---
title: Koopman Operator for Controlled Systems
url: https://www.emergentmind.com/topics/extension-of-the-koopman-operator-to-controlled-systems
type: topic
---

# Koopman Operator for Controlled Systems

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 \( H \) [1602.07647].

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 [1602.07647].

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 [1602.07647, 1611.03537].

## 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 [1602.07647]. 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 [1611.03537]. 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 [1602.07647].

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 [1602.07647]:

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 [1602.07647].

## 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 [1602.07647].

## 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 [1602.07647, 1611.03537].

Source: https://www.emergentmind.com/topics/extension-of-the-koopman-operator-to-controlled-systems