Delayed full-state feedback control of airfoil flutter using sliding mode control method

Journal of Fluids and Structures 61 (2016) 262–273

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Delayed full-state feedback control of airfoil flutter using sliding mode control method Mengxiang Luo, Mingzhou Gao, Guoping Cai n Department of Engineering Mechanics, State Key Laboratory of Ocean Engineering, Shanghai Jiaotong University, Shanghai 200240, China

a r t i c l e in f o

abstract

Article history: Received 21 April 2015 Accepted 30 November 2015

This paper studies the delayed feedback control of flutter of a two-dimensional airfoil using a sliding mode control (SMC) method. The dynamic equation of airfoil flutter is firstly established using the Lagrange method, in which the cubic hardening spring nonlinearity of pitch stiffness is considered. Then, the state equation with time delay is transformed into a standard state equation with implicit time delay by a special integral transformation. Next a nonlinear time-delay controller is designed using the SMC method. Finally the effectiveness of the proposed controller is verified through numerical simulations. Simulation results indicate that time delay in the control system has significant influence on the control performance. Control failure may happen if time delay is not considered in control design. The time-delay controller proposed is effective in suppressing the airfoil flutter with either small or large control time delay. & 2015 Elsevier Ltd. All rights reserved.

Keywords: Airfoil Flutter Sliding mode control Time delay

1. Introduction Airfoil flutter is an aeroelastic phenomenon created by the unsteady aerodynamic pressure loads acting on the airfoil. Beyond flutter speed, the aeroelastic system actually absorbs energy from the free stream flow and an excited response develops. If the flutter is not suppressed in time, the vibration will continue to grow, ultimately leading to aeroelastic instability or catastrophic structural failure. Up to now, many researchers have utilized active control methods to suppress the airfoil flutter. For example, Librescu and Marzocca (2005a) provided a short review of the active aeroelastic control techniques and capabilities. A number of concepts involving various control methodologies, as well as results obtained with such controls are presented. Yu et al. (2007) designed a μ controller to suppress airfoil flutter, and wind tunnel experiments were carried out to verify the effectiveness of the designed controllers. Prime et al. (2010) 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. (2011, 2012) 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. (2012) discussed the modelfree control of aeroelastic vibration of a non-linear wing-flap system. Zhang et al. (2013) designed a partial state feedback continuous adaptive controller in order to suppress the aeroelastic vibrations of the wing section model. Zhang and Behal (2014) proposed a continuous controller to suppress the aeroelastic vibrations of the wing section model. Among active n

Corresponding author. E-mail address: [email protected] (G. Cai).

http://dx.doi.org/10.1016/j.jfluidstructs.2015.11.012 0889-9746/& 2015 Elsevier Ltd. All rights reserved.

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control methods, sliding mode control (SMC) can robustly respond to external disturbance and system uncertainty. As a result, this method has been widely used in airfoil flutter suppression. Lin and Chin (2006) derived an adaptive decoupled fuzzy logic control (FLC) based on SMC to control the plunge and pitch motions of airfoil flutter. Song et al. (2010) focused on the active flutter suppression (AFS) of a typical airfoil by using the state feedback SMC method. Yang et al. (2010) studied the effectiveness of SMC method for AFS and the issues concerning control input constraints using a typical two-dimensional airfoil. Elhami and Narab (2012) used SMC method to suppress airfoil flutter and comparison between SMC method and State-Dependent Riccati Equation (SDRE) approach was carried out. However, time delay exists inevitably in active control system, which mainly results from the follows: (1) the time taken in the online data acquisition from sensors at different locations of the system; (2) the time taken in the filtering and processing of the sensory data for the required control force calculation and the transmission of the control force to the actuator; and (3) the time taken by the actuator to produce the required control force. Due to the time delay, when unsynchronized control force is applied to a structure, it may result in degradation in the control efficiency and instability of the control system (Cai and Huang, 2002; Hu and Wang, 2002). There are some scholars began to focus on the time delay in flutter control system. For example, Zhao (2011) investigated the effect of time delay on the flutter instability of an actively controlled airfoil in an incompressible flow field. Librescu and Marzocca (2005b) provided a systematic research on the effect of time delay on the stability of two-dimensional lifting surfaces and a delayed feedback control is designed to stabilize the system. Yuan et al. (2004) studied the effect of time-delayed feedback control on the flutter instability boundary and its character of a two-dimensional supersonic lifting surface. Ramesh and Narayanan (2001) applied a timedelayed continuous feedback method to control the chaotic motions of a two-dimensional airfoil with cubic pitching stiffness and linear viscous damping. Huang et al. (2012) revealed the effect of input time delay on the stability of a controlled high-dimensional aeroelastic system in an incompressible flow field and presented a new optimal control law to suppress the flutter with an input time delay in the control loop. It is worth being mentioned herein that, although some researches have been done on the effect of time delay on stability of the controlled aeroelastic systems, most of these studies focus on the effect of time delay on dynamic characteristics of the aeroelastic system and the stability of controlled aeroelastic system, little effort has been done on how to deal with the time delay in the active control system. Aircraft generally flies with a high speed, but data sampling period of sensors on the aircraft cannot be infinite small, so system state may vary greatly in a very short time. As a result, even small time delay in the control system may have significant influence on control performance. Therefore, an in-depth exploration of time-delay controller of airfoil flutter is necessary. In this paper, we proposed a method to deal with the time delay in the system, and designed a nonlinear time-delay controller using the sliding mode control (SMC) method. This paper studies time-delay control of airfoil flutter using the sliding mode control method. The dynamic equation of airfoil flutter is established considering the cubic hardening spring nonlinearity of pitch stiffness. A nonlinear time-delay controller is proposed for the airfoil flutter using the SMC method. This paper is organized as follows: Section 2 presents the dynamic model of wing flutter; time-delay controller is designed in Section 3; in Section 4 the results of numerical simulation are shown; and finally Section 5 draws conclusions of this research.

2. Airfoil flutter model In this section, flutter model for a two-dimensional wing including cubic hardening spring nonlinearity is established. As shown in Fig. 1, a two degree-of-freedom (2-dof) wing model is considered herein. The plunge deflection is denoted by h, positive in the downward direction; θ is the pitch angle about the elastic axis, positive nose up; the span of the wing is b; the chord length is c; Q and P are the wing aerodynamic center and elastic axis, respectively; the distance from the leading edge to the elastic axis is xf ; the distance from the aerodynamic center to the elastic axis is ec, e is the eccentricity; and β is the deflection angle of wing flap. For small pitch angle, the displacement (positive downward) of any point of the wing in the vertical direction can be expressed as z ¼ hþ ðx  xf Þθ

ð1Þ

Fig. 1. Two-dimensional airfoil model with wing flap.

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The kinetic energy, potential energy and dissipation of the system can be given by Z Z bZ c 1 m _ 2 dxdy dmz_ 2 ¼ T¼ ½h_ þðx  xf Þθ 2 2 0 0

ð2Þ

1 1 2 U ¼ K h h þ K θ θ2 2 2

ð3Þ

2 1 1 2 Ψ ¼ C h h_ þ C θ θ_ 2 2

ð4Þ

where m is the wing mass per unit area, K h , K θ , C h and C θ are the stiffness coefficient in plunge, torsion stiffness coefficient, damping coefficient in plunge and torsion damping coefficient, respectively. Applying strip theory, the aerodynamic lift and pitching moment acting on wing are (Marzocca et al., 2001; Wang et al., 2011) ! h_ 1 2 þθ ð5Þ dL ¼ ρV cdyaw V 2 ! " # _ θc h_ 1 þ θ þM θ_ dMf a ¼ ρV 2 c2 dy eaw 4V V 2

ð6Þ

where V is the wind speed, ρ is the density of air, aw is the lifting line slope and M θ_ is the non-dimensional aerodynamic derivative. The virtual work done by the aerodynamic lift and pitching moment can be expressed as Z δW ¼ ½ dLδh þ dMf a δθ ð7Þ Substituting Eqs. (5) and (6) into Eq. (7), the aerodynamic generalized force and moment can be obtained as ! Z b h_ ∂ðδWÞ 1 2 þθ ¼ dL ¼  ρV cbaw Qh ¼ V ∂ðδhÞ 2 0 Qθ ¼

∂ðδWÞ ¼ ∂ðδθÞ

Z

b 0

! " # _ h_ 1 θc þ θ þ M θ_ dM f a ¼ ρV 2 c2 b eaw V 2 4V

ð8Þ

ð9Þ

The aerodynamic lift and moment caused by the deflection angle β of wing flap can be expressed as 2 3 1 2 " # ρV ca s  c β  Lc 6 2 7 7β ¼6 p¼ 4 1 2 2 5 Mc ρV c bc sβ 2

ð10Þ

where ac is the coefficient of lift force, bc is the coefficient of the pitching moment and sβ is the span of control surface. The quasi-steady aerodynamic theory is adopted here in building the aerodynamic model. The quasi-steady aerodynamic theory is only available for establishing aerodynamic model of low-speed aircraft. For hypersonic case, this theory is no longer suitable but the piston theory can be used. The quasi-steady aerodynamic theory cannot account for some complex flow phenomena, e.g., vortex-induced vibration, which can frequently happen in the high-aspect-ratio wing design. When the plunge magnitude exceeds much of the chord length or the pitching amplitude reaches a large angle, the flow can be highly unsteady, the quasi-steady aerodynamic theory is also not adequate to describe the flow phenomena. For the study in this paper, the aircraft is low-speed, so we use the quasi-steady aerodynamic theory to build the aerodynamic model of the airfoil. Based on the aerodynamic model built by the quasi-steady aerodynamic theory, the aeroelastic equation of the twodimensional wing system can be deduced using the Lagrange method, given by 2 3( ) 8 2 3 " #9( ) mbc cbaw mbc < 0 € 2 ðc  2xf Þ C h 0 = h_ 2  5 h þ ρV 4 2 4 5 þ 2 mbc c 2 c3 bM θ_ : 0 C θ ; θ_ θ€  ec baw  2 ðc 2xf Þ mbc 3 cxf þxf 2

8 2 < 0 24 þ ρV : 0

cbaw 2 2  ec 2baw

3 5þ

"

Kh 0

8

2 3 1 2 #9( ) 0 = h 6  2 ρV cac sβ 7 7β ¼6 41 2 2 5 Kθ ; θ ρV c bc sβ 2

ð11Þ

Large deformation of the elastic wing may possibly occur due to severe aerodynamic load, resulting in cubic hardening spring nonlinearity of the wing structure. A thin wing or propeller blade which is being twisted will most likely behave as a cubic hardening spring which becomes stiffer as the angle of twist increases (Lee et al., 1999). The moment caused by the

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Fig. 2. Schematic of cubic hardening spring.

cubic hardening spring nonlinearity of the wing can be written as (Chen et al., 2011) MðθÞ ¼ K θ θ þ enl K θ θ3

ð12Þ

where en1 is the nonlinear stiffness coefficient. The relation of cubic nonlinear moment and pitch angle is shown in Fig. 2. Considering Eq. (12) and writing Eq. (11) into matrix form, we have Aq€ þ ðρVB þDÞq_ þðρV 2 C þ EÞqþ fðtÞ ¼ bβ T

where qðtÞ ¼ ½hðtÞ; θðtÞ A ℜ

21

ð13Þ 22

is the generalized displacement vector; and A A ℜ

22

, BAℜ

22

, CAℜ

22

, DAℜ

and

E A ℜ22 are the inertia, aerodynamic damping, aerodynamic stiffness, structural damping and structural stiffness matrices respectively, fðtÞ A ℜ21 and bA ℜ21 . The detailed expressions of parameters in Eq. (13) are 2 3 2 3 cbaw mbc mbc 0 2 ðc  2xf Þ 2 5 4 5; ; B ¼ A ¼ 4 mbc 2 c2 2 c3 bM  ec 2baw  8 θ_ 2 ðc  2xf Þ mbcð 3  cxf þ xf Þ 2 3 2 3 1 2 " # " # " # cbaw 0 0 Ch 0 Kh 0 6  2 ρV cac sβ 7 2 6 7 4 5 ; D¼ ; E¼ ; b¼4 C¼ 2 5; fðtÞ ¼ en1 K θ θ3 1 2 2 0 Cθ 0 Kθ 0  ec 2baw ρV c bc sβ 2 Without considering structural damping, Eq. (13) can be further changed to be ~ ~ _ ¼ AxðtÞ xðtÞ þ BuðtÞ þ f~ ðtÞ

ð14Þ

T _ _ T A ℜ41 is the state space vector, uðtÞ ¼ βðtÞ A ℜ11 is deflection angle of where xðtÞ ¼ ½qT ðtÞ; q_ ðtÞT ¼ ½hðtÞ; θðtÞ; hðtÞ; θðtÞ " #   0 I 0 A ℜ41 and A ℜ44 , B~ ¼ wing flap and acts as the control input here, A~ ¼ 1 1 2  A ðρV C þEÞ  ρVA B A  1b " # 0 ~f ðtÞ ¼ A ℜ41 .  A  1 fðtÞ

3. Design of delayed SMC controller For the aircraft with high flying speed, even in a very short sampling period, system state may change fast. This may induce to time delay, thus to bring great influence to the system stability. Therefore, time delay is an important issue that should be considered in control design. The method used to deal with time delay in this paper has ever been used in the research of active vibration control by the author, Cai Guoping, who took some flexible structures as research object. The research results indicate that this method is effective in suppressing vibration with either small or large control time delay, and the validity of this method has been proved by scientific experiments. Next, we will introduce the time delay treating method. The controller is designed under the assumption that the parameters of nominal system are known and the time delay is known and time-invariant. Considering time delay in control, the system state Eq. (14) changes to be ~ ~ _ ¼ AxðtÞ xðtÞ þ Buðt τÞ þ f~ ðtÞ where τ represents the time delay. Next we design time-delay controller according to Eq. (15). Make the following integral transformation to Eq. (15) Z t ~ ~ HðtÞ ¼ xðtÞ þ e  A ðs  t þ τÞ BuðsÞ ds t τ

ð15Þ

ð16Þ

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where HðtÞ A ℜ41 , the state equation with time delay can be transformed into the following standard state equation without explicit delay (Cai and Huang, 2003) ~ _ HðtÞ ¼ AHðtÞ þ BuðtÞ þ f~ ðtÞ

ð17Þ

where B ¼ e B~ A ℜ41 . Below we design SMC controller based on Eq. (17). The sliding mode control (SMC) method is suitable for nonlinear systems, due to its excellent robustness with respect to system perturbation and external disturbance. As a matter of fact, SMC has received widespread attention and application. The main idea of this method is to design a group of sliding mode surfaces in phase space as well as a series of switching functions. These switches are turned fast by controller to drive all the states moving towards the sliding mode surfaces (also called switching surfaces) and make the states slide on desired trajectories to become stable. In this paper, the SMC method is used to design the time-delay controller of wing flutter. Eq. (17) is a special nonlinear form, in which the nonlinear and linear items are separate. So we can regard the nonlinear item f~ ðtÞ as external disturbance, the linear SMC theory can be used for controller design. The switching function is given by  A~ τ

sðtÞ ¼ Cs HðtÞ

ð18Þ

14

is the coefficient vector of switching function to be determined. sðtÞ ¼ 0 represents the switching surface. where Cs A ℜ The exponential approach law proposed by Gao and Hung (1993) is used here for controller design, and is given by s_ ðtÞ ¼  ηsgn½sðtÞ  ksðtÞ

ð19Þ     where η 4 0 and k 40 are both positive constants. Near the switching surface, s  0, so from Eq. (19) we know s_  η. By choosing a small gain η, the momentum of the motion will be reduced as the system trajectory approaches the switching surface. As a result, the amplitude of the chatter will be reduced. Furthermore, a large value for k increases the reaching rate when the state is not near the switching surface. Chattering can practically be suppressed by using these methods altogether. Furthermore, the sign function sgnðsÞ is often replaced by the saturation function in using SMC to reduce the chattering. The saturation function is given by 8 s4 Δ > <1 jsj r Δ; υ ¼ 1=Δ satðsÞ ¼ υs ð20Þ > : 1 so Δ where Δ is the boundary layer. Considering Eqs. (17)–(20) and the reaching condition ss_ o 0, the active controller can be obtained as ~ þ Cs f~ ðtÞ þηsat½sðtÞ þksðtÞg uðtÞ ¼  ðCs BÞ  1 fCs AHðtÞ

ð21Þ

The coefficient vector Cs of switching function is generally determined using the pole assignment method or the linear quadratic regulator (LQR) method. Here we use the LQR method to determine Cs . The performance index is chosen as Z 1 1 J¼ HT ðtÞQ HðtÞdt ð22Þ 2 0 where Q A ℜ44 is a positive definite symmetric gain matrix. For the system (17), rankðBÞ ¼ 1, there exists an orthogonal " # 0 , where B2 A ℜ11 . After the coordinate transformation zðtÞ ¼ Tr HðtÞ, the state zðtÞ can matrix Tr A ℜ44 such that Tr B ¼ B2 be partitioned as " # z1 ðtÞ zðtÞ ¼ z2 ðtÞ

ð23Þ

where z1 ðtÞ A ℜ31 and z2 ðtÞ A ℜ11 , so Eq. (17) can be written as the following form without considering f~ ðtÞ # " # #" " # " z1 ðtÞ z_ 1 ðtÞ 0 A11 A12 þ uðtÞ ¼ ðtÞ z z_ 2 ðtÞ B A21 A22 2 2 " # A11 A12 ~ r T , in which A11 A ℜ33 , A12 A ℜ31 , A21 A ℜ13 and A22 A ℜ11 . where ¼ Tr AT A21 A22

ð24Þ

The switching function can be changed as sðtÞ ¼ Cs TTr zðtÞ ¼ Ct zðtÞ

ð25Þ

And the coefficient vector Ct ¼   Ct ¼ ½Ct1 ; C t2  ¼ C t2 κ; 1 13

where Ct1 A ℜ

11

, C t2 A ℜ

Cs TTr

14

Aℜ

can be partitioned as

1 and κ ¼ Ct1 A ℜ13 . C t2

ð26Þ

M. Luo et al. / Journal of Fluids and Structures 61 (2016) 262–273

Using zðtÞ ¼ Tr HðtÞ, the performance index can be rewritten as Z 1 1 zT ðtÞQ zðtÞdt J¼ 2 0 2 3 Q 11 Q 12 T 4 5, in which Q 11 A ℜ33 , Q 12 A ℜ31 and Q 22 A ℜ11 . where Q ¼ Tr Q Tr ¼ T Q 12 Q 22

267

ð27Þ

Introduce a pseudo-control term as 1

T

νðtÞ ¼ z2 ðtÞ þ Q 22 Q 12 z1 ðtÞ

ð28Þ

Substituting Eq. (23) into Eq. (27) and rearranging the expression, we have Z 1 1 ðzT1 Q^ z1 þ νT Q 22 νÞdt J¼ 2 0

ð29Þ

1 T where Q^ ¼ Q 11  Q 12 Q 22 Q 12 . Using Eq. (28), the first equation of Eq. (24) can be written as

^ 1 ðtÞ þ A12 νðtÞ z_ 1 ðtÞ ¼ Az 1 T ¼ A11  A12 Q 22 Q 12 .

^ where A mined to be

1

ð30Þ Eqs. (30) and (29) constitute a classical LQR problem, the “optimal controller” can be deter-

T

νðtÞ ¼  ðQ 22 A12 P1 Þz1 ðtÞ

ð31Þ

where P1 satisfies the following Riccati equation ^  P1 A12 Q  1 AT P1 þ Q^ ¼ 0 ^ T P1 þP1 A A 12 22

ð32Þ

On the sliding surface, sðtÞ ¼ 0, so from Eq. (25) and (26) we have z2 ðtÞ ¼  κz1 ðtÞ

ð33Þ

From Eqs. (28), (31) and (33), the vector κ can be finally obtained as 1

T

T

κ ¼ Q 22 ðA12 P1 þ Q 12 Þ

ð34Þ

Once the vector κ has been obtained, the vector Ct can be calculated using Eq. (26), where C t2 can be arbitrarily chosen, and we can obtain Cs ¼ Ct Tr . The architecture of the delayed SMC system is demonstrated in Fig. 3. The deflection angle of wing flap acts as control input here and is limited within [ 5°, þ 5°]. Similar to the results presented by Cai and Huang (2003), the time-delay controller contains not only the current step of state feedback but also the linear combination of some former steps of control. The realization of this controller is the same as the no-delay controller except that there are some former control terms appearing in the time-delay controller. These control terms can be saved in advance and are employed in the later calculation of time-delay controller.

4. Numerical simulations In this section, numerical simulations are carried out to demonstrate the validity of the proposed method in this paper. The two-dimensional airfoil shown in Fig. 1 is considered and the structural parameters of the airfoil are displayed in

Fig. 3. Architecture of the delayed SMC system.

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Table 1 Structural parameters of the two-dimensional airfoil. Geometric parameter Kinetic and dynamic parameter

m ¼ 1320 kg xf ¼ 0:28 m aw ¼ 2π

s¼1m sβ ¼ 0:3 m ac ¼ 3:826

c ¼ 0:54 m e ¼ 0:15

M θ_ ¼  0:22

bc ¼  0:076

K h ¼ 1:085  106 N/m

ρ ¼ 1:225 kg/m3

enl ¼ 10

K θ ¼ 1:1  105 N m/rad

Fig. 4. Relation of wind speed and artificial damping and relation of Wind speed and frequency: (a) full-sized image of V-g; (b) partial enlarged image of V-g and (c) full-sized image of V-ω.

Fig. 5. Time histories of plunge displacement and pitch angle of airfoil with considering cubic hard spring nonlinearity: (a) plunge displacement and (b) pitch angle.

Table 1. At first, the critical flutter speed of the airfoil is calculated using the V-g method. In the V-g method, the speed corresponding to the artificial damping g ¼0 is the flutter speed of the airfoil. Fig. 4 shows the relationship between the wind speed and the artificial damping, as well as the relationship between the wind speed and the frequency. We can obtain from Fig. 4 that the artificial damping crosses the zero point when the wind speed reaches 240 m/s. Therefore, the flutter might occur if wind speed is beyond 240 m/s. We can also see that, when the wind speed reaches 240 m/s, the frequency of pitch angle is 18.1 Hz, and that is the flutter frequency.

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Fig. 6. Time histories of plunge displacement and pitch angle of airfoil without considering cubic hard spring nonlinearity: (a) plunge displacement and (b) pitch angle.

Fig. 7. Time histories of plunge displacement and pitch angle of airfoil and time history of deflection angle of wing flap when using no-delay controller to control the system without time delay: (a) plunge displacement, (b) pitch angle, and (c) deflection angle of wing slap.

Here we investigate the effect of cubic hardening spring nonlinearity on system character. The wind speed 250 m/s which beyond the flutter speed is taken. Assume that the airfoil has an initial plunge displacement h ¼ 0:1 m and an initial pitch angle θ ¼ 31, and no control is applied to the airfoil. The governing ordinary differential equations are solved using the Runge–Kutta method. Figs. 5 and 6 show the time histories of plunge displacement and pitch angle with and without the cubic hardening spring nonlinearity, respectively. We can observe that there exist big differences in those results. When wind speed is larger than flutter speed, large deformation of airfoil will occur, the cubic hardening spring nonlinearity should be considered in dynamics analysis and control design. Fig. 5 also shows that the plunge displacement can reach 15% of the chord length and pitch angle is about 710°. When active control is used to suppress the flutter, the response of airfoil can be nearly controlled to zero which can be observed in Fig. 7, under this condition, the flow will not be highly unsteady. So the quasi-steady aerodynamic theory may be available for this case. It is demonstrated in Fig. 5 that the response of system does not decay along time if no control measure is applied to the airfoil. Here we use the SMC method to design the controller so as to reduce the airfoil response with the same initial conditions as above. The control loop does not contain time delay here. In controller design, the parameters in Eqs. (19) and (20) are chosen as η ¼ 5, k ¼ 1000 and Δ ¼ 0:1. The gain matrix Q in Eq. (22) is chosen as a (4  4) identity matrix, so the coefficient vector Cs of switching function can be determined as Cs ¼[18.9765,  107.1315, 1.1771, 0.3401]. The deflection angle of wing flap is limited within [  5°, þ 5°]. Under

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Fig. 8. Time histories of plunge displacement and pitch angle of airfoil and time history of deflection angle of wing flap when using no-delay controller to control the system with time delay τ ¼ 0:006 s: (a) plunge displacement, (b) pitch angle, and (c) deflection angle of wing slap.

Fig. 9. Time histories of plunge displacement and pitch angle of airfoil and time history of deflection angle of wing flap when using time-delay controller to control the system with time delay τ ¼ 0:006 s: (a) plunge displacement, (b) pitch angle, and (c) deflection angle of wing slap.

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271

Fig. 10. Time histories of plunge displacement and pitch angle of airfoil and time history of deflection angle of wing flap when using time-delay controller to control the system with time delay τ ¼ 0:02 s: (a) plunge displacement, (b) pitch angle, and (c) deflection angle of wing slap.

Fig. 11. Time histories of plunge displacement and pitch angle of airfoil and time history of deflection angle of wing flap when using time-delay controller to control the system with time delay τ ¼ 0:1 s: (a) plunge displacement, (b) pitch angle, and (c) deflection angle of wing slap.

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Fig. 12. Time histories of plunge displacement and pitch angle of airfoil and time history of deflection angle of wing flap when using time-delay controller after LCOs become stable to control the system with time delay τ ¼ 0:02 s: (a) plunge displacement, (b) pitch angle, and (c) deflection angle of wing slap.

active control, Fig. 7 shows the time histories of plunge displacement and pitch angle of the airfoil, and deflection angle of wing flap. We can observe from Fig. 7 that the airfoil responses can be controlled rapidly by the SMC controller. The effect of time delay on control performance is investigated firstly. It is assumed that there exists time delay in control system and the time delay is not treated. The small time delay τ ¼ 0:006 s is considered herein. In the simulations, parameters of the controller and initial conditions of the system are chosen as the same as those in the previous case. Using the controller designed in the case of no time delay to control the system with time delay, the system responses are displayed in Fig. 8. We can see from Fig. 8 that control failure occurs when the time delay is not treated. Our extensive simulations indicate that the controller without considering time delay fails to control the system when the time delay is larger than 0.006 s. Next we use the time-delay controller proposed in this paper to control the system with time delay. The same initial conditions as previously mentioned are considered. The gains of SMC are η ¼ 5, k ¼ 100 and Δ ¼ 0:1. For the case of τ ¼ 0:006 s, the coefficient vector Cs of switching function can be determined as Cs ¼[0.7121,  100.1758, 1.6752,  0.5044], and the simulation results are shown in Fig. 9. We can observe from Fig. 9 that the system can be controlled effectively by the proposed time-delay controller. Then larger time delays τ ¼ 0:02 s and τ ¼ 0:1 s are considered. For these two cases, the gains of SMC control are changed to be k ¼ 60 and k ¼ 62, Cs are updated as Cs ¼[  296.2329, 67.3239, 0.5706,  0.8566] and Cs ¼[  93.2642, 4.3054,  4.1825, 1.2340], respectively. The simulation results are given in Figs. 10 and 11. The same conclusion can be obtained as that in Fig. 9, the time-delay controller has favorable control performance even the system involving large time delay. In order to further prove the validity of the SMC time-delay feedback controller, we turn on the controller after the limit cycle oscillations (LCOs) appear and become stable when there is time delay of τ ¼ 0:02 s. The parameters are the same as the previous τ ¼ 0:02 s case. The results are shown in Fig. 12. We can see From Fig. 12 that the proposed controller is still valid.

5. Conclusion As most of previous studies on the time delay problem in the aeroelastic system pay more attention to the influence of time delay on dynamic characteristics and the stability of controlled aeroelastic system, but few of them focus on dealing

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with the time delay in the active control system, we proposed a method to deal with the time delay in the system, and designed a nonlinear time-delay controller using the sliding mode control (SMC) method. In this paper, the dynamic model of airfoil flutter is established considering cubic hardening spring nonlinearity of pitch stiffness of the airfoil, and a nonlinear time-delay controller is designed. Numerical simulations indicate that cubic nonlinearity of airfoil should be taken into account when the wind speed is larger than the critical flutter speed. The SMC controller fails to control the airfoil flutter if time delay exists in the control system but the time delay is not treated in control design. The proposed time-delay controller can effectively deal with the time delay in the control system and the airfoil flutter can be rapidly suppressed by this controller. Furthermore, the proposed controller is available for small time delay and large time delay as well. Acknowledgments This work is supported by the Natural Science Foundation of China (11132001, 11272202 and 11472171), the Key Scientific Project of Shanghai Municipal Education Commission (14ZZ021), the Natural Science Foundation of Shanghai (14ZR1421000) and the Special Fund for Talent Development of Minhang District of Shanghai.

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