Trajectory Manifolds for Nonlinear Data-Enabled Predictive Control
Abstract
This note establishes a geometric foundation for trajectory-manifold representations of deterministic nonlinear systems in a behavioral setting motivated by data-enabled predictive control. For a discrete-time system xk+1=f(xk,uk) with measured state and a Cr transition map, r≥ 1, we consider the terminal-state-augmented finite-horizon behavior consisting of all admissible state-input trajectories over a prediction horizon N. We prove that this behavior is a Cr embedded submanifold of the ambient trajectory space with intrinsic dimension n+Nm, where n and m are the state and input dimensions. Moreover, the rollout map from the admissible initial-state and input coordinates (x0,u) is a Cr diffeomorphism onto the behavior manifold, providing explicit global smooth coordinates. This yields a canonical exact encoder--decoder representation and implies that any exact differentiable latent representation of the full behavior must have latent dimension at least n+Nm. The geometric result does not require controllability, stabilizability, or invertibility of the dynamics. Corresponding results are given for zero-order-hold sampled continuous-time systems and fixed-step numerical transition maps. These results provide the deterministic geometric foundation for subsequent data-driven approximation and predictive-control development.
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BibTeX
@misc{bayer2026trajectory,
title = {{Trajectory Manifolds for Nonlinear Data-Enabled Predictive Control}},
author = {Arda Bayer},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19079},
eprint = {2609.19079},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19079},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Trajectory Manifolds for Nonlinear Data-Enabled Predictive Control
AU - Arda Bayer
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.19079
UR - https://arxiv.org/abs/2609.19079
AB - This note establishes a geometric foundation for trajectory-manifold representations of deterministic nonlinear systems in a behavioral setting motivated by data-enabled predictive control. For a discrete-time system x_k+1=f(x_k,u_k) with measured state and a C^r transition map, r≥ 1, we consider the terminal-state-augmented finite-horizon behavior consisting of all admissible state-input trajectories over a prediction horizon N. We prove that this behavior is a C^r embedded submanifold of the ambient trajectory space with intrinsic dimension n+Nm, where n and m are the state and input dimensions. Moreover, the rollout map from the admissible initial-state and input coordinates (x_0,u) is a C^r diffeomorphism onto the behavior manifold, providing explicit global smooth coordinates. This yields a canonical exact encoder--decoder representation and implies that any exact differentiable latent representation of the full behavior must have latent dimension at least n+Nm. The geometric result does not require controllability, stabilizability, or invertibility of the dynamics. Corresponding results are given for zero-order-hold sampled continuous-time systems and fixed-step numerical transition maps. These results provide the deterministic geometric foundation for subsequent data-driven approximation and predictive-control development.
ER -