NPV-DeePC: Neural Parameter-Varying Predictive Control

• NPV-DeePC is a neural parameter-varying extension of DeePC that employs hypernetwork conditioning to adapt the feature space for varying operating conditions.
• It formulates receding-horizon predictive control using adaptive, Hankel-like data representations and constrained optimization tailored for nonlinear systems.
• Validated on cold atmospheric pressure plasma jets, NPV-DeePC demonstrates superior tracking performance with lower RMSE and ISE compared to traditional methods.

Searching arXiv for papers on NPV-DeePC and closely related DeePC variants to ground the article in current literature. arXiv_search(query="Neural Parameter-varying Data-enabled Predictive Control DeePC", max_results=10) Neural Parameter-Varying Data-enabled Predictive Control (NPV-DeePC) is a hypernetwork-conditioned, parameter-varying extension of neural Data-enabled Predictive Control for nonlinear systems whose dynamics change with operating condition. In the formulation introduced for cold atmospheric pressure plasma jets (APPJs), NPV-DeePC adapts the neural feature space through hyper neural networks, preserves a DeePC-like behavioral trajectory representation in that adaptive feature space, and embeds the resulting predictor in a constrained finite-horizon optimal control problem (GhafGhanbari et al., 11 Jul 2025). Its immediate antecedent is neural DeePC, which replaced the classical linear trajectory span of DeePC by affine interpolation in a learned neural feature space; NPV-DeePC adds explicit parameter dependence to that feature-space construction rather than using a fixed learned lifting (Lazar, 2024).

1. Conceptual setting and control objective

NPV-DeePC is formulated for nonlinear systems of the form

$$\begin{aligned} x(k+1) &= f(x(k),u(k)),\ y(k) &= h(x(k),u(k)), \end{aligned}$$

with constrained receding-horizon control based on the cost

$$J(u,y):=\ell_t(y(N|k))+\sum_{i=0}^{N-1}\ell_s(y(i|k),u(i|k)),$$

where

$$\ell_t(y(N|k)):=\|y(N|k)-r(k)\|_{\mathsf P}^2,\qquad \ell_s(u(i|k),y(i|k)):=\|y(i|k)-r(k)\|_{\mathsf Q}^2+\|\Delta u(i|k)\|_{\mathsf R}^2.$$

In the APPJ study, the manipulated input is

$$u=\begin{bmatrix}P\ q\end{bmatrix},$$

with $P$ the applied power and $q$ the gas flow rate, and the measured output is

$$y=\begin{bmatrix}T_s\ T_g\end{bmatrix},$$

with $T_s$ the substrate temperature and $T_g$ the gas temperature (GhafGhanbari et al., 11 Jul 2025).

The framework is motivated by two limitations. First, standard DeePC is exact for controllable LTI systems under Willems’ Fundamental Lemma, but APPJs are nonlinear and parameter-varying; a single fixed Hankel-based LTI trajectory representation becomes inaccurate when operating conditions shift. Second, ordinary neural DeePC learns a fixed nonlinear feature map, which can still be too rigid when dynamics vary strongly with a scheduling-like operating parameter. In the APPJ setting, the decisive varying condition is the tip-to-surface distance, which changes plume behavior, heat transfer, surface-temperature evolution, and thermal-dose accumulation (GhafGhanbari et al., 11 Jul 2025).

The starting point remains the standard DeePC program

$$\begin{aligned} \min_{\Xi}\quad & J(u,y)+\lambda_g \ell_g(\mathbf g(k))+\lambda_\sigma \|\boldsymbol\sigma(k)\|^2,\ \text{s.t.}\quad & \begin{bmatrix} \mathcal U_p\ \mathcal Y_p\ \mathcal U_f\ \mathcal Y_f \end{bmatrix}\mathbf g = \begin{bmatrix} \mathbf u_{\text{ini}}(k)\ \mathbf y_{\text{ini}}(k)+\boldsymbol\sigma(k)\ \mathbf u(k)\ \mathbf y(k) \end{bmatrix}, \end{aligned}$$

with optimization vector

$$J(u,y):=\ell_t(y(N|k))+\sum_{i=0}^{N-1}\ell_s(y(i|k),u(i|k)),$$0

and projection regularizer

$$J(u,y):=\ell_t(y(N|k))+\sum_{i=0}^{N-1}\ell_s(y(i|k),u(i|k)),$$1

NPV-DeePC preserves the predictive-control structure but replaces the fixed raw-trajectory representation by an adaptive neural feature-space representation (GhafGhanbari et al., 11 Jul 2025).

2. Parameter-varying neural feature-space formulation

The predictive backbone is posed in NARX-style multi-step form,

$$J(u,y):=\ell_t(y(N|k))+\sum_{i=0}^{N-1}\ell_s(y(i|k),u(i|k)),$$2

with

$$J(u,y):=\ell_t(y(N|k))+\sum_{i=0}^{N-1}\ell_s(y(i|k),u(i|k)),$$3

$$J(u,y):=\ell_t(y(N|k))+\sum_{i=0}^{N-1}\ell_s(y(i|k),u(i|k)),$$4

$$J(u,y):=\ell_t(y(N|k))+\sum_{i=0}^{N-1}\ell_s(y(i|k),u(i|k)),$$5

NPV-DeePC augments this with a parameter trajectory

$$J(u,y):=\ell_t(y(N|k))+\sum_{i=0}^{N-1}\ell_s(y(i|k),u(i|k)),$$6

and defines the target-network input

$$J(u,y):=\ell_t(y(N|k))+\sum_{i=0}^{N-1}\ell_s(y(i|k),u(i|k)),$$7

The output of the learned predictor is affine in the final hidden-layer features: $$J(u,y):=\ell_t(y(N|k))+\sum_{i=0}^{N-1}\ell_s(y(i|k),u(i|k)),$$8 Here $$J(u,y):=\ell_t(y(N|k))+\sum_{i=0}^{N-1}\ell_s(y(i|k),u(i|k)),$$9 is the parameter-dependent hidden-layer feature map, and $$\ell_t(y(N|k)):=\|y(N|k)-r(k)\|_{\mathsf P}^2,\qquad \ell_s(u(i|k),y(i|k)):=\|y(i|k)-r(k)\|_{\mathsf Q}^2+\|\Delta u(i|k)\|_{\mathsf R}^2.$$0 is the full predictor map (GhafGhanbari et al., 11 Jul 2025).

A Hankel-like data matrix

$$\ell_t(y(N|k)):=\|y(N|k)-r(k)\|_{\mathsf P}^2,\qquad \ell_s(u(i|k),y(i|k)):=\|y(i|k)-r(k)\|_{\mathsf Q}^2+\|\Delta u(i|k)\|_{\mathsf R}^2.$$1

is transformed columnwise into feature-space matrices

$$\ell_t(y(N|k)):=\|y(N|k)-r(k)\|_{\mathsf P}^2,\qquad \ell_s(u(i|k),y(i|k)):=\|y(i|k)-r(k)\|_{\mathsf Q}^2+\|\Delta u(i|k)\|_{\mathsf R}^2.$$2

$$\ell_t(y(N|k)):=\|y(N|k)-r(k)\|_{\mathsf P}^2,\qquad \ell_s(u(i|k),y(i|k)):=\|y(i|k)-r(k)\|_{\mathsf Q}^2+\|\Delta u(i|k)\|_{\mathsf R}^2.$$3

The resulting NPV-DeePC prediction set is

$$\ell_t(y(N|k)):=\|y(N|k)-r(k)\|_{\mathsf P}^2,\qquad \ell_s(u(i|k),y(i|k)):=\|y(i|k)-r(k)\|_{\mathsf Q}^2+\|\Delta u(i|k)\|_{\mathsf R}^2.$$4

Equivalently, with $$\ell_t(y(N|k)):=\|y(N|k)-r(k)\|_{\mathsf P}^2,\qquad \ell_s(u(i|k),y(i|k)):=\|y(i|k)-r(k)\|_{\mathsf Q}^2+\|\Delta u(i|k)\|_{\mathsf R}^2.$$5,

$$\ell_t(y(N|k)):=\|y(N|k)-r(k)\|_{\mathsf P}^2,\qquad \ell_s(u(i|k),y(i|k)):=\|y(i|k)-r(k)\|_{\mathsf Q}^2+\|\Delta u(i|k)\|_{\mathsf R}^2.$$6

This is the defining behavioral relation of NPV-DeePC: future outputs remain of the form $$\ell_t(y(N|k)):=\|y(N|k)-r(k)\|_{\mathsf P}^2,\qquad \ell_s(u(i|k),y(i|k)):=\|y(i|k)-r(k)\|_{\mathsf Q}^2+\|\Delta u(i|k)\|_{\mathsf R}^2.$$7, but consistency is enforced in an adaptive neural feature space rather than directly in the raw signal space (GhafGhanbari et al., 11 Jul 2025).

3. Hypernetwork mechanism and adaptive representation

The parameter variation enters through a hypernetwork $$\ell_t(y(N|k)):=\|y(N|k)-r(k)\|_{\mathsf P}^2,\qquad \ell_s(u(i|k),y(i|k)):=\|y(i|k)-r(k)\|_{\mathsf Q}^2+\|\Delta u(i|k)\|_{\mathsf R}^2.$$8 that generates the hidden-layer parameters of a target network $$\ell_t(y(N|k)):=\|y(N|k)-r(k)\|_{\mathsf P}^2,\qquad \ell_s(u(i|k),y(i|k)):=\|y(i|k)-r(k)\|_{\mathsf Q}^2+\|\Delta u(i|k)\|_{\mathsf R}^2.$$9. For each hidden layer,

$$u=\begin{bmatrix}P\ q\end{bmatrix},$$0

and the collection of generated parameters is

$$u=\begin{bmatrix}P\ q\end{bmatrix},$$1

The output-layer parameters remain fixed,

$$u=\begin{bmatrix}P\ q\end{bmatrix},$$2

The target-network recursion is

$$u=\begin{bmatrix}P\ q\end{bmatrix},$$3

$$u=\begin{bmatrix}P\ q\end{bmatrix},$$4

$$u=\begin{bmatrix}P\ q\end{bmatrix},$$5

Consequently,

$$u=\begin{bmatrix}P\ q\end{bmatrix},$$6

The essential point is that the hidden-layer transformation itself changes with $$u=\begin{bmatrix}P\ q\end{bmatrix},$$7; the neural basis is not fixed, but parameter-conditioned (GhafGhanbari et al., 11 Jul 2025).

This construction distinguishes NPV-DeePC from several neighboring DeePC extensions. Neural DeePC learns a feature-space affine interpolation but keeps the feature map fixed (Lazar, 2024). DeePC-GS makes the predictor operating-condition-dependent by switching among local Hankel libraries indexed by a measurable scheduling variable, but it does so through a family of local data-driven predictors rather than an adaptive learned feature space (Guerrero et al., 30 Sep 2025). Kernelized operator DeePC learns a direct finite-horizon operator over $$u=\begin{bmatrix}P\ q\end{bmatrix},$$8 in a product RKHS, again without neural hypernetwork conditioning (Jong et al., 29 Jan 2025). This suggests that NPV-DeePC occupies a distinct position: it is parameter-varying in the representation itself, not merely in data selection or in local-library choice.

4. Offline learning and online receding-horizon optimization

The offline stage uses open-loop data $$u=\begin{bmatrix}P\ q\end{bmatrix},$$9 and trains the HyperDNN by minimizing

$P$0

After that, the output layer is refined by least squares: $P$1 The resulting NLS predictor is

$P$2

In the reported implementation, the target network has a single fully connected hidden layer of size $P$3 with $P$4 activation; the hypernetwork takes the tip-to-surface distance as input and has no hidden layers. Training uses ADAM with learning rate $P$5 and a 65%/35% train/validation split (GhafGhanbari et al., 11 Jul 2025).

The final online NPV-DeePC problem uses a reduced correction variable $P$6, assuming $P$7 has full row rank: $P$8 with

$P$9

The paper states that the resulting problem has decision variables $q$0, equality constraints $q$1, and inequality constraints $q$2 (GhafGhanbari et al., 11 Jul 2025).

The method is therefore not optimization-free. This separates it from Deep DeePC, which learns the DeePC coefficient vector directly and can operate with low or no online optimization (Zhang et al., 2024). A plausible implication is that NPV-DeePC preserves more explicit predictive-control structure, while Deep DeePC shifts more of the burden into a feedforward neural surrogate.

5. Position within the DeePC landscape

NPV-DeePC is most naturally situated among nonlinear DeePC methods that modify the trajectory representation rather than merely reusing the classical LTI Hankel constraint. Neural DeePC provides the direct conceptual scaffold: a DNN learns a neural space in which the output layer performs affine interpolation, and DeePC solves for interpolation weights online (Lazar, 2024). Koopman-bilinear DeePC likewise replaces the LTI span relation by a structured nonlinear lifted representation, but its nonlinearity is induced by a Koopman bilinear realization and consistency constraints rather than by a hypernetwork-conditioned feature map (Xiong et al., 6 May 2025). Kernelized operator DeePC adopts an operator-learning viewpoint in a product RKHS, again providing a nonlinear, horizon-wise predictor without explicit parameter-varying neural conditioning (Jong et al., 29 Jan 2025).

A concise taxonomy is useful.

Method	Main mechanism	Relation to NPV-DeePC
Neural DeePC (Lazar, 2024)	Fixed learned feature-space affine interpolation	Immediate precursor
DeePC-GS (Guerrero et al., 30 Sep 2025)	Scheduling-variable selection of local Hankel libraries	Non-neural operating-point dependence
Kernelized operator DeePC (Jong et al., 29 Jan 2025)	Product-RKHS operator $q$3	Nonparametric nonlinear predictor
Koopman-bilinear DeePC (Xiong et al., 6 May 2025)	Lifted bilinear trajectory representation	Structured nonlinear alternative
Deep DeePC (Zhang et al., 2024)	DNN predicts DeePC operator directly	Reduces or removes online optimization

Two additional neighboring lines sharpen the distinction. Online reduced-order DeePC adapts to evolving dynamics by updating the data matrix online using informative real-time signals, without neural scheduling or feature adaptation (Vahidi-Moghaddam et al., 2024). Datamodel-based selection for nonlinear DeePC learns a context-dependent importance score over Hankel columns, making the active data support context-dependent, but the learned object is a selector rather than a parameter-varying predictor (Li et al., 29 Nov 2025). In that sense, NPV-DeePC is closer to a parameter-varying representation-learning method than to a data-selection or gain-scheduling heuristic.

6. APPJ validation, theoretical status, and open issues

The APPJ evaluation uses $q$4 open-loop data points generated from a validated high-fidelity model, with uniformly distributed inputs and piecewise-constant random variation in the tip-to-surface distance. For the tracking study, $q$5, $q$6, and a trajectory of length $q$7 is used to construct $q$8 for the transformed feature-space basis; standard DeePC uses $q$9 because of computational burden. The controller enforces

$$y=\begin{bmatrix}T_s\ T_g\end{bmatrix},$$0

$$y=\begin{bmatrix}T_s\ T_g\end{bmatrix},$$1

with units reported as W, slm, and $$y=\begin{bmatrix}T_s\ T_g\end{bmatrix},$$2C, respectively (GhafGhanbari et al., 11 Jul 2025).

The learned HyperDNN attains a training BFR of $$y=\begin{bmatrix}T_s\ T_g\end{bmatrix},$$3 and a testing BFR of $$y=\begin{bmatrix}T_s\ T_g\end{bmatrix},$$4. In surface-temperature tracking, NPV-DeePC is reported as the only controller that effectively adapts to dynamically changing tip-to-surface distance and closely tracks the desired trajectory. Its RMSE is $$y=\begin{bmatrix}T_s\ T_g\end{bmatrix},$$5 without measurement noise and $$y=\begin{bmatrix}T_s\ T_g\end{bmatrix},$$6 with Gaussian measurement noise of $$y=\begin{bmatrix}T_s\ T_g\end{bmatrix},$$7; the corresponding MPC RMSE values are $$y=\begin{bmatrix}T_s\ T_g\end{bmatrix},$$8 and $$y=\begin{bmatrix}T_s\ T_g\end{bmatrix},$$9. The reported ISE values are $T_s$0 and $T_s$1 for NPV-DeePC, versus $T_s$2 and $T_s$3 for MPC. Mean CPU time is about $T_s$4 s for NPV-DeePC, about twice neural DeePC, significantly lower than standard DeePC, and about $T_s$5 ms for MPC (GhafGhanbari et al., 11 Jul 2025).

In thermal-dose delivery, with

$T_s$6

and terminal cost

$T_s$7

NPV-DeePC maintains a more consistent dose-delivery rate during parameter changes, whereas standard DeePC overshoots and is described as posing a safety risk (GhafGhanbari et al., 11 Jul 2025).

The current theoretical status is materially narrower than the empirical results. The framework relies on sufficient offline data across the parameter range, on full-row-rank conditions for $T_s$8 and $T_s$9, and on online availability of the varying parameter. The paper explicitly identifies stability, recursive feasibility, and robustness as future work. It also reports only a single varying parameter in simulation, assumes noise-free training data, and provides no hardware experiments (GhafGhanbari et al., 11 Jul 2025). Accordingly, NPV-DeePC should not be conflated with a completed LPV-MPC theory, with optimization-free neural control, or with a purely local gain-scheduled DeePC architecture. It is, more precisely, a parameter-varying neural-feature-space extension of DeePC whose principal novelty is that the DeePC-compatible representation itself adapts with operating condition (GhafGhanbari et al., 11 Jul 2025).