Contraction Certificates for Approximate Koopman Immersions with Nonzero Generator Residuals

Technical Summary: Contraction Certificates for Approximate Koopman Immersions with Nonzero Generator Residuals

Problem Statement
Finite-dimensional Koopman models aim to represent nonlinear dynamics in linear observable coordinates. However, when using finite dictionaries or learned embeddings, the generator relation is rarely exact, resulting in a nonzero generator residual $r(x, u)$. In contraction analysis, the relevant quantity for stability is not the residual itself, but its derivative acting on lifted perturbations. A critical challenge arises in "nonsquare" immersions, where the dimension of the observable map ($N$) exceeds the state dimension ($n$). In such cases, physical state variations generate perturbations that occupy only the immersed tangent space $T_x = \text{im } J(x)$ within the higher-dimensional latent space. Standard contraction tests that evaluate the full ambient latent matrix often fail because they include "normal" directions that are not physically realizable, leading to overly conservative or incorrect stability certificates.

Methodology
The paper establishes a framework for deriving contraction certificates for time-invariant nonlinear control systems $\dot{x} = f(x, u)$ represented by an approximate Koopman immersion $\psi(x)$ with a generator relation:
$$D\psi(x)f(x, u) = A\psi(x) + Bu + r(x, u)$$
where $J(x) = D\psi(x)$ has full column rank.

The core methodology involves:

1. Exact Tangent Dynamics: Differentiating the generator relation yields the exact lifted variational dynamics on the tangent space: $F_\psi J = D_x(Jf)$. The matrix $F_\psi = A + D_x r J_P^\dagger$ represents the true dynamics restricted to the physical tangent space, where $J_P^\dagger$ is the $P$-weighted Moore–Penrose left pseudoinverse.
2. Tangent-Rrestricted Logarithmic Norm: The paper defines a contraction certificate based on the weighted logarithmic norm of $A$ restricted to the tangent space, $\nu_{P, T_x}(A)$, rather than the full ambient norm.
3. Minimum Induced-Norm Extension: A key geometric result proves that the differential residual gain $\kappa_r = \sup_{v \neq 0} \|D_x r v\|_P / \|J v\|_P$ is exactly equal to the induced $P$-norm of the extension $D_x r J_P^\dagger$. This extension is the unique matrix that agrees with the differential residual on the tangent space while annihilating the $P$-orthogonal complement, thereby minimizing the induced norm among all possible extensions.
4. Decomposition of the Certificate: The contraction condition is separated into a nominal margin and a residual gain:
   $$\nu_{P, T_x}(A) + \kappa_r(x, u) \leq -c < 0$$
   This inequality certifies that the true nonlinear flow contracts, even if the ambient matrix $A$ or the full residual extension has a positive logarithmic norm.

Key Contributions

• Geometric Separation of Tangent and Ambient Dynamics: The paper rigorously distinguishes between physical perturbations (lying in $T_x$) and non-physical ambient directions. It proves that contraction of the true flow depends only on the behavior of the system restricted to $T_x$, allowing for stability certificates in cases where ambient tests fail.
• Optimal Residual Extension: The work establishes that the operator $D_x r J_P^\dagger$ provides the minimum induced-norm extension of the differential residual from the tangent space to the full latent space. This allows the residual term to be bounded tightly without penalizing the certificate with arbitrary extensions into non-physical directions.
• Convex Design and Verification: The paper formulates the selection of the nominal matrix $A$ as a convex min-max problem. It also provides a deterministic transfer theorem for $C^1$ model uncertainty, quantifying how vector-field approximation errors propagate into the contraction margin.
• Incremental Input-to-State Stability (δ-ISS): By integrating the tangent-restricted contraction inequality, the paper derives regional δ-ISS estimates that account for input variations.

Results
The theoretical framework is validated through two numerical examples:

1. Nonsquare Immersion Example: A nonlinear two-state plant is represented by a three-observable immersion. In this case, the direct Euclidean logarithmic norm of the system Jacobian and the ambient logarithmic norm of the nominal matrix $A$ are both positive, suggesting instability. However, the tangent-restricted certificate $\nu_{P, T_x}(A) + \kappa_r$ is uniformly negative, successfully certifying contraction of the true nonlinear flow on a forward-invariant region.
2. Fitted-Coordinate Example: A nonlinear system is approximated using a learned coordinate map. While the direct Euclidean test fails at the origin, the fitted coordinates expose a uniform contraction margin. Interval arithmetic is used to provide rigorous, computer-assisted verification of the contraction rate and the differential residual bounds over a compact region.

Significance and Claims
The paper claims to provide a rigorous theoretical foundation for analyzing the stability of approximate Koopman models, specifically addressing the "nonsquare" case where the latent dimension exceeds the state dimension. The primary significance lies in the ability to certify contraction for systems where standard ambient tests are inconclusive or negative. By isolating the physical tangent dynamics from the ambient latent space, the method allows for the use of approximate representations (with nonzero residuals) in control and stability analysis without requiring exact Koopman eigenfunctions. The results demonstrate that a "contractive geometry" can exist in the lifted space even when the ambient matrix measure is positive, provided the contraction is verified strictly on the immersed tangent manifold. The work also offers a convex optimization framework for designing the nominal matrix $A$ to maximize the contraction margin, bridging the gap between data-driven Koopman approximations and rigorous stability guarantees.