An algebraic approach to nonlinear functional expansions
Abstract
A new theory of functional expansion is presented which makes use of formal power series in several noncommutative variables and of iterated integrals. A simple closed-form expression for the solution of a nonlinear differential equation with forcing terms is derived, which enables us to give the corresponding Volterra kernels with utmost precision. The noncommutative variables give birth to a symbolic calculus which generalizes in a nonlinear setting many features of the Laplace and Fourier transforms and which is developed in order to simplify some computations, like the so-called association of variables, related to high-order transfer functions.
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In the identification of nonlinear dynamic systems with known deterministic inputs, there is no state-noise (Σw=0) because the states are known, and the hidden states become a deterministic nonlinear convolution of the causes.
“Note that there is no state-noise; i.e., Σw = 0 because the states are known. In this context, the hidden states become a deterministic nonlinear convolution of the causes [23].”
From Hierarchical Models in the Brain (2008) · cites this paper as [23]
Random fluctuations in biophysical systems are almost invariably the product of dynamical systems, which renders their serial correlations analytic.
“This is because random fluctuations in biophysical systems are almost invariably the product of dynamical systems, which renders their serial correlations analytic ([6], p 83; [23]).”
From Hierarchical Models in the Brain (2008) · cites this paper as [23]
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Hierarchical Models in the Brain
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“In this context, the hidden states become a deterministic nonlinear convolution of the causes [23].”
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Friston, K. (2002) · Annual Review of Neuroscience · doi:10.1146/annurev.neuro.25.112701.142846 · Cited by 465 · see citing works
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Chen, C. C., Kiebel, S. J., & Friston, K. J. (2008) · NeuroImage · doi:10.1016/j.neuroimage.2008.03.026 · Cited by 136 · see citing works
Volterra and generating power series approaches to identifiability testing
Lecourtier, Y., Lamnabhi-Lagarrigue, F., & Walter, E. (1987) · Identifiability of Parametric Models · doi:10.1016/b978-0-08-034929-9.50009-1 · Cited by 37 · see citing works
Self-citation
- Authors
- Fliess, M., Lamnabhi, M., & Lamnabhi-Lagarrigue, F.
- Affiliations
- Centre National de la Recherche Scientifique (FR)
- ORCID
- Fliess, M.
- Venue
- IEEE Transactions on Circuits and Systems
- Published
- 1983-08
- Volume
- 30
- Issue
- 8
- Pages
- 554-570
- ISSN
- 0098-4094 (ISSN-L)
- OpenAlex
- W2127407604
- Cited by
- 274
- References
- 0
- Citations by year
- 2026: 1 · 2025: 2 · 2024: 1 · 2023: 2 · 2022: 3 · 2021: 5 · 2020: 1 · 2019: 2 · 2018: 5 · 2017: 3 · 2016: 3 · 2015: 17 · 2014: 5 · 2013: 3 · 2012: 9
- FWCI
- 31.78
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- IEEE Transactions on Circuits and Systems (publishedVersion)
- Topics
- Physical Sciences › Engineering › Control and Systems Engineering › Control Systems and Identification
- Physical Sciences › Engineering › Control and Systems Engineering › Advanced Control Systems Design
- Physical Sciences › Mathematics › Applied Mathematics › Mathematical Analysis and Transform Methods
- Keywords
- iterated integrals, nonlinear differential equations, Volterra kernels, symbolic calculus, Laplace transform, Fourier transform, variable relationships
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