*Article* **Virtual Sensoring of Motion Using Pontryagin's Treatment of Hamiltonian Systems**

**Timothy Sands**

Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14850, USA; tas297@cornell.edu

**Abstract:** To aid the development of future unmanned naval vessels, this manuscript investigates algorithm options for combining physical (noisy) sensors and computational models to provide additional information about system states, inputs, and parameters emphasizing deterministic options rather than stochastic ones. The computational model is formulated using Pontryagin's treatment of Hamiltonian systems resulting in optimal and near-optimal results dependent upon the algorithm option chosen. Feedback is proposed to re-initialize the initial values of a reformulated two-point boundary value problem rather than using state feedback to form errors that are corrected by tuned estimators. Four algorithm options are proposed with two optional branches, and all of these are compared to three manifestations of classical estimation methods including linearquadratic optimal. Over ten-thousand simulations were run to evaluate each proposed method's vulnerability to variations in plant parameters amidst typically noisy state and rate sensors. The proposed methods achieved 69–72% improved state estimation, 29–33% improved rate improvement, while simultaneously achieving mathematically minimal costs of utilization in guidance, navigation, and control decision criteria. The next stage of research is indicated throughout the manuscript: investigation of the proposed methods' efficacy amidst unknown wave disturbances.

**Citation:** Sands, T. Virtual Sensoring of Motion Using Pontryagin's Treatment of Hamiltonian Systems. *Sensors* **2021**, *21*, 4603. https:// doi.org/10.3390/s21134603

Academic Editors: Javier Cuadrado and Miguel Ángel Naya Villaverde

Received: 5 June 2021 Accepted: 27 June 2021 Published: 5 July 2021

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**Keywords:** virtual sensoring; physical sensors; smart/intelligent sensors; sensor technology and applications; sensing principles; signal processing in sensor systems
