Data-Fused Model Predictive Control

• DFMPC is a predictive control architecture that fuses real-time data with traditional model-based receding-horizon optimization to improve prediction and uncertainty handling.
• It integrates diverse fusion methods such as online learning, stochastic cost design, and hybrid physics/Hankel approaches to address model inaccuracies and noise.
• DFMPC drives practical performance improvements, offering reduced reliance on fixed plant models, enhanced adaptability, and demonstrated tracking precision in robotics.

Searching arXiv for recent and directly relevant papers on Data-Fused Model Predictive Control and closely related predictive-control formulations. Data-Fused Model Predictive Control (DFMPC) denotes a class of predictive-control architectures in which model-based receding-horizon optimization is combined with data-derived information inside the control loop, rather than relying on a fixed a priori plant model alone. In the cited literature, the fusion mechanism is not uniform: it can take the form of online identification of unknown state-space matrices, direct optimization of a control-relevant cost computed from data only, hybrid prediction models that combine physics and Hankel-based trajectory representations, black-box predictions of otherwise unknown costs or constraints, or statistically weighted ensembles of data-trained predictors (Zhou et al., 2019, Chiuso et al., 2023, Gorbani et al., 12 Sep 2025). The central commonality is that data are used not merely for offline modeling, but as an active ingredient in prediction, uncertainty quantification, feasibility handling, or objective construction within MPC itself.

1. Terminology and conceptual scope

The acronym DFMPC is used explicitly in “Data-fused Model Predictive Control with Guarantees: Application to Flying Humanoid Robots” (Gorbani et al., 12 Sep 2025), but closely related formulations appear under different names. “Synthesis of model predictive control based on data-driven learning” presents a learning-based MPC scheme whose stated “fusion” lies in combining data-driven identification with model-based prediction and optimization (Zhou et al., 2019). “Harnessing Uncertainty for a Separation Principle in Direct Data-Driven Predictive Control” does not use the DFMPC acronym, yet its central claim is that control should be designed by optimizing an uncertainty-aware cost computed from data only, which the source describes as a theory-first foundation for DFMPC (Chiuso et al., 2023). “DMPC: A Data-and Model-Driven Approach to Predictive Control” uses the label Data-and Model-Driven Predictive Control, but explicitly frames the method as a form of data-fused predictive control because known dynamics are combined with exogenous predictions of unknown objective or constraint components (Jafarzadeh et al., 2021).

A useful way to organize the literature is by the locus of fusion rather than by nomenclature alone.

Formulation	Fused elements	Representative source
Online learning-based MPC	Measured trajectories + identified linear model + MPC	(Zhou et al., 2019)
Direct stochastic DDPC	Historical data + predictor uncertainty + expected control loss	(Chiuso et al., 2023)
Hybrid parametric/nonparametric DFMPC	Physics model + DeePC-style Hankel representation	(Gorbani et al., 12 Sep 2025)
Data-and model-driven predictive control	Known dynamics + black-box unknown cost/constraints	(Jafarzadeh et al., 2021)
Ensemble predictive control	Multiple data-trained models + statistical weighting + MHE	(Giuli et al., 26 Nov 2025)

This suggests that DFMPC is best understood as an architectural principle rather than a single canonical algorithm. The “fusion” may occur at the level of model construction, cost construction, uncertainty handling, feasibility restoration, or safety supervision.

2. Online identification within receding-horizon control

A foundational DFMPC-style construction is the online-learning MPC scheme for continuous-time plants with unknown matrices $A$ and $B$, written as

$$\dot x(t)=Ax(t)+Bu(t)=\mathcal H(x(t),u(t))\,\Theta,$$

with

$$\Theta=\begin{bmatrix}\mathrm{vec}(A)^\top & \mathrm{vec}(B)^\top\end{bmatrix}^\top.$$

The formulation assumes the standard predictive-control conditions that $(A,B)$ is controllable, $(A,C)$ is observable, and the input lies in a compact convex admissible set $\mathcal U$ containing the origin in its interior. The control objective is finite-horizon reference tracking with

$$J(x(t_k),\hat u_k(\cdot)) = \int_{t_k}^{t_k+T} \left( \|e(\tau)\|_Q^2+\|\hat u_k(\tau)\|_R^2 \right)\,d\tau +\Phi(e(t_k+T)),$$

where $e=x-x_d$ (Zhou et al., 2019).

The data-driven element is an online regression law that estimates the unknown dynamics directly from measured states and inputs. In the full-state case, the dynamics are rearranged into

$\mathcal F(t)=\mathcal G(t)\hat\Theta,$

with

$B$0

and

$B$1

where

$B$2

After stacking multiple samples,

$B$3

and if $B$4 has full column rank, the estimate is obtained by least squares: $B$5 The paper also gives a partial-state construction for plants with the specific block structure

$B$6

using a second integral identity with two sampling windows $B$7 to estimate $B$8.

The MPC itself is atypical in that it optimizes not only the input but a stack of multi-order input derivatives,

$B$9

with $$\dot x(t)=Ax(t)+Bu(t)=\mathcal H(x(t),u(t))\,\Theta,$$0, where $$\dot x(t)=Ax(t)+Bu(t)=\mathcal H(x(t),u(t))\,\Theta,$$1 is the relative degree. State prediction is expressed through derivative stacks $$\dot x(t)=Ax(t)+Bu(t)=\mathcal H(x(t),u(t))\,\Theta,$$2 and polynomial basis vectors $$\dot x(t)=Ax(t)+Bu(t)=\mathcal H(x(t),u(t))\,\Theta,$$3, yielding

$$\dot x(t)=Ax(t)+Bu(t)=\mathcal H(x(t),u(t))\,\Theta,$$4

with

$$\dot x(t)=Ax(t)+Bu(t)=\mathcal H(x(t),u(t))\,\Theta,$$5

Substituting the prediction into the cost produces a quadratic objective in the decision variables, so the resulting control problem is a quadratic program.

The adaptation claim is local and iterative. Because $$\dot x(t)=Ax(t)+Bu(t)=\mathcal H(x(t),u(t))\,\Theta,$$6 are re-estimated at each sampling instant, the learned linear predictor can update with the plant. The paper defines the model mismatch under the applied policy as

$$\dot x(t)=Ax(t)+Bu(t)=\mathcal H(x(t),u(t))\,\Theta,$$7

states that this error is bounded with bounded rate of change, and claims that repeated online updating yields $$\dot x(t)=Ax(t)+Bu(t)=\mathcal H(x(t),u(t))\,\Theta,$$8, implying asymptotic stability under the assumptions summarized in the appendices. The limitations are equally explicit: the method remains a repeated linear approximation rather than a full nonlinear MPC formulation; it requires sufficiently rich data and a full-rank $$\dot x(t)=Ax(t)+Bu(t)=\mathcal H(x(t),u(t))\,\Theta,$$9; the partial-state result needs a specific plant structure; and the stability argument is sketched rather than a full rigorous global nonlinear proof.

3. Direct data-fused predictive control through uncertainty-aware cost design

A distinct line of work places the fusion at the level of the control objective rather than explicit model identification. In the stochastic direct DDPC framework, the standard receding-horizon problem

$$\Theta=\begin{bmatrix}\mathrm{vec}(A)^\top & \mathrm{vec}(B)^\top\end{bmatrix}^\top.$$0

is reformulated so that the controller minimizes the expected finite-horizon loss conditioned on available data $$\Theta=\begin{bmatrix}\mathrm{vec}(A)^\top & \mathrm{vec}(B)^\top\end{bmatrix}^\top.$$1 only. The central quantity is the Final Control Error (FCE),

$$\Theta=\begin{bmatrix}\mathrm{vec}(A)^\top & \mathrm{vec}(B)^\top\end{bmatrix}^\top.$$2

and the control problem becomes

$$\Theta=\begin{bmatrix}\mathrm{vec}(A)^\top & \mathrm{vec}(B)^\top\end{bmatrix}^\top.$$3

For the standard tracking loss

$$\Theta=\begin{bmatrix}\mathrm{vec}(A)^\top & \mathrm{vec}(B)^\top\end{bmatrix}^\top.$$4

the expected cost decomposes as

$$\Theta=\begin{bmatrix}\mathrm{vec}(A)^\top & \mathrm{vec}(B)^\top\end{bmatrix}^\top.$$5

with a certainty-equivalent term

$$\Theta=\begin{bmatrix}\mathrm{vec}(A)^\top & \mathrm{vec}(B)^\top\end{bmatrix}^\top.$$6

and an uncertainty-induced correction

$$\Theta=\begin{bmatrix}\mathrm{vec}(A)^\top & \mathrm{vec}(B)^\top\end{bmatrix}^\top.$$7

The source emphasizes that this regularization is not heuristic: it is induced automatically by predictor uncertainty rather than introduced by ad hoc tuning (Chiuso et al., 2023).

The theory assumes a linear time-invariant stochastic setting with a linear, BIBO-stable one-step predictor

$$\Theta=\begin{bmatrix}\mathrm{vec}(A)^\top & \mathrm{vec}(B)^\top\end{bmatrix}^\top.$$8

and a martingale-difference prediction error $$\Theta=\begin{bmatrix}\mathrm{vec}(A)^\top & \mathrm{vec}(B)^\top\end{bmatrix}^\top.$$9 satisfying

$(A,B)$0

A Gaussian-process prior is placed on the predictor coefficients, and in the non-informative limit the relevant data enter only through sufficient statistics such as

$(A,B)$1

This leads to a separation-principle interpretation: one component of the control objective is driven by the conditional mean predictor, and the second by its conditional variance. In that sense, the framework fuses historical data, stochastic uncertainty, and control design into a single predictive-control criterion.

The same paper establishes formal links to regularized DeePC and $(A,B)$2-DDPC. DeePC-style predictors are recovered after conditioning on a projected predictor, and the regularization structure seen in $(A,B)$3-DDPC emerges asymptotically in the Output Error case. The main empirical study uses a benchmark SISO fourth-order LTI system under white and colored disturbance settings, with $(A,B)$4 dB and $(A,B)$5 dB SNR, $(A,B)$6 training points, horizon $(A,B)$7, and 100 Monte Carlo runs. In the white-noise case all methods perform similarly; in the colored-noise case $(A,B)$8-only regularization performs poorly and can lead to practical instability; DeePC and $(A,B)$9-DDPC are similar; and the proposed tuning-free method $(A,C)$0 matches or outperforms competitors while avoiding offline hyperparameter search.

4. Hybrid physics/data DFMPC with recursive-feasibility and stability guarantees

The most explicit DFMPC formulation in the cited corpus is a hybrid architecture for composite systems with a parametric known subsystem $(A,C)$1 and a data-driven unknown subsystem $(A,C)$2. The known part is modeled as

$(A,C)$3

whereas the unknown subsystem is represented nonparametrically from offline trajectories $(A,C)$4 using Willems’ Fundamental Lemma. If $(A,C)$5 is persistently exciting of suitable order, any length-$(A,C)$6 trajectory can be represented as

$(A,C)$7

The resulting predictive controller is neither purely parametric nor purely data-driven; it is a convex fusion of both within one optimization problem (Gorbani et al., 12 Sep 2025).

A central device is the artificial equilibrium $(A,C)$8, introduced to handle piecewise constant and possibly unreachable references. Rather than enforcing exact reference tracking, the controller computes the closest reachable equilibrium by solving a convex projection problem over the equilibrium manifold induced by the Hankel representation and the admissible sets $(A,C)$9. At each sampling instant, it then solves a finite-horizon problem with stage penalties on deviations from the artificial equilibrium, penalties on the distance between the artificial equilibrium and the external reference, regularization on the DeePC coefficient vector $\mathcal U$0, and penalties on the slack variable $\mathcal U$1 introduced to handle noisy offline data.

The constraints combine physics-based and data-driven dynamics: $\mathcal U$2

$\mathcal U$3

together with initial consistency, a terminal equilibrium tail, and hard input-output constraints. Noise is modeled explicitly through

$\mathcal U$4

and the slack variable preserves feasibility under noisy data, model mismatch, and local nonlinearity.

The paper proves an output prediction-error bound expressed in terms of the noise level $\mathcal U$5, $\mathcal U$6, and the slack variables, and uses this to derive recursive-feasibility and practical-stability results. The formal guarantees hold for a class of piecewise constant references that are feasible or sufficiently close to a feasible equilibrium, sufficiently far from the boundary of the output constraint set, and compatible with a cost-dominance inequality. Practical exponential stability is established around the artificial equilibrium through an offset cost $\mathcal U$7 and a Lyapunov-like function $\mathcal U$8, leading to an exponential decrease up to a residual set proportional to $\mathcal U$9.

The application target is the iRonCub flying humanoid robot. The decomposition is physically meaningful: $$J(x(t_k),\hat u_k(\cdot)) = \int_{t_k}^{t_k+T} \left( \|e(\tau)\|_Q^2+\|\hat u_k(\tau)\|_R^2 \right)\,d\tau +\Phi(e(t_k+T)),$$0 comprises analytical momentum dynamics, while $$J(x(t_k),\hat u_k(\cdot)) = \int_{t_k}^{t_k+T} \left( \|e(\tau)\|_Q^2+\|\hat u_k(\tau)\|_R^2 \right)\,d\tau +\Phi(e(t_k+T)),$$1 represents hard-to-model turbine thrust dynamics through online-updated Hankel matrices. The authors note explicitly that the formal recursive-feasibility and stability proofs do not directly apply to this fully nonlinear adaptive implementation; the practical rationale is instead a local LTI approximation with modeling error absorbed by the slack variables. In simulation, the hybrid DFMPC improves tracking over a purely model-based baseline while remaining real-time feasible at 100 Hz.

Metric	DFMPC	Baseline MPC
Overall CoM position RMSE	$$J(x(t_k),\hat u_k(\cdot)) = \int_{t_k}^{t_k+T} \left( \|e(\tau)\|_Q^2+\|\hat u_k(\tau)\|_R^2 \right)\,d\tau +\Phi(e(t_k+T)),$$2 m	$$J(x(t_k),\hat u_k(\cdot)) = \int_{t_k}^{t_k+T} \left( \|e(\tau)\|_Q^2+\|\hat u_k(\tau)\|_R^2 \right)\,d\tau +\Phi(e(t_k+T)),$$3 m
Roll RMSE	$$J(x(t_k),\hat u_k(\cdot)) = \int_{t_k}^{t_k+T} \left( \|e(\tau)\|_Q^2+\|\hat u_k(\tau)\|_R^2 \right)\,d\tau +\Phi(e(t_k+T)),$$4	$$J(x(t_k),\hat u_k(\cdot)) = \int_{t_k}^{t_k+T} \left( \|e(\tau)\|_Q^2+\|\hat u_k(\tau)\|_R^2 \right)\,d\tau +\Phi(e(t_k+T)),$$5
Pitch RMSE	$$J(x(t_k),\hat u_k(\cdot)) = \int_{t_k}^{t_k+T} \left( \|e(\tau)\|_Q^2+\|\hat u_k(\tau)\|_R^2 \right)\,d\tau +\Phi(e(t_k+T)),$$6	$$J(x(t_k),\hat u_k(\cdot)) = \int_{t_k}^{t_k+T} \left( \|e(\tau)\|_Q^2+\|\hat u_k(\tau)\|_R^2 \right)\,d\tau +\Phi(e(t_k+T)),$$7
Runtime per iteration	about $$J(x(t_k),\hat u_k(\cdot)) = \int_{t_k}^{t_k+T} \left( \|e(\tau)\|_Q^2+\|\hat u_k(\tau)\|_R^2 \right)\,d\tau +\Phi(e(t_k+T)),$$8 ms	about $$J(x(t_k),\hat u_k(\cdot)) = \int_{t_k}^{t_k+T} \left( \|e(\tau)\|_Q^2+\|\hat u_k(\tau)\|_R^2 \right)\,d\tau +\Phi(e(t_k+T)),$$9 ms

The reported CoM improvements are $e=x-x_d$0 in $e=x-x_d$1, $e=x-x_d$2 in $e=x-x_d$3, $e=x-x_d$4 in $e=x-x_d$5, and $e=x-x_d$6 overall; the base-orientation improvements are $e=x-x_d$7 in roll, $e=x-x_d$8 in pitch, and $e=x-x_d$9 in yaw.

5. Related fusion paradigms

Several adjacent formulations broaden the meaning of DFMPC beyond online identification and physics-plus-Hankel hybridization. In DMPC, the known nonlinear dynamics

$\mathcal F(t)=\mathcal G(t)\hat\Theta,$0

and known stage cost $\mathcal F(t)=\mathcal G(t)\hat\Theta,$1 are retained, while an exogenous black-box system provides the unknown term $\mathcal F(t)=\mathcal G(t)\hat\Theta,$2 in the infinite-horizon objective

$\mathcal F(t)=\mathcal G(t)\hat\Theta,$3

Unknown constraints can be converted into an unknown cost via a barrier function. The algorithm is iterative and trajectory-based: each new rollout uses the most recently generated feasible trajectory as a source of terminal candidates and cost-to-go values. The paper proves recursive feasibility, asymptotic stability of the equilibrium $\mathcal F(t)=\mathcal G(t)\hat\Theta,$4, and monotonic improvement across iterations, with

$\mathcal F(t)=\mathcal G(t)\hat\Theta,$5

In the autonomous-vehicle motion-planning case study, DMPC converged after $\mathcal F(t)=\mathcal G(t)\hat\Theta,$6 iterations, with horizon $\mathcal F(t)=\mathcal G(t)\hat\Theta,$7 and time step $\mathcal F(t)=\mathcal G(t)\hat\Theta,$8 s; compared with GP-based approaches, it is reported to need less than $\mathcal F(t)=\mathcal G(t)\hat\Theta,$9 data samples, about half the running time, and no reference trajectory (Jafarzadeh et al., 2021).

A different fusion strategy appears in ensemble predictive control with statistically weighted data-based ensemble models. Here, multiple independently trained dynamic models $B$00 are fused through horizon-varying Mahalanobis-distance weights,

$B$01

The same ensemble structure is paired with a moving horizon estimator that estimates the internal state of each expert model before MPC optimization. In the AROMA district heating system benchmark, the proposed MD-2 strategy achieves the lowest reported cost $B$02, zero constraint violations, and average solve time $B$03 s, outperforming equal weights, least-squares weights, and fixed Mahalanobis weighting in the reported table (Giuli et al., 26 Nov 2025).

A further DFMPC-style variant is the combination of Differentiable Predictive Control (DPC) with a data-driven predictive safety filter. The learned DPC controller approximates receding-horizon MPC offline, while safety is enforced online through a predictive filter triggered only when the state leaves a safe set constructed from DPC training rollouts. The architecture also includes a relative-degree-based system decomposition to address poorly defined vector relative degree. In the quadcopter experiments, DPC + PSF is reported to satisfy constraints in all three experiments, to recover from an adversarial initial condition outside the training distribution, and to reduce computation by one order of magnitude relative to VTNMPC and three orders relative to NMPC. The same source is explicit that safety is not guaranteed formally because the filter uses only an approximate tangent-plane representation of the safe set (Viljoen et al., 2024).

6. Guarantees, limitations, and recurrent points of confusion

A recurrent misconception is that DFMPC is synonymous with a purely data-driven, model-free controller. The cited literature does not support that identification. Some DFMPC-style methods re-estimate a linear model online and then solve a standard MPC problem (Zhou et al., 2019); others optimize a stochastic control loss directly from historical data without first identifying a fixed model (Chiuso et al., 2023); still others merge analytical subsystem dynamics with Hankel-based behavioral representations (Gorbani et al., 12 Sep 2025), or retain known dynamics while outsourcing only unknown cost or constraint components to a black-box predictor (Jafarzadeh et al., 2021). This suggests that “data-fused” refers to how control-relevant information is assembled, not to the absence of models.

A second point of confusion concerns guarantees. The strongest guarantees in the cited corpus are conditional rather than universal. Online learning-based MPC claims asymptotic stability through vanishing model mismatch under repeated re-identification, but its proof structure is sketched and tied to assumptions summarized outside the main text (Zhou et al., 2019). The FCE framework yields optimality with respect to expected finite-horizon loss conditioned on data and provides a principled noise-tolerant regularizer, but its derivation assumes a linear stochastic predictor structure and exact variance calculations under stated assumptions (Chiuso et al., 2023). The hybrid physics/Hankel DFMPC proves recursive feasibility and practical exponential stability only for a specific class of piecewise constant references with sufficient interior safety margin (Gorbani et al., 12 Sep 2025). The DPC + PSF architecture improves safety empirically but does not claim formal safety guarantees (Viljoen et al., 2024).

A third issue is the meaning of adaptability. In the online-identification and iRonCub formulations, adaptability arises from repeated model updates and slack-regularized local approximation of time-varying or nonlinear dynamics (Zhou et al., 2019, Gorbani et al., 12 Sep 2025). In the FCE formulation, adaptability is statistical: the controller changes because the conditional mean and variance of the loss change with the data (Chiuso et al., 2023). In the ensemble formulation, adaptability is achieved by letting model weights vary over the prediction horizon according to the predicted input’s Mahalanobis distance from each expert’s training regime (Giuli et al., 26 Nov 2025).

The principal limitations are similarly consistent across the literature. DFMPC methods frequently require informative or persistently exciting data, careful regularization, explicit handling of measurement noise, and computational structures compatible with real-time receding-horizon optimization. When guarantees are available, they are typically local, practical, or conditional on reference classes, excitation conditions, or structural assumptions. Even so, the surveyed work shows that fusing data with predictive control can reduce dependence on an exact a priori model, incorporate uncertainty or black-box knowledge in a principled way, and preserve much of the optimization structure that makes MPC attractive for constrained control.