Hybrid Data-Driven Predictive Control (HDDPC)

• Hybrid Data-Driven Predictive Control (HDDPC) is a family of methods that fuse data-driven prediction with structured mechanistic insights for enhanced constraint enforcement and interpretability.
• HDDPC encompasses varied architectures including mechanistic/data-driven predictors, partial-model bridges, state-space predictor formulations, and hybrid planners for discrete-continuous systems.
• The approach tackles challenges like noise sensitivity and limited data by integrating robustness mechanisms such as causality enforcement and safeguarding controllers.

Searching arXiv for the cited HDDPC and closely related papers to ground the article in current literature. Hybrid Data-Driven Predictive Control (HDDPC) denotes a family of predictive-control methodologies that combine data-driven prediction with additional structural ingredients drawn from mechanistic modeling, state-space MPC, hybrid-system representations, or safety and robustness layers. Across the recent literature, the term does not refer to a single canonical algorithm. Instead, it covers several related constructions: hybrid mechanistic/data-driven process models embedded in MPC (Caspari et al., 23 Jun 2025), partial-model extensions of data-enabled predictive control that bridge DeePC and MPC (Watson, 18 Feb 2025), trajectory-predictor formulations that recover classical linear MPC from data-driven predictors (Premer et al., 11 Feb 2026), and explicitly hybrid locomotion planners that jointly optimize contact schedules and continuous trajectories (Li et al., 14 Aug 2025). A consistent theme is that prediction is learned from data, but the controller is not purely black-box: it retains structure needed for constraints, interpretability, robustness, or computational tractability.

1. Terminological scope and conceptual boundaries

The recent literature uses HDDPC in both a narrow and a broad sense. In the narrow sense, it denotes a specific framework for exoskeleton locomotion that extends DeePC to hybrid walking by simultaneously planning foot contact schedules and continuous domain trajectories through a Hankel matrix-based representation with step-to-step transitions (Li et al., 14 Aug 2025). In a broader sense, the term is closely related to methods that bridge data-driven prediction and model-based predictive control without committing to a single formulation (Watson, 18 Feb 2025), or to dynamic hybrid modeling approaches that identify mechanistic/data-driven predictors later embedded in NMPC (Caspari et al., 23 Jun 2025).

This non-uniform usage is central to the subject. One line of work treats HDDPC as a DeePC extension for hybrid systems; another treats it as a middle ground between DeePC and MPC when partial model knowledge is available; another uses “hybrid” to describe the combination of offline and online learned predictors inside NMPC (Ma et al., 2024). This suggests that HDDPC is best understood as a research umbrella organized around structured data-driven prediction rather than as a fixed controller architecture.

A common misconception is that HDDPC is simply another name for DeePC. The literature does not support that reduction. DeePC uses measured trajectory data directly instead of an explicit model, whereas HDDPC-style formulations incorporate additional structure such as partial state-space models (Watson, 18 Feb 2025), causal predictor constraints (Sader et al., 2023), mechanistic conservation laws (Caspari et al., 23 Jun 2025), or hybrid contact scheduling (Li et al., 14 Aug 2025). Another misconception is that HDDPC necessarily introduces a new predictive-control optimizer. In the dynamic hybrid modeling literature, the primary contribution may instead be the identification of a hybrid prediction model that is then embedded in a standard NMPC formulation (Caspari et al., 23 Jun 2025).

2. Core architectural patterns

The most common HDDPC architectures in the recent literature fall into four recurring patterns.

Mechanistic/data-driven dynamic predictors combine a mechanistic state evolution with learned surrogates for unknown or hard-to-model quantities. In dynamic hybrid modeling, the plant is written as an index-1 DAE,

$$\frac{\mathrm{d}\boldsymbol{x}}{\mathrm{d}t}(t)=\boldsymbol{f}(\boldsymbol{x}(t),\boldsymbol{y}(t),\boldsymbol{u}(t),\boldsymbol{p}(t)), \qquad \boldsymbol{0}=\boldsymbol{g}(\boldsymbol{x}(t),\boldsymbol{y}(t),\boldsymbol{u}(t),\boldsymbol{p}(t)),$$

where the unknown quantities $\boldsymbol{p}(t)$ are learned from data and then reinserted into the mechanistic DAE as $$\boldsymbol{ML}(\bar{\boldsymbol{x}},\bar{\boldsymbol{y}},\bar{\boldsymbol{u}})$$ (Caspari et al., 23 Jun 2025). The resulting predictor is hybrid in the literal sense: mechanistic balances remain explicit, while difficult terms are supplied by ML surrogates.

Partial-model DeePC/MPC bridges split the plant into known and unknown subsystems. In HDeePC, the data-driven Hankel constraint is kept only for the unknown output component,

$$\begin{bmatrix} U_P & Y_{U,P} & U_F & Y_{U,F} \end{bmatrix}g = \begin{bmatrix} u_{ini} \ y_{u,ini} \ u \ y_u \end{bmatrix},$$

while the known part is enforced through explicit model equations for $x_k$ and $y_k$ (Watson, 18 Feb 2025). In the noiseless LTI case, this yields feasible-set equivalence and equivalent closed-loop behavior relative to DeePC and MPC under the stated assumptions (Watson, 18 Feb 2025).

Indirect data-driven predictors embedded in MPC identify a predictor offline and then solve a standard MPC-like optimization online. Trajectory predictive control represents the future output trajectory as

$y_f(t)=Pz_p(t)+Fu_f(t)+e_f(t),$

and with a particular state-space predictor, the resulting TPC problem becomes a special case of linear MPC with the recent input/output history as the state (Premer et al., 11 Feb 2026). A plausible implication is that some HDDPC variants are best interpreted as data-derived predictor constructions whose control layer is mathematically identical to conventional MPC.

Hybrid planners over discrete and continuous decision variables arise in exoskeleton locomotion, where HDDPC extends DeePC to hybrid systems by separating continuous domain evolution from step-to-step transitions and optimizing both contact schedules and continuous trajectories in a receding-horizon loop (Li et al., 14 Aug 2025). Here the hybrid character refers to hybrid dynamics in the classical sense: alternating contact modes, domain transitions, and impacts.

3. Identification and predictor construction

When HDDPC relies on a mechanistic/data-driven model, predictor construction is often incremental rather than simultaneous. A representative workflow comprises four key steps: regularized dynamic parameter estimation, correlation analysis, data-driven model identification, and hybrid model integration (Caspari et al., 23 Jun 2025). In the first stage, time-varying unknown quantities are inferred by solving a dynamic optimization with a regularization term

$$R(\boldsymbol{p}(t))= \sum_{k=1}^{N_{\mathrm{disc}}-1} \big(\boldsymbol{p}(t_{k+1}^{\mathrm{disc}})-\boldsymbol{p}(t_k^{\mathrm{disc}})\big)^\mathsf{T} \boldsymbol{W}^R_k \big(\boldsymbol{p}(t_{k+1}^{\mathrm{disc}})-\boldsymbol{p}(t_k^{\mathrm{disc}})\big),$$

designed to prevent unrealistic jumps in inferred parameter trajectories (Caspari et al., 23 Jun 2025). Pearson correlation analysis then selects candidate regressors according to the criterion $|c_{x,p}| \ge \tau$, and ANNs are trained outside the DAE optimization using standard ML tools (Caspari et al., 23 Jun 2025).

When HDDPC is based on data-enabled or subspace prediction, predictor construction typically uses Hankel or related factorized data matrices. The exoskeleton formulation retains the DeePC relation

$$\begin{bmatrix} Z_p \ Z_f \end{bmatrix}g = \begin{bmatrix} z_{\mathrm{ini}} \ z_f \end{bmatrix},$$

but generalizes it to hybrid trajectories by coupling continuous-domain motions with step-to-step transitions (Li et al., 14 Aug 2025). In uncertainty-aware predictive control for hybrid power plants, subspace predictive control identifies a multistep map

$\boldsymbol{p}(t)$0

where $\boldsymbol{p}(t)$1 is obtained from a least-squares fit on measured trajectories (Desai et al., 18 Feb 2025).

Predictor structure is a major research issue because implicit multistep predictors can be high variance under finite data. Causality-informed DDPC argues that lack of causality is a main cause for high variance of implicit prediction, and therefore constrains the multistep predictor so that the future-input block is lower-block triangular (Sader et al., 2023). The corresponding causal predictor can be expressed in terms of LQ factors, and the resulting causal $\boldsymbol{p}(t)$2-DDPC preserves the computational advantages of factorized formulations while enforcing a causal prediction structure (Sader et al., 2023).

4. Relation to MPC, DeePC, and sequential decision structure

HDDPC occupies the interface between direct and indirect data-driven predictive control and classical MPC. Some variants remain close to DeePC, in the sense that future trajectories are constrained to lie in the span of recorded data through a behavioral coefficient vector. Others identify a reduced predictor offline and then use a standard MPC optimization online. This distinction is analytically important because it determines whether the controller can be interpreted as a sequential decision-making process.

The literature on predictive-control optimality states that closed-loop optimality is not guaranteed by prediction accuracy alone. What matters is whether the predictive controller is self-consistent in the Bellman sense, namely whether its value function satisfies

$\boldsymbol{p}(t)$3

Generic DDPC formulations need not have this property, because their predictors may not correspond to repeated simulation of an underlying one-step map (Anand et al., 2024). By contrast, TPC with the state-space predictor is literally equivalent to classical linear MPC, and therefore inherits the mature theory of linear MPC (Premer et al., 11 Feb 2026). This supports a useful distinction within HDDPC research: some methods hybridize data and structure strongly enough to recover standard MPC semantics, whereas others remain direct trajectory-optimization schemes whose optimality theory is more delicate.

Bridging results formalize this interface. HDeePC is positioned explicitly as a middle ground between purely data-driven predictive control and fully model-based MPC (Watson, 18 Feb 2025). Data-driven predictive control with estimated prediction matrices likewise estimates MPC prediction matrices directly from data offline, then solves an online quadratic program with similar structure and complexity as linear MPC (Verheijen et al., 2021). A plausible implication is that the most MPC-like HDDPC methods are those that compress data into a predictor class with one-step or state-space semantics.

5. Robustness, noise, and safety layers

Robustness is a defining concern for HDDPC because data-driven predictors are sensitive to finite-sample effects, noise, and operating-regime mismatch. Several lines of work address this problem by imposing structure, separating subspaces, or adding a safeguarding controller.

Noise-tolerant hybrid predictive control focuses on the effect of measurement noise on Hankel matrices. The proposed NTDPC framework uses singular value decomposition to separate dominant dynamics from noise-dominated singular directions in reduced-order Hankel matrices, yielding the compact predictor

$\boldsymbol{p}(t)$4

and introduces the sensitivity index

$\boldsymbol{p}(t)$5

to support horizon selection under different noise levels (Mazare et al., 25 Jun 2025). Simulation studies report improved robustness and efficiency relative to existing hybrid methods, with a reported dominant factorization cost reduction of about 11\% relative to SPC in the simplified SISO comparison (Mazare et al., 25 Jun 2025).

Regularization-based work in stochastic DDPC similarly argues that regularization is useful only when it suppresses the appropriate noise-dominated directions. The two-stage $\boldsymbol{p}(t)$6-DDPC formulation separates initial-condition fitting from future-performance optimization through an LQ decomposition and reduces tuning burden by optimizing only the future-performance subspace online (Breschi et al., 2022). Causality-informed DDPC makes a complementary point: enforcing causal structure reduces variance and improves performance under stochastic noise and process nonlinearity, with no excess computational cost relative to generic $\boldsymbol{p}(t)$7-DDPC in the reported stochastic LTI study (Sader et al., 2023).

A distinct robustness strategy is architectural rather than statistical. The two-component safeguarding framework combines a data-driven or learning-based predictive controller with a model-free high-gain funnel controller:

$\boldsymbol{p}(t)$8

The predictive component provides performance, while the fallback controller guarantees the output-tracking constraint

$\boldsymbol{p}(t)$9

when the predictive controller cannot guarantee safety (Bold et al., 25 May 2025). This is a different sense of “hybrid”: predictive optimization and reactive safety feedback coexist in a supervisory switching architecture.

6. Application domains and representative implementations

The application space of HDDPC-style methods is heterogeneous, and the nature of the hybridization varies by domain.

Domain	HDDPC-style formulation	Reported role
Chemical and biochemical processes	Dynamic hybrid model identification with mechanistic DAE plus ANN surrogates	Predictor construction for NMPC (Caspari et al., 23 Jun 2025)
Exoskeleton locomotion	Hankel-based hybrid planner with contact scheduling and S2S transitions	Robust and reactive walking on Atalante (Li et al., 14 Aug 2025)
Hybrid power plants	Uncertainty-aware SPC supervisory controller for wind, solar, and battery	Load tracking under weather uncertainty (Desai et al., 18 Feb 2025)
Heavy hydraulic robots	Offline LSTM-MLP predictor plus online mismatch-learning MLP in NMPC	Tracking and energy management on a 22-ton excavator (Ma et al., 2024)

In process systems, the control-focused case study in dynamic hybrid modeling embeds the identified hybrid model as the controller model in an NMPC framework for a water-tank research plant with controlled variables given by tank holdups, manipulated variables given by setpoints for FIC 1 and FIC 2, a sampling time of 8 s, and a prediction/control horizon of 180 s (Caspari et al., 23 Jun 2025). In this sense, the work is foundational for HDDPC rather than a new MPC algorithm: it builds the hybrid model that predictive control relies on (Caspari et al., 23 Jun 2025).

In locomotion, HDDPC is an explicit hybrid-system planner. It uses experimentally collected motion segments in a Hankel library to represent feasible continuous-domain motions, feasible step-to-step transitions, and their coupling, while jointly optimizing contact schedules and continuous trajectories in receding horizon (Li et al., 14 Aug 2025). The framework is validated on the Atalante exoskeleton through simulation studies and hardware tests, with the reported outcome of stable walking, improved robustness, and reactive recovery from disturbances (Li et al., 14 Aug 2025).

In power systems, the supervisory controller coordinates a 4 MW wind farm, a 4 MW solar farm, and a 4 MW battery using SPC and a probabilistic lower-confidence wind bound. With prediction horizon $$\boldsymbol{ML}(\bar{\boldsymbol{x}},\bar{\boldsymbol{y}},\bar{\boldsymbol{u}})$$0, initialization horizon $$\boldsymbol{ML}(\bar{\boldsymbol{x}},\bar{\boldsymbol{y}},\bar{\boldsymbol{u}})$$1, data length $$\boldsymbol{ML}(\bar{\boldsymbol{x}},\bar{\boldsymbol{y}},\bar{\boldsymbol{u}})$$2, and a 20 s sampling interval, the reported open-loop normalized prediction errors are about 6.5\% for wind, 8.5\% for solar, 10\% for battery, and 6.1\% for total plant output, with mean solve time about 0.3 s (Desai et al., 18 Feb 2025).

In hydraulic robotics, the proposed HDDPC architecture combines an offline SSMP model with an online mismatch-compensation model. The hybrid predictor

$$\boldsymbol{ML}(\bar{\boldsymbol{x}},\bar{\boldsymbol{y}},\bar{\boldsymbol{u}})$$3

is embedded in NMPC and optimized by gradient descent with an adaptive learning rate (Ma et al., 2024). The paper reports that the online model reduces ARMSE by at least 50\% compared with offline-only prediction under heavy-load interaction, and that the controller runs in real time at about 50 Hz with cycle times roughly 5.5 ms to 15.4 ms depending on horizon and network size (Ma et al., 2024).

7. Open issues, limitations, and research directions

The recent literature identifies several unresolved issues. First, terminology remains unsettled. Some papers use HDDPC for explicitly hybrid dynamical systems such as locomotion (Li et al., 14 Aug 2025), others for partial-model or mechanistic/data-driven bridges (Watson, 18 Feb 2025, Caspari et al., 23 Jun 2025), and others for offline/online learned predictor combinations (Ma et al., 2024). This suggests that comparative taxonomy remains an open editorial and methodological problem.

Second, noise and limited-data effects remain central. Mechanistic/data-driven identification depends strongly on regularization weights and on whether unknown terms can be meaningfully absorbed into latent quantities such as $$\boldsymbol{ML}(\bar{\boldsymbol{x}},\bar{\boldsymbol{y}},\bar{\boldsymbol{u}})$$4 (Caspari et al., 23 Jun 2025). Hankel-based hybrid methods depend on persistent excitation, horizon selection, and spectral separation between dynamics and noise (Mazare et al., 25 Jun 2025). Causal and subspace-structured formulations reduce variance, but they do not eliminate the need for representative data (Sader et al., 2023, Breschi et al., 2022).

Third, optimality theory is incomplete for generic direct data-driven predictors. The literature explicitly concludes that better prediction accuracy alone does not ensure closed-loop optimality and that the decisive property is self-consistent sequential decision structure rather than the model-based/model-free dichotomy (Anand et al., 2024). This favors HDDPC constructions that can be cast as state-space or one-step predictors, as in TPC with the state-space predictor (Premer et al., 11 Feb 2026), but leaves open the general theory for implicit trajectory-based schemes.

Fourth, many practical studies retain strong assumptions. Examples include exact state initialization in HDeePC (Watson, 18 Feb 2025), Gaussian weather uncertainty and unconstrained battery state of charge in hybrid power plants (Desai et al., 18 Feb 2025), or the absence of detailed theoretical validation in the excavator NMPC framework (Ma et al., 2024). In process-control settings, offset-free NMPC is noted as relevant but outside scope (Caspari et al., 23 Jun 2025).

Overall, HDDPC design is converging around a common principle: combine data-driven prediction with just enough structure to recover feasibility, interpretability, robustness, or mature MPC theory. The specific structural prior may be mechanistic equations, partial state-space knowledge, causal multistep prediction, invariant-set certification, hybrid contact logic, or a safety fallback. What unifies the field is not a single algorithmic template, but the systematic use of structure to make data-driven predictive control operational in regimes where purely black-box prediction is insufficient.