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\title{Efficient and accurate parameter estimation for  \\
linear, bilinear and nonlinear systems using Walsh functions}

\author{W.F. Blyth \\Department of Mathematics, RMIT}
\date{}
\maketitle
The Walsh functions were introduced by Walsh in 1923.  Ever since their 
introduction the mathematical properties of these functions have been studied 
extensively. Fifty years after their introduction, they were used for the 
solution of differential equations.  The first such method was due to 
Corrington (1973) where the initial value problem was rewriten as a Volterra 
integral equation.  Recently, Sloss and Blyth have derived a-priori error 
estimates for Corrington's method. Blyth, Sloss and Uljanov have also 
developed related algorithms for the solution of wide classes of linear and 
nonlinear Volterra and Fredholm type integral equations (including some that 
arise from functional differential equations).

\vskip 2truemm

In 1975, Chen and Hsiao used Walsh functions to estimate system parameters 
from the input/ouput records. The algorithm involves rewriting the differential
equations of the system as integral equations and using Walsh functions. In 
this paper, we show how to use the Walsh functions in a more direct spectral 
manner: the Walsh coefficients are used directly to obtain the algebraic 
equations that are solved to find the estimates of the parameters. For a 
problem with $p$ parameters and $n$ derivatives, the problem is rewritten by
integrating $n$ times, but then (unlike Chen and Hsiao) we equate the 
coefficents of the lowest $p$ Walsh functions. A Walsh series with $m$ terms
(where $m$ is  power of 2) is used, with $m \ge p$.

\vskip 2truemm

Our experience with the development of a-priori error estimates and with 
numerical experiments with Walsh functions and integral equations shows that,
usually, Walsh function methods are second order: that is, as $m$ is doubled,
the error is reduced by a factor of four. We propose using Richardson 
extrapolation, so the first two linear problems used by Chen and Hsiao are 
re-visited. We verify, by numerical experiment with $m = 4,\;8,\;16,\;32$, that
the algorithm is second order and that Richardson extrapolation provides 
greatly improved accuracy.

\vskip 2truemm

The Chen and Hsiao third problem (their ``Laboratory test") is 
\[ y'' + a_1 y' + a_2 y = b_1 u' + b_2 u  \;, \qquad u(t) = H(t) - H(t-1/4) \]
where the output is $y(t)$, initially at rest, $u(t)$ is the input and $H(t)$ 
is the Heaviside step function. We achieve high accuracy, although in this 
case (with delta function $u'$), for low $m$, Richardson extrapolation is not
effective - the solution is degraded (but still more accurate than the Chen
and Hsiao results). However we achieve very high accuracy for $m=32$: 
\[ (a_1,\;a_2,\;b_1,\;b_2) = (2,\;2,\;1,\;2) \approx (2.00033,\;
			2.00000,\;1.00016,\;2.00000)\,. \]
A few other papers appeared in the late 1970s and interest in a Walsh function
approach has been rekindled recently - for example, by Jha {\em et al.} (1992)
where a recursive algorithm is developed and applied to a bilinear system and 
to a nonlinear system. We show how to apply and suitably scale the problem to 
use the first 4, 8, 16 input/output records to obtain the estimates of the 
parameters. We obtain results that are orders of magnitude better than those of
Jha {\em et al.}. 

\end{document}

Assoc Prof Bill Blyth,
E-mail: bill.blyth@rmit.edu.au



