  We propose an adaptive algorithm for identifying the unknown parameter in a
linear exponentially stable single-input single-output infinite-dimensional
system. We assume that the transfer function of the infinite-dimensional system
can be expressed as a ratio of two infinite series in s (the Laplace variable).
We also assume that certain identifiability conditions, which include a
persistency of excitation condition, hold. For a fixed integer n, we propose an
update law driven by real-time input-output data for estimating the first n+1
coefficients in the numerator and the denominator of the transfer function. We
show that the estimates for the transfer function coefficients generated by the
update law are close to the true values at large times provided n is
sufficiently large (the estimates converge to the true values as time and n
tend to infinity). The unknown parameter can be reconstructed using the
transfer function coefficient estimates obtained with n large and the algebraic
expressions relating the transfer function coefficients to the unknown
parameter. We also provide a numerical scheme for verifying the identifiability
conditions and for choosing n sufficiently large so that the value of the
reconstructed parameter is close to the true value. The class of systems to
which our approach is applicable includes many partial differential equations
with constant/spatially-varying coefficients and distributed/boundary input and
output. We illustrate the efficacy of our approach using three examples: a
delay system with four unknown scalars, a 1D heat equation with two unknown
scalars and a 1D wave equation with an unknown spatially-varying coefficient.
