Adaptive distributed parameter systems identification with enforceable identifiability conditions and reduced order spatial differentiation

This work presents synthesis of adaptive identifiers for distributed parameter systems (DPS) described by partial differential equations (PDE's) of parabolic and hyperbolic type. The features of the PDE setting are utilized to obtain the not directly intuitive parameter estimation algorithms that use spatial derivatives of the output data with the order reduced from that of the highest spatial plant derivative. The plant parameter estimates are computed in terms of their orthogonal expansion through the integration of the adjustable identifier parameters. In this regard, the approach of the paper is in the spirit of finite dimensional observer realization in terms of integrating rather than differentiating the output data, only applied to the spatial rather than temporal domain. The constructively enforceable identifiability conditions, formulated in terms of the sufficiently rich input signals referred to as generators of persistent excitation, are shown to guarantee the existence of a unique zero steady state for the parameter errors. Under such inputs, the adjustable parameters in the adaptive identifiers proposed are shown to converge in L₂ to their nominal space-varying values guaranteeing the pointwise convergence of their orthogonal expansions, which represent plant parameter estimates, to the actual plant parameters.