Sensor and Actuator Topology for Vibration Control and ParameterEstimation

Michael I. Friswella and Daniel J. Inrnanb

of Wales Swansea, Swansea SA2 8PP, UK.Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0219

ABSTRACTActive vibration control is using increasingly large numbers of sensors and actuators to achieve ever-improvingresults in the control of distributed systems. As the number of actuators and sensors increases, computationaleffort for control purposes increases. As the number of actuators and sensors grows, so too does the frequencyrange over which it is sensible to attempt active control and the time available for control calculations is thereforeshrinking. Notwithstanding the remarkable rate at which processor speeds continue to increase, it is evident thatfull multi-input multi-output control cannot continue to be applied for increasing numbers of sensors andactuators. The requirement for every actuator to have an amplifier and every sensor to have signal conditioning isalso very demanding.

This paper addresses the issue of how best to implement the controller and estimator in smaller sensor Iactuatorgroups and to determine the optimum topology, or grouping, of the sensors and actuators. The demands of boththe control and the parameter estimation are addressed, and the implementation of modal control and selectivesensitivity estimation algorithms are described.

Keywords : actuator topology, sensor topology, vibration control, controllability, parameter estimation,selective sensitivity

1. INTRODUCTIONIn the design of a smart structure the placement of the actuators and sensors is vital for good controllability andobservability, and therefore good performance of the control system. The measures of controllability andobservability are well understood1'2. In siructural dynamics techniques for choosing actuator and sensor locationsare also well established35. Control actuator placement using optimisation techniques is possible6, although forsystems with a large number of degrees of freedom this may be impractical.

In systems with a large number of sensors and actuators it is difficult to implement a single multi-input multi-output (MIMO) controller, sometimes because of communication problems and often because of the limitedcomputation power available, if the system has defined sub-systems, often spatially separated, then deceniralisedcontrol may be applied. In decentralised control the local actuator signals are derived from the local sensormeasurements, thus reducing the order of the local control problem. West-Vukovich et al.7 considered thedecentrailsed control of large flexible space structures, but did not explicitly consider the topology of thecontroller. Unfortunately the lower modes of flexible structures are global in nature, leading to significantinteractions between actuator/sensor groups. This leads to poor perfonnance of decentrailsed controllers based onminimising interactions8.

Further author information -M.I.F. (correspondence): Email: M.I.Friswell@swansea.ac.uk;D.J.I.: Email dinman@vt.edu

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This paper considers the optimum way of physically connecting the actuators and sensors together. Although theresult is similar to deceniralised control in that the actuators are split into a number of subsets, in decentralisedcontrol each member of an actuator group is independent. In this paper we assume that the actuators in a groupare wired together so that they will all receive the same input signal. Thus the number of amplifiers required andthe number controller outputs is determined by the number of groups rather than the number of actuators. Sincethe number of independent inputs to the structure are reduced there will be some reduction in controllability. Asimilar argument applies for connecting the sensors.

2. MODAL CONTROLStructural control is generally only required at the lower frequencies, and thus the control is implemented in termsof a limited number of modal co-ordinates1'9. Here the modal participation factors are identified via a modalfiltering transformation of the system outputs. The damping in each mode may then be increased by the controlsystem which produces the modal forces required. This force is applied to the structure via a secondtransformation. Obviously for continuous structure, and indeed for a discretised structure with a large number ofdegrees of freedom, with a limited number of actuators and sensors, these transformations can only beapproximate, leading to the problems of observation and control spilover10. In terms of physical co-ordinates theequations of motion may be written as

Mq+ D'+ Kq = Bu(1)

y = Cq

where M, D and K are the mass, damping and stiffness matrices. The matrices B and C are the input andoutput matrices for the structure, described by the second order model. Transforming Equation (1) to modal co-ordinates, z, using the lower m mass normalised eigenvectors, '1, gives

z + Az + Az =(2)y=Cz

where A = diag(co),A = ,T and w is the natural frequency of the i th mode. If the damping is

proportional then A is diagonal, and A =diag(21 w) where tj is the damping ratio of the ith mode. A simplecontroller may be obtained by assuming that the number of measurements, the number of actuators and thenumber of modes to be controlled (that is the length of the modal co-ordinate vector z) are all equal. With thisassumption, assuming that the system is controllable and observable, then CCI) may be inverted to obtain the

modal co-ordinates from the measurements, and cI)T B may be inverted to obtain the actuator forces from themodal forces required.

In terms of a state space representation, equation (2) may be written as

9 = FA + [ñ] U

or equivalently in the standard notation, using the state vector x =

x=Ax+Bu(4)y = Cx

where the definitions ofthe matrices A, B and C is obvious by comparing equations (3) and (4).

2.1 Actuator GroupsThe input vector, u, is the input to all of the actuators independently. If the actuators are grouped so that allactuators in the group receive equal control signals, then the number of inputs is reduced to the number ofgroups. If this reduced input vector is ii , then the two input vectors are linked by a transformation B, where

u=fi. (5)

For example, if there are 6 actuators, and they are placed into groups [1 , 3, 6] and [2, 4, 5], then the actuatortransformation mairix is

100110B= (6)

0110

The orientation of the connection to the actuators is also very important. Suppose two piezo-ceramic actuators areplaced either side of a plate to control bending vibration. If the actuators are connected so that they producepositive strains at the same time, there will be no moment created to control bending vibration. However, if theactuators produce opposite strains then the effect of both actuators will be approximately double the effect of asingle actuator. In this simple example it is easy to decide how the actuators should be connected to produce themost effective combined actuator. In general the decision can be made by looking at the scalar products of the

columns of B. In the above example, the first actuator group may be connected differently, with thecorresponding actuator transfonnations,

10 10 10 1001 01 01 0110 —10 10 —10B= 0 1,or 0 1 or 0 1or 0 101 01 - 1 ) 1

10 10 10 10(7)

There are also 4 choices of orientation for the second group, leading to 16 possibilities for the actuatortransfonnation, B, based on the 2 groups. Since there are 20 ways of splitting 6 actuators into 2 groups of 3,there are 320 possible transformations. Obviously for systems of realistic size the number of transformations will

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become enormous. For sensors the equivalent approach may be used. For collocated sensors and actuators theresulting combined input and output mairices are then equivalent. This will be discussed further later in the paper.

3. CONTROLLABILITY AND OBSERVABILITYControllability and observability will be used as a measure of performance for candidate topologies. Theconirollability is based on the state variable description of the equations of motion, and the system is controllableif

ranl4 B AB A2B ... A2m-1B ] = 2m (8)

Note that we are considering a reduced system based on a reduction involving the lower m modes, and thereforethe number of states is 2m. For unthmped systems, or for systems where the damping does not providesignificant coupling between degrees of freedom [2] the structure is controllable if

rani4 Tñ ATB A2TB .. . Am_T}

= m (9)

In a similar way the system is observable if

ranicf CT AT CT (AT)2 CT .. . (AT)2m_l CT ] = 2m (10)

or, for undamped systems, if

rani4 TT ATCT A2TT ... Am_lTCT ] = m (11)

Assuming the actuators and sensors are collocated then B =T and equations (9) and (1 1) are equivalent. Onlythe controllability will be considered further. Since A is diagonal the controllability is determined to a great extent

by

Equation (9) gives a test of whether the structure is controllable, but does not give an indication of close acontrollable system is to being uncontrollable. The controllability gramnilan, We,, for the system described byequation (4) is the solution of the equation,

AW + W A + BBT = 0 (12)

Ordinarily the degree of controllability and observability cannot be determined in isolation [2]. However, forcollocated sensors and actuators, and for the comparison between different topologies, the condition number ofthe controllability grainmian gives a good measure of the quality of the actuator topology.

4. TOPOLOGY SELECTION FOR CONTROLSection 2.1 discussed how actuators and sensors may be grouped to reduce the inputs and outputs in thecontroller design. The controllability and observability grammians may be used to determined the effectiveness ofa given control topology, or grouping of actuators or sensors. Unfortunately it is very difficult to use the

grammians to determine the membership of the groups without performing an exhaustive search. A strategy isrequired to produce an effective, although not necessarily optimal, topology selection. This section outlines one

such strategy based on comparing the columns of T B. A similar strategy applies for the sensors, based on the

rows of CcI: , but for collocated sensors and actuators would give the same result and only the actuator grouping

will be considered. The columns of (1)T B give the modal forces that would be applied to the structure if the a unitinput was applied to the corresponding actuator. The proposed strategy is based on grouping the actuators thathave a siniilar effect on the modal forces, measured by the angles between the modal forces given by the columns

of T B . It is often better to normailse the rows of ciT B so that equal force is applied to all modes. Obviouslygrouping actuators in this way will also have the least detrimental effect on the rank of the controllability matrixgiven by equation (9). This is a sub-optimal procedure because the actuators are allocated to groups sequentially.

The proposed strategy is described based on assigning actuators 1,. ..,p to groups A, B, C,... Let the ith column

of T f themodal force for actuator i, be b. The procedure is as follows:

Step 1. Assign any actuator, say tAl to group A. This choice is arbitrary.

Step 2. Calculate the angle between the modal force for actuator A1 and all the other actuators. Assign theactuator whose modal force is most orthogonal that of actuator A1 (look for minimum, absolute valueofthe cosine ofthe angles) to group B. This is equivalent to finding the minimum of

cos2'q = (b1TbiAl )2/(b1T bj)(bjAlT biAl )(13)

for i = 1,. . . , p , i 1A1 . Lt thechosen actuator be B1

Step 3. For as many groups as required, search the remaining actuators for the one whose modal force is mostorthogonal to the subspace spanned by the modal force of those actuators already allocated. For group Cfind the actuator (labelled iU) whose modal force is most orthogonal to the subspace spanned by

II bjAl biBl ] For group D find the actuator whose modal force is most orthogonal to the subspace

spanned by [ bjAl biBl bj1 ], andso on for the remaining groups.

Step 4. Once one actuator per group has been allocated, the remaining actuators may be allocated to the groups.This is performed sequentially starting with group A. First the remaining actuator whose modal forcethat is most parallel to that of actuator A1 is allocated to group A (labelled iA2). Then the remainingactuator whose modal force that is most parallel to that of actuator B1 is allocated to group B(labelled i1). And so on for the remaining groups.

Step 5. As the actuators are selected their orientation is determined within each group, assuming the firstactuator in each group is positive. The orientation of the remaining actuators are decided by calculatingthe cosine of the angle between modal force of the actuator under consideration and the modal force ofthe first actuator. The orientation of the actuator is the same as the sign of the cosine of this angle.

Step 6. Once a second actuator has been allocated to each group the third actuator is allocated to group A byfinding the remaining actuator whose modal force that is most parallel to the vector bjAl bjA2, themodal force of the already chosen actuators for group A. Whether the vectors are added or subtracted isdetermined by the orientation determined in step 5. The procedure is then repeated for other groups andfor the fourth actuator per group, and so on until all the actuators have been allocated. As each new

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actuator is allocated its orientation is determined as in step 5, by comparison with the first actuator of thegroup.

Step 7. Thus the actuator iransformation matrix B is be generated from the chosen groups, and the orientationofthe actuators within each group.

An alternative to steps 4 and 6 is to choose the actuator whose modal force is most orthogonal to the modal forcescorresponding to the actuators already allocated to the other groups. For each remaining actuator the orthogonalityis assessed group by group and the actuator with the largest angle (or strictly the lowest absolute value of thecosine of the angle) is then chosen. Both of these procedures will be tested in the example. One disadvantage ofthe proposed approach is that the same number of actuators is allocated to each group, although this could bechanged using some criteria based on whether any actuators are sufficiently parallel (or orthogonal) to be includedin a group.

5. AN EXAMPLE PROBLEMWe will now consider the bending vibration of an aluminium cantilever plate which has twelve piezo-ceramicpatches bonded to the surface. The motivation for this example arises from the NASA experiments on fluttersuppression which use a tapered plate containing many actuators1 .The piezo-cerarnic patches are used as bothsensors and actuators, the so called self sensing actuator12. The use of collocated sensors and actuators in thisway reduces the possibility of instabilities due to feedback13. Figure 1 shows the plate which measures 0.8 mlong, 0.3 m wide and is 3 mm thick. The plate is modelled using 24 square plate elements and each piezo-ceramicpatch is assumed to completely cover a single element. The excitation of the plate by the patches is modelled verysimply, namely that a voltage applied to the piezo-ceramic is assumed to produce moments at each node of theelement. This assumption is motivated by the comments of Crawley and de Luis14 for a perfectly bondedactuator. These moments are assumed to be proportional to the applied voltage and for the purposes of illusirationare taken to be 1 Nm/V. Similarly rotations at the nodes are assumed to produce outputs when the patches areused as sensors, and the sensitivity is taken to be 1 V/rad. Although a number of different actuator and sensorpositions could be considered, the odd numbered elements are assumed to include the piezo-ceramic patches forthe purposes of illustration. The positions of these elements are highlighted in Figure 1 .Proportional damping isassumed, with the coefficients of mass and stiffness being 0.5 and 0.0001 respectively. For the design andanalysis ofthe control system the model is reduced by retaining only the 5 lower frequency modes of the system,leading to a 10 state plant model. Thus the controllability and observability of the structure is based only on thelower 5 modes. The natural frequencies and clamping ratios of the plate are given in Table 1.

The success of our selected topologies will be gauged by calculating the condition number of the controllabilitygrammian. A similar measure may be formulated using the observability grammian. Suppose that we wish togroup the 12 actuators into 3 groups of4. Each group is connected electrically so that the same force is applied allactuators in a group. There are 5775 ways of assigning the actuators to the groups. Within each group oneactuator should be assigned a positive orientation, and then there are 8 orientations for the 3 remaining actuators.Thus for all 3 groups there are 512 orientations, and there are 2,956,800 possible topologies even for this simpleexample. To gain some insight into the optimal topology each of the 5775 groupings has been assessed withregard to the resulting controllability. The orientation for each grouping is determined by finding the anglebetween the modal force vector of one reference actuator per group and the force from each of the remainingactuators. In this example the reference actuator is taken as the one with the lowest number, although otherchoices may lead to slightly different relative orientations. In the groups chosen by the methods outlined in thispaper, the reference actuator is taken to be the first one chosen.

Table 2 shows the actuator groups, their orientations and the resulting controllability measure for a number ofcases. The sign of the actuator number gives the orientation of the actuator. The order of the actuators is the orderthat they are assigned to the groups. The first 12 cases use the method of section 4, where the actuator is chosenwhose modal force is most parallel to that ofthe previously chosen actuators in the group. Different actuators areassigned to group A initially in each case. It is quite obvious that this choice is important for the quality of theresulting topology. The next 12 cases are similar to the first except that the actuator is chosen whose modal forceis most orthogonal to that of the previously chosen actuators in the other groups. The best selection is case 19,from the second set based on orthogonality. This best case has a controllability measure that is approximately33% higher than the best topology chosen by the exhaustive search described above (case 33). The methodsdescribed in this paper are obviously sub-optimal, although only 1 10 ofthe 5775 groupings perform better in theexhaustive search than case 19. The other cases (25-32 and 34)are arbitrary topologies shown for comparison.Different orientations of the same groupings are given and show that the actuator orientation within each group isvery important to produce good controllability.

6. TOPOLOGY SELECTION FOR PARAMETER ESTIMATIONOne way of reducing the number of parameters to be identified by model updating15 is to apply excitations whichproduce strong sensitivities to a subset of the parameters whilst causing the sensitivities to other parameters tovanish. The method of selective sensitivity requires the response predictions to a relatively large number ofexcitation forces and only implementations based on piezo-ceramic actuators, or similar, are likely to be feasible.Any subsequent updating method must use frequency response data. In order to provide further explanation, thework of Ben-Haim1619 will be followed.

6.1. Selective Sensitivity in the Frequency Domain

Suppose that the undamped equations of motion are be written in the frequency domain, from equation (1), as,

K(a))q = [_Mco2 + K] q = Bu(14)

y =

where Kd is the dynamic stiffness matrix. The output sensitivity to the change in a stiffness parameter O may be

written, at any frequency, in the form,

S(u) ={_}T {-i--}

(15)

By combining equations (14) and (15) the sensitivity can be expressed as,

Sj(u) = uT jT [FiF1TeT C 1FLLF B u°i ] L - (16)

where F is the dynamic flexibility matrix (the inverse of the dynamic stiffness matrix). Equation (16) may bewritten as,

S(u) = UTDJU. (17)

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The aim of selective sensitivity is to adapt the load system, u, such that,

I = 0 if jE$S(u) . (18)

( 0 if JE3

where = {1, 2, . . . , r} is a subset of r indices representing parameters that do not require updating andS = {r+1, r +2, . . . , s} representing the parameters to be updated. The procedure to be followed in determining

a load system to satisfy equation (18) consists ofthe following steps.

Step 1. At each frequency, select x such that

f = if jE• (19)doj t 0 if JE$

This is often not difficult because the substructure dynamic stiffness matrices, are often sparse

and rank deficient. It should be noted that the determination of x from equation (19) does not dependupon the uncertain parameters O . The existence of an x which satisfies equation (19) is a necessarycondition for selective sensitivity. In general, many vectors x are able to satisfy equation (19), and asubspace may be defined based on these solutions. This subspace is most easily calculated by findingthe intersection of the null spaces of the dynamic stiffness sensitivities for jE S by a singular valuedecomposition.

Step 2. Determine u from the relationship,

B U = KdO + i!j. 0. x = , say, (20)j=1

where KdO is the initial dynamic stiffness matrix. The vector f is calculated from the dynamic stiffnessmatrix sensitivities and the current parameter estimates. Equations (19) and (20) may be combined togive,

u = Kdo + Of x = f, (21)

JE3

In general x is a vector from a subspace of possible solutions, leading to a subspace of vectors f andthe possibility of different solutions u to equation (21) for different vectors x. Indeed the result will be asubspace of solutions u.

It can be seen that the only model parameters implicated in the choice of the input are those to which themeasurements are to be selectively sensitised. If a u can be found from equation (21), whilst x satisfies theconditions in equation (19), then that same u will satisfy the selective sensitivity conditions expressed in equation(18). A solution for equation (21) will only exist if there is a vector f in the space spanned by the columns of B.

If equation (21) has no solution then consideration may be given to altering the location of the actuators,

represented by . Since the values O are unknown, an iterative solution is sought which results in thesimultaneous convergence of the input uand the updating parameters O.

6.2 The Effect of Actuator TopologyThe actuator topology will affect the possibilities of finding a solution to equation (21). For an approximatesolution must be close to the subspace spanned by the columns of B. This subspace has a dimension which isless than or equal to the number of actuators. If we group the actuators in a way determined by the actuatortransformation, f, then f must be close to the subspace spanned by the columns of BB. Unfortunately thismatrix has the same number of columns as actuator groups, and the dimension of the subspace spanned by thecolumns of B is far smaller than the subspace spanned by the columns of B. In general, the vector f is not asingle vector, but may be any vector from a subspace that may be calculated from equations (19) and (21). Wemust choose the actuator transformation so that the angle between the corresponding subspace and the subspaceof vectors f is small. There is a lower bound on this angle, namely the angle between the subspacecorresponding to B and that of the vector f.

The concept of angles between subspaces is a generalisation of the concept of angles between vectors obtained bythe scalar product. In three dimensions, it is easy to visualise the angle between a pair oflines, that is subspacesof dimension one, or indeed the angle between a line and a plane, that is subspaces of dimension one and tworespectively. Bjorck and Golub0 described the definition and calculation of the angles between subspaces of anydimension. These ideas have been applied in structural dynamics in the areas of damage location21 , modelupdating using perturbed boundary condition testing22, mode shape correlation23 and sensor location24.

The calculation of the angle between the subspaces, described above, allows any controller topology to beassessed. Unfortunately it is more difficult to derive a computationally efficient method (even a sub-optimal one)for choosing actuator locations for selective sensitivity. In any event it is likely that parameter estimation will be asecondary requirement and that the topology will be determined primarily from controllability considerations.Therefore topology selection will not be pursued but an example given for the assessment of the topologiesselected for good control.

6.3 An ExampleThe plate example from Section 5 will be used to demonstrate the calculation of the selective sensitivityperformance. The parameters consist of the global stiffness of each element. The sensitivity with respect to thestiffness of the 3 rows of elements closest to the clamped edge of the plate in Figure 1 will be made zero. This ispossible using all 12 actuators as there is a suitable vector f within the subspace obtained from the columns of B.The actuator topologies in Table 2 were tested and the results are shown in the last column of this table. Theangles given are the smallest of the angles between the subspaces of f and spanned by the columns of BI.Clearly a good topology for control is not necessarily a good topology for selective sensitivity in parameterestimation. A different choice of parameters in the selective sensitivity method would, of course, lead to differentperformance measures for the topologies.

7. CONCLUSIONSThis paper has considered the grouping of a large number of actuators for the vibration control of flexiblestructures. It has been demonstrated that the grouping of the actuators, and the orientation of the actuatorsconnected together within each group, has an enormous effect on the quality of the resulting controller, as

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measured by the condition number of the controllability grammian. A procedure has been proposed whichproduces a topology that has a good performance, but the method is sub-optimal. A similar approach may beadopted for the topology of a large number of sensors using observability as a measure of performance. Theeffect of actuator topology on the ability to compute a solution to the selective sensitivity parameter estimationproblem has been discussed and demonstrated. A good topology for control is not necessarily a good topologyfor parameter estimation.

ACKNOWLEDGEMENTSDr Friswell gratefully acknowledges the support of the Engineering and Physical Sciences Research Councilthrough the award of an Advanced Fellowship. Prof. Inman gratefully acknowledges the support of the US AnnyResearch Office, grant number DAALO3-92-G-0180 Under the supervision of Dr. Gary Anderson

REFERENCES1 . D. J. Inman, Vibration: with Control, Measurement, and Stability. Prentice-Hall, 1989.2. P. C. Hughes and R. E. Skelton, "Controllability and Observability ofLinear Matrix-Second-Order

Systems", J Applied Mechanics, 47, pp. 415-420, 1980.3 . J. E. T. Penny, M. I. Friswell and S. D. Garvey, "Automatic Choice of Measurement Locations for

Dynamic Testing", AIAA J., 32(2), pp. 407-414, 1994.4. D. C. Kammer, "Sensor Placements for On-Orbit Modal Identification and Correlation ofLarge Space

Structures", AIAA .1. Guidance, Control and Dynamics, 14(2), pp. 251-259, 1991.5 . T. W. Um, "Acmator/Sensor Placement for MOdal Parameter Identification of Flexible Structures", Modal

Analysis, 8(1), pp. 1-14, 1993.6. H.-D. Chiang, J. S. Thorp, J.-C. Wang, J. Lu and B. Aubert, "Optimal Controller Placements in Large-

Scale Linear Systems", lEE Proc.-D, 139(1), pp. 79-87, 1992.7 . G. S. West-Vukovich, E. J. Davidson and P. C. Hughes, "The Decentralised Control of Large flexible

Space Structures", IEEE Trans. Automatic Control, AC-29(l0), pp. 866-879, 1984.8 . M. I. Friswell, S. D. Garvey, J.E.T. Penny and A. Chan, "Selection of Control Topology for Decentrailsed

Multi-Sensor Multi-Actuator Active Control", 14th International ModalAnalysis Conference, Dearborn,Michigan, pp. 278-284, 1996.

9 . L. Meirovitch, Dynamics and Control of Structures. John Wiley and Sons, New York, 1990.10. M. J. Balas, "Feedback Control ofFlexible Structures", IEEE Trans. Automatic Control, AC-23(4), pp.

673-679, 1978.I I . J. Heeg, A.McGowan, E. Crawley and C. Lin, "The Piezoelectric Aeroelastic Response Tailoring

Investigation", mt. Forum on Aeroelasticity and Structural Dynamics, Manchester, paper 2, 1995.1 2. J. J. Dosch, D. J. Inman and E. Garcia, "A Self-Sensing Piezoceramic Actuator for Collocated Control", J.

IntelligentMaterial Systems and Control, 3, pp. 166-185, 1992.1 3. H. S. Tzou and J. J. Hoilkamp, "Collocated Independent MOdal Control with Self-Sensing Orthogonal

Piezoelectric Actuators (Theory and Experiment)", Smart Materials and Structures, 3, pp. 277-284, 1994.14. E. F. Crawley and J. de Luis, "Use of Piezoelectric Actuators as Elements of Intelligent Structures", AIAA

J., 25(10), pp. 1373-1385, 1987.1 5 . M. I. Friswell and J. E. Mottershead, Finite Element Model Updating in Structural Dynamics, Kiuwer

Academic Publishers, 286 pp., 1995.1 6. Y. Ben-Haim, "Adaptive Diagnosis ofFaults in Elastic Structures by Static Displacement Measurement: The

Method of Selective Sensitivity", Mechanical Systems and Signal Processing, 6, 85-96, 1992.17. Y. Ben-Haim, "Model Updating of Linear Systems by Selective Sensitivity with Deliberate Structural

Modification", Modal Analysis, 9(4), 287-301.18. Y. Ben-Haim and U. Prells, "Selective Sensitivity in the Frequency Domain, -I. Theory", Mechanical

Systems and Signal Processing, 7, 461-475, 1993.

19. U. Prells and Y. Ben-Haim, "Selective Sensitivity in the Frequency Domain -II. Applications", MechanicalSystems and Signal Processing, 7, 551-574, 1993.

20. A. Bjorck and G. H. Golub, "Numerical Methods for Computing Angles Between Linear Subspaces",Mathematics ofComputation, 27(123), pp. 579-594, 1973.

2 1 . A. P. Cherng and M. K. Abeihamid, ??A signal subspace correlation (SSC) index for detection of structuralchanges", 11th International ModalAnalysis Conference, Orlando, florida, pp. 232-239, 1993.

22. M. Yang and D. Brown, "Model Updating Techniques using Perturbed Boundary Condition (PBC) TestingData", 14th InternationalModalAnalysis Conference, Detroit, Michigan, pp. 776-782, 1996.

23. S. D. Garvey, J. E. T. Penny and M. I. Friswell, "Quantifying the Correlation Between Measured andComputed Mode-Shapes", Journal ofVibration and Control, 2(2), pp. 123-144, 1996.

24. S. D Garvey, M. I Friswell and J. E. T. Penny, "Evaluation of a Strategy for Optimal Choice ofMeasurement Locations Based on the Concept of Minimum Angles Between Subspaces", 14th InternationalModalAnalysis Conference, Dearborn, Michigan, pp. 1546-1552, 1996.

NaturalFrequency (Hz)

DampingRatio (%)

1 4.013 1.12•T 22.29 0.88—r 25.17 0.95

70.68 2.28-r 71.10 2.29•T• 129.6 4.10••T 140.3 4.44•T• 190.3 6.00

204.0 6.43.-i- 224.9 7.08

Table 1 . Natural Frequencies and DampingRatios for the Plate Example

Figure 1 . The Plate Example. The Shaded Elements Contain Actuators/Sensorsand the Numbers are the Actuator Numbers

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