KRONIC: Koopman-Based Nonlinear Control

• KRONIC is a data-driven paradigm that constructs reduced-order models by lifting nonlinear dynamics into a space of observables, enabling linear control methods.
• It unifies operator-theoretic identification, dimensionality reduction, and robust control strategies to facilitate real-time applications from network inference to high-dimensional system control.
• KRONIC’s use of Koopman eigenfunctions and bilinear formulations overcomes challenges in noisy or chaotic systems, enhancing stability, safety, and computational efficiency.

Koopman Reduced Order Nonlinear Identification and Control (KRONIC) is a data-driven paradigm that constructs reduced-order linear or bilinear models of complex nonlinear systems by lifting the original dynamics into a space of observables, enabling the application of linear systems and control theory to otherwise challenging nonlinear identification and control problems. The KRONIC approach unifies operator-theoretic system identification, dimensionality reduction, eigenfunction-based embeddings, and modern robust or optimal control strategies, supporting applications from network inference to real-time control of high-dimensional dynamical systems.

1. Koopman Lifting: Identification of Nonlinear Systems as Linear Problems

The KRONIC methodology begins with the insight that nonlinear system identification can be recast as a linear identification problem for the (infinite-dimensional) Koopman operator acting on observables. For a dynamical system $\dot{x} = F(x)$, the Koopman operator $U^t$ acts linearly on any observable $f$ via $U^t f = f \circ \phi^t$, where $\phi^t$ denotes the system's flow (Mauroy et al., 2016). Practically, the system is lifted by evaluating a finite set of basis functions (e.g., monomials up to degree $m$) at snapshot pairs $(x_k, y_k)$ to form data matrices $P_x$ and $P_y$, constructing the finite-dimensional Koopman operator estimate $\tilde{U} = P_x^\dagger P_y$.

The operator is "projected back" to finite dimension for model recovery through the infinitesimal generator:

$L_\text{data} = \frac{1}{T_s} \log(\tilde{U})$

By relating $L_\text{data}$ to the coefficients of the original vector field via $L = F \cdot \nabla$, the nonlinear governing equations can be identified robustly via linear least-squares, avoiding direct numerical differentiation and retaining efficacy in the presence of noise and low sampling rates (Mauroy et al., 2017).

2. Reduced-Order Coordinates: Koopman Eigenfunctions and Invariant Subspaces

A key feature of the KRONIC framework is the identification and utilization of intrinsic coordinates—specifically the eigenfunctions of the Koopman operator—along which dynamics are linear or decouple (Kaiser et al., 2017). These eigenfunctions are discovered using regression techniques on snapshot data, with validation protocols essential due to the closure issue in finite-dimensional approximations (i.e., the risk of finding spurious eigenfunctions). Lightly-damped eigenfunctions typically correspond to nearly-conserved quantities or slow manifolds (e.g., Hamiltonians in mechanical systems) and are particularly valuable for reduced-order modeling and control.

Control strategies are then formulated in the coordinates of these validated eigenfunctions, which form a Koopman-invariant subspace. This leads to more accurate and stable predictive and control models—as opposed to control in arbitrary basis coordinates—and enables the direct application of linear optimal control (e.g., LQR or Riccati-based approaches) in these lifted spaces.

3. Algorithmic and Operator-Theoretic Formulation

The practical KRONIC workflow involves the following procedure:

• Data lifting: Select an appropriate dictionary of observable functions (monomials, trigonometric, radial basis functions, etc.), compute their values at data snapshots.
• Operator estimation: Fit the Koopman operator (or generator) using linear regression in the lifted space, typically via algorithms related to EDMD.
• Model reduction: Identify a reduced-order basis—using sparsity-promoting regression (e.g., LASSO), proper orthogonal decomposition (POD), or Pareto front selection—yielding a compact model that approximates the leading Koopman modes relevant for prediction or control (Bistrian, 5 Aug 2025, Zhang et al., 31 Mar 2024).
• Eigenfunction construction: For high-fidelity reduction, extract eigenfunctions associated with dominant modal dynamics, validating their suitability for control design (Kaiser et al., 2017).
• Model reconstruction: Project the finite-dimensional Koopman generator onto the original basis to reconstruct the polynomial, rational, or even networked nonlinear vector field terms.

For systems where a global Koopman linear realization is not possible, a bilinear form is adopted: $z_{k+1} = Az_k + Bu_k + H(z_k \otimes u_k)$. Such bilinear realizations better capture control-affine nonlinearities and are leveraged directly in control design and Data-Enabled Predictive Control (DeePC), sidestepping explicit model identification (Xiong et al., 6 May 2025, Zhang et al., 2022).

4. Robust and Safe Control via Koopman-Based Surrogates

Once a parsimonious linear or bilinear surrogate is constructed in the lifted space, KRONIC enables the use of state-of-the-art linear control methods for regulation, estimation, and safety-critical control of nonlinear systems. These include:

• Model predictive control (MPC): Reduced-order, computationally efficient robust MPC formulations that utilize the linear Koopman surrogate, with guarantees on solution convergence as the dictionary and data size increase (Peitz et al., 2018, Zhang et al., 31 Mar 2024).
• Robust stabilization: Controller synthesis via LMIs or H∞ theory for bilinear surrogate models that incorporate finite-gain bounds on model mismatch induced by truncation or noise, ensuring robust local or semi-global stability (Strässer et al., 2023, He et al., 16 Jan 2024).
• Safety and adaptivity: Integration with control barrier functions (CBFs), deploying fixed-time parameter adaptation laws for provably safe operation in the presence of unknown disturbances identified through the Koopman generator (Black et al., 2022).
• Dual-loop architectures: Decoupled nominal and robust loops to handle model mismatch due to noise-induced bias, with tractable LMI-based H∞ compensation for bounded error between learned and true dynamics (He et al., 16 Jan 2024).

This approach bridges the gap between operator-theoretic identification and synthesized controllers, enabling real-time, feedback, or switching control schemes for systems previously only amenable to local or approximate linearization.

5. Theoretical Foundations and Connection to Stability Theory

There exists a deep equivalence between the operator-theoretic "lifting" of dynamics and contraction analysis (Yi et al., 2021, Fan et al., 16 Jan 2024). Specifically, if there exists a smooth Koopman embedding $\phi$ and a matrix $A$ (Hurwitz or Schur-stable), such that $\partial \phi/\partial x \cdot f(x) = A\phi(x)$, then the nonlinear system is contracting under the metric $$M(x) = [\partial\phi/\partial x]^\top P [\partial\phi/\partial x]$$ for $P\succ 0$ solving $A^\top P + PA = -I$. This equivalence unifies global linearization (lifted invariance) with strong incremental stability in the original state space and connects observer-based design (e.g., Kazantzis-Kravaris-Luenberger observers) to Koopman embedding construction.

Consequently, enforcing linear structure in the Koopman-lifted space via stability-enforcing parameterizations yields reduced-order models for control that are guaranteed to be robust or even contractive (incrementally stable), facilitating safe learning and control directly from data (Fan et al., 16 Jan 2024).

6. Extensions: Nonlinear Estimators, Learning, and Scalability

KRONIC has been extended to encompass several advanced directions:

• Data-driven nonlinear Koopman estimators: Nonlinear predictors in the lifted space (e.g., models of the form $\gamma_{i+1} = A\gamma_i + C f_n(\gamma_i)$) capture infinite-time behaviors and complex attractors, outperforming purely linear estimators in accuracy and stability for strongly nonlinear regimes (Wilson, 2022).
• Deep learning–based embeddings: End-to-end architectures train parametric (often neural) embeddings and Koopman operators using multi-step losses, incorporating auxiliary control networks to encode state-dependent and nonlinear control effects, with application to high-dimensional and robotics systems (Shi et al., 2022, Tiwari et al., 2023).
• Kernel and randomized orthogonal decompositions: For high-dimensional systems (e.g., fluid flows, PDEs, or digital twins), scalable formulations using kernel methods with Nyström approximation or randomized SVD yield efficient, convergent reduced-order models, with rigorously characterized error scaling and real-time interpretability (Caldarelli et al., 5 Mar 2024, Bistrian, 5 Aug 2025, Bistrian et al., 5 Sep 2024).
• Network reconstruction: Two-step Koopman-based procedures (dual + main method) enable network topology and local nonlinear dynamics identification in complex sparse networks, drastically reducing the data requirements and enabling inference in modular or distributed settings (Anantharaman et al., 24 Dec 2024).

7. Comparative Assessment and Domain Impact

KRONIC distinguishes itself from direct sparse identification (e.g., SINDy), standard EDMD, and classical linearization-based control via several unique advantages:

Aspect	SINDy/Direct Methods	Koopman/KRONIC Approach
Derivative estimation	Required, sensitive to noise	Avoided via lifting and regression
Handling of chaos/instability	Prone to error, local validity	Works efficiently (unstable, chaotic regimes)
Network inference	Possible, limited for high-dimensionality	Dual/main method supports large, sparse nets
High-dimensional PDEs	Intractable, hand-tuned reduction schemes	Data-efficient, real-time reduced models
Model-based control	Local, typically only LTI, dispatchable	Linear/quadratic/robust optimal in lifted space
Robust safe control	Difficult to robustify globally	Finite-time bounds, dual-loop H∞, CBF fusion

This paradigm enables theory-backed, computationally scalable, and noise-robust control solutions for classes of nonlinear, chaotic, high-dimensional, or networked systems previously inaccessible to automated identification and control synthesis.

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The KRONIC framework, drawing on developments in operator-theoretic system identification, linear and bilinear surrogate modeling, deep learning, control theory, and dimensionality reduction, synthesizes a comprehensive and rigorous route from noisy, nonlinear, and high-dimensional system data to robust, validated, real-time feedback control, with provable stability and safety margins—demonstrated across a broad array of contemporary control engineering and scientific domains (Mauroy et al., 2016, Kaiser et al., 2017, Mauroy et al., 2017, Peitz et al., 2018, Yi et al., 2021, Shi et al., 2022, Wilson, 2022, Zhang et al., 2022, Black et al., 2022, Strässer et al., 2023, Tiwari et al., 2023, Fan et al., 16 Jan 2024, He et al., 16 Jan 2024, Caldarelli et al., 5 Mar 2024, Zhang et al., 31 Mar 2024, Bistrian et al., 5 Sep 2024, Zeng et al., 2 Dec 2024, Anantharaman et al., 24 Dec 2024, Xiong et al., 6 May 2025, Bistrian, 5 Aug 2025).