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Gramians for a New Class of Nonlinear Control Systems Using Koopman and a Novel Generalized SVD

Brian Brown, Michael King

Abstract

Certified model reduction for high-dimensional nonlinear control systems remains challenging: unlike balanced truncation for LTI systems, most nonlinear reduction methods either lack computable worst-case error bounds or rely on intractable PDEs. Data-driven Koopman/DMDc surrogates improve tractability, but standard \emph{input lifting} can distort the physical input-energy metric, so $H_\infty$ and Hankel-based bounds computed on the lifted model may be valid only in a lifted-input norm and need not certify the original system. We address this metric mismatch by a Generalized Singular Value Decomposition (GSVD)-based construction that represents general (including non-affine) input nonlinearities in an LTI-like lifted form with a \emph{pointwise norm-preserving} input map $v(x,u)$ satisfying $\|v(x,u)\|_2=\|u\|_2$ and constant matrices $A,B$. This preserves strict causality (constant $B$, no input-history augmentation) and yields computable Hankel-singular-value-based $H_\infty$ error certificates in the physical input norm for reduced-order surrogates. We illustrate the method on a 25-dimensional Hodgkin--Huxley network with saturating optogenetic actuation, reducing to a single dominant mode while retaining certified error bounds.

Gramians for a New Class of Nonlinear Control Systems Using Koopman and a Novel Generalized SVD

Abstract

Certified model reduction for high-dimensional nonlinear control systems remains challenging: unlike balanced truncation for LTI systems, most nonlinear reduction methods either lack computable worst-case error bounds or rely on intractable PDEs. Data-driven Koopman/DMDc surrogates improve tractability, but standard \emph{input lifting} can distort the physical input-energy metric, so and Hankel-based bounds computed on the lifted model may be valid only in a lifted-input norm and need not certify the original system. We address this metric mismatch by a Generalized Singular Value Decomposition (GSVD)-based construction that represents general (including non-affine) input nonlinearities in an LTI-like lifted form with a \emph{pointwise norm-preserving} input map satisfying and constant matrices . This preserves strict causality (constant , no input-history augmentation) and yields computable Hankel-singular-value-based error certificates in the physical input norm for reduced-order surrogates. We illustrate the method on a 25-dimensional Hodgkin--Huxley network with saturating optogenetic actuation, reducing to a single dominant mode while retaining certified error bounds.

Paper Structure

This paper contains 43 sections, 18 theorems, 275 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Linear Certification via Bounds
  3. The Geometric and Projection Alternatives
  4. The Energy Lineage: Rigor vs. Computability
  5. The Data-Driven Era: Koopman and the Control Problem
  6. The Contribution: A Causal, Certified, LTI Synthesis
  7. Background and Notation
  8. Results
  9. Preliminary Results: Generalized SVD
  10. Main Results Part 1: Nonlinear Gramians under Strong Assumptions
  11. A Finite-dimensional Example
  12. Step 1: Exact Koopman closure of the autonomous dynamics.
  13. Step 2: Lifted control-induced contribution.
  14. Step 3: Coordinate gains and a uniform ordering on a compact set.
  15. Step 4: Apply Lemma \ref{['lem:norm_preserve_u']} to construct $B$ and $v(x,u)$.
  16. ...and 28 more sections

Key Result

Lemma 1

Let $f:\mathbb{R}^n\to\mathbb{R}^m$ satisfy $f(0)=0$, and set $l\triangleq n+m$. Assume there exist an orthogonal matrix $U\in\mathbb{R}^{m\times m}$, a diagonal matrix $D=\mathrm{diag}(\sigma_1,\dots,\sigma_m)\succ 0$, and a constant $\beta\in[0,1)$ such that $(U,D)$$\beta$-cages $f$ in the sense o Then there exists an injective mapping $v:\mathbb{R}^n\to\mathbb{R}^l$ satisfying $\|v(x)\|_2=\|x\|

Figures (2)

  • Figure 1: Norm preservation calibrates the lifted input channel, restoring certified error bounds. Left column: models trained with the pointwise constraint $\|v(x_k,u_k)\|_2=\|u_k\|_2$; right column: models trained with unconstrained $v$. (A--B) Eigenvalues of the learned discrete-time matrix $A$ (dashed unit circle) indicate stability in both settings, showing that stability of $A$ alone does not guarantee a meaningful certified reduction. (C--D) Singular values of the learned input matrix $B$ reveal a pronounced scale/identifiability difference: when $v$ is norm-preserving, the required actuation gain must be carried by $B$, whereas unconstrained training can hide effective gain inside $v(x,u)$. (E--F) Hankel singular values (HSVs) of the lifted LTI surrogate suggest apparent reducibility in both cases, but only the norm-preserving construction ensures that these HSVs correspond to the physical input metric. (G--H) Certification check: measured simulation error (vertical axis) versus the predicted $\mathcal{H}_\infty$ reduction bound computed from the lifted surrogate (horizontal axis), across multiple reduced orders $r$. With norm-preserving $v$ (G), the empirical error remains below the bound (conservative but valid); with unconstrained $v$ (H), the bound can underestimate error by orders of magnitude, demonstrating that without input norm calibration the computed $\mathcal{H}_\infty$ bound is not a certificate with respect to the physical input $\|u\|$.
  • Figure 2: Time-domain consequence of input norm calibration: stable prediction and meaningful reduction versus gain-induced blow-up. Left column: norm-preserving $v$; right column: unconstrained $v$. (A--B) Rollout in physical coordinates under the same bounded control input $u_k$ (shown in (E--F)), comparing ground truth (true), the learned lifted surrogate (predicted), and a rank-$r{=}1$ or $2$ reduced surrogate (reduced $r{=}1$ or $2$). Under norm preservation (A), the surrogate and reduced model track the system over the horizon; under unconstrained lifting (B), trajectories diverge catastrophically despite bounded $u_k$, consistent with hidden gain amplification in $v(x,u)$. (C--D) Rollout in the internal (lifted) coordinates shows the same effect: norm-preserving lifting yields bounded internal dynamics, while unconstrained lifting drives the internal state to extreme magnitudes. (E--F) Control signal used in both experiments. Together with Fig. \ref{['fig:hh_certification_spectral']}, these results illustrate why enforcing $\|v\|_2=\|u\|_2$ is essential: it fixes the input-energy gauge so that (i) the lifted LTI surrogate remains physically calibrated and (ii) Hankel/BT-based $\mathcal{H}_\infty$ bounds serve as genuine certificates for reduced-order models in the original input metric.

Theorems & Definitions (66)

  • Definition 1: Diagonal gain cage
  • Remark 1: Geometric interpretation
  • Lemma 1: Gain-caged lift via a support/kernel split
  • proof
  • Definition 2: Directional gains and aggregation constant
  • Remark 2: Zero directional gain implies an identically zero channel
  • proof
  • Corollary 1: Universal bounds for the aggregation constant
  • proof
  • Definition 3: Orthogonal Energy Partition (OEP)
  • ...and 56 more