Frequency-Domain adaptive parameter identification of linear infinite-dimensional systems

In this paper, an adaptive algorithm is proposed for identifying the unknown parameters in a stable linear single-input single-output (SISO) infinite-dimensional system. First, the transfer function of the infinite-dimensional system is assumed to be expressible as a ratio of two infinite series in the Laplace variable s. Additionally, the algebraic expressions relating the coefficients of these infinite series and the unknown parameters of the infinite-dimensional model are assumed to be known. Finally, certain identifiability conditions are also assumed to hold. Then, using an n-th order truncation of the transfer function, an adaptive update law driven by real-time input-output data is proposed for estimating the unknown parameters of the infinite-dimensional system. The estimates for the unknown parameters generated by the update law are shown to converge close to the true values at large times provided n is sufficiently large and the initial guess for the unknown parameters is within an identifiable region. The class of systems to which the proposed approach is applicable includes many partial differential equations (PDEs) with constant/spatially-varying parameters and distributed/boundary input and output. The efficacy of the approach is illustrated using two examples: a 1D heat equation with an unknown constant diffusion coefficient and a 1D wave equation with an unknown linearly (in space) varying parameter.