AbstractParameter estimation in a physical system from observed data is carried out by a method which combines invariant imbedding formalism and the method of characteristics. This avoids the necessity of making use of an initial guess for the parameter necessary if one adopts quasilinearization techniques. Numerical results for a simple one parameter system is presented

Lee, E.S.

Vasudevan, R.

Wang, K.M.

E.S. Lee

R. Vasudevan

K.M. Wang

MUCC (Crossref)

Invariant imbedding, method of characteristics, and parameter estimation

Bellman

Abbott

Crossref

Koch, Karl Heinrich Emil

DigiZeitschriften - OAI Frontend

Die Flachslilie, die Mutterpflanze des neuseeländischen Flachses : Eine Dekorationspflanze 2.

Elsevier - Publisher Connector 

Camp. & Morhs. with Appls. Vol. 2, pp. 95-90. Pergamon Press, 1976. Printed in Great Britain. 
INVARIANT IMBEDDING, METHOD OF 
CHARACTERISTICS, AND PARAMETER ESTIMATION* 
E. S. LEE, R. VAsuDEvANt and K. M. WANGS 
Department of Electrical Engineering, University of Southern California, Los Angeles, CA 90007, U.S.A 
(ReceiuedOcbber 1975) 
Abstract-Parameter stimation i a physical system from observed ata is carried out by a method which 
combines invariant imbedding formalism and the method of characteristics. This avoids the necessity of 
making use of an initial guess for the parameter necessary if one adopts quasilinearization techniques. 
Numerical results for a simple one parameter system is presented. 
1. INTRODUCTION 
Both invariant imbedding and quasilinearization have been used to derive equations for 
parameter estimation (Bellman et al., 1966; Lee, 1%8). When quasilinearization is used, the 
estimation problem is treated as a two-point or multi-point boundary value problem. One 
difficulty with this approach is that the problem is frequently ill-conditioned. Furthermore, in 
order for the problem to converge, the initially guessed values for the parameters must be fairly 
close to the correct value. 
Invariant imbedding has been used to derive the sequential estimators in nonlinear filtering. 
However, in order to obtain the filtering equation, several approximations must be used. (See 
Bellman et al. 1966; Lee, 1%8). In this work, these approximations are avoided by directly 
solving the invariant imbedding equations by the method of characteristics (Abbott, 1%6). 
Furthermore, since the resulting integral equation can be minimized irectly by a suitable search 
technique, the problem with equality or inequality constraints on the parameters can be solved by 
this approach. The resulting integral equations form an ideal sequential estimator. 
Although only one dimensional problem is considered in this work, the approach can be 
extended easily to multidimensional problems. 
2. THE ESTIMATION PROBLEM 
Consider a system whose behavior can be represented by the scalar equation 
$=f(xA (1) 
with the condition 
X(ff) = c. (2) 
Let us assume that although the above equations represent a certain physical system, the 
parameter p and the initial condition c are not known exactly and must be determined from 
experimental data. In most practical situations, only x at various t can be measured. Let these 
measured values be represented by 
r(t) = x(t) + (measurement noise). (3) 
*This report was prepared as an account of work sponsored by the United States Government. Neither the United States 
nor the United States Atomic Energy Commission, or any of their employees, makes any warranty, express or implies, or 
assumes any legal iability or responsibility for the accuracy, completeness or usefulness of any information, apparatus, 
product or process disclosed, or represents hat its use would not infringe privately owned rights. 
ton leave of absence from the Institute of Mathematical Sciences, Madras, India. 
$Graduate Student, Kansas State University, Manhattan, Kansas, U.S.A. 
Supported by the National Science Foundation under Grant No. GP29049, the Atomic Energy Commission, Div. of 
Research under Contract No. AT(O4-3t113, Project 19 and the National Institutes of Health under Grant No. GM 16197-05. 
95 
96 E. S. LEE et al. 
By using these experimental values, we wish to determine the unknown parameters p and c so 
that the following least squares criterion is minimized. 
(4) 
we have assumed that the observation interval is 0~ t I tr. 
3. INVARIANT IMBEDDING 
Let us define a new variable, y(t), with 
y(t)= /‘(x(t)-z(t))‘dt. (5) 
II 
The integral equation (4) can be written as (Lee, 1968) 
$=(x(t)-z(t))‘. (6) 
with 
Y (tf) = J. (7) 
Consider the system represented by equations (1) and (6) and let us first obtain an expression 
for the final condition y (fr) using invariant imbedding. Define 
ec, a) = y(a), (f-3) 
where the process ends at t = a with 
x(a) = c. (9) 
The invariant imbedding equation can be obtained easily 
(10) 
4. THE METHOD OF CHARACTERISTICS 
Various techniques have been developed to solve equation (10) and, at the same time, to 
estimate p and c so that equation (4) is minimized. (See Bellman, 1966; Lee, 1968). In this work, 
the method of characteristics will be used. The characteristic equations of equation (10) are 
dc da dr ----=-= 2 
f(c,a,p) 1 (c-z(a))’ 
(11) 
For the moment, let us assume that the equations In (11) can be integrated. Integrate the 
left-hand equation in (ll), we have 
c = g(c, af, P, ccr) (12) 
where $ is the integration constant and af denotes the a value after integration. The other set of 
the equation from (11) leads to 
r = 
I 
” (c - z(a'))* da’. 
0 
(13) 
Invariant imbedding, method of characteristics, and parameter stimation 
Using equation (12), equation (13) becomes 
91 
r = I o’M c, a, P, IL) - z(a)l* da. (14) 
The problem now reduces to the minimization of the ingegral in equation (14). Notice that if 
equation (14) is minimized, then the least squares expression, equation (4), is also minimized. 
Thus, our problem is to obtain c and p so that equation (14) is minimized. Furthermore, this 
minimization can be carried in a sequential fashion. 
5. AN EXAMPLE 
To illustrate the approach, we shall consider the following simple scalar equation 
d”, -kg 
dt * (19 
With given experimental values for x, we wish to estimate the parameter k. Using the same 
approach discussed in Section 3, the invariant imbedding equation is 
Equation (12) becomes 
(16) 
1 
c = k(@ + 9) (17) 
or 
1 
$ = G - af. 
Equation (13) now becomes 
r(c, a) = )c -z(d))* da’. 
I (19) 
Using equation (17) and (18), equation (19) becomes 
- ~(a’) 1 2 da’. 
(18) 
The noisy measurements for the concentration are generated by the computer in two steps. 
First, equation (15) is integrated numerically. The numerical values used are 
x(0) = 1.0 
k = 0.05 
A?= 0.1 
tf = 50, 
where At is the integration step size. Second, the results from this integration are corrupted with 
noise by the equation 
z(ri)=x(ti)+R(rl), i=1,2 ,..., IV, (21) 
with to=O, rf =N, and tit,- ti = At. The R(h)‘s represent random numbers with Gaussian 
distribution. The mean of the distribution is zero and the standard eviation is 0.05. 
98 E. S. LEE et al. 
60 
0.03 0.05 0.07 
k 
Fig. 1. The invarient imbedding variable as a function of k. 
Using Z(C) and the following numerical values for equation (20) 
af= 50 
c = x(50) 
a=0,0.1,0.2 )...) 50. 
Equation (20) can be minimized with respect to k. Any search technique can be used for this 
minimization. A simple enumeration search is used. The results are shown in Fig. 1. As can be 
seen that at k = 0.05, the function r does reach the minimum. 
If k is not a constant but varies with time, in a suitable flashion, sequential determination of k 
in various regions can be made by the application of dynamic programming methods. The details 
of this procedure and numerical computations will be presented later. 
REFERENCES 
Bellman R., H. Kagiwada and R. Kalaba, Invariant imbedding and nonlinear filtering theory. J. Astronaut. Sci. 13(3), 1 lC-I 15 
(1966). 
Lee E. S., Quasilinearization and Invariant Embedding. Academic Press, New York (1968). 
Abbott, M. B., An Introduction to the Method of Characteristics, Elsevier, New York (1966). 


Invariant imbedding, method of characteristics, and parameter estimation 

https://doi.org/10.1016/0898-1221(76)90009-2

Invariant imbedding, method of characteristics, and parameter estimation

Abstract

Similar works

Full text

Available Versions

MUCC (Crossref)

MUCC (Crossref)

Crossref

MUCC (Crossref)

MUCC (Crossref)

DigiZeitschriften - OAI Frontend

Elsevier - Publisher Connector

MUCC (Crossref)