$Sβh¨+[(d−a)bSβ+Iβ]α¨+Iββ¨+2qdλ(2−d)2b2MAVh˙+2qdλ(d−3a+4)(2−d)2b33MAVα˙+[cβ+4qdλ(2−d)3b33MAV]β˙+2qdλ(2−d)2b2MAα+[Kβ+2qdλ(2−d)2b2MA]β=−qdλ3(k+1)MA(2−d)2b26α3+u$
where $B=K^α/Kα$ denotes the degree of the physical nonlinear of the system [12].

## Airfoil Flutter Damage Estimation Model Via Stress-Damage

If an HFV is subjected to a sustained large amplitude limit cycle oscillations without any timely efficient control measures, the structural damage of the airfoil is certainly inevitable. Once the damage threshold is reached, the airfoil structures may be failed. Thus, it is very important to guarantee the flight safety and reliability of HFVs by accurately estimating the damage caused by the flutter and taking effective mitigation measures.

Based on the results of Refs. [21,22] and Dowling’s amendment [23], the cycle life can be replaced by $1/δ$, where the linear damage $δ$ includes the elastic part $δe$ and the plastic part $δp$, described as Display Formula

(12)
$δe=2[σ−σr2(σf′−σm)]−1/b¯$
Display Formula
(13)
$δp=2[1εf′(σ−σr2K′)1/n′(1−σmσf′)c¯/b¯]−1/c¯$
where $σ=Qα/2b$ is the equivalent stress of the airfoil surface, $σr$ is the minimal value of stress on the airfoil surface, $σm=(σ+σr)/2$ is the main stress, and $σf′$, $εf′$, n′, $b¯$, $c¯$, and K′ are the material parameters based on experimentally determined.

According to the result of Ref. [24], no damage occurs when the equivalent stress $σ$ decreases. Thus, the damage can be ignored when $σ<σr$. Then, the damage rate $dδ/dt$ is computed as a weighted average of $δe$ and $δp$ to improve the accuracy of life prediction, such that Display Formula

(14)
$dδdt={ηdδedσdσdt+(1−η)dδpdσdσdt,σ≥σr0,σ<σr$
where $η=△εe/△ε$ is the weight function, $△εe=(σ−σr)/E$, $△ε=2(σ−σr/2K′)1/n′$, and E is the elastic modulus of the airfoil material.

Since the complex dynamics of airfoil flutter can cause the airfoil surface to subject to loads of varying amplitude, the linear expression of the damage accumulation does not adequately describe this complex situation. Therefore, a nonlinear damage expression is considered here to modify the linear damage model (Eq. (14)). Based on the fatigue crack growth model that the crack growth is dependent on both the stress amplitude and the current level of damage accumulation [1820], the nonlinear damage rate D that is dependent on the current level of damage accumulation $δ$ can be given as follows [22]: Display Formula

(15)
$dDdt=γ1δ(γ1−1)×dδdt+δγ1Inδ×dγ1dt$
where $γ1=(2/3)δ−0.4$ [24]. During the $τ1$ time period, the accumulated damage can be expressed as Display Formula
(16)
$D=∫t1t1+τ1dDdtdτ1$

Obviously, D = 0 means that the structure is an ideal structure with no damage, and D = 1 means that the structure completely fails.

Remark 1. It is important to note that different materials will obtain different structural experimental data; therefore, if different materials are considered, the results and the conclusions may be different, but the analysis processes are similar.

## Damage Predictive of Hypersonic Airfoil Flutter

In order to estimate the fatigue damage of airfoil flutter, the flutter responses of the airfoil motion model should be analyzed first. The nondimensional variables are introduced to simplify the calculation [6,8,9]: $ξ=h/b$ is the nondimensional plunging displacement, $χα=Sα/(mb)$ and $χβ=Sβ/(mb)$ are the nondimensional static unbalance about the elastic axis and nondimensional static unbalance about the hinge, respectively, $τ=Vt/b$ is the nondimensional time, V1 = V/(bwα) is the nondimensional airspeed parameter, $μ=m/(4ρb2)$ is the nondimensional mass parameter, $rα2=Iα/(mb2)$ and $rβ2=Iβ/(mb2)$ are the nondimensional radius of gyration with respect to elastic axis and the nondimensional radius of gyration with respect to the hinge, respectively, $wh2=Kh/m$, $wα2=Kα/Iα$, and $wβ2=Kβ/Iα$ are the decouple plunging, pitching, and flapping frequencies of linearization system, respectively, and $ξh=ch/(2mwh)$, $ξα=cα/(2Iαwα)$, and $ξβ=cβ/(2Iβwβ)$ are the nondimensional plunging, pitching, and flapping damping ratios, respectively. Then, the nondimensional equations describing the airfoil motion can be expressed by the following form Display Formula

(17)
$MZ¨+HZ˙+KZ=F+F1u$
where Display Formula
$Z=[ξαβ]T,M=[1χαχβχαrα2(d−a)χβ+rβ2χβ(d−a)χβ+rβ2rβ2],H=[hij]3×3,K=[kij]3×3,F=α3×[fij]3×1,F1=[0014ρb4μV12wα2]T$
in which Display Formula
$h11=2whξhV1wα+λMAμ,h12=λ(1−a)MAμ,h13=λ(2−d)24MAμh21=λ(1−a)MAμ,h22=2ξαrα2V1+λ2MAμ(83−4a+2a2),h23=λ(d−3a+4)(2−d)212MAμh31=λ(2−d)24MAμ,h32=λ(d−3a+4)(2−d)212MAμ,h33=2ξβrβ2wβV1wα+λ(2−d)36MAμk11=wh2V12wα2,h12=λMAμ,k13=λ(2−d)2MAμk21=0,k22=rα2wα2V12wα2+λ(1−a)MAμ,k23=λ(d−2a+2)(2−d)4MAμk31=0,k32=λ(2−d)24MAμ,k33=rβ2wβ2V12wα2+λ(2−d)24MAμ$
Display Formula
$f11=λ3(k+1)MA12μ,f21=λ3(k+1)MA(1−a)12μ+Brα2V12,f31=λ3(k+1)MA(2−d)248μ$

Defining a state vector $x=[ξ,α,β,ξ˙,α˙,β˙]T=[x1,x2,x3,x4,x5,x6]T$, the nondimensional equations of the airfoil motion can be expressed by a state-space matrix form as follows: Display Formula

(18)
$x˙=Ax+F¯x23+B¯u$
where Display Formula
$A=[0I−M−1K−M−1H]=[000100000010000001a11a12a13a14a15a16a21a22a23a24a25a26a31a32a33a34a35a36]F¯=B1F=[000f¯11f¯12f¯13]T,B¯=B1F1=[000b11b12b13]T,B1=[0M−1]6×3T$

The parameters of the airfoil flutter system are chosen as [6,8,9]: μ = 100, $χα=0.25$, $χβ=0.0125$, $rα2=0.25$, $rβ2=0.00625$, a = 0.6, d = 1.6, wh = 80, wα = 100, $wβ=120$, B = 50, b = 1, $ξh=0.14$, $ξα=0.16$, and $ξβ=0.1$. The TA15 is chosen as the material of the airfoil, and the material parameters of the TA15 are chosen as [2426]: E = 110 GPa, K′ = 1.400 GPa, n′ = 0.214, $εf′=0.201$, $c¯=0.521$, $b¯=−0.281$, and $σf′=1.300GPa$.

Let the control input u = 0, and the initial condition of the airfoil motion system is x(0) = [0, 0.001, 0, 0, 0, 0]T. Based on the Hopf bifurcation theory [27], the nondimensional flutter speed of the airfoil motion model (Eq. (18)) is V1 = 19.5883. The fatigue damage of the airfoil corresponding to the convergence oscillation (see Fig. 2), limit cycle oscillation (see Fig. 3), and divergence oscillation (see Fig. 4) in different flow speeds are indicated in Figs. 2(c), 3(c), and 4(c). From Figs. 3 and 4, we can see that if the undesirable oscillation is not suppressed, airfoil structures will suffer continuing damage. Under this consideration, in order to suppress the airfoil flutter and guarantee the flight safety of the aircraft, a predictive control method will be applied to the airfoil flutter system in Sec. 5.

## Damage-Mitigating Predictive Controller Design

Defining the system output $y(τ)=x2$ and the control input vector u ≠ 0, the nonlinear aeroelastic equation expressed in Eq. (18) can be rewritten as Display Formula

(19)
$x˙(τ)=f(x)+g(x)u+g1(x)d¯$
Display Formula
(20)
$y(τ)=h1(x)=x2$
The specific expressions of f(x), g(x), g1(x), and the disturbance $d¯$ are defined as Display Formula
$f(x)=[000100000010000001a11a12a13a14a15a16a21a22a23a24a25a26a31a32a33a34a35a36][x1x2x3x4x5x6]+[000f¯11f¯21f¯31]x23g(x)=[000b11b21b31]T,g1(x)=[OI]6×3T,d¯=[d1d2d3]$

The aim of the damage-mitigating is to design a control law u such that $limt→∞∥y(τ)−yc(τ)∥=0$, where $yc(τ)$ is the reference signal. It is easy to demonstrate that the airfoil motion equations given by Eqs. (19)(20) satisfy the following assumptions [1417]:

###### Predictive Controller Design.

The objective of the RNPC is to develop an optimal control law such that the difference between the predicted output and reference trajectory is minimum within a moving time frame located at time $τ$. The receding-horizon predictive control performance index is formulated by [1417] Display Formula

(21)
$J=12∫0θ(y^(τ+τ^)−y^c(τ+τ^))T×(y^(τ+τ^)−y^c(τ+τ^))dτ^$
where $y^(τ+τ^)=κ(τ^)Y¯(τ)$ and $y^c(τ+τ^)=κ(τ^)Y¯c(τ)$ are the output and the reference output in the moving time frame, respectively, $θ$ is the predictive period, and $τ^$ is the moving time, $0≤τ^≤θ$.

To predict the output $y^(τ+τ^)$ and reference output $y^c(τ+τ^)$ in the moving time frame, the system output $y(τ)$ is expanded into the $ρ¯th$-order Taylor series using Lie derivative operation, together with repeated substitution of Eqs. (19) and (20), such that Display Formula

(22)
$y˙=Lfh1(x)$
Display Formula
(23)
$y¨=Lf2h1(x)+LgLfh1(x)u+Lg1Lfh1(x)d¯$
For the system Eqs. (19)(20), the Lie derivatives of h1(x) along f(x), g(x), and g1(x) are Lfh1(x) = x5, $Lf2h1(x)=a21x1+a22x2+$$a23x3+a24x4+a25x5+a26x6+f¯21x23$, Lgh1(x) = 0, LgLfh1(x) = b21 ≠ 0, $Lg1h1(x)=0$, and $Lg1Lfh1(x)=[0,1,0]$, so the system relative degree is $ρ¯=2$.

Based on the results of Ref. [17], to improve the system robustness, $κ(τ^)$ and $Y¯(τ)$ are defined as follows: Display Formula

(24)
$κ(τ^)=[1τ^⋯τ^(ρ¯+1)(ρ¯+1)!]=[1τ^τ^22!τ^33!]$
Display Formula
(25)
$Y¯(τ)=[∫0τy(τ^)dτ^y(τ)y˙(τ)⋯y[ρ¯](τ)]=[∫0τy(τ^)dτ^y(τ)y˙(τ)y¨(τ)]$
The integral action of $∫0τy(τ^)dτ^$ designed in here is to consider the error $∫0τe(τ^)dτ^=∫0τ(y(τ^)−yc(τ^))dτ^$ into the predictive controller. Similarly, $Y¯c(τ)$ can be approximated in the same way as Eq. (25).

Now substituting Eqs. (24) and (25) into Eq. (21), the receding-horizon predictive control performance index can be reformulated as follows: Display Formula

(26)
$J=12(Y¯(τ)−Y¯c(τ))Tϑ¯(Y¯(τ)−Y¯c(τ))$
where Display Formula
(27)
$ϑ¯=∫0θκT(τ^)κ(τ^)dτ^=[θθ22θ36θ424θ22θ33θ48θ530θ36θ48θ520θ672θ424θ530θ672θ7252]$
Display Formula
(28)
$Y¯(τ)−Y¯c(τ)=E¯(e)=[∫0τe(τ^)dτ^y(τ)−yc(τ)y˙(τ)−y˙c(τ)y¨(τ)−y¨c(τ)]=[∫0τe(τ^)dτ^ee˙e¨]$
And then, Eq. (26) can be described as Display Formula
(29)
$J=12(Y¯(τ)−Y¯c(τ))Tϑ¯(Y¯(τ)−Y¯c(τ))=θ2(∫0τe(τ^)dτ^)2+θ22∫0τe(τ^)dτ^+e2+θ540(e˙)2+θ7504(e¨)2+θ32∫0τe(τ^)dτ^e˙+θ424∫0τe(τ^)dτ^e¨+θ48ee˙+θ530ee¨+θ672e˙e¨$
To minimize the performance index and obtain the optimal control, the necessary condition is given by Display Formula
(30)
$∂J∂u=θ424∫0τe(τ^)dτ^(LgLfh1(x))+θ530e(LgLfh1(x))+θ672e˙(LgLfh1(x))+θ7252(Lf2h1(x)+LgLfh1(x)u+Lg1Lfh1(x)d¯−y¨c(τ))(LgLfh1(x))=0$
Then, the optimal control law is obtained Display Formula
(31)
$u=−(LgLfh1(x))−1([212θ3425θ272θ][∫0τe(τ^)dτ^ee˙]T+Lf2h1(x)+Lg1Lfh1(x)d^−y¨c(τ))$
and it can be rewritten in another form Display Formula
(32)
$u=−(LgLfh1(x))−1(KθMρ+Lf2h1(x)+Lg1Lfh1(x)d^−y¨c(τ))$
where Display Formula
(33)
$Kθ=[k1k2k3]=[212θ3425θ272θ]$
Display Formula
(34)
$Mρ=[∫0τe(τ^)dτ^ee˙]T$

From Eq. (32) and the elements in the matrix $Kθ$ in Eq. (33), we can see that the control gain matrix $Kθ$ is closely related to the predictive period θ. Therefore, in order to accelerate the control speed and improve the control performance, different from the traditional RNPC with a fixed predictive period, the predictive period θ is designed as follows Display Formula

(35)
$θ(e)=a~expb~|e|+c~$
where $a~>0$, $b~<0$, and $c~>0$, their values depend on the tracking performance requirements of the actual control system. Then, the predictive period $θ$ satisfies: $0, $∀e∈R$. The diagram of the predictive period $θ$ can be depicted in Fig. 5.

According to the analysis of Eqs. (32)(33) and Fig. 5, we can find that the predictive period $θ$ is regulated adaptively by the error e. With the increase of |e|, the predictive period $θ$ is decreased, which means that a higher gain control input is required. In addition, it also can be seen that the parameter $c~$ is the lower bound of $θ$ and a reasonable selection of $c~$ can avoid excessive calculations when the value of $θ$ is small. Similarly, $(a~+c~)$ is the upper bound of $θ$, and a reasonable selection of $(a~+c~)$ can avoid actuator saturation. The parameter $b~$ characterizes the rate of $θ$ changing with the absolute value of the error e, because $θ(e)=a~expb~|e|+c~$ (see Eq. (35)), and a reasonable selection of $b~$ can determine the decline rate of the exponential function.

Remark 2. Different from Refs. [1014], in which the predictive period $θ$ is always set to a fixed value, and the fixed predictive period may reduce the control effect and increase the damage accumulation. In this paper, the predictive period $θ$ is designed to be regulated adaptively by the error e. Based on the analysis of Eq. (33), we can find that the elements in the matrix $Kθ$ are reduced when $θ$ increases, which means the decreases of the control effect input, and this is also in line with engineering practice. Therefore, in order to improve the control performance and reduce the calculation amount, the predictive period $θ$ is designed to be regulated adaptively with e within a certain range to guarantee a faster response when the system error increases.

Remark 3. For the airfoil motion system, since the system is dimensionless, based on the nondimensional time $τ=Vt/b$, the predictive period $θ$ which is nondimensional can be described as $θ=Vθt/b$, where V = V1 · bwα is the airspeed, and $θt$ is the dimensional predictive period.

###### Stability Issues.

The stability analysis of the airfoil motion system (Eqs. (19)(20)) under the predictive controller in Eq. (32) will be discussed in this section. Based on Eq. (32), Eq. (23) can be rewritten as the following form. Display Formula

(36)
$y¨=Lf2h1(x)+LgLfh1(x)u+Lg1Lfh1(x)d¯=Lf2h1(x)+LgLfh1(x)(−(LgLfh1(x))−1(KθMρ+Lf2h1(x)+Lg1Lfh1(x)d^−y¨c(τ)))+Lg1Lfh1(x)d¯=−KθMρ+y¨c(τ)+Lg1Lfh1(x)(d¯−d^)$
Then, we have Display Formula
(37)
$e¨+k3e˙+k2e+k1∫0τe(τ^)dτ^+Lg1Lfh1(x)(d¯−d^)=0$
Equation (37) can be rewritten as Display Formula
(38)
$M˙ρ=A1Mρ+Lg1Lfh1(x)(d¯−d^)$
where Display Formula
(39)
$A1=[010001−212θ3−425θ2−72θ]$
From $0<θmin≤|θ(e)|≤θmax$, $∀e∈R$, we can get A1 is Hurwitz matrix. Consequently, for any symmetric positive definite matrix Q, there exists a symmetric positive definite matrix P being a solution of the Lyapunov equation Display Formula
(40)
$A1TP+PA1=−Q$

Considering the following Lyapunov function: Display Formula

(41)
$V1=12MρTPMρ$
Combining Eqs. (38) and (40) and Assumption 3, the derivation of V1 is Display Formula
(42)
$V˙1=12(M˙ρTPMρ+M˙ρTPM˙ρ)=−12MρTQMρ+12((Lg1Lfh1(x)(d¯−d^))TPMρ+M˙ρTP(Lg1Lfh1(x)(d¯−d^)))≤−12λmin(Q)∥Mρ∥2+ελmax(P)∥Mρ∥$
where $λmin(Q)$ is the minimum eigenvalue of the matrix Q, and $λmax(P)$ is the maximum eigenvalue of the matrix P.

Then, based on the mean inequality (i.e., $a1b1≤z1a12+b12/4z1$, here, $a1=∥Mρ∥$, $b1=ελmax(P)$, $z1=θ1λmin(Q)$, $0<θ<1/2$), we obtain Display Formula

(43)
$V˙1≤−(12−θ1)λmin(Q)∥Mρ∥2+(ελmax(P))24θ1λmin(Q)≤−Λ1V1+Λ2$
where Display Formula
(44)
$Λ1=(1−2θ1)λmin(Q)λmax(P)Λ2=(ελmax(P))24θ1λmin(Q)$
The solution of Eq. (43) is Display Formula
(45)
$V1(τ)≤[V1(0)−Λ2Λ1]exp(−Λ1τ)+Λ2Λ1$

Based on Eqs. (41) and (45), we have Display Formula

(46)
$12MρTPMρ≤12λmax(P)∥Mρ∥2≤[V1(0)−Λ2Λ1]exp(−Λ1τ)+Λ2Λ1≤Λ2Λ1$
so the tracking error matrix $Mρ$ is bounded by Display Formula
(47)
$∥Mρ∥≤ελmax(P)2θ1(1−2θ1)λmin(Q)$
when the disturbance is neglected in the controller, as $τ→∞$, $ε→0$, we will obtain the result that $V˙1$ is negative. A conclusion is drawn that the airfoil motion system (Eqs. (19)(20)) is asymptotic stable. It means that the tracking error converges to zero.

## Simulation

To verify the effectiveness of the proposed controller with the adaptive predictive period, the closed-loop airfoil flutter dynamics with the proposed predictive controller is simulated, while the same controller with the fixed predictive period is employed for comparison. In Sec. 4, we can see that the fatigue damage increases continuously in Figs. 3 and 4, especially in Fig. 4, where the damage shows an exponential increase, which will seriously threaten the flight safety of the HFV. In order to achieve fast convergence rate, small steady-state error, and slow down fatigue damage accumulation, the damage-mitigating predictive controller is designed as Eq. (32) to suppress the limit cycle oscillation (see Fig. 3) and the divergence oscillation (see Fig. 4). According to the characteristics of the airfoil flutter system, we set the dimensional predictive time $θt$ to $0.02sand∼0.2s$, based on Remark 3, and V1 ∈ [14.3000, 21.0000], b = 1, and wα = 100 in this paper, then $θ∈[60,260]$ can be obtained. And from $θ(e)=a~expb~|e|+c~$ and Fig. 5, $a~=200$ and $c~=60$ can be derived. In addition, $b~=−104$ is selected in order to achieve a satisfied descent rate of $θ$, because |e| is about 10−2 when $a~=200$ and $c~=60$. Due to $θ∈[60,260]$, the fixed predictive period is selected as 150. The other parameters are the same as Sec. 4. The airfoil flutter responses and damage evolution under the RNPC with the adaptive predictive period and fixed predictive period are shown in Figs. 6 and 7.

Compared with the limit cycle oscillation responses (see Fig. 3) and the divergence oscillation responses (see Fig. 4), in the same conditions, the airfoil responses with the predictive controller are asymptotically stable (see Figs. 6 and 7). The airfoil flutter responses with the predictive controller using the adaptive predictive period θ have a smaller overshoot and faster convergence speed than the predictive controller with the fixed predictive period.

In addition, different from the increasing damage accumulations in Figs. 3 and 4, the damages in Figs. 6 and 7 tend to stabilize gradually. Compared with the limit cycle oscillation damage being 1.20 × 10−7 (see Fig. 3(c)), the airfoil damage under the predictive controller with the fixed predictive period is reduced to 3.15 × 10−8, and the damage under the proposed predictive controller with the adaptive predictive period is reduced to 9.13 × 10−9. Similarly, compared with the divergent oscillation damage being 3.85 × 10−7 (see Fig. 4(c)), the damage under the traditional predictive control with the fixed predictive period is reduced to 3.66 × 10−8, and the damage under the proposed predictive control is reduced to 7.71 × 10−9.

Additionally, in order to analyze the sensitivity of the control results to the parameters $a~$, $b~$, and $c~$, different values of $a~$, $b~$, and $c~$ are considered for comparative analysis, as shown in Fig. 8.

From Fig. 8(a), it can be seen that the convergence speed of the airfoil flutter gradually slows down with the predictive period interval $(a~+c~)$ increases. And from Fig. 8(b), we can find that as the $b~$ varies from −103 to −104, the convergence speed of the airfoil flutter is obviously faster, although the overshoot has a small increase. As the $b~$ varies from −104 to −105, there is no significant change in the convergence speed of the airfoil flutter.

The simulation results demonstrate the effectiveness of the proposed adaptive predictive control and it can mitigate the airfoil damage effectively. It is worth noting that, the simulation results also demonstrate the effectiveness of the proposed damage accumulation model which can estimate the damage of the airfoil flutter for the HFV online.

## Conclusion

In this paper, a new predictive control with the adaptive predictive period is proposed to mitigate the airfoil flutter damage. Based on the third piston theory, the aerodynamic force is formulated, and the 3DOF airfoil motion model is established. Subsequently, a continuous-time damage model is established to estimate the airfoil flutter damage. Numerical simulations are carried out to analyze the airfoil flutter responses under different flow speeds without control and the airfoil damage accumulation during airfoil flutter. Then, a novel predictive control strategy is employed to achieve the flutter suppression of the 3DOF airfoil motion system, and the adaptive predictive period is regulated by the error. Moreover, based on the Lyapunov theory, the stability of the airfoil system is analyzed. Finally, the simulation results illustrate that the airfoil flutter is suppressed by the predictive controller, and the proposed predictive controller with the adaptive predictive period has a better performance and damage-mitigating effect than the traditional predictive control method with a fixed period.

## Acknowledgements

The authors would like to thank the anonymous referees and the editors for their valuable comments and suggestions leading to an improvement of this article.

## Funding Data

• National Natural Science Foundation of China (Grant No. 61773204)

## References

Dessi, D., and Mastroddi, F., 2004, “Limit-Cycle Stability Reversal Via Singular Perturbation and Wing-Flap Flutter,” J. Fluids Struct., 19(6), pp. 765–783.
Lee, K. W., and Singh, S. N., 2018, “L1 Adaptive Control of an Aeroelastic System With Unsteady Aerodynamics and Gust Load,” J. Vib. Control, 24(2), pp. 303–322.
Hafeez, M. M., and Elbadawy, A., 2018, “Flutter Limit Investigation for a Horizontal Axis Wind Turbine Blade,” ASME J. Vib. Acoust., 140(4), p. 041014.
Bernelli-Zazzera, F., Mantegazza, P., Mazzoni, G., and Rendina, M., 2000, “Active Flutter Suppression Using Recurrent Neural Networks,” J. Guidance Control Dyn., 23(6), pp. 1030–1036.
Lee, B. H., Choo, J., Na, S., Marzocca, P., and Librescu, L., 2010, “Sliding Mode Robust Control of Supersonic Three Degrees-of-Freedom Airfoils,” Int. J. Control Automat. Syst., 8(2), pp. 279–288.
Wang, Y. H., Zhu, L., Wu, Q. X., and Jiang, C. S., 2014, “Fuzzy Approximation by a Novel Levenberg-Marquardt Method for Two-Degree-of-Freedom Hypersonic Flutter Model,” ASME J. Vib. Acoust., 136(4), p. 044502.
Mei, G. H., Zhang, J. Z., and Kang, C., 2017, “Analysis of Curved Panel Flutter in Supersonic and Transonic Airflows Using a Fluid CStructure Coupling Algorithm,” ASME J. Vib. Acoust., 139(4), p. 041004.
Wang, Y. H., Zhang, Q., and Zhu, L., 2015, “Active Control of Hypersonic Airfoil Flutter Via Adaptive Fuzzy Sliding Mode Method,” J. Vib. Control, 21(1), pp. 134–141.
Cao, D. Q., and Zhao, N., 2011, “Active Control of Supersonic/Hypersonic Aeroelastic Flutter for a Two-Dimensional Airfoil With Flap,” Sci. China Technol. Sci., 54(8), pp. 1943–1953.
Song, Z. G., Yang, T. Z., Li, F. M., Carrera, E., and Hagedorn, P., 2018, “A Method of Panel Flutter Suppression and Elimination for Aeroelastic Structures in Supersonic Airflow,” ASME J. Vib. Acoust., 140(6), p. 064501.
Oppenheimer, M., and Doman, D., 2006, “A Hypersonic Vehicle Model Developed With Piston Theory,” AIAA Atmospheric Flight Mechanics Conference and Exhibit, Keystone, CO, Aug. 21–24, Paper No. 2006-6637.
Librescu, L., Chiocchia, G., and Marzocca, P., 2003, “Implications of Cubic Physical/Aerodynamic Non-Linearities on the Character of the Flutter Instability Boundary,” Int. J. Non-Linear Mech., 38(2), pp. 173–199.
Clarke, D. W., 1994, Advances in Model-Based Predictive Control, Oxford University Press, Oxford.
Chen, W. H., Ballance, D. J., and Gawthrop, P. J., 2003, “Optimal Control of Nonlinear Systems: A Predictive Control Approach,” Automatica, 39(4), pp. 633–641.
Hedjar, R., Toumi, R., Boucher, P., and Dumur, D., 2005, “Finite Horizon Nonlinear Predictive Control by the Taylor Approximation: Application to Robot Tracking Trajectory,” Int. J. Appl. Math. Comput. Sci., 15(4), pp. 527–540.
Cheng, L., Jiang, C. S., and Pu, M., 2011, “Online-SVR-Compensated Nonlinear Generalized Predictive Control for Hypersonic Vehicles,” Sci. China Inform. Sci., 54(3), pp. 551–562.
Errouissi, R., Ouhrouche, M., Chen, W. H., and Trzynadlowki, A. M., 2012, “Robust Nonlinear Predictive Controller for Permanent-Magnet Synchronous Motors With an Optimized Cost Function,” IEEE Trans. Ind. Electron., 59(7), pp. 2849–2858.
Caplin, J., Ray, A., and Joshi, S. M., 2001, “Damage-Mitigating Control of Aircraft for Enhanced Structural Durability,” IEEE Trans. Aerosp. Electron. Syst., 37(3), pp. 849–862.
Ray, A., and Patankar, R., 2001, “Fatigue Crack Growth Under Variable-Amplitude Loading: Part II-Code Development and Model Validation,” Appl. Math. Model., 25(11), pp. 995–1013.
Newman, J., and James, C., 1992, “FASTRAN-2: A Fatigue Crack Growth Structural Analysis Program,” Langley Research Center, Hampton, VA, NASA Technical Memorandum 104159.
Li, D., Chen, T., Marquez, H. J., and Gooden, R. K., 2003, “Damage Modeling and Life Extending Control of a Boiler-Turbine System,” IEEE, pp. 2317–2322.
Tangirala, S., Holmes, M., Ray, A., and Carpino, M., 1998, “Life-Extending Control of Mechanical Structures: Experimental Verification of the Concept,” Automatica, 34(1), pp. 3–14.
Dowling, N. E., 2004, “Mean Stress Effects in Stress-Life and Strain-Life Fatigue,” SAE Technical Paper, 32(12), pp. 1004–1019.
Ray, A., Wu, M. K., Dai, X., Carpino, M., and Lorenzo, C., 1993, “Damage-Mitigating Control of Space Propulsion Systems for High Performance and Extended life,” AIAA/SAE/ASME/ASEE 29th Joint Propulsion Conference and Exhibit, Monterey, CA, June 28–30.
Wang, J., Li, Q., Xiong, C., Li, Y., and Sun, B., 2018, “Effect of Zr on the Martensitic Transformation and the Shape Memory Effect in Ti-Zr-Nb-Ta High-Temperature Shape Memory Alloys,” J. Alloys Compd., 737, pp. 672–677.
Ko, W. L., Richards, W. L., and Tran, V. T., 2007, “Displacement Theories for In-Flight Deformed Shape Predictions of Aerospace Structures,” Dryden Flight Research Center, NASA Report No. 214612.
Zhang, Z. H., Li, P. S., and Fu, J. C., 2015, “Analysis on Hopf Bifurcation of the Grid-Connected Small Hydropower System Based on Center Manifold Theory,” J. Vib. Shock 34(2), pp. 50–54.
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## References

Dessi, D., and Mastroddi, F., 2004, “Limit-Cycle Stability Reversal Via Singular Perturbation and Wing-Flap Flutter,” J. Fluids Struct., 19(6), pp. 765–783.
Lee, K. W., and Singh, S. N., 2018, “L1 Adaptive Control of an Aeroelastic System With Unsteady Aerodynamics and Gust Load,” J. Vib. Control, 24(2), pp. 303–322.
Hafeez, M. M., and Elbadawy, A., 2018, “Flutter Limit Investigation for a Horizontal Axis Wind Turbine Blade,” ASME J. Vib. Acoust., 140(4), p. 041014.
Bernelli-Zazzera, F., Mantegazza, P., Mazzoni, G., and Rendina, M., 2000, “Active Flutter Suppression Using Recurrent Neural Networks,” J. Guidance Control Dyn., 23(6), pp. 1030–1036.
Lee, B. H., Choo, J., Na, S., Marzocca, P., and Librescu, L., 2010, “Sliding Mode Robust Control of Supersonic Three Degrees-of-Freedom Airfoils,” Int. J. Control Automat. Syst., 8(2), pp. 279–288.
Wang, Y. H., Zhu, L., Wu, Q. X., and Jiang, C. S., 2014, “Fuzzy Approximation by a Novel Levenberg-Marquardt Method for Two-Degree-of-Freedom Hypersonic Flutter Model,” ASME J. Vib. Acoust., 136(4), p. 044502.
Mei, G. H., Zhang, J. Z., and Kang, C., 2017, “Analysis of Curved Panel Flutter in Supersonic and Transonic Airflows Using a Fluid CStructure Coupling Algorithm,” ASME J. Vib. Acoust., 139(4), p. 041004.
Wang, Y. H., Zhang, Q., and Zhu, L., 2015, “Active Control of Hypersonic Airfoil Flutter Via Adaptive Fuzzy Sliding Mode Method,” J. Vib. Control, 21(1), pp. 134–141.
Cao, D. Q., and Zhao, N., 2011, “Active Control of Supersonic/Hypersonic Aeroelastic Flutter for a Two-Dimensional Airfoil With Flap,” Sci. China Technol. Sci., 54(8), pp. 1943–1953.
Song, Z. G., Yang, T. Z., Li, F. M., Carrera, E., and Hagedorn, P., 2018, “A Method of Panel Flutter Suppression and Elimination for Aeroelastic Structures in Supersonic Airflow,” ASME J. Vib. Acoust., 140(6), p. 064501.
Oppenheimer, M., and Doman, D., 2006, “A Hypersonic Vehicle Model Developed With Piston Theory,” AIAA Atmospheric Flight Mechanics Conference and Exhibit, Keystone, CO, Aug. 21–24, Paper No. 2006-6637.
Librescu, L., Chiocchia, G., and Marzocca, P., 2003, “Implications of Cubic Physical/Aerodynamic Non-Linearities on the Character of the Flutter Instability Boundary,” Int. J. Non-Linear Mech., 38(2), pp. 173–199.
Clarke, D. W., 1994, Advances in Model-Based Predictive Control, Oxford University Press, Oxford.
Chen, W. H., Ballance, D. J., and Gawthrop, P. J., 2003, “Optimal Control of Nonlinear Systems: A Predictive Control Approach,” Automatica, 39(4), pp. 633–641.
Hedjar, R., Toumi, R., Boucher, P., and Dumur, D., 2005, “Finite Horizon Nonlinear Predictive Control by the Taylor Approximation: Application to Robot Tracking Trajectory,” Int. J. Appl. Math. Comput. Sci., 15(4), pp. 527–540.
Cheng, L., Jiang, C. S., and Pu, M., 2011, “Online-SVR-Compensated Nonlinear Generalized Predictive Control for Hypersonic Vehicles,” Sci. China Inform. Sci., 54(3), pp. 551–562.
Errouissi, R., Ouhrouche, M., Chen, W. H., and Trzynadlowki, A. M., 2012, “Robust Nonlinear Predictive Controller for Permanent-Magnet Synchronous Motors With an Optimized Cost Function,” IEEE Trans. Ind. Electron., 59(7), pp. 2849–2858.
Caplin, J., Ray, A., and Joshi, S. M., 2001, “Damage-Mitigating Control of Aircraft for Enhanced Structural Durability,” IEEE Trans. Aerosp. Electron. Syst., 37(3), pp. 849–862.
Ray, A., and Patankar, R., 2001, “Fatigue Crack Growth Under Variable-Amplitude Loading: Part II-Code Development and Model Validation,” Appl. Math. Model., 25(11), pp. 995–1013.
Newman, J., and James, C., 1992, “FASTRAN-2: A Fatigue Crack Growth Structural Analysis Program,” Langley Research Center, Hampton, VA, NASA Technical Memorandum 104159.
Li, D., Chen, T., Marquez, H. J., and Gooden, R. K., 2003, “Damage Modeling and Life Extending Control of a Boiler-Turbine System,” IEEE, pp. 2317–2322.
Tangirala, S., Holmes, M., Ray, A., and Carpino, M., 1998, “Life-Extending Control of Mechanical Structures: Experimental Verification of the Concept,” Automatica, 34(1), pp. 3–14.
Dowling, N. E., 2004, “Mean Stress Effects in Stress-Life and Strain-Life Fatigue,” SAE Technical Paper, 32(12), pp. 1004–1019.
Ray, A., Wu, M. K., Dai, X., Carpino, M., and Lorenzo, C., 1993, “Damage-Mitigating Control of Space Propulsion Systems for High Performance and Extended life,” AIAA/SAE/ASME/ASEE 29th Joint Propulsion Conference and Exhibit, Monterey, CA, June 28–30.
Wang, J., Li, Q., Xiong, C., Li, Y., and Sun, B., 2018, “Effect of Zr on the Martensitic Transformation and the Shape Memory Effect in Ti-Zr-Nb-Ta High-Temperature Shape Memory Alloys,” J. Alloys Compd., 737, pp. 672–677.
Ko, W. L., Richards, W. L., and Tran, V. T., 2007, “Displacement Theories for In-Flight Deformed Shape Predictions of Aerospace Structures,” Dryden Flight Research Center, NASA Report No. 214612.
Zhang, Z. H., Li, P. S., and Fu, J. C., 2015, “Analysis on Hopf Bifurcation of the Grid-Connected Small Hydropower System Based on Center Manifold Theory,” J. Vib. Shock 34(2), pp. 50–54.

## Figures

Fig. 1

Three degrees-of-freedom airfoil motion system

Fig. 2

Flutter responses and damage evolution at V1 = 14.3000: (a) pitching displacement response, (b) phase diagram response, and (c) damage evolution

Fig. 3

Flutter responses and damage evolution at V1 = 19.5883: (a) pitching displacement response, (b) phase diagram response, and (c) damage evolution

Fig. 4

Flutter responses and damage evolution at V1 = 21.0000: (a) pitching displacement response, (b) phase diagram response, and (c) damage evolution

Fig. 5

Diagram of the predictive period

Fig. 6

Flutter responses and damage evolution under the control at V = 19.5883: (a) pitching displacement responses, (b) phase diagram responses, and (c) damage evolution

Fig. 7

Flutter responses and damage evolution under the control at V = 21.0000: (a) pitching displacement responses, (b) phase diagram responses, and (c) damage evolution

Fig. 8

Flutter responses under different predictive period parameters at V = 21.0000: (a) = −104 and (b) = 200, = 60

## Errata

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