Koopman Operator for Controlled Systems

• Extension of the Koopman operator to controlled systems is a framework that lifts nonlinear, input-driven dynamics to an infinite-dimensional space of observables.
• Finite-dimensional approximations using data-driven lifting yield A and B matrices that bridge spectral analysis with techniques like DMDc and model predictive control.
• The method facilitates controller synthesis for diverse applications, as demonstrated by its successful implementation in nonlinear models such as the SIR epidemiological model.

The Koopman operator provides a linear, infinite-dimensional framework for analyzing nonlinear dynamical systems by lifting the system's evolution to a space of observables. Extending Koopman theory to controlled systems—where external inputs and actuation play a central role—requires precise definitions of operator action, observable spaces, and several alternative generalizations to address the specific structure of control and feedback. This extension underpins many data-driven modeling, estimation, and control synthesis algorithms, enabling the application of linear control tools to nonlinear, input-driven dynamical systems.

1. Operator-Theoretic Generalization: Definition and Structure

The foundational generalization of the Koopman operator to controlled systems is based on discrete-time input-driven dynamics of the form

$$x_{k+1} = F(x_k, u_k), \quad x_k \in \mathbb{R}^{n_x}, \ u_k \in \mathbb{R}^{n_u}.$$

The appropriate observable space is the Hilbert space $H=L^2(M\times N)$ of real-valued functions $g:M\times N\to\mathbb{R}$, where $M, N$ are the state and input manifolds. The controlled Koopman operator $\mathcal{K}$ acts via

$\mathcal{K}g(x,u) := g(F(x,u), \star),$

where the placeholder $\star$ encodes how future inputs are treated: if inputs evolve according to exogenous or endogenous dynamics $u_{k+1}=H(u_k)$, set $\star=u_{k+1}$; for exogenous or impulse-like inputs, set $\star=0$. In all cases, $\mathcal{K}$ remains a linear map on $H$ (Proctor et al., 2016).

The spectrum and eigenfunction apparatus is extended directly:

• Eigenfunctions $\varphi_j \in H$ and eigenvalues $\lambda_j$ satisfy $$\mathcal{K}\varphi_j(x,u) = \lambda_j \varphi_j(x,u)$$.
• Any observable $g(x,u)$ in the span of the $\varphi_j$ admits an infinite expansion with Koopman modes $v_j \in \mathbb{R}^{n_y}$, and future outputs propagate linearly via spectral dynamics.

2. Lifting, Function Spaces, and Data-Driven Approximation

For practical modeling, finite-dimensional approximations are essential. A finite dictionary $\{\psi_j(x,u)\}_{j=1}^N$ is chosen to “lift” $(x, u)$ into a (potentially high-dimensional) feature space. Observables may include monomials, kernels, or physics-inspired functions. This finite lifting induces an approximate, closed, finite-dimensional action for $\mathcal{K}$ if the dictionary is rich and closed under the system's dynamics (Proctor et al., 2016).

Given snapshot data $\{ (x_k, u_k, x_{k+1}, u_{k+1}) \}$, construct data matrices capturing lifted observables and input features. A least-squares procedure then yields finite $[A \ B]$ matrices:

$[A \ B] = \Delta \Omega^{\dagger},$

where $\Omega, \Delta$ are data matrices of current and future lifted observables. This approach is the basis of Dynamic Mode Decomposition with control (DMDc), which is shown to be a finite-dimensional, Galerkin-type projection of the controlled Koopman operator (Proctor et al., 2016, Korda et al., 2016).

3. Connections to DMDc, Linear Predictors, and Model Predictive Control

The finite-dimensional surrogate

$y_{k+1} \approx A y_k + B \gamma_k,$

where $y_k, \gamma_k$ are lifted outputs and inputs, respectively, unifies the operator-theoretic view with DMDc (Proctor et al., 2016). This predictor framework is foundational for integrating Koopman models with model predictive control (MPC). The full workflow for synthesis involves:

1. Selection of a dictionary (monomials, RBFs, etc.).
2. Acquisition or simulation of snapshot data.
3. Numerical estimation of lifted system matrices.
4. (Optional) Calculation of Koopman modes/eigenvalues for modal analysis.
5. Synthesis of linear controllers (LQR, MPC) in the lifted space.

The convex optimization structure of MPC is retained, with state and input constraints and quadratic/loss objectives incorporated via the lifted coordinates. Nonlinear constraints can be handled by including additional observables representing the constraints in the dictionary (Korda et al., 2016). This approach achieves computational complexity comparable to that of linear systems of the same dimension.

4. Spectral and Modal Analysis in the Presence of Control

By extending the eigenfunction/eigenvalue framework, future outputs and system behavior are described by spectral decompositions involving the controlled Koopman operator. The modal expansion is formalized as

$$g(x,u) = \sum_{j=1}^{\infty} \varphi_j(x,u) v_j, \quad \mathcal{K}^k g(x_0,u_0) = \sum_{j=1}^{\infty} \lambda_j^k \varphi_j(x_0,u_0) v_j.$$

The presence of inputs alters the spectral structure and can admit input–output mappings not accessible in the autonomous setting. However, the infinite-dimensional linearity and closure of the Koopman operator are preserved (Proctor et al., 2016).

Function-space considerations determine the existence and structure of the eigenbasis, which is assured under mild compactness, smoothness, or measure-preserving dynamics. These spectral formulations underpin Koopman model-based system identification and stabilization algorithms.

5. Algorithmic Implementation: Koopman with Inputs and Control (KIC)

An explicit data-driven algorithm for estimating the controlled Koopman operator is as follows (Proctor et al., 2016):

1. Record time series $\{x_k, u_k\}$ and compute next-step data $\{x_{k+1}, u_{k+1}\}$.
2. Compute chosen observables on all inputs and outputs, forming lifted data vectors for each.
3. Assemble data matrices combining current and next-step outputs and inputs.
4. Compute the Koopman gain matrix as a block matrix, estimate system matrices $A, B$ by least-squares.
5. (Optional) Compute modal decompositions/eigenpairs.
6. Use the fitted model for prediction, state estimation, or control synthesis.

A significant insight is that when observables are inadequately chosen (e.g., SI in SIR without including higher-order terms), the model fails to reconstruct the true dynamics due to non-closure. Accurate Koopman surrogates require the output observables to be restricted to a span compatible with the lift (Proctor et al., 2016).

6. Case Study: Nonlinear SIR Model with Actuation

The approach is validated on a nonlinear SIR (Susceptible-Infectious-Recovered) epidemiological model subject to time-varying vaccination input. After discretization, lifting, and data-driven estimation, the Koopman with Inputs and Control (KIC) operator:

• Accurately tracks the true S, I, R trajectories under exogenous vaccination inputs, given an appropriate input‐output selection and dictionary.
• Demonstrates the necessity of including appropriate nonlinear cross-terms when output observables are more complex than the input lift supports.
• Illustrates that KIC yields interpretable A, B matrices, offering a linear-in-the-lift input–output model directly amenable to analysis and control (Proctor et al., 2016).

7. Theoretical and Practical Implications

Extending the Koopman operator to controlled systems provides:

• A linear input–output framework for otherwise nonlinear forced dynamics.
• Rigorous generalization and formal connection to DMDc, validating the widespread data-driven lifting and regression approach.
• Explicit constraints on function/dictionary selection for closure and prediction accuracy.
• Robust, modular workflows for controller synthesis (LQR, MPC) around nonlinear dynamics, applicable to a diverse range of natural and engineered systems.

Moreover, the technique provides a mathematical foundation for data-driven, equation-free control of complex actuated systems, bridging operator theory with practical, scalable engineering workflows (Proctor et al., 2016, Korda et al., 2016).