---
title: Willems-Koopman Predictive Control
url: https://www.emergentmind.com/topics/willems-koopman-predictive-control
type: topic
---

# Willems-Koopman Predictive Control

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 [1611.03537, 2507.20098].

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 [1611.03537, 2102.05122, 2504.06240].

## 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 [1611.03537, 2103.04978].

### Approximation of the Koopman Operator

The matrices $A\in\mathbb{R}^{N\times N}$ and $B\in\mathbb{R}^{N\times m}$ are computed by least-squares regression to minimize the one-step prediction error in the lifted space. Explicitly, given sample data $\{(x_j, u_j, x_j^+)\}_{j=1}^K$, the optimization problem is
\[
\min_{A, B}\ \sum_{j=1}^K \| \psi(x_j^+) - A \psi(x_j) - B u_j \|_2^2,
\]
which admits a closed-form solution via the normal equations or Moore–Penrose pseudoinverse [1611.03537]. Output or state reconstruction is accomplished by solving for $C$ in
\[
\min_C \sum_{j=1}^K \| x_j - C \psi(x_j) \|_2^2.
\]
This yields the linear predictor with the structure $z_{k+1} = A z_k + B u_k$, $\hat y_k = C z_k$ [1611.03537, 2103.04978].

## 3. Predictive Control Formulation and Algorithmic Anatomy

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

\[
\begin{align*}
\min_{\{u_{k+i}\}_{i=0}^{N_p-1}} \ & \sum_{i=0}^{N_p-1} (z_{k+i}^T Q z_{k+i} + u_{k+i}^T R u_{k+i}) + z_{k+N_p}^T Q_f z_{k+N_p} \\
\text{s.t.} \quad & z_{k+i+1} = A z_{k+i} + B u_{k+i},\\
& E_i z_{k+i} + F_i u_{k+i} \leq b_i, \\
& z_k = \phi(x_k).
\end{align*}
\]

Nonlinear constraints and cost terms in the original state-space, $g(x) \le 0$, $\ell(x)$, become linear in the lifted space if $g,\ell$ are included in the dictionary of observables: $g(x_k) = L\phi(x_k) \le 0$; $\ell(x) = L'\phi(x)$ [1611.03537].

### Algorithmic Workflow

The standard online control loop follows:
1. Measure current state $x_k$.
2. Compute lifted state $z_k = \phi(x_k)$.
3. Solve the quadratic program for optimal control sequence $\{u_{k+i}^\star\}$.
4. Apply $u_k = u_{k}^\star$ to the system [1611.03537, 2103.04978].

A "dense form" (i.e., eliminating the lifted states as optimization variables) yields a QP in the input sequence $U$ of dimension $mN_p$, with all matrices involving $A,B,Q,R$ precomputed offline. Computational complexity is similar to linear MPC with the lifted dimension $N$ not affecting online solution time [1611.03537].

## 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 $\{\psi_i\}$, thus alleviating the model-selection bottleneck and reducing the need for iterative retraining when the system equilibrium changes [2504.06240]. 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:
\[
z_{k+1} = A z_k + \sum_{i=1}^m u_{i,k} B_i z_k
\]
with observables $\Psi$ chosen so that this realization is (approximately) valid [2505.03346]. The corresponding Hankel-based Fundamental Lemma is extended to this setting, and, crucially, the DeePC optimization incorporates consistency constraints on bilinear terms $v_k = z_k \otimes u_k$, 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 [1611.03537, 2102.05122, 2505.03346]:
- **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 $N=385$ lift on a spatial PDE, 6.9 ms for $N=103$ for a DC motor) [1611.03537, 2504.06240].

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 [1611.03537, 2102.05122]. 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 [2103.04978].

Limitations include:
- Sensitivity to the choice and richness of observables for finite-dimensional lifts (yet dictionary-free techniques mitigate this bottleneck) [2504.06240, 2507.20098].
- Growth of Hankel matrix dimensions and data demands with the size of the dictionary and horizon, which impacts computation [2507.20098].
- 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 [2603.17089].

## 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 [2603.17089].
- Scalability improvements via randomized sketching, adaptive online updates, or distributed/decentralized Hankel matrix construction for large networks [2504.06240, 2505.03346].
- Extension to time-varying and stochastic systems, and integration with purely behavioral data-driven frameworks such as Data-enabled Predictive Control (DeePC) [2505.03346, 2604.02251].

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 [2507.20098].

---

**Key References:**
- "Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control" [1611.03537]
- "Model Predictive Control of a Vehicle using Koopman Operator" [2103.04978]
- "Dictionary-free Koopman Predictive Control for Autonomous Vehicles in Mixed Traffic" [2504.06240]
- "Stability Guarantees for Data-Driven Predictive Control of Nonlinear Systems via Approximate Koopman Embeddings" [2603.17089]
- "Comparative Analysis of Data-Driven Predictive Control Strategies" [2507.20098]
- "Koopman based data-driven predictive control" [2102.05122]
- "Data-Driven Koopman Predictive Control for Frequency Regulation of Power Systems using Black-Box IBRs" [2604.02251]
- "Data-Enabled Predictive Control for Nonlinear Systems Based on a Koopman Bilinear Realization" [2505.03346]

Source: https://www.emergentmind.com/topics/willems-koopman-predictive-control