{'fraction_non_alphanumeric': 0.013758599124452783, 'fraction_numerical': 0.0031269543464665416, 'mean_word_length': 5.751054852320675, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Reduced spatial order model reference adaptive control of spatially varying distributed parameter systems of parabolic and hyperbolic types

This paper presents control laws for distributed parameter systems of parabolic and hyperbolic types with unknown spatially varying parameters. These laws, based on the model reference adaptive control approach, guarantee asymptotic tracking of the output of the reference model by the output of the plant for arbitrary time invariant, but spatially varying reference input. The novel capabilities of the algorithms proposed are providing reduced sensitivity to measurement noise due to the reduced order of the spatial differentiation of the output data and permitting on‐line estimation of the spatially varying plant parameters, constructively enforceable through the reference input and/or boundary conditions. The parameter estimation is carried out by means of an auxiliary system with the time‐varying parameters that simultaneously converge in L2 to plant parameters when appropriate input signals in the reference model are used. The orthogonal expansions of these time‐varying parameters, which can be computed by passing the auxiliary system parameters through the integrator block, converge to the plant parameters pointwise if the latter are sufficiently smooth. The parameter convergence is obtained by combining the adaptation laws with sufficiently rich input signals, referred to as generators of persistent excitation, which guarantee the existence of a unique steady state for the parameter errors. Copyright © 2001 John Wiley & Sons, Ltd.