Multivariable feedback active structural acoustic control using adaptive ...

Multivariable feedback active structural acoustic control using adaptive piezoelectric sensoriactuators Jeffrey S. Vippermana) Department of Mechanical Engineering, 5711 Boardman Hall, University of Maine, Orono, Maine 04401-5711

Robert L. Clark Department of Mechanical Engineering and Materials Science, Box 90300, Duke University, Durham, North Carolina 27708-0300

~Received 9 May 1997; accepted for publication 25 August 1998! An experimental implementation of a multivariable feedback active structural acoustic control system is demonstrated on a piezostructure plate with pinned boundary conditions. Four adaptive piezoelectric sensoriactuators provide an array of truly colocated actuator/sensor pairs to be used as control transducers. Radiation filters are developed based on the self- and mutual-radiation efficiencies of the structure and are included into the performance cost of an H 2 control law which minimizes total radiated sound power. In the cost function, control effort is balanced with reductions in radiated sound power. A similarity transform which produces generalized velocity states that are required as inputs to the radiation filters is presented. Up to 15 dB of attenuation in radiated sound power was observed at the resonant frequencies of the piezostructure. © 1999 Acoustical Society of America. @S0001-4966~98!04812-7# PACS numbers: 43.40.Vn @PJR#

INTRODUCTION

Control of structures and their associated sound radiation has been a topic of interest for both civil and military applications. Active control techniques have been found to be complimentary to passive control methods, which are best suited for high frequency ~.1 kHz! noise and vibration. Applications for control include reduction of interior and exterior noise in aircraft, submarines, appliances, automobiles, and machinery. In active noise control ~ANC!, acoustic sensors are used to make measurements at points in the acoustic field and the information is then used to form a control signal. Loudspeakers are typically used to create the secondary field which is intended to cancel the primary or disturbance field. The same concept applies for active vibration control ~AVC!, except that structural based sensors and actuators are used. For ANC, it is not always feasible to measure the acoustic field, particularly for underwater or aviation applications. For these cases, the radiated acoustic power can often be related to structural-based measurements which then form the basis of the acoustic control objective.1,2 This concept of controlling the radiated sound power through the realization of adaptive structures was proposed by Clark and Fuller3,4 and is frequently denoted as active structural acoustic control, or ASAC. ASAC can also be applied to increase the transmission loss across panels. Early implementations of ASAC used feedforward control topologies. The objective function minimized by the adaptive feedforward algorithm was typically composed of measurements of the far-field acoustic pressure3 or from near field sensors5 which naturally emphasized control of the modes which are efficient at radiating sound. Maillard and a!

Work performed as a Graduate Research Assistant at Duke University.

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J. Acoust. Soc. Am. 105 (1), January 1999

Fuller1 developed real-time estimation filters for feedforward control of sound radiation that were based on an array of structural measurements which were considered to be discrete monopole radiators. Feedback ASAC has been largely confined to analytical studies.6–8 The cost functionals typically provided an ad hoc method of more heavily weighting the modes of the structure that are known to be efficient acoustic radiators and similarly provided diminished weighting of the inefficiently radiating modes. Baumann et al.2 used frequency-shaping concepts9 to develop radiation filters which augment the structural states of the plant to proportionally weight each mode proportional to it’s radiation efficiency. An analytical study2 showed that radiation filters allow more control energy to be focused into the modes most responsible for the sound radiation. In this work, a new transducer technology known as adaptive piezoelectric sensoriactuators ~APSAs!10,11 will be used to experimentally demonstrate a multi-input multioutput ~MIMO! feedback active structural acoustic control ~ASAC! system. The testbed is a simply supported plate with four adaptive piezoelectric sensoriactuators as control transducers. These self-sensing actuators provide truly collocated sensor/actuator pairs from each piezoceramic patch which greatly enhances stability robustness of the feedback controller. The chronological steps taken to implement the experimental ASAC control system were: ~1! Implement a 434 array of APSAs on a simply supported plate testbed; ~2! Experimentally measure all 16 frequency response functions ~FRFs! between each possible APSA pairs; ~3! Fit a 20-state, four-input, four-output (4i4o) state-space model to the 16 measured FRFs;

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~4! Perform a similarity transform on the state space model such that the generalized velocities appear as internal states; ~5! Append analytical radiation filters to the transformed state space model of the piezostructure; ~6! Formulate an H 2 cost functional based on acoustic radiation output of the model and design the control system that will minimize this cost; ~7! Experimentally implement the control system and measure open- and closed-loop acoustic power to evaluate the control system performance. In Sec. I, the basic H 2 control design philosophy is presented. Following in Sec. II is a description of the experimental setup used to demonstrate ASAC using adaptive piezoelectric sensoriactuators. Section III contains information pertaining to modeling the structural acoustic system, including the system identification of the plate piezostructure, appending analytical radiation filters to the plate model, and performing a similarity transform that produces generalized velocity states that are required as inputs to the radiation filters. Finally the results of the experiments are presented in Sec. IV and a summary of the findings is presented in Sec. V.

F F

J5E lim T→`

J5E lim T→`

1 T 1 T

E E

T

0 T

0

G

z~ t ! T z~ t ! dt ,

~3!

G

~ xv ~ t ! T Q~ v ! xv ~ t ! 1u~ t ! T Ru~ t !! dt ,

~4!

where Q~v! is a positive semi-definite, frequency-dependent weighting matrix which determines the error or performance penalty that is proportional to radiated sound power, and R is a positive definite control penalty weighting matrix. Note that this cost functional is based upon the generalized velocity states, xv(t), of the piezostructure such that the analytical radiation filters can be appended to each mode. This result will require a similarity transform be performed on the statespace model obtained from system identification of the piezostructure testbed. Note from Fig. 1 that the H 2 control problem has been cast in the standard two-port form with LQG-style weightings. There are optimization methods readily available to determine the optimal compensator for this configuration.14 II. EXPERIMENTAL SETUP A. Simply-supported plate testbed

I. CONTROLS

H 2 control design techniques in conjunction with linear quadratic Gaussian ~LQG! weightings were used for the feedback active structural acoustic control design. Static feedback ASAC designs have been previously studied.12 The cost functionals include frequency-shaping9 that is proportional to the modal radiation efficiencies, as demonstrated analytically by Baumann2 and later incorporated by others.11–13 In this method, time-domain radiation filters are used to augment the states of the state-space piezostructure model such that the root-mean-square error variable is proportional to radiated sound power. In this way, the control system is able to focus more effort into modes which are efficient at radiating sound and less effort into those modes which are inefficient radiators. The generalized plant model is of the form: x˙ 5Ax1Bu1wd,

~1!

y5Cx1wn,

~2!

where x is the state vector, x˙ is the derivative of the state vector, and u is the 431 vector of control voltages to the piezoelectric sensoriactuators. The system matrix, A, and input and output influence matrices, B and C, respectively will be determined by experimental system identification of the piezostructure control paths. The process noise, wd , and sensor noise vectors, wn , are Gaussian processes that satisfy: E@ wd(t)wd( t ) T # 5Wd (t2 t ), E@ wn(t)wn( t ) T # 5Vd (t2 t ), E@ wn(t)wd( t ) T # 50. Each state of the plate model was excited by the process noise, wd , and sensor noise, wn , was used to corrupt the four sensoriactuator outputs. Experimental characterization of the process and sensor noises was not conducted, but rather the magnitudes of these noise processes served as design parameters that were determined heuristically. The LQG cost is of the form: 220

J. Acoust. Soc. Am., Vol. 105, No. 1, January 1999

A classic simply supported plate test bed was designed and assembled. The x-y dimensions of the 2 mm thick steel plate are given in Fig. 2. Details of the construction can be found in the dissertation by Vipperman.11 The locations of the four arbitrarily placed PZTs used for the control transducers are shown in Fig. 2. Also shown in Fig. 2 is the input location of a Ling Dynamics Systems V203 shaker, which was attached to the back of the plate frame and used for the disturbance excitation source. B. Electronic hardware

Real-time signal processing was performed with a Spectrum Signal Processing, Inc. 50-MHz TMS320C40-based DSP system in conjunction with the Spectrum PC16IO8 16input and 8-output analog I/O Card. Custom DSP software conveniently allowed the user to alternate between off-line training of the APSAs10 or implementing the fixed feedback control as desired. Although you can theoretically train the APSAs simultaneously with control, this was found unnecessary since the APSAs remained stationary within the time scale of the system identification and control experiments. A sampling rate of 4 kHz was used for both the training of the APSAs and the active control experiments. Many other peripheral items were needed to perform tasks such as filtering, adding or removing gain, and generating, measuring and analyzing signals, etc. A customdesigned sound intensity probe15 was used to obtain sound intensity readings near the surface ~70 mm! of the plate, allowing the sound power generated by the plate to be calculated by performing a discrete integration of the intensity across the measurement grid. Two high-quality Realistic electret tie-clip microphones spaced a distance of 43 mm in a handheld aluminum support comprised the sound intensity probe. Positioning errors between measurements were elimiJ. S. Vipperman and R. L. Clark: Multivariable feedback ASAC

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FIG. 1. Generalized plant used in two-port control design.

FIG. 3. Schematic diagram of hardware setup used for a single channel APSA with the plate test bed.

C. Adaptive piezoelectric sensoriactuators

nated by performing the open-loop and three closed-loop measurements sequentially while the probe was positioned at each point. The high quality Bru¨el and Kjær 4190 microphone was used to calibrate the magnitude of each transducer. Transfer functions between the two electret microphones permitted phase correction factors for the intensity probe to be computed across the bandwidth in which measurements would be made.15,16 Statistical tests for accuracy and repeatability were performed to verify the operation of the intensity probe. Far-field pressure measurements were taken using a Bru¨el and Kjær type 4190 microphone with type 5935 power supply. Although the pressure measurements are not presented, they were used to corroborate the sound power calculations. Additional hardware items include Ithaco 4302 Low pass Filters, Krohn-Hite 7600 Wideband Amplifiers, AVC Series 790 Wide band Amplifiers, DSPT, Inc. SigLab 2022 spectrum analyzers, and Ithaco 453 Gain Preamplifiers. A schematic of the experimental setup is depicted in Fig. 3.

Adaptive piezoelectric sensoriactuator ~APSA! technology is mature and has been demonstrated for complex structural control problems.11 The analog hardware was designed by the Adaptive Systems and Structures Laboratory at Duke University and constructed by NASA-Langley Research Center. The current capability of the hardware is to simultaneously adapt four channels of the piezoelectric sensoriactuator concurrently10 using four separate, uncorrelated, white noise training signals generated by the DSP hardware. The adaptive piezoelectric sensoriactuators were typically trained once and then fixed for the duration of the system identification or control phases of the experiments.

FIG. 2. Schematic diagram of experimental simply supported plate showing arbitrary locations of four PZT bending actuators ~dimensions in mm!.

FIG. 4. Picture of simply supported plate test bed mounted in baffle wall inside anechoic chamber.

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D. Test facilities

Acoustic experiments were performed in the Duke University anechoic chamber test facility. The chamber has a volume of approximately 27 m3 and a cutoff frequency of approximately 250 Hz. A baffle which spanned the width and height of the chamber was constructed from 13 mm thick plywood and 503100 mm framing studs. A picture of the simply supported plate installed flush with the baffle wall is shown in Fig. 4. Figure 4 also includes a view of the sound

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TABLE I. Modal properties of experimental plate test bed.

Mode

Modal indices

1 2 3 4 5 6 7 8 9 10

~1,1! ~2,1! ~1,2! ~2,2! ~3,1! ~1,3! ~3,2! ~2,3! ~4,1! ~3,3!

Strong acoustic radiator?

Theory natural freq. ~Hz!

Measured resonant freq. ~Hz!

*

33.6 76.2 91.6 135 148 189 205 231 247 302

32.0 70.8 85.9 125 136 173 191 212 227 279

* *

*

Measured damping ratio 2.56% 1.56% 1.06% 0.64% 0.41% 0.92% 0.25% 0.42% 0.39% 0.20%

intensity probe which was used to measure intensity near the surface of the plate. The chamber is fitted with a radial microphone array seen in the figure which was not used in this experimental setup. III. SYSTEM MODEL A. System identification

Once the four APSAs were trained, all sixteen possible frequency response functions ~FRFs! between the four APSAs were experimentally computed between 0 and 500 Hz. FRFs were measured in groups of four whereby one piezoceramic patch was driven with a random noise signal and the responses of all four patches were measured simultaneously by the Siglab spectrum analyzer. A four-input, four-output (4i4o) state-space model having 40 states was then fit to the 16 FRFs using the SmartID17 software package. The statespace model corresponds to the matrices APR40340, B PR4034 , CPR4340 which are presented in Eqs. ~1! and ~2!. A typical comparison of measured and modeled FRFs is shown in Fig. 5 which corresponds to the PZT located at normalized plate coordinates $0.23, 0.20%. Similar results were achieved for all control paths of the multivariable system resulting in an accurate plant model which is essential for good control system performance. The curve-fit of the FRFs allowed the resonant, modal properties of the structure to be extracted. The analytical and experimental modal properties of the plate are presented in Table I, where the modes are numbered consecutively in order of their resonant frequencies as well as listed by the corresponding ~x,y! modal indices. In addition, Table I indicates whether a particular mode is an efficient sound radiator or not. Analytical resonant frequencies are slightly overestimated due to the weak torsional-spring boundary conditions of the experimental test rig which were not modeled analytically. B. Appending radiation filters to experimentally determined models

Next, the analytical radiation filters for the plate were designed2,11–13 and appended to the state-space model of the plate in order to create the acoustic cost functional for the control system. The inputs to the analytical radiation filters are the plate generalized velocity states ~modal velocities for 222


FIG. 5. Analytical and experimental frequency response functions across the adaptive piezoelectric sensoriactuator centered at normalized coordinates $0.23, 0.20%.

an ideal plate with no included mass or stiffness!.2 The internal states resulting from the system identification process unfortunately do not correspond to generalized plate velocities, but rather correspond to one of the infinitude of other valid state-space realizations for the system model. Two different state space realizations of the same system model, ¯ ,B ¯ ,C ¯ % , each having state vectors, x(t) and $A,B,C% and $ A ¯x(t), respectively are related through the similarity transform, Tsim , as18 ¯x~ t ! 5Tsimx~ t ! ,

~5!

¯ 5T21 A sim ATsim ,

~6!

¯ 5T21 B sim B,

~7!

¯ 5CTsim , C

~8!

provided that Tsim exists. Analytical piezostructures models based on Galerkin’s method whereby analytical mass, stiffness, and damping matrices are determined results in a state-space model having physical states that correspond to generalized plate velocities, h˙ (t), and displacements, h(t), 13 as: x~ t ! 5

H J

h~ t ! , h˙ ~ t !

~9!

and the following corresponding system matrix, designated Ad/ v : Ad/ v 5

F

0 ~ Mtot!

21

I ~ Ktot!

~ Mtot! 21 Cs

G

,

~10!

where Mtot is the sum of the mass contributions from the structure and the piezoceramic patches, Ktot is the sum of the structural and piezoceramic stiffness contributions, Cs is the assumed viscous structural damping used to bound the model, and I is the identity matrix. Since the modal properties that comprise the system matrix, A, for any realization can be easily extracted from an eigenvalue decomposition, the desired matrix, Ad/ v , can be found from two successive similarity transforms of the arbitrary system matrix, A, that is obtained from the system identification process. J. S. Vipperman and R. L. Clark: Multivariable feedback ASAC

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First, the system identification model is transformed into a Modal canonical form, where the system matrix, AJ , becomes a tri-diagonal matrix consisting of 232 submatrices composed of the eigenvalues for each second-order mode of the piezostructure. The transform matrix required to realize this form is simply the eigenvector matrix, F~•!, of the current, arbitrary system matrix, or Tsim5F(A). The transform exists and is nonsingular, since there will be N linearly independent eigenvectors resulting from an accurate system identification of the piezostructure. Once in the Modal form, we ´ 21 can then find the inverse transform, T sim , that yields the displacement/velocity realization of the plant model. First, the resonant frequencies, v n , and damping ratios, z n , are extracted from the eigenvalues of the system. These values can then be manually sorted and formed into a displacement/ velocity form of the system matrix, Eq. ~10!, since @ v 2n # 5(Mtot)21(Ktot) and @ 2 z n v n # 5(Mtot)21Cs . The eigenvectors of this matrix then define the inverse of the similarity transform between Modal form and displacement/velocity ´ 21 form, T sim 5F(AJ ). To summarize, the steps to find Ad/ v are: ~1! Find $ AJ ,BJ ,CJ % using the transform Tsim5F(A); ~2! Extract and sort the natural frequencies, v n and damping ratios, z n from AJ and manually form matrix Ad/ v . ~3! Find T´sim5F(Ad/ v ) 21 and use it to transform $ AJ ,BJ ,CJ % into $ Ad/ v ,Bd/ v ,Cd/ v % . ~4! Once in displacement/velocity form, the following output matrix, Cgv , can be appended to Cd/ v such that the generalized velocities can be observed Cgv5 @ 0 u I# PR20340;

~11!

~5! The analytical mass matrix, Ms , could be used to form a generalized modal input influence matrix to be appended to Bd/ v which is computed from Bgv5

F G 0

M21 s I

PR40320.

~12!

Typical analytical and experimentally obtained generalized velocity outputs for individual generalized force inputs for the plate are shown in Fig. 6. The analytical and experimental curves of Fig. 6 compare well. Some small discrepancies in magnitude exist due to the difference in modal damping ratios between the analytical model ~0.45% for all modes! the experimental model ~see Table I for values!. Note that if these were modal velocity outputs for a structure with no mass or stiffness discontinuities, each peak in in Fig. 6 would have uniform magnitude since the response would correspond to the true eigenfunctions. IV. EXPERIMENTAL RESULTS

Control system performance is evaluated by experimentally measuring radiated sound power from the structure in the anechoic chamber for the open- and closed-loop cases. For the purpose of this study, three different ASAC systems were experimentally investigated. Each case differs by the number of modes which have radiation filters appended ~either 4, 6, or 8!. Increasing the number of radiation filters in 223


FIG. 6. Response between the generalized force inputs and generalized velocities for the first six plate modes for the analytical and experimental models.

the control system increases the number of plate modes that experience radiation control. However, the order of the controller will increase nearly quadratically with respect to the increase in radiation filters. Two results are presented: the spectrum of the highest order ASAC controller ~filters appended to eight plate modes! and a bar chart summarizing the amount of control achieved by each ASAC controller at each plate resonance. Figure 7 presents the closed-loop performance of the most complex ASAC system used in this study ~radiation filters appended to the first eight structural modes!. The solid line represents the measured closed-loop acoustic power and the measured open-loop power is given by the dotted line. Also depicted in the figure are modal indices corresponding with each structural plate resonance as well as the cutoff frequencies corresponding to the system model truncation and the inclusion of the radiation filters. From inspection of Fig. 7, good reductions in acoustic power ~>15 dB! are observed at the ~3,1! and ~1,3! modes at 136 and 173 Hz, respectively, which are excellent acoustic radiators. Note that despite the radiation efficiency of the ~3,1! mode, the measured open-loop power is relatively low because it is only marginally excited by the disturbance shaker which was arbitrarily located near a nodal line of that mode. Excellent reduction is also observed in the ~2,3! mode at 212 Hz which does not radiate as efficiently as the ~odd,odd! modes, but for the particular arbitrary actuator placement, experiences significant coupling to the ~1,3! and ~3,1! radiation modes. Control of the ~1,1! mode ~34 Hz! is very limited since PZTs do not couple well to modes having a trace structural wavelength much longer than the dimension of the piezoceramic transducer.19 The least amount of control is observed in the weakly radiating ~2,2! mode at 125 Hz, as expected. Note that some control is observed beyond the bandwidth of the radiation filters and the structural plate model which is likely due to some coupling between the experimentally determined generalized velocities which is caused by the piezoceramic actuators. Finally, a summary of the power reductions at each plate resonance for the open-loop and three closed-loop ASAC J. S. Vipperman and R. L. Clark: Multivariable feedback ASAC

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FIG. 7. Acoustic power measurements for multivariable H 2 active structural acoustic control with eight radiation filters.

cases is presented in Fig. 8. Note that increasing the number of plate modes ~4, 6, or 8! that have radiation filters appended increases the frequency range over which radiation control is observed, at the expense of increased controller complexity. For example, note that the acoustic response at the resonance frequency of the fifth or ~3,1! mode is not controlled by the ‘‘ASAC, 4 Filters’’ closed-loop case, but is well-controlled by the 6- and 8-filter ASAC cases. For the latter two cases, the ~3,1! mode is included into the

FIG. 8. Comparison of reductions achieved in acoustic power at each modal resonance for all control systems. 224


structural-acoustic controller cost function and hence sound radiation at the resonance frequency of that mode is reduced by approximately 12 dB.

V. CONCLUSIONS

Multivariable feedback active structural acoustic control ~ASAC! has been experimentally demonstrated using adaptive piezoelectric sensoriactuators as collocated control transducers. A performance cost which minimizes acoustic radiation from the baffled, simply supported plate test structure is accomplished by augmenting the experimentally obtained structural model with radiation filters that are designed using the self- and mutual-radiation efficiencies of the plate modes. A similarity transform to achieve an experimental state-space model having generalized velocity states is presented since the radiation filters require these velocity states as inputs. Acoustic control can thus be achieved without the need for sensors in the acoustic field which is consistent with ASAC. Reductions in acoustic power at the structural resonances from 1 to 15 dB were achieved within the frequency range of the appended radiation filters. The effects of increasing the number of modes ~4, 6, or 8! that are appended with radiation filters was investigated. Since including more radiation filters effectively increases the frequency range of the control system, better overall performance was achieved ~more modes are controlled and less spillover is observed!. J. S. Vipperman and R. L. Clark: Multivariable feedback ASAC

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ACKNOWLEDGMENTS

The authors gratefully acknowledge the Structural Acoustics Branch at NASA Langley Research Center for funding this research under Grant No. NAG-1-1570. The authors also acknowledge Dave Cox of the Guidance and Controls Branch at NASA Langley Research Center for help with the control system design as well as Ran Cabell of the Structural Acoustics Branch at NASA Langley Research Center for the TMS320C40 port of the sensoriactuator/ control code. 1

J. P. Maillard and C. R. Fuller, ‘‘Advanced time domain sensing for structural acoustic systems I. Theory and design,’’ J. Acoust. Soc. Am. 95, 3252–3261 ~1994!. 2 W. T. Baumann, W. R. Saunders, and H. H. Robertshaw, ‘‘Suppression of acoustic radiation from impulsively excited structures,’’ J. Acoust. Soc. Am. 90, 3202–3208 ~1991!. 3 R. L. Clark and C. R. Fuller, ‘‘Control of sound radiation with adaptive structures,’’ J. Intell. Mater. Syst. Struct. 2, 431–452 ~1991!. 4 R. L. Clark and C. R. Fuller, ‘‘Experiments on active control of structurally radiated sound using multiple piezoceramic actuators,’’ J. Acoust. Soc. Am. 91, 3313–3320 ~1991!. 5 R. L. Clark, ‘‘Advanced sensing techniques for active structural acoustic control,’’ Ph.D. thesis, VPI&SU, Blacksburg, VA 24061, 1992. 6 L. Meirovitch and S. Thangjitham, ‘‘Active control of sound radiation pressure,’’ Journal of Vibration and Acoustics 112, 237–244 ~1990!. 7 O. Fluder and R. Kashani, ‘‘Robust control of structure-borne noise using ^ ` methods,’’ in Proceedings of the ASME Winter Annual Meeting, Vol. DSC-38, pp. 191–204, November 1992.

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H. T. Banks, R. J. Silcox, and R. C. Smith, ‘‘Approximation methods for control of structural acoustics models with piezoceramic actuators,’’ J. Intell. Mater. Syst. Struct. 4, 98–116 ~1993!. 9 N. K. Gupta, ‘‘Frequency-shaped cost functionals: Extension of linearquadratic-Gaussian design methods,’’ J. Guid. Control Dyn. 3, 529–535 ~1980!. 10 J. S. Vipperman and R. L. Clark, ‘‘Implementation of an adaptive piezoelectric sensoriactuator,’’ AIAA J. 34, 2102–2109 ~1996!. 11 J. S. Vipperman, ‘‘Adaptive piezoelectric sensoriactuators for active structural acoustic control,’’ Ph.D. thesis, Duke University, Durham, NC 27708, 1997. 12 R. L. Clark and D. E. Cox, ‘‘Multi-variable structural acoustic control with static compensation,’’ J. Acoust. Soc. Am. 102, 2747–2756 ~1997!. 13 R. L. Clark, G. P. Gibbs, and W. R. Saunders, Adaptive Structures, Dynamics and Control ~Wiley, New York, 1998!. 14 J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis, ‘‘Statespace solutions to standard H 2 /H ` control problems,’’ IEEE Trans. Autom. Control. 34, 831–847 ~1989!. 15 N. M. An, ‘‘Acoustic intensity probe,’’ Technical Report M.E. 198 Graduation With Departmental Distinction, Duke University, Durham, NC, 1995. 16 J. Y. Chung, ‘‘Cross-spectral method of measuring acoustic intensity without error caused by instrument phase mismatch,’’ J. Acoust. Soc. Am. 64, 1613–1616 ~1978!. 17 Active Control eXperts, Inc., 215 First Street, Cambridge, MA 12142. Smart ID System Identification Software. 18 T. Kailath, Linear Systems ~Prentice-Hall, Englewood Cliffs, NJ, 1980!. 19 J. S. Vipperman and R. L. Clark, ‘‘Implications of using colocated strainbased transducers for output active structural acoustic control,’’ J. Acoust. Soc. Am. ~submitted!.

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