Koopman Eigenfunctions for Control

• Koopman eigenfunctions for control are a method that transforms nonlinear dynamics with inputs into linear systems using observable functions.
• They utilize lifting techniques and data-driven approaches like EDMD to approximate infinite-dimensional operators in a finite-dimensional setting.
• Applications span epidemic modeling, fluid flow, and power grid monitoring, while challenges remain in observable selection and computational validation.

Koopman eigenfunctions for control constitute a foundational methodology for embedding nonlinear dynamical systems with control or input signals into lifted, typically infinite-dimensional, linear frameworks. By recasting the evolution of nonlinear dynamics as linear dynamics in the space of appropriately chosen observables—i.e., Koopman eigenfunctions—the approach enables the application of linear systems theory to nonlinear prediction, estimation, and, critically, control design. The following sections provide a detailed exposition of theoretical foundations, mathematical structures, methodological advances, application exemplars, implications for control, and the limitations and challenges in practical deployment.

1. Theoretical Extension: Koopman Operator with Inputs and Control

The classical Koopman operator defines a linear evolution on the space of scalar-valued observables for autonomous systems of the form $x_{k+1} = f(x_k)$. For such systems, the operator $K$ acts on observables $g$ as $K g(x) = g(f(x))$. To incorporate control, the dynamics generalize to $x_{k+1} = f(x_k, u_k)$, where $u_k$ encodes the control action or exogenous forcing.

The generalized Koopman operator for systems with inputs operates on augmented observables $g(x, u)$ as:

$K g(x, u) := g(f(x, u), *),$

where $*$ denotes a choice reflecting the evolution of the input:

• $* = u$ when the control has its own dynamics (e.g., feedback or evolving actuator state),
• $* = 0$ when $u$ acts as an instantaneous exogenous disturbance.

This generalized operator projects the nonlinear, controlled system into a lifted space where the evolution remains linear in the observable functions, provided an appropriate function library.

The connection to Dynamic Mode Decomposition with control (DMDc) is made explicit: DMDc builds a finite-dimensional approximation by collecting state and input data $(Y, U)$ and constructing a non-square operator $[A \; B]$ such that:

$$Z \approx AY + BU, \qquad [A~B] = Z \Omega^{\dagger},$$

with $\Omega = [Y; U]$. The Koopman operator with inputs and control (KIC) recovers this structure when the observable space is restricted to state observables, thus subsuming DMDc (Proctor et al., 2016).

2. Mathematical Formulation and Spectral Properties

Given a nonlinear control system $x_{k+1} = f(x_k, u_k)$, one considers a suitable Hilbert space $H$ of scalar observables $g(x, u)$. The generalized Koopman operator $K$ defines the evolution:

$K g(x, u) = g(f(x, u), *),$

with eigenfunctions $$\varphi_j:\mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{C}$$ and eigenvalues $\lambda_j$ satisfying:

$$K \varphi_j(x, u) = \lambda_j \varphi_j(x, u), \quad j=1,2,\dots$$

These eigenfunctions serve as intrinsic coordinates along which the dynamics are (exactly, or nearly) linear.

Any observable $g(x, u)$ can be expanded as:

$g(x, u) = \sum_j \varphi_j(x, u) v_j,$

where $v_j$ are associated Koopman modes. For linear systems $x_{k+1} = Ax_k + Bu_k$, the full Koopman operator on the [state; input] vector is simply a block matrix, and the spectral decomposition yields the system’s modes and eigenfunctions directly. The operator can be further restricted to subspaces—e.g., state-only observables—by partitioning $H$ into $H_x$, $H_u$, and $H_{xu}$, and writing spectral expansions accordingly.

3. Role of Observables and Library Selection

A critical issue in constructing Koopman eigenfunctions for control is the choice of observables. For the approach to yield a closed, linear (finite or infinite-dimensional) model, the selected observables must span an invariant subspace under the dynamics:

• For linear or bilinear systems, monomials or specific nonlinear observables may suffice.
• For more complex systems, including observables that do not “close”—i.e., whose evolution depends on terms outside the span—produces inaccurate or ill-posed Koopman approximations.

The SIR model with vaccination illustrates this point: augmenting the observable library with the nonlinear mixed term $SI$ does not close under the dynamics. Restricting output observables to the basic states $[S, I, R]$ yields an operator that correctly captures the system’s evolution (Proctor et al., 2016).

This outcome demonstrates that improper selection of observables leads to operator predictions that cannot reconstruct the true system, emphasizing the need for careful calibration of the basis.

4. Data-driven Computation and Model Construction

Koopman eigenfunctions for control are rarely available in closed form for general nonlinear systems. Thus, data-driven, regression-based methods—such as Extended Dynamic Mode Decomposition (EDMD) and sparse regression—are central:

1. Collect time-series data $(x_k, u_k, x_{k+1})$ from the controlled system.
2. Select a dictionary of observables (monomials, radial basis functions, mixed terms) depending on application constraints and prior knowledge.
3. Assemble the lifted snapshot data matrix and solve for the best-fit linear operator (either via least squares or with regularization).
4. Extract leading eigenfunctions and validate their suitability as intrinsic coordinates.

Crucially, closure validation (i.e., ensuring approximate invariance under the lifted dynamics) is necessary, as finite-dimensional approximations cannot always represent the infinite-dimensional operator exactly.

5. Application Examples

The paper provides several exemplary applications:

• Nonlinear SIR Epidemiological Model: Demonstrates both the necessity and power of appropriate observable selection. Including only the primary states as observables facilitates successful prediction of vaccination-driven epidemic dynamics.
• Linear Systems with Random Disturbance, Feedback, or Autonomous Inputs: KIC accurately identifies underlying system dynamics and distinguishes between the effects of state evolution and exogenous inputs.
• Systems with Quadratic Nonlinearity: Strategic inclusion of quadratic observables produces a closed, finite-dimensional Koopman, enabling precise linear modeling.
• Systems with Periodic Trajectories: The expansion of the Koopman operator in terms of Fourier or Z-transforms for systems where both state and input are periodic demonstrates the adaptability of the framework.

These examples span both synthetic and real-world-relevant scenarios, illustrating the breadth of control problems addressable via Koopman eigenfunctions (Proctor et al., 2016).

6. Implications for Control System Design

Integrating Koopman eigenfunctions into control offers several system-theoretic and practical benefits:

• Unified Linear Paradigm: Model predictive control (MPC), LQR, and Riccati-based schemes, which are analytically and computationally efficient but typically limited to linear systems, can now be extended to high-dimensional nonlinear systems—provided Koopman-invariant subspaces are well-constructed.
• Input-output Characterization: Data-driven Koopman models enable lifted representations that explicitly encode the impact of actuation and disturbances.
• System Identification and Model Reduction: Methods such as DMDc are subsumed as special cases of KIC, facilitating more expressive and accurate low-dimensional approximations of input–output systems.
• Forecasting and Feedback: Once a linear operator is identified in the lifted space, standard control theory is immediately applicable for prediction and feedback control, even for originally nonlinear and high-dimensional systems.

In domains such as fluid flow control, epidemic management, power grid monitoring, and neural engineering, this operator-theoretic approach enables control design directly from data—circumventing reliance on high-fidelity first-principles models.

7. Limitations and Practical Challenges

Despite its promise, the Koopman eigenfunction approach for control presents several challenges:

• Closure Problem: The finite set of observables may not yield a closed Koopman-invariant subspace, introducing error in model predictions.
• Observable Selection: No universal recipe exists; improper choice limits approximation quality and may obscure key nonlinear phenomena.
• Computational Complexity: High-dimensional or highly nonlinear systems can require large or specialized observables sets, which may present computational and interpretational barriers.
• Validation: Prediction and control performance must be empirically validated, as theoretical guarantees are generally asymptotic or local.

These considerations necessitate a problem-specific and iterative approach to observable library selection, error quantification, and model deployment.

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In summary, the generalization of the Koopman operator to include inputs and control (KIC) provides a principled and mathematically rigorous pathway to lift nonlinear, forced dynamical systems into a framework amenable to linear system analysis and control synthesis. The successful application of Koopman eigenfunctions in control hinges on the judicious selection of invariant observable subspaces and validates through concrete demonstrations that, with appropriate architecture, complex input–output nonlinear systems can be effectively modeled, predicted, and controlled using operator-theoretic strategies (Proctor et al., 2016).