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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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This content will become publicly available on October 1, 2026
Scalable computation of input-normal/output-diagonal balanced realization for control-affine polynomial systems
We present a scalable tensor-based approach to computing input-normal/output-diagonal nonlinear balancing transformations for control-affine systems with polynomial nonlinearities. This transformation is necessary to determine the states that can be truncated when forming a reduced-order model. Given a polynomial representation for the controllability and observability energy functions, we derive the explicit equations to compute a polynomial transformation to induce input-normal/output-diagonal structure in the energy functions in the transformed coordinates. The transformation is computed degree-by-degree, similar to previous Taylor-series approaches in the literature. However, unlike previous works, we provide a detailed analysis of the transformation equations in Kronecker product form to enable a more scalable implementation. We derive the explicit algebraic structure for the equations, present rigorous analyses for the solvability and algorithmic complexity of those equations, and provide general purpose open-source software implementations for the proposed algorithms to stimulate broader use of nonlinear balanced truncation model reduction. We demonstrate that with our efficient implementation, computing the nonlinear transformation is approximately as expensive as computing the energy functions using state-of-the-art methods.
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- Award ID(s):
- 2130727
- PAR ID:
- 10657687
- Publisher / Repository:
- Elsevier
- Date Published:
- Journal Name:
- Systems & Control Letters
- Volume:
- 204
- Issue:
- C
- ISSN:
- 0167-6911
- Page Range / eLocation ID:
- 106178
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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