Adaptive identification of linear infinite-dimensional systems

We propose an adaptive algorithm for identifying the unknown parameter (possibly vector-valued) in a linear 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 adaptive update law (a differential equation in time) 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 linearly-varying (in space) coefficient.