Data-enabled Predictive Control (DeePC)

• DeePC is a data-driven framework that uses empirical Hankel matrices and behavioral systems theory to predict system trajectories and compute optimal control commands.
• It employs regularization techniques and slack variables to enhance robustness against noise, nonlinearity, and practical constraints in control applications.
• Scalability and computational tractability are achieved through strategies like PCA, adaptive sampling, and online recursive updates, enabling real-time implementation.

Data-enabled Predictive Control (DeePC) is a data-driven, nonparametric predictive control framework that synthesizes optimal control policies solely from input-output measurement data, circumventing explicit model identification. DeePC is founded on behavioral systems theory and the Willems Fundamental Lemma, representing all system trajectories compatible with collected data in terms of empirical Hankel matrices. This approach has led to rigorous connections with model predictive control (MPC), distributionally robust control, and reinforcement learning–augmented tuning strategies. Recent research has substantially advanced DeePC to address practical issues including computational tractability, robustness to noise and nonlinearity, and scalability to real-time or nonlinear settings.

1. Fundamental Formulation and Theoretical Basis

DeePC operates by constructing block-Hankel matrices from single or multiple recorded I/O trajectories of the unknown system. For a discrete-time LTI system

$$x(t+1) = A\,x(t) + B\,u(t), \quad y(t) = C\,x(t) + D\,u(t),$$

data sequences are partitioned into "past" (initialization) and "future" (prediction) windows. If the input sequence $u^d$ is persistently exciting, Willems' Fundamental Lemma guarantees that any trajectory of length $L=T_{\rm ini}+N$ can be written as a linear combination of columns of the Hankel matrices: $$\left[ U_p \; Y_p \; U_f \; Y_f \right] g = \left[ u_{\mathrm{ini}} ; y_{\mathrm{ini}} ; u ; y \right]$$ where $g$ is a free coefficient vector. The core DeePC finite-horizon optimization thus searches for future control inputs $u$ and outputs $y$ by solving

$$\begin{aligned} &\min_{g,u,y} \sum_{k=0}^{N-1} (\|y_k - r_k\|_Q^2 + \|u_k\|_R^2 ) \ &\text{s.t.} \quad \begin{bmatrix} U_p & Y_p & U_f & Y_f \end{bmatrix} g = \begin{bmatrix} u_{\mathrm{ini}}; y_{\mathrm{ini}}; u; y \end{bmatrix}, \ &\quad u_k \in \mathcal{U},\; y_k \in \mathcal{Y}, \quad \forall k = 0, ..., N-1, \end{aligned}$$

where $Q \succ 0$, $R \succ 0$ are stage cost weights and $\mathcal{U}, \mathcal{Y}$ encode input/output constraints. Once $g^*$ is computed, the control action $u_0$ is applied, and the process recedes with updated measurements (Coulson et al., 2018).

Remarkably, for deterministic noiseless LTI systems with sufficiently rich data, DeePC is provably equivalent to state-space MPC in both feasible set and closed-loop performance (Coulson et al., 2018, Fiedler et al., 2020).

2. Noise, Robustness, and Regularization

In realistic scenarios with measurement noise or nonlinearities, direct enforcement of the Hankel equality constraint often renders the problem infeasible or unreliable. Regularized DeePC introduces slack variables and regularization penalties to promote feasibility and robustness: $$\begin{aligned} \min_{g, u, y, \sigma_y} \; &\|y-r\|_Q^2 + \|u\|_R^2 + \lambda_g \|g\|_1 + \lambda_y \|\sigma_y\|_1 \ \text{s.t.}\quad & [ U_p\; Y_p\; U_f\; Y_f ]\,g = [ u_{\mathrm{ini}}; y_{\mathrm{ini}} + \sigma_y; u; y ], \end{aligned}$$ with penalty weights $\lambda_g, \lambda_y$ selected by cross-validation or heuristic rules (Coulson et al., 2018, Coulson et al., 2019). Distributionally robust DeePC further formalizes this by minimizing a worst-case expectation over a Wasserstein ambiguity set around the empirical data distribution, yielding closed-form regularization interpretations and finite-sample performance guarantees (Coulson et al., 2019, Coulson et al., 2020). This robustification ensures out-of-sample reliability, especially in high-noise settings or for safety-critical applications.

3. Computational Tractability and Complexity Reduction

The computational complexity of DeePC is linked to the number of columns in the Hankel matrix, which increases with the volume and granularity of data. For large $T$, this leads to a high-dimensional quadratic program with decision variable $g \in \R^{T-T_{\rm ini}-N+1}$. Several strategies target complexity reduction:

• Principal Component Analysis (PCA): Dimension reduction via SVD-based truncation of the Hankel (or concatenated constraint) matrix, solving the control problem in a lower-dimensional subspace while retaining predictive accuracy (Zhang et al., 2022).
• Contextual or Adaptive Sampling: Dynamic selection of a subset of data most relevant to the current "context" (i.e., similarity to recent I/O history), drastically reducing QP size without significant performance loss (Beerwerth et al., 31 Mar 2025).
• Recursive and Online Hankel Updates: Efficient recursive SVD or rank-1 updates allow DeePC to absorb new informative data online, focusing on columns that increase the minimum nonzero singular value of the data matrix. This maintains persistency of excitation and adaptivity to time-varying systems while keeping computation tractable (Vahidi-Moghaddam et al., 2024).
• Learning-Based Approximations: Offline-trained, size-invariant convex programming surrogates for the data likelihood scoring function decouple computational cost from data size, speeding up real-time implementation (Zhou et al., 2024).
• Projection Methods and Bilevel Reformulations: Approaches that project onto least-squares identified subspaces or eliminate regularization weight tuning (e.g., OP DeePC, BL DeePC, IV DeePC) streamline tuning and reduce solve times for noisy or biased data (Koch et al., 2023, Fiedler et al., 2020).

4. Extensions for Nonlinear, Nonstationary, and Stochastic Systems

While canonical DeePC is theoretically sound for LTI systems, recent developments have broadened applicability:

• Nonlinear DeePC: Koopman subspace lifting, basis function transformations (e.g., via Michaelis–Menten/Hill functions in biochemical systems), and neural-feature liftings enable DeePC to leverage nonlinearities implicitly or explicitly (Perreault et al., 4 Jan 2026, Lazar, 2024, Zhang et al., 2024). Neural DeePC, for example, uses deep networks to generate a nonlinear basis and either (a) combines DeePC with online optimization over neural features, or (b) directly learns the DeePC operator for rapid inference.
• Gain Scheduled DeePC: For regime-dependent nonlinear systems, DeePC can switch among multiple local Hankel matrices, each indexed by measurable scheduling variables (e.g., speed, load), thus "gain scheduling" DeePC for performance across disparate operating points (Guerrero et al., 30 Sep 2025).
• Distributionally Robust/Chance Constrained DeePC: DeePC is extended with data-driven ambiguity sets and chance constraints, enforcing probabilistic guarantees on constraint satisfaction and robust tracking under data uncertainty (Coulson et al., 2019, Coulson et al., 2020, Huang et al., 2019).
• Reinforcement Learning–Augmented DeePC: Offline RL agents (e.g., tabular SARSA) are trained to select optimal regularization hyperparameters online based on system performance, balancing control effort and noise robustness (Wang et al., 30 May 2025).

These methods collectively address practical challenges in process industry, grid-connected converters, biological regulation, and mobile robotics (Perreault et al., 4 Jan 2026, Zieglmeier et al., 29 Oct 2025, Beerwerth et al., 31 Mar 2025).

5. Application Domains and Case Studies

DeePC has been validated in diverse domains, highlighting its versatility:

• Aerial and Marine Vehicles: DeePC has been deployed for guidance and trajectory control of quadcopters (Coulson et al., 2019), marine vessels, and autonomous underwater vehicles, achieving superior robustness to ocean currents and model mismatch compared to classical PI/PID control (Zieglmeier et al., 29 Oct 2025).
• Power Systems: Centralized and decentralized DeePC architectures have been used for oscillation damping in VSC-HVDC grids, even under measurement noise, nonlinear loads, and communication delays (Huang et al., 2019, Huang et al., 2019). Extension to robust and disturbance feedback DeePC enables resilience to inter-area disturbances.
• Energy Storage and Battery Charging: DeePC has been applied to fast-charging protocols for lithium-ion batteries, using PCA-based Hankel dimension reduction for real-time feasibility and enforcing temperature and voltage safety constraints (Zhang et al., 2022).
• Biological Systems: DeePC, with nonlinear basis lifting, has achieved competitive sample efficiency and closed-loop control in synthetic gene-expression systems, outperforming deep RL in the data-scarce regime (Perreault et al., 4 Jan 2026).

Performance metrics typically include tracking RMSE, constraint violation rates, average computation time per step, and, where applicable, energy or resource utilization.

6. Current Limitations, Guidelines, and Future Research

While DeePC presents a unifying, nonparametric MPC paradigm, several limitations persist:

• Scalability: The online QP's size scales with the richness of the data; while mitigation techniques exist (PCA, contextual sampling, neural surrogates), there remains a trade-off between predictive accuracy and computational load (Zhang et al., 2022, Beerwerth et al., 31 Mar 2025, Zhou et al., 2024).
• Noise Sensitivity and Tuning: Regularization weights are crucial for robustness yet often require careful tuning. RL-augmented tuning or noise-informed bilevel reformulations address some challenges, but further automation is desirable (Wang et al., 30 May 2025, Koch et al., 2023).
• Nonlinear/Nonstationary Systems: While lifted, gain-scheduled, and neural DeePC approaches extend coverage, no formal global guarantees exist for general nonlinear systems. Empirical results suggest strong performance, but theoretical guarantees are limited (Lazar, 2024, Guerrero et al., 30 Sep 2025).
• Data Requirements: DeePC requires persistently exciting data spanning the trajectory subspace of interest; insufficient data may limit controllability or prediction fidelity.
• Implementation Details: Stability proofs for the closed-loop DeePC-MPC system in the regularized/noisy/nonlinear setting are limited; in practice, empirical performance has been robust, but further theoretical development is warranted (Zhang et al., 2024, Zhou et al., 2024).

Best-practice guidelines compiled across studies include careful selection of horizon lengths, regularization weights via cross-validation or RL, use of PCA or SVD where computational scaling is critical, and incorporating problem-domain prior structure (e.g., nonlinear basis lifting) wherever applicable.

7. Relationship to Other Data-Driven Predictive Control Methods

DeePC is closely connected to Subspace Predictive Control (SPC) and convex multi-step ARX/MPC approaches. In the deterministic LTI case, DeePC and SPC/MPC are formally equivalent: DeePC’s implicit nonparametric predictor can be replaced by a multi-step least-squares fit, achieving identical closed-loop trajectories at lower computational cost (Fiedler et al., 2020). In stochastic or regularized settings, the equivalence is lost except in special cases, and DeePC retains the advantage of not requiring explicit identification or model structure. Several DeePC variants also bridge to robust and distributionally robust model predictive control via principled regularization (Coulson et al., 2019, Coulson et al., 2020). Decentralized and hierarchical DeePC architectures have been proposed for large-scale interconnected systems (Huang et al., 2019).

---

In summary, Data-enabled Predictive Control unifies system identification, estimation, and trajectory optimization in a single data-driven formulation. It is grounded in behavioral systems theory and has catalyzed cross-fertilization with robust and nonlinear control, model-free learning, and real-time optimization. Continued research is extending its applicability to nonlinear, nonstationary, and resource-constrained domains, with a focus on computational tractability, robust performance, and theoretical guarantees.