Virtual absorbed energy in decentralized velocity feedback control of a plate with piezoelectric patch actuators

Applied Acoustics 74 (2013) 909–919

Contents lists available at SciVerse ScienceDirect

Applied Acoustics journal homepage: www.elsevier.com/locate/apacoust

Virtual absorbed energy in decentralized velocity feedback control of a plate with piezoelectric patch actuators Yin Cao, Hongling Sun ⇑, Fengyan An, Xiaodong Li Key Laboratory of Noise and Vibration Research, Institute of Acoustics, Chinese Academy of Sciences, No. 21 North 4th Ring Road West, Beijing 100190, China

a r t i c l e

i n f o

Article history: Received 3 August 2012 Received in revised form 5 November 2012 Accepted 7 January 2013

Keywords: Decentralized feedback control Virtual energy absorption Piezoelectric patch actuators Robust self-tuning algorithm

a b s t r a c t Feedback gain is a key factor in decentralized velocity feedback control. It is known that an optimal feedback gain value exists with which the active damping effect could be maximized. In this paper, a method of finding suitable feedback gains for decentralized velocity feedback control with piezoelectric patch actuators is investigated. The energy absorption of piezoelectric patch actuators is calculated. The concept of virtual energy absorption of the piezoelectric patch actuator is proposed, through which the optimal feedback gain can be got conveniently. Numerical investigations are performed to explore the relationships between the virtual energy absorption by piezoelectric patch actuators and the kinetic energy of the structure. The results show that maximizing the broadband virtual energy absorption is nearly equivalent to minimizing the kinetic energy. A robust self-tuning algorithm is also proposed with maximizing the broadband virtual energy absorption. This algorithm can simultaneously update feedback gains of more than one control units, and find the optimal feedback gains automatically. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Multi-channel feedback control is one major control strategy in active structural acoustic control when the disturbance is broadband in nature and appears mainly at relatively low frequencies [1–4]. There are two kinds of multi-channel feedback controls, namely, the centralized control and the decentralized control. For centralized control, all the actuators are driven by a single controller which deals with the information coming from all the sensors. It is a relatively complicated control method which needs to be designed carefully. Decentralized feedback control is a less complicated control strategy since it uses several independent control units where each unit is typically composed of one sensor, one actuator and one controller. Recently, the decentralized velocity feedback control received extensive attention due to its less complexity, good stability and it gives nearly the same performance as the centralized control [5–13]. When an actuator and a sensor transducer pairs are dual and collocated, the velocity feedback control strategy is unconditionally stable [5,6], i.e. the failure of one control unit has no effect on the stability of the whole control system. Hence, the decentralized velocity feedback control is a relatively robust control method. Feedback gain is a key factor, which directly determines the stability and performance of the decentralized feedback control. In general, there is an optimal feedback gain for controlling elastic ⇑ Corresponding author. Tel./fax: +86 010 82547519. E-mail address: [email protected] (H.L. Sun). 0003-682X/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.apacoust.2013.01.001

structure which can be obtained using vibration or sound radiation information of the structure offline [6–10]. However, it is usually not convenient to get the physical information of the structure, hence, difficult to derive the optimal feedback gain. To solve the problem, several researchers studied an easy way of using maximized energy absorption of actuators instead of minimized kinetic energy to find the optimal feedback gain [14–19]. The results showed that by maximizing the energy absorption of the actuator, the kinetic energy of the structure is generally reduced as a result [16,17]. However for low-frequency narrow-band disturbances, maximizing the energy absorption of the actuator may lead to an increase in the kinetic energy of the structure [18]. Sharp [19] further verified that with broadband disturbances, maximizing the energy absorption of the actuator can indeed reduce the total kinetic energy, but this relation does not hold when the disturbances are narrow-band. Compared to the kinetic energy or the sound radiation power, the energy absorption of the actuator is much easier to measure. For inertial actuator, the energy absorption is the product of the force produced by the actuator and the velocity at the mounting point of the actuator. When the actuator and sensor are collocated, the abovementioned energy absorption can be measured directly without the knowledge of the physical path, hence it is more convenient to adjust the feedback gain using the energy absorption as the cost function. Zilletti [20,21] further checked the relationship between the absorbed energy of force actuator and the kinetic energy of the plate, and then proposed a self-tuning algorithm to automatically find the optimal feedback gain using energy absorption of the inertial actuator as the cost function for decentralized

910

Y. Cao et al. / Applied Acoustics 74 (2013) 909–919

vibration control of the plate. This method does not require the need to measure the vibration of the whole plate. Numerical and experimental results showed that maximizing the energy absorption of the inertial actuator has a similar feedback gain to that minimizing the kinetic energy. Later on, Gardonio [22] further analyzed the energy absorption of force actuator both in frequency domain and time domain, numerically. It was proved that the feedback gain obtained during the maximization of the energy absorption of the actuator does not vary significantly with the following parameters including the number of feedback control loops, the structural damping in the plate, the mutual distance of a pair of control loops and the mutual gains implemented in a pair of feedback loops. The energy absorption of the force actuator seems to be an effective cost function. However the inertial actuator needs some large mass blocks to generate reaction forces and will be relatively bulky and heavy when large forces are required in low frequency range. It would be more convenient in practice to use piezoelectric patch actuators that are fully integrated with the panel [7–9,23,24]. The energy absorption of an inertial actuator is easy to measure, but it is hard to get the energy absorption of a moment actuator such as a piezoelectric patch actuator which requires the knowledge of the angular velocity at the actuator mounting point. In addition, the self-tuning algorithm proposed by Zilletti can only update the feedback gain for a single control unit at one time, which might be inefficient. In this paper, the energy absorption of piezoelectric patch actuators for decentralized velocity feedback control of plate vibrations is investigated numerically. The concept of virtual energy absorption of the piezoelectric patch actuator is proposed, which is the product of the torque produced by the piezoelectric patch actuator and the velocity of the actuator at the mounting point. Numerical investigations are performed to explore the relationships between the energy absorption or the virtual energy absorption of piezoelectric patch actuators and the kinetic energy of the structure. The results show that maximizing the broadband energy absorption or virtual energy absorption is nearly equivalent to minimizing the kinetic energy, while the virtual energy absorption is easier to measure. An alternative self-tuning algorithm for finding the optimal feedback gain is also proposed, which is capable of updating more than one control unit at a time. This paper is organized into six sections. Section 2 briefly presents the dynamic model of the plate and the model of the decentralized velocity feedback control considered in this study. Section 3 introduces the mathematical model of the energy expressions. The effectiveness of using the proposed virtual energy absorption as a cost function is discussed in Section 4, where the valid conditions are also examined. Section 5 supplies a feasible multi-channel self-tuning algorithm to find the optimal feedback gain, in which the virtual energy absorption is maximized. Moreover the convergence behavior is also verified. Finally, the conclusions are then presented in Section 6. 2. Dynamic model of the decentralized control of the plate The diagram of decentralized velocity feedback control of a plate using piezoelectric patch actuator is shown in Fig. 1, in which a thin panel is subject to an incident plane acoustic wave. The plane wave is assumed to be incident at azimuthal and lateral angles of both 45°, thus excites all the structural modes of the panel [25]. A number of feedback control units which are composed of piezoelectric patch actuators and velocity sensors are used to control the vibration and sound radiation of the plate. The plate is divided into a grid of rectangular elements whose dimensions have been taken to be lxe = Lx/(4M) and lye = Ly/(4N), where Lx and Ly are length and width of the plate, respectively, M and N are the highest plate modal orders used in the simulation [7–9].

Sound wave Velocity Piezoelectric patch sensor actuator

Fig. 1. Decentralized feedback control of plate with piezoelectric patch actuators.

It can be assumed that all points on a rectangular element experience the same transverse velocity and therefore, the overall elemental velocity can be represented by the velocity at the center of each rectangular element. The transverse velocity at the center _ ei ðxÞ and at the velocity sensor of a rectangular element, w _ ck ðxÞ, have been, respectively, grouped into following vectors w

8 9 _ e1 ðxÞ > w > > > > > > _ e2 ðxÞ > . >; .. > > > > > > : ; _ eP ðxÞ w

8 9 _ c1 ðxÞ > w > > > > > > _ c2 ðxÞ > .. > > . > > > > > : ; _ cQ ðxÞ w

ð1Þ

where P = 16MN is the total number of elements, Q is the total number of control units. The excitation induced by the piezoelectric patch actuator can be deemed as 4 line moments acting along the edges of the actuator [23,26] and the magnitude can be written as

C ¼ msl ðxÞ ¼ C o epe

ð2Þ

where Co are constants determined by both the characteristics of the piezoelectric patch actuator and the size of the plate [1] 3

Co ¼

Ep hp f K 1  mp 12

ð3Þ 2

Kf ¼

3Epe ½ð0:5h þ ha Þ2  0:25h ð1  mp Þ 3

3

2Epe ½ð0:5h þ ha Þ3  hb ð1  mp Þ þ 0:25Ep h ð1  mpe Þ

ð4Þ

and epe can be expressed as

epe ¼ d31 V=ha

ð5Þ

where Ep, Epe are Young’s modulus of the plate and piezoelectric patch actuator, respectively, vp, vpe are Poisson’s ratio of plate and piezoelectric patch actuator, respectively, hp is the thickness of the plate, ha is the thickness of the piezoelectric patch actuator, d31 is the piezoelectric material strain constant. It can be known from Eqs. (2)–(5) that the excitation from piezoelectric patch actuator can be equivalent to 4 line moments which are proportional to the input voltage applied to the piezoelectric patch actuator. So the control output can be grouped into a vector of moments as follows

8 9 ms1 ðxÞ > > > > > > > < ms2 ðxÞ > = Fs ðxÞ ¼ . > > .. > > > > > > : ; msQ ðxÞ

ð6Þ

The acoustic excitation can be expressed as Sp(x). So the vibration of the panel can be expressed using mobility functions as [7–9]

Ve ðxÞ ¼ Hep ðxÞSp ðxÞ þ Hes ðxÞFs ðxÞ

ð7Þ

Vc ðxÞ ¼ Hcp ðxÞSp ðxÞ þ Hcs ðxÞFs ðxÞ

ð8Þ

911

Y. Cao et al. / Applied Acoustics 74 (2013) 909–919

where the transfer functions of the acoustic pressure to the velocity at the center of the element, Hep and the transfer functions of the acoustic pressure to the velocity at the control point, Hcp have the same form, and they are given by [23] 1 X 1 X _ i ðxÞ 8ImIn sinðkm xk Þ sinðkn yk Þ w ¼ jx Sp ðxÞ qhðx2mn  x2 þ 2jfmn xmn xÞ m¼1 n¼1

H k ð xÞ ¼

ð9Þ

where (xk, yk) is the coordinate at the middle point of the element or the control point, km = mp/Lx, kn = np/Ly, Im and In are the constants which are determined by the azimuthal and lateral angles, q and h are the density and thickness of the plate, respectively, xmn is the natural frequency of the (m, n)th bending mode, fmn is the modal damping ratio. The transfer function of the control moment to the velocity at the center of the element, Hes, and the transfer functions of the control moment to the velocity at the control point, Hcs are given by [23] _ i ðxÞ w msk ðxÞ 2 2 1 X 1 X 4ðkm þ kn Þðcos km x1i  cos km x2i Þðcos kn y1i  cos kn y2i Þ ¼ jx qhmnp2 ðx2mn  x2 þ 2jfmn xmn xÞ m¼1 n¼1

Hi;k ðxÞ ¼

 sinðkm xk Þ sinðkn yk Þ

ð10Þ

where the piezoelectric patch actuator is at x e [x1i, x2i] and y e [y1i, y2i]. When the decentralized feedback control is implemented, the feedback force can be expressed as

Fs ðxÞ ¼ HðxÞv c ðxÞ

ð11Þ

where H(x) is the diagonal feedback gain matrix. Hence the controlled transverse velocity can be obtained through Eqs. (7), (8), (11) as

v c ðxÞ ¼ ½I þ Hcs ðxÞHðxÞ1 Hcp ðxÞSp ðxÞ

ð12Þ

v e ðxÞ ¼ Hep ðxÞSp ðxÞ  Hes ðxÞHðxÞ½I þ Hcs ðxÞHðxÞ1 Hcp ðxÞSp ðxÞ

Usually, the minimization of the kinetic energy is used to determine the suitable feedback gain off-line. However it is not a convenient method as mentioned in Section 1. While maximizing the virtual energy absorption proposed in this paper is not only easy to implement, but also it is nearly equivalent to minimizing the kinetic energy, which will be proved in next section. In this section, the expressions of various energy terms are presented. 3.1. Kinetic energy With the excitation at a certain frequency, the total kinetic energy can be expressed as [7–9]

qh

Z

4

Lx

0

Z

Ly

_ jwðx; y; xÞj2 dxdy

qhlxe lye 4

fmax

EðxÞdf

ð16Þ

fmin

where fmin and fmax are the starting and ending frequency of the broadband excitation. It has been shown that, the total kinetic energy of a panel is a better representation of the near-field pressure than the radiated sound power [25]. The radiated sound power is a measurement mainly for far-field pressure. So both the kinetic energy and the radiated sound power are important cost functions. 3.2. Radiated sound power The total sound power radiation by a baffled panel can be obtained by integrating the product of the near-field sound pressure on the radiating side and the transverse velocity of the panel [7–9]

WðxÞ ¼

1 2

Z

Lx

Z

0

Ly

_ Re½pðx; y; xÞ wðx; y; xÞdxdy

ð17Þ

0

where  denotes the complex conjugate, p(x, y, x) is the sound pressure in front of the panel, which is related to the transverse velocity _ wðx; y; xÞ of the panel

_ pðx; y; xÞ ¼ Zðx; y; xÞwðx; y; xÞ

ð18Þ

where Z(x, y, x) is the radiation impedance of the plate. Similarly to the kinetic energy, the sound radiation power of the panel can be approximated to the summation of the sound radiation power of each element, so

WðxÞ ¼

DS Re½v He ðxÞZðxÞv e ðxÞ ¼ v He ðxÞRðxÞv e ðxÞ 2

ð19Þ

where R(x) is the radiation matrix [27], and can be expressed as

2

1

6 6 sinðk r Þ   0 21 DS x2 q0 DS2 6 6 k0 r21 Re½ZðxÞ ¼ RðxÞ ¼ 6 2 4pc0 6 .. 6 . 4 sinðk0 r I1 Þ k0 r I1

sinðk0 r 12 Þ k0 r12



1

...

.. .

.. .

..

.

.. .





1

PSound ¼

Z

7 7 7 7 7 7 7 5

fmax

WðxÞdf

ð21Þ

fmin

3.3. Energy absorption of the piezoelectric patch actuator Suppose the piezoelectric patch actuator acting between the area x e [x1, x2], y e [y1, y2] as shown in Fig. 2. The excitation of the actuator can be equivalent to 4 line moments acting along the edges of piezoelectric patch actuator as

Y

ð14Þ y2 y1

ð15Þ

If the excitation is broadband, according to Parseval’s formula, the total kinetic energy can be expressed as the integration in the frequency domain, as

3

where k0 = x/c0 is the wave-number in air, rij is the distance between the ith and jth elements. When the excitation is broadband in nature, the total sound radiation power can be expressed as

0

vHe ðxÞve ðxÞ

sinðk0 r 1I Þ k0 r 1I

ð20Þ

This expression can be approximated to the summation of the kinetic energies of each element, so that

EðxÞ ¼

Z

ð13Þ

3. Energy expressions

EðxÞ ¼

Pkinetic ¼

0

x1

x2

X

Fig. 2. Piezoelectric patch actuator acting on the panel.

912

Y. Cao et al. / Applied Acoustics 74 (2013) 909–919

mx1 ¼ Cdðx  x1 Þ½Hðy  y1 Þ  Hðy  y2 Þ

ð22Þ

ðmx2 ¼ Cdðx  x2 Þ½Hðy  y1 Þ  Hðy  y2 Þ

ð23Þ

my1 ¼ Cdðy  y1 Þ½Hðx  x1 Þ  Hðx  x2 Þ

ð24Þ

my2 ¼ Cdðy  y2 Þ½Hðx  x1 Þ  Hðx  x2 Þ

ð25Þ

The energy absorption of line moments can be expressed as the multiplication of line moments and the corresponding angular velocity, that is

Eabsorb ¼

Z

y2

y1

ðmx1 h_ x1 þ mx2 h_ x2 Þdy þ

Z

x2

x1

ðmy1 h_ y1 þ my2 h_ y2 Þdx

ð26Þ

easy to obtain in practice. It can be known from Section 3.3 that the energy absorption of the piezoelectric patch actuator is proportional to the amplitude of the line moment and the modal velocity. Here if the velocity signal from the velocity sensor is substituted for the angular velocity, and a physical quantity which is the product of the amplitude of the line moment and the velocity signal from the sensor is used, then this quantity is also proportional to the amplitude of the line moment and the modal velocity. Hence there may be a possibility for this quantity to be used to represent the energy absorption, and this quantity is termed the virtual energy absorption. It can be expressed as

E0absorb ¼ jxC

1 X 1 X W mn sin km x0 sin kn y0

ð35Þ

m¼1 n¼1

The angular velocity can be expressed as

 2  _hx ¼ @ w  1 @x@t 

1 X 1 X ¼ jx W mn km cos km x1 sin kn y

 2  _hx ¼ @ w  2 @x@t 

¼ jx

 @ 2 w  h_ y1 ¼  @y@t 

¼ jx

h_ y2

1 X 1 X W mn km cos km x2 sin kn y

ð28Þ

m¼1 n¼1

x¼x2

 @ 2 w  ¼  @y@t 

ð27Þ

m¼1 n¼1

x¼x1

1 X 1 X W mn kn sin km x cos kn y1

ð29Þ

m¼1 n¼1

y¼y1

¼ jx

1 X 1 X W mn kn sin km x cos kn y2

ð30Þ

m¼1 n¼1

y¼y2

After some simplifications using Eqs. (22)–(30), the energy absorption can be expressed as

Eabsorb

  1 X 1 X km kn ðcos km x1  cos km x2 Þ ¼ jxC  W mn þ kn km m¼1 n¼1  ðcos kn y1  cos kn y2 Þ

ð31Þ

It can be known from Eq. (31) that the energy absorption of piezoelectric patch actuator is proportional to the amplitude of the line moment M (which is proportional to the input voltage V), and the modal velocity jxWmn. Denote x0 = (x1 + x2)/2, y0 = (y1 + y2)/2 as the coordinates of the center of the piezoelectric patch actuator, and lpe_x = (x2  x1)/2, lpe_y = (y2  y1)/2 as the length and width of the piezoelectric patch actuator, respectively. Eq. (31) can be rearranged into

Eabsorb ¼ 4jxC 

  1 X 1 X km kn sin km x0 sin kn y0 W mn þ kn km m¼1 n¼1

 sin km lpe x sin kn lpe

y

ð32Þ

Considering the feedback control, the feedback gain is denoted as h, so that the magnitude of the line moment excited by the actuator can be written as

C ¼ C 0 epe ¼ jxh 

1 X 1 X W m0 n0 sin km0 x0 sin kn0 y0

ð33Þ

Comparing Eqs. (32) and (35), it can be found that the modal amplitude of the energy absorption equals the modal amplitude of the virtual energy absorption multiply by a factor of A = 4(km/kn + kn/km) sin kmlpe_x sin knlpe_y. In the low frequency range, km lpe x 1; kn lpe y 1, A 4ðm2 =L2x þ n2 =L2y Þp2 lpe x lpe y , it means that A is always positive, so the virtual energy absorption and energy absorption will have the same trend. The condition above can also be expressed as m Lx =ðplpe x Þ; n Ly =ðplpe y Þ. Usually it is only valid in low frequency range for decentralized feedback control. So this condition is easy to satisfy. When the excitation is broadband, the virtual energy absorption can be expressed as

P0absorb ¼

Z

fmax

fmin

E0absorb ðxÞdf

ð36Þ

4. Numerical results In order to study the relation between the kinetic energy of the plate and the virtual energy absorption of the actuator, double control units is considered and is shown in Fig. 3. There are two control units placed on the panel, and two velocity sensors placed at the center of the piezoelectric patch actuators. The parameters used in this simulation are listed in Table 1. 4.1. Control performance of single control unit with fixed gain When only control unit 1 is set to work, the feedback gain of control unit 2 is zero, and the control result with broadband disturbance is shown in Fig. 4a. With the increase of the feedback gain, the kinetic energy in low frequency range decreases at first and then increases. When the feedback gains are 0.1, 100 and 105, the frequencies of the first peak are 66 Hz, 69 Hz and 70 Hz, respectively. This indicates that the feedback control unit may gradually change the dynamic characteristics of the plate with the increase of the feedback gain. When the feedback gain is very large, new peaks emerge at high frequency range. It is due to the pin effect of the feedback control [6]. When the feedback gain is very large, the velocity at the control point is nearly zero, which means the new

m0 ¼1n0 ¼1

When the excitation is broadband, the energy absorption of piezoelectric patch actuator can be expressed as

Pabsorb ¼

Z

Y

fmax

Eabsorb ðxÞdf

-H

ð34Þ

fmin

-H

3.4. Virtual energy absorption of the piezoelectric patch actuator

0 The energy absorption of the piezoelectric patch actuator needs to have the information along the border of the patch which is not

X

Fig. 3. The diagram of two control units implemented on a simply supported plate.

Y. Cao et al. / Applied Acoustics 74 (2013) 909–919 Table 1 Parameters of the plate and control units. Parameter

Value

Dimensions of the plate Mass density of the plate Young’s modulus of the plate Poisson ratio of the plate Modal damping ratio Dimensions of the actuator Mass density of the actuator Young’s modulus of the actuator actuator Poisson ratio of the actuator d31 Control position 1 Control position 2

300 mm  250 mm  1 mm 2700 kg/m3 7  1010 N/m2 0.3 0.01 25 mm  25 mm  0.2 mm 4700 kg/m3 23.3  106 N/m2 0.3 23  1012 m/V (0.25Lx, 0.25Ly) (0.7Lx, 0.73Ly)

913

broadband white noise. It can be shown from the figure that when the feedback gain increases from 0.1 to 105, the energy absorption will increase at first and then decrease when the feedback gain is large. Moreover there is at least one maximum value in the energy absorption curve against the feedback gain. Fig. 4c shows the virtual energy absorption of the piezoelectric patch actuator when the feedback gain increases from 0.1 to 105. Similarly the virtual energy absorption is not monotonic and there exists at least one maximum value of the virtual energy absorption against the feedback gain. The results above indicate that, when only a single control unit is working, the variation of the kinetic energy, the energy absorption and the virtual energy absorption against feedback gain are not monotonic. There exists a maximum value of the virtual energy absorption over the entire frequency band (1 Hz–1 kHz). 4.2. Control performance of single control unit with varying gains

(a)

(b)

(c) Fig. 4. Control results when the feedback gain of control unit 1 is 0.1 (red solid line), 100 (blue dashed line), 105 (green dot-dashed line). (a) The kinetic energy of the plate; (b) the energy absorption of the actuator; (c) the virtual energy absorption of the actuator. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

boundary conditions form at the control point, hence, the control unit loses its control effect. Fig. 4b shows the energy absorption of the piezoelectric patch actuator when the excitation is a

4.2.1. Broadband excitation When the excitation is a broadband white noise (bandwidth is 1 Hz–1 kHz), and only control unit 1 is working, the control gain of control unit 2 is zero, the kinetic energy of the plate, the energy absorption and the virtual energy absorption of the actuator against the control gain of unit 1 are plotted in Fig. 5. It can be shown that when the feedback gain is nearly 18 dB (which is calculated by 20 log 10(Gain)), the kinetic energy of the plate reaches the minima. And when the energy absorption and the virtual energy absorption of the actuator reach the maxima, the feedback gains are approximately 18 dB and 19 dB respectively. So when the excitation is a white noise (bandwidth is 1 Hz–1 kHz), the feedback gains which maximizes the kinetic energy and minimizes the energy absorption or the virtual energy absorption are nearly the same. The similar results are analyzed by Zilletti and Gardonio [20–22]. Hence the energy absorption or the virtual energy absorption may replace the kinetic energy to be a cost function. Now consider the case where the feedback gain of control unit 2 is set to a constant value of 20 dB. The variation of the kinetic energy, the energy absorption and the virtual energy absorption against feedback gain of control unit 1 are shown in Fig. 6. It is clear that when the kinetic energy is at the minima, the feedback gain is nearly 19 dB, while when the energy absorption and the virtual energy absorption reaches their maxima, the feedback gains are respectively nearly 20 dB and 19 dB. It can be seen from Fig. 6b and c that when the energy absorption passes the maximum point against feedback gain, it changes little compared with the virtual energy absorption. This is mainly because the energy absorption uses the information of angular velocity, while the virtual energy absorption uses the information of velocity at the control point. When the feedback gain is large, the velocity at the control point is very small but the angular velocity may not be small according to Eqs. (32) and (35). In general, the energy absorption and the virtual energy absorption can both be used as the cost function. 4.2.2. Narrowband excitation When the excitation is concentrated at one single frequency, and the force or moment actuator is used to control the vibration of the plate, maximizing the absorbed energy of the actuator can usually reduce the input energy of the primary source and the secondary source, thus reducing the vibration of the structure overall. But for the low-frequency narrow band excitation, maximizing the absorbed energy may increase the kinetic energy. When the excitation bandwidth is narrow, the energy absorption of the actuator is not suited to be used as the cost function. Fig. 7a shows the difference of the feedback gains against bandwidth in two situations. The one with blue solid line is the

914

Y. Cao et al. / Applied Acoustics 74 (2013) 909–919

(a)

(b)

(c)

(a)

(b)

(c)

Fig. 5. Broadband excitation (bandwidth 1 Hz –1 kHz). (a) The kinetic energy of the plate against feedback gain; (b) the energy absorption of the actuator against feedback gain; (c) the virtual energy absorption of the actuator against feedback gain.

Fig. 6. Broadband excitation (bandwidth 1 Hz–1 kHz). The feedback gain of control unit 2 is fixed at 20 dB. (a) The kinetic energy of the plate against the feedback gain of control unit 1; (b) the energy absorption of the actuator against the feedback gain of control unit 1; (c) the virtual energy absorption of the actuator against the feedback gain of control unit 1.

difference of the feedback gains between minimizing the kinetic energy and maximizing the energy absorption. The other one with red dashed line is the difference of the feedback gains between minimizing the kinetic energy and maximizing the virtual energy absorption. Fig. 7b shows the difference of the kinetic energy against bandwidth in the same two situations used in Fig. 7a. The bandwidth is from 0 Hz to 1 kHz, the starting frequency is 1 Hz. It can be seen that when the bandwidth is smaller than 75 Hz, the difference of the feedback gain and the kinetic energy in Fig. 7a and b are both large; when the bandwidth exceeds 75 Hz, however, the difference of the feedback gain and the kinetic energy are small. Fig. 4a shows that the 75 Hz is just beyond the first peak of the kinetic energy. When the excitation bandwidth is between 0 and 75 Hz, the plate can be treated as single degree of freedom system, at which the actuator has enough controllability, which means that the variation of the kinetic energy is monotonically decreasing with the increase of the feedback gains, i.e. there is no minimum in the kinetic energy. On the other hand,

when the feedback gain is zero, the energy absorption and the virtual energy absorption are also zero. When the feedback gain is very large, the velocity at the control point approaches zero, and the angular velocity around the piezoelectric patch actuator also decreases to nearly zero. Therefore, the energy absorption and the virtual energy absorption always have a maximum value, and it is independent from the bandwidth used. With the increase of the feedback gain, the kinetic energy will finally decrease to 0, and so will the energy absorption of the actuator. So, when the bandwidth is reduced so as not to include any peaks of kinetic energy, then minimizing the kinetic energy will no longer be equivalent to maximizing the energy absorption or the virtual energy absorption. When the bandwidth exceeds 75 Hz, the energy absorption or the virtual energy absorption may replace the kinetic energy to act as the cost function. The following discussion will focus on the scenario where both control units are working at the same time: the control gain of unit 2 is fixed to 20 dB, only the control gain of unit 1 is tuned. Fig. 8a

915

Y. Cao et al. / Applied Acoustics 74 (2013) 909–919

(a)

(a)

(b) Fig. 7. Control unit 1 on, control unit 2 off. (a) The difference of the feedback gains against bandwidth between minimizing the kinetic energy and maximizing the energy absorption (blue solid line), between minimizing the kinetic energy and maximizing the virtual energy absorption (red dashed line); (b) the difference of the kinetic energy against bandwidth in the same two situations as (a). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

shows the difference of the feedback gains against bandwidth in the same two situations used in Fig. 7a. Fig. 8b shows the difference of the kinetic energy against bandwidth in the same two situations. The bandwidth and the starting frequency are the same as those shown in Fig. 7. It can be shown from Fig. 8 that when the bandwidth exceeds 75 Hz, the energy absorption or the virtual energy absorption may act as the cost function. However, there is some tiny error. This is mainly because that maximizing the energy absorption replacing minimizing the kinetic energy is only an approximation result [18], there is no fixed relation between them. In addition, the wider the bandwidth is, the smaller the error is. The results of another case are shown in Fig. 9 where the starting frequency of the bandwidth is 100 Hz, and now the bandwidth is increased from 0 Hz to 900 Hz. The starting frequency exceeds the first vibration resonance frequency of the plate due to Fig. 4a. It can be seen that when the bandwidth exceeds 51 Hz, the energy absorption or the virtual energy absorption can still act as the cost function. A frequency of 151 Hz is just beyond the second vibration resonance frequency of the plate according to the Fig. 4a. Hence the bandwidth should at least include one vibration resonance peak of the plate. 4.3. Control performance of double control unit with varying gains It can be known from previous analysis that when the gain of only one single control unit is tuned, the energy absorption or the virtual energy absorption can replace the kinetic energy to act as the cost function. In this section, the control performance of double control units with varying gains is discussed. The excitation is a broadband (1 Hz–1 kHz) white noise. Fig. 10 show the variation of the kinetic energy, the total virtual energy

(b) Fig. 8. Both control units working, control gain of unit 2 is 20 dB. (a) The difference of the feedback gains against bandwidth between minimizing the kinetic energy and maximizing the energy absorption (blue solid line), between minimizing the kinetic energy and maximizing the virtual energy absorption (red dashed line); (b) the difference of the kinetic energy against bandwidth in the same two situations as (a). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

absorption and the total energy absorption against the variation of the feedback gain of control unit 1 h1 and of control unit 2 h2. It can be seen from the figure that when both the feedback gains approach nearly 19 dB, there is a minimum value of the kinetic energy. At nearly the same feedback gains, the total energy absorption or the total virtual energy absorption approaches their maxima. Hence, when the excitation is a broadband white noise, a suitable feedback gain which makes the kinetic energy to approximately reach its minima can be found by maximizing the total energy absorption or the total virtual energy absorption. Fig. 11 shows that the variation of the virtual energy absorption and the energy absorption of each control unit against the variation of the feedback gains h1 and h2. It can be shown that if the feedback gain of one control unit is kept constant, and that of the other one is tuned, then the virtual energy absorption or the energy absorption of the corresponding control unit will have only one maximum value. Furthermore, the optimal feedback gain of this control unit is not changing as the feedback gain of the other control unit. Taking Fig. 11a as an example, if the feedback gain of the control unit 2 is set to 10 dB, the optimized feedback gain of the control unit 1 is nearly 19 dB. If the feedback gain of the control unit is changed to another value, the optimal feedback gain of the control unit 1 will remain to be nearly 19 dB. With these characteristics, the optimal feedback gain of each control unit can be found with their own virtual energy absorption. The analyses above indicate that, firstly, the minimization of the kinetic energy can be substituted by the maximization of the virtual energy absorption of the actuator to find an optimal gain; secondly, only if the bandwidth includes at least one vibration resonance peak, can the virtual energy be a valid cost function to

916

Y. Cao et al. / Applied Acoustics 74 (2013) 909–919

(a) (a)

(b) Fig. 9. Both control units working, control gain of unit 2 is 20 dB, the starting frequency is 100 Hz. (a) The difference of the feedback gains against bandwidth between minimizing the kinetic energy and maximizing the energy absorption (blue solid line), between minimizing the kinetic energy and maximizing the virtual energy absorption (red dashed line); (b) the difference of the kinetic energy against bandwidth in the same two situations as (a). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

(b)

find the optimal feedback gain, and under this circumstances the feedback gain derived from either maximizing the virtual energy absorption or minimizing the kinetic energy are approximately the same, and the wider the bandwidth is, the more stable the approximation becomes; thirdly, either tuning one control gain at a time or tuning multiple control gains simultaneously may be used according to Fig. 10 and Fig. 11; and finally, the virtual energy absorption uses only the information of the velocity signals and the feedback gains, so it is easy to measure. 5. Multi-channel robust self-tuning algorithm Maximizing the virtual energy absorption can be used to find the optimal feedback gain due to its convenience to measure. Maximizing the virtual energy absorption can be approximately equal to minimizing the kinetic energy. In this section, a multi-channel robust self-tuning algorithm based on maximizing the virtual energy absorption is proposed. This algorithm can update feedback gains of more than one control unit at a time. Fig. 12 is the diagram of decentralized velocity feedback control of plate using 9 piezoelectric patch actuator control units. The energy absorption of each control unit can be written as

1 2

P0i ¼ Refmi  v ci g ¼ C 0

d31 1 hi jv ci j2 / hi jv ci j2 2 ha

ð37Þ

where hi is the feedback gain of ith control unit, vci is the velocity signal at the ith velocity sensor, gi = 10 log 10(hi). A robust gradient updating method can be used to obtain the optimal feedback gain iteratively. With this algorithm, feedback gains of more than one

(c) Fig. 10. The variation of the kinetic energy, the total virtual energy absorption and the total energy absorption against the variation of the feedback gain of control unit 1 h1 and the feedback gain of control unit 2 h2. (a) The kinetic energy; (b) the total virtual energy absorption; (c) the total energy absorption.

control unit can be updated at a time. G(n) = [g1(n), g2(n), . . . , gi(n)]T is the feedback gains vector, P0absorb ¼ f ðGÞ is the total virtual energy absorption of the whole updating control unit, which is a function of the feedback gains vector, Dg is the increment of the feedback gain that is used to calculate the gradient of the virtual energy absorption against the feedback gain, l = [l1, l2,    , li]T is the step-size of the algorithm, glow and ghigh are the lower limit and the upper limit of the optimal feedback gain respectively, which should be settled down before the updating, d is the smallest gradient when g e [glow, ghigh], it is used to control when to stop the updating, the detailed algorithm is shown in Table. 2.

917

Y. Cao et al. / Applied Acoustics 74 (2013) 909–919

(a)

(b)

(c)

(d)

Fig. 11. The variation of the virtual energy absorption and the energy absorption of each control unit against the variation of the feedback gain of control unit 1 h1 and the feedback gain of control unit 2 h2. (a) The virtual energy absorption of control unit 1; (b) the virtual energy absorption of control unit 2; (c) the energy absorption of control unit 1; (d) the energy absorption of control unit 2.

Fig. 12. The diagram of decentralized velocity feedback control of plate using nine piezoelectric patch actuator control units.

Using the algorithm described in Table. 2, the optimal feedback gains of 9 control units can be found. The initial values of feedback gains are chosen randomly and the remaining parameters are shown in Table. 3. Fig. 13 shows the updating process of feedback

gains of the control unit 1, 2, 5 and 9. It can be seen that even if the initial value are chosen randomly, the updating of the feedback gains can converge to the optimal value with the number of iterations to be less than 15. Fig. 14 shows the corresponding feedback gains of each control unit shown in Fig. 12 after the updating. It can be seen that the feedback gains of every control units are between 17.5 dB and 19.5 dB. If the feedback gains are obtained off-line by minimizing the kinetic energy and it is assumed that all of the feedback gains are the same, the feedback gains are 18.14 dB. Fig. 15 shows the controlled performance of the decentralized control system with the optimized feedback gains. It can be seen that the kinetic energy is reduced significantly at low frequencies especially near the frequency of the resonance peaks. However, the kinetic energy after control is raised at the relatively high frequencies, which may be due to insufficient number of actuators. Nine actuators cannot control the vibration of the panel at relatively high frequencies. When the control strategy is implemented in practice, more actuators can be used or the relatively narrow frequency band can be controlled. Meanwhile, the sound radiation can be reduced significantly at the frequency of resonance peak, but raised at the frequency of anti-resonance since this type of control method is equivalent to increasing the damping. Overall, Fig. 15 shows that the proposed algorithm is able to find the optimal feedback gains.

918

Y. Cao et al. / Applied Acoustics 74 (2013) 909–919

Table 2 Multi-channel robust self-tuning algorithm. Initialization stage 1. Initialize the feedback gains of each control units with random initial value 2. Set the value of Dg, l, glow, ghigh, d properly according to the practical system. For decentralized control of plate with piezoelectric patch actuator, Dg can be chosen to be 1 dB, d can be chosen to be a very small number Direction search stage 3. Calculate the gradient of the total virtual energy absorption against the feedback gain according to the central difference formula with accuracy of O(h2), when calculating the total virtual energy against feedback gain of one control unit, the feedback gains of other control units need to be constant @ P0absorb ðnÞ @g i ðnÞ

ðnÞÞ ðg i ðnÞDgÞ ¼ @f@gðgiðnÞ ffi f ðg i ðnÞþDgÞf 2Dg i

gradðP0absorb ðnÞÞ

@ P0absorb ðnÞ @ P0absorb ðnÞ @g 1 ðnÞ ; @g 2 ðnÞ ;

¼ð

;

4. When gi R [glow, ghigh], check whether li 

li 

@ P0absorb ðnÞ @g i ðnÞ

@ P0absorb ðnÞ T @g 9 ðnÞ Þ @ P0absorb ðnÞ @g i ðnÞ

< 1, if it is true, make li = 2li; check whether li 

@ P0absorb ðnÞ @g i ðnÞ

> 5, if it is true, make li = li/2. Repeat the step 4 until

2 ½1; 5, then go to step 6

5. When gi e [glow, ghigh], repeat to check whetherli 

@ P0absorb ðnÞ @g i ðnÞ

> 2, if it is true, make li = li/2. And then check whether

@ P0absorb ðnÞ @g i ðnÞ

< d, if it is not true, go to step 6, if it is true,

stop the updating Updating stage 6. Update the feedback gain using the formula below: Gðn þ 1Þ ¼ GðnÞ þ lðnÞ  gradðP0absorb ðnÞÞ Return to step 3

Table 3 The parameters of the algorithm. Parameter

Value

Dg

1 2000 22 15 0.00001

l ghigh glow d

(a)

Fig. 13. The process of iteration.

(b) Fig. 15. The performance of the decentralized control system, controlled (solid line); uncontrolled (dashed line). (a) The kinetic energy; (b) the sound radiation power.

Fig. 14. The optimized feedback gains.

6. Conclusion In this paper, the on-line selection of the optimal feedback gain using piezoelectric patch actuator to implement the decentralized velocity feedback control is discussed. The concept of virtual energy absorption of piezoelectric patch actuator is proposed. The

numerical results have shown that, when the excitation is a broadband white noise, the feedback gain obtained from either maximizing the virtual energy absorption or minimizing the kinetic energy are almost the same. However, when the bandwidth is reduced to not include any vibration resonance peak, the virtual energy absorption and the kinetic energy have no direct relations. So, when the excitation is broadband, maximizing the virtual energy absorption can be used to find the optimal feedback gain. The virtual energy of the piezoelectric patch actuator is easy to measure

Y. Cao et al. / Applied Acoustics 74 (2013) 909–919

since only the velocity signal and the feedback gains are needed. Meanwhile, a type of multi-channel robust self-tuning algorithm which is able to find the multi-channel optimal feedback gains simultaneously is proposed. The convergence of the algorithm is checked and is assured to be robust to random initial values. Acknowledgements This work is supported by National Natural Science Foundation of China (Grant Nos. 11004216, 11004217) and the Knowledge Innovation Program of Institute of Acoustics, Chinese Academy of Sciences. References [1] Fuller CR, Elliott SJ, Nelson PA. Active control of vibration. London: Academic Press; 1996. [2] Fahy FJ, Gardonio P. Sound and structural vibration. London: Elsevier; 2007. [3] Elliott SJ. Signal processing for active control. London: Academic Press; 2001. [4] Preumont A. Vibration control of active structures. Dordrecht: Kluwer Academic Publishers Group; 2002. [5] Jayachandran V, Sun JQ. Unconditional stability domains of structural control systems using dual actuator–sensor pairs. J Sound Vib 1997;208:159–66. [6] Elliott SJ, Gardonio P, Sors TC, Brennan MJ. Active vibroacoustic control with multiple local feedback loops. J Acoust Soc Am 2002;111:908–15. [7] Gardonio P, Bianchi E, Elliott SJ. Smart panel with multiple decentralized units for the control of sound transmission. Part I: Theoretical predictions. J Sound Vib 2004;274:163–92. [8] Gardonio P, Elliott SJ. Smart panels with velocity feedback control systems using triangularly shaped strain actuators. J Acoust Soc Am 2005;117:2046–64. [9] Gardonio P, Elliott SJ. Smart panels for active structural acoustic control. Smart Mater Struct 2004;13:1314–36. [10] X. Pan, A.J. Forrest, Active Structural Acoustic Control Radiation from Flat Plates, ICA2010, Sydney, Australia.

919

[11] Paulitsch C, Gardonio P, Elliott SJ. Active vibration control using an inertial actuator with internal damping. J Acoust Soc Am 2006;119:2131–40. [12] Frampton KD, Baumann ON, Gardonio P. A comparison of decentralized distributed and centralized vibro-acoustic control. J Acoust Soc Am 2010;128:2798–806. [13] Hill SG, Tanaka N, Iwamoto H. Local sensing/control or global error sensing/ control in structural acoustics: a comparison of two techniques influence over sound power. Appl Acoust 2010;71:965–78. [14] White WR, Nelson PA, Curtis ARD. Experiments on the active control of flexural wave power flow. J Sound Vib 1987;112:187–91. [15] Guicking D, Melcher J, Wimmel R. Active impedance control in mechanical structures. Acoustics 1989;69:39–52. [16] Hirami N. An active maximum power absorber for the reduction of noise and vibration. J Sound Vib 1997;200:261–79. [17] Hirami N. Optimal energy absorption as an active noise and vibration control strategy. J Sound Vib 1997;200:243–59. [18] Bardou O, Gardonio P, Elliott SJ, Pinnington RJ. Active power minimization and power absorption in a plate with force and moment excitation. J Sound Vib 1997;208:111–51. [19] Sharp SJ, Nelson PA, Koopmann GH. A theoretical investigation of optimal power absorption as a noise control technique. J Sound Vib 2002;251:927–35. [20] Zilletti M, Elliott SJ, Gardonio P. Self-tuning control systems of decentralized velocity feedback. J Sound Vib 2010;329:2738–50. [21] Zilletti M, Elliott SJ, Gardonio P, Rustighi E. Experimental implementation of a self-tuning control system for decentralized velocity feedback. J Sound Vib 2012;331:1–14. [22] Gardonio P, Miani S, Blanchini F, Casagrande D, Elliott SJ. Plate with decentralized velocity feedback loops: power absorption and kinetic energy considerations. J Sound Vib 2012;331:1722–41. [23] Wang BT, Fuller CR, Dimitriadis EK. Active control of structurally radiated noise using multiple piezoelectric actuators. AIAA J 1990;29:1802–9. [24] Choi SB. Active structural acoustic control of a smart plate featuring piezoelectric actuators. J Sound Vib 2006;294:421–9. [25] Johnson ME, Elliott SJ. Active control of sound radiation using volume velocity cancellation. J Acoust Soc Am 1995;98:2174–86. [26] Clark RL, Fuller CR, Wicks A. Characterization of multiple piezoelectric actuators for structural excitation. J Acoust Soc Am 1991;90:346–57. [27] Elliott SJ, Johnson ME. Radiation modes and the active control of sound power. J Acoust Soc Am 1993;94:2194–204.