Active vibration control of flexible cantilever plates using piezoelectric materials and artificial neural networks

Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Active vibration control of flexible cantilever plates using piezoelectric materials and artificial neural networks Osama Abdeljaber a, Onur Avci a, Daniel J. Inman b,n a b

Department of Civil and Architectural Engineering, Qatar University, P.O. Box 2713, Doha, Qatar Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI, USA

a r t i c l e i n f o

abstract

Article history: Received 18 March 2015 Received in revised form 14 September 2015 Accepted 23 October 2015 Handling Editor: L. Huang

The study presented in this paper introduces a new intelligent methodology to mitigate the vibration response of flexible cantilever plates. The use of the piezoelectric sensor/ actuator pairs for active control of plates is discussed. An intelligent neural network based controller is designed to control the optimal voltage applied on the piezoelectric patches. The control technique utilizes a neurocontroller along with a Kalman Filter to compute the appropriate actuator command. The neurocontroller is trained based on an algorithm that incorporates a set of emulator neural networks which are also trained to predict the future response of the cantilever plate. Then, the neurocontroller is evaluated by comparing the uncontrolled and controlled responses under several types of dynamic excitations. It is observed that the neurocontroller reduced the vibration response of the flexible cantilever plate significantly; the results demonstrated the success and robustness of the neurocontroller independent of the type and distribution of the excitation force. & 2015 Elsevier Ltd. All rights reserved.

1. Introduction Engineers are employing lightweight materials and thinner structures in their designs in an attempt to reduce the construction cost and increase the efficiency of the design. However, flexible structures with smaller damping and lighter masses are more vulnerable to dynamic loading induced by environment conditions and human activities [1,2]. The control of vibrations due to the dynamic forces has always been a concern for engineers. Therefore, it is necessary to design and implement efficient vibration control techniques to enhance the serviceability and extend the life-cycle of structures [3]. Vibration control systems are classified according to their dynamics and energy requirements into passive, semi-active, and active systems [4]. Active vibration control systems utilize a network of sensors and actuators to measure the structural response and produce control forces in a prescribed manner to dissipate the energy and reduce the response of the host structure [5]. Flexible plates, particularly cantilever plates, are among the most commonly used flexible members in aerospace structures and aircrafts. Consequently, extensive research has been conducted on vibration control of flexible plates. One of the most interesting and feasible active control configurations in this field includes the implementation of the excellent sensing and actuating properties exhibited by piezoelectric materials [6,7].

n

Corresponding author. E-mail addresses: [email protected] (O. Abdeljaber), [email protected] (O. Avci), [email protected] (D.J. Inman).

http://dx.doi.org/10.1016/j.jsv.2015.10.029 0022-460X/& 2015 Elsevier Ltd. All rights reserved.

Please cite this article as: O. Abdeljaber, et al., Active vibration control of flexible cantilever plates using piezoelectric materials and artificial neural networks, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.10.029i

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O. Abdeljaber et al. / Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Several researchers have utilized piezoelectric materials in the form of piezoelectric patches as both actuators and sensors to develop different active control schemes for vibration control of flexible plates. The most pronounced difference between these schemes is the control algorithm that is implemented to determine the appropriate control voltage signal applied on the piezoelectric actuators. Caruso et al. [8] have studied the active vibration control of cantilever plates employing multiple piezoelectric sensor/actuator pairs. They applied H 2 control techniques to create multiple controllers to control the voltage applied on the actuators. Similarly, Qiu and Haraguchi [9] employed piezoelectric patches along with a controller that utilizes finite impulse response filter and the filtered-X LMS algorithm to control the response of flexible plates. Sadek et al. [10] used piezoelectric patch actuators for active control of simply supported flexible plates. An optimal control law was derived explicitly using the maximum principle theory to compute the voltage applied on the actuators. In an experimental study, Qiu et al. [11] investigated the optimal placement of piezoelectric sensors and actuators for active control of a cantilever plate. After that, they combined positive position feedback with a PD controller to suppress the plate's response. The efficiency of this control methodology was demonstrated analytically and experimentally. In another study [12], the same authors have integrated piezoelectric patches with gyroscope sensor to control bending and torsional vibrations of cantilever plates. Sensors and actuators were optimally placed such that the vibration modes can be decoupled. A discrete-time sliding mode variable structure control algorithm was used to drive the piezoelectric actuators. Additionally, other control algorithms have been utilized for active control of plates using piezoelectric actuators such as proportional iterative learning algorithm [13] and nonlinear fuzzy control [14].

1.1. Neural networks for active vibration control Artificial neural networks are black-box models consisting of processing units interconnected according to an architecture that is based on human's central nervous system [15]. Neural networks have been utilized in several areas of engineering applications such as medical diagnoses, computer vision, pattern recognition, and speech pronunciation. In civil, mechanical and aerospace engineering, neural networks have been applied in design optimization, structural health monitoring, structural system identification, and finite element mesh generation [16]. Neural networks are also widely used in the vibration control field. Several applications of neural networks for active and semi-active vibration control of civil and mechanical systems can be found in the literature. For instance, Bani-Hani and Sheban [17] have developed a neural network based controller (i.e. neurocontroller) for semi-active base isolation of a frame structure equipped with magnetorheological (MR) dampers. The neurocontroller was trained based on an LQG controller to compute the optimal voltage command applied on the MR dampers. In another analytical study, Xu et al. [18] used neural networks for active control of a hypothetical cable-stayed bridge. The stay cables were modeled as active tendons attached to actuators that provide the control forces. The force generated by each single actuator was controlled by a decentralized neurocontroller. Recently, neural networks have been used in applications related to active control of flexible plates using piezoelectric sensors and actuators. Qiu et al. [19] have utilized a proportional-derivative (PD) controller for vibration suppression of a smart beam consisting of a cantilever flexible beam featuring a number of piezoelectric sensor/actuator pairs. A backpropagation neural network (BPNN) was used to update the parameters of the PD controller online. This BPNN-PD algorithm was applied to compute the optimal voltage signal applied on the piezoelectric actuator. Similarly, Mohit et al. [20] studied the control of a cantilever plate using a neurocontroller and piezoelectric patches. This neurocontroller was trained and tuned based on an LQR controller. Clearly, the aforementioned two studies have used control algorithms that integrate neural networks with conventional controllers (PD and LQR controllers). Alternatively, there are a few intelligent control algorithms available in the literature that depend only on neural networks and do not require other controllers such as model-reference neural control [21], neural network predictive control [22], and NARMA-L2 algorithm [23]. Another interesting example here is the neural network based control algorithm which has been developed by Ghaboussi and Joghataie [24] and used for active control of a multi-story shear frame using hydraulic actuators. This algorithm was improved later by Bani-Hani [25] and used for the same application. As will be explained in Section 3, this algorithm requires training a neurocontroller based on a set of emulator neural networks that are designed to predict the future response of the controlled system. The study presented in this paper utilizes a modified version of the abovementioned neural network based control algorithm to introduce a new intelligent methodology for vibration suppression in smart cantilever plates. Based on a set of emulator neural networks, a neurocontroller is designed to compute the optimal voltage applied on the piezoelectric patches. In order to formulate the model required for simulations and controller design, a finite difference method is applied on the equations governing the response of a cantilever plate under the effect of both piezoelectric patches and external excitations. Finally, numerical simulations are carried out to verify the model and examine the efficiency of the proposed active control technique.

Please cite this article as: O. Abdeljaber, et al., Active vibration control of flexible cantilever plates using piezoelectric materials and artificial neural networks, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.10.029i

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2. Problem formulation According to the classical theory of thin plates, the governing equation of a rectangular plate under the effect of a timevarying dynamic load function P ðx; y; t Þ is given as [26] h4 i 4 ð 2 Þ Þ w x;y;t Þ ∂4 wðx;y;t Þ x;y;t Þ þ 2∂ ∂x þ ρh ∂ w∂tðx;y;t D ∂ w∂xðx;y;t þC s ∂wð∂t 4 2 ∂y2 þ 2 ∂y4 ¼ P ðx; y; t Þ

(1)

where D is the flexural rigidity of the plate, wðx; y; t Þ represents the deflection map of the plate at time t, C s is the structural damping operator, ρ is the density of the plate's material, and h denotes the thickness of the plate. The internal moments of the plate can be written in terms of the deflection as follows:  2  ∂ wðx; y; t Þ ∂2 wðx; y; t Þ þ v (2.a) M x ðx; y; t Þ ¼  D ∂x2 ∂y2  2  ∂ wðx; y; t Þ ∂2 wðx; y; t Þ þ v M y ðx; y; t Þ ¼  D ∂y2 ∂x2 M xy ðx; y; t Þ ¼  Dð1  vÞ

(2.b)

∂2 wðx; y; t Þ ∂x∂y

(2.c)

Therefore, Eq. (1) can be rewritten in terms of the internal moments as ∂2 M xy ðx;y;t Þ ∂2 M y ðx;y;t Þ ∂2 Mx ðx;y;t Þ þ 2 ∂x∂y þ ∂y2 ∂x2

x;y;t Þ  C s ∂wð∂t  ρh ∂

2

wðx;y;t Þ ∂t 2

¼  P ðx; y; t Þ

(3)

When N piezoelectric transducers are perfectly bonded to the plate, they will produce moments that will affect the internal moments of the plate. Since the piezoelectric patches are typically much smaller and lighter than the host plate, their effects on the dynamics of the plates are assumed to be negligible. Therefore, only the first 3 terms of the left hand side of Eq. (3) are affected by the addition of the patches. This can be described as ∂2 ðM xy mxy Þ ∂2 ðM y my Þ ∂2 ðM x mx Þ þ2 þ ∂x∂y ∂y2 ∂x2

x;y;t Þ  C s ∂wð∂t  ρh ∂

2

wðx;y;t Þ ∂t 2

¼  P ðx; y; t Þ

(4)

where mx ðx; y; t Þ, my ðx; y; t Þ, and mxy ðx; y; t Þ represent the bending and twisting moments produced by the N piezoelectric patches. Eq. (4) can be simplified as Þ Þ ρh ∂ w∂ðtx;y;t þ C s ∂wð∂x;y;t þD∇4 wðx; y; t Þ 2 t 2

Þ ¼ P ðx; y; t Þ  ∂ m∂xxðx;y;t 2 2 2

∂2 mxy ðx;y;t Þ ∂2 m ðx;y;t Þ  ∂yy2 ∂x∂y

(5)

∂ ∂ ∂ where the term ∇4 is a differential operator equivalent to ∂x 4 þ2∂x∂y þ ∂y4 . According to Qiu et al. [11], the moments produced by N piezoelectric patches having the same properties and thickness under a unified voltage signal can be computed using the following equations. 4

N

i¼1

4

mx ðx; y; t Þ ¼ C 0

d31 V ðt ÞRðx; yÞ hPZT

(6a)

my ðx; y; tÞ ¼ C 0

d32 V ðt ÞRðx; yÞ hPZT

(6b)

d36 V ðt ÞRðx; yÞ ¼ 0 hPZT

(6c)

mxy ðx; y; t Þ ¼ C 0

Rðx; yÞ ¼ Σ

2

h    ih    i H y  yi1  H y  yi2 H x  xi1  H x  xi2

(7)

where d31 , d32 , and d36 ¼ 0 are the piezoelectric material strain constants; hPZT is the thickness of the patches; H is the Heaviside function; xi1 and yi1 denote the coordinates of the lower left corner of the ith piezoelectric patch; xi2 and yi2 denote the coordinate of the upper right corner of the ith piezoelectric patch, V ðt Þ is the voltage applied at the current time step, Please cite this article as: O. Abdeljaber, et al., Active vibration control of flexible cantilever plates using piezoelectric materials and artificial neural networks, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.10.029i

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a

b L H

Fig. 1. Original nodes (black) and additional nodes (gray).

a

b L H

Set 1

Set 2

Set 3

Set 4

Set 5

Set 6

Fig. 2. Node sets.

and the terms C 0 is given as 1 1 þ νPZT Eh K 6 1  ν 1 þ ν  ð1 þ νPZT ÞK 2

C0 ¼ 

(8)

where ν is the Poisson ratio of the plate's material, νPZT is the is the Poisson ratio of the patch's material, E is the modulus of elasticity of the plate, and K is given by the following equation: K¼

EPZT 1  v2 1:5hPZT hðh þhPZT Þ   2 E 1  v2PZT 2 0:125h3 þh3 þ1:5h  h PZT

(9)

PZT

where EPZT is the modulus of elasticity of the patch's material. Next, the objective is to utilize the finite difference method to obtain a solution for Eq. (5) in continuous state space form. This solution will address the case of cantilever plates only since it is the only case focused in this study herein. 2.1. Finite difference solution The first step toward applying finite difference method is to divide the plate into a finite mesh of nodes. The horizontal distance between two adjacent nodes is denoted by H while the vertical distance is denoted by L. Also, in order to express the boundary conditions of the cantilever plate in finite difference form, additional nodes should be added outside the plate, as shown in Fig. 1. Next, the nodes are divided into six sets as shown in Fig. 2. Each set is assigned by a number of equations as described in the next subsections. 2.1.1. Set 1 nodes Each node in this set will be assigned by one equation which can be obtained by translating Eq. (5) terms into their finite difference approximation according to the following formulas.  ∇4 wði;j;t Þ ffi

6

6

H

L

þ 4

þ 4

# " # " #   " w wði1;j1;t Þ þ wði1;j þ 1;t Þ wði;j þ 2;t Þ þ wði;j2;t Þ ði;j1;t Þ þ wði;j þ 1;t Þ 4 4 2 1 þ  w þ þ ð i;j;t Þ H 2 L2 H4 H 2 L2 þ wði1;j;t Þ þ wði þ 1;j;t Þ H2 L2 þwði þ 1;j1;t Þ þ wði þ 1;j þ 1;t Þ H 4 þ wði2;j;t Þ þ wði þ 2;j;t Þ 8

(10) Please cite this article as: O. Abdeljaber, et al., Active vibration control of flexible cantilever plates using piezoelectric materials and artificial neural networks, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.10.029i

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 ∂2 mx ði;j;tÞ C 0 d31 ffi V ðt Þ Rði;j1Þ  2Rði;jÞ þ Rði;j þ 1Þ 2 ∂x2 hPZT H

(11)

 ∂2 my ði; j; t Þ C 0 d32 ffi V ðt Þ Rði1;jÞ  2Rði;jÞ þ Rði þ 1;jÞ ∂y2 hPZT L2

(12)

∂2 mxy ¼0 ∂x∂y

(13)

where i denotes the index of the row in which the node is located, j denotes the index of the column, t represents the current time step. Substituting the Eqs. (10)–(13) into Eq. (5), the following is obtained: 9 8h i 6 6 8 > > > > 4 þ 4 þ 2 2 wði;j;t Þ þ > > L H L H > > > > " # > > > > h i > > þ w w ð i;j1;t Þ ð i;j þ 1;t Þ > > = < 4  4 þ 4 2 2 ρhw€ ði;j;tÞ þ C s w_ ði;j;tÞ þ H H L þwði1;j;tÞ þ wði þ 1;j;tÞ > " # " #> > > > > > > wði1;j1;tÞ þwði1;j þ 1;tÞ wði;j þ 2;t Þ þ wði;j2;tÞ > > > > 2 1 > > þ > > 2 2 4 > H þwði2;j;t Þ þ wði þ 2;j;t Þ > ; : H L þwði þ 1;j1;t Þ þ wði þ 1;j þ 1;tÞ ¼ F ðt ÞP ði;jÞ  V ðt ÞZ ði;jÞ Z ði;jÞ ¼

C 0 d32  C 0 d31  Rði;j1Þ  2Rði;jÞ þ Rði;j þ 1Þ  Rði1;jÞ  2Rði;jÞ þ Rði þ 1;jÞ hPZT H 2 hPZT L2

(14) (15)

2.1.2. Set 2-to-Set 6 nodes The nodes constituting Set 2 are simply the nodes along the fixed edge of the cantilever plate. The equation of each node in this set is straightforward since the displacement is restrained at the fixed edge: wði;j;tÞ ¼ 0

(16)

The nodes corresponding to Set 3 are the extra nodes to the left of the fixed edge of the plate. To obtain the equation of these nodes, the rotation around the fixed edge is restricted per boundary conditions. Therefore, the equation of each node in this set can be written as wði;j;tÞ ¼ wði;j þ 2;tÞ

(17)

Meanwhile, Set 4 includes the unsupported nodes along the upper and lower free edges. The boundary conditions at these edges are given as  2  ∂ wðx; y; t Þ ∂2 wðx; y; t Þ M y ðx; y; t Þ ¼  D þ v ¼0 (18) ∂y2 ∂x2 V y ðx; y; t Þ ¼  D

 3  ∂ wðx; y; t Þ ∂3 wðx; y; t Þ ¼0 þ ð2  vÞ ∂y3 ∂x2 ∂y

(19)

Therefore, each node in this set will be assigned by two equations which correspond to the finite difference approximations of Eq. (17) and (18). These equations are given as  ð2  2υÞwði;j;tÞ þ υ wði;j1;tÞ þwði;j þ 1;tÞ þ wði1;j;tÞ þwði þ 1;j;tÞ ¼ 0 (20) h i  ð2υ  6Þ wði1;j;t Þ  wði þ 1;j;tÞ þ ð2  υÞ wði1;j1;t Þ þ wði1;j þ 1;tÞ  wði þ 1;j1;tÞ  wði þ 1;j þ 1;tÞ þwði2;j;t Þ  wði þ 2;j;tÞ ¼ 0

(21)

Set 5 includes the nodes along the free edge on right hand side. The boundary conditions along this edge are given as  2  ∂ wðx; y; t Þ ∂2 wðx; y; t Þ Mx ðx; y; t Þ ¼  D þv ¼0 (22) 2 2 ∂x ∂y  3  ∂ wðx; y; t Þ ∂3 wðx; y; t Þ þ ð2  vÞ ¼0 V x ðx; y; t Þ ¼  D ∂x3 ∂x∂y2

(23)

Therefore, each node in this set will be assigned by two equations which corresponds to the finite difference approximations of Eqs. (21) and (22). These equations are given as  ð2  2υÞwði;j;tÞ þ υ wði1;j;tÞ þwði þ 1;j;tÞ þ wði;j1;tÞ þwði;j þ 1;tÞ ¼ 0 (24) h i  ð2υ  6Þ wði;j þ 1;tÞ  wði;j1;tÞ þ ð2  υÞ wði1;j þ 1;tÞ þ wði þ 1;j þ 1;tÞ  wði1;j1;tÞ  wði þ 1;j1;tÞ þwði;j þ 2;tÞ  wði;j2;tÞ ¼ 0

(25)

Please cite this article as: O. Abdeljaber, et al., Active vibration control of flexible cantilever plates using piezoelectric materials and artificial neural networks, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.10.029i

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Finally, Set 6 includes the two unsupported corner nodes only. These nodes must satisfy the following equation [27] ∂2 wðx; y; t Þ ¼0 R ¼ 2Dð1  υÞ ∂x∂y

(26)

The finite difference translation for Eq. (25) can be written as wði1;j þ 1;t Þ  wði1;j1;t Þ þ wði þ 1;j1;tÞ  wði þ 1;j þ 1;tÞ ¼ 0

(27)

2.2. State-space representation Each node in Set 1 is assigned by an index from 1 to n, where n is the total number of nodes in Set 1. Then, the value of Rðx; yÞ is computed at each node according to Eq. (7). Next, these values are substituted in Eq. (15) to compute the value of Z at each node. After that, each node in this set is assigned by Eq. (14) which relates the motion of the node to the properties of the plates, the displacements of the adjacent nodes, the mesh size, the applied load, the applied voltage across the PZT patches, and the Z values at this node and also the neighboring nodes. Additionally, the nodes in the Sets 2–6 are assigned by indices from n þ 1 to nt , where nt is the total number of nodes. The displacement of each node in these sets is written in terms of the displacements of the adjacent nodes according to the equation(s) of the set in which the node is located as described above. This procedure results in a linear system which can be written as 2

"

ρhIðnnÞ 0ðna nÞ

€1 3 w 6 w 7 6 €2 7 6 7 #6 ⋮ 7 " 7 CðnnÞ 0ðnna Þ 6 6 w 7 6 € n 7þ 0 0ðna na Þ 6 7 ðna nÞ 6w 7 € 6 nþ1 7 6 ⋮ 7 4 5 € nt w

2

0ðnna Þ 0ðna naÞ

2

P1

_1 3 w 6 w 7 6 _2 7 6 7 #6 ⋮ 7 " 6 7 K11ðnnÞ 6 w 7 6 _ n 7þ K21ðna nÞ 6 7 _ 6w 7 6 nþ1 7 6 ⋮ 7 4 5 _ nt w

3

2

Z1

2

w1

3

6 w 7 2 7 6 6 7 #6 ⋮ 7 6 7 K12ðnna Þ 6 7 6 wn 7 K22ðna na Þ 6 7 6 wn þ 1 7 6 7 6 ⋮ 7 4 5 wnt

3

6P 7 6Z 7 6 27 6 27 6 7 6 7 6 ⋮ 7 6 ⋮ 7 6 7 6 7 6 7 6 7 ¼ F ðt Þ6 P n 7  V ðt Þ6 Z n 7 6 7 6 7 6 0 7 6 0 7 6 7 6 7 6 7 6 7 4 ⋮ 5 4 ⋮ 5 0 0

(28)

where I denotes the identity matrix, 0 denotes zero matrix, and na ¼ nt  n is the total number of additional nodes. Then, static condensation is conducted to reduce the number of degrees of freedom in the above system by condensing out the dependent equations (i.e. the equations of the additional nodes) [28]. The reduced system is obtained according to the following equations: MðnnÞ ¼ ρhIðnnÞ " T¼ " T

Knn ¼ T 

(29) #

IðnnÞ

(30)

K221  K21

K11ðnnÞ

K12ðnna Þ

K21ðna nÞ

K22ðna na Þ

# T

3 2 _ 3 2 3 2 3 2 3 €1 w1 w1 P1 Z1 w 6 € 7 6w 7 6 7 6 7 6 7 _ w P Z 6 w2 7 6 27 6 27 6 27 6 27 7 MðnnÞ 6 7 þKnn 6 7 ¼ F ðt Þ6 7  V ðt Þ6 7 6 ⋮ 7 þ Cnn 6 ⋮ ⋮ ⋮ ⋮ 4 5 4 5 4 5 4 5 4 5 _n €n w wn Pn Zn w

(31)

2

(32)

Please cite this article as: O. Abdeljaber, et al., Active vibration control of flexible cantilever plates using piezoelectric materials and artificial neural networks, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.10.029i

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_ 1 ; …; w_n T , the reduced system can be rewritten in a conDefining a 2n-dimensional state vector xðtÞ ¼ ½w1 ; …; wn ; w tinuous state-space form: 8 2 3 2 39 P1 Z1 > > > >  >   = < 6 7 6 7> P In 0n 0n 6 27 6 Z2 7 F ð t Þ x_ ðtÞ ¼  V ð t Þ x ð t Þ þ 6 7 6 7 1 1 1 > > ⋮ ⋮ 4 5 4 5 M K M C M > > > > ; : Pn Zn 9 8 (33) 2 3 2 3 P1 Z1 > > > > > > < 6 P2 7 6 Z 2 7=  6 7 6 7 yðtÞ ¼ 0n In xðtÞ þ ½0n  F ðt Þ6 7  V ðt Þ6 7 > 4 ⋮ 5 4 ⋮ 5> > > > > ; : Pn Zn where yðt Þ is the state-space output, and it is adjusted here to compute the velocity at each node in set 1 (i.e. _ 1 ; …; w _ n T ). In this study, the damping matrix C is computed based on the assumption of modal damping according yðt Þ ¼ ½w to the following equation: 2 3 2ζ 1 ω1 ⋯ 0 6 ⋮ 1 ⋱ ⋮ 7 C ¼ MΦ4 (34) 5Φ 0 ⋯ 2ζ n ωn where ζ i and ωi are the hand-picked modal damping ratio and is the natural frequency of the ith mode, respectively, and Φ is the modal matrix. In Section 2.3, the equations governing the response of piezoelectric sensors are introduced. Then, the state-space model defined by Eq. (33) will be manipulated to give the voltage measured by the piezoelectric sensors as an output. 2.3. Formulation of piezoelectric sensor equations The voltage generated by a piezoelectric sensor is given as [11,29] V s ðt Þ ¼ Rp r

 Zy2 Zx2  _ _ _ ∂2 w ∂2 w ∂2 w dx dy e31 2 þ e32 2 þ 2e36 ∂x∂y ∂x ∂y

(35)

y1 x1

where Rp is the constant of the current amplifier, e31 , e32 , and e36 ¼ 0 are the piezoelectric material stress constants for the _ ðx; y; t Þ is the velocity map of PZT patch, r is the distance between the middle plan of the sensor and that of the host plate, w the plate, and x1 , x2 , y1 , and y2 denote the coordinates of the lower left corner and the upper right corner of the piezoelectric sensor. By dividing the plate according to the same mesh used to form the state-space model and applying finite difference method on the terms inside the integral, the previous equation can be rewritten as Zy2 Zx2

V s ðt Þ ffiRp r y1 x1

e31  H

2

  _ ði;j;tÞ þ w _ ði;j þ 1;tÞ þ e32 w _ ði;j;t Þ þ w _ ði þ 1;j;tÞ _ ði;j1;tÞ  2w _ ði1;j;tÞ  2w w

(36)

This equation can be simplified by assigning the nodes with indices from 1 to n as described in the Section 2.2 and expressing the coefficients obtained by finite difference method in a matrix S. Zy2 Zx2 V s ðt Þ ffiRp r

 _1 SðnnÞ w

_2 w

…

_n w

T

dxdy

(37)

y1 x1

Using a numerical double integration method such as trapezoidal or Simpson's rule, the piezoelectric sensor response can be written as  _n T _1 w _2 … w V s ðt Þ ffi Rp rNð1nÞ SðnnÞ w (38) In the constant vector N, the elements corresponding to the nodes covered the piezoelectric sensor are assigned by the weights associated with the numerical integration method employed [30]. The other elements in the vector N are assigned by zeros. Finally, the state-space system introduced in the Section 2.2 is redefined to change its output from nodal velocities to piezoelectric sensor output.    

0n In 0n x_ ðtÞ ¼ F ðt ÞP  V ðt ÞZ xðtÞ þ 1 1 1 M K M C M (39)

 yv ðtÞ ¼ Rp rNð1nÞ SðnnÞ 0n In xðtÞ þ 0n  F ðt ÞP  V ðt ÞZ where yv ðtÞ is the modified state-space output which is equivalent to V s ðt Þ. The resulting state space model is expected to have a large number of states and high-frequency dynamics, especially when the plate is divided into a fine mesh. Therefore, Please cite this article as: O. Abdeljaber, et al., Active vibration control of flexible cantilever plates using piezoelectric materials and artificial neural networks, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.10.029i

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it is necessary to apply a model reduction technique to obtain a reduced model that is more suitable for control design without affecting the accuracy of the solution. 2.4. Model reduction and design of Kalman filter The first step toward reducing the model is to apply balanced realization on the state-space defined by Eq. (33) in order to obtain an equivalent model for which the observability and controllability matrices are equal and diagonal. Next, the reduced-order state-space is formed by eliminating the states with relatively small balanced Gramians. The reduced model retains the most important characteristics of the full system since the eliminated states are weakly coupled to the inputs and outputs. The reduced-order model is given as

x_ r ðtÞ ¼ Ar xr ðtÞ þ Br F ðt ÞP  V ðt ÞZ

(40) yvr ðtÞ ¼ Cr xr ðtÞ þDr  F ðt ÞP  V ðt ÞZ where xr ðt Þ is the reduced state vector, Ar and Br are the system matrices, Cr and Dr ¼ 0 are the mapping matrices of the reduced-order system, and yvr ðtÞ is the reduced-order system estimate of the piezoelectric sensor output V s ðt Þ. It is worth noting here that the states in the balanced state vector are arranged in a decreasing order of significance. Consequently, the controller must be designed such that the first states of the reduced-order model are controlled. The neurocontroller designed in this study will be devoted to the control of the 1st state of the model which corresponds to the lowest natural frequency and the 1st bending mode of the plate as illustrated in Fig. 3. As will be discussed in Section 3, the neurocontroller will be designed based on a set of emulator neural networks trained to simulate the behavior of the reduced-order model to estimate the response of the 1st state according to the control voltage applied on the piezoelectric actuator. Therefore, in order to generate the data required to train the emulator neural networks, a regulated output vector zðtÞ is introduced to compute the 1st state response:

zðtÞ ¼ Cz xr ðtÞ þ Dz  F ðt ÞP  V ðt ÞZ (41) where Cz ¼ ½1; 0; ; 0 and Dz ¼ 0 are the mapping matrices of the regulated system. On the other hand, in real life applications, the states of the reduced-order system cannot be measured directly. The only measurement available for the feedback controller is the piezoelectric sensor output. Therefore, it is necessary to introduce a Kalman Filter optimal estimator that estimates the state vector xr ðtÞ according to the measurement yvr ðtÞ. The Kalman filter

Fig. 3. Preliminary implementation of the neurocontroller.

Fig. 4. Implementation of Kalman filter.

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is given as  x^_ r ðtÞ ¼ Ar x^ r ðtÞ þ Br uðtÞ þ L yvr ðt Þ  Cr x^ r ðtÞ  Dr uðtÞ

(42)

where x^ r ðtÞ is the Kalman filter estimate of the state vector xr ðtÞ, uðtÞ ¼ V ðt ÞZ are the control forces applied on the plate's nodes by the piezoelectric actuators, Dr ¼0, and L is the Kalman gain filter. Accordingly, the preliminary schematic of the control algorithm introduced in Fig. 3 is modified as shown in Fig. 4.

3. Neurocontroller design As introduced in Section 1.1, this study utilizes a modified version of the neural network based algorithm developed Ghaboussi and Joghataie [24] and improved later by Bani-Hani [25] to train the neurocontroller to drive the piezoelectric actuators. This algorithm is stable, relatively simple, and suitable for the problems where the control signal is bounded between upper and lower bounds. The objective of this algorithm, which can be considered as an iterative search method, is to find the best control signal that will result in the minimum predicted response over a predefined time horizon. The training of the neurocontroller depends mainly on a set of multi-layer feedforward emulator neural networks that are designed and trained to predict the system's response after a certain number of time steps. As explained in Fig. 4, the neurocontroller is integrated with the Kalman filter designed earlier. This algorithm is described in detail in the following sections. 3.1. Multi-layer feedforward neural networks The most commonly used neural networks are multilayer feedforward neural networks. These models consist of a mesh of interconnected nonlinear processing units arranged in layers. The first layer is called the input layer and the last one is called the output layer. The layers lying between these two layers constitute the hidden layers of the neural network. In feedforward neural networks, the processing units in each layer of the neural network are connected to all units in the preceding layer. In addition, connections between the nodes in the same layer and bridging layer connections are not allowed. The interconnections between the processing units in a neural network are assigned by weights. These weights are adjusted through a training process to optimize the neural network output. A typical multilayer neural network with two hidden layers is shown in Fig. 5. In this study, the following notation will be used to represent multilayer neural networks with k hidden layers symbolically. 



(43) Outputs ¼ NN Inputs ; ni ; nh ¼ 1 ; …; nh ¼ k ; no where ni is the number of neural network's inputs, no is the number of outputs, and nh ¼ i is the number of nodes in the ith hidden layer. 3.2. Emulator neural networks The control algorithm adopted in this study utilizes a set of multilayer feedforward neural networks to predict the current and future values of the 1st state of the reduced-order system. Each network is responsible for predicting the Hidden Layers

Output Layer

Input Signals

output Signals

Input Layer

Signal Propagation Direction Fig. 5. A typical multilayer feedforward neural network.

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response of the 1st state after a certain number of time steps. The inputs to these neural networks include the immediate past values of the 1st state as well as the current and the past values of the command voltage signal applied on the piezoelectric actuator. The emulator neural networks can be viewed as transfer functions from the applied voltage to the controlled state variable. For instance, a symbolic representation of an emulator neural network with l hidden layers created to predict the 1st state after 1 time step ahead can be written as



(44) xr1;k ¼ NN xr1;k1 ; xr1;k2 ; …; xr1;kc ; V k ; …; V kd ; c þ d þ1; nh ¼ 1 ; …; nh ¼ k ; 1 where k is the current time step, xr1;k1 ; xr1;k2 ; …; xr1;kc are the history values of the 1st state response, and V k ; V kd ; …; V kd are the current and the past control voltage values. The data required for the training of emulator neural networks are generated by recording the regulated response of the reduced-order model under the effect of a band-limited white noise load and voltage signals, which will be explained later. After that, the recorded data is arranged in input–output patterns. More information on the training of emulator neural networks will be provided within the numerical demonstration. After the training process is completed, the resulting set of emulators are utilized to compute the appropriate control voltage signal that will achieve a specific control criterion as explained in the Section 3.3. 3.3. The neurocontrol algorithm This section explains the procedure of utilizing the formulated state-space model along with the emulator neural networks to generate the data required for training the neurocontroller. As mentioned previously, the main objective of the neurocontrol algorithm is to obtain the control voltage at each time step that would result in the minimum predicted 1st state response over a certain time horizon. This can be achieved through the following steps: 1. The response of the reduced-order system with regulated output is simulated under a band-limited white noise force signal distributed over the plate according to a certain distribution. The 1st state value computed by the state-space model is collected after each sampling period. The 1st state value computed at the time step t k ¼ kΔt s is denoted by xr1;k . 2. If the magnitude of xr1;k exceeds a predefined threshold value T, the algorithm will follow steps 3 and 4 in order to determine the appropriate control value. Otherwise, the algorithm will issue zero control voltage for this time step. 3. At each time step k, the voltage V k is varied m times within the upper and lower voltage bounds by an increment of ΔV according to the following equation V jk ¼ V j1 þ jΔV; V 0k ¼ V min ; j ¼ 1; …; m k

(45)

This process is conducted in attempt to find the best control voltage V jk that will result in the minimum performance index P which is calculated using the emulator neural networks according to the following sidesteps: a. At each voltage increment V jk , the emulator neural networks trained earlier are utilized to predict the 1st state value  over a prediction horizon q ¼ 1; 2; …; np . The command voltage V jk is assumed to be constant over the prediction horizon q. The predicted future values are denoted by xjr;k þ 1 , xjr;k þ 2 ,…,xjr;k þ np . b. The predicted future values are assigned with importance factors. Since the prediction quality deteriorates considerably as moved further into the future, the importance factor w1 corresponding to the future 1st state valuexjr;k þ 1 will be less than the factor w2 assigned to the future relative velocity xjr;k þ 2 and so on. This can be accomplished by using the following equation Wiþ1 ¼ Wi 

i np þ i

(46)

given that W 1 ¼ 1. c. Next, the performance index P j is calculated as the weighted average of the predicted values according to the following equation np P j

P ¼

i¼1

jxjr;k þ i j  W i np P i¼1

(47) Wi

4. The voltage V jk that produces the least value of performance index P j is selected to be the command voltage at the current time step V k . 5. The algorithm is iterated until the simulation is terminated. Fig. 6 shows a schematic for the neurocontrol algorithm. In this Figure, it is assumed that 3 emulator neural networks are used for generating the data required for training the neurocontroller. Please cite this article as: O. Abdeljaber, et al., Active vibration control of flexible cantilever plates using piezoelectric materials and artificial neural networks, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.10.029i

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Fig. 6. The neurocontrol algorithm.

3.4. Training of the neurocontroller Section 3.3 described the procedure of utilizing emulator neural networks to compute the control voltage signal that will minimize the response of the controlled state under a particular load signal. Next, a new multilayer feedforward neural network referred to as the neurocontroller is trained. The objective of the neurocontroller is to issue the current control voltage directly according to the immediate past values of the controlled state. This means that the neurocontroller is trained to learn the transfer function from the controlled state to the control voltage. The data required for the training process can be generated by exciting the plate by a band-limited white-noise load signal and compute the corresponding control voltage signal according the neurocontrol algorithm. Then, the resulting data is arranged as input–output patterns, and a neural networking learning algorithm is utilized to train the neurocontroller according to the generated patterns. The neurocontroller can be written symbolically in neural network notation as 



V k ¼ NN xr1;k ; xr1;k1 ; …; xr1;ke ; e þ1; nh ¼ 1 ; …; nh ¼ k ; 1 (48) Nevertheless, in the real implementation of the neurocontroller, the previous values of the controlled state are not directly measurable. Therefore, the Kalman Filter introduced earlier is utilized to compute an estimate x^ t ðtÞ for the state vector xr ðt Þ. As a result, the current and previous values of the 1st state in Eq. (48) are substituted by the their corresponding Kalman estimates as given in the following equation 



V k ¼ NN x^ r1;k ; x^ r1;k1 ; …; x^ r1;ke ; e þ 1; nh ¼ 1 ; …; nh ¼ k ; 1 (49)

4. Analysis and results A numerical example of a cantilever plate featuring a single piezoelectric sensor/actuator pair is utilized to verify the model introduced in this study and demonstrate the neural network based control methodology. The plate selected for the simulation is a square thin plate made of steel. The dimensions of the plate are a ¼0.5 m, b¼0.5 m, and h¼1.78 m. The density, modulus of elasticity, and Poisson's ratio are ρ ¼ 7800 kg/m3, E ¼200 GPa, and v ¼ 0:3, respectively. The flexural  3 rigidity D ¼ Eh =ð12 1  υ2 ÞÞ ¼ 103:3N Um. For all vibration modes the modal damping ratio was assumed as ζ ¼ 0:006. The square PZT patch dimensions are 6.67 cm  6.67 cm and a thickness of hPZT ¼1 mm. The material properties of the S/ A pair are ρPZT ¼7650 kg/m3, EPZT ¼63 GPa, and νPZT ¼ 0:30. The PZT strain constants are taken as Please cite this article as: O. Abdeljaber, et al., Active vibration control of flexible cantilever plates using piezoelectric materials and artificial neural networks, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.10.029i

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12 -3

1.5

x 10

Displacement at (0.5m,0.25m) (m)

Full state-space Reduced-order state space 1

0.5

0

-0.5

-1

-1.5 0

0.5

1

1.5

2

2.5

3

Time (s)

Fig. 7. Comparison between the responses of the full state-space model and the reduced-order model under a band-limited white noise excitation.

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

Fig. 8. The mode shapes extracted from Abaqus model.

Table 1 Model verification. Mode Frequencies of the state-space model (Hz) Natural frequencies estimated by Abaqus (Hz)

Natural frequencies estimated analytically (Hz) [33]

1 2 3 4 5 6

6.0601 14.846 37.229 47.682 54.124 –

6.0207 14.741 36.776 47.005 53.491 93.472

6.0278 14.769 37.047 47.294 53.833 94.223

d31 ¼ d32 ¼ 166  1012 m=V, and the PZT stress constants are taken as e31 ¼ e32 ¼ 10:46 m=V. Both of the sensing and the actuating patch were placed on the same location on the x–y plane, but on opposite sides of the plate. According to an analytical study conducted by Quek et al. [31], the optimal placement of a single piezoelectric S/A pair to control 1st bending mode of a cantilever plate is along the plate's central line and close to the cantilever's fixed end support. Therefore, the coordinates of the S/A pair were selected as x1 ¼0.033 m, y1 ¼0.217 m, x2 ¼0.100 m, and y2 ¼0.283 m. Please cite this article as: O. Abdeljaber, et al., Active vibration control of flexible cantilever plates using piezoelectric materials and artificial neural networks, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.10.029i

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Table 2 Emulator neural networks. Name

Symbolic representation

xr1;k1 ; …; xr1;k4 ; V k ; …; V k3 ;f8; 6; 6; 1g

¼ NN xr1;k2 ; …; xr1;k5 ; V k ; …; V k3 ;f8; 6; 6; 1g 

¼ NN xr1;k3 ; …; xr1;k6 ; V k ; …; V k3 ;f8; 6; 6; 1g

ENN1

xr1;k ¼ NN

ENN2

xr1;k

ENN3

xr1;k

 

Fig. 9. Emulator neural network ENN1.

Fig. 10. Simulink model for the generation of input–output patterns required for training the 3 emulator neural networks.

4.1. System formulation A Matlab code was written to utilize finite difference method to formulate the state-space system and the Kalman Filter optimal estimator (the mathematical formulation has been described previously in this text). The program asks the user to Please cite this article as: O. Abdeljaber, et al., Active vibration control of flexible cantilever plates using piezoelectric materials and artificial neural networks, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.10.029i

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enter the dimensions and properties of the host plate and the piezoelectric patch(s). Also, the user is asked to input the desired size of the finite difference mesh. Accordingly, the program writes the nodal equations and arranges them in a matrix form. Then, the mass, stiffness, and damping matrices are formulated and stored. Additionally, the vector Z that governs the effect of the control voltage on the cantilever plate is computed according to the coordinates and properties of the S/A pair(s). Trapezoidal rule is utilized to estimate the integration weights in the vector N required for the sensor equation. After that, the program defines the full state space system given in Eq. (39). Next, the program applies the state realization and model reduction tools available in Matlab to form a balanced realization of the full system and eliminate the states with relatively small controllability and observability Gramians. Finally, the Kalman Filter gain L is computed using the routine lqew.m within the Matlab Control System Toolbox [32]. For this numerical simulation, the plate was divided into a mesh of H  L ¼ 0:0167 m  0:0167 m elements, giving a total number of nodes nT ¼ 1180, and a total number of set 1 nodes n ¼ 930. The number of states in the full system is equal to 2n ¼ 1860 states. However, after applying model reduction techniques, the system was reduced into a state-space with 12 states only. It is worth noting that these states correspond to the natural frequencies of the first 6 vibration modes of the cantilever plate. Fig. 7 shows a comparison between the response of the full state space model and the response of the reduced-order model under a band-limited white noise excitation.

2.5 ENN1 Response

2

Actual Response

1.5 1 0.5 0 -0.5 -1 -1.5

0

0.5

1

1.5

Time (sec)

Fig. 11. Comparison between the actual response and the response predicted by ENN1.

Fig. 12. Simulink model for utilizing the neurocontrol algorithm to generate input–output patterns required for the training of the neurocontroller.

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4.2. System verification In order to verify the reduced-order state-space system obtained for the cantilever plate, the frequencies of the state space model were compared to the natural frequencies of the same plate estimated by an analytical method reported by Plunkett [33], as well as to the natural frequencies approximated by finite element method. Matlab was used to obtain the poles of the reduced-order state-space and their corresponding frequencies. The finite element commercial software Abaqus 6.12 [34] was used to compute the finite element approximation of the mode shapes and corresponding natural frequencies. The mode shapes from Abaqus model are presented in Fig. 8. The results reported in Table 1 reveal excellent agreement between the system frequencies and the plate's estimated natural frequencies.

Fig. 13. Content of the neurocontrol algorithm Simulink block.

Fig. 14. Simulink model for neurocontroller evaluation.

Please cite this article as: O. Abdeljaber, et al., Active vibration control of flexible cantilever plates using piezoelectric materials and artificial neural networks, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.10.029i

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0.7 0.6

Force (N)

0.5 0.4 0.3 0.2 0.1 0

0

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1

1.5

2

2.5

3

3.5

4

4.5

5

Time (s) Fig. 15. Load signal for case 1.

-4

Displacement at (0.5m,0.25m) (m)

8

x 10

Uncontrolled Neurocontrolled

6 4 2 0 -2 -4 -6 -8

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time (s)

Fig. 16. Comparison between the uncontrolled and neurocontrolled displacement response under load case 1.

80

Actuator Voltage (V)

60 40 20 0 -20 -40 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time (s) Fig. 17. Command voltage signal under load case 1.

4.3. Training of the emulator neural networks Three emulator neural networks have been trained in order to be included in the neurocontrol algorithm. The inputs, outputs, and the architecture of these neural networks are shown in Table 2. Additionally, a graphical representation of the emulator neural network ENN1 is shown in Fig. 9. The Simulink model shown in Fig. 10 was built to generate the data Please cite this article as: O. Abdeljaber, et al., Active vibration control of flexible cantilever plates using piezoelectric materials and artificial neural networks, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.10.029i

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4 3

1

2

Load (N/m )

2

0 -1 -2 -3 -4 -5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time (s) Fig. 18. Load signal for case 2.

-3

1.5

x 10

Displacement at (0.5m,0.25m) (m)

Uncontrolled Neurocontrolled 1

0.5

0

-0.5

-1

-1.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time (s)

Fig. 19. Comparison between the uncontrolled and neurocontrolled displacement response under load case 2.

required for training. For all simulations conducted hereafter, the sampling period is taken as Δt ¼ 0:001 s. The response of the plate's model was simulated under the following excitations: 1. In the 1st simulation, the plate was excited for 10 s by a uniformly distributed load with a magnitude that varies according to a 0–60 Hz band-limited white noise signal. 2. In the 2nd simulation, the plate was excited for 10 s by applying a 0–5 Hz band-limited white noise voltage signal on the piezoelectric patch. 3. Finally, in the 3rd simulation, the system was subjected to a combination of random load distributed uniformly over the plate and random voltage applied on the piezoelectric actuator for 10 s. The abovementioned simulations resulted in a total of 30=ð0:001 ¼ 30000Þ input–output samples for each one of the three emulators. The neural networks were chosen to have 2 hidden layers with 6 nodes in each layer. The training process was carried out using Levenberg–Marquardt backpropagation training algorithm included in Matlab Neural Network Toolbox under the routine “trainlm” To illustrate the efficiency of the emulator neural networks in predicting the future response, Fig. 11 shows a comparison between the actual response and the approximate response predicted by the emulator ENN1. 4.4. Implementation of the neurocontrol algorithm for neurocontroller training The three emulator neural networks along with the neurocontrol algorithm discussed earlier have been implemented to generate the data required to train the neurocontroller. The Simulink model described in Figs. 12 and 13 has been employed Please cite this article as: O. Abdeljaber, et al., Active vibration control of flexible cantilever plates using piezoelectric materials and artificial neural networks, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.10.029i

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80

Actuator Voltage (V)

60

40

20

0

-20

-40

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

3.5

4

4.5

5

Time (s) Fig. 20. Command voltage signal for load case 2.

0.1 0.08 0.06

Force (N)

0.04 0.02 0 -0.02 -0.04 -0.06 -0.08 -0.1

0

0.5

1

1.5

2

2.5

3

Time (s) Fig. 21. Load signal for case 3.

for simulation and data generation. Again, the plate was subjected to a uniformly distributed load with a magnitude that varies according to a 0–60 Hz band-limited white noise signal for 10 s. Using the Simulink model, the optimal voltage signal that will minimize the response of the plate under the random load signal was computed. This process generated a total of 10000 input–output patterns which were utilized to train the neurocontroller according to Levenberg–Marquardt backpropagation training algorithm. The output, inputs, and the architecture of the resulting neurocontroller are expressed symbolically as 

V k ¼ NN xr1;k1 ; xr1;k2 ; …; xr1;k7 ;f7; 10; 10; 1g (50) 4.5. Evaluation of the neurocontroller To demonstrate the efficiency and robustness of the neurocontroller in mitigating the response of the cantilever plate, the Simulink model shown in Fig. 14 was built. This model utilizes the neurocontroller along with the Kalman filter designed earlier to compute the current control voltage value directly from the current and previous piezoelectric sensor Please cite this article as: O. Abdeljaber, et al., Active vibration control of flexible cantilever plates using piezoelectric materials and artificial neural networks, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.10.029i

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-4

x 10

5

Uncontrolled Neurocontrolled

Displacement at (0.5m,0.25m) (m)

4 3 2 1 0 -1 -2 -3 -4 -5 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time (s)

Fig. 22. Comparison between the uncontrolled and neurocontrolled displacement response under load case 3.

80

Actuator Voltage (V)

60

40

20

0

-20

-40

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time (s) Fig. 23. Command voltage signal for load case 3.

measurements. The displacement response of the cantilever plate controlled by the neurocontroller was simulated under the following loading cases 1. Concentrated pulse load applied over the point x ¼0.4 m, y ¼0.25 m (Fig. 15). 2. Uniformly distributed load with a magnitude varying according to a 0–40 Hz band-limited white noise signal (Fig. 18). 3. Two concentrated loads applied on the points x¼0.4 m, y¼0.1 m and x¼0.4 m, y¼0.4 m with a magnitude that varies according to a pseudo random binary sequence (PRBS) (Fig. 21). For the three loading cases, the displacement response was computed at the point x ¼0.5 m, y¼0.25 m (i.e. the center of the unconstrained end). Similarly, the response of the uncontrolled plate under the same loading conditions was simulated. Figs. 16, 19 and 22 show comparisons between the neurocontrolled and uncontrolled responses in the time domain. The control voltage commands generated by the neurocontroller under the 3 loading cases are given in Figs. 17, 20 and 23.

5. Discussions The response of the cantilever plate featuring a piezoelectric S/A pair was computed under the aforementioned load cases. The 1st load case was introduced to examine the efficiency of the neurocontroller in mitigating the free vibrations Please cite this article as: O. Abdeljaber, et al., Active vibration control of flexible cantilever plates using piezoelectric materials and artificial neural networks, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.10.029i

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induced by a pulse concentrated load. Fig. 16 shows clearly that applying the proposed active control methodology has reduced the displacement response of the cantilever plate significantly. Furthermore, the 2nd and 3rd load cases were investigated to evaluate the performance of the neurocontroller under forced vibrations. The results illustrated in Figs. 19 and 22 demonstrate the efficiency of the neurocontroller in both cases. The maximum absolute response was reduced by 76.7% for the 2nd case and 70.0% for the 3rd case. Also, the RMS value of the response over 5 s was reduced by 80.9% for the 2nd case and 81.9% for the 3rd case. This shows clearly that the neurocontroller is successful, effective and robust regardless to the type and distribution of the applied excitation. In this numerical evaluation of the neurocontrol methodology, the neurocontroller was designed to control the 1st bending mode of the cantilever plate. It is very important to mention here that the same methodology can be employed to design other neurocontrollers to control other bending modes and torsional vibration modes.

6. Conclusions This research proposed a new intelligent methodology for active control of flexible cantilever plates using piezoelectric sensor/actuator pairs. A neural network based algorithm was developed to control the voltage signal applied on the piezoelectric patches. This algorithm relies on a set of emulator neural networks to train a neural network based controller (i.e. neurocontroller). First, finite difference method was applied to extract the equations governing the response of a cantilever plate featuring piezoelectric patches. Then, the equations were formulated in a state-space form. Also, a model reduction technique was applied to reduce the number of state variables in the resulting model. In the next step, a Kalman Filter was designed to estimate the state variables from the piezoelectric sensors measurements. Following, a neural network based algorithm to control the vibration response of cantilever plates was introduced. This algorithm requires training of a set of emulator neural networks to predict the future response of the controlled state variable. The procedure of employing this algorithm to design and train a neurocontroller was discussed. Finally, the proposed control methodology was examined analytically on a cantilever square plate featuring one piezoelectric sensor/actuator pair. A neurocontroller was trained and used along with the Kalman Filter to control the vibration response of the plate. The neurocontroller was evaluated by comparing the uncontrolled and controlled responses under several dynamic loading conditions. The comparison shows that the neurocontroller trained by this intelligent methodology was able to significantly reduce the vibration response of the cantilever plate. The neurocontroller is proved to be successful in mitigating the accelerations of the flexible cantilever plate under various types of dynamic excitations.

Acknowledgments The financial support for this research was provided by Qatar National Research Fund, QNRF (a member of Qatar Foundation) via the National Priorities Research Program (NPRP), Project no. NPRP 6-526-2-218. The statements made herein are solely the responsibility of the authors.

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Please cite this article as: O. Abdeljaber, et al., Active vibration control of flexible cantilever plates using piezoelectric materials and artificial neural networks, Journal of Sound and Vibration (2015), http://dx.doi.org/10.1016/j.jsv.2015.10.029i