NASA

Technical w,s_-TP-237_ _9_50022829

Paper

2371

January 1985

Adaptive Identification

and Control of Structural

Dynamics Systems Using

Recursive Lattice Filters

N. Sundararajan;

Raymond C. Montgomery,

and Jeffrey p. Williams r_ " _] L! RARYCO?_,,

LANGLEYRESEARCH!jENnER

LIBRARYNASA

BAMPION, VIRGINIA

NASA

iiil_ii[iiiiini

I

' 3 1176 01449 6195 NASA

Technical

Paper

2371

1985

Adaptive Identification

and Control of Structural

Dynamics Systems Using

Recursive Lattice Filters

N. Sundararajan,

Raymond C. Montgomery,

and Jeffrey p. Williams

Langley Research Center

Hampton, Virginia

N

National

I

Aeronautics

_NA

and Space Administration

Scientific and Technical

Information Branch

Symbols random variable

A1, A2, B1, B2 autoregressive moving average _ damping ratio

(ARMA) model parameters scalar weighting factor in equationB Bayes risk function (see eq. (9)) error identification method

C weighting matrix a standard deviation of measurement

noise

Cij cost of selecting hypothesis Hi when

/-/3.is true aconv parameter-convergence threshold

C1, C2, D1, D2 controller gains aerror fit-error threshold

E expectation operator ffinf information threshold

orthogonal projection operator T sampling period, sec

e equation error Ti decision function

_e normalized forward residual • mode-shape matrix, NS × NM

ei,n forward residual (with i representing w natural frequency, rad/sec

time and n representing order)

f actuator force vector, lb Superscript:

Hi hypothesis i

T transpose

i, j, k, n time instants

ki,n+l reflection coefficient

N number Abbreviations:

NM number of modes ARMA autoregressive moving average

NS number of sensors A/D analog/digital (see fig. 1)

NW data-window size D/A digital/analog (see fig. 1)

nk random noise vector, NS × 1 DFT discrete Fourier transform

p probability LMS least mean squares

q modal amplitude vector, NM × 1 LQG linear quadratic Gaussian method

R noise-covariance matrix LSS large space structures

ri,,_ backward residual (with i represent- VLSI very large scale integration

ing time and n representing order)

r normalized backward residual Notation:

s Laplace variable

det determinant

u modal control force vector, lb

/., ./ inner product

Y_ subspaee spanned by vectors

(Yi, Yi--1,''', Yi--n.l) III[ Euclidean norm

Yk measurement vector, NS × 1 _1_ is orthogonal to

_'n linear least-squares estimate of Yn • direct sum

Zi decision region in observation space C is contained in

z -1 time-delay operator ^ estimate of

oo.

111

Summary includes these capabilities. The system order, mode

shapes, and modal amplitudes are estimated on line by

This report presents the application of recursive, using an identifcation method based on recursive, least- least-squares lattice filters that are widely used in sirsquares lattice filters (ref. 5). By using the identified

nal processing for adaptive identification and control of model parameters, a modal control law based on a

structural dynamics systems. The theory behind identification using lattice filters, the method for control pole-placement method with the objective of vibration

suppression (ref. 6) is employed. by modal pole placement, and on-line validation proThe closed-loop distributed adaptive control stud- cedures for properly integrating the identification and

led in this report is shown in figure 1. The function

control are presented. The methods are illustrated by

computer simulation for both a one-dimensional free- corresPonding to each block of figure 1 is described as

free beam and a two-dimensional flexible-grid structure, follows: The lattice filter on the front end provides an

on-line estimate of the system order, mode shapes, and The results are then verified by using the experimental facility for both structures at the Langley Research modal amplitudes. This is carried out by using the modCenter. ified Gram-Schmidt procedure involving both forward

and backward residuals, in which the backward residuIntroduction als form an orthonormal basis for the entire observation

sequence. Also, the orthonormal basis functions generThe dynamics and control of large space structures ated by the lattice filter provide the mode-shape infor-

(LSS) is essentially a distributed parameter system mation. Although the basis functions from the lattice

problem. However, for a practical control-systems de- filter form an orthonormal set, they are not the natusign, modal and finite-element analyses are used with ral modes but are linear combinations of them. Hence,

the subsequent application of modern control-theory the modal amplitudes constructed by using these batools to a truncated, finite-order design model. Prob- sis functions will generally have spectral components of

lems which arise in this process include the following: more than one natural mode. Since the ultimate objective is to provide a method of obtaining the natu-

(1) Design-model order: Since the original system ral mode-shape functions required for implementation

has infinite dimensions, the number of modes to be se- of the modal control method of reference 6, a method

lected in the design model for adequate control design is of decoupling the modal amplitudes by transforming

difficult to determine. If a high order is selected, com- the basis functions obtained from the lattice filter has

putational and hence onboard implementation problems been developed (ref. 7). After obtaining the decoupled

arise. Conversely, ira low order is selected, the resulting modes, the control-law design is carried out indepensystem may not satisfy performance requirements, dently for each mode.

(2) Knowledge of model parameters: With detailed

analysis and ground testing, the dynamic characteristics Toaccommodate digital implementation, an autoregressive moving average (ARMA) description of the

of a structure may be estimated; however, once it is in modal dynamics is used. In order to calculate conorbit, its characteristics may be quite different because troller gains, the parameters of this model must be of environment, construction anomalies, and other unidentified based on the time series of decoupled modal foreseen factors. Different adaptive control methods

displacements generated by the lattice filter. This is

have been proposed to overcome these problems (refs. 1 accomplished through an equation-error identification

to 3). method for determining the ARMA parameters (ref. 8).

(3) Number of required configurations: A large hum- In order to make the closed-loop adaptive control staber of configurations may be required for LSS in orbit, ble and meet the performance requirements, a proceand the design model required for each configuration dure for integrating the identification and control methmay be different (ref. 4). Hence, the controller .may be ods is developed. The identified parameters are pro- different both in order as well as gains. Since both order

and gains may change, a gain scheduling method may cessed through a series of test and validation procenot be suitable, dures. The tests consist of checking for the fit error,

convergence of parameters, and the amount of informaFor these reasons, an adaptive control system for LSS is tion in the measurement data. In figure 1, the block

highly desirable. The system should be capable of on- marked "Parameter identification and testing" repreline identification of the required design-model order, sents these functions.

mode shapes, frequencies, and damping. Based on these When the identified parameters pass all the tests,

identified parameters, the system should also be capable the controller gains are updated and the modal control

of on-line controller design and implementation that can forces are computed. For each mode, the control law

guarantee stability and desired performance. In this used is based on a discrete-time pole-placement method

report, an adaptive control system is proposed that that allows exact pole placement of system poles when
the modal parameters are known and the number of basis for the entire observation sequence. Hence, any

actuators is equal to the number of controlled modes least-squares estimate is the orthogonal projection onto

(ref. 6). Even though pole-placement control law is used this basis.

here, any other control method like the linear quadratic The immediate interest herein is to provide an efliGaussian (LQG) method can be used for design. By cient means for order and mode-shape determination for

using the estimated mode shapes from the lattice filter, the adaptive control methods set forth in references 1,

these modal control forces are converted to actuator 3, and 14. For that purpose, assume that the ith meaforces and the entire adaptive control loop is closed, surement sample is of the form

The adaptive identification and control using the

lattice filters proposed previously are illustrated first yT = [yl(i), y2(i),.. .,yNS(i)]

by simulations for a free-free beam and then for a more where the superscript NS represents the number of

complex two-dimensional free-free grid structure. The

sensors. It is assumed that y is generated from a model

identification results obtained for these two structures wherein

by simulations are then compared with those obtained

yi = (I)qi + ni (1) from the experimental data for these structures at the

Langley Research Center. The full closed-loop adaptive Here, (I) is a NS × NM mode-shape matrix, qi is

control method as shown in figure 1 is also studied by the NM × 1 modal-amplitude vector, and n_ is a

using simulations for both structures. NS × 1 Gaussian random variable with zero mean and a

covariance matrix R. Typically, for structural dynamics

Elements of Lattice-Filter Theory applications, each component of qi is the output of an

uncoupled second-order process. The task here is to

Lattice filters have found wide acceptance in the estimate the order and obtain the least-squares estimate

fields of adaptive signal and speech processing (ref. 9). of qi and • from N + 1 measurement samples Y0

Reference 10 is a comprehensive tutorial on the the- through YN depicted in figure 2. For the purpose of

ory and applications of lattice filters. Lattice-filter the- analysis, we will find it expedient to embed the problem

ory simultaneously provides the tools required to se- in a set of problems of finding the estimate of Yi based lect model order appropriately and determine the mode

on the measurements (that is, yi_,_, ..., Yi-1). Let Yi'_

shapes. In all cases familiar to the authors, the order be the space spanned by the n measurements up to and

of the model is assumed a priori and the lattice filter is including y_. (See fig. 2.) The forward and backward

then used for the identification of the parameters of an residuals are the errors in estimating the measurements

autoregressive (AR) model. The novelty of the present adjacent to the sample spaces defined in figure 2. For

report is that the model order is determined, on line, by example, by referring to figure I, the estimation of Yi

using statistical multiple-decision theory (ref. 11). The based on the space Yi_-I is a forward estimation and

main idea in this approach is that the system outputs produces an error

contain all the information about its natural modes and

that the complete system behavior can be characterized yn

by a set of orthonormal functions derived from the out- ei,_ = Yi - _ (Yil i-1) (2)

puts through a modified Gram-Schmidt procedure. The where $ (yilY_n_l) is the orthogonal projection of y_ on

lattice filter provides an order as well as a time recursive the Yi_-I subspace. Correspondingly, the estimation of

algorithm for generating an orthonormal basis for the Yi-_, based on the space y/n, produces a backward

measurement data by using the most recent measure- estimate with an estimation error (residual)

ment samples. Herein, a method is presented to add an

orthogonal function to the already existing basis func- ri,n = Yi-n - _ (yi IYin_l) (3)

tions if the measurement samples contain new signifi- Clearly, the backward residuals ri,_ for n -- 0, 1, ..., i

cant information. The decisions to increase, maintain, are orthogonal and form a basis for the space Yi'_. Note

or decrease the model order are based upon hypothesis- however that this is not true for the forward errors ei,_.

testing criteria. Such testing methods have been used We will now seek formulas that relate order n and

for failure accommodation in the design of an aircraft time i recursions for the forward and backward residdigital flight control system (ref. 12). After the identifi- uals. Rewriting equation (2) for the case where n is

cation of the appropriate ARMA model is accomplished replaced by n . 1 results in

via recursive filters, the model is used for control purposes. The concept of orthogonal projections in a den- e_,_+l --Yi - _ (YilYin-+l1)

eral real Hilbert space setting has been presented in ref- Solving for Yi in equation (2) and substituting in the

erence 13. The procedure is a modified Gram-Schmidt last equation results in

procedure involving both forward and backward residuals, in which the backward residuals form an orthogonal ei,,_+l = ei,n + C(yi]Yin_ 1) - _ (yi]Yi__+ 1)

2
Now, the space Yin_+llcan be written as Equations (5), (6), and (7) represent the lattice-filter

algorithm in a general real Hilbert space setting. A

y/n_+1 = y/n_1 G Y_-n--1 = Y/n--1O ri-l,n normalized form of these equations may be derived by

using the inner-product form of the projection operator where ® represents the direct sum of two spaces. This

(a,b). follows since ri-l,n is the component of Yi-n-1 that is $(aIb ) = b

not in the space Yi_-I' Hence, -_-

g'(Y_IYin-+I1) = $(YilYin-1) + _'(y_[ri-l,,_) where a and b are nontrivial elements of the nilbert

space and where (a, b/ is their inner product. Normalso that izing the e and r residuals results in unit vectors e and

ei,n+l = el,n - _(yilri-l,_) (4) _r,respectively, satisfying

We may substitute for yi from equation (2) to simplify ei,_

the last term and obtain e--i"_- (ei,_, ei,n)

yn __ ri, n

_(yilri-l,n) -= _ {[ei,n + _'(Yil i-1)] Iri-l,n} ri'n (ri,n,ri,n)

and since $(yi[Yin_l) C Y?-I and ri_i, n .J_ Yin_l , the

last term on the right-hand side vanishes and Equations (5) and (6) can be written in terms of_e and

r_resulting in

_(yilri_l,n) ----_(ei,nlri_l,n)

e-i,n -I-1 = (1 --k i2,n.t_l)--l/2(ei,n -- ki,n_blr_i_l,n )

Hence, equation (4) becomes

ri,n+l ---- (1 k 2 _-l/2{r -- i,n.l} k--i--l,n -- ki,n.le-i,n) (8)

ei,n+l = ei,n -- g(ei,n]ri-l,n) (5) wherein

A similar argument may be used to derive a recursion ki,n+l = (ei,n, r__i_l,n)

formula for r yielding The symmetry of the recursion formulas is now apparri,n+l = ri-l,n- _'(ri-l,n]ei,_) (6) ent. The equations are coupled by the term ki,n+l,

which is customarily called the reflection coefficient.

Note the similarities in the last two equations and the The structure of this equation is depicted in figure (4)

time shift in the latter, in which we have used the symbol z-1 to represent the

Equations (5) and (6) are the basic recursion for- time-shift operator. That is,

mulas needed for constructing a lattice-filter algorithm.

By referring to figure 3 we may, in fact, write the least- z-lri,n = ri-l,n

squares estimate of YN based on the space Y_-I by The lattice filter derives its name from this structure.

noting that rg_l, j for j = 0, 1, ..., n -- 1 is a basis The lattice-filter form has the following advantages: for Y_-I so that

1. Given a basis for order N, a basis for order N + 1

_rN ----_rN(Y_-I) ---- _ (YN{Y_-I) can be obtained by using the recursive formulas given

n-1 previously in equation (8).

= E _(yNlrN-I,j) 2. Because of the modified Gram-Schmidt procej=0 dure, the basis for all orders n between 0 and N are the

first n elements of the basis of order N.

Substitution for YN from equation (2) with n replaced 3. The estimate assuming any order n between 0

by N yields and N + 1 can be computed by using equation (7).

Thus, the lattice filter provides the information needed N--1

(YNIY_ 1) to determine the residual sequence for any model order YN = E _ {[eN,n + - ] IrN-l'n } between 0 and N + 1 inclusive. This information

n=0 provides the basis for the model-order determination

Again, since g(yNIY__I) C Y_-I and rN-l,n J- Y_r-1, presented in the next section.

the last term on the right-hand side vanishes and we

have Model-Order Determination

N--1

_rN = E _(eN'n{rN--l'n) (7) In this section we develop a method for selecting

_=o the model order (between 0 and N + 1) that is optimal
with respect to a Bayes risk function. When a new of a Gaussian noise process so that

measurement is processed, the order of the system can

bereduction increased,in decreased, order can orbe maintained of any size,as whereas before. The the p(alHj)- (detR)l1/2 exp ( - _ rTR-lr )

increase in order is by 1.

The method is based on multiple hypothesis testing where r = a-6 for each measurement sample. Further,

of N + 2 hypotheses where hypothesis Hi corresponds since the noise process is assumed to be white, the chain

to a system of order i for i values between 0 and N + 1. rule of probabilities reduces to

Thus, the hypothesis Hi corresponds to

where • is of rank i. Based on a set of measurements, p(alHj ) =

NW11/2 exp --21eT'jR-lek'j ) (det l-Ik=l R) k=l

Y(NW) = [y(1), y(2), ..., y(NW)] where NW represents the data-window size. The most probable hypoth- in which a data window of NW samples is used in the

esis for the system is selected by minimizing the Bayes decision process. In the last equation we have used

risk function defined by the forward residual e taken from equation (2) where

N+I Y+l r (i, n) is replaced by (k, j). Note that the minimization

B = E E CijPH_. ] PylH(a]Hj) doL (9) of T and, hence, the optimal decision for the system

i=0 j=0 Jzi order depends on the choice of weighting matrix C and

on the a priori probabilities. These parameters can

where PHi represents the a priori probability of hypoth- be selected depending on one's objectives. If one has

esis Hj being true, Cij is the cost of selecting Hi when a priori knowledge about the system (say, its order),

//3. is true, and PyIg(a]Hj) is the conditional proba- then C and PH can be selected to reflect this. Also,

bility density of measurement y when//3, is true. The if one wants the selected model order to be greater

symbol fzi implies that the integral is carried out over than or equal to the true model order, the C matrix

the decision region Zi in the observation space. Deei- can be selected with the upper diagonal elements being

sion regions Zi are subsets of observation space such greater than the respect!ve lower ones. These options

that if y is in Zi, then the hypothesis Hi is to be se- are illustrated in simulation examples to follow.

lected. Note that the integral in equation (9) represents Clearly, in this approach one may "fit the noise" by the probability of making the incorrect decision of selecting hypothesis Hi when/-/3 is true for i -fi j. So, the continually increasing the order of the system; however,

once the order of the estimator has increased beyond the

Bayes risk function B represents the sum of probabilities corresponding to different decisions weighted by the correct order, then the residual errors should lie within

a noise band which can be predicted based on assumed

a priori probabilities PH_ and the design weights Cij. noise characteristics. A threshold can be selected based

The problem then is to choose the boundary of decision

regions Zi that will result in the minimum Bayes risk on this predicted noise band, and the order can be

function. This minimization can be accomplished by determined by a test of whether the residuals have been reduced to lie within the noise band. The residuals

selecting the hypothesis that produces the minimum: will generally consist of signal and noise parts, with

g+l r the signal part being reduced as the correct order is

B = E ] Ti(a) da reached until the residuals essentially consist only of

i=o Jz_ noise. Thus, for sufficiently large n,

where NW

Nq-1

Vi(a) = E CiJPgJPyIH(aIHj) (10) E Ei=leTnei'n = NW E(nTni)

j=0 NS

The Bayes risk function is minimized by selecting Hi = NW tr E(nin T) = NW E a_.

at each point a in the observation space so that vi(a) j=l

is the smallest of its N + 1 possible values. Hence, the where E is the expectation operator and tr represents

optimal decision regions are, for all j, the trace. This can be used as the la threshold for

Zi = [alri(a) < Tj(_)] the order-determination test. In the last equation, aj is the standard deviation of the noise process for the jth

The computation ofp(a]Hj) is based on the assumption sensor.
ComputationalRequirements Spectral Decoupling of Lattice-Filter Basis

In order to give an insight into the computational Functions

complexities involved in using the lattice-filter algorithms, a comparison with other least-squares meth- The adaptive control method proposed in this study

ods is given in the following table (ref. 10). A rough (see fig. 1) is based on modal decomposition and recomparison of the different algorithms can be achieved quires a set of uncoupled modes for implementation.

by counting the number of arithmetic operations (in- Lattice filters form an orthonormal basis from the meacluding multiplication, division, and addition) needed surement samples by using a modified Gram-Schmidt

to carry out the operations. The count is made for an procedure. Since the basis functions are orthonormal,

Nth-order predictor, they may be easily added or deleted, and hence, the

model order can be easily increased or decreased. Although the basis functions form an orthonormal set,

they are not the natural modes but are linear combiNumber of nations of them. Hence, the modal amplitudes conMethod operations structed by using these basis functions will generally

Block processing; prediction error method 5N have spectral components of more than one natural

mode. Thus, the modal amplitudes obtained through a

Fast Kalman; recursive least squares 8N direct application of lattice filters are coupled.

In this section a method is presented for decoupling

Gradient least mean squares (LMS) 2N the modal amplitudes by transforming the basis functions obtained from the lattice filter (ref. 7). The ultiGradient lattice (two coefficients) I0N mate objective is to provide a method of obtaining the

mode-shape functions required for implementation of

Unnormalized least-squares lattice 10N the adaptive control method of figure 1. The previous

(3N division) section presented a method for estimating the number

of independent modes present in the output measureNormalized least-squares lattice 12N ments. Lattice filters were used to obtain the least-

(3N square roots; squares estimate of the measurements based on the last

4N division) n time samples, with n being the order estimate. The

criterion used for the order estimate was that the estimation residuals lie within a band determined from

Compared with other methods, the lattice method assumed sensor noise characteristics. The lattice filter

generates estimates for all orders up to and including was operated at a fixed order but continued updating

N at the same time. If there is a need for an on- the orthonormal basis at each time sample so that the

line order determination, the lattice filter offers an current basis corresponds to that required for the last

ideal choice. For example, in comparison with the n time samples. Note that this approach produces a

fast Kalman method, if the order N is greater than different set of basis functions (or modes) at each time

2 and one needs estimates for all orders up to N at the sample.

same time, then the lattice-filter method offers a smaller In the present work, when the order estimate is

number of computations, constant, a fixed set of basis functions that are specThe comparison based on number of operations trally deeoupled is required to carry out identification

should be considered with some caution since there are and control. Because of this requirement, a fixed orother issues involved. For example, the normalized lit- thonormal basis is used during intervals when the ortice filter has the potential of fixed-point implementa- der estimate is constant. (However, the order estimate

tion because of the fact that all internal variables are is checked at each measurement sample based on the

less than 1 in magnitude. Also, the normalized lattice threshold test.) The method is a frequency domain

recursions can actually be expressed as a sequence of cir- approach that utilizes the discrete Fourier transform

cular and hyperbolic rotations. Efficient computation of (DFT). The lattice filter uses the current measurement

these rotations can be achieved by the use of the Co- as the first mode shape and, by using a modified Gramordinate Rotation Digital Computer (CORDIC). The Schmidt orthonormalization procedure, generates addipossibility of performing the basic lattice recursions by tional basis functions from estimation residuals. Conusing special-purpose hardware (e.g., a very large scale sequently, the output of the lattice filter produces couintegration (VLSI) chip) makes the number of opera- pled mode shapes and corresponding modal amplitudes

tions less important than, say, the architecture of the in which the first coupled modal amplitude will contain

computations. (See ref. 10.) all significant natural modes.
The theory of reference 5 is used to provide an es- tion error is given by

timate of the order of the process n. Thus, the first

coupled mode OFT amplitude spectrum is searched for e(k - 1) = q(k - 1) - _l(k - 1)

the n most significant peaks and corresponding frequencies. Because the spectrum contains n peaks for the n _l(k - 1) -- -41q(k - 2) + .42q(k - 3)

frequencies, a transformation matrix can be obtained +/_au(k - 2) + !_2u(k - 3)

that decouples the spectrum. This transformation ma- where Clis the modal amplitude estimated by the lattice

trix is the inverse of the matrix whose elements are the filter, k is the sample number, and -41, fi_2,/_1, and B2

real parts of the transforms of the n coupled modal are the ARMA coefficients. The ARMA coefficients are

amplitude channels (rows) evaluated at the n peak fre- then updated by

quencies (columns). After the transformation matrix

is determined, the coupled mode shapes are decoupled [-Al(k) ] [41(k - 1) [q(k - 2)

by using the same transformation matrix. Some of the [A2(k) | |A2(k - 1) |q(k - 3)

limitations associated with obtaining the natural modes |/_1 (k) _ = _1 (k - 1) + e(k - 1)# _ q(k - 1)

by decoupling from lattice basis functions are presented 1-/_2(k) J I.B2 (k - 1) I.u(k - 2)

in the following discussion. The weight/z assures stability if (ref. 8)

To provide a fine frequency resolution, the number

of points in the DFT data base have to be increased or 0 </_ < 2[ [q2(k - 2) + q2(k - 3) + u_(k - 2)

the sampling rate has to be increased. If the modes are +u2(k - 3)] closely spaced, this may pose a problem.

Since the decoupling is based on the real part of the This identifier performs well in a low-noise environFourier transform, the phase information between the ment; but when the information content of the signal

different lattice modes is not considered in the trans- is small, it attempts to fit the noise (ref. 14). Also, the

formation. Incorporation of this phase information, al- ideal ARMA model for the beam has input parameters

though straightforward, will result in a complex trans- (B) which are three orders of magnitude smaller than

formation. This involves scaling and time shifting of the model parameters (A). This causes a very high senthe different lattice modes in time domain to obtain sitivity to noise in the identification of B; and when the

decoupling. Neglecting the phase information for struc- input force is applied, it tends to cause the identifier

tures with near-zero damping is reasonable. An alter- gain on A to decrease significantly. Although these efnate method of avoiding this transformation is to carry fects are evident in the results presented here, they did

out an eigenanalysis of the ARMA model obtained from not prevent successful identification.

the lattice filter to obtain the natural modes. A better If one is interested in determining the damping rasolution is to use the ARMA model estimated from the tios and natural frequencies of the modes, they can

lattice filter directly into the col_troller design. These be obtained in a straightforward manner from equaaspects are not considered in this report. In this study, tion (11). However, it should be noted that this is not

the adaptive control method is tested by using the mode unique because of the foldover phenomenon due to samshapes obtained previously from the lattice filter and pling. By finding the roots of equation (11) and using

also from the analytic prediction of the natural modes the relation z = e s_ in the primary strip, where r is the

of the structural dynamics system. Based on the decou- sampling period, the following equations for natural fiepling transformation, the decoupled modal amplitude quency and damping ratio are obtained, respectively:

time series is then analyzed for each mode to identify

the parameters of its ARMA model, w -_ O/27rT

= C(C2+ 02)1/2

ARMA Parameter Identification where

The objective of the identifier is to determine 0 = tan -1 b/a

the coefficients in the second-order ARMA model of C =-_-!ln(a 2 +b 2)

the lattice-filter decoupled modal-amplitude time series

2

q(k). The method is based on an equation-error method and

described in references 1 and 8. For each mode in fig- a = Real(zr)

ure 1, the model is described by the equation b - Imaginary(zr)

q(k - 1) = Alq(k - 2) + A2q(k - 3) + B1u(k - 2) where z_ pertains to the roots of equation (11) in the z-domain. The behavior of this overall system-

+ B2u(k- 3) (11) identification methodology with experimental data is

where u represents the modal control force. The equa- discussed in the subsequent sections.

6
Modal Control Law be used and the exact closed-loop poles will not be reall The control method used here is based on a modal ized. Since the control gains are dependent on the ideadecomposition of the sensor outputs. The closed-loop tiffed parameters, improper identification Will prevent

design objective is to suppress the vibrations of each the design goals from being met. Sensitivity of closedcontrolled mode. To achieve this, the modal control loop stability to parameter error was studied in refermethod described in references 1 and 6 is used to ence 6 to determine parameter-accuracy requirements

calculate control gains that will place the roots of the for the identifier. The study of reference 6 showed that

open-loop undamped system onto a constant damping when the solution of equation (12) is exact, stability

line. This pole-placement objective is specified in the will be maintained for parameter errors of +12 percent

frequency domain and then converted to the discrete z in mode 3 and up to +30 percent in mode 4. These

domain by the transformation z = es_, where T is the studies were made for an ideal case with no noise and

sampling period. The control law that will place the no mode-shape errors; thus, these limits should be conpoles at the desired locations is calculated by assuming sidered absolute maximums.

a control law that takes the same form as the equations

of motion: Parameter Testing

u(k) = Clq(k-1)+C2q(k-2)+Dlu(k-1)+D2u(k-2) In the earlier sections, a method for identifying the

ARMA model parameters from modal amplitude time

The control gains C1, C2, D1, and 02 are calculated series and a method for modal control using these paon line. Note that this law requires only the feedback rameters were described. Serious problems may arise

of modal amplitudes. The velocity feedback needed to if one directly integrates the identification and conprovide proportional viscous-type damping is embedded trol methods if the identification method produces erin the difference of successive displacements. By trans- roneous parameter values. Hence, before attempting

forming the equations of motion and control to the z- control, one should check the validity and accuracy of

domain, the closed-loop characteristic equation can be the identified parameters. Also, if the signal from which

written as identification is carried out does not contain enough information, it is unreasonable to expect the identified

z 4 + (-A1 - D1)z 3 + (D1A1 - A2 - D2 - B1C1)z 2 parameters to be accurate. One method for testing pa-

+ (D1A2 + D2A1 - B1C2 - B2C1)z rameters would be simply to turn on the controller and

+ (D2A2 - B2C2) = 0 observe the result. This is not a desirable approach in

an on-line adaptive system. To overcome these probwhich we note is a fourth-order algebraic equation. Two lems, a series of test procedures which ensure the adeof its roots are specified by the closed-loop design re- quacy of identification before control is attempted are

quirements, whereas the remaining two roots are discre- described subsequently. The technique used here is to

tionary. These discretionary roots may be selected by test the parameter-identifier input signal and the identithe designer to improve the closed-loop characteristics fled parameters against some a priori criteria. The three

relative to noise suppression or robustness to parameter tests used in this procedure pertain to the following:

uncertainty. In this study, these two discretionary roots

had the same frequency as the design roots but addi- (1) Model fit error

tional damping was introduced to avoid repeated roots. (2) Parameter convergence

Given the four characteristic roots, one may construct (3) Signal information

a desired characteristic equation and, by equating its Although we do not propose that this set is sufficient

coefficients with those of the last equation, obtain four to guarantee proper identification, it does serve to

equations in the four unknown control gains C1, C2, illustrate the basic concepts of parameter testing and

D1, and D2. has been used successfully in experimental work.

Having calculated the modal control forces accord- The fit-error test uses a fixed parameter set to

ing to the aforementioned control law, the actuator calculate an estimated modal amplitude for the past

forces are determined by solving NW sample. These tests were run with NW arbitrarily

set to 10. The equation for this test is

oTf = u (12)

NW

where • is the mode-shape matrix determined by the aerror > E q(k- n)- [-41q(k 1) [

n

transformation algorithm and f and u are the vectors _z0

of actuator and generalized modal forces, respectively.

Note that if the number of modes controlled is more + A2q(k - n - 2) + !_lU(k - n - 1)

than the number of actuators, a least-squares fit must +B2u(k-n- 2)] I (k > NW) (13)
If the absolute sum of the error between the mod- both simulation and experimental hardware, first for a

eled amplitude and the displacement calculated by the free-free beam and then for a free-free grid structure.

ARMA model exceeds a given threshold, the fixed pc- Results of these studies are presented in the following

rameter set is updated with the present identified pc- sections.

rameter set. This process is repeated until the parameter set fits the data. The convergence test runs concur- Identification and Adaptive-Control Studies

rently with the fit test, and it simply checks the magni- for a Free-Free Beam

tude of the changes in successive estimated parameters. In this section, the lattice-filter theory developed

For h = A1, A2, 81, and/_2, the equation is earlier is illustrated for the identification and adapNw tire control of a free-free beam. The identification

O'conv> _ [hn - hn-11 (14) method yields the structural dynamic characteristics of

n=o the beam, and the objective of control is vibration supression. For the identification, results are presented

If the absolute sum of 10 successive changes in parame- by using both digital simulation and the experimental

ter estimates is above a specified level of ffconv, the test hardware for the beam available at the Langley Reindicates a failure and a logical switch is set to false, search Center. The adaptive control results are preThe final test is on information content of the esti- sented by using simulations.

mated modal amplitude signals from the lattice filters. The experimental apparatus for the free-free beam,

The purpose of this test is to check whether enough which is shown in figure 5, consists of a 12-ft beam

information is present in the signal for proper identi- of rectangular cross section that is suspended from the

fication of the parameters. If this test fails, the con- ceiling by two cables and is attached to four electrotroller gains are not updated based on the identified magnetic force actuators. There are nine noncontacting

parameters but are frozen at the last values before the deflection sensors that measure the translational deflectest failed. Here, the estimated modal amplitudes and tion of the beam. The arrangement of the equipment is

velocities from the lattice filter are checked for suffi- schematically indicated in figure 6, which also shows the

cient excitation by summing over 10 samples. Thus, first three flexible modes and frequencies of the beam.

the equation is The sensors are numbered 1 to 9 from left to right in

NW figure 6. The actuators are compensated to eliminate

ffinf < _ (IClnl + ICln+l -- qnl) (15) the effects of friction as much as possible. This compen-

_=o sation is nonlinear and produces a force in the direction

of the beam motion at the actuator attachment points

If the sum is below a threshold of ainf, the updating of that is designed to equalize the effect of friction. Furthe control gains based on the identified parameters is ther details of the experimental apparatus can be found

stopped. The control gains are kept frozen at the values in reference 15.

obtained just before the tests failed.

When all tests for parameters of a given mode have Simulation Results

passed, control gains are calculated according to the

previously described pole-placement method. The in- The digital simulaton used herein is based on a

formation and fit-error tests (eqs. (13) and (15)) consti- finite-element analysis of the beam. The finite-element

tute one test for each mode, and the convergence tests analysis was made with 25 equally spaced joints with

(eq. (14)) constitute four tests for each mode. Thus, 1 at each end of the beam. The analysis was made

six tests must be passed before control is applied to a by constraining the motions to translation and rotation

given mode. Note that if any one of the tests fail, the along an axis perpendicular to the axis of suspension.

control-law design parameters are not changed. Hence, Figure 6 presents the modal frequencies obtained and

the control gains are kept frozen at the values obtained the mode shapes for the vibration modes of interest as

just before the tests failed. The actual stability and determined by the digital sample rates (32 Hz)that are

performance of the controller is directly affected by the available in the experimental apparatus. The hypothcriterion chosen for passing a test. If the test criterion esis testing and threshold approaches presented in this

paper were tested in the digital beam simulation for is too stringent, system noise and nonlinearities may

preclude initiation of control. However, if the tests are identifying the system order along with its parameters

not adequate, it is possible that an error in the esti- (mode shapes).

mated parameters could result in gain calculations that Hypothesis-Testing Simulations

produce an unstable system.

This theory developed for identification and adap- In these studies of hypothesis-testing simulations,

rive control by using lattice filters is tested by using the effects of two different choices for the cost matrix C
on order estimation are illustrated. The a priori proba- noise may be modeled as an infinite dimensional probilities were assumed equal and the diagonal elements of cess. Hence, when a large amount of noise is present,

the C matrix were assumed to be zero in these studies, the hypothesis-testing method will tend to increase the

In the first case, C was selected as system order continually and "fit the noise." This tendency can be avoided if one implements the threshold

0 1 2 3 ... test previously presented. Results of simulations un0.5 0 1 2 ... dertaken by using this test are the subject of the next

C = 1 0.5 0 1 ... section.

1.5 1.0 0.5 0 ...

: : : : Threshold-Test Simulations

This choice of C gives preference to having a modelorder estimate greater than that of the true system Nominal case. For the nominal case two rigidrather than risking a lower one. Based on the theory body and three flexible modes were used. The mendeveloped in the section entitled "Model-Order Deter- surements were sampled at 32 Hz and were corrupted

mination," the TL matrix (consisting of likelihood func- with Gaussian white noise that had a standard deviations Ti in eq. (10)) at measurement samples 2, 3, 4, and tion of 0.005 in. In all studies, the initial conditions for

5 is, respectively, all modal amplitudes were 0.05 and the modal velocities

were 0. The data-window size was taken to be 4, and

7T = [1.43, 0.69, 0.62] threshold tests were not started until enough measurement samples were taken to fill the window. This results

TTL = [2.73, 1.61, 0.85, 0.84] in a continual increase in order estimate for the first

__T= [4.92, 3.19, 2.17, 1.81, 2.32] NW samples. Figure 7 shows simulation results for the nominal case. Note that the order estimate increases

TT = [4.92, 3.12, 2.03, 1.65, 2.03] for the first four samples and then oscillates between

4 and 5. Even though the system has five modes, with

Based on the minimization of T_, it is clear that the the initial conditions selected, the two rigid-body modes

correct order is indicated as 4 after the fourth sample, appear as a single zero-frequency mode and, hence, the

For the second case, the weightings of C_j were correct order estimate should be 4. The: second graph

reversed. It was assumed that it is better to have a in figure 7 is the norm of all nine measurement residulower model-order estimate than the true system order, als and indicates the fit of the model. The next curve is

This philosophy might be evoked if the order estimate the residual of measurement 4. This is followed by the

approaches saturation of the computational system, output of sensor 4. Sensor 4 data were selected for ilThe C matrix for this case is lustration and are typical of the remaining eight sensors

C_3 ---0.5(j - i) (i < j) shown in figure 6. In data processing, all nine sensors

were used to determine the model order and the leastCij = 0 (i = j) squares estimates on which the residual computations

Cij = (i - j) (i > j) are based. From detailed simulation, it was found that

the order oscillation seen in the figures occurs because

The corresponding TL matrix at the same sample in- of the inadequacy of sampling frequency compared with

stants as those given earlier are that of the highest mode natural frequency. For examTT = [0.25, 0.39] ple, figure 8 shows the order variation for the cases with three and four modes in the simulation. From these sim-

_T = [0.25, 0.23] ulations, it seems that at least one order of magnitude

separation should exist between sampling frequency and

rLT = [0.74, 0.69, 1.35] the highest mode natural frequency.

TT = [0.73, 0.54, 1.05] With nominal results available, the effect of thresh- old selection on the order determination was studied.

It is seen from these data that the model order is Instead of assuming a 3a threshold as in the nominal

indicated as 2 even though the true system order was 4. case, the threshold was relaxed to 2a and to la levels.

Thus, the choice of cost matrix C reflects the weights The results are summarized in figure 9. As anticipated,

one wants to put on the hypothesis testing. If one has a the 3a threshold is most conservative, whereas for 2a

priori knowledge, the Cij and PHi elements can be se- and la the order shifts toward 5 and 6, respectively.

lected to reflect this. In applying this algorithm a corn- A value of 3a was selected for the nominal threshold.

plication exists when there is a large amount of men- For all further studies, the nominal parameters are used

surement noise. It is well-known that Gaussian white except for the particular parameter being studied.

9
Effects of data-window size. In order to avoid the from these figures that the technique works well for this

problem of sudden variations in order estimate, the test case.

for order is carried out after accumulating the residuals

for a data-window size of NW. Figure !0 presents the Experimental Results For Free-Free Beam order time histories for the two cases in which the datawindow size was 1 and 8. As anticipated, the order time In this section, the experimental results for the idenhistory for NW -- 8 is smoother that that for 1. The tification for the dynamics free-free beam are presented

choice of window size for a particular problem is closely (ref. 15). Testing was done by manually exciting the

related to the measurement noise level and also to the beam approximately in its first flexible mode and samsampling rate. pling the 9 sensors at 64 samples per second. A total

Effects of noise level. In the noise-free case, the of 5 sec of data were stored on a tape which was post

method should lead to exact-order determination as processed with the algorithm.

indicated in figure 11. The figure also shows the case in Figure 16 shows a time history of some of the meawhich the noise standard deviation a is 0.01 in., which surement data processed by the algorithm. The senis twice that of the nominal case. This increase in sors are numbered sequentially from the left, as shown

noise predictably affects the model-order estimate as in figure 6. The innovations (INOV) sequence for senshown. This effect can be reduced by increasing the sor 4 is shown just below its time history. Also shown

data-window size, which essentially acts as an averaging is the estimation norm (ENORM) of the forward residprocess, ual, which includes all components of the measurement

Effects of modal damping. The effects of including vector. Below the norm is the estimate of model order.

modal damping are that higher modes are caused to This was obtained by using a data window of eight samdecay faster and hence reduce the order as time evolves, ples. Initially, the order estimator fills the data window

Figure 12 shows the effect of including damping for all and, hence, the indicated order estimate increases to 8.

five modes. The damping ratio for all modes was taken After this, the order estimator settles to 2, indicating

as 0.005. The resulting modal amplitude history is also that even though we attempted to excite only one mode,

shown. It is seen from the figure that the order estimate there were, in fact, two significant modes excited. Note

has a tendency to decrease, also that the norm of the forward estimation error is

Model-order tracking. In order to study the effec- small compared with the value at the start of the protiveness of the proposed method, the system order was cess when the order estimate was settling.

increased by 1 at time increments of 2 sec, starting from The modal amplitudes obtained from the lattice fill sec. (The initial order was 2.) The system order was ter are spectrally decoupled, by using the aforementhen checked as to whether the method correctly tracks tioned procedure, after enough data are taken to obtain

it. Results of increasing the model order in this man- the DFT accurately (for 64 time samples, about 1 see).

ner are shown in figure 13. It is seen that the method This occurs at about 1.75 sec, with the first 0.75 sec betracked the simulated system behavior. For the decreas- ing used for the identification of mode shapes and model

ing order study, the initial order was set at 5; and at order. (See fig. 17.) Figure 17 shows the modal amplievery time increment of 2 sec, starting from 1 sec, the tudes for both of the identified modes. These are the

order was reduced by 1. From figure 13, it is seen that signals that are inputs to the parameter-identification

the method tracks the order reduction as well as the method used to identify the parameters of the ARMA

order increase, model of the modes. The identified ARMA parameters

are shown in figure 17 for each of the two modes identiDecoupling Application to Free-Free Beam fled. The a priori parameter estimates are initially offset

The decoupling technique presented has been tested from the values predicted by a finite-element analysis,

by using a simulation of a free-free beam described pre- which are also indicated in figure 17. These parameters

viously. One rigid body and three flexible modes were track the instantaneous value required to minimize the

used in the simulation studies. Figure 14 shows time output error. One possible explanation of the oscillahistories of the first two coupled modal amplitudes ob- tory behavior of the mode 2 parameter estimates is the

tained from the lattice filters. Figure 15 shows the nonlinearity of the actuator compensation. Nonlinearcorresponding spectrum for all modes simulated with ity is apparent in the sensor 6 data in figure 16. Note

peaks at frequencies corresponding to the rigid (zero) that the lattice filter produces a linear least-squares fit

and first three elastic modes. The deeoupling transfor- of the data to the measurements and, in so doing, promations were obtained by using a DFT data base of duces a predominantly linear first-mode estimate and

256 time samples at 32 Hz. The resulting decoupled lumps the nonlinear dynamics into the higher modes.

modal amplitudes and the corresponding spectrum are Thus, the parameter tracking is more stable in mode 1

also shown in figures 14 and 15, respectively. It is seen and produces estimates of an undamped (A2 = -1) os10
eillation at nearly 2.7 Hz. If the algorithm is constrained histories are then generated. From the lattice-filter

to an order estimate of 1, the predominant response is mode shapes and modal amplitudes, natural modes

linear; however, the fit error is increased by an order of and modal amplitudes are obtained through the linear

magnitude, transformation. The application of the transformation

The mode-shape estimates obtained from the lattice is delayed for 2 sec because the on-line transformation

filter are shown in figure 18 in which we compare the technique of reference 3 requires 2 sec of data for a

estimates obtained by three methods (one analytical fast Fourier transform data base to obtain the required

and two experimental). The analytical result is the transformation. The natural modal amplitudes are

primary mode shape of the beam using the Euler- input to the equation-error parameter identifier that

Bernoulli theory. The two experimental results, which identifies the ARMA parameters. The identification

are in substantial agreement, are the nonlinear least- results are then tested by using the test procedures

squares algorithm of reference 17 and the lattice-filter described previously. When the tests are passed, the

algorithm of this report. Again, note that there is control is turned on. Results of the simulations are

apparently an effect of the four attached actuators on presented in figures 20 to 22.

the dynamics of the test article. The lattice filter Figure 20 shows the estimated modal displacement

produces two modes, one near the mode of reference 17 for the first lattice-filter mode. The order-estimate

(mode 2 of the lattice-filter output in fig. 6) and another plot shows that the correct order of 4 is obtained in

that is shown in figure 18. This other estimated mode 0.3 sec. After the parameter identification, when all

does not resemble any mode analytically predicted by the tests are passed, the control is turned on at 5.5 sec

using the linear Euler-Bernoulli theory; rather, it can and the modes are damped. Figure 21 shows when

perhaps be attributed to the effect of nonlinearities in the tests for each mode are passed and when they

the apparatus, are failed. The result of the adaptive control on the

natural modes is shown in figure 22. It is evident that

Adaptive Control Simulation Results when the identification is validated by passing the tests

The closed-loop adaptive control method of figure 1 and the control is turned on, the vibration suppression

has been tested 'in the digital simulation for the 12-ft, is achieved. The control forces applied by the four

flexible free-free beam at the Langley Research Center. actuators are shown in figure 23. When the modes are

The simulation contains one rigid-body mode and the damped out, the lattice-filter order estimate drops from

first three flexible modes. Nine deflection sensors were 4 to 1, indicating that the flexible modes are damped

used for obtaining the measurement data and four out. Although the lattice-filter order decreased, the

actuators were employed for control purposes. The control design order was maintained at 4 throughout

natural mode shapes and their frequencies are shown the time that the per'iod control was on. Allowing the

along with the sensor and actuator locations in figure 6. order to vary in real time and updating the control order

The initial conditions on the modal amplitudes were set are topics for further studies.

to 0.05 in., and the modal velocities were set to 0. The The main results of the identification and the test

modal damping was also set to 0. A digital sampling procedures are summarized in figure 24. It shows the

rate of 32 Hz was selected for the simulation, and the identified frequency parameter A1 for the first flexible

standard deviation for measurement noise was assumed mode along with time histories of the fit error, converto be 0.005, based on the noise observed in the available gence, and information plotted over the same time scale.

hardware. The lattice-filter estimates are based on a When all the tests are passed, the corresponding pass

data-window size of 4 (ref. 5). The testing procedures parameter (plotted as a binary logical variable) is set to

are all carried out based on a data window NW of 10 1. The various thresholds for the tests are also marked

samples. Initial parameter guesses are offset from the to indicate when the tests pass. The initial estimate of

true values to demonstrate the rapid convergence of the A1 is offset from the finite-element prediction by about

identification algorithm. An arbitrary delay of 2 sec was 100 percent so that the capability of the identification

added between the time that identification starts and method can be demonstrated. When the identifier is

the time that the control would be applied to show the turned on, the estimate converges to the true value of

behavior of the identification method. 1.8 from 3. The thresholds indicate that the fit erThe parameter-testing method is shown in figure 19. ror test is passed first, and then the convergence test.

Note that the inputs are both modal displacements and With enough signal in the measurements, the informaparameter estimates. This procedure is run at each time tion test is always passed. When all tests are passed

step; and if any one test fails, no control will be applied, at 5.5 sec, the control is turned on. This is immediAt the start of the simulation, the lattice filter ately seen from the history of A1 as it starts deviating,

determines the number of modes in the simulation indicating that the dynamics are being altered. When

along with the mode shapes. Modal amplitude time control is fully effective, that is, when the modes are

11
damped out, the measurement data will contain only believed to be numerically accurate since the change in

the noise and then the information test will fail. Also, the eigenvalue iterate of the higher frequency mode is

if the parameter excursions are large, the convergence 0.1(10) -1° in the final iteration. The highest frequency

tests will also fail, indicating a failure for the binary eigenvalue iteration was 0.1565(10) 4.

variable (tests pass). Once this happens, the control

gain updating is stopped. TABLE I. MODAL FREQUENCIES OBTAINED FROM FINITE-ELEMENT ANALYSIS OF GRID

Identification Studies for a Flexible Grid

In this section, the lattice-filter identification method Mode Frequency, Hz

is tested in a structure that is more complex than that 1 0.364

of the beam. The candidate structure considered is that 2 .625

of a two-dimensional flexible grid. Identification results 3 1.398

are given by using both simulation and the experimen- 4 2.299 5 3.07

tal data obtained from the laboratory apparatus. 6 4.791

Figure 25 shows the flexible-grid experimental apparatus currently being built at the Langley Research 7 5.933 8 6.297

Center (ref. 18). The grid is a 7-ft by 10-ft planar structure made by overlaying aluminum bars of rectangular 9 7.337

cross section. The bars are centered every foot so that 10 10.352

there are 8 vertical and 11 horizontal bars. As shown

in figure 25, the grid is suspended by cables at two lo- A simulation was developed that accommodates the

cations on the top horizontal bar. The motions of the first 15 modes of the analysis, but only 4 modes were

grid perpendicular to the plane of figure 25 are the ones used herein for the identification study. The modes

of interest in this study. There are nine noncontacting used were modes 4, 6, 7, and 8. A sampling rate of

deflection sensors mounted on a back frame that give a 32 Hz was simulated with a standard deviation for the

9 × 1 measurement vector. The sensor data are linked

measurement noise of 0.005 in., which was based on

to the main Control Data CYBER 175 real time tomactual sensor characteristics. Modes were simulated

puter system at the Langley Research Center so that with modal amplitude initial conditions of 0.1. The the identification can be carried out in real time. For data window for order determination included eight

the experimental tests, the locations of the sensors are samples. In this work, the sensor locations were chosen

indicated in figure 26. based on several simulations. These locations differ

Simulation Studies from those of the experimental apparatus because they were selected to maximize the effect that simulated

In order to carry out the simulation of the lattice- modes have on the sensors. This was accomplished

filter identification of the flexible-grid facility, a finite- by visual examination of the simulated sensor outputs.

element analysis of the grid was made that included the The selected locations are indicated in figure 26, in

suspension cables. For this analysis, nodes were placed which an asterisk is used to distinguish simulation

at each overlapping joint on the grid, at the ceiling sensor locations from experimental ones. One may

attachment points of the cable, and at every 1/2 ft expect that location 5 would be preferable to location

along the cable. The grid elements connecting the nodes 5*; but since some simulated modes had little input to

were modeled as bending elements, whereas the cable a sensor at location 5, location 5* proved to be a better

elements were modeled as two-force members. Thus, a location.

total of 165 elements were included in the model. Four Based on the entire measurement vector, the latticedegrees of freedom appropriate for motion normal to filter order estimate is shown in figure 28. Also shown

the plane of the grid were considered. No damping was in figure 28 are sensor 5* data, which are typical of the

included in the model. Thirty modes were obtained other sensors. After estimating the order, a transformafrom this analysis. The frequencies of the first eight tion based on the discrete Fourier transform (DFT) was

modes are listed in table I. (See ref. 18.) The first three carried out by using 128 samples in order to obtain the

modes are the pendulum modes, the fourth is the first natural modes; and the equation error method was used

bending mode, and the fifth is the first torsional mode. to identify associated modal frequencies and damping.

The first four flexible mode shapes obtained from the The resulting (identified) modal frequencies, damping,

finite-element analysis are given in figure 27. The finite- and mode shapes are compared with those predicted

element analysis was an iterative method to calculate (simulated) by finite-element analysis in table II. The

mode frequencies. The ones used in simulation are identification of frequencies and damping are close for

12
TABLE II. MODE-SHAPE COMPARISON BETWEEN SIMULATED AND IDENTIFIED RESULTS

Shape of mode 4 with Shape of mode 6 with Shape of mode 7 with Shape of mode 8 with

frequency of-- frequency of-- frequency of-- frequency of--

Sensor 2.299 Hz 2.4 Hz 4.791 Hz 4.8 Hz 5.933 Hz 6.0 Hz 6.3 Hz 6.4 Hz

(a) (simulated) (identified) (simulated) (identified) (simulated) (identified) (simulated) (identified)

1" 0.29 0.30 -0.16 -0.17 0.31 0.32 0.59 0.45

2* -.40 -.41 -.12 -.08 -.43 -.43 0 -.43

3* .29 .30 -.16 -.18 .31 .32 -.16 ,21

4* .30 .31 .38 .37 -.06 -.06 .01 -,09

5* -.39 -.36 -.72 -.74 .49 .49 .01 .49

6* .30 .30 .37 .36 - .06 -'.06 - .01 - .06

7* .31 .31 -.20 -.15 -.32 -.31 .39 -.19

8* -.39 -.38 -.26 -.29 .41 .41 0 .37

9* .31 .31 -.20 -.14 -.32 -.31 -.39 -.38

aAsterisk denotes simulated sensor.

13
all four simulated modes. However, the mode-shape about 0.8 sec, the order estimate was fixed at 3 and a

estimates agree with simulation for only three modes, data collection of the 64 time samples required for the

One possible explanation for this is the limitation im- DFT was started at 32 Hz. The DFT was accomplished

posed by sampling rate and the number of samples used at about 2.8 sec, and then the decoupling transformato decouple the lattice-filter modes. By sampling at tion matrix was calculated. The modal amplitudes af32 Hz and including 128 data points in the DFT, a fre- ter this time should contain a single frequency, and the

quency resolution of only 0.25 Hz is obtained. Since the transformed mode shapes should correspond to the natexpected frequency separation between modes 7 and 8 ural modes of the structure which were excited. In that

is only 0.4 Hz, good decoupling cannot be achieved, manner, three modes were extracted from experimenTo summarize, the lessons learned from the simula- tal data tape 5. These have frequencies near 0.5, 2.5,

tion studies are given as follows: and 5 Hz. Table III presents the mode-shape estimates

obtained from the experiment. Also presented are se1. The least-squares lattice filter gives good identi- lected mode-shape predictions taken from finite-element

fication of simulated modal frequencies, damping, and analyses. The modes selected were those whose frequenmode shapes in the presence of sensor noise expected in cies bracket the experimentally derived ones. The folthe experimental apparatus, lowing discussion deals with the data from table III in

2. The DFT method of obtaining natural modes order of increasing frequency.

from the lattice modes is inaccurate if the modes hap- A good comparison does not exist between either

pen to be closely spaced in frequency. This may be bracketing finite-element-analysis mode and the first eximproved by adding more samples to the DFT. perimental mode. Additionally, there is some bending

3. Sensor locations should be properly selected to in the experimental mode as evidenced by sensors 4,

ensure good identification of simulated mode shapes. 5, and 6. The amplitude for this mode is shown in

figure 30(a) along with the ARMA parameters As and

Experimental Results A2 for the mode and their primary strip counterparts of

Experiments were conducted by using the grid ap- damping and frequency. Figures 30(b) and (c) show the

paratus described previously. The procedure for con- same information for the second and third modes, reducting the experiments was to excite the grid by using spectively. For the second mode, good agreement does

an air shaker. The shaker was capable of periodically exist between experiment and the finite-element results

exhausting a jet of air that impinged on the grid at sen- for the 3.07-Hz finite-element-analysis mode. Note,

sor location 1. The frequency of the jet was adjustable however, that the output of sensor 4 is opposite in sign

from 0 to 50 Hz. The resulting grid excitation was not and reduced in amplitude from the finite-element prepurely sinusoidal but was rich in harmonics. Because of diction. This means that a feedback on that sensor,

the range limits of the deflection sensors (0 to approx- based on the finite-element analysis, will be destabilizimately 2 in.), the maximum peak-to-peak deflections ing near the 2.5-Hz frequency. The validity of this deof the grid were limited to about 1 in. When the peak- duction can be established by examining the outputs of

to-peak deflection neared this limit, the air shaker was sensors 1 and 4 (fig. 29). According to the finite-element

turned off and the grid was allowed to vibrate freely analysis, the 2.5-Hz content of the sensor outputs should

with only air and material damping. A CYBER 175 be opposite in sign. However, they are in phase, that

real time computer system sampled the deflection sen- is, in agreement with the identification results.

sor data at 32 Hz for 5 sec. The data were stored on Concluding Remarks

a system data file for further analysis. This test procedure was repeated for several shaker frequencies in the This report has presented a theory for identifying

range from 1 to 10 Hz. This range has an upper limit be- the dynamics of flexible structures. The theory has

cause of the sampling frequency (32 Hz), and the lower been applied to a one-dimensional beam and a twolimit is selected to include the predicted lowest vibra- dimensional grid structure by using both simulated and

tion mode (2.2 Hz). Eight data sets corresponding to experimental data, and it was also used for the idendifferent shaker excitation frequencies were created and tification portion of a parameter adaptive control sysstored on tapes. The following discussion pertains to tem. Identification is done based on the assumption of a

results extracted from data tape 5. modal representation of structural dynamics. First, by

Figure 29 presents data from sensors 1 and 4 as well using least-squares lattice filters, an algorithm is develas the lattice-filter order estimate. For this case, the or- oped for identification of the number of modes present

der estimate was based on a data window of 8 samples in sensor data and for the calculation of the mode

and spectral decoupling was done with 64 time samples, shapes. The algorithm uses the least-squares lattice illFrom this figure, the order estimate is seen to converge ter that constructs an orthonormal basis for representto an oscillation between 2 and 3 at about 0.5 sec. At ing the measurement data. The basis is orthonormal in

14
TABLE III. MODE-SHAPE COMPARISONS BETWEEN ANALYSIS AND EXPERIMENT

Experimental mode 1 comparisons Experimental mode 2 comparisons Experimental mode 3 comparisons

for a frequency of-- for a frequency of-- for a frequency of--

0.364 Hz 0.5 Hz 0.625 Hz 2.299 Hz 2.5 Hz 3.07 Hz 4.791 Hz 5 Hz 5.933 Hz

(analysis) (experiment) (analysis) (analysis) (experiment) (analysis) (analysis) (experiment) (analysis)

Sensor (a) (b) (a) (a) (b) (a) (a) (b) (a)

1 -0.509 0.256 0.451 0.353 -0.452 -0.474 -0.359 -0.035 0.427

2 -.509 .461 0 -.254 .107 -.001 -.340 -.380 -.350

3 -.509 .656 -.451 .353 .436 .476 -.357 -.139 .425

4 -.260 .243 .409 .375 -.080 .133 .220 .176 -.074

5 -.260 -.09 0 -.249 .05 0 .243 .104 .072

6 -.260 .405 -.409 .375 -.030 -.133 .219 -.010 -.073

7 -.078 .098 .360 .375 .536 .505 '-.376 -.646 -.446

8 -.077 .199 0 -.244 -.068 0 -.434 .291 .318

! 9 -.078 .090 -.360 .375 -.540 -.507 -.375 -.539 -.443

aprediction based on finite-element analysis.

bCalculation based on identification from experimental data.

15
the space of the data and not in the physical coordinate Regulator for Distributed Parameter Systems. Auspace of the structure. Hence, a transformation is tomatica, vol. 14, no. 5_ Sept. 1978, pp. 453-463.

developed to obtain "natural" modes from lattice-filter 3. Montgomery, R. C.; and Thau, F. J.: Adaptive

modes, and Learning Control of Large Space Structures.

After obtaining the natural modes, the sensor data A Collection of Technical Papers--AIAA Guidance

are processed to obtain the modal amplitude time series and Control Conference, Aug. 1980, pp. 154-162.

and an autoregressive, moving-average model is used to (Available as AIAA-80-1739.)

represent the dynamics of each mode. The coefficients 4. Mendel, J. M.: Feasiblity and Design Study of Adapof this model are the difference-equation counterparts tire Control of Flexible, Highly Variable Spacecraft.

of modal frequency, damping, and control effectiveness NASA CR-111781, 1970.

of actuators. The coefficients of this model are identi- 5. Sundararajan, N.; and Montgomery, R. C.: Identififled by using an algorithm for equation error identifi- cation of Structural Dynamics System Using Leastcation. Based on these identified coefficients, a modal Square Lattice Filters. J. Guid., Control and Dyn.,

pole-placement control law is used to calculate the con- vol. 6, no. 5, Sept.-Oct. 1983, pp. 374-381.

troller gains for an adaptive control method for flexible 6. Williams, Jeffrey P.; and Montgomery, Raymond C.:

structures. Before the identification results are used for Simulation and Testing of Digital Control on a

control-law design, they are validated first by a series Flexible Beam. A Collection of Technical Papersof test procedures and the controller gains are updated AIAA Guidance and Control Conference, Aug. 1982,

only when all tests are passed, pp. 403-409. (Available as AIAA-82-1569.)

Based on the simulation studies of the flexible beam 7. Sundararajan, N.; and Montgomery, R. C.: Decouand the grid structure, the lattice filter has been shown pling the Structural Modes Estimated Using Reto provide a good on-line method for identifying the cursive Lattice Filters. Proceedings of the 21st

number of modes excited, mode shapes, modal damp- IEEE Conference on Decision (_ Control, Volume 3,

ing, and modal frequencies. The results from experi- 82CH1788-9, Dec. 1982, pp. 998-999.

mental data on both the beam and the grid differ sub- 8. Mendel, Jerry M.: Discrete Techniques of Paramstantially from the finite-element-analysis prediction of eter Estimation--The Equation Error Formulation.

the structural dynamics characteristics. The results in- Marcel Dekker, Inc., 1973.

dicate that reliance on finite-element analysis without 9. Lee, Daniel T. L.; Morf, Martin; and Friedlander,

ground testing may pose a problem and thus highlight Benjamin: Recursive Least Squares Ladder Estithe need for adaptive control or on-orbit testing, mation Algorithms. IEEE Trans. Acoust., Speech,

Simulation results on adaptive control of the beam _ Signal Process., vol. ASSP-29, no. 3, June 1981,

indicate that the lattice-filter identification method pp. 627-641.

along with validation procedures can provide a good on- 10. Friedlander, Benjamin: Lattice Filters for Adaptive

line model for control-law design and that the adaptive Processing. Proc. IEEE, vol. 70, no. 8, Aug. 1982,

control can be effectively used for structural dynam- pp. 829-867.

ics systems. This study also indicates that the lattice 11. Sage, Andrew P.; and Melsa, James L.: Estimation

method can be used for both identification and con- Theory With Applications to Communications and

trol in a single algorithm that sequentially accomplishes Control. McGraw-Hill, Inc., c.1971.

identification followed by closed-loop control. 12. Montgomery, Raymond C.; and Caglayan, Alper K.:

Failure Accommodation in Digital Flight Control

Systems by Bayesian Decision Theory. J. Aircr.,

Langley Research Center vol. 13, no. 2, Feb. 1976, pp. 69-75.

National Aeronautics and Space Administration 13. Benveniste, Albert; and Chaure, Christian: AR and

Hampton, VA 23665 ARMA Identification Algorithms of Levinson Type:

July 11, 1984 An Innovations Approach. IEEE Trans. Autom.

Control, vol. AC-26, no. 6, Dec. 1981, pp. 1243-

1261.

14. Thau, F. E.; Montgomery, R. C.; and Horner,

References G.C.: On-Line Structural Parameter Identification.

A Collection of Technical Papers--AIAA Guidance

1. Johnson, C. Richard, Jr.; and Montgomery, Ray- and Control Conference, Aug. 1981, pp. 530-539.

mond C.: A Distributed System Adaptive Control (Available as AIAA-81-1846.)

Strategy. IEEE Trans. Aerosp _ Electron. Syst., 15. Sundararajan, N.; and Montgomery, R. C.: Expervol. AES-15, no. 5, Sept. 1979, pp. 601-612. iments Using Least Square Lattice Filters for the

2. Hamza, M. H.; and Sheirah, M. A.: A Self-Tuning Identification of Structural Dynamics. A Collection

16
of Technical Papers--AIAA/ASME/ASCE/AHS 24th mation for Large Space Structures. Proceedings of

Structures, Structural Dynamics and Materials Con- the 1982 American Control Conference, 82CH1772-

ference, Part 2, May 1983, pp. 206-210. (Available 3, American Automatic Control Council, June 1982,

as AIAA-83-0880.) pp. 16-21.

16. Montgomery, Raymond C.; Homer, Garnett C.; and 18. Homer, Garnett C.: Optimum Aatuator Plaaement,

Cole, Stanley R.: Experimental Research on Struc- Gain, and Number for a Two-Dimensional Grillage.

rural Dynamics and Control. Dynamics and Con- A Collection of Technical Papers--AIAA/ASME/

trol of Large Flezible Spacecraft--Proceedings of the ASCE/AHS 24th Structures, Structural DynamThird VPI _ SU/AIAA Symposium, L. Meiroviteh, ics and Materials Conference, Part 2, May 1983,

ed., June 1981, pp. 365-372. pp. 179-184. (Available as AIAA-83-0854.)

17. Thau, F. E.; Eliazov, T.; and Montgomery, R. C.:

Least-Squares Sequential Parameter and State Esti17
18
n

ri,n = Yi-n - _(Yi-nlYi )

1

A

Yo y Yi-I YN

--- ' ' I I----_-_

Yi-n+ 1 yn ]/ Yi

ei,n = Yi - E(YilYn-i)

Figure 2. Schematic diagram ordering the measurement sequence.

yn ^ n n

N-1 YN = _'(YN]YN- 1)

y3 •

N-I •

2 YN^2 = g (YNiy2 - 1)

! _ ^1 = YNI-1) E(YN

YN _:(YN I = I YN- 1)

^0

YN = 0

i t I I

YN-n YN-5 YN -2 YN-1 YN

Figure 3. Nesting of estimation.

1,n+l <ei ,n' ri-l,n>

°

--z,n (1 - k2) -1⁄2

ri_l,n +

Figure 4. Flow diagram of a lattice section.

19
L-80-8918

Figure 5. Photograph of beam apparatus.

20
Sensor

1 2 3 4 5 --_6 7 8 9

'i'i" i "'i "

Actuator

1 Rigid body

Mode 1 0

-iL

i.89 Hz

Mode 3 0 _ _'_----, ,,

-i

4.99 Hz

Mode 4 0

--i

Mode 5 0

I ,, ,i ...... ! I I

0 3 6 9 12

Position, ft

Figure 6. Natural mode shapes and their frequencies along with sensor and actuator locations.

21
.i

r4, in. 0

-.i

1

Y4' in. 0

-.I

I I I I I I

0 2 4 6 8 i0

Time, see

Figure 7. Simulation results for nominal case.

22
Three modes

8 --

Order, n 4

0

.i

r4

in. 0

, _v.'-_- lP _'1_' '_' q'_ ' _'lt'' <r":'.....'i"_Y'l'i "''w'<''_'_'7 '_'''r '__'f'''_'"_°

-.1

Four modes

8-

Order, n 4 _________,_,_.,p/_]l _/,__!_j,-._r_L__/_d_r,k.___tl_ltL___

0

.i --

r4,in. 0 il""'l]l"""l'g'iif''' 'vT"'_"_r"'O'"t"""r" ....''"°(' ""*_"i""'7'"-_"t"'''°'#"t"'" "t"7'"'t'

--e "1 --

I I I I I

0 2 4 6 8 i0

Time, sec

Figure 8. Comparison of cases with three and four modes.

23
I_ threshold

8

Order, n 4

°2 ---

I eli, in..1 -- t I I I ,

20 threshold \

Order, n 4 _-

.2

llell, in..i

I I I I I 1

0 2 4 6 8 10

Time, sec

Figure 9. Effects of threshold on order determination.

24
NW= 1

8j--

Order, n 4

0

I I I I I I

0 2 4 6 8 i0

Time, sec

Figure 10. Effects of data-window size on order determination.

25
No noise

8 --

Order, n 4 /

I

0

.2 --

lle11, in..i--

Order, n 4

0

llell, in..i

I

0

l l l I I l

0 2 4 6 8 i0

Time, sec

Figure 11. Effects of additive sensor noise on order determination.

26
.O5

q5 0

-.05

I I I I I I

0 2 4 6 8 i0

Time, sec

Figure 12. Effects of damping on order determination.

27
Increasing order

8 --

Order, n 4 -- ._-_'-J_f_I_7_!'-J_i'qI_f_J_

____/

0

1 --

o -- " v V V '_' v° I1 _ _ y' ',] v v v

-1 --

I 1 1 l I I

0 2 4 6 8 10

Time, sec

Figure 13. Model order tracking.

28
Coupled Decoupled

1.0 .05 --

Modal amplitude

for mode 1 .5 0

1.0 .05 --

Modalformodeamplitude2.5 0 _]/_']_/_/_I_/_ /_]_,_._/

0 -.05 --

i I I i L t I l

0 2 4 6 0 2 4 6

Time, see

Figure14.Comparison ofcoupledand decoup]edmodalamplitudetime histories.
Coupled Decoupled

Modal amplitudespectra for 40 FA _ i- 2"5 F mode i 0 0

I

Modal amplitude 80 5.0 F _

spectra for

mode 2

0 0 -

80 5.0 [_ Modal amplitude

spectra for

mode 3

o _ A o -'

80 [ 5-0 I Modal amplitude _

spectra for

mode 4 0 _ - 0

t I J I I I

0 8 16 0 8 16

Frequency, Hz

Figure 15. Comparison of coupled and decoupled modal amplitude spectra.
.2

1

y , in. 0

-.2

.2 --

y , in. 0 I\V,, k, \/ '%/ \.¢" "V ",.I k.; ",J ,.d "..I \¢ _° _,/ _._

--.2 L

y , in. 0 I":' " \'/ \l ",j '_,, .j '.,,_.,, _ ',./ ',_" ",,.,I \-/ ",--

--.2 L_

<P',', ,..A_ ,_,.. ,.',,..,\._.i\-,. .^...._.-_. 2% .1_,_o+.A_,-A

(INOV)4 , in. 0 F_-_I '_-,%d _"v">" '_d 'k,P"" \"4 '..'d 'I'- nq _,.</-'_v W V v_''-v_

I -.05 --_

Order 2

estimate

0

I I .... I I I I

0 1 2 3 4 .5

Time, sec

Figure 16.- Data relevant to real-time processing of algorithm.

31
Figure 17. Estimated modal amplitudes and identified ARMA parameters resulting from algorithm.

32
O Mode 1

Least-squares lattice

i r] Mode 2

Nonlinear least squares

F Analytic _1

D O

[]

[]

Deflection, in. 0

121

-1 ---

I..... I, , . l

0 6 12

Position, ft

Figure 18. Mode-shape estimates obtained from the algorithm and a comparison with an analytic prediction and

the estimates obtained by using the nonlinear least-squares technique of reference 15.

nlmn IIII III I
Lattice

filter

Parameter

identification

A and B

Fail

Fit error

Pass

Fail

Fail

Pass gence

Pass

Stop

control gain

Calculate update

control

gains

Modal

control

on

Figure 19. Flow diagram for adaptive control parameter testing.

34
0rder determination

F Lattice-filter decoupling

f Parameter identification and testing

_ _ _ _ _ C:ntrol redesign and application

Lattice "5 II_ !III! i

filter- 0 --

moeIIVl ,

Control

design 3 --

order

J J I J

0 2 4 6 8 i0

Time, sec

Figure 20. Typical time history of an adaptive control run using on-line identification, testing, and control design.
Pass i --

Mode 1 0 A 27

-1 --

Pass 1 --

Mode 2 0

-1 --

Pass i-- _ _ _ Mode 3 0

-i --

Pass 1 --

Mode 4 0 /.

-I --

I I I I I I

0 2 4 6 8 I0

Time, sec

Figure 21. Test status for each controlled mode.

36
.O5

Mode 3 0

-.05

M°de40iVVVIV

.O5

Mode 5 0

-.05

6 --

Latticefilter 3 --

order

0

I I l f l J

0 2 4 6 8 i0

Time, sec

Figure 22. Time histories of three natural modes with lattice-filter order indicated.

3?
fl' ib 0 ll_y;_./"

f3' ib 0

-i--

f4' ib 0

-i --

l I I I I I

0 2 4 6 8 i0

Time, see

Figure 23. Actuator force commands.

38
5 --

Parameter A1

for mode 3

0

.75

.25 Pass below this threshold

.... _ - 0

.02 --

-- this old

Fit error .01

0 I I

Information .25 ass above his threshold

O ........ -- ....

Tests pass 01 _

i l I I l f

0 2 4 6 8 i0

Time, sec

Figure 24. Time histories of test variables for one mode with test thresholds and logic sum of tests indicated.

39
L-82-12,048

Figure 25. Photograph of flexible-grid experimental apparatus.

40
s Cable /-Cable

7 f Grid 8,5 , 9

7 8 9

4 4 5 6

6

1 2 3

1 2

3

Figure 26. Schematic diagram of grid structure with sensor and actuator locations. Asterisks indicate simulation.

I-a
(a) First flexible mode shape. Frequency, 2.299 Hz.

(b) Second flexible mode shape. Frequency, 4.791 Hz.

Figure 27. First four flexible mode shapes for the grid.

42
(c) Third flexible mode shape. Frequency, 5.933 Hz.

(d) Fourth flexible mode shape. Frequency, 6.297 Hz.

Figure 27. Concluded.

.43
y 5*, in. 0

-2

Order, n 3

o

.io

llellin..05 ,

o

I I i I I I 0 2 4 6 8 l0

Time, sec

Figure 28. Simulation time histories of sensor 5* as well as lattice-filter order estimate and norm of estimation error

for entire measurement vector.

1 nl_", _'., m r_ _., .._/\ /7 _'._ .... _ r

-i

/4 0 .... _---_ _-_--_-_ "* _-'-_-'"-_'"_ ""--'-" _--_- "-_---'_

y , in.

-i

Order, n 2

0

I I I I I I

0 i 2 3 4 5

Time, sec

Figure 29. Sensors 1 and 4 from experimental tape 5 and estimate of signal order obtained by lattice filter.

44
r-J \ i,, i_ .r-i p. ._ ._.

F" i.I _.J i__ ,, i i ,' ,, i i -_. ,-_

L _,,, , _l _ _

AI "--_'_._----_--

0

0

A2 L --" " '

-2

0'F I! :

_ 0 ,

-. 2 IL_I_-]_ --_'-!nL,_-

I I I I I 1

0 1 2 3 4 5

Time, see

(a) First mode.

Figure 30. Characteristics of modes identified from experimental tape 5.

45
ir

_.#"'.,; ',f "_i - _" "%" "-' " -'- "---"°_-'-'- ---"_'_--'_" -"'-" --

°L

A2 ..... -_.....-_...... __-__

-2

3 2, rq_',"'t_. "_.........

tO, l-lz ,_.

0 .... I

.2[ ,f_'_,

II _. ;_ .

0 ! " - L-/ ........

-.2

[ 1 I I I I

0 I 2 3 4 5

Time, sec

(b) Second mode.

Figure 30. Continued.

46
q 3 '-....-"-:"ir..=, _ " _'J " ' _"

'_4F,_:,'%" _._,4i" "1\" -"-'-._.---,_-__.-..._._ .,_..-'.....:;...__.._. _..--- _

A1 k

0

A2 ,"i..I ,-. J',.. _,_.,'_....... _,-'...,"

-2

6 _ ....... _,_. ._. ._.... ,, ,'..., "

tO, Hz ] -,

I

I

o i

• iI i . .r,,.,', ''V" I[l_ !'%1 .! 'fl,.Ii1=1] _,, ,,_..._|li

" i,,',r'; "L/"V" W "

--.2

I I I I I I

0 i 2 3 4 5

Time, sec

(c) Third mode.

Figure 30. Concluded.

47
1. Report No. / 2. Government Accession No. 3. Recipient's Catalog No.

NASA TP-2371

4. Title and Subtitle 5. Report Date

ADAPTIVE IDENTIFICATION AND CONTROL OF STRUCTURAL January 1985

DYNAMICS SYSTEMS USING RECURSIVE LATTICE FILTERS

6. Performing Organization Code

7. Author(s) 506-57-13-03

N. Sundararajan, Raymond C. Montgomery, and Jeffrey P. Williams 6. Performing Organization Report No.

L-15737

9. Performing Organization Name and Address 10. Work Unit No.

NASA Langley Research Center

Hampton, VA 23665 11.' Contract or Grant No.

12. Sponsoring Agency Name and Address 13. Type of Report and Period Covered

National Aeronautics and Space Administration Technical Paper

Washington, DC 20546 14. Sponsoring Agency Code

15. Supplementary Notes

N. Sundararajan: NRC-NASA Resident Research Associate, now on leave from the Indian Space Research

Organization.

16. Abstract

This report presents a new approach for adaptive identification and control of structural dynamic systems by

using least-squares lattice filters that are widely used in the signal processing area. Testing procedures for

interfacing the lattice-filter identification methods and modal control method for stable closed-loop adaptive

control have been presented. The methods have been illustrated for a free-free beam and for a complex flexible

grid, with the basic control objective being vibration suppression. The approach has been validated by using

both simulations and experimental facilities available at the Langley Research Center.

17. Key Words (Suggested by Authors(s)) 18. Distribution Statement

Adaptive control Unclassified--Unlimited

Lattice filters

ARMA identification

Free-free beam

Flexible grid

Experimental results Subject Category 18

19. Security Classif.(ofthis report) 120. Security Classif.(of this page) 21. No. of Pages 22. Price

Unclassified I Unclassified 50 A03

For sale by the NationM Technical Information Service, Springfield,Virginia 22161 NASA-Langley,1985

National Aeronautics .and THIRD-CLASSBULKRATE II//I d r'_-- _

Space Administration o 3 1176 01449 6195

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