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We present a scalable approach to computing nonlinear balancing energy functions for control-affine systems with polynomial nonlinearities. Al’brekht’s powerseries 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.
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Scalable Computation of $\mathcal{H}_{\infty}$ Energy Functions for Polynomial Drift Nonlinear Systems
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.
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- Award ID(s):
- 2130727
- PAR ID:
- 10552207
- Publisher / Repository:
- IEEE
- Date Published:
- ISBN:
- 979-8-3503-8265-5
- Page Range / eLocation ID:
- 2521 to 2526
- Format(s):
- Medium: X
- Location:
- Toronto, ON, Canada
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.23919/ACC60939.2024.10644363
