Koopman Operator Theory in Data-Driven Control

• Koopman operator theory is a framework that represents nonlinear dynamics as linear evolution in an observable space, bridging the gap between nonlinear systems and linear control methods.
• Data-driven finite-dimensional Koopman models use neural network-based lifting functions to map measurements into a latent space, facilitating accurate predictions and efficient control strategies.
• In economic MPC applications, Koopman-based surrogates transform complex nonconvex problems into convex quadratic programs, achieving significant speedups and robust performance.

Koopman operator theory provides a framework for representing nonlinear dynamical systems by linear evolution in a (possibly infinite-dimensional) space of observables, thus enabling the application of linear analysis and control techniques to inherently nonlinear systems. In contemporary engineering and control literature, "finite-dimensional Koopman models" typically refer to data-driven constructions, often leveraging deep neural networks, which approximate the infinite-dimensional Koopman operator within a learned embedding. This approach is finding a central role in advanced model predictive control (MPC) and economic MPC (EMPC) applications across domains such as water treatment, process control, carbon capture, and energy systems.

1. Mathematical Foundations of the Koopman Operator

The Koopman operator is a linear (though infinite-dimensional) operator $\mathcal{K}$ acting on the space of observables of a dynamical system. For a discrete-time, possibly nonlinear, system

$x_{k+1} = f(x_k, u_k),$

the Koopman operator acts on a scalar observable $g$ as

$[\mathcal{K}g](x, u) = g(f(x, u)).$

A core insight is that dynamics in the lifted observable space are linear: $g(x_{k+1}, u_k) = [\mathcal{K}g](x_k, u_k).$ However, explicit computation of $\mathcal{K}$ is not tractable in general. Therefore, recent work focuses on finite-dimensional approximations: embedding the dynamics into a latent space where they can be represented as

$z_{k+1} = A z_k + B u_k,$

$\hat{y}_k = C z_k + D u_k,$

with $z_k = \psi_\theta(y_k, d_k)$, where $\psi_\theta$ is a neural network-based encoder that "lifts" measurements and known disturbances (and possibly controls) into a latent space, and $(A, B, C, D)$ are (potentially learned) matrices.

2. Deep Koopman Architectures for Input-Output Modeling

The state-of-the-art employs neural networks to learn both the lifting function $\psi_\theta$ and the output (prediction) maps. For instance, in the DIOKO (Deep Input–Output Koopman) approach for water treatment EMPC (Han et al., 21 May 2024), the following structure is adopted:

• Inputs: only sensor measurements $y_k \in \mathbb{R}^{41}$, control $u_k \in \mathbb{R}^2$, and possibly known disturbances $d_k$ (e.g., inflow or weather).
• Encoder: $\psi_\theta(y_k, d_k) \in \mathbb{R}^{60}$, implemented as an MLP with two hidden layers (width 128, ELU activation).
• Lifted-space dynamics:

$$\psi_{k+1} = A \psi_k + B u_k \quad (A \in \mathbb{R}^{60 \times 60}, B \in \mathbb{R}^{60 \times 2})$$

• Output mapping for economic cost (quadratic-affine form):

$\hat{c}_k = \psi_k^\top Q \psi_k + P \psi_k + b$

where $Q$ is diagonal, $P$ a row vector, $b$ a scalar; all are learned.

The full model can thus be summarized as: $$\begin{cases} \psi_{k+1} = A \psi_k + B u_k \ \hat{c}_k = \psi_k^\top Q \psi_k + P \psi_k + b \end{cases}$$ This parameterization enables accurate prediction of both process outputs and (convex) surrogate economics based strictly on available partial output information, not requiring full state observation.

Training typically employs a penalized loss: $$\mathcal{L} = \mathbb{E}_\mathcal{D} \left[ \sum_{j=k}^{k+T_f} \|\psi_\theta(y_j, d_j) - \psi_{j|k}\|_2^2 + \|c_j - \hat{c}_{j|k}\|_2^2 \right],$$ where the first term enforces adherence to Koopman-invariant structure and the second encourages output (cost) accuracy over a multi-step horizon $T_f$.

3. Convex Economic MPC Formulation via Koopman Surrogates

With the learned Koopman model, the EMPC problem is formulated over a prediction horizon $N$: $$J = \sum_{j=0}^{N}\ \left(\psi_{k+j}^\top Q \psi_{k+j} + P \psi_{k+j} + b\right) +\sum_{j=0}^{N-1} \Delta u_{k+j}^\top R \Delta u_{k+j}$$ subject to lifted linear dynamics, affine feasibility constraints (from process and actuator limits), and box constraints on $u_k$.

The resulting optimization problem,

$$\min_{U} \quad \frac{1}{2} U^\top H U + f^\top U \quad \text{s.t.} \quad G U \leq h$$

where $U$ stacks future controls and all constraint matrices ($E$, $F$, $H$, $f$, $G$, $h$) are constructed from the lifted dynamics and cost parameters, is a convex quadratic program (QP). This allows exploitation of efficient QP solvers (e.g., OSQP, CPLEX) with warm starts and tight tolerances for real-time feasibility.

Crucially, the nonconvexity of the original large-scale EMPC (caused by the underlying plant nonlinearities) is bypassed by the latent linearization in Koopman space, enabling orders-of-magnitude faster resolves: for example, DIOKO-EMPC achieves $\sim 0.034$ s per step vs. $334$ s for the first-principles nonlinear EMPC baseline—over $9,800\times$ speedup (Han et al., 21 May 2024).

4. Model Training, Generalization, and Robustness

Training deep Koopman models relies on extensive datasets from the target process. For the water treatment controller (Han et al., 21 May 2024), $10^5$ samples are collected across different weather regimes and split into train/validation/test sets. All parameters (encoder network $\theta$ and linear/subspace matrices) are trained jointly via stochastic gradient descent on the above loss, with batch size 128, 400 epochs, ADAM optimizer, and $\ell_2$ regularization.

Robustness is assessed by retraining the model under process and measurement noise. The DIOKO variant trained under such conditions ("DIOKO-noisy") achieves similar economic performance and computation times, indicating strong generalization. Even a DIOKO model trained on a restricted regime ("DIOKO-dry," i.e., dry-weather data only) outperforms nonlinear EMPC baselines under previously unseen weather types ("rainy," "stormy"), highlighting the framework’s ability to encapsulate relevant process phenomenology within the learned embedding.

However, the approach is currently deterministic; the inclusion of a stochastic Koopman model is suggested for capturing unmodeled uncertainties and handling unknown disturbances.

5. Comparative Performance and Practical Impact

Simulation across operational regimes evidences pronounced gains compared to both nonlinear-EMPC and tracking-MPC solutions:

• Under "dry" conditions, DIOKO-EMPC yields:
  • Stage cost reduction by 18.1% ($1.582\times 10^7$ to $1.295\times 10^7$)
  • Effluent quality (EQ) improvement by 33.4% ($8.893\times 10^6$ to $5.926\times 10^6$)
  • Solver time reduction from $334.5$ s to $0.0339$ s.

Closed-loop trajectories of cost, EQ, and overall cost index (OCI) remain strictly below those of nonlinear baselines and always respect feasibility constraints.

Robust performance under measurement/process noise and extrapolation to unseen regimes is consistently observed, evidencing that the Koopman-based surrogate—when properly trained—retains high-fidelity input-output matching relevant for economic optimization, not merely open-loop prediction.

6. Limitations and Future Directions

Current Koopman architectures for EMPC (exemplar: DIOKO) assume deterministic dynamics and are fully reliant on the quality and coverage of the training data. Model generalization is effective within the operational envelope represented in the data but may degrade with systemic nonstationarities or rare events. Integration with physics-based models (“hybridization”) is proposed to improve extrapolative and data efficiency, especially where first-principles models or partial state knowledge is available.

Further, while QP-based solves are orders of magnitude faster than their nonlinear EMPC counterparts, overall resource requirements for model training (network optimization, data acquisition) remain substantial.

Adapting stochastic Koopman formulations, robust constraint handling, and incorporating domain knowledge are viewed as promising directions for improving data efficiency and resiliency against plant/model mismatch. Scaling to high-dimensional plants and time-varying dynamics has been demonstrated in recent studies, but formal analysis and systematic benchmarking remain subjects of ongoing research.

7. Broader Context and Ongoing Developments

The Koopman operator perspective is now a foundational element in data-driven control for complex nonlinear systems. Recent literature extends these methods to reinforcement learning-driven policy refinement, physics-informed training, large-scale partial observability, and industrial-scale EMPC deployment (Mayfrank et al., 21 Mar 2024, Mayfrank et al., 24 Mar 2025, Dony, 12 May 2025, Mayfrank et al., 6 Nov 2025, Valábek et al., 6 Nov 2025, Han et al., 9 Apr 2025). Key methodological themes include:

• End-to-end joint training of encoders and linear predictors for task-driven observability.
• Economic MPC problem casting directly in latent Koopman coordinates for convexity.
• Neuro-symbolic and hybrid approaches for embedding physical constraints and prior knowledge.
• Demonstrated real-time feasibility in high-dimensional systems with only partial state measurement.

The main practical impact is the effective transformation of nonlinear, large-scale optimal control problems into tractable convex surrogates—achieving not only speed but also higher reliability and enforceability of system constraints in real-world operation.