Design of cascaded PDE-ODE observers for LTI systems with numerical differentiators as fictitious sensors with distributed delay

Unmeasured states of an observable LTI system without sensor delay can be determined by use of a state observer or calculated directly from the input, the output, and their successive time derivatives. The inherent ill-posedness of numerical differentiation renders the latter approach unsuitable for most control applications. Design objectives like the need to balance convergence rates and steady-state noise suppression, on the other hand, pose significant challenges for state observation. This work proposes a systematic framework for the combination of numerical differentiation and output error injection to leverage benefits of both methods. By interpreting estimates of the output derivatives of an LTI system without sensor delay as additional, but delayed, outputs, a cascaded PDE-ODE formulation of the augmented system is derived. In it, the derivative estimates, which are treated as readings from fictitious sensors, are modeled by transport equations. An observer for the resulting system is developed. We prove that additional information inferred from output derivatives improves observability properties of the underlying system in terms of the observability Gramian. Moreover, incorporating the output derivatives introduces additional degrees of freedom to the design. We show how such degrees of freedom can be exploited; in particular, they can be used to improve noise attenuation due to a reduction of the injection gains. An analysis of the observer for the PDE-ODE cascade and a comparison to the Luenberger observer are provided. The efficacy of the proposed method is illustrated experimentally.