Robust active sound radiation control of a piezo-laminated composite circular plate of arbitrary thickness based on the exact 3D elasticity model

Abstract

A multi-objective mixed $\mathcal{H}$ ₂ / $\mathcal{H}$ _∞ robust output feedback control synthesis with regional pole placement constraints in a linear matrix inequalities framework is adopted for active low-frequency sound radiation control of an arbitrarily thick, rigidly baffled, simply supported, multi-layered piezo-composite circular panel. The adopted control system concurrently captures the benefits of both $\mathcal{H}$ ₂ transient control performance and $\mathcal{H}$ _∞ robust stability in the face of external disturbances and system uncertainties. Also, the implemented volumetric sensing/actuation configuration avoids the typical problems associated with conventional (spatially discrete) piezoelectric sensor/actuator patches, where the total volume velocity can be effectively cancelled with the main contribution being to the long wavelength acoustic power emission. The elasto-acoustic analysis is based on the spatial state-space method in the context of exact 3D elasticity theory along with the Rayleigh integral formula where Neumann’s addition theorem is incorporated in the associated Hankel transform representation to arrive at a computationally efficient expression for the nonaxisymmetric pressure field within the acoustic half-space, valid in both near and far fields. Subspace system identification of the fully coupled structure–fluid interaction problem is performed, and the truncated modes are considered as multiplicative uncertainties in synthesis of the mixed-norm controller. Numerical simulations establish the ability of the implemented volumetric sensing/actuation methodology in cooperation with the multi-objective robust active control scheme for restraining low-frequency sound radiation from a $\text{Ba}2\text{NaNb}5\text{O}15$ /steel/PZT4 circular piezo-laminated plate, without provoking instability of the closed-loop system. Also, superior bandwidth frequency and tracking performance in comparison to the $\mathcal{H}2$ and $\mathcal{H}\infty$ controllers are observed. This work is believed to be the first such attempt to exactly model (and actively control) the 3D nonaxisymmetric acousto-elastodynamic frequency response of an arbitrarily thick, smart piezo-laminated circular plate in heavy fluid loading condition (i.e. without using any kind of far-field, low-frequency, and/or light fluid coupling approximations), with straightforward extensibility for any arbitrary through-thickness variation of distributed material properties.

Introduction

Active abatement of low-frequency noise and vibration emitted from vibro-acoustic systems, as a complimentary approach to the inactive restraining techniques that are normally appropriate in the medium- and high-frequency range,^1,2 has been a subject of great interest during the past few decades. Two main strategies are generally adopted, namely, active noise control ³ and the active structural acoustic control (ASAC).^4,5 In the more viable and practical ASAC method, error sensors together with a minimization procedure associated with a judiciously designated cost function are generally employed to manage sound radiation by specific actuation of tractable auxiliary forces (i.e. via shakers, electro/magneto-rheological, and piezoelectric materials) on the resonating structure. The piezoelectric materials are particularly used in new-generation high-performance smart structures for vibration and noise control owing to their low power consumption, lightweight, flexibility, wide dynamic range, and fast response time.^6,7 Also, a wide variety of control strategies, such as velocity feedback control, ⁸ optimal control,^9,10 neural network control, ¹¹ adaptive control, ¹² and $\mathcal{H}2$ ¹³ and $\mathcal{H}\infty$ robust control^14,15 theories, have been developed for active vibro-acoustic response control of oscillating fluid-coupled structures. Important among these control schemes are the $\mathcal{H}\infty$ controllers that have the key advantage of systematically assuring the robust performance/stability along with optimized vibration eradication in the face of uncertainties in the nominal system model and external disturbances.^16,17 As a further step, the $\mathcal{H}2$ and $\mathcal{H}\infty$ control objectives can be unified in a single (mixed) time–frequency domain design framework, in order to simultaneously optimize system performance and enforce robustness constraints.^18,19 Synthesis of such multi-objective controllers can expediently be achieved by employment of the linear matrix inequalities (LMIs)^18,20 that provide algebraic representations of various control specifications and allow computationally efficient and systematic design of robust controllers by reducing the controller design into a standard convex optimization problem. Furthermore, application of extra robust regional pole placement constraints can cause ample satisfaction of both the stability robustness and the temporal performance requirements. ²¹

Plates are one of the most extensively used structural components in numerous engineering, civil, martial, oceanic, and aerospace/aeronautical applications (e.g. building walls and floors, ship hulls, machine elements, ground-based or aerospace vehicle panels, and aircraft sidewalls). These structures can practically be modelled, to a first approximation, as a finitely bounded baffled panel.^22,23 The literature review presented here will largely focus on the most pertinent contributions regarding the vibro-acoustic response control of flexible planar radiators of circular planform. In comparison to the rectangular plate geometry, ²⁴ far fewer authors have considered active acoustic radiation suppression of circular plates. For example, Fuller ²⁵ used directly applied oscillating point forces from electrodynamic shakers to study active suppression of acoustic transmission through a rigidly baffled clamped thin circular elastic plate excited on one side by a plane acoustic wave. He demonstrated that global attenuation of broadband radiated sound levels for low to mid frequencies can be attained with one or two control forces, regardless of system resonance condition. Subsequently, Metcalf et al. ²⁶ performed experiments to find good correlation between the analytical predictions of Fuller ²⁵ and their experimental measurements. Dimitriadis and Fuller ²⁷ theoretically explored the prospects of actively controlling sound radiation (transmission) from (through) a vibrating thin circular plate using surface-bonded piezoelectric actuator elements. They demonstrated that these types of actuators have great potential for controlling the vibration in distributed systems and consequently the control of sound radiation. Van Niekerk et al. ²⁸ presented an active ( $\mathcal{H}$ ₂ optimal feedback/feedforward) control methodology that reduces transient noise transmission through a thin circular plate set in a circular duct by employing two circular piezoceramic patches as actuators. Leniowska and Leniowski ²⁹ considered active extinction of the far-field acoustic pressure radiated by a fluid-loaded thin circular plate set in a finite baffle, through application of a point control force from an electromechanical shaker directly to the plate surface. Rdzanek and Zawieska ³⁰ investigated far-field acoustic radiation reduction from a vibrating clamped-guided annular plate under an external pressure, by direct application of a shaker clamped into the internal edge of the panel. Leniowska ³¹ analytically studied active vibration Linear-quadratic regulator (LQR) control of a fluid-coupled baffled circular plate excited on one side by a uniform periodic force by using centrally placed circular-shaped piezoelectric PZT (Lead Zirconium Titanate, Pb[ZrxTi1-x]O3) actuator patch pairs. The results indicated that while the adopted control law provided a significant drop in the plate vibration, it is rather ineffective for noise attenuation. Leniowska ³² examined the effect of the active vibration control strategy on the sound emission from a fluid-coupled circular plate set in a finite baffle using a point control force. Leniowska ³³ subsequently employed Simulink/MATLAB to investigate the effects of viscous damping, structural internal damping, and fluid loading on active acoustic radiation attenuation from a baffled, thin circular plate, under a constant-amplitude periodic force, by using symmetrically located circular piezoelectric actuators. Wiciak^34,35 used numerical and experimental techniques to investigate active suppression of acoustic wave passage through a thin clamped circular plate baffled in one face of a water-filled rigid-walled aquarium, by using distributed surface-bonded square-shaped piezoelectric actuator elements. More recently, Johnson et al. ³⁶ studied acoustic radiation mode shapes of circular plates and used objective functions to target the most efficient radiation modes for reducing the radiated sound power.

The above brief literature assessment evidently shows that, although there are a number of research works that employ various control methodologies to investigate active acoustic radiation extinction from thin panels of circular planform, rigorous analytical or numerical solutions dealing with a (heavily) fluid-coupled smart circular piezo-laminated plate of arbitrary thickness appears to be missing. Consequently, in the current manuscript, we shall utilize the linear three-dimensional exact piezo-elasticity theory and the classical Rayleigh integral formulation, ²² along with a multi-objective mixed $\mathcal{H}2/\mathcal{H}\infty$ robust output feedback control synthesis with regional pole placement constraints in a LMI framework, ¹⁸ to fill this important gap in the literature. In this regard, the main scientific novelty of the suggested study with respect to the previous research works is not in the applied techniques themselves, but rather it resides in presenting a multi-disciplinary (fully coupled) theoretical approach that effectively combines the strategic advantages of these methods to tackle a currently unsolved practical problem in the field of ASAC. More specifically, the key contributions/advantages of the proposed model may be highlighted as follows:

Exploiting the linear elasticity theory along with the classical Rayleigh integral approach to arrive at an exact 3D acousto-elastodynamic solution for describing the forced nonaxisymmetric sound radiation from a fluid-coupled multi-layered smart piezoelectric circular panel of arbitrary thickness. Such solution is known to be much more accurate than the those based on the approximate 2D (Kirchhoff, Mindlin) plate theories^22,23 that lose accuracy as the panel thickness to radius ratio increases.

Making use of proper Hankel transform representation and Neumann’s addition theorem to obtain a very useful computationally efficient and inclusive exact (single integral) analytical expression for the nonaxisymmetric pressure field within the coupled acoustic half-space that is valid in both near and far fields over the entire frequency range of interest, while avoiding the serious disadvantages of the semi-analytic and numerical methods (e.g. inaccuracy, computational intensity, mathematical ill-posedness, fictitious frequencies). To the author’s knowledge, all existing (thin-panel) models either treat only the axisymmetric sound radiation problem^28,30–33 or study the nonaxisymmetric sound radiation based on various simplifying assumptions (e.g. far-field, low-frequency, and/or light fluid loading approximations).^25,27,29

Applying the spatial state-space approach in tandem with a laminate model that allows for an arbitrary through-thickness variation (gradient) of distributed constituent material properties (FGM, FGPM), with no restrictions on the variations of stresses or displacements.

Implementing the LMI-based multi-objective mixed $\mathcal{H}$ ₂ / $\mathcal{H}$ _∞ robust active control synthesis to simultaneously and effectively capture the benefits of both the $\mathcal{H}$ ₂ transient control performance (i.e. satisfaction of temporal requirements) and the $\mathcal{H}$ _∞ robust stability (i.e. fulfilment of the frequency domain constraints) as a convex optimization problem in the face of external disturbances and system uncertainties (unmodelled dynamics).

Using a uniformly distributed piezoelectric volume velocity sensor layer perfectly matched with a constant force actuator layer in order to primarily sense and excite the first (piston-like) radiation mode of the oscillating plate. ²⁴ In this control configuration, the total volume velocity that is known to have the largest contribution to the low-frequency acoustic power emission can effectively be cancelled.^37–39 Furthermore, the common problems frequently associated with the conventional spatially discrete sensors and actuators, such as the weight/volume/support requirements, line moment excitation effects, ⁸ observability/controllability problems, and control spillover effects,^4,40 are avoided.

Furthermore, the suggested model, which takes full account of the mechanical interaction between the sensor/host/actuator layers, is of imperative interest because of its intrinsic importance as a standard problem in vibro-acoustics. It is also of applied benefit for structural acoustics engineers interested in improvement of low-frequency sound radiation characteristics of thick piezo-composite smart circular plates in the face of uncertainties.^{15,19,41–42} These elements are of realistic concern in advancement of intelligent micro-electromechanical systems,^43,44 for which precise movement control is essential. It can readily be combined with the classical passive control methods^1,2,45,46 and/or expediently be augmented with multiple pairs of electroded piezoelectric actuator/sensor segments in a multi-input multi-output (MIMO) active control framework^47–49 to effectively deal with the medium- and high-frequency acoustic radiation problems.^1,2 Finally, the accessible set of numerical simulations can aid as the reference for evaluation of results acquired by the merely algorithmic or asymptotic methods.

Formulation

Consider a flat composite circular panel set in a rigid baffle, and in contact with neighbouring upper and lower acoustic fluid mediums of densities $(\rho \text{U},\rho \text{L})$ , as shown in Figure 1(a). It consists of a transversely isotropic circular base plate (radius $r\text{o}$ , thickness $h\text{c}$ ) bonded on its bottom boundary to an entirely electroded perfect volume velocity sensor lamina of thickness $h\text{s},$ while it is perfectly matched on its top boundary with a constant-force evenly spread piezoelectric actuator lamina of thickness $h\text{a}$ (see Figure 1(b)). The piezoelectric actuator lamina is basically the reciprocal of the volume velocity sensor lamina, capable of applying a uniform distributed force over the top boundary of the base layer. Such matched distributed actuator/sensor pair configuration will be seen later (section ‘System Identification and Controller Design’) to primarily measure and excite the first radiation mode of the composite panel.^37–40 Also, the associated control system, which essentially reduces the total volume velocity, is then anticipated to attain decent mitigations in the low-frequency sound power radiation, while avoiding the control spillover effects.^4,23,40 Before outlining the control method, we shall briefly describe the basic elasto-acoustic model^50,51 (i.e. the governing equations for the piezo-laminated circular panel structure coupled with the surrounding fluid mediums) in the next two subsections.

Figure 1.

(a) Problem geometry and (b) the smart sandwich plate.

Piezo-composite panel

With the lack of free charge density and bulk forces, the basic linear constitutive equations, along with the equations of motion and electric equilibrium, are specified in the cylindrical coordinate system as

$\sigma =\mathbf{cS}-\mathbf{e}\text{T}\mathbf{E},\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathbf{D}=\mathbf{eS}+\epsilon \mathbf{E}$

(1a)

$\nabla .\Xi =\rho \text{p}\mathbf{r}\mathrm{··},\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\nabla .\mathbf{D}=0$

(1b)

where $\mathbf{r}=[ur,u\theta ,w]\text{T}$ is the movement vector, $\rho \text{p}$ is the piezo-material density, $\Xi$ is the Cauchy stress matrix, and $\mathbf{S}=[\varepsilon \mathrm{rr},\varepsilon \theta \theta ,\varepsilon \mathrm{zz},2\varepsilon \theta z,2\varepsilon \mathrm{rz},2\varepsilon r\theta ]\text{T}$ , $\sigma =[\sigma \mathrm{rr},\sigma \theta \theta ,\sigma \mathrm{zz},\sigma \theta z,\sigma \mathrm{rz},\sigma r\theta ]\text{T}$ , $\mathbf{D}=[Dr,D\theta ,Dz]\text{T},$ and $\mathbf{E}=-[\partial \varphi \partial r,1r\partial \varphi \partial \theta ,\partial \varphi \partial z]\text{T}$ , are the strain, stress, electric displacement, and electric field vectors, correspondingly. Also, ϕ is the electric potential, and the piezoelectric coupler matrix, e, and the elastic/dielectric constant material matrices $(\mathbf{c},\epsilon )$ , and as well as expressions for the pertinent movement and stress elements with regard to proper stress $(\psi 1,\psi 2)$ and movement $(\chi 1,\chi 2)$ functions are given in the Appendix 1. By straight use of the these expressions in the governing equation (1), one attains the ensuing two totally uncoupled state equations

$\partial \partial \text{z}\mathbf{V}\text{N}=\mathbf{N}\text{p}\mathbf{V}\text{N},\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\partial \partial \text{z}\mathbf{V}\text{M}=\mathbf{M}\text{p}\mathbf{V}\text{M}$

(2)

in which $\mathbf{V}\text{N}=[\chi 1,\psi 1]\text{T}$ and $\mathbf{V}\text{M}=[\chi 2,\sigma \mathrm{zz},Dz,\psi 2,w,\varphi ]\text{T}$ , and the matrix coefficients $(\mathbf{N}\text{p},\mathbf{M}\text{p})$ are given in Appendix 1. Also, adopting the simple elastic support conditions at $r=r\text{o}$ , as proposed by Ding and Xu^52,53 (i.e. $w=u\theta =0,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}2c66ur+r\text{o}\sigma r=0$ ), one can assume the subsequent vector results for the piezoelectric layers

$\mathbf{V}\text{N}=\left\{\chi 1\psi 1\right\}=\sum n=1\infty \sum m=1\infty \{H2\chi ¯1(\zeta )\mathrm{Hc}44\text{s}\psi ¯1(\zeta \left)\right\}\mathrm{nm}\mathcal{J}m(\alpha \mathrm{nm}\xi )\sin \left(m\theta \right)e\text{i}\omega t$

(3a)

$\mathbf{V}\text{M}=\left\{\chi 2\sigma \mathrm{zz}Dz\psi 2w\varphi \right\}=\sum n=1\infty \sum m=0\infty \{H2\chi ¯2(\zeta )c44\text{s}\sigma ¯\mathrm{zz}(\zeta )e33\text{s}D¯z(\zeta )\mathrm{Hc}44\text{s}\psi ¯2(\zeta )Hw¯(\zeta )\mathrm{He}33\text{s}\varphi ¯(\zeta )/\epsilon 33\text{s}\}\mathrm{nm}\mathcal{J}m(\beta \mathrm{nm}\xi )\cos \left(m\theta \right)e\text{i}\omega t$

(3b)

where $\text{i}=\sqrt{-1}$ , $\mathcal{J}m$ denotes the mth-order cylindrical Bessel Function of the first kind, $\alpha \mathrm{nm}$ and $\beta \mathrm{nm}$ are the nth roots of $\mathcal{J}m\text{'}\left(\alpha \right)=0$ and $\mathcal{J}m\left(\beta \right)=0$ , respectively, ω is the frequency, $\xi =r/r\text{o}$ , $\zeta =z/H$ , and $(c44\text{s}$ , $e33\text{s}$ , $\epsilon 33\text{s})$ are piezoelectric parameters of the sensor lamina. Straight replacement of results from equations (3) in the state equations (2), leads to the subsequent modal state relations

$\text{dd}\zeta \mathbf{V}¯\text{N}\left(\zeta \right)=\mathbf{N}¯\text{p}\mathbf{V}¯\text{N}\left(\zeta \right),\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{dd}\zeta \mathbf{V}¯\text{M}\left(\zeta \right)=\mathbf{M}¯\text{p}\mathbf{V}¯\text{M}\left(\zeta \right)$

(4)

where $\mathbf{V}¯\text{N}\left(\zeta \right)=[\chi ¯1,\psi ¯1]\text{T}$ and $\mathbf{V}¯\text{M}\left(\zeta \right)=[\chi ¯2,\sigma ¯\mathrm{zz},D¯z,\psi ¯2,w,\varphi ¯]\text{T}$ are modal state vectors. Also, the matrix coefficients $(\mathbf{N}¯\text{p},\mathbf{M}¯\text{p})$ are given in Appendix 1, and the index ‘ $\text{p}=\text{s},\text{a}$ ‘ is associated with the sensor and actuator lamina, respectively.

Next, by using the approximate laminate model assumption, one can advantageously split piezoelectric actuator/sensor laminae into $l\text{a}$ and $l\text{s}$ amply thin identical-width sublayers, correspondingly. ⁵⁴ Also, as a result of solving state equations (4), the state parameters at the subjacent boundary of sensor and actuator laminae can be linked to those on the top boundary in the form ²⁴

$\mathbf{V}¯\text{N}(\zeta =\zeta \text{s})=\mathbf{T}\text{N}\text{s}\mathbf{V}¯\text{N}(\zeta =0),\mathbf{V}¯\text{M}(\zeta =\zeta \text{s})=\mathbf{T}\text{M}\text{s}\mathbf{V}¯\text{M}(\zeta =0)$

(5a)

$\mathbf{V}¯\text{N}(\zeta =1)=\mathbf{T}\text{N}\text{a}\mathbf{V}¯\text{N}(\zeta =\zeta \text{a}),\mathbf{V}¯\text{M}(\zeta =1)=\mathbf{T}\text{M}\text{a}\mathbf{V}¯\text{M}(\zeta =\zeta \text{a})$

(5b)

where $\zeta \text{s}=h\text{s}/H$ , $\zeta \text{a}=(h\text{s}+h\text{c})/H$ , $\mathbf{T}\text{N}\text{s},\text{a}=\Pi j=l\text{s},\text{a}1\text{exp}(\mathbf{N}¯\text{s},\text{a}/l\text{s},\text{a})$ , and $\mathbf{T}\text{M}\text{s},\text{a}=\Pi j=l\text{s},\text{a}1\text{exp}(\mathbf{M}¯\text{s},\text{a}/l\text{s},\text{a})$ are partial transfer matrices connected with sensor/actuator laminae.

Following a similar procedure for the core layer, with the associated constitutive relation $\Sigma \text{c}=\mathbf{CS}\text{c}$ (see Appendix 1), the associated state equations can be expressed using suitable movement $(X1,X2)$ and stress $(\Psi 1,\Psi 2)$ functions as

$\partial \partial z\mathbf{Y}\text{N}=\mathbf{N}\text{c}\mathbf{Y}\text{N},\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\partial \partial z\mathbf{Y}\text{M}=\mathbf{M}\text{c}\mathbf{Y}\text{M}$

(6)

where $\mathbf{Y}\text{N}=[\text{X}1,\Psi 1]\text{T}$ , $\mathbf{Y}\text{M}=[\text{X}2,\Sigma \mathrm{zz},\Psi 2,W]\text{T}$ , W is the transverse movement constituent, the matrix coefficients $(\mathbf{N}\text{c},\mathbf{M}\text{c})$ are provided in Appendix 1, and the associated state vectors are expanded as ⁵⁵

$\mathbf{Y}\text{N}=\left\{\text{X}1\Psi 1\right\}=\sum n=1\infty \sum m=1\infty \{H2\text{X}¯1(\zeta )\mathrm{Hc}44\text{s}\Psi ¯1(\zeta \left)\right\}\mathrm{nm}\mathcal{J}m\left(\alpha \mathrm{nm}\xi \right)\sin \left(m\theta \right)e\text{i}\omega t\mathbf{Y}\text{M}=\left\{\text{X}2\Sigma \mathrm{zz}\Psi 2W\right\}=\sum n=1\infty \sum m=0\infty \{H2\text{X}¯2(\zeta )c44\text{s}\Sigma ¯\mathrm{zz}(\zeta )\mathrm{Hc}44\text{s}\Psi ¯2(\zeta )HW¯(\zeta \left)\right\}\mathrm{nm}\mathcal{J}m\left(\beta \mathrm{nm}\xi \right)\cos \left(m\theta \right)e\text{i}\omega t$

(7)

Direct substitution of above expansions into the state equations (6) leads to the following modal state equations

$\text{dd}\zeta \mathbf{Y}¯\text{N}\left(\zeta \right)=\mathbf{N}¯\text{c}\mathbf{Y}¯\text{N}\left(\zeta \right),\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{dd}\zeta \mathbf{Y}¯\text{M}\left(\zeta \right)=\mathbf{M}¯\text{c}\mathbf{Y}¯\text{M}\left(\zeta \right)$

(8)

where $\mathbf{Y}¯\text{N}\left(\zeta \right)=[\text{X}¯1,\Psi ¯1]\text{T}$ and $\mathbf{Y}¯\text{M}\left(\zeta \right)=[\text{X}¯2,\Sigma ¯\mathrm{zz},\Psi ¯2,W¯]\text{T}$ are the associated modal vectors, and matrix coefficients $(\mathbf{N}¯\text{c}$ , $\mathbf{M}¯\text{c})$ are given in Appendix 1. Also, by adopting the laminate approximate model once again, and with direct solution of state equations (8), the state parameters at the subjacent boundary of the transversely isotropic layer can be linked to those on the top boundary in the form ²⁴

$\mathbf{Y}¯\text{N}(\zeta =\zeta \text{a})=\mathbf{T}\text{N}\text{c}\mathbf{Y}¯\text{N}(\zeta =\zeta \text{s})\mathbf{Y}¯\text{M}(\zeta =\zeta \text{a})=\mathbf{T}\text{M}\text{c}\mathbf{Y}¯\text{M}(\zeta =\zeta \text{s})$

(9)

in which $\mathbf{T}\text{N}\text{c}=\Pi j=l\text{c}1\text{exp}(\mathbf{N}¯\text{c}h\text{c}/\mathrm{Hl}\text{c})$ and $\mathbf{T}\text{M}\text{c}=\Pi j=l\text{c}1\text{exp}(\mathbf{M}¯\text{c}h\text{c}/\mathrm{Hl}\text{c})$ denote the partial transfer matrices connected with the transversely isotropic core layer, with $l\text{c}$ being the number of fictitious layers.

Next, the electrical surface conditions on the topmost and lowermost boundaries of the piezo-actuator lamina are correspondingly given as

$\varphi (\xi ,\theta ,\zeta =1,t)=\Phi (\xi ,\theta )e\text{i}\omega t,\varphi (\xi ,\theta ,\zeta =\zeta \text{a},t)=0$

(10)

with the prescribed electrical potential magnitude being expressed as (see equation (3b))

$\Phi (\xi ,\theta )=He33\text{s}\epsilon 33\text{s}\sum n=1\infty \sum m=0\infty \varphi ¯\mathrm{nm}(\zeta =1)\mathcal{J}m\left(\beta \mathrm{nm}\xi \right)\cos \left(m\theta \right)$

(11)

Successive grouping of equations (5b) and (10), after some mathematical managements, leads to

$\mathbf{V}¯\text{M}\text{m}(\zeta =1)=\mathbf{T}\wedge \text{M}\text{a}\mathbf{V}¯\text{M}\text{m}(\zeta =\zeta \text{a})+\mathbf{B}¯\text{a}$

(12)

where $\mathbf{V}¯\text{M}\text{m}=[\chi ¯2,\sigma ¯\mathrm{zz},\psi ¯2,w¯]\text{T}$ denotes the ‘mechanical constituent’ of vector $\mathbf{V}¯\text{M}$ , and

$\mathbf{T}\wedge \text{M}\text{a}=\left[\mathbf{C}\text{a}\right(1):\mathbf{C}\text{a}(2):\mathbf{C}\text{a}(4):\mathbf{C}\text{a}(5\left)\right]-1T63\text{a}\mathbf{C}\text{a}\left(3\right)\mathbf{G}\text{a}\left(6\right),\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathbf{B}¯\text{a}=\varphi ¯(1)T63\text{a}\mathbf{C}\text{a}\left(3\right)$

where $T\mathrm{ij}\text{a}$ refer to the entries of the partial transfer matrix $\mathbf{T}\text{M}\text{a}$ , $\mathbf{C}\text{a}\left(i\right)=[T1i\text{a},T2i\text{a},T4i\text{a},T5i\text{a}]\text{T}$ , and $\mathbf{G}\text{a}\left(6\right)=[T61\text{a},T62\text{a},T64\text{a},T65\text{a}]$ . Similarly, the electrical surface conditions associated with the piezo-sensor lamina are written in the form ⁵⁴

$Dz(\xi ,\theta ,\zeta =0,t)=0,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\varphi (\xi ,\theta ,\zeta =\zeta \text{s},t)=0$

(13)

Consequently, by benefit of equations (5a) and (13), one has

$\mathbf{V}¯\text{M}\text{m}(\zeta =\zeta \text{s})=\mathbf{T}\wedge \text{M}\text{s}\mathbf{V}¯\text{M}\text{m}(\zeta =0)$

(14)

in which $\mathbf{T}\wedge \text{M}\text{s}=[\mathbf{C}\text{s}(1):\mathbf{C}\text{s}(2):\mathbf{C}\text{s}(4):\mathbf{C}\text{s}(5\left)\right]-1T66\text{s}\mathbf{C}\text{s}\left(6\right)\mathbf{G}\text{s}\left(6\right),$ where $T\mathrm{ij}\text{s}$ refers to the entries of the transfer matrix $\mathbf{T}\text{M}\text{s}$ , $\mathbf{C}\text{s}\left(i\right)=[T1i\text{s},T2i\text{s},T4i\text{s},T5i\text{s}]\text{T}$ , and $\mathbf{G}\text{s}\left(6\right)=[T61\text{s},T62\text{s},T64\text{s},T65\text{s}]$ . Lastly, the state results (9), (12), and (14), may be united to arrive at the final overall transfer relation for the entire laminated smart structure

$\mathbf{V}¯\text{M}\text{m}(\zeta =1)=\mathcal{T}\mathbf{V}¯\text{M}\text{m}(\zeta =0)+\mathbf{B}¯\text{a}$

(15)

where $\mathcal{T}=\mathbf{T}\wedge \text{M}\text{a}\mathbf{T}\text{M}\text{c}\mathbf{T}\wedge \text{M}\text{s}$ represents the general $4\times 4$ transfer matrix.

Sound radiation and fluid/structure coupling

Following the standard procedure, ²² for a rigidly baffled flat area, S, oscillating with the transverse acceleration, $w··$ , in a harmonic manner (see Figure 1(a)), one can make use of the classical Rayleigh integral formula for obtaining the sound pressure at the field location R within the neighbouring acoustic half-space

$P\left(\mathbf{R}\right)=\rho \text{f}2\pi \int Se\text{i}k|\mathbf{R}-\mathbf{R}\text{srf}||\mathbf{R}-\mathbf{R}\text{srf}|w\mathrm{··}\left(\mathbf{R}\text{srf}\right)\text{d}S\left(\mathbf{R}\text{srf}\right)$

(16)

where $\rho \text{f}$ refers to the liquid density, $k=\omega /c$ denotes the acoustic wave number, c signifies the speed of sound in the half-space fluid, $\mathbf{R}\text{srf}=[\xi \text{srf},\theta \text{srf},\zeta \text{srf}]\text{T}$ is an arbitrary point on the vibrating surface ( $\zeta \text{srf}=0$ , $1$ ), and $|\mathbf{R}-\mathbf{R}\text{srf}|$ indicates the separation between the receiver and source locations (see Figure 1(a)). Subsequently, by making use of the fifth relation in equation (3b), the complex harmonic pressure fields at any field point $\mathbf{R}=[\xi ,\theta ,\zeta ]\text{T}$ in upper ( $\zeta \ge 1$ ) and lower ( $\zeta \le 0$ ) acoustic mediums (after some manipulations) can be obtained in the form^22,24

$P\left(\mathbf{R}\right)=-\rho \text{f}\omega 2r\text{o}H2\pi \sum n=1\infty \sum m=0\infty w¯\mathrm{nm}\left(\zeta \text{srf}\right)\kappa \mathrm{nm}\left(\mathbf{R}\right)$

(17a)

in which

$\kappa \mathrm{nm}\left(\mathbf{R}\right)=\int 02\pi \int 01e-\text{i}kgg\mathcal{J}m\left(\beta \mathrm{nm}\xi \text{srf}\right)\text{cos}\left(m\theta \text{srf}\right)\xi \text{srfd}\xi \text{srfd}\theta \text{srf}$

(17b)

where $k=r\text{o}k,\hspace{1em}g=[\xi 2+\xi \text{srf}2-2\xi \xi \text{srfcos}(\theta -\theta \text{srf})+t02(\zeta -\zeta \text{srf}\left)2\right]12$ , and $t0=H/r\text{o}$ .

Now, supposing that the piezo-composite plate is acted upon its top surface by a distributed harmonic normal mechanical load of general form, $q\text{m}(\xi ,\theta ,t)=Q\text{m}(\xi ,\theta )e\text{i}\omega t$ , as depicted in Figure 1(a), the structure/fluid coupling conditions that must hold at the subjacent and top plate/fluid boundaries are written in the form

$\sigma \mathrm{zz}=-P\left(\mathbf{R}\right),\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}(\zeta =0)\sigma \mathrm{zz}=Q\text{m}-P\left(\mathbf{R}\right),\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}(\zeta =1)\sigma \mathrm{rz}=0,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\sigma \theta z=0,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}(\zeta =0,1)$

(18)

where the load amplitude, $Q\text{m}(\xi ,\theta )$ , is expanded as

$Q\text{m}(\xi ,\theta )=c44\text{s}\sum n=1\infty \sum m=0\infty Q¯\mathrm{nm}\text{m}\mathcal{J}m(\beta \mathrm{nm}\xi )\cos \left(m\theta \right)$

(19)

Subsequent use of expansions (3b), (17), and (19), in the surface conditions (18), after using the classical orthogonality relation in cylindrical coordinates, and with some tedious manipulations, leads to

$\sigma ¯\mathrm{zz}\left(\zeta \text{srf}\right)=-\mathcal{K}\mathrm{nm}(\text{L})\sum r=1\infty \sum q=0\infty \mathcal{Z}\mathrm{nmrq}w¯\mathrm{rq}\left(\zeta \text{srf}\right),\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}(\zeta \text{srf}=0)\sigma ¯\mathrm{zz}\left(\zeta \text{srf}\right)=Q¯\mathrm{nm}\text{m}+\mathcal{K}\mathrm{nm}(\text{U})\sum r=1\infty \sum q=0\infty \mathcal{Z}\mathrm{nmrq}w¯\mathrm{rq}\left(\zeta \text{srf}\right),\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}(\zeta \text{srf}=1)\psi ¯1\left(\zeta \text{srf}\right)=0,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\psi ¯2\left(\zeta \text{srf}\right)=0,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}(\zeta \text{srf}=0,1)$

(20)

where $\mathcal{K}\mathrm{nm}(\text{L},\text{U})=1\Delta m\mathcal{J}m+12\left(\beta \mathrm{nm}\right)\rho \text{L},\text{U}\omega 2r\text{o}3H2\pi c44\text{s}$ , $\Delta m=\{\pi m=0,12\pi m=1,2,\dots$ , and

$\mathcal{Z}\mathrm{nmrq}=\int 02\pi \int 02\pi \int 01\int 01e-\text{i}kg\text{s}g\text{s}\mathcal{J}m\left(\beta \mathrm{nm}\xi \text{srf}\right)\mathcal{J}q\left(\beta \mathrm{rq}\xi \right)\text{cos}\left(m\theta \text{srf}\right)\text{cos}\left(q\theta \right)\xi \text{srf}\xi \text{d}\xi \text{srfd}\xi \text{d}\theta \text{srfd}\theta$

(21)

in which $g\text{s}=[\xi 2+\xi \text{srf}2-2\xi \text{srf}\xi \text{cos}(\theta -\theta \text{srf}\left)\right]12.$ Thus, by exploiting the surface conditions (20) in equation (15), one ultimately obtains the linear matrix system

$\mathbf{H}\Theta =\Gamma$

(22)

in which the matrix $\mathbf{H}$ is provided in Appendix 1, and

$\Theta =[\Theta 10,\Theta 20,\dots ,\Theta N0;\Theta 11,\Theta 21,\dots ,\Theta N1;\dots ;\Theta 1M,\Theta 2M,\dots ,\Theta \mathrm{NM}]\text{T},\Gamma =[\Gamma 10,\Gamma 20,\dots ,\Gamma N0;\Gamma 11,\Gamma 21,\dots ,\Gamma N1;\dots ;\Gamma 1M,\Gamma 2M,\dots ,\Gamma \mathrm{NM}]\text{T},$

where the constants $(N,M)$ are truncation numbers, $\Gamma \mathrm{nm}=[-B¯\text{a}(1),Q¯\text{m}-B¯\text{a}(2),-B¯\text{a}(3),-B¯\text{a}(4\left)\right]\text{T},$ and $\Theta \mathrm{nm}=[\chi ¯2(\zeta =0),w¯(\zeta =0),\chi ¯2(\zeta =1),w¯(\zeta =1)]\text{T},$ where $B¯\text{a}\left(i\right)$ (i = 1,2,3,4) refer to the entries of electric load vector $\mathbf{B}¯\text{a}$ .

Evaluation of quadruple integral and the radiated sound power

The Rayleigh integral has been adopted as an alternative to the finite and boundary element solutions of acoustic boundary value problems.^56–58 Unfortunately, due to presence of singularities,^59,60 the Rayleigh integral (equation (16)) is not actually responsive to direct numerical calculation of the near-field pressure, especially at high frequencies. ³⁹ This difficulty can be circumvented by following the mathematical treatment presented for example in the literatures,^59,61,62 where the traditional Hankel transform (cylindrical wave) representation of the singular term (spherical wave) is adopted; ⁶³ i.e.

$e-ikgsgs=\int 0\infty \lambda (\lambda 2-k2)12\mathcal{J}0(\lambda gs)\text{d}\lambda$

(23)

where, when $\lambda <k,\hspace{1em}(\lambda 2-k2)12$ should be replaced by $\text{i}(k2-\lambda 2)12$ . Also, by virtue of Neumann’s addition theorem in cylindrical coordinates ⁶⁴

$\mathcal{J}0\left(\lambda g\text{s}\right)=\sum j=0\varepsilon j\mathcal{J}j\left(\lambda \xi \right)\mathcal{J}j\left(\lambda \xi \text{srf}\right)\cos [j(\theta -\theta \text{srf})]$

(24)

equation (23) is transformed into

$e-\text{i}kg\text{s}g\text{s}=\sum j=0\varepsilon j\cos [j(\theta -\theta \text{srf})]\int 0\infty \mathcal{J}j\left(\lambda \xi \right)\mathcal{J}j\left(\lambda \xi \text{srf}\right)\hspace{1em}\lambda \text{d}\lambda (\lambda 2-k2)12$

(25)

with $\varepsilon 0=1,$ and $\varepsilon j=2(j=1,2,\dots )$ . Thus, by making use of above approximation in equation (21), changing the order of integration, and using the classical orthogonality relations of transcendental functions, one ultimately arrives at the computationally efficient single integral expression

$\mathcal{Z}\mathrm{nmrq}=\{0,m\ne q4\pi \Delta m\int 0\infty Hm\mathrm{nm}(\lambda \left)Hm\mathrm{rm}\right(\lambda \left)\lambda \text{d}\lambda \right(\lambda 2-k2)12,m=q$

(26)

where $Hj\mathrm{nm}\left(\lambda \right)=\int 01\mathcal{J}m\left(\beta \mathrm{nm}\xi \right)\mathcal{J}j\left(\lambda \xi \right)\xi \text{d}\xi$

The integral within the limits $(0,\infty )$ in equation (26) should be recognized as the Cauchy principal value integral along the real axis in the complex variable plane. ⁶⁵ Accordingly, the integral in the right hand side of equation (26) can advantageously be decomposed into^59,60

$14\pi \Delta m\mathcal{Z}\mathrm{nmrq}=\int 0kHm\mathrm{nm}\left(\lambda \right)Hq\mathrm{rq}\left(\lambda \right)\lambda \text{d}\lambda \text{i}(k2-\lambda 2)12+\int k\infty Hm\mathrm{nm}\left(\lambda \right)Hq\mathrm{rq}\left(\lambda \right)\lambda \text{d}\lambda (\lambda 2-k2)12$

(27)

The first integration $(\lambda \le k)$ can readily be evaluated using a standard Gaussian quadrature scheme. ⁶⁶ However, the second integration interval $(\lambda >k)$ should be truncated at a certain wave number, $\lambda \text{max}$ , beyond which the kernel of integral decays very rapidly, and the contribution to the integral is negligible. ⁶⁰

The radiated sound power could readily be determined by integrating the acoustic intensity on the vibrating circular panel surface in the form ²²

$\Pi =12\int Sw·*P\left(\mathbf{R}\text{srf}\right)\text{d}S$

(28)

where ‘*’ denotes the complex conjugate of a complex number. Thus, employing equations (3b) and (17) in equation (28), results in the expression for the radiated acoustic power in convenient matrix form as

$\Pi =\rho \text{f}\omega 3H2r\text{o}34\pi wHMw$

(29)

where $w=[w¯10w¯20\dots w¯N0;w¯11\dots w¯N1;\dots w¯\mathrm{NM}]\text{T}$ is the modal displacement coefficient vector, ‘H’ denotes the Hermitian transpose, and the entries of the $\left[\right(M+1\left)N\right]\times \left[\right(M+1\left)N\right]$ radiation impedance matrix, $M$ , are denoted by $\mathcal{M}i,j=\mathcal{M}(n+\mathrm{mN}),(r+\mathrm{qN})=\text{i}\mathcal{Z}\mathrm{nmrq}$ . Also, by making use of the so-called modal reference sound power, $\Pi n(\infty )=limk\to \infty 12\rho \text{f}c\int Sw·n\text{d}S$ (where S is panel area, and $w·n$ denotes normal velocity of the nth panel mode), the standardized modal sound power, $\Pi ¯,$ can be written in terms of the radiated modal acoustic power, $\Pi n$ , in the form ⁶⁵

$\Pi ¯=\Pi n\Pi n(\infty )=k\mathcal{M}i,i2\pi \Delta m\mathcal{J}m-12\left(\beta \mathrm{nm}\right)=\Pi ¯\text{a}-\text{i}\Pi ¯\text{r}$

(30)

where, by keeping decompositions (27) and (30) in mind, the standardized active and reactive sound powers are respectively defined as ⁶²

$\Pi ¯\text{a}=\text{Re}[\Pi ¯]=k\text{Re}[\mathcal{M}i,i]2\pi \Delta m\mathcal{J}m-12\left(\beta \mathrm{nm}\right)=2k\mathcal{J}m-12\left(\beta \mathrm{nm}\right)\int 0kHm\mathrm{nm}\left(\lambda \right)Hq\mathrm{rq}\left(\lambda \right)\hspace{1em}\lambda \text{d}\lambda (k2-\lambda 2)12\Pi ¯\text{r}=\text{Im}[\Pi ¯]=k\text{Im}[\mathcal{M}i,i]2\pi \Delta m\mathcal{J}m-12\left(\beta \mathrm{nm}\right)=2k\mathcal{J}m-12\left(\beta \mathrm{nm}\right)\int k\infty Hm\mathrm{nm}\left(\lambda \right)Hq\mathrm{rq}\left(\lambda \right)\hspace{1em}\lambda \text{d}\lambda (\lambda 2-k2)12$

(31)

where ‘Re’ and ‘Im’ denote the real and imaginary parts of a complex number, respectively. Furthermore, by using the trigonometric substitution $\lambda =k\text{sin}\phi$ , the square root singularity in equations (31) can advantageously be removed, giving rise to the computationally efficient final expressions ⁶²

$\Pi ¯\text{a}=2k\mathcal{J}m-12\left(\beta \mathrm{nm}\right)\int 0\pi /2Hm\mathrm{nm}\left(k\text{sin}\phi \right)Hq\mathrm{rq}\left(k\text{sin}\phi \right)\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{sin}\phi \text{d}\phi \Pi ¯\text{r}=2k\mathcal{J}m-12\left(\beta \mathrm{nm}\right)\int \pi /2\pi 2-\text{i}\infty Hm\mathrm{nm}\left(k\text{sin}\phi \right)Hq\mathrm{rq}\left(k\text{sin}\phi \right)\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\text{sin}\phi \text{d}\phi$

(32)

Volume velocity control and sensor voltage

Technological advances in production of smart panels with collocated and dual sensor–actuator transducer pairs has stimulated the development of light, compact, and noninvasive control systems that are very effective for eradication of tonal or broadband random acoustic disturbances.^23,67–69 Also, the dominant contribution to sound power radiation from the panel at very low frequencies is known to be linked with the (strongly radiating) first radiation mode (characterized by a piston-type or monopole behaviour) which includes a combination of structural modes with nonzero volume velocities, commonly referred to in the literature as the ‘volumetric modes’.^{36–39,68,70} A fairly effective method in active suppression of low-frequency acoustic radiation for prediction of the first radiation mode magnitude is to use uniformly spread piezoelectric sensor materials which essentially respond to the velocity distribution associated with such radiation pattern.^37,38,40 Therefore, a single-input single-output (SISO) volume velocity control system^38,40,71 employing an integrated piezoceramic sensor layer collocated with a constantly distributed volume velocity (uniform force) piezoelectric actuator layer (see Figure 1(b)) can be advantageously configured to effectively reduce the total volume velocity vibration of the panel at such low-excitation frequencies.^{23,24,37–40}

The electric charge accessible through the closed-circuit electrodes of the sensor lamina may be determined by simple integration of z-component of electrical displacement upon the electrode surface area, ⁷² i.e.

$q\text{s}(t)=\int SDz\text{d}S$

(33)

Also, the voltage signal generated by the sensor lamina is linked to the volume velocity of emitting surface (output current of piezoelectric sensor, $I\text{s})$ through the simple relation ⁷³

$V\text{s}\left(t\right)=R\text{L}I\text{s}=R\text{Ld}q\text{sd}t$

(34)

where $R\text{L}$ is the leakage resistor of the current amplifier (see Figure 1(b)). This output voltage is returned back to the control unit which produces a feedback voltage for the piezo-actuator lamina.

System identification and controller design

At this point, we shall apply the N4SID subspace-based estimation algorithm ⁷⁴ provided in MATLAB® System Identification Toolbox ⁷⁵ to identify the system and approximate state matrices for the complete coupled fluid–structural problem from sensor/actuator (output/input) voltage data of our numerical simulations, in the nth-order state-space form

$\mathbf{x}·\left(t\right)=\mathbf{A}\mathbf{x}\left(t\right)+\left[\mathbf{B}\text{d}\right|\mathbf{B}\text{u}\left]\right\{\mathbf{d}\left(t\right)\mathbf{u}\left(t\right)\left\}\mathbf{y}\right(t)=\mathbf{C}\mathbf{x}(t)+[\mathbf{D}\text{d}\left|\mathbf{D}\text{u}\right]\left\{\mathbf{d}\right(t\left)\mathbf{u}\right(t\left)\right\}$

(35)

where the $(1\times 1)$ vector $\mathbf{u}\left(t\right)=[\varphi (\xi ,\theta ,1,t)]$ is the actuator voltage, $\mathbf{y}(t)=[V\text{s}(t)]$ is the sensor voltage, $\mathbf{x}(t)$ is the $(n\times 1)$ state vector, $\mathbf{d}\left(t\right)=q\text{m}(\xi ,\theta ,t)$ represents the mechanical disturbance, and the matrices $\mathbf{A}(n\times n)$ , $\mathbf{B}\text{d}(n\times 1)$ , $\mathbf{B}\text{u}(n\times 1)$ , $\mathbf{C}(1\times n)$ , $\mathbf{D}\text{d}(1\times 1)$ , $\mathbf{D}\text{u}(1\times 1)$ can be readily recognized based on the input/output vectors $(\mathbf{u}(t),\mathbf{y}(t))$ .

Next, we shall follow the procedure in Skogestad and Postlethwaite ⁷⁶ to formulate the mixed-norm controller synthesis problem in the general control framework. Here, it should be noted that, as $\mathcal{H}2$ and $\mathcal{H}\infty$ control methodologies are standard tools in the control community, there is not a vital need for detailed re-description of this mathematically intensive subject, where the interested reader is referred to the literatures.^{13–21,77–83} Figure 2(a) shows the standard configuration of the multi-objective $\mathcal{H}2/\mathcal{H}\infty$ robust output feedback controller where $\mathbf{P}\left(s\right)$ is the generalized plant comprising the arrangement of all dynamics required for design and synthesis of the generalized controller $\mathbf{K}\left(s\right)$ based on the measurable error signal e, while w denotes the exogenous input vector to the generalized plant (e.g. reference signal, r, disturbances, d, noise,n, etc., see Figure 2(b)). The regulated output channel $\mathbf{z}\infty$ is associated with the $\mathcal{H}\infty$ performance and the regulated output channel $\mathbf{z}2$ is related to $\mathcal{H}2$ performance. Also, to get a meaningful controller synthesis problem, one generally has to include the frequency-dependent weights $\mathbf{W}\text{S}$ (reflecting the requirements on control objective), $\mathbf{W}\text{u}$ (imposing restrictions on the actuator signal), and $\mathbf{W}\text{T}$ (known as the multiplicative uncertainty weighting function) in the generalized plant. Moreover, the nominal plant, $\mathbf{G}\text{r}\left(s\right)$ , as the approximate model of the coupled fluid/piezolaminate panel system, which relates the actuator input voltage to sensor output voltage, can readily be obtained from the estimated system matrices (see the state equations 35) in the form

$\mathbf{G}\text{r}\left(s\right)=\mathbf{C}(s\mathbf{I}-\mathbf{A})-1\mathbf{B}\text{u}+\mathbf{D}\text{u}$

(36)

where I is the identity matrix. Consequently, for generalized plant $\mathbf{P}\left(s\right)$ in Figure 2(a), the relation between the regulated outputs and the input of interest (selected as a reference-like signal) is given by

$\left[\mathbf{z}\infty \mathbf{z}\text{S}\mathbf{z}\text{u}\right]=[\mathbf{W}\mathbf{T}(\mathbf{s})\mathbf{T}(\mathbf{s})\mathbf{W}\mathbf{S}(\mathbf{s})\mathbf{S}(\mathbf{s})\mathbf{W}\mathbf{u}(\mathbf{s})\mathbf{K}(\mathbf{s})\mathbf{S}(\mathbf{s})]\left[\mathbf{w}\right]$

(37)

where $\mathbf{S}(s)=1/[1+\mathbf{G}\text{r}(s)\mathbf{K}(s)$ ] is the closed-loop sensitivity function defined as the transfer function from w to error e, and $\mathbf{T}\left(s\right)=\mathbf{G}\text{r}\left(s\right)\mathbf{K}\left(s\right)/[1+\mathbf{G}\text{r}(s\left)\mathbf{K}\right(s\left)\right]$ is the complementary sensitivity function which is the transfer function from w to output y. Subsequently, a multi-objective output-feedback synthesis for our SISO linear time-invariant (LTI) system (plant) may be considered with the following state-space realization ¹⁸

$\mathbf{x}·\left(t\right)=\mathbf{A}\mathbf{x}\left(t\right)+\mathbf{B}\text{w}\mathbf{w}\left(t\right)+\mathbf{B}\text{u}\mathbf{u}\left(t\right)\mathbf{z}\infty \left(t\right)=\mathbf{C}\infty \mathbf{x}\left(t\right)+\mathbf{D}\infty \text{w}\mathbf{w}\left(t\right)+\mathbf{D}\infty \text{u}\mathbf{u}\left(t\right)\mathbf{z}2\left(t\right)=\mathbf{C}2\mathbf{x}\left(t\right)+\mathbf{D}2\text{w}\mathbf{w}\left(t\right)+\mathbf{D}2\text{u}\mathbf{u}\left(t\right)\mathbf{y}\left(t\right)=\mathbf{Cx}\left(t\right)+\mathbf{D}\text{w}\mathbf{w}\left(t\right)+\mathbf{D}\text{u}\mathbf{u}\left(t\right)$

(38)

Figure 2.

(a) The multi-objective mixed $H2/H\infty$ output feedback control problem and (b) block diagram of the closed-loop acousto-elastic system.

Our main objective here is to design a dynamic output feedback controller that minimizes the $\mathcal{H}2$ norm of the closed-loop transfer function from w to $\mathbf{z}2$ and keeps the $\mathcal{H}\infty$ norm of the transfer function from w to $\mathbf{z}\infty$ under some appropriate level. Accordingly, the state-space representation of the synthesized output-feedback controller $\mathbf{K}\left(s\right)$ is described by

$\varsigma ·\left(t\right)=\mathbf{A}\text{K}\varsigma \left(t\right)+\mathbf{B}\text{K}\mathbf{e}\left(t\right)\mathbf{u}\left(t\right)=\mathbf{C}\text{K}\varsigma \left(t\right)+\mathbf{D}\text{K}\mathbf{e}\left(t\right)$

(39)

where $\varsigma \left(t\right)$ is the controller state variable. Thus, the closed-loop state-space representation containing performance and robustness channels for the generalized plant and controller is presented as

$\mathbf{x}·\text{cl}\left(t\right)=\mathcal{A}\mathbf{x}\text{cl}\left(t\right)+\mathcal{B}\mathbf{w}\left(t\right)\mathbf{z}\infty \left(t\right)=\mathcal{C}\infty \mathbf{x}\text{cl}\left(t\right)+\mathcal{D}\infty \mathbf{w}\left(t\right)\mathbf{z}2\left(t\right)=\mathcal{C}2\mathbf{x}\text{cl}\left(t\right)+\mathcal{D}2\mathbf{w}\left(t\right)$

(40)

in which $\mathbf{x}\text{cl}=[\mathbf{x}(t),\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\varsigma (t)]\text{T}$ denotes the closed-loop state vector, and

$\mathcal{A}=[\mathbf{A}+\mathbf{B}\text{u}\mathbf{D}\text{K}\mathbf{C}\mathbf{B}\text{u}\mathbf{C}\text{K}\mathbf{B}\text{K}\mathbf{C}\mathbf{A}\text{K}],\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathcal{B}=[\mathbf{B}\text{w}+\mathbf{B}\text{u}\mathbf{D}\text{K}\mathbf{D}\text{w}\mathbf{B}\text{K}\mathbf{D}\text{w}]\mathcal{C}\infty =[\mathbf{C}\infty +\mathbf{D}\infty \text{u}\mathbf{D}\text{K}\mathbf{C}\mathbf{D}\infty \text{u}\mathbf{C}\text{K}],\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathcal{C}2=[\mathbf{C}2+\mathbf{D}2\text{u}\mathbf{D}\text{K}\mathbf{C}\mathbf{D}2\text{u}\mathbf{C}\text{K}]\mathcal{D}\infty =[\mathbf{D}\infty \text{w}+\mathbf{D}\infty \text{u}\mathbf{D}\text{K}\mathbf{D}\text{w}],\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathcal{D}2=[\mathbf{D}2\text{w}+\mathbf{D}2\text{u}\mathbf{D}\text{K}\mathbf{D}\text{w}].$

$\mathcal{H}\infty$ and $\mathcal{H}2$ control performances

Let $\mathbf{T}\mathbf{wz}\infty$ be the transfer function from w to $\mathbf{z}\infty$ with the realization ( $\mathcal{A},\mathcal{B},\mathcal{C}\infty ,\mathcal{D}\infty$ ). Also, let $\mathbf{T}\mathbf{wz}\infty (s)\infty$ represent the $\mathcal{H}\infty$ norm of the stable transfer matrix $\mathbf{T}\mathbf{wz}\infty (s)$ , defined in the frequency domain as

$\Vert \mathbf{T}\mathbf{wz}\infty (s)\Vert \infty \Delta =sup\sigma \text{max}[\mathbf{T}\mathbf{wz}\infty (\text{i}\omega )]$

(41)

where $\sigma \text{max}$ is the maximum singular value. The constraint $\Vert \mathbf{T}\mathbf{wz}\infty (s)\Vert \infty <\gamma$ (where $\gamma >0$ ) can be supposed as a disturbance rejection performance as well robust stability of closed-loop system. This particularly ensures that the closed-loop system remains stable for all perturbations w. Based on the Bounded Real Lemma, ¹⁷ there exists a unique positive definite matrix $X\infty$ such that the following matrix inequality is satisfied

$[\mathcal{A}X\infty +X\infty \mathcal{A}\text{T}\mathcal{B}X\infty \mathcal{C}\infty \text{T}\mathcal{B}\text{T}-\mathbf{I}\mathcal{D}\infty \text{T}\mathcal{C}\infty X\infty \mathcal{D}\infty -\gamma \mathbf{I}]<0$

(42)

Next, let $\mathbf{T}\mathbf{wz}2$ be the transfer function from w to $\mathbf{z}2$ with the closed-loop realization ( $\mathcal{A},\mathcal{B},\mathcal{C}2,\mathcal{D}2$ ). Supposing that $\mathcal{A}$ is stable and $\mathcal{D}2=0$ , the $\mathcal{H}2$ norm of $\mathbf{T}\mathbf{wz}2$ is defined as

$\Vert \mathbf{T}\mathbf{wz}2\Vert 22\Delta =12\pi \int -\infty \infty \text{trace}(\mathbf{T}\mathbf{wz}2(\text{i}\omega )\text{H}\mathbf{T}\mathbf{wz}2(\text{i}\omega ))\text{d}\omega$

(43)

Also, recall that this norm can be computed as $\Vert \mathbf{T}\mathbf{wz}2\Vert 22=\text{trace}\left(\mathcal{C}2\mathcal{L}\text{c}\mathcal{C}2\text{T}\right)$ , where $\mathcal{L}\text{c}$ solves the following Lyapunov equation ²¹

$\mathcal{A}\mathcal{L}\text{c}+\mathcal{L}\text{c}\mathcal{A}\mathbf{T}+\mathcal{B}\mathcal{B}\text{T}=0$

(44)

It is readily verified that $\left|\right|\mathbf{T}\mathbf{wz}2\left|\right|22<v2$ ( $v>0$ ), if and only if there exists $X2>\mathcal{L}\text{c}$ satisfies

$\mathcal{A}X2+X2\mathcal{A}\mathbf{T}+\mathcal{B}\mathcal{B}\text{T}<0$

(45)

Consequently, using Schur complements for above inequality, $\left|\right|\mathbf{T}\mathbf{wz}2\left|\right|2<v$ is equivalent to the existence of two symmetric matrices $X2>0$ and $\mathcal{Q}$ such that

$[X2\mathcal{A}\mathbf{T}+\mathcal{A}X2\mathcal{B}\mathcal{B}\text{T}-\mathbf{I}]<0\left[X2\mathcal{C}2X2X2\mathcal{C}2\text{T}\mathcal{Q}\right]>0,\text{trace}\left(\mathcal{Q}\right)<v2$

(46)

Regional pole constraints and multi-objective output feedback problem

The transient response of the system may be tuned by closed-loop pole placement in the convex left-half region $\text{D}$ of the complex plane $ℂ$ within the framework of LMI, described as

$\text{D}=\{k\in ℂ:\mathbf{L}+\mathbf{M}z+\mathbf{M}Tz*<0\}$

(47)

where $\mathbf{L}=\mathbf{L}\text{T}$ and M are fixed real matrices, and the ‘star’ superscript denotes complex conjugate. Also, the pole placement constraint is satisfied if and only if there exists a symmetric matrix $X\text{p}>0$ such that ²¹

$[l\mathrm{ij}X\text{p}+m\mathrm{ij}\mathcal{A}X\text{p}+m\mathrm{ji}X\text{p}\mathcal{A}\text{T}]<0$

(48)

where $\{l\mathrm{ij}\}$ and $\{m\mathrm{ij}\}$ denote the entries of the matrices Land M. Here, the primary objective is to minimize the radiated sound power which will be described by the $\mathcal{H}2$ norm of $\mathbf{T}\mathbf{wz}2$ over all output-feedback stabilizing gains $\mathbf{K}\left(s\right)$ that enforce the $\mathcal{H}\infty$ and pole placement constraints (i.e. $min\left|\right|\mathbf{T}\mathbf{wz}\left|\right|\infty \left(s\right)\infty <1\left|\right|\mathbf{T}\mathbf{wz}2\left(s\right)\left|\right|2$ ). This is equal to minimizing the $\text{trace}\left(\mathcal{Q}\right)$ over all matrices $X\infty$ , $X2$ , $\mathcal{Q}$ , and $X\text{p}$ satisfying (42), (46), and (48) objectives. To recover convexity in the LMI framework, all specifications must be enforced by seeking a single closed-loop Lyapunov function $\mathbf{V}\left(\mathbf{x}\text{cl}\right(t\left)\right)=\mathbf{x}\text{cl}\left(t\right)\text{T}\mathcal{P}\mathbf{x}\text{cl}\left(t\right),$ where the symmetric matrix $\mathcal{P}>0$ satisfies ¹⁸

$\mathcal{P}=X\infty =X2=X\text{p}$

(49)

With the above equation imposed, our multi-objective synthesis problem can be further reduced to minimize $\text{trace}\left(\mathcal{Q}\right)$ over $\mathcal{P}$ and $\mathcal{Q}$ subject to the LMI constraints

$[\mathbf{L}\mathcal{P}+\mathbf{M}\mathcal{A}\mathcal{P}+\mathbf{M}T\mathcal{P}\mathcal{A}\text{T}]<0,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}[\mathcal{A}\mathcal{P}+\mathcal{P}\mathcal{A}\mathbf{T}\mathcal{B}\mathcal{P}\mathcal{C}\infty \text{T}\mathcal{B}\mathbf{T}-\mathbf{I}\mathcal{D}\infty \text{T}\mathcal{C}\infty \mathcal{P}\mathcal{D}\infty -\gamma \mathbf{I}]<0,[\mathcal{P}\mathcal{A}\mathbf{T}+\mathcal{A}\mathcal{P}\mathcal{B}\mathcal{B}\text{T}-\mathbf{I}]<0,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\left[\mathcal{P}\mathcal{C}2\mathcal{P}\mathcal{P}\mathcal{C}2\text{T}\mathcal{Q}\right]>0.$

(50)

Robust stability design and closed-loop performance specification

Modelling errors are unavoidable when a plant is identified either theoretically or experimentally. A multiplicative unstructured uncertainty model seems to be a good choice for treating the model truncation error due to the negligence of high-order residual modes (i.e. avoiding spillover effects). Thus, the relation between the actual model of the system, $\mathbf{G}(s)$ and the reduced-order nominal model, $\mathbf{G}\text{r}(s)$ can be written as

$\mathbf{G}\left(s\right)=\mathbf{G}\text{r}\left(s\right)(1+\Delta \text{m}(s\left)\mathbf{W}\text{T}\right(s))$

(51)

where $\mathbf{W}\text{T}(s)$ is the frequency-dependent multiplicative uncertainty weighting function which is used to normalize uncertainty band, and $\Delta \text{m}\left(s\right)$ represents the dynamics of neglected higher modes with $\left|\right|\Delta \text{m}\left(s\right)\left|\right|\infty \le 1.$ In order to assure the latter inequality, the weighting function $\mathbf{W}\text{T}(s)$ should be selected to cover the upper bound of uncertainty, which comprises the difference between the actual system FRF and the nominal model, in the from

$\left|\mathbf{W}\text{T}\right(\text{i}\omega \left)\right|\ge \left|\mathbf{G}\right(\text{i}\omega )-\mathbf{G}\text{r}(\text{i}\omega \left)\mathbf{G}\text{r}\right(\text{i}\omega \left)\right|,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\forall \omega$

(52)

According to the small gain theorem, to ensure the robust stability of the closed-loop system against multiplicative uncertainty, the following norm inequality must be satisfied^17,76

$\left|\right|\mathbf{T}\mathbf{wz}\infty (s)\left|\right|\infty =\left|\right|\mathbf{W}\text{T}\left(s\right)\mathbf{T}(s)\left|\right|\infty <1$

(53)

In the mixed-norm $\mathcal{H}2/\mathcal{H}\infty$ robust controller synthesis, the closed-loop performance can be obtained by minimizing $\left|\right|\mathbf{T}\mathbf{wz}2(s)\left|\right|2$ . Here, the objective is to achieve a high disturbance rejection by limiting the closed-loop sensitivity function, $\mathbf{S}(s)$ over the control bandwidth. ⁷⁷ Also, noting that the actuator control signal cannot be boundless, the control effort u should be appropriately confined for proper controller realization. Thus, one should limit the corresponding control effort described by $\mathbf{K}(s)\mathbf{S}(s)$ . In particular, to minimize the effect of the exogenous disturbances in the controller bandwidth frequency interval, a low-pass filter $\mathbf{W}\text{S}$ should be used to weight $\mathbf{S}(s)$ . Furthermore, to avoid controller saturation as well as reduce the possibility of residual mode excitation, a high-pass filter $\mathbf{W}\text{u}$ can be used for weighting the function $\mathbf{K}\left(s\right)\mathbf{S}\left(s\right)$ . Therefore, the desired performance of the output feedback controller can be achieved if the following mixed sensitivity $\mathcal{H}2$ norm $\left|\right|\mathbf{T}\mathbf{wz}2\left|\right|2=\left|\right|\mathbf{W}\text{S}\left(s\right)\mathbf{S}(s)\mathbf{W}\text{u}\left(s\right)\mathbf{K}(s)\mathbf{S}(s)\left|\right|2$ is minimized.

Numerical examples

At this point, we shall consider some numerical simulations. Noting the numerous input variable presented here, besides our computational restrictions, we shall focus on a simple problem. Therefore, the core material is supposed to be fabricated from a single lamina $(l\text{c}$  = 1) of isotropic steel with $h\text{c}=0.05\text{m},\hspace{1em}r\text{o}=1\text{m},\text{and}\hspace{1em}\rho \text{c}=7850\text{Kg}/\text{m}3$ , and the physical properties as given in Table 1. The sensor and actuator laminae are similarly supposed as single-lamina $(l\text{a}=l\text{s}=1$ , $h\text{a}=0.02\text{m},h\text{s}=0.01\text{m}$ ), made of $\text{Ba}2\text{NaNb}5\text{O}15$ /PZT4, $(\rho \text{s}=7500\text{Kg}/\text{m}3,\rho \text{a}=5300\text{Kg}/\text{m}3)$ , with their physical properties as provided in Table 1. The acoustic mediums are assumed to be water at ambient temperature ( $\rho \text{U}=\rho \text{L}=1000\text{kg}/\text{m}3,\hspace{1em}c\text{U}=c\text{L}=1500\text{m}/\text{s}$ ). An expansive Mathematica® program was written to calculate the response spectra of the structure–fluid interaction problem through solving the linear system of equations (22) with a maximal truncation parameter of $N=M=5$ . Since the functions in the integrands of equations (17b), (27), and (32) are known analytically, the classical Gaussian quadrature ⁶⁶ appears to be the most suitable (accurate) numerical method of integration, keeping in mind the speed of modern computers. Accordingly, the double integral in equation (17b) can be numerically evaluated by using the two-dimensional Gauss–Legendre quadrature with $60\times 60$ quadrature points, while the first and the second parts of the single integrals in equations (27) and (32) may be evaluated by employing maximums of $15$ and $80$ quadrature points, respectively.

Table 1.

Physical properties of the constituent materials.

		Actuator layer	Sensor layer		Core layer
$c\mathrm{ij},C\mathrm{ij}(\mathbf{N}/\mathbf{m}2)$	$c11$	$23.9\times 1010$	$13.9\times 1010$	$C11$	$26.92\times 1010$
	$c12$	$10.4\times 1010$	$7.8\times 1010$	$C12$	$11.53\times 1010$
	$c13$	$5.2\times 1010$	$7.43\times 1010$	$C13$	$11.53\times 1010$
	$c22$	$24.7\times 1010$	$13.9\times 1010$	$C22$	$26.92\times 1010$
	$c23$	$5.2\times 1010$	$7.43\times 1010$	$C23$	$11.53\times 1010$
	$\text{c}33$	$13.5\times 1010$	$11.5\times 1010$	$C33$	$26.92\times 1010$
	$c44$	$6.5\times 1010$	$2.56\times 1010$	$C44$	$7.69\times 1010$
	$c55$	$6.6\times 1010$	$2.56\times 1010$	$C55$	$7.69\times 1010$
	$c66$	$7.6\times 1010$	$3.06\times 1010$	$C66$	$7.69\times 1010$
$e\mathrm{ij}$ $(\mathbf{C}/\mathbf{m}2)$	$e15$	$2.8$	$12.7$
	$e24$	$3.4$	$12.7$
	$e31$	$-0.4$	$-5.2$
	$e32$	$-0.3$	$-5.2$
	$e33$	$4.3$	$15.1$
$\epsilon \mathrm{ij}$ $(\mathbf{F}/\mathbf{m})$	$\in 11$	$196\times 10-11$	$650\times 10-11$
	$\epsilon 22$	$201\times 10-11$	$650\times 10-11$
	$\epsilon 33$	$28\times 10-11$	$560\times 10-11$

Here, it should be noted that fortunately the (weak) singularity present in the integrand of equation (17b) is well behaved enough so that numerical instability is not an issue when using the Gauss–Legendre quadrature algorithm (i.e. the denominator function g does not practically vanish identically). Furthermore, as the upper bounds of the second integrations in equations (27) and (32) are the symbolic ‘∞’, one should adopt a simple trial and error procedure to select proper fixed large numbers for the upper limits (i.e. by increasing the numerical values of the upper bounds, and look for stability of results). Lastly, Figure 3 plots the radiated sound power level ²² as calculated from $\text{LW}=10\log 10\left(\Pi \Pi 0\right)$ based on equation (29) with $\Pi 0=10-12\text{watt}$ , and the volume velocity sensor voltage $V\text{s}\left(t\right),$ as calculated from equation (34) with $R\text{L}=100\text{k}\Omega .$ Good correlation is obtained, which implies the effectiveness of the selected volume velocity sensor configuration in sensing the radiated power.

Figure 3.

The correlation between the calculated radiated sound power level and the volume velocity sensor voltage.

Before presenting the main results, we shall rigorously verify the overall accuracy of the proposed elasto-acoustic model. Accordingly, we initially treat a sandwich $\text{Ba}2\text{NaNb}5\text{O}15$ /steel/PZT4 water-immersed thick functionally graded piezolaminate circular plate $(r\text{o}=1\text{m},h\text{c}=0.1\text{m},h\text{a}=h\text{s}=0.025\text{m}$ with a material gradient exponent of $\gamma \text{g}=$ 0.75, and $l\text{c}$  =  $l\text{a}=l\text{s}=1$ 0), under any one of the subsequent external excitations: a transverse harmonic (nonaxisymmetric) sectorial mechanical distributed load $(Q\text{m}(\xi ,\theta )=-103[\text{H}(\xi -0.35)-\text{H}(\xi -0.65)\left]\right[\text{H}(\theta -50\circ )-\text{H}(\theta -100\circ )]$ N/m²), a uniform harmonic voltage imposed on an annular region on the top surface of actuator lamina $(\varphi (\xi ,\theta ,\zeta =1,t)=200[\text{H}(\xi -0.2)-\text{H}(\xi -0.4)]e\text{i}\omega t\text{volt})$ , and a nonaxisymmetric electro-mechanical harmonic excitation composed of simultaneous applications of the latter two loads. Figure 4(a) (i.e. the first row subplots in Figure 4) displays the good agreements obtained with the finite element method (FEM) ⁸⁴ results for the calculated displacement response and the electric potential at the observation point ( $\xi =0.35,\theta =\pi /12,\zeta =0$ ) on the bottom boundary of the intelligent plate. Also seen is the good agreement obtained for the radiated sound pressure level (SPL) at the observation point ( $\xi =0.5,\theta =0,\zeta =5$ ) in the upper acoustic medium. In the finite element model, about 3000 20-node piezo-brick elements (C3D20RE) were used for the top and bottom piezoelectric laminae, while about 2000 20-node brick elements (C3D20R) were employed to model the steel sheet. Also, a total of about 150,000 10-node tetrahedron elements (AC3D10) in hemispherical regions were utilized to treat the top and subjacent acoustic half-spaces. Here, it is interesting to note that while there is not much distinction observed between the SPLs of the mechanically, electrically, and the combined electro-mechanically excited panels in the low-frequency range ( $f<100\text{Hz}$ ), the purely mechanical (electrical) excitation dominates the pressure response in the mid- (high-) frequency regime. Furthermore, although the latter observation is also true for the calculated displacement response and the electric potential in mid and high frequencies ( $f>400\text{Hz}$ ), the mechanical and electrical excitations appear to somewhat cancel each other out in case of the electro-mechanically excited panel in the low-frequency range. Moreover, it should be mentioned here that the presented exact solution methodology imposes no restrictions on the panel thickness and/or excitation frequency, while the coupled FEM model was seen in various simulations performed by the present authors to break down when the panel thickness considerably increases, i.e. it is observed that the frequency signatures of the coupled hemispherical acoustic region show up within (and deteriorate) the frequency spectrum of the thick composite panel.

Figure 4.

(a) The displacement, electric potential, and radiated sound pressure level spectrums of a three layered functionally graded piezo-laminated water-submerged circular panel in three loading configurations, and comparisons with FEM data. (b) The standardized active and reactive modal radiated sound powers for both axisymmetric and nonaxisymmetric structural modes of a simply supported circular piezo-laminated plate. (c) Selected modal radiation efficiencies of a single layer isotropic circular panel. (d) The normalized radiation pattern plots of a simply supported circular piezo-laminated plate for the (1,0) axisymmetric and the (1,1) nonaxisymmetric modes at two selected wavelengths.

As a further check, Figure 4(b) displays the standardized active and reactive radiated (modal) sound powers, $\Pi ¯\text{a}$ and $\Pi ¯\text{r},$ for both axisymmetric and nonaxisymmetric structural modes of the simply supported circular piezo-laminated plate, as calculated from equations (32), where excellent agreements are obtained with the axisymmetric results presented in Figures 2 and 3 by Rdzanek et al. ⁸⁵ (note that the numerical results presented in Rdzanek et al. ⁸⁵ are for axisymmetric loading, and based on certain asymptotic approximations). Next, we let the z-dimensions of the piezo-actuator/sensor laminae in our general code nearly zero $(r\text{o}=1\text{m},h\text{c}=0.1\text{m},h\text{a}=h\text{s}\approx 0)$ and calculated selected modal radiation efficiencies of the circular panel based on the following relationship $\sigma n=2\mathcal{M}n,n\pi \rho \text{f}\mathrm{cr}\text{o}2(n=1,2,5)$ ³⁷ (see equation (29)). Good agreements are obtained with the results provided in Figure 1 by Zawieska et al. ⁸⁶ as shown in Figure 4(c) (note that the numerical data in Zawieska et al. ⁸⁶ are for axisymmetric loading, and based on the low-frequency assumption). Subsequently, Figure 4(d) presents the normalized radiation pattern plots of the simply supported circular piezo-laminated plate for the (1,0) axisymmetric and the (1,1) nonaxisymmetric modes (see equations (17)) at both long and short wavelengths $(\mathrm{kr}\text{o}=10-5,10).$ Excellent agreements are obtained with the results presented in Figure 2 by Christiansen et al. ⁵⁷ (note that the plots have been normalized with respect to their peak values). Here, it should be noted that at the low-frequency selection $(\mathrm{kr}\text{o}=10-5),$ radiation patterns of the symmetric and asymmetric modes are very similar to those of a monopole and a dipole radiator, respectively. Also, it is noteworthy that, although the numerical results presented by Christiansen et al. ⁵⁷ include the nonaxisymmetric loading situation, they are based on the simplifying far-field approximation. Lastly, we used our main code to compare the calculated nondimensional natural frequencies (i.e. as read from the associated displacement spectra plots) ²⁴ with the exact results based on the 3D elasticity theory presented in Refs.^53,55,87 for various axisymmetric $(n=1,m=0)$ and nonaxisymmetric $(n=1,2,3,m=1,2)$ vibration modes of simply supported thick laminated circular plates in the nonexistence of fluid coupling $(\rho \text{U},\text{L}=0)$ for designated thickness parameters $(t0=H/r\text{o}=0.1,0.5,1).$ The outcome, as displayed in Table 2, demonstrate very good accuracies.

Table 2.

Nondimensional natural frequencies for various simply supported thick circular plates without fluid loading.

	Xu 55			Ding et al. 53			Wang et al. 87
	$m=1$			$m=2$			$m=0,\hspace{1em}n=1$
$t0$	$n=1$	$n=2$	$n=3$	$n=1$	$n=2$	$n=3$	$\gamma g=0$	$\gamma g=1$	$\gamma g=2$
0.1
Present	0.0776	0.2446	0.4744	0.0676	0.1698	0.2993	0.0133	0.0130	0.0122
Literature	0.0776	0.2446	0.4744	0.0676	0.1698	0.2993	0.0133	0.0130	0.0122
0.5
Present	1.2988	2.9080	4.4971	1.0299	1.9746	2.8813	0.2699	0.2644	0.2494
Literature	1.2988	2.9080	4.4971	1.0299	$-$	$-$	0.2700	0.2645	0.2495
1.0
Present	4.8754	6.4049	9.4762	2.5039	4.2764	5.9110	0.7673	0.7533	0.7172
Literature	4.8754	6.4049	$-$	2.5039	$-$	$-$	0.7673	0.7536	0.7174

At this point, before considering the full active sound radiation control problem, it is worthy to briefly study some important sound radiation and mechanical characteristics of the water-immersed circular piezo-laminated $\text{Ba}2\text{NaNb}5\text{O}15$ /steel/PZT4 panel $(l\text{c}=l\text{a}=l\text{s}=1$ , $h\text{c}=0.05\text{m},h\text{a}=0.02\text{m},h\text{s}=0.01\text{m},\text{o}=1\text{m})$ . For a comprehensive study on the mechanical properties of the thick piezo-composite panel, the reader is referred to the literatures.^53–55 Figure 5 presents the frequency spectrums of off-axis radiated surface/far-field sound pressure levels, SPL (dB), at the observation points ( $\xi =0.35,\theta =75\circ ,\zeta =1,1+10r\text{o}H$ ) as well as the normal stress magnitude at the interface between the piezoelectric sensor and the core layers, $\sigma \mathrm{zz}(\xi =0.35,\theta =75\circ ,\zeta =\zeta \text{s})$ (Pa), in two basic (nonaxisymmetric) loading configurations. In the first loading case (i.e. the first column subplots), four separate transverse pie-shaped homogenous distributed mechanical loads $(Q\text{m}(\xi ,\theta )=-102[\text{H}\left(\xi \right)-\text{H}(\xi -1)\left]\right[\text{H}\left(\theta \right)-\text{H}(\theta -\alpha \text{o})]$ N/m²; $\alpha \text{o}=30,90,180,360\circ )$ are considered, while in the second loading case (i.e. the second column subplots), unit amplitude point loads at four different positions along the panel radius $(Q\text{m}(\xi ,\theta )=-\delta (\xi -\xi \text{o})\delta (\theta )/2\pi \xi$ N; $\xi \text{o}=0,0.25,0.5,0.75)$ are applied. The most interesting observations are as follows. As the action area (wedge angle $\alpha \text{o}$ ) of the applied distributed surface load increases, the radiated surface/far-field pressure levels as well as the normal interface stress magnitudes notably rise, especially in the mid- to high-frequency range, which is as expected. In the second loading configuration, the smallest sound radiation and interface stress levels are observed when the circular panel is excited exactly at its centre point, which is clearly linked to excitation of only the axisymmetric modes. As the excitation position of the point load is changed away from the panel centre point, the nonaxisymmetric modes also get involved, ultimately leading to constructive/destructive interference effects, depending on the frequency of excitation. Moreover, simple comparisons of the Figure 5(a) and (b) subplots (i.e. the off-axis radiated surface/far-field sound pressure level spectrums) indeed demonstrate that, while the effects of the nonaxisymmetric structural modes are strong near the surface of the panel, they seem to nearly disappear in the far-field. Lastly, it is clear from the first subplot in Figure 5(c) that increasingly more nonaxisymmetric structural modes show up in the interface stress response curve, as the asymmetry of loading action area increases (e.g. for $\alpha \text{o}=30\circ$ ).

Figure 5.

(a) The off-axis radiated surface sound pressure level spectrums of a water-immersed circular piezo-laminated panel in two basic loading configurations. (b) The off-axis radiated far-field sound pressure level spectrums of a water-immersed circular piezo-laminated panel in two basic loading configurations. (c) The normal interface stress magnitude spectrums for a water-immersed circular piezo-laminated panel in two basic loading configurations.

Now, as stated earlier, MATLAB System Identification Toolbox N4SID function is used here to identify the sixth-order state-space representation $(\mathbf{A},\mathbf{B}\text{u},\mathbf{C},\mathbf{D}\text{u})$ of the smart structure from the calculated input/output (actuator/sensor voltage) frequency response data. This estimated model is subsequently used as our nominal system, followed by application of MATLAB Robust Control Toolbox and Simulink® software for design and simulation of the control system. Figure 6(a) displays the magnitude and phase of the calculated normalized sensor output voltage for the nominal model with $N=M=3$ , and the sixth-order identified model for a harmonic actuator input voltage of 200 volts. Here, it should be noted that the output voltage normalization is done with respect to that of an extended model at $\omega =0$ with $N=M=7.$ Obviously, the estimated model represents the dynamics of the system very well within the targeted frequency range $0<\omega \le 9000\text{rad}/\text{s}$ . Figure 6(b) compares the normalized frequency response functions of the identified nominal model, $\mathbf{G}\text{r}\left(s\right)$ , and the identified extended (14th order with $N=M=7$ ) model, $\mathbf{G}\left(s\right)$ . Also shown is the normalized frequency response function of the identified 14th-order model, $\mathbf{G}\text{d}\left(s\right)$ (see Figure 2(b)), for an external mechanical disturbance of annular form, $Q\text{m}(\xi ,\theta )=103\left[\text{H}\right(\xi -0.35)-\text{H}(\xi -0.65\left)\right]$ N, applied on the top surface of the piezo-laminated panel, where ‘H’ denotes the Heaviside function. Clearly, the nominal model $(N=M=3)$ ), deviates from the extended model $(N=M=7$ ) outside the targeted frequency range $(\omega >9000\text{rad}/\text{s}$ ). In other words, the identified nominal model is not suitable for frequencies higher than 9000 rad/s, and thus the modes outside this frequency band will be assumed as uncertainty of truncated system. Figure 6(c) displays the modelling error (i.e. the relative difference between the latter two models, or $\left|\right[\mathbf{G}\left(\text{i}\omega \right)-\mathbf{G}\text{r}\left(\text{i}\omega \right)]/\mathbf{G}\text{r}(\text{i}\omega \left)\right|,$ shown with the solid curve) in addition to the weighting function, $\mathbf{W}\text{T}(s)$  =  $(80s+800)/(s+800)$ (designated by the dashed curve), which is designed to characterize an upper bound for the modelling error in the entire f6requency range and reduce the effects of the neglected high-frequency residual modes, to keep away from the spillover problems (see equation (52)). ⁷⁸

Figure 6.

(a) The magnitude and phase of the calculated sensor output voltage for the real and identified models. (b) The frequency response functions of the identified nominal model, $\mathbf{G}\text{r}$ , the identified extended model, G, and the disturbance model, $\mathbf{G}\text{d}$ . (c) The modelling error and the upper bound weighting function $\mathbf{W}\text{T}$ .

As noted previously, in order to reject the effect of the external disturbance, d, on the error signal, e (see Figure 2(b)), the magnitude of sensitivity function, $\mathbf{S}(s)$ , must be kept small in low-frequency range. ⁷⁶ This can be done by choosing a proper a low-pass filter in the form $\mathbf{W}\text{S}=\left[\right(s/\text{Ms}+\omega \text{b})/(s+\omega \text{b}\epsilon \text{e}\left)\right]$ where $\text{Ms}=150$ , $\omega \text{b}=50$ , $\epsilon \text{e}=0.005$ sets requirements on the control performance (e.g. the bandwidth frequency and steady-state tracking error ⁷⁹ ). In addition, to restrict the control signal amplitudes and to avoid depolarization of the piezoelectric actuator (i.e. by exceeding the maximum 200 V regulation voltage limit), as well as reducing the possibility of residual mode excitation, a high-pass filter $\mathbf{W}\text{u}=\left[\right(s+\omega \text{bc}/\text{Mu})/(\epsilon \text{u}s+\omega \text{bc}\left)\right]$ with $\text{Mu}=1$ , $\omega \text{bc}=1200$ and $\epsilon \text{u}=0.01$ , is used for weighting the function $\mathbf{K}(s)\mathbf{S}(s)$ . After choosing the weighting functions, $(\mathbf{W}\text{T},\mathbf{W}\text{S},\mathbf{W}\text{u})$ , as explained above, the generalized plant $\mathbf{P}(s)$ can be constructed and subsequently the controller can be synthesized using the adopted controller design procedure (see Figure 2(a)). Figure 7(a) compares the closed-loop frequency domain performance of our mixed-norm $\mathcal{H}2/\mathcal{H}\infty$ controller with that of a $\mathcal{H}2$ controller and a $\mathcal{H}\infty$ controller utilizing the above selected weighting functions. Here, for the mixed-norm $\mathcal{H}2/\mathcal{H}\infty$ controller, the robust stability constraint is set as $\mathbf{W}\text{T}\left(s\right)\mathbf{T}(s)\infty \le 0.95$ in conjunction with an additional regional pole placement constraint selected as the open left half plane. This way, the synthesized controller is found to satisfy the mixed sensitivity $\mathcal{H}2$ norm $\left|\right|\mathbf{W}\text{S}\left(s\right)\mathbf{S}(s)\mathbf{W}\text{u}\left(s\right)\mathbf{K}(s)\mathbf{S}(s)\left|\right|2\le 0.87$ . Also, the $\mathcal{H}2$ and $\mathcal{H}\infty$ controllers are, respectively, found to satisfy the following stacked sensitivity norms $\left|\right|\mathbf{W}\mathbf{T}\mathbf{T}\mathbf{W}\mathbf{S}\mathbf{S}\mathbf{W}\mathbf{u}\mathbf{KS}\left|\right|2\le 1.92$ , $\left|\mathbf{W}\mathbf{T}\mathbf{T}\mathbf{W}\mathbf{S}\mathbf{S}\mathbf{W}\mathbf{u}\mathbf{KS}\right||\infty \le 1.75.$ Furthermore, all three controllers are found to be of ninth order. Here, it is clear from the figure that the first three response peaks in the open-loop (proper) system are effectively annihilated, while the remaining two peaks are greatly attenuated. Moreover, the bandwidth frequency associated with the mixed-norm $\mathcal{H}2/\mathcal{H}\infty$ controller is at least about 50% higher than that corresponding to either of the $\mathcal{H}2$ or $\mathcal{H}\infty$ controllers. Consequently, the mixed-norm controller is expected to display a faster time response. This fact is illustrated in Figure 7(b), where the extended system closed-loop responses to a step reference signal, $\mathbf{r}\left(t\right)=\text{H}\left(t\right),$ for the adopted controllers are presented. It is clear that all three control systems promptly track the reference signal $(t\le 0.2\text{s},$ with less than 10% overshoot), and with the best performance being linked to the mixed-norm $\mathcal{H}2/\mathcal{H}\infty$ controller $(t<0.12\text{s}$ with $tr\approx 0.04\text{s}).$ Also shown are the corresponding actuator regulation voltages, $\varphi (\xi ,\theta ,1,t)$ , which all gradually converge towards the 200 volts asymptote. Here, it is clear from the last subplot that, owing to the relatively large thickness of the adopted PZT4 actuator layer $(h\text{a}=0.02\text{m}$ ), the maximum applied actuator control voltages remain well below the associated $1000\text{V}/\text{mm}$ saturation limit noted in the literatures.^88,89

Figure 7.

(a) Comparison of the frequency domain performance of the closed-loop systems. (b) Comparison of the closed-loop time domain responses to a step reference signal, and the associated applied regulation voltages.

Figure 8 displays the disturbance rejection performance of the smart circular plate in the time domain for a broadband random (Gaussian white noise) mechanical disturbance applied over an annular region designated by $\left[\text{H}\right(\xi -0.35)-\text{H}(\xi -0.65\left)\right],$ on the top panel surface, while applying the mixed-norm $\mathcal{H}2/\mathcal{H}\infty$ , the $\mathcal{H}2$ , and the $\mathcal{H}\infty$ controllers to the identified extended model $\mathbf{G}\left(s\right)$ (see Figure 2(b)). The associated regulation voltages are shown in the second column of the figure. Also shown are results using a second mixed-norm strategy which excludes the regulated output channel $\mathbf{z}\text{u}$ associated with the $\mathcal{H}2$ performance (i.e. there is no restrictions on the actuator signal in the generalized plant), denoted in the figure by $[\mathcal{H}2/\mathcal{H}\infty ]*$ , with the robust stability constraint set as $\left|\right|\mathbf{W}\text{T}\left(s\right)\mathbf{T}(s)\left|\right|\infty \le 0.95$ and the mixed sensitivity $\mathcal{H}2$ norm obtained as $\left|\right|\mathbf{W}\text{S}\left(s\right)\mathbf{S}(s)\left|\right|2\le 0.71$ . The success of both mixed-norm robust active control strategies with a matched distributed actuator–sensor pair for proper rejection of the external random disturbance without causing any instabilities is clear. The time domain performance of the mixed-norm $[\mathcal{H}2/\mathcal{H}\infty ]*$ is slightly superior, however, at the expense of a higher actuator control effort.

Figure 8.

The disturbance rejection performance of the smart panel in the time domain for a broadband random mechanical disturbance, along with the associated regulation voltages.

Conclusions

The 3D exact piezoelasticity theory and the multi-objective mixed $\mathcal{H}2/\mathcal{H}\infty$ control strategy are employed for robust active sound radiation reduction of an optionally thick, rigidly baffled, piezo-composite circular panel with constantly distributed matched sensor-actuator laminae, under broadband general (nonaxisymmetric) electromechanical excitations. In the first part of the numerical results, the important sound radiation and mechanical characteristics of the piezo-composite panel are briefly studied in two basic loading configurations. It is seen that, as the action area of the applied distributed load increases, the radiated sound pressure and interface normal stress levels notably increase, especially in the mid- to high-frequency range. Also, as the asymmetry of distributed loading is increased, increasingly more nonaxisymmetric structural modes appear in the radiated surface pressure and interface stress response spectrums. In the point loading configuration, the smallest effects are observed when the panel is excited exactly at its centre point, simply due to excitation of only the axisymmetric modes. As the excitation position is moved away from the panel centre point, the nonaxisymmetric modes also get involved, ultimately leading to observation of constructive/destructive interference effects, depending on the frequency of excitation. Moreover, it is seen in both (nonaxisymmetric) loading configurations that, while the sound radiation effects of nonaxisymmetric structural modes are very strong near surface of the panel, they markedly decline in (nearly disappear from) the far-field radiation spectrums. In the second part of the paper, system identification is conducted based on the low-frequency-coupled fluid–structure modes, and design/simulation of the control system is performed using reduced order (nominal) and extended models which are found to estimate the dynamics of the system very well within the targeted frequency range. An uncertainty profile (weighting) function sets an upper bound on the modelling error in the entire frequency range and reduces the effects of the neglected high-frequency residual modes in the closed-loop system. A low-pass filter is designed to reject the effect of the external disturbance on the error signal and set requirements on the control performance, while a high-pass filter is used to restrict the control signal amplitudes in order to avoid depolarization of the piezoelectric actuator as well as likelihood of residual mode excitation. It is found that the bandwidth frequency associated with the mixed-norm $\mathcal{H}2/\mathcal{H}\infty$ controller is at least about 50% larger than that of the $\mathcal{H}2$ or $\mathcal{H}\infty$ controllers. Also, the best tracking performance is observed for the closed-loop step response of the mixed-norm controller. Furthermore, the effectiveness of the adopted LMI-based multi-objective robust control strategy with a matched distributed actuator–sensor pair for proper rejection of the external disturbance (in the time domain), without causing any instabilities or actuator saturation problems, is demonstrated for the thick smart circular panel. Lastly, the main novelties/advantages of the proposed methodology may be summarized as follows:

Using the 3D linear piezoelasticity model in conjunction with the Rayleigh integral formula to develop an exact 3D analysis for the forced nonaxisymmetric fully coupled acousto-elastodynamic frequency response of a multi-layered piezo-composite circular plate of arbitrary thickness under heavy fluid loading, without applying any limiting (e.g. far-field, light fluid loading, and low-frequency) approximations, extendable for any arbitrary through-thickness variation of distributed material properties.

Applying an LMI-based multi-objective mixed-norm ( $\mathcal{H}2/\mathcal{H}\infty$ ) robust control strategy for active low-frequency sound radiation control of the arbitrarily thick piezo-composite circular panel with satisfactory transient control performance and robust stability in the face of external disturbances and unmodelled dynamics (truncated modes).

Implementation of continuously distributed perfectly matched piezoelectric uniform force actuator and volume velocity sensor layers to effectively cancel the total volume velocity with the main contribution to the long wavelength acoustic power emission.