Active monitoring and vibration control of smart structure aircraft based on FBG sensors and PZT actuators

JID:AESCTE AID:3876 /FLA

[m5G; v1.194; Prn:5/01/2017; 15:09] P.1 (1-9)

Aerospace Science and Technology ••• (••••) •••–•••

1

Contents lists available at ScienceDirect

2

67 68

3

Aerospace Science and Technology

4 5

69 70 71

6

72

www.elsevier.com/locate/aescte

7

73

8

74

9

75

10

76

11 12 13

Active monitoring and vibration control of smart structure aircraft based on FBG sensors and PZT actuators ✩

14 15 16 17

81

School of Mechatronic Engineering and Automation, Shanghai University, Shanghai 200072, China

83

24 25

85

a r t i c l e

i n f o

a b s t r a c t

86 87

Article history: Received 5 May 2016 Received in revised form 18 October 2016 Accepted 27 December 2016 Available online xxxx

26 27 28 29 30 31

82 84

21 23

79 80

19

22

78

Zhiyuan Gao ∗ , Xiaojin Zhu, Yubin Fang, Hesheng Zhang

18 20

77

Keywords: Smart structure Active monitoring Vibration control Shape reconstruction Adaptive control

Active vibration monitoring and control system becomes a very hot topic in recent years with the rapid development of smart materials, sensing technology and actuating technology. A smart aircraft model is constructed using fiber Bragg grating (FBG) sensors and piezoelectric ceramics. Vibration shape reconstruction of the aircraft model is achieved using discrete fiber Bragg grating sensors based reconstruction method. Vibration control is achieved by employing modified multi input multi output (MIMO) hybrid filtered-x least mean square (FXLMS) control algorithm with online identification and reference signal self-extraction using distributed piezoelectric patches. An experimental platform was constructed, and experimental verifications were done. The results show that the proposed reconstruction method is effective for real-time shape reconstruction of the aircraft model, and the structural vibration response is suppressed to a great extent by the proposed vibration control method. © 2017 Elsevier Masson SAS. All rights reserved.

88 89 90 91 92 93 94 95 96 97

32

98

33

99

34

100

35 36

101

1. Introduction

37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

Using composite structures, modern aircrafts are becoming more lighter, which means aircrafts are more flexible and their aerodynamics is more complex and more vibration failures could happen. Thus, real-time accurate shape reconstruction and vibration control plays an important role in maintaining the healthy operation for them. Recent years, smart structure applications have been introduced to this field [25,14,21,6]. It becomes possible to carry out shape reconstruction and vibration control for reentry vehicles [1,8,18] with the development of high-temperature optical fiber technology and piezoelectric ceramic technology. Light intensity modulation fiber-optic sensor to measure curvature directly has been developed by Fu [10]. Long-gauge FBG has already been used for monitoring the dynamic response of the composite structure utilizing the collected dynamic response data [23]. The effectiveness of piezoelectric actuators is investigated in reducing

53 54 55 56 57 58 59 60 61 62 63 64 65 66

✩ The Project was supported by National Natural Science Foundation of China (Grant Nos. 61503232, 51575328, 51375293), “Chen Guang” project supported by Shanghai Municipal Education Commission and Shanghai Education Development Foundation (No. 15CG44), key program of Shanghai Municipal Education Commission (No. 13ZZ075), key program for Basic Research of the Science and Technology Commission of Shanghai Municipality (No. 12JC1404100), Mechatronics Engineering Innovation Group project from Shanghai Education Commission and Shanghai Key Laboratory of Power Station Automation Technology. Corresponding author. E-mail address: [email protected] (Z. Gao).

*

http://dx.doi.org/10.1016/j.ast.2016.12.027 1270-9638/© 2017 Elsevier Masson SAS. All rights reserved.

vibratory response due to buffet loads on the F-16 ventral fin [5]. Li reviews recent publications regarding strain-related techniques on structural damage identification [28]. Song reviews the vibration suppression techniques of civil structures using piezoceramic smart materials [25]. Lopez gives a comprehensive review of uncertainties involved in flight vehicle structural damage monitoring, diagnosis, prognosis and control [20]. Barlas provides an overview of smart rotor control for wind turbines [3]. Jenkins makes a historical review of membrane vibration experiments [15]. Landau reviews various techniques proposed for unknown multiple narrow band disturbances rejection [9]. The existing monitoring methods are using camera based technology, or sensor signal analyzing technology, which makes it very difficult to apply these methods for aircraft or spacecraft applications, as illumination is not possible, and installation is inconvenient. Meanwhile signal analyzing technology only works for healthy conditions, and cameras should be installed in a place that would allow to take a panoramic photograph, which is almost impossible for aircraft or spacecraft applications. Non-visual shape reconstruction method is a promising way to solve this problem, and non-visual reconstruction method has many advantages, including small amount data needed to acquire, and high acquisition precision, it has become a research hot spot. Non-visual perception method can be divided into strain data based displacement mode superposition method [16,19], and curvature data based geometrical iteration method [27]. The basic idea of displacement mode superposition method is using strain data and strain-displacement transformation matrix to achieve shape

102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

JID:AESCTE

AID:3876 /FLA

2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

[m5G; v1.194; Prn:5/01/2017; 15:09] P.2 (1-9)

Z. Gao et al. / Aerospace Science and Technology ••• (••••) •••–•••

reconstruction by obtaining the free structural vibration equation. The basic idea of geometrical iteration method is: converting the obtained strain data to curvature and achieving curvature continuity by interpolation method, and employing moving coordinate system to achieve iterative recursion of surface coordinates, and employing curved surface fitting algorithm to achieve shape reconstruction. Displacement mode superposition method proposed in [16,19] is only suitable for shape reconstruction with small amplitude distortion as the strain direction on the measuring surface is uncertain. Geometrical iteration method proposed in [27] uses curvature data along a single direction and plane curve fitting algorithm to achieve shape reconstruction, and the method is not suited for real time reconstruction as it needs to solve complex non-linear equations. To overcome the shortcomings of the above algorithms, a new 3-D reconstruction algorithm using orthogonal curvature is proposed. Recursive computation is employed for real time implementation instead of solving non-linear equations. With strong adaptability, adaptive control is a promising control algorithm. Also, it is easy to implement without accurate model. The most famous one is filter-x least mean square (FXLMS) algorithm which has been independently proposed by different researchers [26,4,22], the convergence analysis of this algorithm is given by [2]. In recent years many modified FXLMS algorithm is proposed to improve the control performance and convergence speed of the FXLMS algorithm [7,13,12,11]. However, the above modified FXLMS based control algorithms need a reference signal, but in most practical situations, it is not possible to add a reference sensor into the active vibration control system. To overcome these problems, a novel MIMO hybrid FXLMS active vibration control algorithm with online identification is proposed in this paper. This paper is organized as follows. Section 2 introduces the active monitoring method using FBG sensors, including how to convert FBG sensing signal into structural curvature, and shape reconstruction of the aircraft frame and wing. Section 3 proposes an hybrid active vibration control method based on FXLMS algorithm, in which online identification of secondary path and reference signal self-extraction is achieved. Section 4 introduces experimental platform construction process, including the smart aircraft model construction, and the active monitoring system and active vibration control system hardware composition. Section 5 gives the experimental test results of the proposed active monitoring and control method. Section 6 comes to the conclusion.

46 47 48

2. Active monitoring method

49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

Real-time shape sensing and control are important to maintain dynamic performance of structures. However, sometimes we can not measure structural deformation directly due to operating conditions of the structures, for example, we can not use camera based and displacement sensor based method as these equipments are relatively large and not suit for aeronautical environment or aerospace environment. When displacements cannot be measured directly, estimation of the displacements using strain data can be an alternative solution. In which FBG sensors are suitable due to its multiplexing capability as many strain sensors are needed to estimate the structural deformation more accurately. This section would develop a non-visual reconstruction method based on embedded FBG arrays. The distributed curvatures are obtained by the strain distribution information measured by FBG sensors. A 3-D reconstruction algorithm using orthogonal curvature is developed as well as a recursive surface fitting based space surface reconstruction algorithm.

67 68 69 70 71 72 73 74 75 76 77 78

Fig. 1. FBG sensing principle.

79 80

2.1. Converting FBG sensing signal into curvature

81 82

The Bragg central wavelength for one FBG sensor is as follows:

λ B = 2ne f f · 

(1)

where ne f f is the effective refractive index and  is the grating space. Assuming the sensor measurements are related only to isothermal conditions and a mechanical axial strain ε is applied to the FBG, the grating period  would change and a shift λ B of the Bragg wavelength can be detected:

λ B = λ B · (1 − P e ) · ε

λ B λ B · (1 − P e ) · h

85 86 87 88 89 90 92 93 94 95 96 97 98 99 100 101 102

(3)

103 104

While the FBG sensor is bonded to a structure, the wavelength shift detected by FBG sensor array, can be converted into curvature k as follows:

k=

84

91

(2)

where P e is the effective photoelastic coefficient of the fiber. The FBG detection unit can be modeled as a circular cross-section flexible beam. While a mechanical axial strain is applied, it will be subjected to tensile and compressive stresses at the same time and it forms an arc, as shown in Fig. 1. Here ρ is the curvature radius, dθ is the angle of the arc, h is the distance from the neutral surface to the sensing position. According to theoretical mechanics principles,

(ρ + h)dθ − ρ dθ h ε= = = hk ρ dθ ρ

83

105 106 107 108

(4)

2.2. Aircraft frame reconstruction The aircraft frame reconstruction can be considered as space curve reconstruction for flexible slender rod. Moving coordinate system is built on the bending curve using moving frame method, as shown in Fig. 2. O 1 , O 2 , O 3 are three points on the space curve. Fixed coordinate system σ (0) = [ O 1 ; x, y , z] can be built taking O 1 as the origin. While tangent of the curve at O 1 is taking as axis c 1 and curvature component ka1 and kb1 is taking as axis a1 and b_1, moving coordinate system σ (1) = [ O 1 ; a1 , b1 , c 1 ] can be built. Cosystem σ (0) . ordinate system σ (1) is overlapping with coordinate 

k1 is the composite vector of ka1 and kb1 , k1 = ka1 2 + kb1 2 . Osculating plane π1 is consisted of k1 axis and c 1 . Similarly, moving coordinate system σ 2 , σ 3 , . . . , σ i , . . . can be built, taking O 2 , O 3 , . . . , O i , . . . as the origin. While O 2 is close to O 1 , arc segment O 1 , O 1 O 2 is very small and can be treated as a curve in osculating plane π1 . Location of O 2 on osculating plane π1 can be calculated by arc length d s and its curvature k1 . Then the space coordinates of O 2 could be obtained using fixed coordinate system σ (0) .

109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

JID:AESCTE AID:3876 /FLA

[m5G; v1.194; Prn:5/01/2017; 15:09] P.3 (1-9)

Z. Gao et al. / Aerospace Science and Technology ••• (••••) •••–•••

3

1

67

2

68

3

69

4

70

5

71

6

72

7

73

8

74

9

75

10

76

Fig. 3. FBG detection unit.

11

77

While coordinate system σ ( R ) is moved to O R , it is consistent with σ (2) . The transform matrix is M 2R .

12 13 14

⎡

15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

Fig. 2. Schematic diagram of space curve reconstruction and moving coordinate system.

Let σ (0) = [ O 1 ; x, y , z], then the moving coordinate system corresponding to O 1 and O 2 is σ (1) = [ O 1 ; a1 , b1 , c 1 ] and σ (2) = [ O 2 ; a2 , b2 , c 2 ]. Coordinate of O 2 can be obtained by coordinate transform.

⎡

37 38 39 40 41 42 43 44 45 46 47 48

⎡ ⎤

a2 x σ (0) → σ (2) : ⎣ b2 ⎦ = M 20 ⎣ y ⎦ z c2

(5)

⎡

1 ⎢0 ⎢ M 10 = ⎣ 0 0

0 0 1 0 0 1 0 0

⎤

0 0⎥ ⎥ 0⎦ 1

(6)

cos(αi ) − sin(αi ) ⎢ sin(αi ) cos(αi ) ⎢ M p1 = ⎣ 0 0 0 0

⎤

0 0 0 0⎥ ⎥ 1 0⎦ 0 1

(7)

49 50 51 52

Coordinate system σ ( Q ) = [ O 1 ; a Q , b Q , c Q ] can be obtained by turning coordinate system σ ( P ) about b P axis with angular θ1 . The transform matrix is M Q P .

⎡

53 54 55 56

MQ P

57

cos(θi ) ⎢ 0 =⎢ ⎣ sin(θi ) 0

0 1 0 0

− sin(θi ) 0 0 cos(θi ) 0

⎤

60 61

⎡

63 64 65 66

MRQ

cos(−αi +1 ) ⎢ sin(−αi +1 ) =⎢ ⎣ 0 0

85

dai , db i , dc i is the coordinate of O 2 in coordinate system it can be obtained by (11) and (12).

⎧ ⎪ ⎨dai = cos(αi ) ∗ (1 − cos(θi )/ki ki = 0 db i = sin(αi ) ∗ (1 − cos(θi )/ki ⎪ ⎩ dc i = sin(θi )/ki

σ (2) . And

86 87 88 89

(11)

90 91 92

⎧ ⎪ ⎨dai = 0 ki = 0 db i = 0 ⎪ ⎩ dc i = ds

94 95

(12)

96 97 98

ki is point i’s curvature in osculating plane πi . θi = ds is central angle of the curve segment between point i and i + 1. Thus, M 20 could be obtained.

99 100 101

0⎥ ⎥ 0⎦ 1

(8)

⎤

− sin(αi +1 ) 1 0 cos(−αi +1 ) 0 0 ⎥ ⎥ 0 1 0⎦ 0

M 20 = M 2R M R Q M Q P M P 1 M 10

(13)

According to equation (13), O 2 can be calculated. The rest discrete points can be calculated in the same way. And the reconstruction can be realized. The FBG detection units are comprised of four optical fibers which are distributed around the outside surface of a shape memory alloy (SMA) wire with an angle of 90 degree from each other. The FBG sensors are placed along the length of the SMA wire, as shown in Fig. 3(a). And the arrangement of the four fibers along the circumferential direction is shown in Fig. 3(b). A, B and A , B are in pairs. The shape reconstruction performance of such FBG detection unit is shown in Fig. 4. This FBG detection unit’s function is like commercially-available sensor, called ShapeTapeTM, but has more flexibility.

103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119

Coordinate system σ ( R ) = [ O 1 ; a R , b R , c R ] can be obtained by turning coordinate system σ ( Q ) about c Q axis with angular −α2 . The transform matrix is M R Q .

62

83 84

2.3. Aircraft wing reconstruction

58 59

82

(10)

102

New coordinate system σ ( P ) = [ O 1 ; a P , b P , c P ] can be obtained by turning coordinate system σ (1) about c 1 axis with angular α1 . The transform matrix is M P 1 .

⎡

81

93

As is shown in equation (5), σ (0) → σ (2) represents coordinate transform from σ (0) to σ (2) . M 20 represents coordinate transform matrix. By introducing intermediate transform matrix M 2R , M R Q , M Q P , M P and M 10 , the transformation process can be accomplished. While the initial point O 1 = (0, 0, 0),

35 36

⎤

M 2R

79 80

⎤

1 0 0 dai ⎢ 0 1 0 dbi ⎥ ⎥ =⎢ ⎣ 0 0 1 dc i ⎦ 0 0 0 1

78

0

1

(9)

120 121

The aircraft wing reconstruction can be considered as a 3 dimension space surface reconstruction using surface fitting and joining technique, as shown in Fig. 5. Firstly, the wing is decomposed into curves on space surface vertically. Then 3 dimension space surface reconstruction could be realized using smooth surface fitting method. Curve reconstruction on space surface could be implemented as shown in Fig. 6. Assume the origin of a micro arc sn is O n , the end is O n+1 , the corresponding curvature is kn and kn+1 . The corresponding coordinate is (xn , yn ) and (xn+1 , yn+1 ). αn and αn+1 is the corresponding tangent vector. ln is the chord length. θn is central

122 123 124 125 126 127 128 129 130 131 132

JID:AESCTE

AID:3876 /FLA

[m5G; v1.194; Prn:5/01/2017; 15:09] P.4 (1-9)

Z. Gao et al. / Aerospace Science and Technology ••• (••••) •••–•••

4

1

67

2

68

3

69

4

70

5

71

6

72

7

73

8

74

9

75

10

76

11

77

12

78

13

79

14

80

15

81

16

82

Fig. 4. Shape reconstruction performance of the FBG detection unit.

17

83

Fig. 7. Schematic diagram for coordinate data fusion.

18

84

19

85

20 21 22 23

While the distance between O n and O n+1 is small, the curvature can be considered linear to arc length

86

k(s) = M × s + N

(17)

88

A curve could be reconstructed using s1 ˜s2 s2 ˜s3 · · · , sn ˜sn+1 · · · .

90

24



25

kn+1 = M n × sn+1 + N n

27



28 29

Fig. 5. Schematic diagram of wing reconstruction.

M n = (kn+1 − kn )/(sn+1 − sn ) N n = (kn × sn+1 − kn+1 × sn )/(sn+1 − sn )

30 31

(18)

33

θ(s) =

34 35

1 2

(19)

37 38 39 40

(20)

49 50 51 52 53 54 55 56 57 58

angle. θn is the angle between x axis and αn , θn+1 is the angle between x axis and αn+1 . As shown in Fig. 6,

⎧ ⎪ ⎪θn = θn+1 − θn ⎪ ⎪ ⎪ ln = 2 · sin(θn /2)/kn , (kn = 0) ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ln = sn , (kn = 0) xn = 2 · ln · cos(θn − θn /2) ⎪ ⎪ ⎪ yn = 2 · ln · sin(θn − θn /2) ⎪ ⎪ ⎪ ⎪ ⎪ xn+1 = xn + xn ⎪ ⎪ ⎩ y n +1 = y n +  y n

(14)

k(s) =

62 63

66

dθ

(15)

ds

We can get

64 65

102 103 104 105 106

θ(s) =

k(s)ds

108 110 111 112 113 114 115 116 117 118 119 120 121

According to curvature definition,

60

To reconstruct the whole smart aircraft model, coordinate data fusion is required. Its process is shown in Fig. 7. Firstly, set up the fixed coordinate for the whole smart aircraft model ( O ; x, y , z). Then set up the corresponding coordinate ( O 1 ; x1 , y 1 , z1 ), ( O 1 ; x1 , y 1 , z1 ) for aircraft frame and ( O 2 ; x2 , y 2 , z2 ), ( O 2 ; x2 , y 2 , z2 ) for aircraft wing. L 1 , L 2 and L 3 is the distance between the coordinates and the horizontal plane. Using these coordinates data fusion could be done. Finally all the data can be converted into the fixed coordinate ( O ; x, y , z). 3. Active vibration control method

59 61

101

109

Fig. 6. Curve reconstruction on space surface.

44

48

100

107

43

47

96

99

· M × s2 + N × s + c

2.4. Coordinate data fusion

42

46

95

98

41

45

93

97

Then equation (14) can be used to calculate the corresponding coordinates of each FGB sensor, and curve reconstruction can be achieved on the space surface. And 3 dimension space surface reconstruction for the aircraft wing can be realized by smooth surface fitting.

36

92 94

Substitute equation (16) into (17), we can get:

32

89 91

kn = M n × sn + N n

26

87

(16)

The famous filter-x least mean square (FXLMS) algorithm which has been independently proposed by different researchers [26,4, 22], is an single input single output control algorithm developed for active noise control. While it is applied to multiple input multiple output active vibration control,the diagram is shown in Fig. 8 [17]. Here P ( z) is the primary path from the disturbance to PZT sensors. H ( z) is the secondary path from the PZT actuators to PZT ˆ is the identification model of the secondary path. W sensors. H is an finite impulse response (FIR) filter controller using WidrowHoff LMS algorithm. X (k) is the reference signal which is picked by

122 123 124 125 126 127 128 129 130 131 132

JID:AESCTE AID:3876 /FLA

[m5G; v1.194; Prn:5/01/2017; 15:09] P.5 (1-9)

Z. Gao et al. / Aerospace Science and Technology ••• (••••) •••–•••

5

1

67

2

68

3

69

4

70

5

71

6

72

7

73

8

74

9

75

10

76

11

77

Fig. 8. FXLMS algorithm for active vibration control.

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

78

ˆ . E (k) is residual reference PZT sensors. X (ˆk) is X (k) filtered by H signal. d(k) is vibration disturbance. B (k) is d(k) filtered by P ( z). T

E (k) = B (k) − H ( z) ∗ [ W X (k)]

(21)

Here k is time index. If there are M inputs and N outputs of the MIMO controller,

W (k) = [ W 1 (k), W 2 (k), · · · , W m (k), · · · , W M (k)] T

(22)

W m (k) = [ W m1 (k), W m2 (k), · · · , W mn (k), · · · , W mN (k)] T

(23)

Performance objective function can be defined as follows:

   J = E E (k) E (k) = E e 21 (k) + e 22 (k) + · · · + e 2M (k) 

T

(24)

The point at the minimum value of J (n) is corresponding to the following W :

30 31 32

1 W ∗ = R− Pf f

(25)

33

Here,

34

P f = E [ Xˆ (k)d(k)]

(26)

R f = E [ Xˆ (k) Xˆ T (k)]

(27)

35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57

P f is the cross-correlation vector between Xˆ (k) and d(k). R f is

the correlation matrix of Xˆ (k). Xˆ (k) is obtained by filtering Xˆ (k) ˆ . This optimum solution could be calculated from R f through H and P f , but it requires a large amount of calculation, which makes it is impossible for real time control. By using LMS algorithm, we can get the following MIMO FXLMS used for active vibration control.

W (k + 1) = W (k) − μ E (k) Xˆ T (k)

(28)

60

63 64 65 66

81 82 83

using the residual signal extracted directly from the vibrating structure using the error sensor rather than reference sensor. And for time varying process, an online identification of H ( z) should be performed. The block diagram of the proposed MIMO hybrid FXLMS active vibration control algorithm with online identification is shown in Fig. 9. Reference signal self-extraction and secondary path online identification is employed in this adaptive controller. In this controller, P D is a performance discriminator that determine identification process. Online identification is achieved by adding white noise signal to the controller output as identification input. X˜ (k) is control output. V (k) is the white noise signal that is involved into the identification process. The input–output relation can be expressed as:

⎤ ⎡ ⎡ ˜ X 1 (k) w (k) w 12 (k) ⎢ X˜ 2 (k) ⎥ ⎢ 11 ⎥ ⎢ w 21 (k) w 22 (k) ⎢ ⎢ . ⎥=⎣ ··· ··· ⎣ .. ⎦ w M1 (k) w M2 (k) ˜X M (k) ⎤ ⎡ x(k) ⎢ x(k − 1) ⎥ ⎥ ⎢ ×⎢ ⎥ .. ⎦ ⎣ . x(k − N + 1)

⎤

∗ H 11 ⎢ H∗ ⎢ 21

ˆ = H ⎣ ··· H ∗M1

∗ H 12 ∗ H 22

···

∗ H 1N ∗ ⎥ H 2N ⎥

···

H ∗M N

∗ H mn

··· ⎦

86 87 88 89 90 91 92 93 94 95 96 98 99 100 101 102 103 104

(31)

105 106 107

⎤

··· ···

85

97

· · · w 1N (k) · · · w 2N (k) ⎥ ⎥ ··· ··· ⎦ · · · w M N (k)

The identification process would end while the performance disˆ is shown as follows: criminator gives a negative signal. H

⎡

84

108 109 110 111

(32)

112 113

The aim of the MIMO adaptive controller is to find an optimal solution of W to minimize the vibration signal obtained through the piezoelectric sensors or FBG sensors. The proposed hybrid MIMO adaptive control algorithm can be summarized as

ˆ ( z)Y (k) + E (k) Bˆ (k) = H

(33)

121

2 − μ P f ( M + 2eq ) > 0

X (k) = Bˆ l (k)

(34)

123

Y (k) = W (k) X (k) + V (k)

(35)

E (k) = B (k) − H ( z)Y (k)

(36)

(29)

And it can be stated that, the sufficient condition for FXLMS algorithm is

61 62

80

Fig. 9. Proposed MIMO hybrid FXLMS active vibration control algorithm with online identification.

FXLMS algorithm is very tolerant of model identification errors of H ( z) [22]. Within the limit of slow adaptation, the adaptive filˆ (z). The convergence analysis of ter W could be commuted with H FXLMS algorithm given by [2] is as follows: While J (k) is a positive definite function, according to Lyapunov stability theorems, if for some μ, the difference of J (k) is negative. J (k) is a Lyapunov function and the whole control system is uniformly asymptotically stable in the sense of Lyapunov. As shown in [2], it can be concluded that J (k) is a Lyapunov function for

58 59

79

μ<

2

( M + 2eq ) P f

(30)

While a reference sensor is required by traditional filtered-X algorithm and it is impossible for a structure that is excited by unknown disturbance, the reference signal could be constructed

H ∗M2

114 115

W (k + 1) = W (k) − μ E (k) Xˆ (k)

(37)

ˆ ( z) V (k) Vˆ (k) = H

(38)

E s (k) = E (k) − Vˆ (k)

(39)

ˆ ( z) = Hˆ ( z) + β E s (k) V T (k) H

(40)

T

116 117 118 119 120 122 124 125 126 127 128 129 130 131 132

JID:AESCTE

AID:3876 /FLA

[m5G; v1.194; Prn:5/01/2017; 15:09] P.6 (1-9)

Z. Gao et al. / Aerospace Science and Technology ••• (••••) •••–•••

6

1

67

2

68

3

69

4

70

5

71

6

72

7

73

8

74

9

75

10

76

11

77

12

78

13

79

14

80

15

81

16

82

17

83

18

84

19

85

20

86

21

87

22

88

23

89

24

90

25

91

26 28 29 30 31 32 33 34 35 36

ˆ (z) = H (z), X (k) = B (k). If the step size of From Fig. 9, while H LMS algorithm is small, the adaptive filter W could be commuted ˆ (z). The proposed algorithm converges, while the optimal with H performance objective function should be obtain as n → ∞. μ is LMS step factor for control, β is LMS step factor for identification. If,

μ<

37 38 39

β<

2

42 43 44

(41)

( M + 2eq ) P f 2

(42)

( M + 2eq ) P f

40 41

Here, eq 

τs2 , σs2

τs2 

M N i =1

i =1

j H i2j ,

σs2 

M N i =1

i =1

H i2j .

J (k) would be a Lyapunov function. Under this sufficient condition, the proposed MIMO hybrid FXLMS active vibration control system is uniformly asymptotically stable in the sense of Lyapunov.

45 46

4. Experimental platform design

47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

92

Fig. 10. Monitor and control system diagram.

27

The experimental aircraft frame is made by aluminium alloy. The aircraft wing is made by epoxy resin board. The length of the experimental model is 1500 mm, the height is 160 mm, while the aircraft nose width is 500 mm, aircraft tail width is 350 mm. The experimental platform is constructed by aircraft model, FBG sensors, piezoelectric sensors and actuators, optical signal analyzer, piezoelectric charge amplifier, computer with AD card and DA card as well as other auxiliary measurement and control equipments. The designed monitor and control system is shown in Fig. 10. Four FBG cables each contains sixteen surface-bonded FBG sensors are fixed in the aircraft frame and another 16 FBG sensors are bonded on the surface of the airfoil. Meanwhile eight groups of PZT sensors and actuators are bonded on the aircraft surface. The FBG is manufactured by Shenzhen Sinomaster Investment Group, it should be designed according to the measurement range and the limitation of the optical signal analyzer. The PZT actuator type is PZT-5H, the PZT sensor type is P51, they are manufactured by Zibo Yuhai Electronic Ceramic Company. JZK-10 exciter is used to excite vibration. While the FBG sensors are used to obtain the vibration

93

information. And FONA-2008C optical signal analyzer is used to demodulate the FBG sensor signal, which can be processed by the shape reconstruction program. And the piezoelectric power amplifier is used to supply control voltage for the piezoelectric actuators. While piezoelectric charge amplifier is used to amplify the piezoelectric sensors’ signal, which is acquired by PCI-1712 AD card. The piezoelectric change amplifier type is YE5852A, manufactured by Sinocera Piezotronics, Inc. The sampling frequency of the control system is 1000 Hz. The gain of the PZT amplifier is 60. The photograph of the system is shown in Fig. 11. The software design diagram is shown in Fig. 12. Three threads are employed, two of them are focusing on the data acquisition and shape reconstruction, one is used for real time vibration control.

94 95 96 97 98 99 100 101 102 103 104 105 106 107 108

5. Experiment result

109 110

Experimental research is carried out for the smart structure aircraft. As static deformation is difficult to impose on the framework, only aircraft wing structure deformation reconstruction is done. Reconstruction effect of static wing deformation is shown in Fig. 13. Real-time reconstruction experiment is done to verify dynamic reconstruction effect of the shape reconstruction algorithm. While the model aircraft is excited continuously by the vibration exciter, side view and front view of the aircraft model in the first-order vibration mode is shown in Fig. 14. Relative root-mean-square error E R R M S E can be defined to evaluate the reconstruction performance.



E RRMSE =

n

i =1



wi − wi n

2  /

n 

 w i /n

111 112 113 114 115 116 117 118 119 120 121 122 123

× 100%

(43)

i =1

ˆ is the reconHere w is the actual deformation displacement, w structed deformation displacement. The relative root-mean-square error under different vibration mode can be shown in Table 1. The reconstruction effect would be deteriorated as the exciting frequency rises, the reason is the real time computing of the algorithm could not be guaranteed as the computer’s performance

124 125 126 127 128 129 130 131 132

JID:AESCTE AID:3876 /FLA

[m5G; v1.194; Prn:5/01/2017; 15:09] P.7 (1-9)

Z. Gao et al. / Aerospace Science and Technology ••• (••••) •••–•••

7

1

67

2

68

3

69

4

70

5

71

6

72

7

73

8

74

9

75

10

76

11

77

12

78

13

79

14

80

15

81

16

82

17 18

83

Fig. 11. Photo of monitor and control system.

84

19

85

20

86

21

87

22

88

23

89

24

90

25

91

26

92

27

93

28

94

29

95

30

96

31

97

32

98

33

99

34

100

35

101

36

102

37

103

38

104

39 40

105

Fig. 12. Software design diagram of the active monitoring and vibration control system.

106

41

107

42

108

43

109

44

110

45

111

46

112

47

113

48

114

49

115

50

116

51

117

52

118

53

119

54 55

Fig. 13. Static wing reconstruction.

120 121

56

122

57

123

58

124

59

125

60

126

61

127

62

128

63

129

64

130

65 66

131

Fig. 14. Front view and side view of real-time reconstruction result.

132

JID:AESCTE

AID:3876 /FLA

[m5G; v1.194; Prn:5/01/2017; 15:09] P.8 (1-9)

Z. Gao et al. / Aerospace Science and Technology ••• (••••) •••–•••

8

1 2 3 4

67

Table 1 Root-mean-square error under different vibration mode.

68

Exciting frequency (Hz)

5

10

20

30

40

RRMES %

3.26

3.75

4.68

7.31

14.7

69 70

5

71

6

72

7

73

8

74

9

75

10

76

11

77

12

78

13

79

14

80

15

81

16

82

Fig. 17. Overall control performance.

17 18

84

19 20 21 22 23

Fig. 15. Control performance measured by the first sensor on the framework.

24 25 26 27 28 29 30 31 32 33 34 35 36

CPI is negative. While the controller failed to suppress the vibrating system, CPI is positive. Another four algorithms proposed in [7,13,12,11] are implemented on the same experiment platform, as well as FXLMS algorithm. In [7], it is a FXLMS algorithm with online secondary-path modeling. In [13], it is a FXLMS algorithm with vibration residual signal extracted directly from the vibrating structure. In [12], it is a FULMS algorithm. In [11], it is a normalized FXLMS algorithm. The CPI of each sensor can be shown in Table 2. As shown in Table 2, the control performance index of the proposed is much better than the four algorithms proposed in [7,13, 12,11] and FXLMS algorithm. In actual vibration control situations, using normalized LMS algorithm [11] could improve the control performance of FXLMS. And using IIR structure [12] may cause control system crash. Just using online identification method [7], may worse the control performance. And vibration residual signal extracted directly from the vibrating structure [13] plays an important role in vibration control tests.

37 39

Fig. 16. Control performance by the eighth sensor on the left wing.

41 42

45 46 47 48 49 50 51 52 53 54 55 56 57 58 59

could not process so many data in time. But the overall reconstruction performance is still acceptable. 8 groups of piezoelectric patches are used to control the whole aircraft’s vibration response. While the model aircraft is excited continuously by the vibration exciter, vibration control experiments are done using the proposed adaptive control algorithm. The experiment results are shown in Fig. 15, Fig. 16, and Fig. 17. Fig. 15 shows the signal time history of the piezoelectric sensor attached on the model aircraft framework. Fig. 16 shows the sensor signal time history attached on the model aircraft wings, and Fig. 17 shows the overall vibration suppression performance. Vibration of the model aircraft is suppressed to a great extent using proposed method. The control performance on the wing is better. The reason is the wing is made using epoxy resin board, which makes it more flexible, and easier to control. To compare the control performance of different control algorithms, a control performance index (CPI) is defined as follows:

N

60 61 62 63 64 65 66

86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 104 105

40

44

85

103

6. Conclusion

38

43

83

CPI =

2

|d(i )| 20log10 i =1 2 N i =1 |e (i )|

(44)

d(i ) is the output of the error sensor without active vibration control. And e (i ) is the output of the error sensor under active vibration control. While the vibration of the system is suppressed,

Using fiber Bragg grating (FBG) and piezoelectric ceramics, a smart aircraft model is constructed. And a kind of 3 dimension shape reconstruction method which is based on orthogonal curvature and recursive curve fitting technique and an improved MIMO hybrid FXLMS algorithm with online identification function are proposed for active monitoring and vibration control of the aircraft model. Experimental platform is constructed to test the proposed methods. The results show that the proposed methods are effective with good reconstruction effect and good vibration control performance. To the best knowledge of the authors, it is the first time that active shape monitoring of a complex object is realized using nonvisual reconstruction method based on embedded FBG arrays, and the first time that a multi input multi output stable vibration control with online identification is implemented using distributed PZT patches for an experimental aircraft model structure. In this work, the measurement is carried out in a laboratory environment, so the structure is isothermal. While the temperature changes at each measurement point, temperature compensation FBG should be bonded beside each measurement FBG to make equation (2) true [24]. Due to our experiment restriction, all the experiments were carried out in laboratory environment. The PZT and FBG used in our experiment can not be applied to real aeronautical environment or aerospace environment directly. The readers who are interested should use special high temperature FBG and PZT for engineering implementation.

106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

JID:AESCTE AID:3876 /FLA

[m5G; v1.194; Prn:5/01/2017; 15:09] P.9 (1-9)

Z. Gao et al. / Aerospace Science and Technology ••• (••••) •••–•••

1 2 3 4 5 6 7 8 9

9

67

Table 2 Control performance index for each sensor. Sensor CPI CPI CPI CPI CPI CPI

(−dB) (−dB) (−dB) (−dB) (−dB) (−dB)

of of of of of of

proposed method [7] [13] [12] [11] FXLMS

68

1

2

3

4

5

6

7

8

14.9 5.5 14.2 4.5 6.8 6.5

15.8 8.2 13.9 4.7 8.6 8.7

15.4 8.1 14.5 5.8 8.7 8.8

15.5 8.6 15.1 5.9 9.8 8.9

16.4 8.4 15.7 3.4 9.2 9.4

17.7 8.9 16.8 5.5 9.3 8.5

19.8 8.6 17.5 inf 12.5 8.4

19.9 8.7 19.1 inf 13.6 8.3

10 11

None declared.

14 15

References

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

70 71 72 73 74 75 76

Conflict of interest statement

12 13

69

[1] F. Abdi, K. Bowcutt, C. Godines, J. Bayandor, Collision provoked failure sequencing in space reentry vehicles, Comput. Struct. 89 (11–12) (2011) 930–939. [2] I.T. Ardekani, W. Abdulla, Theoretical convergence analysis of fxlms algorithm, Signal Process. 90 (12) (2010) 3046–3055. [3] T.K. Barlas, G.A.M. van Kuik, Review of state of the art in smart rotor control research for wind turbines, Prog. Aerosp. Sci. 46 (1) (2010) 1–27. [4] J.C. Burgess, Active adaptive sound control in a duct: a computer simulation, J. Acoust. Soc. Am. 70 (3) (1981) 715–726. [5] R.A. Canfield, S.D. Morgenstern, D.L. Kunz, Alleviation of buffet-induced vibration using piezoelectric actuators, Comput. Struct. 86 (3–5) (2008) 281–291. [6] G. Cazzulani, S. Cinquemani, L. Comolli, A. Gardella, F. Resta, Vibration control of smart structures using an array of fiber Bragg grating sensors, Mechatronics 24 (4) (2014) 345–353. [7] S.-C. Chan, Y. Chu, Performance analysis and design of fxlms algorithm in broadband anc system with online secondary-path modeling, IEEE Trans. Audio Speech Lang. Process. 20 (3) (2012) 982–993. [8] C.E. Cockrell, S.R. Davis, K. Robinson, M.L. Tuma, G. Sullivan, NASA crew launch vehicle flight test options, Acta Astronaut. 61 (1–6) (2007) 438–449. [9] I. Doré Landau, M. Alma, A. Constantinescu, J.J. Martinez, M. No, Adaptive regulation–rejection of unknown multiple narrow band disturbances (a review on algorithms and applications), Control Eng. Pract. 19 (10) (2011) 1168–1181. [10] Y. Fu, H. Di, R. Liu, Light intensity modulation fiber-optic sensor for curvature measurement, Opt. Laser Technol. 42 (4) (2010) 594–599. [11] M.T.A. Hindustani, P.S.S. Rao, S. Raja, C. Maheshan, Performance analysis and comparison of fxlms and nfxlms algorithms for active vibration control, in: 2015 International Conference on Applied and Theoretical Computing and Communication Technology, iCATccT, IEEE, 2015, pp. 54–57. [12] Q. Huang, J. Luo, H. Li, X. Wang, Analysis and implementation of a structural vibration control algorithm based on an IIR adaptive filter, Smart Mater. Struct. 22 (8) (2013) 085008. [13] Q. Huang, X. Zhu, Z. Gao, S. Gao, E. Jiang, Analysis and implementation of improved multi-input multi-output filtered-x least mean square algorithm for active structural vibration control, Struct. Control Health Monit. 20 (11) (2013) 1351–1365.

[14] S. Hurlebaus, L. Gaul, Smart structure dynamics, Mech. Syst. Signal Process. 20 (2) (2006) 255–281. [15] C.H.M. Jenkins, U.A. Korde, Membrane vibration experiments: an historical review and recent results, J. Sound Vib. 295 (3–5) (2006) 602–613. [16] H.-I. Kim, J.-H. Han, H.-J. Bang, Real-time deformed shape estimation of a wind turbine blade using distributed fiber Bragg grating sensors, Wind Energy 17 (9) (2014) 1455–1467. [17] S.M. Kuo, D.R. Morgan, Active noise control: a tutorial review, Proc. IEEE 87 (6) (1999) 943–973. [18] F. Leleu, P. Watillon, J. Moulin, A. Lacombe, P. Soyris, The thermo-mechanical architecture and tps configuration of the pre-x vehicle, Acta Astronaut. 56 (4) (2005) 453–464. [19] P. Li, Y. Liu, J. Leng, A new deformation monitoring method for a flexible variable camber wing based on fiber Bragg grating sensors, J. Intell. Mater. Syst. Struct. 25 (13) (2014) 1644–1653. [20] I. Lopez, N. Sarigul-Klijn, A review of uncertainty in flight vehicle structural damage monitoring, diagnosis and control: challenges and opportunities, Prog. Aerosp. Sci. 46 (7) (2010) 247–273. [21] A.R. Mehrabian, A. Yousefi-Koma, A novel technique for optimal placement of piezoelectric actuators on smart structures, J. Franklin Inst. 348 (1) (2011) 12–23. [22] D. Morgan, An analysis of multiple correlation cancellation loops with a filter in the auxiliary path, in: Acoustics, Speech, and Signal Processing, IEEE International Conference on ICASSP ’80, vol. 5, Apr 1980, pp. 457–461. [23] A. Panopoulou, T. Loutas, D. Roulias, S. Fransen, V. Kostopoulos, Dynamic fiber Bragg gratings based health monitoring system of composite aerospace structures, Acta Astronaut. 69 (7–8) (2011) 445–457. [24] Y.-J. Rao, Recent progress in applications of in-fibre Bragg grating sensors, Opt. Lasers Eng. 31 (4) (1999) 297–324. [25] G. Song, V. Sethi, H.-N. Li, Vibration control of civil structures using piezoceramic smart materials: a review, Eng. Struct. 28 (11) (2006) 1513–1524. [26] B. Widrow, J.R. Glover, J. McCool, J. Kaunitz, C. Williams, R. Hearn, J. Zeidler, J. Eugene Dong, R. Goodlin, Adaptive noise cancelling: principles and applications, Proc. IEEE 63 (12) (Dec 1975) 1692–1716. [27] J. Xu, H. Zhang, X. Zhu, L. Li, P. Ding, Curve surface fitting based on an improved genetic algorithm, in: 6th International Congress on Image and Signal Processing, CISP, vol. 2, IEEE, 2013, pp. 747–752. [28] L. Y.Y, Hypersensitivity of strain-based indicators for structural damage identification: a review, Mech. Syst. Signal Process. 24 (3) (2010) 653–664.

77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109

44

110

45

111

46

112

47

113

48

114

49

115

50

116

51

117

52

118

53

119

54

120

55

121

56

122

57

123

58

124

59

125

60

126

61

127

62

128

63

129

64

130

65

131

66

132