Publications

You can also find my articles on my Google Scholar profile and preprints on arXiv.

Scalable Computation of H∞ Energy Functions for Polynomial Control-Affine Nonlinear Systems

Published in In review, 2024

We present a scalable approach to computing nonlinear balancing energy functions for control-affine systems with polynomial nonlinearities. Al’brekht’s power-series method is used to solve the Hamilton-Jacobi-Bellman equations for polynomial approximations to the energy functions. The contribution of this article lies in the numerical implementation of the method based on the Kronecker product, enabling scalability to over 1000 state dimensions. The tensor structure and symmetries arising from the Kronecker product representation are key to the development of efficient and scalable algorithms. We derive the explicit algebraic structure for the equations, present rigorous theory for the solvability and algorithmic complexity of those equations, and provide general purpose open-source software implementations for the proposed algorithms. The method is illustrated on two simple academic models, followed by a high-dimensional semidiscretized PDE model of dimension as large as n=1080.

Recommended citation: Recommended citation: N. A. Corbin and B. Kramer, “Scalable computation of $\mathcal{H}_\infty$ energy functions for polynomial control-affine systems,” Aug. 2024, doi: 10.48550/ARXIV.2408.08970. https://arxiv.org/abs/2408.08970

Scalable Computation of H∞ Energy Functions for Polynomial Drift Nonlinear Systems

Published in 2024 American Control Conference (ACC), 2024

This paper presents a scalable tensor-based approach to computing controllability and observability-type energy functions for nonlinear dynamical systems with polynomial drift and linear input and output maps. Using Kronecker product polynomial expansions, we convert the Hamilton-Jacobi-Bellman partial differential equations for the energy functions into a series of algebraic equations for the coefficients of the energy functions. We derive the specific tensor structure that arises from the Kronecker product representation and analyze the computational complexity to efficiently solve these equations. The convergence and scalability of the proposed energy function computation approach is demonstrated on a nonlinear reaction-diffusion model with cubic drift nonlinearity, for which we compute degree 3 energy function approximations in n=1023 dimensions and degree 4 energy function approximations in n=127 dimensions.

Recommended citation: N. A. Corbin and B. Kramer, “Scalable computation of $\mathcal{H}_\infty$ energy functions for polynomial drift nonlinear systems,” in 2024 American Control Conference (ACC), Jul. 2024, pp. 2506–2511. https://arxiv.org/abs/2408.08387

Computing Solutions to the Polynomial-Polynomial Regulator Problem

Published in 2024 Conference on Decision and Control (CDC), 2024

We consider the optimal regulation problem for nonlinear control-affine dynamical systems. Whereas the linear-quadratic regulator (LQR) considers optimal control of a linear system with quadratic cost function, we study polynomial systems with polynomial cost functions; we call this problem the polynomial-polynomial regulator (PPR). The resulting polynomial feedback laws provide two potential improvements over linear feedback laws: 1) they more accurately approximate the optimal control law, requiring lower control costs, and 2) for some problems they can provide a larger region of stabilization. We derive explicit formulas for the polynomial approximation to the value function that solves the optimal control problem; we also provide scalable algorithms and software that exploit the tensor structure of the equations for general purpose practical use. The method is illustrated first on a low-dimensional aircraft stall stabilization example, for which PPR control recovers the aircraft from more severe stall conditions than LQR control. Then we demonstrate the scalability of the approach on a semidiscretization of a PDE, for which the PPR control reduces the control cost by approximately 75\% compared to LQR.

Recommended citation: N. A. Corbin and B. Kramer, “Computing Solutions to the Polynomial-Polynomial Regulator Problem,” in 2024 Conference on Decision and Control (CDC), Dec. 2024. https://arxiv.org/abs/2410.22291

Reference-free Longitudinal Rail Stress and Neutral Temperature Measurement Utilizing Multidirectional Elastic Waves

Published in Federal Railroad Administration, 2021

This Federal Railroad Administration technical report documents the development and testing of a rail neutral temperature measurement technique based on low frequency flexural wave measurements. This report includes much of my masters work.

Recommended citation: Corbin, N., Albakri, M., Tarazaga, P., “Reference-free longitudinal rail stress and neutral temperature measurement utilizing multidirectional elastic waves” Federal Railroad Administration, Washington, D.C., Tech. Rep., submitted 2021, available upon request.

Overcurvature induced multistability of linked conical frusta: How a ‘bendy straw’ holds its shape

Published in Soft Matter, 2018

This paper is about the stability of linked conical frusta, i.e. bendy straws. We investigate the effects of material, geometry, and prestress on the stability of such structures.

Recommended citation: Bende, N. P., Yu, T., Corbin, N. A., Dias, M. A., Santangelo, C. D., Hanna, J. A., Hayward, R. C., “Overcurvature induced multistability of linked conical frusta: How a ‘bendy straw’ holds its shape”, Soft Matter, vol. 14, no. 42, pp. 8636–8642, 2018, issn: 1744-683X. https://doi.org/10.1039/C8SM01355A

Impact-induced acceleration by obstacles

Published in New Journal of Physics, 2018

This paper is about a phenomenon in which a falling chain impacting a surface accelerates faster than gravitational acceleration. The paper gives experimental results confirming the effect and a mathematical model which captures the behavior.

Recommended citation: Recommended citation: Corbin, N. A., Hanna, J. A., Royston, W. R., Singh, H., Warner, R. B., “Impact-induced acceleration by obstacles”, New Journal of Physics, vol. 20, no. 5, p. 053031, May 2018, issn: 1367-2630. https://doi.org/10.1088/1367-2630/aac151

Nicholas Corbin