Willems-Koopman Predictive Control

• Willems-Koopman Predictive Control is a data-driven methodology that combines Willems’ Fundamental Lemma with Koopman operator theory to build finite-dimensional linear predictors for nonlinear systems.
• It constructs lifted observables and employs regression techniques to transform complex nonlinear dynamics into a framework compatible with standard linear MPC.
• Recent advances, including dictionary-free formulations and bilinear realizations, have enhanced its robustness and broadened its applicability in fields like robotics, power grids, and physical systems.

Willems-Koopman Predictive Control (WKPC) is a data-driven model predictive control methodology that uniquely combines the behavioral, Hankel-matrix–based approach of Willems’ Fundamental Lemma for linear time-invariant (LTI) systems with the functional lifting framework provided by the Koopman operator for nonlinear dynamical systems. The integration of these paradigms yields a class of predictive control algorithms that utilize finite-dimensional, approximately linear predictors constructed in a lifted observable space, enabling the application of standard linear MPC techniques to complex nonlinear and high-dimensional systems. WKPC is strictly data-driven: all system identification, predictor synthesis, and controller construction proceed from experimental time-series or simulation data, with no requirement for an explicit parametric model or analytic derivation of the system’s governing equations. Recent advances in dictionary-free lifting, approximate bilinear Koopman realizations, and robust stability theory have expanded the applicability and reliability of WKPC across diverse domains, including complex physical systems, high-performance robotics, and large-scale power grids.

1. Theoretical and Mathematical Foundations

WKPC leverages two foundational principles from the control literature:

(a) Willems’ Fundamental Lemma and Behavioral Prediction.

For an LTI system, every input-output trajectory of finite length can be expressed as a linear combination of columns of a Hankel matrix constructed from a single persistently exciting data record. This property—enforced via block Hankel matrices partitioned into "past" and "future" segments—enables direct trajectory prediction and open-loop control synthesis from empirical data, bypassing explicit system identification.

(b) Koopman Operator Lifting for Nonlinear Dynamics.

The Koopman operator provides a linear (but typically infinite-dimensional) description of nonlinear dynamics by acting on a space of observable functions. Finite-dimensional approximations (typically via the Extended Dynamic Mode Decomposition, EDMD) generate linear time-invariant or bilinear models in a set of chosen observables ("lifting functions"), yielding a surrogate system: $z_{k+1} = A z_k + B u_k, \qquad y_k = C z_k,$ with $z_k = \phi(x_k)$, where $\phi$ is the lifting map constructed from a selected dictionary of observables (Korda et al., 2016, Rezaei et al., 27 Jul 2025).

WKPC combines these approaches by using the lifted data $Z = H_L(z)$ in the Hankel constructions, so that future states, controls, and outputs are enforced to lie in the column span of past lifted trajectories. The resulting finite-horizon optimization problem is formulated entirely in terms of trajectory data and lifted observables, allowing the control of nonlinear systems under constraints by solving a quadratic program similar to that for standard LTI MPC but operating in the (potentially high-dimensional) lifted space (Korda et al., 2016, Lian et al., 2021, Shang et al., 8 Apr 2025).

2. Construction of Koopman Linear Predictors

The central modeling step in WKPC is the construction of a finite-dimensional linear predictor for the nonlinear system:

Lifting Map and Observable Selection

A nonlinear lifting map $\phi: \mathbb{R}^n \rightarrow \mathbb{R}^N$, with $\phi(x) = [\psi_1(x), \ldots, \psi_N(x)]^T$, is selected, where each $\psi_i$ is an observable function. Guidelines for selecting observables include:

• Inclusion of raw state coordinates $x_i$ to ensure the original state is present in the span,
• Addition of nonlinearities present in the dynamics (monomials, radial basis functions, Fourier terms),
• Introduction of functions to capture anticipated constraints or cost terms in MPC,
• Balancing the expressivity of the dictionary (large $N$) with sample complexity and computational burden (Korda et al., 2016, Cibulka et al., 2021).

Approximation of the Koopman Operator

The matrices $A\in\mathbb{R}^{N\times N}$ and $z_k = \phi(x_k)$0 are computed by least-squares regression to minimize the one-step prediction error in the lifted space. Explicitly, given sample data $z_k = \phi(x_k)$1, the optimization problem is

$z_k = \phi(x_k)$2

which admits a closed-form solution via the normal equations or Moore–Penrose pseudoinverse (Korda et al., 2016). Output or state reconstruction is accomplished by solving for $z_k = \phi(x_k)$3 in

$z_k = \phi(x_k)$4

This yields the linear predictor with the structure $z_k = \phi(x_k)$5, $z_k = \phi(x_k)$6 (Korda et al., 2016, Cibulka et al., 2021).

3. Predictive Control Formulation and Algorithmic Anatomy

Given the Koopman-based linear predictor, WKPC formulates the constrained finite-horizon MPC problem as

$z_k = \phi(x_k)$7

Nonlinear constraints and cost terms in the original state-space, $z_k = \phi(x_k)$8, $z_k = \phi(x_k)$9, become linear in the lifted space if $\phi$0 are included in the dictionary of observables: $\phi$1; $\phi$2 (Korda et al., 2016).

Algorithmic Workflow

The standard online control loop follows:

1. Measure current state $\phi$3.
2. Compute lifted state $\phi$4.
3. Solve the quadratic program for optimal control sequence $\phi$5.
4. Apply $\phi$6 to the system (Korda et al., 2016, Cibulka et al., 2021).

A "dense form" (i.e., eliminating the lifted states as optimization variables) yields a QP in the input sequence $\phi$7 of dimension $\phi$8, with all matrices involving $\phi$9 precomputed offline. Computational complexity is similar to linear MPC with the lifted dimension $Z = H_L(z)$0 not affecting online solution time (Korda et al., 2016).

4. Advanced Data-Driven WKPC: Dictionary-Free and Bilinear Formulations

Emerging research recognizes two important extensions of WKPC:

(a) Dictionary-Free WKPC

The Dictionary-Free Koopman MPC (DF-KMPC) paradigm bypasses explicit dictionary selection entirely. Instead, a Hankel matrix–based behavioral approach is used to infer an optimal low-order linear time-invariant predictor directly from data by imposing rank, causality, and Hankel-structure constraints through an alternating projection algorithm. This methodology identifies the best data-driven Koopman realization that fits the trajectory data without specifying $Z = H_L(z)$1, thus alleviating the model-selection bottleneck and reducing the need for iterative retraining when the system equilibrium changes (Shang et al., 8 Apr 2025). The DF-KMPC optimization embeds the behaviorally identified predictor into an MPC formulation over an input-output Hankel span.

(b) Koopman Bilinear Realization

For nonlinear control-affine systems, the standard linear predictor can be generalized to a bilinear form: $Z = H_L(z)$2 with observables $Z = H_L(z)$3 chosen so that this realization is (approximately) valid (Xiong et al., 6 May 2025). The corresponding Hankel-based Fundamental Lemma is extended to this setting, and, crucially, the DeePC optimization incorporates consistency constraints on bilinear terms $Z = H_L(z)$4, yielding a data-driven predictive control scheme robust to model-approximation error and bypassing explicit EDMD system identification.

5. Practical Implementation and Computational Considerations

WKPC entails the following practical steps (Korda et al., 2016, Lian et al., 2021, Xiong et al., 6 May 2025):

• Data Collection: Offline sampling or recording of state and control data with persistently exciting inputs, ensuring identifiability in the lifted space.
• Observable/Lifting Function Choice: Dictionary selection balancing representation fidelity and computational cost; in dictionary-free methods, this selection is circumvented.
• Predictor Identification: Offline computation of the predictor matrices via regression or, in behavioral approaches, via direct Hankel matrix projections.
• Controller Synthesis: Precomputation of QP matrices for the online MPC problem.
• Online Loop: Rapid lifting, QP solution, and actuation, with computational performance often below real-time sampling thresholds (e.g., 0.28 ms for $Z = H_L(z)$5 lift on a spatial PDE, 6.9 ms for $Z = H_L(z)$6 for a DC motor) (Korda et al., 2016, Shang et al., 8 Apr 2025).

High-dimensional systems benefit from the separation of offline and online computation, since only the lifted state for the current measurement needs to be computed online, not the full predictor identification.

6. Properties, Limitations, and Performance

WKPC enables the use of advanced constraints (state, input, output), accommodates nonlinear cost/penalty terms, and scales to high-dimensional nonlinear systems, as demonstrated on benchmark examples such as the forced Van der Pol oscillator, bilinear DC-motor, and Korteweg–de Vries PDE (Korda et al., 2016, Lian et al., 2021). When compared to local-linearization-based MPC and Carleman linearization, the Koopman-lifted predictor achieves lower RMSE and constraint robustness. In vehicle dynamics, WKPC-based MPC outperforms local-linear MPC in aggressive maneuvers where local linearization breaks down (Cibulka et al., 2021).

Limitations include:

• Sensitivity to the choice and richness of observables for finite-dimensional lifts (yet dictionary-free techniques mitigate this bottleneck) (Shang et al., 8 Apr 2025, Rezaei et al., 27 Jul 2025).
• Growth of Hankel matrix dimensions and data demands with the size of the dictionary and horizon, which impacts computation (Rezaei et al., 27 Jul 2025).
• Weakness under non-persistent excitation, measurement noise, and limited data coverage for rare or extreme regimes.
• For finite dictionaries, the embedding is only approximate, although theoretical work now provides robust stability guarantees in the presence of bounded embedding/reconstruction error (Taghieh et al., 17 Mar 2026).

7. Future Directions and Open Research Problems

Key directions for extension and refinement include:

• Systematic methods for minimal observable dictionary selection or neural network–based learning of optimal lifting functions.
• Enhanced robustness to noise and quantification of closed-loop stability under finite-dimensional Koopman approximation error, exploiting proportional error structure for tighter ultimate bounds (Taghieh et al., 17 Mar 2026).
• Scalability improvements via randomized sketching, adaptive online updates, or distributed/decentralized Hankel matrix construction for large networks (Shang et al., 8 Apr 2025, Xiong et al., 6 May 2025).
• Extension to time-varying and stochastic systems, and integration with purely behavioral data-driven frameworks such as Data-enabled Predictive Control (DeePC) (Xiong et al., 6 May 2025, Rezaei et al., 2 Apr 2026).

Comparative studies position WKPC between DeePC (purely LTI data–driven) and model-free adaptive controllers, achieving superior steady-state accuracy and rise time at the cost of higher data requirements and solve times than the most adaptive alternatives (Rezaei et al., 27 Jul 2025).

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Key References:

• "Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control" (Korda et al., 2016)
• "Model Predictive Control of a Vehicle using Koopman Operator" (Cibulka et al., 2021)
• "Dictionary-free Koopman Predictive Control for Autonomous Vehicles in Mixed Traffic" (Shang et al., 8 Apr 2025)
• "Stability Guarantees for Data-Driven Predictive Control of Nonlinear Systems via Approximate Koopman Embeddings" (Taghieh et al., 17 Mar 2026)
• "Comparative Analysis of Data-Driven Predictive Control Strategies" (Rezaei et al., 27 Jul 2025)
• "Koopman based data-driven predictive control" (Lian et al., 2021)
• "Data-Driven Koopman Predictive Control for Frequency Regulation of Power Systems using Black-Box IBRs" (Rezaei et al., 2 Apr 2026)
• "Data-Enabled Predictive Control for Nonlinear Systems Based on a Koopman Bilinear Realization" (Xiong et al., 6 May 2025)