Robust fault-tolerant control for wing flutter under actuator failure

Robust fault-tolerant control for wing flutter under

actuator failure

Gao Mingzhou, Cai Guoping

Department of Engineering Mechanics, State Key Laboratory of Ocean Engineering, Shanghai Jiaotong University, Shanghai 200240, China

Received 24 September 2015; revised 5 January 2016; accepted 7 March 2016 Available online 23 June 2016

KEYWORDS Actuator fault; Actuator saturation; Aeroservoelasticity; Fault-tolerant control; Flutter; Observer

Abstract Many control laws, such as optimal controller and classical controller, have seen their applications to suppressing the aeroelastic vibrations of the aeroelastic system. However, those con-trol laws may not work effectively if the aeroelastic system involves actuator faults. In the current study for wing flutter of reentry vehicle, the effect of actuator faults on wing flutter system is rarely considered and few of the fault-tolerant control problems are taken into account. In this paper, we use the radial basis function neural network and the finite-timeH1adaptive fault-tolerant control technique to deal with the flutter problem of wings, which is affected by actuator faults, actuator saturation, parameter uncertainties and external disturbances. The theory of this article includes the modeling of wing flutter and fault-tolerant controller design. The stability of the finite-time adaptive fault-tolerant controller is theoretically proved. Simulation results indicate that the designed fault-tolerant flutter controller can effectively deal with the faults in the flutter system and can promptly suppress the wing flutter as well.

Ó2016 Chinese Society of Aeronautics and Astronautics. Production and hosting by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

With the rapid development of aerospace technology, modern aircraft perform their characteristics such as high velocity, lightweight structures, flexible and low damping, which makes

the aeroelasticity phenomenon more and more prominent. Flutter is one of such problems. Flutter instability may decrease aircraft performance or even lead to the catastrophic failure of the structure.1The traditional passive techniques are usually inefficient (because they add weight to the structure), and they do not always succeed. In order to overcome the inadequacy of passive techniques, the active flutter suppression techniques were developed in early 1970s. In active flutter sup-pression, we carry it out by utilizing multiple steerable control surfaces laid out on the surface of the wing.

The technique of active flutter suppression has drawn much attention over the past decade.2–6 For example, Yu et al.2 designed a l controller to suppress airfoil flutter, and wind tunnel experiments were carried out to verify the effectiveness * Corresponding author. Tel.: +86 21 34204798.

E-mail addresses:jsycgaomingzhou@163.com(M. Gao),caigp@sjtu. edu.cn(G. Cai).

Peer review under responsibility of Editorial Committee of CJA.

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of the designed controllers. Prime et al.3 synthesized a state-feedback controller using linear matrix inequalities (LMIs) to control the vibration of an improved three-degree-of-freedom aeroelastic model and this controller could effectively suppress limit-cycle oscillations over a range of airspeeds. Wang et al.4,5 considered a class of aeroelastic systems with an unmodeled nonlinearity and external disturbance and proposed a full-state feedforward/feedback controller with a high-gain observer; they also designed a continuous robust controller to suppress the aeroelastic vibrations of a nonlinear wing-section model. Zhang et al.6_{designed a partial state feedback} continuous adaptive controller in order to suppress the aeroe-lastic vibrations of the wing section model.

Although a number of flutter controller design approaches,2–6most of the researches assume that there exists no actuator fault or failure during the entire flutter suppres-sion. This assumption is rarely satisfied in practice because some catastrophic faults may occur due to the malfunction of actuators. As a result, if the flutter controller is designed without any fault tolerance capability, an abrupt occurrence of an actuator fault could ultimately fail the flutter control. Therefore, we must give priority to considering the faults of actuators and sensors in the design of the flutter controller. Therefore, the fault-tolerant control (FTC)7 should be taken into consideration for the flutter control. In general, FTC methods are classified into two types: passive fault-tolerant control (PFTC) and active fault-tolerant control (AFTC) schemes.7The PFTC designed by limited faults and fixed con-troller will not be able to guarantee the performance of the sys-tem.8–10 Correspondingly, active method11–13 may provide more powerful fault-tolerant capability for compensating for faults of the systems in terms of reconfiguring control strate-gies online or switching to a more suitable control law based on the fault information. Therefore, in this paper, the investi-gation of an active fault-tolerant controller for a flutter control system with the occurrence of unexpected faults or failures. It is worth mentioning that the above results2–6are derived from the assumption that the actuators are able to provide any requested outputs. However, in almost every physical applica-tion, the actuator has bounds on its input. Therefore, the phe-nomenon of actuator saturation has to be considered when the controller is designed in practical industrial process control field. In addition, the flutter will destroy the vehicle in a short time, we must control the flutter within a certain range in a finite time. However, to the best of our knowledge, the studies on finite-time adaptive fault-tolerant control of wing flutter are very limited in the published literature.

In order to reveal the negative effect of the conventional control on the stability of aeroelastic system and considering the influence of faults, time-varying parameter uncertainties and external disturbances, this paper focuses on the design

of finite-timeH₁adaptive fault-tolerant controller for flutter of wing. A 2D cubic structure nonlinearity wing system is adopted as structure model. The actuator fault is considered in the controller design. This paper is organized as follows. The dynamic equation of wing flutter and the control problem of flutter system with faults are established in Section 2. Section 3presents a finite-time adaptive fault-tolerant flutter controller based on observer. Numerical simulations are given in Section4. Section5briefs the conclusions of the research.

2. Flutter model of 2D wing and fault-tolerant control problem In this section, we briefly recall the mathematical model for the flutter of a reentry vehicle with actuators fault-free. Based on this nominal flutter system, the state equation with actuator faults and saturation, parameter uncertainties and external dis-turbances are established.

2.1. Wing flutter model under actuators fault-free

In this section, flutter problem for a 2D wing including cubic hard spring nonlinearity is analyzed. As shown in Fig. 1, a two degree-of-freedom (2-DOF) wing system model is consid-ered herein. The plunge deflection is denoted byh, positive in the downward direction;his the pitch angle about the elastic axis, positive nose up;Vdenotes the air speed; the chord length isc;Q,pandCare the aerodynamic center, elastic axis and center of mass, respectively; the distance from the leading edge to the elastic axis isxp, and that from the leading edge to the

mass center is xC;dLEoutanddLEin(ordREoutanddREin) are

the control surface angles.

From Fig. 1, the velocity of mass center of wing can be expressed as

_

z¼h_þ ðxCxpÞh_ ð1Þ

The kinetic energy, potential energy and dissipation of the system can be given by

T¼1 2mWz_ 2_þ1 2meh_ 2_þ1 2ICh_ 2 U¼1 2Khh 2_þ1 2Khh 2 f¼1 2Chh_2þ12Chh_2 8 > > < > > : ð 2Þ

whereIC,mW,me,Kh,Kh,ChandChare the moment of inertia

about center of mass, wing mass, wing extra-mass, stiffness coefficient in plunge, torsion stiffness coefficient, damping coefficient in plunge and torsion damping coefficient, respectively.

For supersonic and hypersonic flow, the piston theory is widely used to calculate the aerodynamics acting on a lifting surface.14_{Applying the piston theory, the aerodynamic force} and moment acting on the wing can be obtained as

L¼2q_MVcc 1 0:5cð1x0Þh_þh_þVhþ 1 12Vc 2_ðjþ1ÞM2 1h3 h i MT¼qVcc 2 M1 1 6cð46x0þ3x 2 0Þh_þ ð1x0Þh_þVð1x0Þhþ₁₂1c2ðjþ1ÞM2₁ð1x0ÞVh3 h i 8 > < > : ð3Þ

where q is the atmospheric density, M₁ the Mach number, c¼M1= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M2 11 q

the aerodynamic correction factor, j the ratio of specific heat, and x0 the non-dimensional distance

from the leading edge to the elastic axis.

The aerodynamic lift and moment caused by control sur-face can be expressed as

LdLEout¼LdLEin ¼ 1 2qV 2_ca CsbdLEout MdLEout¼MdLEin¼ 1 2qV 2 c2_b CsbdLEout ( ð4Þ where aC is the coefficients of lift force, bC the ratio of the

pitching moment to the deflection angle, andsbthe span of

control surface.

The moment caused by the cubic hard spring nonlinearity of the wing can be written as15

MðhÞ ¼Khhþen1h3 ð5Þ

whereen1is the nonlinear stiffness coefficient.

Without considering structural damping, the aeroelastic equation of the two-dimensional wing system can be deduced using the Lagrangian method and written into matrix form, we have

~

A€qðtÞ þqVB~q_ðtÞ þ ðqV2_C~_þD~_{ÞqðtÞ þ}~_{fðtÞ ¼bu}~_{ðtÞ ð6Þ}

where q(t) = [h(t),h(t)]Tis the generalized displacement vec-tor;A~,B~,C~andD~are the inertia, aerodynamic damping, aero-dynamic stiffness and structural stiffness matrices respectively;

u(t) = [dLEout,dLEin]Tis the control input. The expressions of

parameters in Eq.(6)are

~ A¼ mWþme mWðxCxpÞ mWðxCxpÞ mWðxCxpÞ 2 þIC " # ; ~ B¼ 2cc M1 cc2 M1ð1x0Þ cc2 M1ð1x0Þ cc3 6M1ð46x0þ3x 2 0Þ 2 4 3 5_; ~ C¼ 0 2cc M1 0 _Mcc2 1ð1x0Þ " # ; ~ D¼ Kh 0 0 Kh ; ~ b¼ 1 2qV 2_ca Csb 12qV 2_ca Csb 1 2qV 2 c2_b Csb 1₂qV2c2bCsb " # ; ~_f¼ 16qV 2_c3_cðjþ1ÞM 1h3 1 12qV 2_c3_c2_ðjþ1ÞM 1ð1x0Þh3þen1h3 " #

Eq.(6)can be further changed to be

_

xðtÞ ¼AxðtÞ þBuðtÞ þfðtÞ ð7Þ where x¼ ½hðtÞ;hðtÞ;h_ðtÞ;h_ðtÞT is the state space vector;

A¼ _A~1_ðqV02_C~_þD~_{Þ A}~I1_qVB~ , B¼ _A~01~_b , and f¼ _A~01~_fðtÞ .

The parameters in the matrices A and B are determined based on the computing results for the most serious moment of reentry flight of the vehicle given in Ref.16_{. In Ref.}16_{, the} reentry will reach its maximum velocity of 1406 m/s at the moment 699.19 s; the altitude, longitude, latitude and atmo-spheric density of the reentry vehicle at this serious moment can be determined to be 21.87 km, 3907.11 km, 66.05 km and 0.0644 kg/m3, respectively.

2.2. Wing flutter model under actuator failures

Note that the dynamic equation of flutter in Eq.(7)assumes that all of the actuators are fault-free, and it is called the nom-inal system. For an active flutter control system, it is difficult to ensure the actuator in the ideal working condition. They are likely to have some problems, such as loss of effectiveness, float or saturation. For the hypersonic reentry vehicle, the air-craft will face hostile aerodynamic environment in the reentry process, such as high temperature, high pressure. The actuator may have faults, such as loss of effectiveness and float, which are likely to cause catastrophic accidents of the reentry vehicle. Therefore, fault tolerance capability should be considered in flutter controller design.

In this section, we consider two types of actuator fault simultaneously, namely the float fault (actuators generate additional torque under zero control command caused by volt-age or current intermittent fault) and the loss of effectiveness of the actuators as well as actuator saturation, parameter uncertainties and external disturbances. Hence, the flutter dynamic model given by Eq.(7)can be rewritten as

_

xðtÞ ¼ ðAþDAðtÞÞxðtÞ þBqvðtÞ þBusðtÞ þfðt;xÞ þB1wðtÞ

yðtÞ ¼c1xðtÞ

ð8Þ whereDArepresents the parameter uncertainties of wing flut-ter system, v(t) the actual actuator control vector, q¼diagðq₁;q₂Þthe effectiveness factor matrix for the reentry vehicle actuators with 0<qi61ði¼1;2Þ; the case when

qi= 1 means that the ith actuator works normally, and

0 <qi< 1 denotes the case in which theith actuator partially

loses its effectiveness.usðtÞ ¼ ½us1;us2 T₂

R21 corresponds to theith actuator float fault of reentry vehicle;fðt;xÞis a nonlin-ear term;w(t) represents the external disturbances; andy(t) is the measured output. The parameters B1 andc1 are known matrices with appropriate dimensions.DAis assumed to satisfy the following matching condition of Eq.(9).

DA¼BNðtÞ ð9Þ

where kk represents Euclidean norm of vectors or matrices;

N(t) is an unknown matrix withkNk6landl*_{is an unknown}

constant.

Remark 1. The assumptionDA=BN(t) in this paper is made for the convenience of design is the controller (Eq. (20)). Through the assumption DA= BN(t), we altered the uncer-tainties ofDAinto the uncertainties ofN(t). In our paper, we made compensation forN(t) by the adaptive lawk^5(Eq.(23)).

vis the actual actuator control vector defined as

v¼ vmax uðtÞ>vmax uðtÞ þduðxÞ vmax 6uðtÞ6vmax vmax uðtÞ<vmax 8 > < > : ð10Þ

wherevmaxanddu(x) are the actuator saturation level and aux-iliary variable;u(t) is the control signal to be designed.

In nonlinear control problem, the radial basis function (RBF) network is usually used as a tool for modeling nonlinear system because of its good capabilities in function approxima-tion. In this paper, the unknown_du(x) is approximated by the RBF network17

Fig. 2 Time responses of wing states and observer errors in case of fault.

hNðxÞ ¼exp kxcNk 2 2b2 N duðxÞ ¼WThðxÞ þeuðxÞ x2Dx 8 > > < > > : ð 11Þ

Herein,xis the input of neural networks and_du(x) the output of neural networks; the subscript N means theNth input of neural networks; hN(x) is the Gaussian function; h(x) is the

output of Gaussian function;cNandbNare the centre vector

and width; W* _{is the ideal weight matrix. Define}

^

duðxÞ ¼W^T_{hðxÞ denotes the estimate of}

du(x), and

duðxÞ ¼duðxÞ du^ðxÞ,W^ the estimate ofW*_,

eu(x) the approx-imation error, andD_xeR41a sufficiently large compact set. 3. Observer-based finite-timeH‘adaptive fault-tolerant flutter controller design

In this section, an active fault-tolerant flutter controller design for reentry vehicle is presented, including the design of the observer and the design of the fault-tolerant flutter controller. The observer design is first presented to achieve precise esti-mated flutter states without states measurement. A fault-tolerant flutter controller is then designed using the flutter states information from the observer to perform the flutter sta-bilization control.

3.1. Design of observer

In this section, an observer is designed, which can provide the information of flutter states. These information is sent to the controller to obtain the control law, which is sent back to the actuator.

Considering the form of the model Eq.(8), the following observer is designed:

Fig. 4 Output of neural networks.

Fig. 5 Adaptive parameters ofK11^ ðtÞandK12^ ðtÞ.

Table 1 Structural parameters of 2D wing.

Parameter Value Parameter Value

Mh_ 1.2 aC 3.82 c(m) 0.7 Kh(N/m) 2106 mW(kg) 1320 xp(m) 0.28 me(kg) 490 V(m/s) 1406 j 1.4 q(kg/m3_{) 0.0644} IC(kgm2_{) 13,205 e} n1 10 sb(m) 1.6 Kh(Nm/rad) 2104 bC 0.076 xC(m) 0.525

_^

xðtÞ ¼ ðAþDAðtÞÞx^ðtÞ þBvðtÞ þfðt;x^Þ þLðyðtÞ ^yðtÞÞ ð12Þ By using the definitioneðtÞ ¼xðtÞ x^ðtÞ, the dynamics of the errore(t), between the actual statex(t) and its estimate values

^

xðtÞ, can be obtained from Eqs.(8) and (12)as

_

eðtÞ ¼ ðAþDAðtÞ Lc1ÞeðtÞ þBðqIÞvðtÞ þBusðtÞ

þfðt;xÞ fðt;^xÞ þB1wðtÞ ð13Þ

Theorem 1. If there exists a positive-definite matrix biP_ and considering the estimation error Eq.(13),we suppose that the observer gains are chosen as

P _ ðALc1Þ þ ðALc1ÞT_P__þ e1 f P _ P _ þðefL2gþeAÞI þe1 A P _ BBT_P__< 0 kP_BðIqÞkvðtÞ>k_PBkk_rkusðtÞ þ kP_B1kwðtÞ 8 > > > < > > > : ð14Þ

Then the estimation error dynamics systeme(t) is stable. .

Proof. Select the following Lyapunov function V1ðxðtÞÞ ¼

eT_ðtÞP__{e. Differentiating the above Lyapunov function yields}

_ V1ðxðtÞÞ ¼e_TðtÞP _ eðtÞ þeT_ðtÞP__{e_ðtÞ} ¼eT_ðtÞh_P__{ðALc1Þ þ ðALc1Þ}T_P_i_eð tÞ þ2eT_ðtÞP__{DAðtÞeðtÞ 2e}T_ðtÞP__BðI qÞvðtÞ þ2eTðtÞ_PBusðtÞ þ2eTðtÞP_ðfðt;xÞ fðt;^xÞÞ þ2eTðtÞP_B1wðtÞ ð15Þ

LetNTðtÞNðtÞ6I, if there exist scalarsefP0 andeAP0,

fðt;xÞ fðt;x^Þ satisfies the Lipschitz condition kfðt;xÞ fðt;^xÞk6LgkxðtÞ ^xðtÞk ¼LgkeðtÞk, Lg> 0, note

the fact that for any positive constantc>0, 2ab61

ca

2_þ

cb2 8a;b>0 ð16Þ Using Eqs.(9) and (16), we have

2keT_ðtÞP__L geðtÞk6ef1keTðtÞP _ k2_þ efL2gkeðtÞk 2 2keT_ðtÞP__{DAðtÞeðtÞk6e}1 A ke T_ðtÞP__Bk2_þ eAkeðtÞk 2 8 < : ð17Þ

Using Eq.(17), Eq.(15)can be written as

_ V1ðxðtÞÞ ¼eTðtÞ P _ ðALc1Þ þ ðALc1ÞT_P__þ e1 f P _ P _ þefL2gI h þe1 A P _ BBT_P__þe AI i eðtÞ þ2eT_ð tÞP_B1wðtÞ2eT_ð tÞP_BðqIÞvðtÞ þ2eT_ðtÞP__{BusðtÞ ð18Þ}

According toTheorem 1, Eq.(18)can be written as

_

V1ðxðtÞÞ60 ð19Þ

then the error dynamics system Eq.(13)is stable. Thereby the proof is completed. h

3.2. Finite-time fault-tolerant flutter controller design

In this section, a novel finite-time fault-tolerant flutter control algorithm is investigated to suppress the wing flutter based on the observer Eq.(12).

Considering Eqs.(8) and (11), the adaptive flutter control law is designed as

uðtÞ ¼ K1^ ðtÞ þK2ðtÞ þK3ðtÞ

^

xðtÞ ^duðxÞ ð20Þ whereK1^ ðtÞis given by an adaptive gain for the guarantee of the flutter system stability;K2(t) andK3(t) are by an adaptive gain for eliminating the effects of float, external disturbances and parameter uncertainties. The expressions of K1^ ðtÞ, K2(t) andK3(t) are given below. We can prove that these expressions may guarantee the stability of the flutter control system. The proof procedure is given below too.

Below we give the expressions ofK1^ ðtÞ,K2(t) andK3(t) and then utilize the Lyapunov method to prove that the flutter con-troller given by Eq.(20)can guarantee the stability of the sys-tem. The adaptive gainK1^iðtÞis chosen as

dK1^iðtÞ

dt ¼ Ci^xðtÞ^x

T_ðtÞP~j_b

i i¼1;2 ð21Þ

whereCiis any positive constant;biis theith column ofB; and

~

Pj_{:¼ P}~j_:max jðkP~jkÞ

.

K2(t) andK3(t) are given as

K2ðtÞ ¼BTP~jb1k^xTðtÞP~jBkk^4ðtÞ a1kx^TðtÞP~jBk2 K3ðtÞ ¼1 2gB T_P~j_k^ 5ðtÞ 8 < : ð22Þ

wherea1,b1andgare given positive constants;k^4 andk^5 are

updated by the following adaptive equations of Eq.(23).

dk^4ðtÞ dt ¼ r1k^x T_ðtÞP~j_Bk d^k5ðtÞ dt ¼ r2gk^xTðtÞP~jBk 2 8 < : ð23Þ

wherer1andr2are given positive constants.

^ Wis given as dW^i dt ¼r3hðxÞ^x T_ðtÞP~j_b i i¼1;2 ð24Þ

wherer3is a given positive constant.

Denote ~ K1ðtÞ ¼K1K1^ ðtÞ ~ k4ðtÞ ¼k4k^4ðtÞ ~ k5ðtÞ ¼k5k^5ðtÞ ~ WT_¼WT_W^T 8 > > > > < > > > > : ð25Þ

where k4 and k5 are positive constants for eliminating the

effects of float, external disturbances and parameter uncertainties.

Substituting Eqs.(10) and (20)into Eq.(8), the closed-loop flutter system can be written as

_ xðtÞ ¼ ðAþDAðtÞÞxðtÞ þBq½ðK1^ ðtÞ þK2ðtÞ þK3ðtÞÞ^xðtÞ ^duðxÞ þduðxÞ þBusðtÞ þfðt;xÞ þB1wðtÞ yðtÞ ¼c1xðtÞ 8 > < > : ð26Þ

Assumption 1. For any given positive numberds,dw,ddand the

actual working timeTf, the float faultus(t), nonlinear vector fðt;xÞ, external disturbancesw(t) and duðxÞare time-varying and satisfy RTf 0u T sðtÞusðtÞdt6ds dsP0 RTf 0 w T_{ðtÞwðtÞdt6d} w dwP0 RTf 0d T uðxÞduðxÞdt6dd ddP0 8 > > < > > : ð 27Þ

Remark 2. The actual output torque generated is bounded due to practical physical limitations of the actuators, and thus, the float faultus(t) and the auxiliary variable (du(x) =v(t)u(t)) are also bounded. In addition, the external disturbancesw(t), in Eq.(8)include center of mass migration, moment of inertia error, gravitational perturbation, atmospheric density devia-tion, and are also bounded.Assumption 1is, therefore, reason-able for flutter system.

Definition 1 18. For given positive constants c1,ds,df,dw,Tf

and a symmetric matrixR>0, the resulting closed-loop flutter system Eq. (26) is said to be robustly finite-time bounded (FTB) with respect to (c1,c2,Tf,R,ds,df,dw), if there exists a constantc2(c2>c1), such that

xT

0Rx0^ 6c1)xTðtÞRxðtÞ<c2 8t2 ½0;Tf ð28Þ

Definition 2 19. If there exists feedback controller in form Eq. (20), such that the resulting closed-loop flutter system Eq.(26)is FTB with respect to (c1,c2,Tf,R,ds,dw,dd) under

the assumed zero initial condition, the flutter system output satisfies the following inequality forTf> 0 and for all

admis-siblew(t) which satisfyAssumption 1. Z Tf 0 yT_ð tÞyðtÞdt6c2 Z Tf 0 wT_ð tÞwðtÞdt ð29Þ wherecis a constant. Then the flutter control law Eq.(20)is called the robust finite-timeH₁flutter controller of the nonlin-ear flutter systems Eq.(26).

Lemma 1 20. For given symmetric matrixS¼ S11 S12

S21 S22

,the following three conditions are equivalent.

ð1ÞS<0 ð2ÞS11<0;S22ST₁₂S₁₁1S12<0 ð3ÞS22<0;S11S12S₂₂1ST₁₂<0 8 > < > : ð30Þ .

Theorem 2. For given positive constantsa0,c1,Tf,ds,dw,ddand

a symmetric matrix R >0, the closed-loop flutter control system Eq.(26)is FTB with respect to (c1,c2,Tf,R,ds,dw,dd), if there

exist positive constant c2and symmetric positive-definite matrix

~ P,such that X P~j_B q P~j_{B P}~j_B1 I 0 0 I 0 I 2 6 6 6 4 3 7 7 7 5<0 ð31Þ c1 kminðPjÞ þ ðdsþdwþddÞ< c2ea0t kmax ðPjÞ ð32Þ where _X¼P~j_AþAT_P~j_a 0P~jþef1P~ j_P~j_þe fL2gI, P~ j_¼R12PjR 1 2, andA¼AþDAðtÞ þB_q K1^ ðtÞ þK2ðtÞ þK3ðtÞ .

Proof. For the closed-loop flutter system Eq. (26), we first

define a Lyapunov functional candidate as

V2ðxðtÞÞ ¼xTðtÞP~jxðtÞ. Then _ V2ðxðtÞÞ ¼x_TðtÞP~jxðtÞ þxTðtÞP~jx_ðtÞ ¼xT_ðtÞP~j_AþAT_P~j_{xðtÞ þ2x}T_ðtÞP~j_B qduðxÞ þ2xT_ðtÞP~j_{BusðtÞ þ2x}T_ðtÞP~j_fðt;xÞ þ2xT_ðtÞP~j_{B1wðtÞ ð33Þ}

Using Eq.(17), Eq.(33)can be written as _ V2ðxðtÞÞ ¼x_TðtÞP~jxðtÞ þxTðtÞP~jx_ðtÞ ¼xT_{ðtÞ P}~j_AþAT_P~j_þe1 f P~ j_P~j_þe fL2gI xðtÞ þ2xT_ð tÞP~jBqduðxÞ þ2xTðtÞP~jBusðtÞ þ2xT_ðtÞP~j_{B1wðtÞ ð34Þ}

Define the following function

J1¼V_2ðxðtÞÞ a0xTðtÞP~jxðtÞ d_uTðxÞduðxÞ

uT

sðtÞusðtÞ w

T_{ðtÞwðtÞ ð35Þ}

The condition inequality(31) implies J1< 0. Multiplying

the above inequality by ea0t_{, we derive} d dt e a0t_Vðxð_tÞÞ ð Þ<ea0t dTuðxÞduðxÞ þuTsðtÞusðtÞ þw T_ð tÞwðtÞ ð36Þ Note that P~j_¼R1

2PjR12_{. By integrating the aforementioned} inequality between 0 andt, we get

V2ðxðtÞÞ<ea0tVðx0Þ þea0t Zt 0 ea0sð_dT uðsÞduðsÞ þuT sðsÞusðsÞ þw T_ð sÞwðsÞÞds <ea0txTð₀ÞP~jxð₀Þ þ ð_d sþdwþddÞea0t 6kmaxðPjÞc1ea0tþ ðdsþdwþddÞea0t ð37Þ

On the other hand, the following condition holds:

V2ðxðtÞÞ ¼xTðtÞR

1

2PjR12xð_tÞP_k

minðPjÞxTðtÞRxðtÞ ð38Þ

From Eqs.(37) and (38), we can get

xT_{ðtÞRxðtÞ<}ðkmax ðPjÞc1þdsþdwþddÞea0t

kminðPjÞ

ð39Þ Condition Eq.(32)implies that for8t2 ½0;Tf,xT(t)Rx(t)

<c2. According toDefinition 1, this completes the proof. h

Theorem 3. For given positive constantsa0,c1,Tf,ds,dw,dd,l,

a1, b1, k4,k5 and a symmetric matrix R >0, the closed-loop

flutter control system Eq.(26)is FTB with respect to (c1,c2,

Tf,R,ds,dw,dd) and satisfies the Eq.(29)for all admissible

w(t),if there exist positive constants c2,cf>cn,g,ef,symmetric

positive-definite matrix P~j _{for any q and any appropriately}

dimensioned matrices Z,Jjand L,which satisfy X3 ZT 0 v1ZTcT1 0 0 0 X4 B1 0 BqL 0 P~j c2 0 0 0 0 0 X^ 0 0 0 ZZT _P~j_{Z 0} Jj 0 X 2 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 5 <0 ð40Þ c1 kminðPjÞ þ ðdsþdwþddÞ< c2ea0t kmaxðPjÞ ð41Þ kxT_ðtÞP~j_B qBTP~j_xðtÞk6lkxT_ðtÞP~j_Bk2 ð42Þ a1xTðtÞP~jB 2 Pb₁ xT_ðtÞP~j_B q12 2 ð43Þ xT_ðtÞP~j_B k4þxTðtÞP~jBusðtÞþxTðtÞP~jBeumax 60 ð44Þ where X3¼v1ðP~jZZTÞ, X4¼AP~jþP~jAT ðvþa0ÞP~j þef1IþJ j_þB qLþ ðBqLÞT, X^¼ Iv1_c1P~j_cT 1, X¼ 1 gþefL 2 g 1

I;e_umax(x) is the maximum value of

approx-imation error e_u(x) (Eq. (11)), c0=cnandc0=cf denote the

adaptive H₁performance bounds for the normal case and fault cases of the closed-loop flutter FTC system Eq. (26), respec-tively, then the adaptive closed-loop flutter system Eq.(26)will exists a H₁FTC controller.

Proof. Select the same Lyapunov function candidate as Theorem 2and define the following function.

J2¼V2ðxðtÞÞ þqiK~ T 1iðtÞC 1 i K1~iðtÞ þr11k~ 2 4ðtÞ þ1 2lr 1 2 k~ 2 5ðtÞ þqir 1 3 W~ T iW~i X2 i¼1 X2 i¼1 ð45Þ J3¼J_2a0xTðtÞP~jxðtÞ þyTðtÞyðtÞ c20w T_ð tÞwðtÞ ð46Þ

Then, according to Eqs.(9), (11), (26), (22) and (25),J3can

be written as J36xTðtÞ P~j AþBqK1^ ðtÞ þ ðAþBqK1^ ðtÞÞTP~j_a 0P~j h i xðtÞ þ2lkxT_ðtÞP~j_BkkxðtÞk þ2xT_ðtÞP~j_B qBT_P~j_xðtÞb 1kxTðtÞP~jBkk^4ðtÞ= a1kxTðtÞP~jBk2 þxT_ðtÞP~j_B qgBT_P~j_xðtÞk^ 5ðtÞ þ2xTðtÞP~jBqðW~ThðxÞ þeuðxÞÞ þ2kxT_ð tÞP~jBkusðtÞ þ2xT_ð tÞP~jfðt;xÞ þ2xT_ðtÞP~j_{B1wðtÞ þx}T_ðtÞcT 1c1xðtÞ c 2 0wT_ðtÞwðtÞ 2qiK~T1iðtÞC 1 i K1_^ iðtÞ 2r11k~4ðtÞk_^4ðtÞ lr1 2 k~5ðtÞk_^5ðtÞ 2r31qiW~ T iW_^i X2 i¼1 X2 i¼1 ð47Þ Note the fact that for any positive constantc0> 0,

2xT_ðtÞP~j_B1wðtÞ6c2 0 x T_ðtÞP~j_B1BT 1P~ j_{xðtÞ þc}2 0w T_{ðtÞwðtÞ ð48Þ}

J36xTðtÞ P~j AþBqK1^ ðtÞ þ ðAþBqK1^ ðtÞÞTP~j_a 0P~j h i xðtÞ þ2lkxT_ðtÞP~j_{BkkxðtÞk þ2kx}T_ðtÞP~j_Bkk^ 4ðtÞ þlgkxT_ðtÞP~j_Bk2^ k5ðtÞ þ2xTðtÞP~jBqðW~ThðxÞ þeuðxÞÞ þ2kxT_ðtÞP~j_{BkusðtÞ þ2x}T_ðtÞP~j_fðt;xÞ þc2 0 x T_ðtÞP~j_B1BT 1P~ j_{xðtÞ þc}2_0wT_{ðtÞwðtÞ þx}T_ðtÞcT 1c1xðtÞ c2_0wT_{ðtÞwðtÞ 2q} iK~ T 1iðtÞC 1 i K1_^ iðtÞ 2r11k~4ðtÞk_^4ðtÞ lr1 2 k~5ðtÞk_^5ðtÞ 2r31qiW~ T iW_^i X2 i¼1 X2 i¼1 ð49Þ Using Eqs.(16) and (17), we have

2lkxT_ðtÞP~j_{BkkxðtÞk6l glkx}T_ðtÞP~j_Bk2_þ 1 glkxðtÞk 2 2kxT_ðtÞP~j_{fðt; xÞk6e}1 f kx T_ðtÞP~j_k2 þefL2gkx T_ðtÞk2 8 < : ð50Þ Using Eq. (50), lglkxT_ðtÞP~j_Bk2_¼ lgl2 lkx T_ðtÞP~j_Bk2 and let k5¼l 2

l, Eq.(49)can be written as

J36xTðtÞ P~jðAþBqK1^ ðtÞÞ þ ðAþBqK1^ ðtÞÞ T_~ Pj_a 0P~j h þe1 f P~ j_P~j_þe fL2g 1 gIþc 2 0 P~ j_B1BT 1P~ j_þcT 1c1 xðtÞ lgkxT_ðtÞP~j_Bk2 k5þ2kxTðtÞP~jBkk^4ðtÞ þlgkxT_ð tÞP~jBk2k^5ðtÞ þ2xTðtÞP~jBqðW~ThðxÞ þeuðxÞÞ þ2kxT_ðtÞP~j_{BkusðtÞ 2q} iK~ T 1iðtÞC 1 i K1_^ iðtÞ 2r1 1 k~4ðtÞk_^4ðtÞ lr21k~5ðtÞk_^5ðtÞ 2r31qiW~ T iW_^i X2 i¼1 X2 i¼1 ð51Þ Using Eqs.(25) and (44), Eq.(51)becomes

J36xTðtÞ½P~jðAþBqK1Þ þ ðAþBqK1ÞTP~ja0P~j þe_f1P~j_P~j_þe fL2gIþ 1 gIþc 2 0 P~ j_B1BT 1P~ j_þcT 1c1xðtÞ 2xT_ðtÞP~j_B qK1~ ðtÞxðtÞ 2kxT_ðtÞP~j_Bkk~ 4ðtÞ lgkxTðtÞP~jBk2k~5ðtÞ þ2xTðtÞP~jBqW~ThðxÞ 2q_iK~T 1iðtÞC 1 i K1_^ iðtÞ 2r11k~4ðtÞk_^4ðtÞ lr1 2 k~5ðtÞk_^5ðtÞ 2r31qiW~ T iW_^i X2 i¼1 X2 i¼1 ð52Þ Set ~ Pj_ðAþB qK1Þ þ ðAþBqK1ÞTP~j_a 0P~jþef1P~ j_P~j_þe fL2gI þ1 gIþc 2 0 P~ j_B1BT 1P~ j_þcT 1c1<0 ð53Þ

Using Eq.(30), Eq.(53)can be written as X1 P~jB1 cT1 I c2_{I 0 0} I 0 X 2 6 6 6 4 3 7 7 7 5<0 ð54Þ whereX1¼P~jAþATP~ja0P~jþef1P~ j_P~j_þP~j_B qK1þðBqK1ÞTP~j_.

Pre- and post-multiplying the inequality (54) by block-diagonal matrix diagðP~j_{;I;. . .;IÞ, Eq. (54) can be further}

changed to be X2 B1 P~jcT1 P~ j c2_{I 0 0} I 0 X 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5 <0 ð55Þ whereX2¼AP~j_þP~j_AT a0P~jþef1IþBqK1P~jþðBqK1P~jÞ T . For any positive constantv> 0, Eq.(55)is equivalent to X2vP~j B1 0 P~j c2_{0I 0 0} X^ 0 X 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 þKT_vP~j K<0 ð56Þ whereK¼ ½I; 0; v1_cT

1; 0. Then, applying Eq.(30), Eq.

(56)can be written as v1_P~j _{I 0 v}1_cT 1 0 X2vP~j B1 0 P~j c2_{0I 0 0} X^ 0 X 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 <0 ð57Þ

Pre- and post-multiplying both sides of Eq. (57) by diagðZT_{;I;. . .;IÞ, Eq.(57)can be further changed to be}

v1_ZT_P~j_{Z Z}T 0 v1_ZT_cT 1 0 X2vP~j B1 0 P~j c2_{0I 0 0} X^ 0 X 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 <0 ð58Þ

Due toZT_P~j_ZPZþZT_P~j_{, Eq.(58)can be written as}

X3 ZT 0 v1ZTcT1 0 X2vP~j B1 0 P~j c2_{0I 0 0} X^ 0 X 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 <0 ð59Þ

where_X3¼v1ðP~jZZTÞ. LetK1=LZ1, from Eq.(59)

we have X2vP~j¼ ðAþBqK1ÞP~jþP~jðAþBqK1ÞT vP~j_a 0P~jþef1I ¼ ðAþBqLZ1ÞP~jþP~jðAþBqLZ1ÞT vP~j_a 0P~jþef1I ¼AP~j_þP~j_AT _ð vþa0ÞP~jþef1IþJ j þBqLþ ðBqLÞTþBqLZ1ðP~jZÞ þ ½BqLZ1_ðP~j_ZÞT_Jj _ð60Þ

Due to

BqLZ1ðP~jZÞ þ ½BqLZ1ðP~jZÞTJj

6BqLZ1ðP~j_ZÞJj_½B

qLZ1ðP~j_ZÞT _ð61Þ

Then, applying Eq.(30), Eq.(60)can be written as

X4 BqL 0 ZZT _P~j_Z Jj 2 6 4 3 7 5<0 ð62Þ where X4¼AP~jþP~jAT ðvþa0ÞP~jþef1IþJ j_þB qLþ ðBqLÞT

From Eq.(62), Eq.(59)can be written as X3 ZT 0 v1ZTcT1 0 0 0 X4 B1 0 BqL 0 P~j c2_{0I 0 0 0 0} X^ 0 0 0 ZZT _P~j_{Z 0} Jj 0 X 2 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 5 <0 ð63Þ From Eq.(63), Eq.(52)can be written as

J3<2xTðtÞP~jBqK1~ ðtÞxðtÞ 2kxTðtÞP~jBkk~4ðtÞ lgkxT_ðtÞP~j_Bk2~ k5ðtÞ þ2xTðtÞP~jBqW~ThðxÞ 2qiK~ T 1iðtÞC 1 i K1_^ iðtÞ 2r11k~4ðtÞk_^4ðtÞ lr1 2 k~5ðtÞk_^5ðtÞ 2r31qiW~TiW_^i X2 i¼1 X2 i¼1 ð64Þ From Eqs.(21), (23) and (24), Eq.(64)can be written as

J3<0 ð65Þ

which implies that the flutter system is ultimately uniformly bounded, and the flutter statex(t) converges to zero.h

Remark 3. Assume that LMIs (40) and Eqs. (41)–(44) are satisfied, control gainK2(t),K3(t), adaptive update lawsK^1iðtÞ,

^

k4ðtÞ,k^5ðtÞandW^ are given by Eqs.(22), (21), (23) and (24) then the closed-loop flutter system Eq.(26) is stable, thencn

andcfare minimized if the following optimization problem is

solvable min anc2nþafc2f

n o

s:t: Eqs:ð40Þandð41Þ ð66Þ whereanandafare weighting coefficients.

4. Numerical simulations

In this section, numerical example is provided to illustrate the validity of our proposed approaches. The flutter of two-dimensional nonlinear wing is used as numerical example. Table 1 shows the structural parameters of 2D wing. The matricesAandBin Eq.(7)are given by

A¼ 0 0 1 0 0 0 0 1 1109:7 952:5 0:67 0:00003 27:01 26:24 0:021 0:0002 2 6 6 6 4 3 7 7 7 5 B¼ 0 0 0 0 30:37 28:47 0:73 0:67 2 6 6 6 4 3 7 7 7 5 8 > > > > > > > > > > > > > < > > > > > > > > > > > > > :

In the simulations, the parametersbN,g,Ci,r1,r2andr3in

Eqs.(11), (22), (21), (23) and (24)are chosen as

bN¼5; g¼100;Ci¼0:45; r1¼0:25; r2¼0:25; r3¼4:5

The parametersN(t),cNand the initial values of the state

x(t) in Eqs.(9), (11) and (7)are

NðtÞ ¼ 0:5 sint 0 0 0 0 0 0 0 cN¼ 1 0:5 0 0:5 1 1 0:5 0 0:5 1 xð0Þ ¼ ½0;0:5;0;0:5T 8 > > > > > > < > > > > > > :

The reentry vehicle is assumed to experience the following faulty case during its operation: before 4 s, the system operates in normal case, that is, all of the two actuators are normal. Between the 4th and the 10th second, the first actuator is float at us(t) = 30 + 30 sin (0.1t) + 20 cos (0.5t) and the second

actuator is loss of effectiveness, that is, q2= 10.05t until

loss effectiveness of 50%. After the 10th second, the second actuator has float fault and other actuator is loss of effective-ness, that is, q1= 10.01t until loss effectiveness of 70%.

The perturbationswðtÞ ¼ ½10 sinð0:1tÞ;15Tenter the systems at the beginning (tP0 s).

Using algorithm withan= 5,af= 1, we obtainH1 perfor-mances of the closed-loop flutter system as 0.3923 (normal) and 1.3553 (fault).Fig. 2shows that the state variablex(t) of wing flutter and observer errorein case of fault.Fig. 3 illus-trates the control signals’ variations under different faults, which is, the parameteru(t) in Eq.(7). In the simulation of this paper, we assume that vehicles’ two actuators from the 4th sec-ond to the 10th secsec-ond have some faults, while external distur-bances exist all the time. From Fig. 2we can see that in the fault situations mentioned above, by using the controller (as shown in Fig. 3) which we have put forward in this article, the flutter can be controlled within 1 s, which verifies the reli-ability and the robustness of the control method proposed in this article. Fig. 4 displays the output of neural networks

^

duðxÞ. From Fig. 4, we can see that the neural networks

^

duðxÞ will change parameters to adjust the controller when the actuator is under saturation situation.

Fig. 5shows the variation of control gainK1^ ðtÞin different situations, whichK1^ ðtÞensures the stable of the flutter system. We also know that K1^ ðtÞ varies with the flutter state in Eq. (21). The faults in the actuator might cause changes in the flutter state (as shown inFig. 2). FromFig. 5, control gain

^

K1ðtÞ will change parameters to adjust the controller when faults occur, so that the actuator of the vehicle would receive effective compensation and return to stable state. From

Fig. 5, when wing flutter is back to stable, the control lawK1^ ðtÞ will not change any more.

In order to effectively compensate the actuator float fault and uncertainties of flutter dynamic model mentioned above and ensure the stability of the flutter system, we design the con-trol lawk^4 andk^5 in this article. From Eq.(23), we can find

thatk^4 andk^5 change with the flutter state variation. Under

the circumstances of the faults of actuator and parameters uncertainty, the state of flutter system may have corresponding changes. FromFig. 6, we can see that the parameters of con-trol gain k^4 andk^5 may justify with the flutter system state

when the state of flutter changes. Then the control law of con-troller (Fig. 3) will be changed and will effectively suppress the uncertainty of actuator fault and flutter dynamic model. After the reentry vehicle flutter being stable, we can see fromFig. 6 that parameters of control gain k^4 and k^5 will keep

immutability.

The finite-time closed-loop flutter FTC system with actua-tor saturation, external disturbances and parameter uncertain-ties can be ensured to be asymptotically stable in the presence of actuator faults. The results show that the proposed con-troller performs very well and accomplishes the flutter suppres-sion despite these undesired effects in the closed-loop flutter system.

5. Conclusions

(1) In this paper, a novel finite-time H₁ adaptive fault-tolerant flutter control design scheme is proposed for wing flutter subject to actuator saturation, external dis-turbances and parameter uncertainties.

(2) The actuator saturation is approximated by a radial basis function. Actuator faults are considered, including loss of effectiveness and float.

(3) The proposed finite-time H₁ adaptive fault-tolerant flutter controller is proved to adaptively adjust con-troller parameters to compensate the faults, actuator saturation, disturbances and parameter uncertainties within the flutter system.

(4) Numerical simulation results further illustrate the effec-tiveness of the presented approach.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos. 11132001, 11272202 and 11472171), the Key Scientific Project of Shanghai Municipal Education Commission (No. 14ZZ021) and the Natural Science Foundation of Shanghai (No. 14ZR1421000). References

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Gao Mingzhouis a Ph.D. candidate of Shanghai Jiaotong University, China. His major is engineering mechanics. His current research interests focus on structural dynamics and control.

Cai Guoping is a professor in the Department of Engineering Mechanics, Shanghai Jiaotong University, China. He received the Ph. D. degree in Engineering Mechanics from Xi’an Jiaotong University in 2000. His current research interests focus on structural dynamics and control, delayed system dynamics and control, and coupled system dynamics and control.