---
title: 'NPV-DeePC: Neural Parameter-Varying Predictive Control'
url: https://www.emergentmind.com/topics/neural-parameter-varying-data-enabled-predictive-control-npv-deepc
type: topic
---

# NPV-DeePC: Neural Parameter-Varying Predictive Control

Searching arXiv for recent 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 [2507.08259]. 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 [2406.08003].

## 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 [2507.08259].

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 [2507.08259].

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
\[
\Xi:=\operatorname{col}\big(\mathbf u(k),\mathbf y(k),\mathbf g(k),\boldsymbol\sigma(k)\big),
\]
and projection regularizer
\[
\ell_g(\mathbf g(k)):=\|(I-\Pi)\mathbf g(k)\|^2,\qquad
\Pi=
\begin{bmatrix}
\mathcal U_p\\ \mathcal Y_p\\ \mathcal U_f
\end{bmatrix}^\dagger
\begin{bmatrix}
\mathcal U_p\\ \mathcal Y_p\\ \mathcal U_f
\end{bmatrix}.
\]
NPV-DeePC preserves the predictive-control structure but replaces the fixed raw-trajectory representation by an adaptive neural feature-space representation [2507.08259].

## 2. Parameter-varying neural feature-space formulation

The predictive backbone is posed in NARX-style multi-step form,
\[
\mathbf y(k)=\mathcal F\big(\mathbf u_{\text{ini}}(k),\mathbf y_{\text{ini}}(k),\mathbf u(k)\big),
\]
with
\[
\mathbf u_{\text{ini}}(k):=\operatorname{col}\big(u(k-T_{\text{ini}}),\dots,u(k-1)\big),
\]
\[
\mathbf y_{\text{ini}}(k):=\operatorname{col}\big(y(k-T_{\text{ini}}),\dots,y(k-1)\big),
\]
\[
\mathbf u(k):=\operatorname{col}\big(u(k),\dots,u(k+N-1)\big),
\qquad
\mathbf y(k):=\operatorname{col}\big(y(k),\dots,y(k+N-1)\big).
\]
NPV-DeePC augments this with a parameter trajectory
\[
\mathbf p(k):=\operatorname{col}\big(p(k-T_{\text{ini}}),\dots,p(k-1)\big),
\]
and defines the target-network input
\[
\mathbf u_{NN}(k):=\operatorname{col}\big(\mathbf u_{\text{ini}}(k),\mathbf y_{\text{ini}}(k),\mathbf u(k)\big),
\qquad
\mathbf y_{NN}(k):=\mathbf y(k).
\]
The output of the learned predictor is affine in the final hidden-layer features:
\[
\phi_{NN}(\mathbf u_{NN},\mathbf p)=W_o\,\phi_{HL}(\mathbf u_{NN},\mathbf p)+b_o.
\]
Here \(\phi_{HL}\) is the parameter-dependent hidden-layer feature map, and \(\phi_{NN}\) is the full predictor map [2507.08259].

A Hankel-like data matrix
\[
\mathscr H:=\operatorname{col}(\mathcal U_p,\mathcal Y_p,\mathcal U_f)
\]
is transformed columnwise into feature-space matrices
\[
\boldsymbol\Phi_{HL}:=
\begin{bmatrix}
\phi_{HL}(\mathscr H_{:1}) & \cdots & \phi_{HL}(\mathscr H_{:L})
\end{bmatrix},
\]
\[
\boldsymbol\Phi_{NN}:=
\begin{bmatrix}
\phi_{NN}(\mathscr H_{:1}) & \cdots & \phi_{NN}(\mathscr H_{:L})
\end{bmatrix}.
\]
The resulting NPV-DeePC prediction set is
\[
\mathbf y^{\text{NPV-DeePC}}(k)\in
\left\{
\mathcal Y_f\,\mathbf g(k):
\begin{bmatrix}
\boldsymbol\Phi_{HL}\\ \mathbf 1^\top
\end{bmatrix}
\mathbf g(k)=
\begin{bmatrix}
\phi_{HL}(\mathbf u_{NN}(k),\mathbf p(k))\\ 1
\end{bmatrix}
\right\}.
\]
Equivalently, with \(\phi_{HL}(k):=\phi_{HL}(\mathbf u_{NN}(k),\mathbf p(k))\),
\[
\begin{bmatrix}
\boldsymbol\Phi_{HL}\\ \mathbf 1^\top\\ \mathcal Y_f
\end{bmatrix}
\mathbf g(k)=
\begin{bmatrix}
\phi_{HL}(k)\\ 1\\ \mathbf y(k)
\end{bmatrix}.
\]
This is the defining behavioral relation of NPV-DeePC: future outputs remain of the form \(\mathcal Y_f\mathbf g\), but consistency is enforced in an adaptive neural feature space rather than directly in the raw signal space [2507.08259].

## 3. Hypernetwork mechanism and adaptive representation

The parameter variation enters through a hypernetwork \(\mathfrak H_\psi\) that generates the hidden-layer parameters of a target network \(\mathfrak T_\theta\). For each hidden layer,
\[
\theta_i:=[W_i\ \ b_i]=\mathfrak h_i(\mathbf p),
\]
and the collection of generated parameters is
\[
\boldsymbol\theta_{\mathfrak H}:=\{\theta_i\}_{i=1}^l=\mathfrak H_\psi(\mathbf p).
\]
The output-layer parameters remain fixed,
\[
\theta_o:=[W_o\ \ b_o].
\]
The target-network recursion is
\[
\mathbf z_1:=\boldsymbol\sigma\big(W_1(\mathbf p)\mathbf u_{NN}+b_1(\mathbf p)\big),
\]
\[
\mathbf z_i:=\boldsymbol\sigma\big(W_i(\mathbf p)\mathbf z_{i-1}+b_i(\mathbf p)\big),\qquad i=2,\dots,l,
\]
\[
\hat{\mathbf y}_{NN}:=W_o\mathbf z_l(\mathbf p)+b_o.
\]
Consequently,
\[
\phi_{NN}(\mathbf u_{NN},\mathbf p)=W_o\phi_{HL}(\mathbf u_{NN},\mathbf p)+b_o.
\]
The essential point is that the hidden-layer transformation itself changes with \(\mathbf p\); the neural basis is not fixed, but parameter-conditioned [2507.08259].

This construction distinguishes NPV-DeePC from several neighboring DeePC extensions. Neural DeePC learns a feature-space affine interpolation but keeps the feature map fixed [2406.08003]. 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 [2509.26334]. Kernelized operator DeePC learns a direct finite-horizon operator over \((x,\mathbf u)\) in a product RKHS, again without neural hypernetwork conditioning [2501.17500]. 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 \(\{u(i),y(i),p(i)\}_{i=0}^{N_d-1}\) and trains the HyperDNN by minimizing
\[
(\theta_o^*,\boldsymbol\theta_{\mathfrak T}^*,\boldsymbol\psi^*)
=
\arg\min
\left\|
\mathcal Y_f-\boldsymbol\Phi_{NN}(\theta_o,\boldsymbol\theta_{\mathfrak T},\boldsymbol\psi)
\right\|_F^2.
\]
After that, the output layer is refined by least squares:
\[
\theta_o^{LS}
=
\arg\min
\left\|
\mathcal Y_f-
\theta_o
\begin{bmatrix}
\boldsymbol\Phi_{HL}(\boldsymbol\theta_{\mathfrak T}^*,\boldsymbol\psi^*) & \mathbf 1_{L\times 1}^\top
\end{bmatrix}
\right\|_F^2.
\]
The resulting NLS predictor is
\[
\mathbf y^{\text{NLS}}(k)=
\phi_{NN}\big(\mathbf u_{NN}(k),\mathbf p(k),\boldsymbol\theta_{\mathfrak T}^*,\boldsymbol\psi^*,\theta_o^{LS}\big).
\]
In the reported implementation, the target network has a single fully connected hidden layer of size \(\nu_L=30\) with \(\tanh\) activation; the hypernetwork takes the tip-to-surface distance as input and has no hidden layers. Training uses ADAM with learning rate \(10^{-3}\) and a 65%/35% train/validation split [2507.08259].

The final online NPV-DeePC problem uses a reduced correction variable \(\tilde{\mathbf g}(k)\in\mathbb R^{n_yN}\), assuming \(\mathcal Y_f\) has full row rank:
\[
\begin{aligned}
\min_{\Xi}\quad & J(u,y)+\lambda_g\|\tilde{\mathbf g}(k)\|^2,\\
\text{s.t.}\quad &
\begin{bmatrix}
\boldsymbol\Phi_{HL}\\ \mathbf 1^\top
\end{bmatrix}
\mathcal Y_f^\dagger \tilde{\mathbf g}(k)=\mathbf 0,\\
&
\mathcal Y_f
\begin{bmatrix}
\boldsymbol\Phi_{HL}\\ \mathbf 1^\top
\end{bmatrix}^\dagger
\begin{bmatrix}
\phi_{HL}(k)\\ 1
\end{bmatrix}
+\tilde{\mathbf g}(k)=\mathbf y(k),\\
&
(\mathbf u(k),\mathbf y(k))\in\mathbb U^N\times\mathbb Y^N,
\end{aligned}
\]
with
\[
\Xi:=\operatorname{col}\big(\mathbf u(k),\mathbf y(k),\tilde{\mathbf g}(k)\big).
\]
The paper states that the resulting problem has decision variables \((n_u+2n_y)N+\nu_L+1\), equality constraints \(n_yN+\nu_L+1\), and inequality constraints \(2(n_u+n_y)N\) [2507.08259].

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 [2408.16338]. 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 [2406.08003]. 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 [2505.03346]. 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 [2501.17500].

A concise taxonomy is useful.

| Method | Main mechanism | Relation to NPV-DeePC |
|---|---|---|
| Neural DeePC [2406.08003] | Fixed learned feature-space affine interpolation | Immediate precursor |
| DeePC-GS [2509.26334] | Scheduling-variable selection of local Hankel libraries | Non-neural operating-point dependence |
| Kernelized operator DeePC [2501.17500] | Product-RKHS operator \( (\mathbf u,x)\mapsto \mathbf y \) | Nonparametric nonlinear predictor |
| Koopman-bilinear DeePC [2505.03346] | Lifted bilinear trajectory representation | Structured nonlinear alternative |
| Deep DeePC [2408.16338] | 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 [2407.16066]. 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 [2512.00276]. 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 \(N_d=20{,}000\) 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, \(T_{\text{ini}}=5\), \(N=10\), and a trajectory of length \(K=1000\) is used to construct \(\mathscr H\) for the transformed feature-space basis; standard DeePC uses \(K=300\) because of computational burden. The controller enforces
\[
1.5\le P\le 8.0,\qquad 1.0\le q\le 6.0,
\]
\[
25\le T_s\le 42.5,\qquad 20\le T_g\le 80,
\]
with units reported as W, slm, and \(^{\circ}\)C, respectively [2507.08259].

The learned HyperDNN attains a training BFR of \(88.66\%\) and a testing BFR of \(88.31\%\). 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 \(0.14^\circ\mathrm C\) without measurement noise and \(0.48^\circ\mathrm C\) with Gaussian measurement noise of \(\sigma_T=0.2^\circ\mathrm C\); the corresponding MPC RMSE values are \(0.68^\circ\mathrm C\) and \(0.84^\circ\mathrm C\). The reported ISE values are \(9.57\) and \(39.98\) for NPV-DeePC, versus \(60.97\) and \(74.70\) for MPC. Mean CPU time is about \(0.2\) s for NPV-DeePC, about twice neural DeePC, significantly lower than standard DeePC, and about \(5\) ms for MPC [2507.08259].

In thermal-dose delivery, with
\[
\mathrm{CEM}(k+1)=\mathrm{CEM}(k)+\kappa^{(43-T_s(k))}\delta t,
\qquad
\kappa=
\begin{cases}
0.5,& T_s\ge 35^\circ\mathrm C,\\
0,& \text{otherwise},
\end{cases}
\]
and terminal cost
\[
J=\|\mathrm{CEM}_T-\mathrm{CEM}(N)\|_2^2,
\]
NPV-DeePC maintains a more consistent dose-delivery rate during parameter changes, whereas standard DeePC overshoots and is described as posing a safety risk [2507.08259].

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 \(\operatorname{col}(\boldsymbol\Phi_{HL},\mathbf 1^\top)\) and \(\mathcal Y_f\), 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 [2507.08259]. 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 [2507.08259].

Source: https://www.emergentmind.com/topics/neural-parameter-varying-data-enabled-predictive-control-npv-deepc