Data-Enabled Economic Predictive Control

• DE-EPC is a data-driven predictive control paradigm that optimizes economic performance using input/output data without relying on explicit state-space models.
• It leverages Willems’ Fundamental Lemma, neural-lifting maps, and robust optimization to provide closed-loop performance guarantees and ensure constraint satisfaction.
• Practical implementations in industrial systems such as water treatment and battery storage demonstrate its ability to balance economic objectives with operational constraints.

Data-Enabled Economic Predictive Control (DE-EPC) is a family of methods that optimize economic performance of dynamic systems using model predictive control (MPC) formulations derived solely from input/output data, rather than explicit state-space or first-principles models. This paradigm leverages recent advances in data-driven control theory, notably Willems’ Fundamental Lemma, deep neural surrogate modeling, and robust optimization, to deliver closed-loop performance guarantees and constraint satisfaction for linear, nonlinear, and stochastic systems. DE-EPC unifies the objectives of minimizing operational cost (including non-quadratic, non-positive-definite criteria) with rigorous data-based prediction, and is applicable to a wide range of industrial and infrastructure systems.

1. Theoretical Foundations and Problem Formulation

The architecture of DE-EPC is grounded in the theoretical results of data-driven system identification and predictive control. For unknown linear time-invariant (LTI) systems, all possible length-$L$ input–output trajectories can be reconstructed from a single persistently exciting run via the Fundamental Lemma:

$$\begin{bmatrix} u_{pred} \ y_{pred} \end{bmatrix} = \mathcal{H}_L(u^d, y^d) g$$

where $\mathcal{H}_L(u^d, y^d)$ is the concatenated Hankel matrix formed from offline data $\{u_k^d, y_k^d\}_{k=0}^{N-1}$. The coefficient vector $g$ encodes all initial condition and future prediction information. This construction allows direct trajectory prediction without explicit model identification, provided the offline input is persistently exciting of suitable order (Xie et al., 2022).

The general DE-EPC objective is to minimize a finite-horizon sum of stage costs: $J_N(g) = \sum_{k=0}^{N-1} \ell(y_k(g), u_k(g))$ for arbitrary (including economic) stage cost $\ell$. Hard constraints on predicted inputs/outputs are naturally imposed in the optimization over $g$.

In the nonlinear system case, modern DE-EPC introduces neural-lifting maps—parametric transformations learned from data—that embed measured outputs (and optionally inputs) into higher-dimensional "latent" coordinates where the Hankel-based linear structure and a quadratic surrogate for the economic cost are approximately valid. This enables the extension of DE-EPC formulations to the domain of nonlinear or input-nonlinear plants (Yan et al., 12 May 2025, Yan et al., 29 Dec 2025).

2. Economic Objectives and Generalized Terminal Constraints

Unlike set-point tracking MPC, DE-EPC is explicitly economic: its stage cost $\ell(y_k, u_k)$ may encode profit, energy use, emissions, or arbitrary operational metrics, not necessarily positive-definite or centered at a reference trajectory. A critical feature is the use of a generalized terminal constraint via an artificial equilibrium $(u^e, y^e)$:

$(u_N, y_N) = (u^e, y^e)$

where $(u^e, y^e)$ is any data-consistent equilibrium (need not be the true optimal one). By adding a weighted terminal cost term $\beta \ell(u^e, y^e)$, the controller implicitly drives convergence of the average stage cost to the optimal equilibrium cost, while maintaining recursive feasibility and hard constraint satisfaction (Xie et al., 2022).

The asymptotic average performance can be tuned arbitrarily close to the true optimum by increasing the penalty $\beta$, i.e.,

$$\limsup_{T\to\infty} \frac{1}{T+1} \sum_{t=0}^T \ell(u_t, y_t) \leq \ell(u^{oe}, y^{oe}) + \varepsilon$$

for any $\varepsilon > 0$ and sufficiently large $\beta$.

3. Robustness and Regularization

The practical implementation of DE-EPC in the presence of noise, stochasticity, or disturbance is addressed through robust and regularized extensions. In stochastic settings, distributionally robust optimization (DRO) is employed via Wasserstein ambiguity sets. The resultant formulation yields a regularized DeePC objective:

$$\min_{g}\; f(U_{pred}, Y_{pred}) + \lambda_{ini} \|Y_p g - y_{ini}\|_1 + \epsilon \|g\|_\ast$$

where $\|g\|_\ast$ is the dual norm induced by the DRO geometry. This encapsulates a principled tradeoff between trajectory complexity (overfitting) and robustness to sampling uncertainty. Probabilistic guarantees ensure that with high confidence, the true expected economic cost is upper-bounded by the worst-case cost over the ambiguity set (Coulson et al., 2019). In practice, regularization parameters must be tuned to balance conservativeness and performance, with typical regimes showing a U-shaped tradeoff curve for economic cost versus $\epsilon$.

4. Extensions: Nonlinear Systems and Machine Learning Integration

Recent DE-EPC frameworks extend to nonlinear systems by synthesizing virtual linear representations ("Koopman embeddings") via deep learning. Here, two neural-lifting maps are jointly trained: one transforming measured outputs $y_k$ to latent $z_k = F_\theta(y_k)$, and one mapping each input $u_k$ to $v_k = N_\gamma(u_k)$ (if input nonlinearity is present). The controller then predicts future $z_k$ and $v_k$ via Hankel-matrix mechanics and optimizes a quadratic approximation to the economic cost:

$$\hat{\ell}_e(z_k, v_k) = z_k^{T} Q_z z_k + P_z z_k + b_z + v_k^T Q_v v_k + P_v v_k + b_v$$

Constraints on measured outputs are enforced by learning an affine reconstruction $G z_k$ for those variables. Critically, all online optimization is performed as a convex quadratic program (QP), with regularization and slack variables ensuring feasibility in the event of model mismatch or disturbance (Yan et al., 29 Dec 2025, Yan et al., 12 May 2025).

Establishment of sufficient conditions for the validity of the lifted representations (finite-dimensional truncations of C.O.N.S.) and bounds on the approximation error are provided. Empirically, these methods require only moderate amounts of open-loop data and permit tuning of prediction horizon and lift dimension for computational tractability.

5. Practical Implementations and Application Case Studies

DE-EPC has been implemented for large-scale industrial and infrastructure systems covering chemical reactor optimization (Xie et al., 2022), connected open water systems with mixed-integer actuators (Chen et al., 3 Oct 2025), shipboard carbon capture plants (Han et al., 9 Apr 2025, Yan et al., 29 Dec 2025), pasteurization units (Valábek et al., 6 Nov 2025), battery storage grids (Lipka et al., 2024), and wastewater treatment processes (Han et al., 2024). Each case leverages Hankel-based prediction, economic objectives reflecting domain-specific costs, and hard or soft constraints as appropriate.

For instance, in water systems, the optimization hierarchy integrates zone-tracking (convex QP or MILP for water-level constraints) and lexicographic energy minimization, with Bayesian optimization for meta-tuning of control target zones under external disturbance—a structure enabled by the modularity of DE-EPC with data-driven prediction (Chen et al., 3 Oct 2025).

In battery storage and power systems, explicit handling of input nonlinearities (Hammerstein structures) is achieved by appropriately lifting the input via known nonlinearities and constructing Hankel matrices on these auxiliary variables (Lipka et al., 2024).

6. Performance Guarantees, Limitations, and Future Directions

DE-EPC offers several formal properties: recursive feasibility, constraint satisfaction (as determined by the imposed constraints on the Hankel-predicted trajectories), and the ability to achieve average performance arbitrarily close to the true economic optimum under mild assumptions (sufficiently persistent excitation and data length, appropriate regularization/tuning) (Xie et al., 2022, Coulson et al., 2019, Yan et al., 29 Dec 2025).

Practical limitations include the dependence on noise-free or low-noise data in the basic formulation, the computational scaling with prediction horizon and latent dimension (primarily in convex QP or MIQP solvers), and the necessity of sufficiently rich excitation in offline data gathering. Extensions to robust constraint tightening, tube-based formulations for uncertainty, and neural-embedded Koopman or virtual linear models are active areas of research. Data requirements are moderate in most reported industrial case studies (typically $10^3$–$10^4$ samples), with real-time deployment feasible at control rates of under 1 second per QP solve for large-scale systems (Yan et al., 29 Dec 2025, Han et al., 2024).

Research trajectory suggests continued integration of DE-EPC with differentiable programming for efficient policy synthesis (King et al., 2022), meta-learning for regularization parameter selection, and expansion to broader classes of hybrid and networked systems.

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• "Linear Data-Driven Economic MPC with Generalized Terminal Constraint" (Xie et al., 2022)
• "Regularized and Distributionally Robust Data-Enabled Predictive Control" (Coulson et al., 2019)
• "Economic zone data-enabled predictive control for connected open water systems" (Chen et al., 3 Oct 2025)
• "Deep Neural Koopman Operator-based Economic Model Predictive Control of Shipboard Carbon Capture System" (Han et al., 9 Apr 2025)
• "Deep Koopman Economic Model Predictive Control of a Pasteurisation Unit" (Valábek et al., 6 Nov 2025)
• "Economic data-enabled predictive control using machine learning" (Yan et al., 12 May 2025)
• "Data-driven model predictive control of battery storage units" (Lipka et al., 2024)
• "Efficient Economic Model Predictive Control of Water Treatment Process with Learning-based Koopman Operator" (Han et al., 2024)
• "Learning-based data-enabled economic predictive control with convex optimization for nonlinear systems" (Yan et al., 29 Dec 2025)
• "Koopman-based Differentiable Predictive Control for the Dynamics-Aware Economic Dispatch Problem" (King et al., 2022)