Data-Driven Predictive Control (DDPC)

• Data-driven predictive control is a family of methods that bypass explicit model identification by synthesizing control laws directly from measured trajectories.
• It integrates behavioral and predictor-based formulations to achieve constrained tracking while addressing challenges like noise, persistence of excitation, and closed-loop bias.
• Regularization, robustness, and LPV extensions highlight DDPC’s practical applicability in complex systems such as renewable energy and robotics.

Data-driven predictive control (DDPC) denotes a family of predictive control methods that synthesize receding-horizon control laws from measured trajectories rather than from an explicitly identified parametric plant model. Across the literature, DDPC encompasses direct behavioral formulations based on Hankel or trajectory libraries, predictor-based formulations such as subspace predictive control (SPC), and broader trajectory-predictor frameworks in which future outputs are represented as linear functions of recent input-output history and planned future inputs (Mattsson et al., 2024, Liu et al., 16 Dec 2025, Premer et al., 11 Feb 2026). In the standard finite-horizon setting, DDPC seeks MPC-like constrained tracking or regulation performance while replacing model propagation by a data-derived prediction map; its distinctive technical issues are persistence of excitation, finite-sample noise, closed-loop identification bias, and the relation between direct trajectory parameterizations and indirect predictor estimation (Breschi et al., 2022, Chiuso et al., 2023).

1. Generic predictive-control formulation

At time $t$, DDPC methods typically assume access to a finite window of past inputs and outputs,

$$\zeta_t = \begin{bmatrix} u_{t-\rho}^\top & \cdots & u_{t-1}^\top & y_{t-\rho}^\top & \cdots & y_{t-1}^\top \end{bmatrix}^\top,$$

together with a future input trajectory $\mathbf u_t$ to be optimized and a predicted future output trajectory $\hat{\mathbf y}_t$. A generic DDPC problem is

$$\begin{aligned} \min_{\mathbf{u},\hat{\mathbf{y}}}\quad & J(\hat{\mathbf{y}},\mathbf{u}) \ \text{s.t.}\quad & \hat{\mathbf{y}}=\text{Prediction}(\zeta,\mathbf{u}), \ & \mathbf{u}\in\mathcal U,\quad \hat{\mathbf{y}}\in\mathcal Y, \end{aligned}$$

where the essential modeling question is how the prediction operator is obtained from data (Mattsson et al., 2024).

A broad indirect formulation writes the future trajectory as

$y_f(t)=P z_p(t)+F u_f(t)+e_f(t),$

with $z_p(t)$ the recent input-output history, $u_f(t)$ the planned future input sequence, and $P,F$ predictor matrices learned from training data (Premer et al., 11 Feb 2026). In SPC, the same idea appears as an explicit multi-step regression, for example

$\hat{\mathbf y}=K\varphi(\zeta,\mathbf u)$

or, in lifted form,

$$\zeta_t = \begin{bmatrix} u_{t-\rho}^\top & \cdots & u_{t-1}^\top & y_{t-\rho}^\top & \cdots & y_{t-1}^\top \end{bmatrix}^\top,$$0

where $$\zeta_t = \begin{bmatrix} u_{t-\rho}^\top & \cdots & u_{t-1}^\top & y_{t-\rho}^\top & \cdots & y_{t-1}^\top \end{bmatrix}^\top,$$1 is identified by least squares from previously measured trajectories (Mattsson et al., 2024, Desai et al., 18 Feb 2025).

This generic formulation places DDPC close to MPC in online structure and close to system identification in offline construction. A plausible implication is that most disputes in the field concern not the receding-horizon optimization itself, but the geometry, causality, and statistical properties of the predictor extracted from data.

2. Behavioral and trajectory-based representations

A central DDPC line descends from Willems’ Fundamental Lemma and behavioral system theory. In the direct formulation studied in the equivalence literature, historical trajectories are arranged in data matrices $$\zeta_t = \begin{bmatrix} u_{t-\rho}^\top & \cdots & u_{t-1}^\top & y_{t-\rho}^\top & \cdots & y_{t-1}^\top \end{bmatrix}^\top,$$2 and $$\zeta_t = \begin{bmatrix} u_{t-\rho}^\top & \cdots & u_{t-1}^\top & y_{t-\rho}^\top & \cdots & y_{t-1}^\top \end{bmatrix}^\top,$$3, and prediction is parameterized by a trajectory coefficient vector $$\zeta_t = \begin{bmatrix} u_{t-\rho}^\top & \cdots & u_{t-1}^\top & y_{t-\rho}^\top & \cdots & y_{t-1}^\top \end{bmatrix}^\top,$$4 through

$$\zeta_t = \begin{bmatrix} u_{t-\rho}^\top & \cdots & u_{t-1}^\top & y_{t-\rho}^\top & \cdots & y_{t-1}^\top \end{bmatrix}^\top,$$5

This is the canonical DeePC-style representation: the future trajectory is selected directly from the span of stored trajectories rather than from an explicitly identified state-space model (Mattsson et al., 2024).

The same literature shows that direct parameterization does not eliminate predictor estimation; instead, it can be rewritten as

$$\zeta_t = \begin{bmatrix} u_{t-\rho}^\top & \cdots & u_{t-1}^\top & y_{t-\rho}^\top & \cdots & y_{t-1}^\top \end{bmatrix}^\top,$$6

where $$\zeta_t = \begin{bmatrix} u_{t-\rho}^\top & \cdots & u_{t-1}^\top & y_{t-\rho}^\top & \cdots & y_{t-1}^\top \end{bmatrix}^\top,$$7 is the least-squares multi-step predictor and $$\zeta_t = \begin{bmatrix} u_{t-\rho}^\top & \cdots & u_{t-1}^\top & y_{t-\rho}^\top & \cdots & y_{t-1}^\top \end{bmatrix}^\top,$$8 is an additive slack constrained to the residual subspace generated by the training data (Mattsson et al., 2024). This algebraic reduction is significant because it explains why direct methods and predictor-based methods often display nearly identical closed-loop behavior.

For stochastic systems, a subspace-identification view replaces raw future-output blocks $$\zeta_t = \begin{bmatrix} u_{t-\rho}^\top & \cdots & u_{t-1}^\top & y_{t-\rho}^\top & \cdots & y_{t-1}^\top \end{bmatrix}^\top,$$9 by their projection onto the row space generated by past trajectories and future inputs. In that setting, the projected predictor

$\mathbf u_t$0

yields a deterministic future-output estimate $\mathbf u_t$1, where $\mathbf u_t$2 solves

$\mathbf u_t$3

This reframing underlies the stochastic DDPC framework and the later $\mathbf u_t$4-DDPC decomposition (Breschi et al., 2022).

The key structural assumption behind these trajectory-based methods is data informativeness, usually expressed as persistence of excitation or a rank condition on the constructed Hankel or Page-type matrices. In deterministic LTI settings this condition recovers the classical trajectory-spanning property; in noisy settings it determines whether the data-derived predictor is merely feasible or statistically meaningful.

3. Predictor-based DDPC, reduced coordinates, and theoretical unification

An important development in the literature is the systematic reduction of DDPC to explicit predictor coordinates. In $\mathbf u_t$5-DDPC, an LQ decomposition of the stacked data matrix yields reduced coordinates $\mathbf u_t$6, where $\mathbf u_t$7 is fixed by the current past trajectory and $\mathbf u_t$8 parameterizes future performance. The resulting predictor takes the form

$\mathbf u_t$9

with $\hat{\mathbf y}_t$0. This two-stage scheme separates initial-condition fitting from future optimization and reduces the online decision dimension from the full trajectory coefficient vector to a smaller performance coordinate (Breschi et al., 2022).

Several papers subsequently showed that distinctions between direct and indirect DDPC are often algebraic rather than fundamental. One result rewrites a broad class of direct methods exactly as an indirect predictive control problem with predictor $\hat{\mathbf y}_t$1, plus covariance-weighted regularization on the regressor and on an additive prediction slack $\hat{\mathbf y}_t$2 (Mattsson et al., 2024). Another shows that quadratic-regularized DeePC and $\hat{\mathbf y}_t$3-DDPC are equivalent up to coordinate transformation: in particular, $\hat{\mathbf y}_t$4 in DeePC corresponds to $\hat{\mathbf y}_t$5, while projection-based regularization corresponds to penalizing only $\hat{\mathbf y}_t$6 (Klädtke et al., 2024).

A later unification enlarges this point from specific reformulations to an entire class of indirect methods. Trajectory predictive control (TPC) defines DDPC by the predictor

$\hat{\mathbf y}_t$7

and shows that SPC, closed-loop SPC, $\hat{\mathbf y}_t$8-DDPC, causal-$\hat{\mathbf y}_t$9-DDPC, transient predictive control, and related methods differ mainly by the choice of $$\begin{aligned} \min_{\mathbf{u},\hat{\mathbf{y}}}\quad & J(\hat{\mathbf{y}},\mathbf{u}) \ \text{s.t.}\quad & \hat{\mathbf{y}}=\text{Prediction}(\zeta,\mathbf{u}), \ & \mathbf{u}\in\mathcal U,\quad \hat{\mathbf{y}}\in\mathcal Y, \end{aligned}$$0 and $$\begin{aligned} \min_{\mathbf{u},\hat{\mathbf{y}}}\quad & J(\hat{\mathbf{y}},\mathbf{u}) \ \text{s.t.}\quad & \hat{\mathbf{y}}=\text{Prediction}(\zeta,\mathbf{u}), \ & \mathbf{u}\in\mathcal U,\quad \hat{\mathbf{y}}\in\mathcal Y, \end{aligned}$$1 (Premer et al., 11 Feb 2026). In that framework, a newly introduced state-space predictor uses the recent input-output history itself as state, making TPC a special case of linear MPC. This suggests that much of DDPC theory can be transferred from classical MPC once the predictor is embedded in a causal linear state-space realization.

The unification theme reappears in convex-relaxation analyses of regularization. A bi-level view interprets direct DDPC as an outer predictive-control problem coupled to an inner identification or trajectory-library preprocessing problem; projection-based, $$\begin{aligned} \min_{\mathbf{u},\hat{\mathbf{y}}}\quad & J(\hat{\mathbf{y}},\mathbf{u}) \ \text{s.t.}\quad & \hat{\mathbf{y}}=\text{Prediction}(\zeta,\mathbf{u}), \ & \mathbf{u}\in\mathcal U,\quad \hat{\mathbf{y}}\in\mathcal Y, \end{aligned}$$2-, and causality-based regularizers then appear as convex relaxations of different subsets of the inner identification constraints (Shang et al., 10 Sep 2025). This is significant because it turns regularization from a heuristic device into an implicit system-identification mechanism.

4. Noise, regularization, and closed-loop data

The noisy-data regime is a defining difficulty of DDPC. In a stochastic setting, the finite-sample future-output predictor contains an uncertainty term of order $$\begin{aligned} \min_{\mathbf{u},\hat{\mathbf{y}}}\quad & J(\hat{\mathbf{y}},\mathbf{u}) \ \text{s.t.}\quad & \hat{\mathbf{y}}=\text{Prediction}(\zeta,\mathbf{u}), \ & \mathbf{u}\in\mathcal U,\quad \hat{\mathbf{y}}\in\mathcal Y, \end{aligned}$$3, and the associated variance depends on the DDPC decision coordinates. For $$\begin{aligned} \min_{\mathbf{u},\hat{\mathbf{y}}}\quad & J(\hat{\mathbf{y}},\mathbf{u}) \ \text{s.t.}\quad & \hat{\mathbf{y}}=\text{Prediction}(\zeta,\mathbf{u}), \ & \mathbf{u}\in\mathcal U,\quad \hat{\mathbf{y}}\in\mathcal Y, \end{aligned}$$4-DDPC, the prediction-error analysis shows that the average uncertainty scales with $$\begin{aligned} \min_{\mathbf{u},\hat{\mathbf{y}}}\quad & J(\hat{\mathbf{y}},\mathbf{u}) \ \text{s.t.}\quad & \hat{\mathbf{y}}=\text{Prediction}(\zeta,\mathbf{u}), \ & \mathbf{u}\in\mathcal U,\quad \hat{\mathbf{y}}\in\mathcal Y, \end{aligned}$$5, where $$\begin{aligned} \min_{\mathbf{u},\hat{\mathbf{y}}}\quad & J(\hat{\mathbf{y}},\mathbf{u}) \ \text{s.t.}\quad & \hat{\mathbf{y}}=\text{Prediction}(\zeta,\mathbf{u}), \ & \mathbf{u}\in\mathcal U,\quad \hat{\mathbf{y}}\in\mathcal Y, \end{aligned}$$6, which motivates regularization directly on $$\begin{aligned} \min_{\mathbf{u},\hat{\mathbf{y}}}\quad & J(\hat{\mathbf{y}},\mathbf{u}) \ \text{s.t.}\quad & \hat{\mathbf{y}}=\text{Prediction}(\zeta,\mathbf{u}), \ & \mathbf{u}\in\mathcal U,\quad \hat{\mathbf{y}}\in\mathcal Y, \end{aligned}$$7 or the introduction of an output slack shaped by $$\begin{aligned} \min_{\mathbf{u},\hat{\mathbf{y}}}\quad & J(\hat{\mathbf{y}},\mathbf{u}) \ \text{s.t.}\quad & \hat{\mathbf{y}}=\text{Prediction}(\zeta,\mathbf{u}), \ & \mathbf{u}\in\mathcal U,\quad \hat{\mathbf{y}}\in\mathcal Y, \end{aligned}$$8 (Breschi et al., 2022). A companion study interprets these regularizers statistically: $$\begin{aligned} \min_{\mathbf{u},\hat{\mathbf{y}}}\quad & J(\hat{\mathbf{y}},\mathbf{u}) \ \text{s.t.}\quad & \hat{\mathbf{y}}=\text{Prediction}(\zeta,\mathbf{u}), \ & \mathbf{u}\in\mathcal U,\quad \hat{\mathbf{y}}\in\mathcal Y, \end{aligned}$$9 suppresses predictor variance induced by noisy finite data, while $y_f(t)=P z_p(t)+F u_f(t)+e_f(t),$0 regulates a slack accounting for projection mismatch (Breschi et al., 2023).

A more general stochastic formulation replaces heuristic tuning by optimization of the conditional expected control loss. The Final Control Error,

$y_f(t)=P z_p(t)+F u_f(t)+e_f(t),$1

admits the decomposition

$y_f(t)=P z_p(t)+F u_f(t)+e_f(t),$2

where $y_f(t)=P z_p(t)+F u_f(t)+e_f(t),$3 is a certainty-equivalent cost based on the posterior mean predictor and $y_f(t)=P z_p(t)+F u_f(t)+e_f(t),$4 is an uncertainty penalty derived from the conditional variance of the predictor coefficients (Chiuso et al., 2023). This establishes a separation principle for direct DDPC in the sense that prediction and uncertainty enter through distinct terms, and it provides a tuning-free interpretation of regularization as uncertainty weighting rather than ad hoc shrinkage.

Closed-loop data introduce an additional issue: identification bias. Closed-loop DeePC was proposed precisely because standard DeePC can suffer from closed-loop identification bias when data are collected under feedback and affected by noise; the CL-DeePC framework uses instrumental variables to synthesize consistent single-step or multi-step predictors and reveals an equivalence with closed-loop SPC (Dinkla et al., 2024). A related instrumental-variable DDPC method constructs the instrument

$y_f(t)=P z_p(t)+F u_f(t)+e_f(t),$5

from past data, a controller-informed future instrument $y_f(t)=P z_p(t)+F u_f(t)+e_f(t),$6, and the exogenous reference $y_f(t)=P z_p(t)+F u_f(t)+e_f(t),$7, thereby mitigating feedback-induced bias in closed-loop data sets (Wang et al., 2023).

Bias analyses sharpened this observation. For subspace-based DDPC trained on closed-loop data, the prediction error was decomposed into Subspace Bias, caused by the correlation between future innovations and future inputs, and Optimism Bias, caused by the extra output-adjustment directions used by DeePC and $y_f(t)=P z_p(t)+F u_f(t)+e_f(t),$8-DDPC (Moffat et al., 3 Jul 2025). This result explains why projection regularization and explicit suppression of residual directions often improve performance: they reduce an output-trajectory optimism that is not causally implementable.

Alternative predictor constructions respond to this critique by abandoning the fundamental-lemma parameterization. SSARX is explicitly fundamental-lemma-free, causal, and closed-loop consistent: it first estimates predictor/observer Markov parameters from a high-order ARX model, then learns a multi-step past-to-future map by regression, optionally with a reduced-rank constraint (Liu et al., 16 Dec 2025). TPC and SSARX therefore represent a shift within DDPC from raw-trajectory coordinates toward causal multi-step predictors with clearer statistical semantics.

Robustness has also been addressed directly. Min-max robust DDPC treats the non-uniqueness of the behavioral coefficient $y_f(t)=P z_p(t)+F u_f(t)+e_f(t),$9 under noisy data as an uncertainty source, constructs the uncertainty set

$z_p(t)$0

and optimizes against the worst admissible output deviation (Wang et al., 28 Jan 2025). A plausible implication is that robust DDPC and regularized DeePC are best viewed as different ways of managing the same underlying uncertainty geometry.

5. Extensions beyond LTI systems

The most developed generalization beyond LTI is the LPV setting. For unknown LPV systems with measurable scheduling signal $z_p(t)$1, one paper introduced a data-driven predictive control scheme using measured input-output-scheduling trajectories directly, without identifying an explicit LPV model, under the assumptions that the plant admits an LPV representation, the scheduling dependence is affine, the data are informative, and the future scheduling trajectory over the horizon is known or predicted (Verhoek et al., 2021). The underlying input-output model is

$z_p(t)$2

with

$z_p(t)$3

This construction is an LPV generalization of Willems-lemma-based trajectory prediction and DeePC-style predictive control.

A later LPV development added output-feedback and state-feedback LPV-DPC schemes with terminal ingredients and data-based computation of terminal costs, controllers, and invariant sets, yielding recursive feasibility and exponential stability under LPV persistence of excitation and known future scheduling over the horizon (Verhoek et al., 2023). This is significant because it brings DDPC closer to the mature guarantees of model-based LPV-MPC while remaining direct-from-data at the predictor level.

These LPV results matter because the LPV framework is routinely used as a surrogate for nonlinear or time-varying dynamics. This suggests that LPV-DDPC is a principled route from LTI DDPC to constrained predictive control for nonlinear systems, provided a measurable scheduling description is available.

6. Applications, boundaries, and open issues

DDPC has been deployed as a supervisory controller for a hybrid power plant composed of wind, solar, and battery storage. In that setting, SPC is used to coordinate component setpoints $z_p(t)$4 so that total delivered power

$z_p(t)$5

tracks a demand profile $z_p(t)$6 under uncertain renewable availability, with an uncertainty-aware relaxation of the predictor equation and constraints on wind, solar, and battery power (Desai et al., 18 Feb 2025). The reported interpretation is pragmatic rather than foundational: DDPC serves both as a dispatcher and as a forecaster for the hybrid plant.

In robotics, DDPC has been used as the planning layer for robust exoskeleton locomotion. A multi-layer architecture places a DDPC gait planner at the top, using Hankel matrices and a state transition matrix to generate reduced-order trajectories, and inverse kinematics plus passivity-based control at the lower layer to realize those trajectories on the Atalante lower-body exoskeleton (Li et al., 2024). The comparison target is LIP-based MPC, and the stated advantage of DDPC is improved robustness to user and payload variability, especially at higher walking speeds.

Industrial closed-loop-only scenarios motivated instrumental-variable DDPC for a simulated tubular furnace, where the proposed closed-loop predictor improves outlet-temperature tracking relative to generic SPC and the pre-existing stabilizing controller (Wang et al., 2023). By contrast, some methods labeled “data-driven predictive control” occupy a different conceptual position. For continuous-time industrial processes with completely unknown dynamics, one paper estimates $z_p(t)$7 and $z_p(t)$8 online from integral state-input regressions and then computes a Taylor-series-based predictive controller from the estimated model (Zhou et al., 2020). That approach is data-driven in the sense of online model learning, but it is not direct DDPC in the DeePC/SPC sense because it reconstructs a parametric state-space model before control.

Several open issues remain consistent across the literature. Persistence of excitation and data richness remain indispensable, and for LPV systems the joint excitation of input and scheduling is still an open design problem (Verhoek et al., 2023). Closed-loop data continue to demand either instrumental-variable corrections or causal predictor constructions because standard subspace-based DDPC can remain asymptotically biased even with large data sets (Dinkla et al., 2024, Moffat et al., 3 Jul 2025). Regularization is now better understood—as statistical variance control, as uncertainty penalization, or as a convex relaxation of implicit identification—but its best form still depends on predictor structure and data-collection conditions (Chiuso et al., 2023, Shang et al., 10 Sep 2025). More broadly, the field has moved toward a consensus that “direct” and “indirect” DDPC are often different coordinate systems for the same finite-horizon control problem rather than fundamentally different paradigms (Mattsson et al., 2024, Klädtke et al., 2024).