Simulation study of active control of vibro-acoustic response by sound pressure feedback using modal pole assignment strategy

Abstract

The active control of vibro-acoustic response using sound pressure feedback is numerically studied. An output feedback approach based on sound pressure measurement for modal pole placement is proposed. The control performance is evaluated for the case of a baffled plate. The finite element method and the Rayleigh integral are used to model the structural vibration and sound radiation. Measures of observability of a mode in pressure outputs and the effect of time delay in the pressure feedback loop are discussed. Numerical results show that the chosen poles may be assigned to predetermined values by the active control with complex gains. It is also demonstrated that the pressure feedback control may make a very large reduction in acoustic radiation and structural vibration of the controlled modes in a wide frequency band when using constant gain at a lower frequency.

Introduction

There has been much work concerned with active vibration control (AVC) of flexible structures,^1–5 and the great majority of the work has been concerned with feedback control. Fuller et al. ¹ also deals with the field of active structural acoustic control (ASAC) in which structurally radiated sound is directly controlled by active structural inputs. In a general arrangement of ASAC, microphones are used as the error transducers, which directly measure the radiated pressure to be minimized and then provide the error signals. Burdisso and Fuller ⁶ obtained the error information by placing a single microphone in the acoustic far field and used a feedforward control approach to derive the optimum control input applied to the actuator. Clark and Fuller ⁷ used microphones as error sensors, while the control approach was based upon a filtered-X version of the adaptive least-mean-squares algorithm. Wang and Fuller ⁸ used a single microphone as an error sensor and defined the mean square of the error sensor signal as the quadratic cost function. Wang ⁹ investigated the optimal placement of microphones and piezoelectric transducer actuators for far-field sound radiation control with microphones used as error sensors and radiated sound power as the objective function. Berry et al. ¹⁰ investigated the acoustic near-field sensing strategies for active control of sound radiated from a plate. It is clear that in the general ASAC analysis, a quadratic cost function is formed based on the observed pressures, and the quadratic optimization theory is used to find the optimal control inputs that minimize the cost function. Besides the acoustic sensing and acoustic objective functions, structural sensing approaches^11,12 and structural objective functions^13–15 such as squared velocity, volume velocity, and weighted sum of spatial gradients were studied for ASAC. Berkhoff ¹⁶ discussed sensor scheme design for the active reduction of sound radiation from plates based on error signals derived from spatially weighted plate velocity or near-field pressure. In this paper, microphones are not used as error sensors; the acoustic pressures from the microphones are not sought to be minimized but to feedback for pole assignment for the active control of vibro-acoustic response. As is known, the sound pressure measurement by microphones is non-contact measurement and thus can eliminate mass loading due to use of vibration sensors such as accelerometers in AVC or ASAC implementations.

The problem of pole assignment (pole placement or eigenvalue assignment) has received considerable attention from the active-control and vibration communities over several decades. ⁵ The pole placement is about to set natural frequencies and damping to specified values and capable of changing their behaviour to respond in a desirable way to a varying demand. ¹⁷ A variety of pole assignment algorithms such as eigenvalue assignment by state feedback, eigenvalue assignment by output feedback, and robust eigenvalue assignment problem have been developed. ¹⁸ Ram and Mottershead ⁵ developed a new theory known as receptance method for pole assignment. The receptance method is based entirely upon measured vibration data in the form of receptances, there being no need to know or evaluate the mass, stiffness and damping matrices, no need to estimate the unmeasured state using an observer or a Kalman filter and no need for model reduction. ¹⁸ The independent modal space control, formulated by Meirovitch, ¹⁹ was based on the principle of modal decomposition and provides independent control over the natural frequencies and damping of structural vibration modes. The modal control in general refers to the procedure of decomposing the dynamic equations of a structure into modal coordinates and designing the control system in this modal coordinate system. As noted by Inman, ²⁰ modal control can be cast either in ‘state space’ form or ‘physical space’ form (in terms of the physical modes of the structure). From the open literature, it may be concluded that most researchers used acceleration, velocity, and displacement feedback to assign eigen data of vibration systems. ²¹ In this paper, a modal pole assignment control method is proposed, which is based entirely upon sound data, not vibration data.

As stated before, in the present paper, structural sensing approaches are not used and also microphones are not used as error sensors, acoustic pressures from the microphones are used to feedback for pole assignment for the active control of vibro-acoustic response. The goal of this work is only to numerically study the possibility of active control by sound pressure feedback using modal pole assignment strategy. An output feedback approach based on sound pressure outputs is proposed for pole placement. The finite element method and the Rayleigh integral are used to model the structural vibration and sound radiation.

Theory

Conventional feedback control for pole assignment

The system considered here has the form

$\mathbf{M}\stackrel{\mathrm{··}}{\mathbf{q}}+\mathbf{C}\stackrel{·}{\mathbf{q}}+\mathbf{Kq}=\mathbf{f}f+\mathbf{f}$

(1)

where M, C, and K are the structural mass, proportional damping ( $\mathbf{C}=\alpha \mathbf{M}+\beta \mathbf{K}$ , where α and β are real scalars), stiffness matrices respectively, q is the displacement vector, the vector f represents external disturbance forces, and the vector $\mathbf{f}f$ represents the control force derived from the action of force actuators represented by

$\mathbf{f}f=\mathbf{B}f\mathbf{u}$

(2)

where the $m\times 1$ vector u denotes the m inputs, one for each control device (actuator), and $\mathbf{B}f$ denotes the $nd\times m$ matrix of influence coefficients (weighting factors), n_d is the total number of structural degrees of freedom (DOF). In order to be able to feedback the acceleration, velocity or displacement, let y be an $s\times 1$ vector of sensor outputs denoted and defined by

$\mathbf{y}=\mathbf{C}a\stackrel{\mathrm{··}}{\mathbf{q}}+\mathbf{C}v\stackrel{·}{\mathbf{q}}+\mathbf{C}d\mathbf{q}$

(3)

where $\mathbf{C}a$ , $\mathbf{C}v$ and $\mathbf{C}d$ are $s\times nd$ matrices of acceleration, velocity or displacement influence coefficients, respectively.

Let $\Phi$ be the matrix of eigenvectors (mode shapes) of K normalized with respect to M, then the commutivity of the matrices yields

$\Phi \mathbf{T}\mathbf{M}\Phi =\Lambda \mathbf{M}=\text{diag}(mi)=\mathbf{I}$

(4)

$\Phi \mathbf{T}\mathbf{K}\Phi =\Lambda \mathbf{K}=\text{diag}(ki)=\text{diag}({\omega }_i^2)$

(5)

$\Phi T\mathbf{C}\Phi =\Lambda \mathbf{C}=\text{diag}(ci)=\text{diag}(2\zeta i\omega i)$

(6)

where $\omega i$ are the modal frequencies and $\zeta i$ are the modal damping ratios, $\omega i=\sqrt{ki/mi}$ , $ci=2\zeta i\sqrt{miki}$ , $i=1,2,\dots ,nd$ .

Substituting $\mathbf{q}=\Phi z$ in equation (1) and premultiplying by $\Phi T$ to transform the control problem in physical coordinates into modal coordinates yields

If the output feedback is used, the input vector u is of the form

$\mathbf{u}=\mathbf{G}f\mathbf{y}$

(9)

where the $m\times s$ matrix, $\mathbf{G}f$ is the feedback gain matrix. Then, the preceding becomes

$(\mathbf{I}-\Phi T\mathbf{B}f\mathbf{G}f\mathbf{C}a\Phi )\stackrel{\mathrm{··}}{\mathbf{z}}+(\Lambda \mathbf{C}-\Phi T\mathbf{B}f\mathbf{G}f\mathbf{C}v\Phi )\stackrel{·}{\mathbf{z}}+(\Lambda \mathbf{K}-\Phi T\mathbf{B}f\mathbf{G}f\mathbf{C}d\Phi )\mathbf{z}=\Phi T\mathbf{f}$

(10)

Now, equation (1) is posed for modal pole assignment problem.

To cause the closed-loop system to have the desired eigenvalues and, hence, a desired response, the $\mathbf{B}f$ , $\mathbf{C}a$ , $\mathbf{C}v$ , $\mathbf{C}d$ and $\mathbf{G}f$ must be chosen appropriately. The choice of these gains is not independent but rather coupled through the equations

$\Phi T\mathbf{B}f\mathbf{G}f\mathbf{C}a\Phi =\mathrm{diag}(-\gamma i)$

(11)

$\Phi T\mathbf{B}f\mathbf{G}f\mathbf{C}v\Phi =\mathrm{diag}(-\alpha i)$

(12)

$\Phi T\mathbf{B}f\mathbf{G}f\mathbf{C}d\Phi =\mathrm{diag}(-\beta i)$

(13)

By doing this, equation (10) reduces to the n_d single DOF control problems of the form

$(1+\gamma i)\stackrel{\mathrm{··}}{z}+(2\zeta i\omega i+\alpha i)\stackrel{·}{z}+({\omega }_i^2+\beta i)z=fi$

(14)

In the case as acceleration feedback or velocity feedback or displacement, feedback is used independently, equation (14) represents an independent set of equations that can be solved for $\gamma i$ or $\alpha i$ or $\beta i$ .

For the case of harmonic excitation of frequency ω, equations (1) and (10) can be written as

$(-\omega 2\mathbf{M}+\text{i}\omega \mathbf{C}+\mathbf{K})\mathbf{x}=\mathbf{F}f+\mathbf{F}$

(15)

$-\omega 2(\mathbf{I}-\Phi T\mathbf{B}f\mathbf{G}f\mathbf{C}a\Phi )\mathbf{w}+\text{i}\omega (\Lambda \mathbf{C}-\Phi T\mathbf{B}f\mathbf{G}f\mathbf{C}v\Phi )\mathbf{w}+(\Lambda \mathbf{K}-\Phi T\mathbf{B}f\mathbf{G}f\mathbf{C}d\Phi )\mathbf{w}=\Phi T\mathbf{F}$

(16)

where x is the complex amplitude of the displacement vector for all structural DOF, and w is the complex amplitude of the modal displacement vector, $\mathbf{x}=\Phi w$ . From the above equations, it is clear that the modal matrix $\Phi$ is required for the control. In the present study, the modal matrix is obtained based on the finite element method. To avoid the uncertainties, assumptions, and errors associated with the finite element model, a more reliable modal matrix could also be obtained by experimental modal analysis.

The theory above began with the assumption of a multi-DOF lumped parameter system, flexible structures are distributed parameter systems which, in principle, have an infinite number of DOFs. In practice, flexible structures are discretized by a finite number of coordinates (e.g. finite elements), and this leads to a finite number of modes and is in general quite sufficient to account for the low-frequency dynamical behaviour in most practical situations. It is clear that both observation spillover and control spillover will inevitably be present when a distributed parameter system is controlled using the above modal control method which assumes a finite number of modes. The spillover phenomenon arises from the excitation of the residual modes (modes which are not accounted for by the control system) by the control (control spillover) and the contamination of the sensor output by the residual modes (observation spillover). It has been shown, however, that a small amount of damping greatly reduces the possibility of instability due to spillover. ²² The spillover problem will not be addressed in this paper.

Acoustic pressures at field points

For a time harmonic $(ei\omega t)$ acoustic field, the Helmholtz integral equation is expressed as follows

$C(P)p(P)=\int S(\frac{\partial G(Q,P)}{\partial n}p(Q)-G(Q,P)\frac{\partial p(Q)}{\partial n})\text{d}S(Q)$

(17)

where the function $G(Q,P)=e-\text{i}\mathrm{kR}/4\pi R$ is the free-space Green’s function in which $R=|Q-P|$ , Q is any point on S, and P maybe outside, inside, or on S; $k=\omega /c$ is the wavenumber, where c is the speed of sound and n is the outward unit normal on S. The coefficient C(P) can be calculated by

$C(P)=1,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}P\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{outside}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}S$

(18a)

$C(P)=0,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}P\text{inside}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}S$

(18b)

and

$C(P)=1-\int S\frac{\cos \beta }{4\pi R2}\text{d}S(Q),\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}P\text{on}S$

(18c)

where β is defined as the angle between the normal n and the vector R. On the boundary, the normal derivative of acoustic pressure is related to the normal velocity v_n through the momentum equation

$\frac{\partial p}{\partial n}=-\text{i}\omega \rho vn$

(19)

where ρ is the density of the acoustic medium.

The discretization of the Helmholtz integral equation $(P\text{on}S)$ and $(P\hspace{1em}\text{outside}\hspace{1em}S)$ leads to ²³

$\mathbf{E}s\mathbf{p}s=\mathbf{D}s\mathbf{v}\mathrm{ns}$

(20)

$\mathbf{p}f=\mathbf{E}f\mathbf{p}s+\mathbf{D}f\mathbf{v}\mathrm{ns}$

(21)

where $\mathbf{E}s$ and $\mathbf{D}s$ are dipole and monopole matrices on the surface, $\mathbf{E}f$ and $\mathbf{D}f$ are those corresponding to field pressures, $\mathbf{p}s$ and $\mathbf{v}\mathrm{ns}$ denote the acoustic pressure and normal velocity vectors at the nodal locations of a grid defining the surface of the structure, and $\mathbf{p}f$ represents the field pressure vector, that is, the acoustic pressure at field points. If equation (20) is substituted into equation (21), the field pressure $\mathbf{p}f$ could be expressed by only $\mathbf{v}\mathrm{ns}$ as

$\mathbf{p}f=(\mathbf{D}f+\mathbf{E}f{\mathbf{E}}_s^{-1}\mathbf{D}s)\mathbf{v}\mathrm{ns}=\mathbf{D}v\mathbf{v}\mathrm{ns}$

(22)

If a planar surface extends over an infinite half-space, then the acoustic pressure at any field point P according to the Rayleigh integral can be described as follows

$p(P)=\text{i}\omega \rho \int Se-\text{i}\mathrm{kR}vn(Q)/2\pi R\text{d}S$

(23)

where $p(P)$ is the acoustic pressure at the field point P, $vn(Q)$ is the normal velocity of the vibrating surface at a point Q on the plate surface, S is the plate surface. According to equation (23), the field pressures $\mathbf{p}f$ also can be related to the structural normal velocity $\mathbf{v}\mathrm{ns}$ by

$\mathbf{p}f=\mathbf{D}v\mathbf{v}\mathrm{ns}$

(24)

The vector of normal velocity $\mathbf{v}\mathrm{ns}$ in equation (22) or equation (24) is related to the vector of the structural velocity v, and the vector of the structural displacement x by a transformation matrix G

$\mathbf{v}\mathrm{ns}=\mathbf{G}T\mathbf{v}=\text{i}\omega \mathbf{G}T\mathbf{x}$

(25)

Pole assignment using acoustic pressure feedback

In order to be able to feedback the field pressures, let y be an $s\times 1$ vector of acoustic pressure outputs $\mathbf{p}f$ denoted and defined by

$\mathbf{y}=\mathbf{p}f=\text{i}\omega \mathbf{D}v\mathbf{G}T\mathbf{x}=(\mathbf{C}\mathrm{pr}+\text{i}\mathbf{C}\mathrm{pi})\mathbf{x}=\mathbf{C}p\mathbf{x}$

(26)

where $\mathbf{C}\mathrm{pr}$ and $\mathbf{C}\mathrm{pi}$ are $s\times nd$ matrices of pressure influence coefficient.

If the output feedback is used, the input vector u is of the form

$\mathbf{u}=\mathbf{G}f\mathbf{y}=\mathbf{G}f\mathbf{p}f=\mathbf{G}f\mathbf{C}p\mathbf{x}$

(27)

where the $m\times s$ matrix $\mathbf{G}f$ is the pressure feedback gain matrix. Then, the preceding equation (16) becomes

$-\omega 2\mathbf{w}+\text{i}\omega (\Lambda \mathbf{C}-\Phi T\mathbf{B}f\mathbf{G}f\mathbf{C}\mathrm{pi}\Phi /\omega )\mathbf{w}+(\Lambda \mathbf{K}-\Phi T\mathbf{B}f\mathbf{G}f\mathbf{C}\mathrm{pr}\Phi )\mathbf{w}=\Phi T\mathbf{F}$

(28)

To cause full pole placement, complex gain is needed, that is

$\Phi T\mathbf{B}f\mathbf{G}f(\mathbf{C}\mathrm{pr}+\text{i}\mathbf{C}\mathrm{pi})\Phi =\mathrm{diag}(-\beta i-\text{i}\omega \alpha i)$

(29)

where $\mathbf{G}f$ is a complex gain matrix. As is known, a feedback control system is to feed each output back to each input via one element in the matrix of feedback gains. In practice, the input is the signals fed back from the output weighted by the elements of the gain matrix. It is clear that a real gain means that the phase difference between the input and the output is 0° (positive real gain) or 180° (negative real gain), and a complex gain indicates that not only the signals from the output should be weighted but also a phase shift should be needed.

It should be noted that when using real control gain for the sound pressure feedback, the following holds

$\Phi T\mathbf{B}f\mathbf{G}f\mathbf{C}\mathrm{pi}\Phi /\omega =\mathrm{diag}(-\alpha i)$

(30a)

$\Phi T\mathbf{B}f\mathbf{G}f\mathbf{C}\mathrm{pr}\Phi =\mathrm{diag}(-\beta i)$

(30b)

That is, the conditions given by equations (30a) and (30b) should be satisfied simultaneously by a real gain matrix $\mathbf{G}f$ , and so the control of $\alpha i$ and $\beta i$ is not always capable of being independently implemented in the case of using real gain. One may simply use only one matrix $\mathbf{C}\mathrm{pr}$ or $\mathbf{C}\mathrm{pi}$ to solve the real gain matrix $\mathbf{G}f$ by equations (30a) or (30b). As is known, for example, when using velocity feedback, real controller gain leads to active damping control and complex gain leads to active damping/mass control. ²⁴ It is clear that when using sound pressure feedback, even a real control gain leads to active damping( $\alpha i$ )/mass( $\beta i$ ) control; however, due to $\mathbf{C}\mathrm{pr}$ and $\mathbf{C}\mathrm{pi}$ are not independent and only one equation (30a) or (30b) is satisfied, the control of damping and mass are not independent for the sound pressure feedback using a real gain.

The effect of time delays in the feedback loop

One of the most important effects which limit the performance of feedback controllers in practical mechanical systems is unmodelled phase shift. ¹ Such phase shift may arise because of the dynamic response of sensors or actuators used or may be due to time delays in the controller and sound propagation. Such time delay in the feedback path can be considered as $e-\text{i}\omega \tau$ in the frequency domain. If the delay τ is small, $e-\text{i}\omega \tau$ can be expressed as ¹

$e-\text{i}\omega \tau \approx 1-\text{i}\omega \tau \mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{for}\hspace{1em}\omega \tau \ll 1$

(31)

With such a delay, equation (29) becomes

$(1-\text{i}\omega \tau )\Phi T\mathbf{B}f\mathbf{G}f(\mathbf{C}\mathrm{pr}+\text{i}\mathbf{C}\mathrm{pi})\Phi =\mathrm{diag}[(-\beta i-\omega 2\tau \alpha i)-\text{i}\omega (\alpha i-\tau \beta i)]$

(32)

It can be seen that for the pole assignment using acoustic pressure feedback, the effect of the delay, to a first approximation, is to change the natural frequency and the damping, especially the natural frequency is affected in a frequency-dependent manner.

Modal controllability and observability

Controllability and observability are structural properties that carry useful information for structural testing and control. A structure is controllable if the installed actuators excite all its structural modes. It is observable if the installed sensors detect the motions of all the modes. ²⁵ A measure of modal controllability of the ith mode from the jth input could be defined by the consines of angles^17,26

$\cos \theta (c)\mathrm{ij}=(\frac{|{\varphi }_i^{T}b\mathrm{fj}|}{\Vert \varphi i\Vert \Vert b\mathrm{fj}\Vert })$

(33)

where $\varphi i$ is the ith open-loop eigenvector, $b\mathrm{fj}$ is the jth column of $\mathbf{B}f$ , and $\Vert •\Vert$ is the l₂ norm $\Vert •\Vert 2$ . The ith mode is then said to be uncontrollable from the jth input when ${\varphi }_i^{T}b\mathrm{fj}=0$ . A measure of modal observability of the ith mode in the kth output could be defined by the consines of angles^17,26

$\cos \theta (o)\mathrm{ki}=(\frac{|{\varphi }_i^T(-{\omega }_i^{2}c\mathrm{ak}+\text{i}\omega ic\mathrm{vk}+c\mathrm{dk})|}{\Vert \varphi i\Vert \Vert -{\omega }_i^{2}c\mathrm{ak}+\text{i}\omega ic\mathrm{vk}+c\mathrm{dk}\Vert })$

(34)

where $c\mathrm{ak},c\mathrm{vk},c\mathrm{dk}$ are the kth rows of $\mathbf{C}a,\mathbf{C}v,\mathbf{C}d$ . Likewise, the ith mode is said to be unobservable in the kth output when ${\varphi }_i^T(-\omega 2c\mathrm{ak}+\text{i}\omega c\mathrm{vk}+c\mathrm{dk})=0$ . As is known, the matrices $\mathbf{B}f$ , $\mathbf{C}a$ , $\mathbf{C}d$ and $\mathbf{C}v$ are dependent on the actuator and sensor locations. Modal controllability and observability mean not to put all the actuators and sensors at the nodal points of the ith mode if the ith mode is to be controllable and observable.

For pressure feedback, the measure of modal observability could be defined as

$\cos \theta (o)\mathrm{ki}=(\frac{|{\varphi }_i^T(\text{i}c\mathrm{pik}+c\mathrm{prk})|}{\Vert \varphi i\Vert \Vert \text{i}c\mathrm{pik}+c\mathrm{prk}\Vert })$

(35)

where $c\mathrm{pik}\text{and}c\mathrm{prk}$ are the kth rows of $\mathbf{C}\mathrm{pi}\text{and}\mathbf{C}\mathrm{pr}$ . Likewise, the ith mode is said to be unobservable in the kth output when ${\varphi }_i^T(\text{i}c\mathrm{pik}+c\mathrm{prk})=0$ . That is, when a microphone lies on a nodal line in the radiation field of the ith mode, ${\varphi }_i^T(\text{i}c\mathrm{pik}+c\mathrm{prk})=0$ , and the mode cannot be measured because the acoustic waves corresponding to that mode are out of phase and the resulting pressure at the position of the microphone vanishes. ²⁷

Active control simulation and vibro-acoustic response

For a structure and a multiple-input-multiple-output control system using acceleration/velocity/displacement feedback or pressure feedback, the modal pole assignment problem is to solve the gain $\mathbf{G}f$ from equations (11) to (13) based on $\Phi$ , $\mathbf{B}f$ , $\mathbf{C}a$ , $\mathbf{C}d$ and $\mathbf{C}v$ for the acceleration/velocity/displacement feedback or from equation (29) based on $\Phi$ , $\mathbf{B}f$ , $\mathbf{C}\mathrm{pr}$ and $\mathbf{C}\mathrm{pi}$ for the sound pressure feedback, to get desired modal response information.

Once $\mathbf{G}f$ is chosen, the eigenvalues of the controlled system and the corresponding vibro-acoustic response x, $\mathbf{v}\mathrm{ns}$ and $\mathbf{p}s$ are determined. Further, the mean-square normal velocity ${\overline{v}}_n^2$ and the acoustic power Π of the structure can be calculated from the following formulas

${\overline{v}}_n^2=\frac{1}{2S0}\int S|vn|2\text{d}S$

(36)

where S₀ is the area of the structure surface.

$\Pi =\frac{1}{2}\int S\text{Re}({\mathrm{pv}}_n^{*})\text{d}S$

(37)

in which the asterisk denotes complex conjugate.

Numerical results

The dimensions of a simply supported plate in a baffle considered ( $\mathrm{xoy}$ plane with $z=0$ ) are $Lx=0.38\text{m}$ , $Ly=0.30\text{m}$ , and thickness $h=0.002\text{m}$ . The plate is made of steel with density $\rho s=7800\text{kg}/\text{m}3$ , Young’s modulus $E=2.0\times 1011\text{N}/\text{m}2$ , and Poisson’s ratio $\nu =0.28$ . The proportional damping coefficients are $\alpha =7.362$ and $\beta =1.177\times 10-5$ . The plate is assumed to be vibrating in air with a density $\rho =1.21\text{kg}/\text{m}3$ and a sound speed $c=343\text{m}/\text{s}$ . The plate and the selected actuator/sensor configuration are shown in Figure 1, where the configuration (F, u1, u2, VS1, VS2, VS3) is based on Rodriguez and Burdisso. ²⁸ The disturbance is assumed to be a point force with amplitude $F=1\text{N}$ and located at $(x,y)=(0.23750\text{m},0.13125\text{m})$ .

Figure 1.

Plate element mesh and sensor/actuator configuration.

To control the structural vibration and acoustic radiation of the plate, a 2I3O control configuration is used, where “I” and “O” denote input and output channels, that is, $m=2$ and $s=3$ . The inputs (control forces) u1 and u2 are located at $(0.26125\text{m},0.11250\text{m})$ and $(0.26125\text{m},0.20625\text{m})$ . For the acceleration, velocity, and displacement feedback control, the vibration sensors VS1, VS2, and VS3 are located at $(0.23750\text{m},0.15000\text{m})$ , $(0.07125\text{m},0.24375\text{m})$ , and $(0.04750\text{m},0.01875\text{m})$ . For the sound pressure control, the microphones PS1, PS2, and PS3 are located at the same $(x,y)$ as the vibration sensors VS1, VS2, and VS3 but on a plane $z=1.5\text{m}$ (or $z=0.01\text{m}$ ) above the surface of the plate. To investigate the effect of the sensor and actuator positions on the control performance, locations $(0.095\text{m},0.15\text{m})$ , $(0.190\text{m},0.15\text{m})$ , and $(0.285\text{m},0.15\text{m})$ are also considered as the sensor/actuator locations as shown in Figure 1 (VSa, VSb, VSc for outputs and ua, ub, uc for inputs, respectively, the microphones PSa, PSb, and PSc are located at the same $(x,y)$ as the vibration sensors VSa, VSb, and VSc but on a plane $z=1.5\text{m}$ or $z=0.01\text{m}$ above the surface of the plate).

The plate is modelled by $16\times 16$ four-noded quadrilateral isoparametric elements, and the element mesh is also shown in Figure 1. The finite element is based on the Mindlin plate theory (the first-order shear deformation theory). The Rayleigh integral on the plate surface is calculated by discretising the plate surface with the same mesh as the plate finite element mesh. The total number of structural DOFs is $nd=803$ . Based on the developed finite element method and Rayleigh integral method, the matrices M and K are obtained by the finite element model, and then the matrices C and $\Phi$ are obtained; the matrices $\mathbf{C}\mathrm{pr}$ and $\mathbf{C}\mathrm{pi}$ are determined from the Rayleigh integral. From the actuator/sensor configuration, the input matrix $\mathbf{B}f$ is $803\times 2$ with two non-zero elements, the output matrices $\mathbf{C}a$ , $\mathbf{C}v$ and $\mathbf{C}d$ are $3\times 803$ matrices with three non-zero elements, the output matrices $\mathbf{C}\mathrm{pr}$ and $\mathbf{C}\mathrm{pi}$ are also $3\times 803$ matrices but with $3\times 225=675$ non-zero elements, the feedback gain matrix $\mathbf{G}f$ is $2\times 3$ .

The modal natural frequencies and the modal damping ratios of the uncontrolled plate in the frequency band (5, 400) Hz are listed in Table 1. The modal controllability measures for inputs (u1, u2), (ua, uc), and (ub, uc) are shown in Table 2. The modal observability measures for displacement outputs (VS1, VS2, VS3) and (VSa, VSb, VSc) are shown in Table 3. The modal observability measures for pressure outputs (PS1, PS2, PS3) and (PSa, PSb, PSc) at 20 Hz and 300 Hz with $z=1.5\text{m}$ and $z=0.01\text{m}$ are shown in Tables 4 and 5. From Table 2, it is noted that the third mode and the fourth mode are uncontrollable from ua, ub, and uc, and the second mode is uncontrollable from ub. From Table 3, the third mode and the fourth mode are unobservable in VS1, VSa, VSb, VSc, and the second mode is unobservable in VSb. From Tables 4 and 5, the third mode and the fourth mode are unobservable in PS1, PSa, PSb, PSc, and the second mode is unobservable in PSb. From Tables 3 to 5, it can be seen that when the pressure measurement locations are very near the plate, for example, on the plane $z=0.01\text{m}$ , the modal observability measures from the pressure outputs and from the displacement outputs have similar trends, but when the pressure measurement locations are a little far from the plate, for example, on the plane $z=1.5\text{m}$ , it seems that pressure outputs have better observability for the first mode (1,1) and the fifth mode (3,1), which have the highest sound radiation efficiencies. From Table 4, it can also be seen that the observability measures of the second mode, the third mode and the fourth mode in PS1, PS2, and PS3 decrease significantly when the measurement locations move from $z=0.01\text{m}$ to $z=1.5\text{m}$ , especially for the four mode (2,2) which has the lowest sound radiation efficiency. Tables 4 to 5 show that the excitation frequency has only a small effect on the observability measures.

Table 1.

Modal natural frequencies and modal damping ratios of the plate.

Mode (nx, ny)	(1,1)	(2,1)	(1,2)	(2,2)	(3,1)
Frequency (Hz)	86	185	245	341	352
Damping ratio	0.0099	0.0100	0.0114	0.0143	0.0147

Table 2.

Modal controllability measures.

Mode	Inputs (u1, u2)		Inputs (ua, uc)		Inputs (ub, uc)
	u1	u2	ua	uc	ub	uc
1	0.0960	0.0864	0.0884	0.0884	0.1250	0.0884
2	0.1067	0.0960	0.1250	0.1250	0.0000	0.1250
3	0.0735	0.0960	0.0000	0.0000	0.0000	0.0000
4	0.0816	0.1067	0.0000	0.0000	0.0000	0.0000
5	0.0225	0.0203	0.0884	0.0884	0.1250	0.0884

Table 3.

Modal observability measures of displacement outputs.

Output		Mode
		1	2	3	4	5
(VS1, VS2, VS3)	VS1	0.1154	0.0884	0.0000	0.0000	0.0478
	VS2	0.0386	0.0642	0.0642	0.1067	0.0681
	VS3	0.0093	0.0173	0.0183	0.0338	0.0225
(VSa, VSb, VSc)	VSa	0.0884	0.1250	0.0000	0.0000	0.0884
	VSb	0.1250	0.0000	0.0000	0.0000	0.1250
	VSc	0.0884	0.1250	0.0000	0.0000	0.0884

Table 4.

Modal observability measures of pressure outputs at PS1, PS2, and PS3.

Output			Mode
			1	2	3	4	5
20 Hz	$z=0.01\text{m}$	PS1	0.8775	0.2995	0.0000	0.0000	0.0122
		PS2	0.5740	0.3142	0.2601	0.2931	0.2820
		PS3	0.4970	0.2165	0.1903	0.2006	0.2299
	$z=1.5\text{m}$	PS1	0.8628	0.0020	0.0000	0.0000	0.2757
		PS2	0.8628	0.0049	0.0030	0.0000	0.2757
		PS3	0.8627	0.0058	0.0042	0.0001	0.2757
300 Hz	$z=0.01\text{m}$	PS1	0.8505	0.3310	0.0000	0.0000	0.1079
		PS2	0.5288	0.3689	0.2973	0.3118	0.2418
		PS3	0.4523	0.2764	0.2363	0.2234	0.1854
	$z=1.5\text{m}$	PS1	0.8717	0.0121	0.0000	0.0000	0.2668
		PS2	0.8712	0.0301	0.0183	0.0007	0.2663
		PS3	0.8708	0.0360	0.0256	0.0012	0.2661

Table 5.

Modal observability measures of pressure outputs at PSa, PSb, and PSc.

Output			Mode
			1	2	3	4	5
20 Hz	$z=0.01\text{m}$	PSa	0.7667	0.4560	0.0000	0.0000	0.3009
		PSb	0.9179	0.0000	0.0000	0.0000	0.1477
		PSc	0.7667	0.4560	0.0000	0.0000	0.3009
	$z=1.5\text{m}$	PSa	0.8628	0.0039	0.0000	0.0000	0.2757
		PSb	0.8628	0.0000	0.0000	0.0000	0.2757
		PSc	0.8628	0.0039	0.0000	0.0000	0.2757
300 Hz	$z=0.01\text{m}$	PSa	0.7297	0.5111	0.0000	0.0000	0.2663
		PSb	0.8949	0.0000	0.0000	0.0000	0.2328
		PSc	0.7297	0.5111	0.0000	0.0000	0.2663
	$z=1.5\text{m}$	PSa	0.8715	0.0242	0.0000	0.0000	0.2666
		PSb	0.8718	0.0000	0.0000	0.0000	0.2669
		PSc	0.8715	0.0242	0.0000	0.0000	0.2666

First, it is desired to modify the first modal frequency from 86 Hz to 100 Hz and the fifth modal frequency from 352 Hz to 375 Hz and to modify the first modal damping ratio from 0.0099 to 0.03 and the fifth modal damping ratio from 0.0147 to 0.03 by active control with inputs (ub, uc) and outputs (PSa, PSb, PSc) of $z=1.5\text{m}$ . For this inputs/outputs configuration, based on Tables 2 and 5, the influences of modes 3 and 4 are removed since these two modes are completely unobservable and uncontrollable. It should be noted that since the pressure influence coefficient matrices $\mathbf{C}\mathrm{pr}$ and $\mathbf{C}\mathrm{pi}$ are frequency dependent, the complex gains are also frequency dependent. To achieve full pole assignment, the complex gain matrix for each frequency is obtained from equation (29) in the frequency band (5, 400) Hz with a frequency increment of 1 Hz and the 2-norm of the complex gain matrix is shown in Figure 2 as a function of the frequency. The sound power level (dB, $\mathrm{re}:10-12\text{W}$ ), the mean-square normal velocity level (dB, re: $1\text{m}2/s2$ ) and the sound pressure level (dB, $\mathrm{re}:20\mu \text{Pa}$ ) at PSb $(0.190\text{m},0.15\text{m},1.50\text{m})$ of the plate before and after control in the frequency band (5, 400) Hz are shown in Figures 3 to 5. Figures 3 to 5 visually confirm the excellent realization of the pole assignment.

Figure 2.

The 2-norm of the complex gain matrix of inputs (ub, uc) and outputs (PSa, PSb, PSc).

Figure 3.

Sound power level before and after control using pressure feedback with inputs (ub, uc) and outputs (PSa, PSb, PSc) for controlling mode 1 and mode 5.

Figure 4.

Mean-square normal velocity level before and after control using pressure feedback with inputs (ub, uc) and outputs (PSa, PSb, PSc) for controlling mode 1 and mode 5.

Figure 5.

Sound pressure level at field point $(0.190\text{m},0.15\text{m},1.50\text{m})$ before and after control using pressure feedback with inputs (ub, uc) and outputs (PSa, PSb, PSc) for controlling mode 1 and mode 5.

Second, it is desired to modify the second modal frequency from 185 Hz to 200 Hz, and the second modal damping ratio from 0.01 to 0.03 by active control with inputs (ua, uc) and outputs (PS1, PS2, PS3). To investigate the effect of the distance between the microphone and the plate on the sound pressure feedback control, (PS1, PS2, PS3) with $z=1.5\text{m}$ and (PS1, PS2, PS3) with $z=0.01\text{m}$ are considered. Figures 6 and 7 show the sound power level and the mean-square normal velocity level before and after control. Figures 8 and 9 show the sound pressure level at field point PS1 and PS2 of $z=1.5\text{m}$ and $z=0.01\text{m}$ before and after control. The two norms of the feedback gain matrices used for the control with different distances of $z=1.5\text{m}$ and $z=0.01\text{m}$ as a function of the frequency are shown in Figure 10. Figures 6 to 9 visually show that the poles are assigned to prescribed values. From Figure 10, it can be seen that the gain of $z=0.01\text{m}$ is much smaller than the gain of $z=1.5\text{m}$ .

Figure 6.

Sound power level before and after control using pressure feedback with inputs (ua, uc) and outputs (PS1, PS2, PS3) for controlling mode 2.

Figure 7.

Mean-square normal velocity level before and after control using pressure feedback with inputs (ua, uc) and outputs (PS1, PS2, PS3) for controlling mode 2.

Figure 8.

Sound pressure level at field point $(0.2375\text{m},0.15\text{m},1.50\text{mor}0.01\text{m})$ before and after control using pressure feedback with inputs (ua, uc) and outputs (PS1, PS2, PS3) for controlling mode 2.

Figure 9.

Sound pressure level at field point $(0.07125\text{m},0.24375\text{m},1.50\text{mor}0.01\text{m})$ before and after control using pressure feedback with inputs (ua, uc) and outputs (PS1, PS2, PS3) for controlling mode 2.

Figure 10.

The 2-norms of the complex gain matrices of inputs (ua, uc) and outputs (PS1, PS2, PS3) with different distances of $z=1.5m$ and $z=0.01m$ .

Third, to see the effect of delays, $e-\text{i}\omega \tau$ with $\tau =0.1s$ , $\tau =0.001s$ , and $\tau =0.00001s$ are considered in the simulation for the first case. The sound power level, the mean-square normal velocity level, and the sound pressure level for control with delay are shown in Figures 11 to 13. Figures 11 to 13 show that only the feedback control system with a very small delay $\tau =0.00001s$ makes the excellent realization of the pole assignment, and the delay $\tau =0.1s$ significantly changes the controlled modes and fails to assign poles. It seems that if pressure feedback is implemented in a control loop, the effect of unmodelled phase shifts should be considered carefully and all the electronics (filter and amplification of sensor and actuator), the controller and the sound propagation are responsible for the delays and have to be compensated.

Figure 11.

Sound power level with control delay using pressure feedback with inputs (ub, uc) and outputs (PSa, PSb, PSc) for controlling mode 1 and mode 5.

Figure 12.

Mean-square normal velocity level with control delay using pressure feedback with inputs (ub, uc) and outputs (PSa, PSb, PSc) for controlling mode 1 and mode 5.

Figure 13.

Sound pressure level at field point $(0.190\text{m},0.15\text{m},1.50\text{m})$ with control delay using pressure feedback with inputs (ub, uc) and outputs (PSa, PSb, PSc) for controlling mode 1 and mode 5.

Fourth, from Figures 2 and 10, it can be seen that the gains are decreasing with increasing frequency. Thus, for practical purposes, it is attempted to use complex gains at 20 Hz for the whole frequency range for the above two control cases. That is, the pressure influence coefficient matrices at 20 Hz are used to get $\mathbf{G}f$ from equation (29), and further the active control simulations with this constant complex gain matrix $\mathbf{G}f$ are made in the whole frequency band (5, 400) Hz. The feedback gain matrix used for the first control case of modes 1 and 5 is given in Table 6. Figures 14 to 16 show the sound power level, the mean-square normal velocity level, and the sound pressure level before and after control. The feedback gain matrix used for the second control case of mode 2 is given in Table 7. Figures 17 to 19 show the sound power level, the mean-square normal velocity level, and the sound pressure level before and after control. It can be seen that the poles are not assigned to prescribed values, but the control using constant gain matrix at 20 Hz can achieve a very large reduction in acoustic radiation and structural vibration of the controlled modes in a very wide frequency band. The explanation for the large reduction performance is: if one uses gains at 20 Hz for the whole frequency range, it means for frequency higher than 20 Hz, bigger control gains are used than needed, because $\mathbf{C}p$ is frequency dependent and increasing with increasing frequency. It should be pointed out that using constant complex gain matrix for pressure feedback cannot achieve modal pole assignment due to the frequency dependence of the pressure influence coefficient matrices, and also it may cause negative effect on the uncontrolled modes such as the second mode in Figures 14 and 15.

Figure 14.

Sound power level before and after control using pressure feedback with inputs (ub, uc) and outputs (PSa, PSb, PSc) for controlling mode 1 and mode 5 with complex gains at 20 Hz.

Figure 15.

Mean-square normal velocity level before and after control using pressure feedback with inputs (ub, uc) and outputs (PSa, PSb, PSc) for controlling mode 1 and mode 5 with complex gains at 20 Hz.

Figure 16.

Sound pressure level at field point $(0.190\text{m},0.15\text{m},1.50\text{m})$ before and after control using pressure feedback with inputs (ub, uc) and outputs (PSa, PSb, PSc) for controlling mode 1 and mode 5 with complex gains at 20 Hz.

Figure 17.

Sound power level before and after control using pressure feedback with inputs (ua, uc) and outputs (PS1, PS2, PS3) for controlling mode 2 with complex gains at 20 Hz.

Figure 18.

Mean-square normal velocity level before and after control using pressure feedback with inputs (ua, uc) and outputs (PS1, PS2, PS3) for controlling mode 2 with complex gains at 20 Hz.

Figure 19.

Sound pressure level at field point $(0.07125\text{m},0.24375\text{m},1.50\text{mor}0.01\text{m})$ before and after control using pressure feedback with inputs (ua, uc) and outputs (PS1, PS2, PS3) for controlling mode 2 with complex gains at 20 Hz.

Table 6.

Feedback gain matrix (20 Hz) of pressure feedback control for mode 1 and mode 5.

$\mathbf{G}f$ (1.0e+07)
−4.9291 − 0.7749i	9.8384 + 1.5353i	−4.9291 − 0.7749i
3.1668 + 0.4945i	−6.3209 − 0.9797i	3.1668 + 0.4945i

Table 7.

Feedback gain matrix (20 Hz) of pressure feedback control for mode 2.

$z=1.5\text{m}$ $\mathbf{G}f$ (1.0e + 05)	− 1.2367 − 0.0892i	0.7232 + 0.0545i	0.5212 + 0.0395i
	1.2367 + 0.0892i	− 0.7232 − 0.0545i	− 0.5212 − 0.0395i
$z=0.01\text{m}$ $\mathbf{G}f$ (1.0e + 01)	− 4.3478 − 0.0672i	3.0608 + 0.1162i	5.0768 + 0.1815i
	4.3478 + 0.0672i	− 3.0608 − 0.1162i	− 5.0768 − 0.1815i

Finally, it is attempted to use real control gain for sound pressure feedback with inputs (u1, u2) and outputs (PS1, PS2, PS3) of $z=1.5\text{m}$ . The pressure influence coefficient matrices $\mathbf{C}\mathrm{pr}$ at 20 Hz and 86 Hz are used to get $\mathbf{G}f$ by equation (30a) to modify the first modal frequency from 86 Hz to 100 Hz and 75 Hz, respectively, and the pressure influence coefficient matrices $\mathbf{C}\mathrm{pi}$ at 20 Hz and 86 Hz are used to get $\mathbf{G}f$ by equation (30b) to modify the first modal damping ratio from 0.0099 to 0.03, and further the active control simulations with the $\mathbf{G}f$ s are made in the frequency band (5, 400) Hz, respectively. The feedback gain matrices used for the control are given in Tables 8 and 9. Figures 20 to 25 show the sound power level, the mean-square normal velocity level and the sound pressure level before and after control. It can be seen that the pressure feedback with a real gain at a lower frequency may result in a very large reduction in acoustic radiation and structural vibration of the controlled mode in a wide frequency band, and also it may cause negative effect on the fifth mode as shown in Figures 20 and 21. It should be pointed out that the pressure feedback with real gain cannot achieve the pole assignment.

Figure 20.

Sound power level before and after control using pressure feedback (20 Hz).

Figure 21.

Mean-square normal velocity level before and after control using pressure feedback (20 Hz).

Figure 22.

Sound pressure level at field point $(0.23750\text{m},0.15000\text{m},1.50\text{m})$ before and after control using pressure feedback (20 Hz).

Figure 23.

Sound power level before and after control using pressure feedback (86 Hz).

Figure 24.

Mean-square normal velocity level before and after control using pressure feedback (86 Hz).

Figure 25.

Sound pressure level at field point $(0.23750\text{m},0.15000\text{m},1.50\text{m})$ before and after control using pressure feedback (86 Hz).

Table 8.

Feedback gain matrix of pressure feedback (20 Hz).

$\mathbf{G}f$
Pressure feedback based on $\mathbf{C}\mathrm{pr}$	100 Hz	857.7324	77.4416	−109.7410
		656.5729	59.2796	−84.0040
	75 Hz	−598.1332	−54.0033	76.5270
		−457.8561	−41.3382	58.5795
Pressure feedback based on $\mathbf{C}\mathrm{pi}$	0.03	−26.2857	−6.1207	−6.0215
		−20.1210	−4.6853	−4.6093

Table 9.

Feedback gain matrix of pressure feedback (86 Hz).

$\mathbf{G}f$
Pressure feedback based on $\mathbf{C}\mathrm{pr}$	100 Hz	32.3056	−20.1685	−70.2717
		24.7291	−15.4385	−53.7913
	75 Hz	−22.1705	13.8411	48.2257
		−16.9710	10.5950	36.9156
Pressure feedback based on $\mathbf{C}\mathrm{pi}$	0.03	−3.8008	−0.9555	−1.0407
		−2.9094	−0.7314	−0.7967

Conclusions

In this paper, the active control of vibro-acoustic response using sound pressure feedback has been studied. An output feedback approach based on sound pressure inputs has been proposed and the feedback gains determined by assigning poles. The control performance of structural vibration and acoustic radiation of a baffled plate has been numerically evaluated based on the proposed method. Numerical simulations of the active control with complex gains confirm the excellent realization of the pole assignment. Numerical results also show that the pressure feedback control using a constant gain at a lower frequency may result in large reduction in vibro-acoustic response in a wide frequency band but may cause negative effect on uncontrolled modes. Finally, it should be noted that the results presented here are theoretical and numerical in nature and the associated practical problems related to implementation, applications and performances of the method are left for future research and investigation.