Approximation techniques for use in numerical schemes for estimating spatially varying coefficients in continuum models such as those for Euler-Bernoulli beams are discussed. The techniques are based on quintic spline state approximations and cubic spline parameter approximations. Both theoretical and numerical results are presented

Banks, H. T.

Crowley, J. M.

English

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Produced by the NASA Center for Aerospace Information (CASI) 
https://ntrs.nasa.gov/search.jsp?R=19830020663 2020-03-21T01:59:03+00:00Z
H. T. Banks
and
J. M. Crowley
Contract Nos. NAS1-15810 6 NAS1-16394L
May 1983
(NASA-C6-172132) PARAMETER 1DENTIFICAiiUDi	 N83-28934
IN CUNTINUUM AUDELS Final Report (NASA)
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INSTITUTE FOR COMPUTER APPLICATIONS IN SCIENCE AND ENGi	 NASA Langley Research Center, Hmoton, V•.rginia 23665
Operated by the Universities Space Research Association
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PARAMETER IDENTIFICATION
IN
CONTINUUM MODELS+
H. T. BANKS*
Lefschetz Center for Dynamical Systems
Division of Applied Mathematics
Brown University
Providence, RI 02912
and
J. M. CROWLEY*
Department of Mathematical Sciences
U. S. Air Force Acade,ny
Academy, CO 80840
ABSTRACT
We discuss approximation techniques for use in numerical schemes for
estimating spatially varying coefficients in continuum models such as those
for Euler-Bernoulli beams. The techniques are based on quintic spline state
approximations and cubic spline parameter approximations. Both theoretical
and numerical results are presented.
+Invited lecture, 1983 American Control Conference, June 22-24, 1983, San
Francisco, to appear in Prc,ceeding 1983 A.C.C.
*The research reported here was supported in part by NSF Grant MCS-8205335, by
I AFOSR Contract No. 81-0198, and ARO Contract No. ARO-DAAG-29-79-C-0161. Parts
of the research were carried out while the authors were in residence at the
Institute for Computer Applications in Science and Engineering, NASA Langley
Research Center, Hampton, Vh 73665, which is operated under NASA Coutracts No.
NAS1-15810 and No. NAS1-16394.
i
L	 i^
PAGE E^
OF POOR QUALITY
PARAMETER IDENTIFICATION IN CONTINUUM MODELS
H.T. Banks
Lefschetz Center for Dynamical systems
Division of Applied Mathematics
Brown University
and
Department of Mathematics
Southern Methodist University
Dallas, Texas 75275
ABSTRACT
We discuss approximation techniques for use in
numerical schemes for estimating spatially varying
coefficients in continuum models such as those for
Euler-Bernoulli beams. The techniques are based on
quintic spline state approximations and cubic spline
parameter approximations. Both theoretical and
numerical results are presented.
Introduction
We discuss here our efforts on the development of
numerical algorithms for the estimation of parameters
in variable structure elastic models. The class of
problems we investigate is of fundamental importance
in the use of continuum models (such as those for
beams, plates. thin shells, etc.) to represent large
flexible space structures [1-4]. Our own research has
been motivated by the need to detect and/or estimate
structural/ material property changes in such struc-
tures while they are on orbit. In this regard the
usefulness of parameter estimation techniques (i.e.,
inverse algorithms) for spatially varying parameters
in models such as
P( X ) 1 
z 
t D
2	 z
(EI(x)	 )	 f( B,t.x)	 (1)
at	 ax	 ax
from the Euler-Bernoulli theory or in more sophisti-
cated models arising in the Timoshenko theories should
be rather obvious. In this presentation we describe
spline-based methods that use and extend in a non-
trivial way the ideas developed in [5-7] for constant
parameter systems. These extensions allow one to treat
variable parameters (such as mass density p, elastic
modulus EI, load parameters S) in models such as (1)
and were first developed in the context of variable
coefficient first order parabolic systems in [8].
Although our ideas have much wider applicability
in elastic structures, to illustrate the basic princi-
ples we, for simplicity, restrict our considerations
here to a simply supported beam with constant mass
density and variable stiffness, which is under a known
load f. We assume that we have normalized our model
and thus consider (with D - ax)
utt + 02 (a(x)D2u) = f(t.x)	 0<x<1. 1:10,
	
(2)
with boundary conditions
u(t,O) - u(t,l) - u xx (t,0)
 - uxx (t.l) - 0	 (3)
and initial conditions
u(O,x) - O(x)
(4)
u t (O.x) - OX) -
The methods presented here readily extend to more
general cases (e.g., unknown parameters in more general
J.M. Crowley
Department of Mathematical Sciences
U.S. Air Force Academy
Academy, Colorado 80840
boundary conditions, in initial conditions or load,
variable mass and damping, etc.) as will be discussed
in a forthcoming paper.
The problem we consider is that of estimating
functions a in (2) from observations of the state u.
To be more specific. we consider a least squares fit-
to-data for (2). That is, given observations y ij for
for u(t i ,xpa), where u is the solution to (2)-(4)
corresponding to the parameter a from the set Q of
admissible parameters, we seek a parameter a ; that
minimizes
J(a) _ £ lu(t i ,xj ;a) - yijl2	 (5)
i, j
over Q.
We outline briefly the ideas of our approach to
this problem. We first rewrite (2)-(4) as an abstract
system in an appropriately chosen Hilbert space Z =
Z(a), our state space. The resulting abstract iden-
tification problem is then approximated on finite
dimensional state subspaces Z N (a) which in our case
here are generated by quintic spline elements. We
then obtain estimates aN
 from minimizing a fit-to-data
criterion using the approximate states.
The problem of obtaining aN is, however, not
computationally feasible since it requires that a min-
imization procedure be carried out over an infinite
dimensional function space or set Q. We therefore
introduce a further approximation by emp l oying sets QN
(in the results reported below these involved cubic
spline interpolations to the elements in Q) in the
minimization problems. This results in a double
approximation procedure: a state approximalt on (spaces
Z  to approximate Z) and a parameter approximation
(sets ON
 
to approximate Q); thus computationally we
obtain least squares parameters aN,M and any conver-
gence discussions must involve a double limit process
for N +-, M - -. As we shall note below. the Trotter-
Kato approximation theorem for linear semigroups, which
we have successfully used in a i,rmber of other problems
[5.6.7.9], can be used in the present case to establish
a convergence theory. (For the convenience of the
reader, we note that we are using the notation a for
the unknown parameter function here whereas in the
general formulations of [5-9] the symbol q is used to
denote a vector of unknown parameters.) We proceed to
outline the schemes we have used, discuss briefly con-
vergence results. and present some of our numerical
findings.
uintic/Cubic _Spline Approximation Schemes
Given (2)-(4), we rewrite this system in abstract
form in the parameter dependent state space
Z(at) _ (Ha i1 H0) x H0 . Here H2 is the space of
elements in H2 (0,1)
 with inner product
ORIGINAL PAGE 19
OF POOR QUALITY
<z,w>a 
n 
CQD2zl9D2w1>0 + <z
2
,1
2> 0 where < . > 0 is the
usual inner product in H D(0,1) n L 2 (0.1). Define the
operator A(m) by
0	 1
A(a)
\ -D2 (a02 )	 0
on dom(A) n (z E H4 x H4Iz1(n) - z2(n) - D2z l (n) •
D2z2 (rJ n 0. n n 0.1). Then putting z- (u,ut)T,
z0 = (a.t) T , and F - (f,0) T , we can rewrite the system
(2)-(4) as
i(t) - A(a)z(t) + F(t)	 t > 0,
(6)
z(0) - z0.
The estimation problem is then one of minimizing
J(s) _ I	 lz l (t i ;aa)(x j ) - " J1 2
	
(7)
i,J
over Q, where we assume Q is a given subset of H2(0,1)
such that 0 <a < a(x) < b for each a E Q. (Here a
and b are given bounds for elements in Q.)
Turning to approximations for (6), we let SS(aN)
denote the set of quintic splines (C 4 functions that
are piecewise polynomials of degree 5 -- see [6,10]
for notation) corresponding to the partition
aN = {x.}
N
	, x. = j/N, of [0,1]. We then define
J j.0	 J
ZN (a) as the subspace of Z(a) given by
Z  = S5(aN ) . S5(AN ) where S 5 (AN ) _ (s E S5(aN)1s(n)
02 s(n) - 0, n - 0.1). Let PN (a): Z ; Z  be the
orthogonal projection of Z(a) onto Z N (a) and define
approximations to A by AN (a) = PN ( a)A(a)PN (a). The
approximations to (6) in ZN (a) are then taken to be
zN (t) = AN (a)zN (t) + P NF(t)	 t > 0,
z N (0)	 P 
N 
z 0 .	
(8)
The approximate estimation problem is that of mini-
mizing
JN (a) • £ I z 1( t i ;a )( xj) - yijl2
	
(9)i, j
over Q.
In order to establish a convergence theory for
such Galerkin-type estimation schemes, it is not dif-
ficult (see [5,8.9]) to argue that it suffices to show
that zN (t;ak ) _ z(t;a *). N +	 k +	 for any
sequence (a k } in Q converging to a * in Q. The topology
for this convergence depends on the problem at hand
and in the present case, it is desirable to use the H2
topology on Q. Using fundamental estimates from spline
approximation theory (in particular, minor modifica-
tions of estimates given in Lemmas 1.18 and 1.19 of
[11] iuffice) and fundamental properties of dissipative
operators, one can readily employ the Trotter-Kato
theorem (see [5]) to establish the desired convergence
zN (t;ak ) --- z(t;a ) whenever a + a
*
 in Q. Further-
more, letting oN be a solution of niMmiting (9) over
Q, if we have nN (or a subsequence a j ) converging to
some a in Q (this requires a compactness property for
Q), it can be argued that a is a solution to the
original parameter estimation problem involving (6) and
(7). Hence assuming that Q is compact in H2 , we have
a sequence of problems which approximate in a desired
sense our original problem.
As we have already observed, the problem of
minimizing (9) over Q subject to (8) involves mini-
mization over an infinite dimensional parameter space
and a further approximation is necessary to obtain an
implementable computational procedure. For this we
use cubic interpolatory spline approximations to the
elements of Q. For a given partition a  n 
Iyj1M,iso
yj n j/M, of [0,1]. let IM be the cubic spline inter-
polation map (see [12, p.48]) corresponding to a  and
define QM = IN(Q). Then for a given function a E Q,
IM (a) involves the values a(y ), j n O.1,....M, and
Da(0), Da(1), so that I M :Q ^ QM is continuous in the
C l topology on Q and the C O topology on QN. If Q is
compact in the H2 topology, then it is therefore easily
seen that QM is compact in C(0.1) --id furthermore has
a representation
(	 M+3
M^
QM n {a: [0,11-R'J a n j^l yisi Yj E r 
where IBM} are the cardinal cubic basis elements and
the r  are compact subsets of R l . It follows that QN
is compact in the H 2 topology and hence the problem of
minimizing J N over Q can be readily replaced by the
finite dimensional state space - finite dimensional
parameter space problem of minimizing J N over QN.
Under additional smoothness assumptions on the elements
of Q, one can employ standard estimates for inter-
polatory splines to argue that Q M approximates Q in an
appropriate sense and that solutions of the problem for
JN over QM approximate those of the problem of mini-
mizing J over Q. More precisely one can establish the
following convergence results.
Theorem.	 Suppose Q C H4 (0,1) with JID4010Ea4EQ bounded
and suppose further that Q is compact in the H2(0,1)
topology. Let QM . IM(Q) be the cubic spline inter-
polatory approximations to Q and let, for each N and M,
;N 
IM note a solution of the finite state-finite
parameter Apace problem of minimizing J N of (9) over QM.
Let In k, j } be any convergent (in the H 2
 topology)
subsequence with limit a* ; then a*
 is a solution to the
problem of minimizing J of (7) over Q.
Numerical Results
We have developed and tested software packages
based on the quintic/cubic spline scheme discussed
above. The packages were modifications of those
(described in some detail in [6]) we have used for con-
stant parameter estimation problems (e.g., the IMSL
i►
t
2
ORIGINAL PAGir' iS
OF POOR QUALITY
References
[1] C.T. Sun, B.J. Kim. and J.L. Bagdanoff, On the
derivation of equivalent simple models for beam-
and plate-like structures in dynamic analysis.
Proceedings  AIM lReSiglists Cge!2Cjn ce , Atlanta,
Georgia. April 5-3. 1931. pp .	 .
package DGEAR is used to solve the ordinary differen-
tial equations (8). while the IMSL version - ZXSSQ -
of the Levenberg-Marquardt algorithm is used in mini-
mizing JN over QN). To test the method and software.
"data" was generated in the
*
 following manner: A
"true" parameter function a was chosen and an
independent numerical algorithm (finite differences)
was used to generate corresponding solution values
yi' - u(ti ,x
i
). a subset of which was then used as
observations or data in our inverse algorithms. We
present here the results from one of our test examples.
Additional numerical results along with a complete
theoretical treatment will be given in a paper cur-
rently in preparation.
Example.	 We take (2) with a• (x) - .15 + .10
tan x-.5)),a constant load f n 10 and assume that
the beam is initially at rest (4-poi. Observations
at points x i - .1..2......9 and times t i n .1..2, ... ,
1.0, were used in the least squares criterion (9).
For a given value of M. Q  involves M+ 3 cubic splines
for the approximation of coefficients in (2) while the
index N for the state involves N+ 1 quintic spline
elements in the state approximation. We present in
tabular form some of the results we obtained for
several values of N and M. The values denoted by J in
this table are JN(;N,M) while E denotes the L2 norm
of the error in the parameter estimates, i.e.,
E - ! a* - -N,M 1 O . Several graphs of aN,M vs. a* ollow
the references.
N-2
	
N-4	 N-6
J	 .21x10-3
	
.43x10 4	 .20x10-4
M-1
E	 .0207	 .0106	 .0055
J	 .18x10-3	 .13x10-4	 .16x10-4
M-2
E	 .0213	 .0115	 .0060
J	 .18x10-3	 .91x10-5	 .34x10-5
M-3
E	 .0048	 .0146	 .0010
Acknowledgements
The research reported here was supported in part
by NSF grant MCS-8205335. •Oy AFOSR contract 81-0198.-
and ARO contract ARO-DMG-29-79-C-0161. Parts of the
research were carried out while the authors were
visitors at the Institute for Computer Applications in
Science and Engineering (ICASE), NASA Langley Research
Center, Hampton, Va., which is operated under NASA
contracts No. NAS1-15810 and No. NAS1-16394. The
authors would like to express appreciation to B. Hanks.
NASA Langley Research Center, for fruitful discussions
in the course of this research.
[2] C.C. Chen, and C.T. Sun, Transient analysts of
large from* structures by simple models, Proceed-
ings of Symposium on Engineering Science and
Mechanics,
,
 (National Chen Kung University, Dec.
28-31. 19). pp. 753-775; also to appear in J.
Astronautical Sciences.
[3] A.K. Noor and C.M. Anderson. MalyEis of beam-like
lattice trusses, Comm. Meth. Appl. Mich. Enorg.,
20 (1979), pp. 53-X.
[4] J.N. Juang and C.T. Sun. System identification of
large flexible structures by using simple con-
tinuum models. J. Astronautical Sciences. to appear.
[5] H.T. Banks. J.M. Crowley and K. Kunisch. Cubic
spline approximation techniques for parameter
estimation in distributed systems, LOOS Tech. Rep.
81-25, Nov.. 1981, Brawn Univ.. IEEE Trans. Auto.
Control. to appear.
[6] H.T. Banks and J.M. Crowley. Parameter estimation
for distributed systems arising in elasticity,
Proc. S	 sium on Engineering Sciences and
Mechanics,  at once ng Kung University, Dec.
28-31. 11i81) pp. 158-177; LCDS Tech. Rep. 81-24,
November, 1981, Brown University.
[1] H.T. Banks and J.M. Crowley. Parameter estimation
in Timoshenko beam models. LCDS #82-14, Brown
Univ., June, 1982; J. Astronautical Sci., to
appear.
[8] H.T. Banks and P.L. Daniel. Estimation of variable
coefficients in parabolic distributed systems,
LCDS Rep. /82-22, Sept. 1982, Brown Univ.
[9] H.T. Banks and K. Kunisch, M approximation theory
for nonlinear partial differential equations with
applications to identification and control, LCDS
Tech. Rep. 81-7, Brown Univ., April, 1981, SIAM J.
Control and Optimization. 20 (1982). pp. 813=$E97
[10]P.M. Prenter, Spline and Variational Methods,
Wiley-Interscience. New York. 1975.
[11]J.M. Crowley, Numerical Methods of Parameter
Identification for Problems Arising in Elasticity.
Ph.D. Thesis, Brown University. May, 1982.
0 2] M.H. Schultz. __Sppl^i neAnal si^s. Prentice-Hall.
Englewood Cliff, li J . 1973.
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Parameter identification in continuum models

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