Limitations of LTI Koopman Modeling for Nonlinear Control Systems
Pith reviewed 2026-05-07 15:51 UTC · model grok-4.3
The pith
Nonlinear control systems admit exact LTI Koopman models only if they are affine linear, under mild controllability and full-state observables.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We show that, assuming a mild controllability condition and full-state observables, the dynamics of the underlying control system must be affine linear. The paper derives this result for continuous-time nonlinear control systems and further studies the modeling bias from the LTI structure and its dependency on the choice of observables.
What carries the argument
The exact LTI Koopman representation that lifts the nonlinear control system to linear dynamics, whose existence under the given conditions forces the original vector field to be affine linear.
If this is right
- Linear control techniques such as LQR and convex MPC apply exactly via Koopman lifting only when the system is already affine linear.
- Imposing the LTI structure on nonlinear systems always introduces modeling bias whose size depends on the selected observables.
- Exact LTI Koopman representations cannot exist for general nonlinear control systems under the stated assumptions.
- The result bounds the domain where Koopman-based linear control design works without approximation error.
Where Pith is reading between the lines
- For systems far from affine linear, control design may require time-varying Koopman operators or nonlinear lifting techniques instead.
- Engineers should first check whether their plant satisfies the affine linear condition before adopting LTI Koopman models for guaranteed performance.
- The limitation may motivate hybrid approaches that combine Koopman lifting with feedback linearization or other nonlinear methods.
- Analogous restrictions could appear in discrete-time settings or when only partial-state observables are available.
Load-bearing premise
The mild controllability condition together with the choice of full-state observables.
What would settle it
A concrete non-affine nonlinear control system that still admits an exact LTI Koopman representation with full-state observables and satisfies the mild controllability condition would falsify the central claim.
Figures
read the original abstract
Koopman operator theory yields powerful tools for modeling, analysis, and control of nonlinear dynamical systems. Prominently, linear time-invariant (LTI) Koopman representations have been proposed to enable the application of linear control techniques, such as LQR and convex MPC. In this work, we investigate the implications of exact LTI Koopman representations for continuous-time nonlinear control systems. In particular, we show that, assuming a mild controllability condition and the inclusion of the coordinate maps, the dynamics of the underlying control system must be affine linear. Furthermore, we study the modeling bias introduced by the LTI structure and analyze its dependency on the choice of observables.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proves that exact LTI Koopman representations of continuous-time nonlinear control systems are possible only for affine-linear dynamics when a mild controllability condition holds and the observables include the full state (identity map). It further quantifies the modeling bias that arises when the LTI structure is imposed on non-affine systems, showing its dependence on the choice of observables.
Significance. The result is significant because it supplies a sharp, assumption-minimal characterization of when LTI Koopman lifting can be exact for control systems, thereby clarifying the boundary beyond which linear control techniques applied to lifted models necessarily introduce structural error. The derivation is direct from the definition of an exact Koopman operator together with the inclusion of the identity map among the observables, and the controllability assumption is used only to rule out confinement to a proper invariant subspace; this parameter-free logical implication is a clear strength.
minor comments (3)
- §2.2: the precise statement of the 'mild controllability condition' (rank condition on the Lie algebra generated by the vector fields) appears only after the main theorem; a one-sentence forward reference in the abstract or introduction would improve readability.
- §4, Figure 2: the bias plots for the non-affine example would benefit from an explicit statement of the observable dictionary used and the numerical integration tolerance, to allow direct reproduction.
- Notation: the lifted state vector is denoted both as z and as ψ(x) in different sections; a single consistent symbol would reduce minor confusion.
Simulated Author's Rebuttal
We thank the referee for their positive assessment of the manuscript, recognition of its significance, and recommendation to accept. We appreciate the clear articulation of how the result provides a sharp, assumption-minimal boundary for exact LTI Koopman lifting in control systems.
Circularity Check
No significant circularity; derivation is direct logical implication from definitions
full rationale
The paper proves that exact LTI Koopman representations (lifted dynamics satisfy linear ODE) with full-state observables (identity map included) force the underlying control system to be affine-linear, using controllability only to rule out invariant subspaces. This follows immediately from the definitions of the Koopman operator and exact representation without any fitted parameters, self-referential definitions, or reduction to prior results. No load-bearing self-citations or ansatzes are invoked for the central claim; the argument is self-contained deductive mathematics.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption Mild controllability condition
- domain assumption Full-state observables
discussion (0)
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