Source-linked AI summary
Trajectory Manifolds for Nonlinear Data-Enabled Predictive Control
Arda Bayer
TL;DR
The note asks when deterministic nonlinear systems have smooth finite-horizon behavior and what its dimension is. It characterizes terminal-state-augmented trajectories geometrically and establishes a smooth manifold representation with global coordinates. The resulting exact decoder is theoretical because it evaluates the unknown dynamics.
Problem
The note asks under what conditions finite-horizon behavior of a deterministic nonlinear system is a smooth finite-dimensional manifold, and what its dimension is.
Source: §1 · p. 2Method
Results
The finite-horizon behavior is a Cr embedded submanifold, and the free variables provide global smooth coordinates.
Source: §1 · p. 2, §3.3 · p. 5Takeaways & Limitations
A smooth finite-dimensional decoder exists theoretically before any neural-network approximation, but the exact decoder evaluates the unknown dynamics.
Source: §4.2 · p. 10The exact maps are not proposed as a practical model-free controller because the decoder evaluates the unknown dynamics.
Source: §4.2 · p. 10
Abstract
from arXiv · showhide
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, $r\geq 1$, 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 $(x_0,\mathbf u)$ 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.
1 Introduction
This note asks when deterministic nonlinear systems have smooth finite-horizon behavior and what its dimension is. It presents that behavior as a finite-dimensional embedded manifold with global smooth coordinates, within a behavioral viewpoint related to DeePC and prior trajectory-manifold work.
1 Introduction
- The behavioral viewpoint also underlies DeePC, while the cited linear Fundamental Lemma requires controllability and persistent excitation for finite-data representation.The note instead studies the geometry of finite-horizon trajectories of deterministic nonlinear systems.Source: §1 · p. 2
- Earlier continuous-time work studied smooth trajectory manifolds, their tangent spaces, and their second-order geometry.The cited work also connected trajectory manifolds to optimization through a projection-operator viewpoint.Source: §1 · p. 2
- Finite-horizon trajectories of deterministic nonlinear systems form a finite-dimensional Cr embedded manifold, with the initial state and all N inputs providing global smooth coordinates.The supplied passage states that the natural free coordinates are (x0, u), but does not give the dimension value.Source: §1 · p. 2
- The note asks under what conditions deterministic nonlinear finite-horizon behavior is a smooth finite-dimensional manifold and what its dimension is.Source: §1 · p. 2
2 Behavioral notation for a nonlinear state-space system
The section adopts behavioral notation for a deterministic nonlinear discrete-time state-space system with measured state and input signals. Its finite-horizon behavior retains the terminal state and includes trajectories only when the required dynamics evaluations are defined.
2 Behavioral notation for a nonlinear state-space system
- In the behavioral framework, a system is described by its time axis, signal space, and the admissible trajectories compatible with its laws.Here the time axis is nonnegative discrete time.Source: §2 · p. 2
- The input and measured state are the external variables of interest, matching the input/output partition used in the LTI behavioral formulation and DeePC.The paper assumes full-state measurement.Source: §2 · pp. 2–3
- The terminal-state-augmented N-step behavior retains the terminal state, distinguishing it from an ordinary restriction of trajectories to a finite time interval.The trajectories satisfy xk+1 = f(xk, uk) over the horizon.Source: §2 · p. 3
- When the transition map is not globally defined, the behavior includes only trajectories for which every required evaluation of the map is well-defined.Source: §2 · p. 3
3 Existence of the trajectory manifold
Under smooth deterministic dynamics, admissible initial states and future inputs provide global coordinates for the terminal-state-augmented finite-horizon behavior. Its rollout image is an embedded manifold, and its tangent space consists exactly of trajectories of the dynamics linearized along each nominal trajectory.
3.1 Assumptions
- The result assumes a C^r transition map for r ≥ 1, the minimum regularity needed here for a differentiable embedded manifold.The state-input domain is open, the standard setting for differentiable maps.Source: §3.1 · p. 3
3.2 The admissible free coordinates
- The admissible rollout domain is open, and each rollout state is C^r on that domain.Free coordinates are restricted when necessary so every state-input pair lies in the dynamics’ domain.Source: §3.2 · pp. 4–5
3.3 Main manifold theorem
- The finite-horizon behavior is a C^r embedded submanifold of dimension n + Nm, and the rollout map is a C^r diffeomorphism onto it.The free coordinates are the initial state and all N inputs; controllability, stabilizability, stability, and invertibility are not required.Source: §3.3 · pp. 5, 7
3.4 Equivalent regular-level-set viewpoint
- An independent regular-level-set argument gives the same embedded-manifold dimension by showing the dynamic constraints have full rank at every feasible trajectory.The constraint Jacobian is block triangular with identity blocks on the diagonal, yielding codimension Nn in signal space.Source: §3.4 · pp. 7–8
3.5 Tangent space and the linearized finite-horizon behavior
- At each nonlinear trajectory, the tangent space is exactly the finite-horizon behavior of the dynamics linearized along that trajectory.Any initial-state and input variation uniquely determines the state variations through the linearized recursion.Source: §3.5 · p. 8
4 Consequences for predictive control coordinates
Fixing the measured initial state restricts the full behavior manifold to an Nm-dimensional slice. The rollout coordinates also give an exact encoder–decoder pair, while exact differentiable reconstruction of the full behavior requires at least n + Nm latent dimensions.
4.1 Behavior conditioned on the measured current state
- With a measured initial state, feasible behavior is an Nm-dimensional slice of the full n + Nm-dimensional behavior manifold.The online constraint x0 = x(t) restricts the optimizer to this fixed-initial-state submanifold.Source: §4.1 · p. 9
4.2 Canonical exact encoder and decoder
- The rollout coordinates form an exact C^r encoder–decoder pair, but the decoder evaluates the unknown dynamics and is not proposed as a practical model-free controller.Its role is theoretical: it establishes that a smooth finite-dimensional decoder exists before neural-network approximation.Source: §4.2 · p. 10
4.3 A lower bound on exact latent dimension
- An exact differentiable reconstruction of the full behavior requires latent dimension p ≥ n + Nm, without ruling out useful lower-dimensional approximations on restricted data distributions.The bound is an exact C1 reconstruction statement.Source: §4.3 · p. 10
5 Consistency with the LTI behavior used in DeePC
In the LTI special case, the behavior has familiar linear-subspace geometry. This geometric characterization is distinct from the Fundamental Lemma’s finite-data spanning result used by DeePC.
5 Consistency with the LTI behavior used in DeePC
- For LTI systems, the trajectory map is linear and injective, so the behavior is a linear subspace.The nonlinear theorem recovers this familiar linear-behavior geometry as a special case.Source: §5 · p. 11
- The Fundamental Lemma says sufficiently long Hankel data span the controllable LTI behavior under persistent excitation; the geometric theorem makes no finite-data spanning claim.DeePC uses the Fundamental Lemma to represent relevant finite-horizon trajectories directly from measured data.Source: §5 · p. 11
- For output-only behavior, distinct internal states can produce the same finite output window, so input-output coordinates require additional observability assumptions.The full-state setting avoids this issue because the initial state is retained in the behavior vector.Source: §5 · p. 11
6 Sampled continuous-time systems
A zero-order-hold sampled continuous-time system has a smooth discrete transition map under the stated vector-field and solution-existence assumptions. Its finite-horizon behavior therefore has the same manifold structure, while fixed-step numerical maps describe the discretized dynamics used to generate data.
6 Sampled continuous-time systems
- The sampled transition map is C^r when the vector field is C^r and the considered initial-state/input pairs admit a unique solution over the sampling interval.Smooth dependence of ODE solutions on initial conditions and parameters supplies the transition-map regularity.Source: §6 · pp. 11–12
- Under Assumption 6.1, the zero-order-hold sampled system has a C^r embedded finite-horizon behavior manifold.The result follows by applying the discrete-time theorem to the smooth sampled transition map.Source: §6 · p. 12
- Zero-order hold replaces the continuous-time input function with a finite vector, yielding intrinsic dimension n + Nm; other finite-dimensional input parameterizations can be treated similarly.With unrestricted function-space inputs, the corresponding continuous-time trajectory space is naturally an infinite-dimensional Banach manifold.Source: §6 · p. 12
- Applying the theorem to a fixed-step RK4 map characterizes the trajectory manifold of the discretized dynamics that generated the training trajectories.For a C^r vector field, the RK4 update is itself C^r wherever its evaluations are defined.Source: §6 · p. 12
7 How much smoothness should be assumed?
C1 dynamics suffice for the manifold theorem, while higher smoothness is inherited by the behavior. C2 dynamics support second-order geometry, but a decoder’s raw Hessian is not itself coordinate-invariant curvature.
7 How much smoothness should be assumed?
- C1 dynamics suffice for a C1 embedded behavior manifold with well-defined tangent spaces that vary continuously.The theorem does not require C∞ regularity merely to establish the manifold structure.Source: §7 · pp. 12–13
- For C2 dynamics, the behavior and canonical rollout parameterization are C2, supporting second-derivative decoder regularization.C2 is presented as a natural working assumption when decoder second derivatives are used as a smoothness or local-nonlinearity regularizer.Source: §7 · p. 12
- For C2 dynamics, the Euclidean embedding has a well-defined second fundamental form, whereas tangential components of a decoder’s raw Hessian depend on the chosen coordinates.Thus the raw Hessian alone is not a coordinate-invariant notion of manifold curvature.Source: §7 · pp. 12–13
- Notarstefano and Hauser’s trajectory-manifold curvature is extrinsic and metric-dependent, based on a weighted L2 inner product and a nearest-trajectory optimal-control condition.Unprojected decoder-Hessian penalties are more accurately read as local-nonlinearity regularizers unless linked to invariant geometry.Source: §7 · p. 13
8 Boundaries, constraints, and noise
The main theorem concerns unconstrained deterministic behavior on an open admissible domain; constraints and noise require distinguishing the resulting trajectory-set geometry from the deterministic manifold.
8 Boundaries, constraints, and noise
- Restricting free inputs to a closed box preserves the manifold structure in its interior, while including the boundary gives a manifold-with-corners structure.General nonlinear state constraints can create more complicated boundary geometry and require separate regularity conditions.Source: §8 · p. 13
- When process noise is an additional free coordinate, the graph argument still applies and the intrinsic dimension grows by the noise-coordinate dimension.Without noise as a coordinate, measured trajectories lie near a deterministic nominal manifold or are samples from a trajectory distribution.Source: §8 · p. 13
- The trajectory-manifold radius is a local, direction- and metric-dependent second-order construction, not generally the global reach.Reach is the largest tubular radius on which nearest-point projection onto an embedded subset is unique.Source: §8 · p. 13
- Measurement noise leaves deterministic state behavior unchanged, although observed samples may not lie exactly on it.This is analogous to slack-variable and regularized DeePC formulations outside the noiseless LTI setting.Source: §8 · p. 14
9 What has been established
For deterministic nonlinear dynamics with measured state, the finite-horizon behavior is an embedded manifold with explicit global coordinates. This makes the next learning step approximation of the rollout map, while data-driven approximation and control analysis are deferred to Part II.
9 What has been established
- The deterministic finite-horizon behavior is a C^r embedded manifold.Source: §9 · p. 14
- Unlike earlier continuous-time Banach-manifold descriptions, finite horizon and finite-dimensional input parameterization expose global coordinates explicitly.Source: §9 · p. 14
- The initial state and future input sequence provide an explicit global chart, so learning can target the rollout map on a compact operating set.Data-driven approximation, predictive control, and closed-loop stability analysis are deferred to Part II.Source: §9 · p. 14