%======================================================================
% GLOBAL CONSISTENCY CLEAN VERSION (Five-stand, thickness+tension only,
% actuators: roll gap + stand speed; Δx = deviation, Δu = increment)
%======================================================================

%========================
\section{Construction of Dataset}
%========================

\subsection{Five-stand tandem mill setting and unified notation}
\label{subsec:notation_clean}

Consider a five-stand tandem cold rolling mill indexed by $i\in\{1,2,3,4,5\}$.
Let $t_n$ denote the $n$-th sampling instant and $\delta_n=t_{n+1}-t_n$ the corresponding sampling interval length.
Define the local within-interval time variable $\tau=t-t_n\in[0,\delta_n]$.

\paragraph{Thickness and tension signals (what is controlled).}
Let $h_i(t)$ denote the exit thickness of stand $i$ ($i=1,\dots,5$),
and let $T_i(t)$ denote the inter-stand strip tension between stand $i$ and $i+1$ ($i=1,\dots,4$).
The system is strongly coupled because the inter-stand tensions propagate along the mill line and are affected by neighboring stands' actions.

\paragraph{Reference trajectories and deviation-state definition (meaning of $\Delta x$).}
Let $h_i^{\mathrm{ref}}(t)$ and $T_i^{\mathrm{ref}}(t)$ be the desired references (setpoints) given by process requirements
(e.g., schedule-based references or constant setpoints).
We define deviation (tracking-error) variables
\begin{equation}
\Delta h_i(t)\triangleq h_i(t)-h_i^{\mathrm{ref}}(t),\qquad
\Delta T_i(t)\triangleq T_i(t)-T_i^{\mathrm{ref}}(t).
\label{eq:dev_def}
\end{equation}
Throughout this paper, the symbol ``$\Delta$'' attached to \emph{states} always means \emph{deviation from reference}.

\paragraph{Local state vector (where thickness and tension appear).}
For each stand $i$, we choose the local deviation state as
\begin{equation}
\Delta x_i(t)\triangleq
\begin{bmatrix}
\Delta h_i(t)\
\Delta T_{i-1}(t)\
\Delta T_i(t)
\end{bmatrix}\in\mathbb{R}^{d},\qquad d=3,
\label{eq:xi_def_clean}
\end{equation}
with the boundary convention $\Delta T_0(t)\equiv 0$ and $\Delta T_5(t)\equiv 0$ to keep a unified dimension $d=3$ for all stands.
(Equivalently, one may remove nonexistent boundary tensions from the state and use varying dimensions; here we keep a unified form.)

\paragraph{Neighbor sets and coupling representation (five-stand chain).}
For a five-stand tandem mill, the dominant coupling is between adjacent stands, hence we define
\begin{equation}
Z_1={2},\quad
Z_i={i-1,i+1}\ (i=2,3,4),\quad
Z_5={4}.
\label{eq:Zi_clean}
\end{equation}
Define the neighbor-state stack
\begin{equation}
\Delta x_{Z_i}(t_n)=\mathrm{col}{\Delta x_k(t_n),|,k\in Z_i}.
\label{eq:xZi_clean}
\end{equation}

\paragraph{Actuators and increment inputs (meaning of $u$ and $\Delta u$).}
Each stand $i$ is manipulated by \emph{roll gap} (screw-down/hydraulic gap) $s_i(t)$ and \emph{stand speed} $v_i(t)$:
\begin{equation}
u_i(t)=
\begin{bmatrix}
s_i(t)\
v_i(t)
\end{bmatrix}\in\mathbb{R}^{n_u},\qquad n_u=2.
\label{eq:ui_clean}
\end{equation}
To ensure smooth actuation and match industrial practice, we optimize \emph{discrete input increments}:
\begin{equation}
\Delta u_i(t_n)\triangleq u_i(t_n)-u_i(t_{n-1})

\begin{bmatrix}
\Delta s_i(t_n)\
\Delta v_i(t_n)
\end{bmatrix}.
\label{eq:du_discrete_clean}
\end{equation}
Throughout this paper, the symbol ``$\Delta$'' attached to \emph{inputs} $\Delta u_i(t_n)$ means \emph{sample-to-sample increment}.
Thus, $\Delta x$ (deviation state) and $\Delta u$ (input increment) are conceptually different, and this is fixed by definition.

\paragraph{Disturbance.}
Let $d_i(t)$ denote exogenous disturbances (e.g., entry thickness fluctuation, friction variation, material parameter drift, etc.).
We denote the interval-level equivalent disturbance by $\Delta d_i(t_n)$ (defined via interval averaging below).

\paragraph{Basic matrix notation.}
$I_d$ denotes the $d\times d$ identity matrix; $0_{a\times b}$ denotes the $a\times b$ zero matrix.

\subsection{Discrete interval mapping and data-driven learning objective}
\label{subsec:mapping_clean}

The stand-wise deviation-state evolution over $[t_n,t_{n+1}]$ can be expressed by a discrete-time mapping
\begin{equation}
\Delta x_i(t_{n+1})

\Phi_i\Big(\Delta x_i(t_n),,\Delta x_{Z_i}(t_n),,\Delta u_i([t_n,t_{n+1}]),,\Delta d_i([t_n,t_{n+1}])\Big),
\label{eq:true_mapping_clean}
\end{equation}
where $\Phi_i(\cdot)$ is generally nonlinear and coupled due to rolling deformation and tension propagation.
A commonly used \emph{conceptual} equivalent discrete linear form is
\begin{equation}
\Delta x_i(t_{n+1})

M_d,\Delta x_i(t_n)
+
N_d,\Delta u_i(t_n)
+
F_d,\Delta d_i(t_n),
\label{eq:linear_form_concept}
\end{equation}
where $M_d,N_d,F_d$ represent equivalent discrete-time matrices around operating conditions.
In a practical five-stand cold rolling mill, accurately deriving/identifying these matrices and disturbance models from first principles is difficult,
due to strong coupling, unmodeled nonlinearities, and time-varying operating regimes.
Therefore, this paper aims to learn a high-fidelity approximation of the interval evolution from data and then embed it into distributed MPC.

\begin{remark}
In fact, due to the existence of complex coupling relationships, it is difficult to directly and accurately establish \eqref{eq:linear_form_concept}
based on first principles. Therefore, in this paper, we learn an approximate mapping of \eqref{eq:true_mapping_clean} from data.
\end{remark}

%========================
\subsection{Interval-level parameterization and one-step dataset}
%========================

\paragraph{Why interval-level parameterization is reasonable in the five-stand setting.}
Although decisions are updated at discrete instants $t_n$, the hydraulic gap and drive systems evolve continuously inside each interval,
and abrupt within-interval changes may excite tension oscillations and deteriorate thickness stability.
Thus, parameterizing the within-interval increment trajectory by a low-order polynomial:
(i) yields a compact finite-dimensional decision representation;
(ii) enforces smooth profiles inside the interval;
(iii) enables enforcing increment constraints for all $\tau\in[0,\delta_n]$.
This is appropriate when $\delta_n$ is not excessively large relative to actuator bandwidth and the within-interval evolution is well approximated by a low-order basis.

\paragraph{Vector quadratic polynomial parameterization (two inputs).}
On the interval $[t_n,t_{n+1}]$, parameterize the control increment trajectory as
\begin{equation}
\Delta u_{i,n}(\tau;\Gamma_{i,n})

\Gamma_{i,n0}+\Gamma_{i,n1}\tau+\Gamma_{i,n2}\tau^2,
\qquad \tau\in[0,\delta_n],
\label{eq:du_poly_vec_clean}
\end{equation}
where $\Gamma_{i,n0},\Gamma_{i,n1},\Gamma_{i,n2}\in\mathbb{R}^{n_u}$ are coefficient vectors ($n_u=2$).
Component-wise, \eqref{eq:du_poly_vec_clean} corresponds to
\begin{equation}
\begin{aligned}
\Delta s_{i,n}(\tau) &= \gamma^{(s)}{i,n0}+\gamma^{(s)}{i,n1}\tau+\gamma^{(s)}{i,n2}\tau^2,\
\Delta v{i,n}(\tau) &= \gamma^{(v)}{i,n0}+\gamma^{(v)}{i,n1}\tau+\gamma^{(v)}{i,n2}\tau^2.
\end{aligned}
\label{eq:du_components_clean}
\end{equation}
Define the stacked parameter vector
\begin{equation}
\Gamma{i,n}\triangleq
\big[
(\Gamma_{i,n0})^\top,,
(\Gamma_{i,n1})^\top,,
(\Gamma_{i,n2})^\top
\big]^\top
\in\mathbb{R}^{p},
\qquad
p=3n_u=6.
\label{eq:Gamma_clean}
\end{equation}
Here, $\Gamma_{i,n0}$ is the baseline increment at $\tau=0$, while $\Gamma_{i,n1}$ and $\Gamma_{i,n2}$ describe the linear and quadratic variation rates.

\paragraph{Equivalent discrete-time (interval-averaged) increments.}
Define the interval-averaged equivalent increments as
\begin{equation}
\begin{aligned}
\Delta u_i(t_n) &\triangleq \frac{1}{\delta_n}\int_0^{\delta_n}\Delta u_{i,n}(\tau),d\tau,\
\Delta d_i(t_n) &\triangleq \frac{1}{\delta_n}\int_0^{\delta_n}\Delta d_i(\tau),d\tau.
\end{aligned}
\label{eq:avg_def_clean}
\end{equation}
With \eqref{eq:du_poly_vec_clean}, the input average has a closed form:
\begin{equation}
\Delta u_i(t_n)=
\Gamma_{i,n0}
+\Gamma_{i,n1}\frac{\delta_n}{2}
+\Gamma_{i,n2}\frac{\delta_n^2}{3}.
\label{eq:avg_closed_clean}
\end{equation}

\paragraph{Sampling domains for offline data generation.}
Let $\mathcal{I}_x$ denote the sampling domain (ranges) of deviation states $\Delta x_i(t_n)$ and neighbor stacks $\Delta x_{Z_i}(t_n)$,
and let $\mathcal{I}_\Gamma$ denote the sampling domain of polynomial parameters $\Gamma_{i,n}$ (covering both gap and speed channels).
These domains specify the operating envelope used to generate supervised training data.

\paragraph{One-step sample generation (five-stand coupled simulation).}
Given the above parameterization, one training sample is generated on each interval $[t_n,t_{n+1}]$.
In addition to the local deviation state, the neighbor deviation states are included to represent inter-stand coupling.
The process is summarized in Table~\ref{tab:interval_sample_generation_en}.

\begin{table}[t]
\centering
\small
\renewcommand{\arraystretch}{1.15}
\caption{Procedure for generating one interval-level sample on $[t_n,t_{n+1}]$ (five-stand coupled mill).}
\label{tab:interval_sample_generation_en}
\begin{tabularx}{\linewidth}{>{\centering\arraybackslash}p{0.09\linewidth} X}
\toprule
\textbf{Step} & \textbf{Operation} \
\midrule
1 & \textbf{State sampling:} sample $\Delta x_i(t_n)$ and $\Delta x_{Z_i}(t_n)$ from $\mathcal{I}_x$. \
2 & \textbf{Parameter sampling:} draw $\Gamma_{i,n}\sim\mathcal{I}_\Gamma$ (coefficients for both $\Delta s_{i,n}(\tau)$ and $\Delta v_{i,n}(\tau)$). \
3 & \textbf{Control construction:} compute $\Delta u_{i,n}(\tau)$ via \eqref{eq:du_poly_vec_clean}. \
4 & \textbf{State propagation:} integrate the \emph{five-stand coupled} mill model on $[t_n,t_{n+1}]$ (e.g., RK4) using the within-interval control trajectory, and record $\Delta x_i(t_{n+1})$. \
\bottomrule
\end{tabularx}
\end{table}

Accordingly, an interval sample for subsystem $i$ can be represented as
\begin{equation}
\mathcal{D}{i,n}=\big{\Delta x_i(t_n),\ \Delta x{Z_i}(t_n),\ \Delta u_{i,n}(\tau),\ \Delta x_i(t_{n+1})\big}.
\label{eq:interval_sample_clean}
\end{equation}
Note that $\Delta u_{i,n}(\tau)$ is fully determined by $(\Gamma_{i,n},\delta_n)$ via \eqref{eq:du_poly_vec_clean},
therefore it is sufficient to store $(\Gamma_{i,n},\delta_n)$ as the learning input.

For each subsystem $i$, by repeating the above procedure across multiple intervals and randomized draws,
the local one-step training dataset is formed as
\begin{equation}
\begin{split}
S_i=\Big{&
\big(\Delta x_i^{(j)}(t_n),,\Delta x_{Z_i}^{(j)}(t_n),,\Delta x_i^{(j)}(t_{n+1});,
\Gamma_{i,n}^{(j)},,\delta_n^{(j)}\big)
\ \Big|\ j=1,\ldots,J
\Big}.
\end{split}
\label{eq:S_i_clean}
\end{equation}
Here $J$ is the number of one-step samples for subsystem $i$.
The overall dataset for the five-stand mill is denoted by $\{S_i\}_{i=1}^{5}$.
The point-cloud visualization of the training dataset is shown in Figure~\ref{2}.

\begin{figure*}[htbp]
\centering
\includegraphics[scale=0.5]{picture/Fig2.pdf}
\caption{Point cloud map of the training dataset.}\label{2}
\end{figure*}

%========================
\subsection{Multi-step rollout segment dataset}
%========================

The one-step set $S_i$ is sufficient for one-step regression, but it is not sufficient for training with multi-step rollout loss
and reciprocal-consistency regularization, because these objectives require ground-truth deviation-state trajectories over a horizon of $K$ consecutive intervals.
Therefore, without changing the single-interval sampling mechanism above, we additionally organize the offline-simulated samples
into $K$-step trajectory segments.

Specifically, during offline simulation, for each starting time $t_n$ we generate a segment of length $K$ by consecutively sampling
$\{\Gamma_{i,n+s},\delta_{n+s}\}_{s=0}^{K-1}$ (and the corresponding inputs/disturbances),
and integrating the five-stand coupled mill model over $[t_{n+s},t_{n+s+1}]$ for $s=0,\ldots,K-1$.
Hence, we obtain the deviation-state sequence $\{\Delta x_i(t_{n+s})\}_{s=0}^{K}$ as well as the neighbor stacks
$\{\Delta x_{Z_i}(t_{n+s})\}_{s=0}^{K}$.

Define a $K$-step segment sample for subsystem $i$ as
\begin{equation}
\begin{aligned}
\mathcal{W}{i,n}=
\Big{&
\big(\Delta x_i(t{n+s}),,\Delta x_{Z_i}(t_{n+s}),,\Gamma_{i,n+s},,\delta_{n+s}\big){s=0}^{K-1}; \
&\big(\Delta x_i(t{n+s+1})\big){s=0}^{K-1}
\Big}.
\end{aligned}
\label{eq:segment_clean}
\end{equation}
By repeating the above segment generation, we form the multi-step training set
\begin{equation}
S_i^{(K)}=\Big{\mathcal{W}{i,n}^{(j)}\ \Big|\ j=1,\ldots,J_K\Big},
\label{eq:S_i_K_clean}
\end{equation}
where $J_K$ is the number of $K$-step segment samples.
Note that $S_i$ can be viewed as the marginal one-step projection of $S_i^{(K)}$ (keeping only $s=0$),
thus the original dataset design is preserved, and only an additional \emph{segment organization} is introduced for multi-step training.

%========================
\section{Construction of Residual Neural Network}
%========================

\subsection{Network architecture (what is learned, why residual, why include control)}
\label{subsec:net_clean}

\paragraph{Learning target (one-step controlled deviation-state evolution).}
Given the dataset, the neural network model is trained to learn a stand-wise, control-dependent one-step evolution law of deviation states:
\begin{equation}
\Delta x_i(t_{n+1})
\approx
\Delta x_i(t_n)+
\mathcal{N}i!\Big(\Delta x_i(t_n),,\Delta x{Z_i}(t_n),,\Gamma_{i,n},,\delta_n;,\Theta_i\Big),
\label{eq:learned_dyn_clean}
\end{equation}
where $\mathcal{N}_i(\cdot)$ outputs the one-step deviation-state change and $\Theta_i$ are trainable parameters.

\paragraph{Why the model must include $u$ (difference between with $u$'' and without $u$'').}
If $\mathcal{N}_i$ does not take control information as input (here $\Gamma_{i,n}$ and $\delta_n$),
the predictor becomes an autoregressive model that only reproduces trajectories under the training input patterns
and cannot answer the counterfactual question: ``what will happen if we choose a different roll gap/speed trajectory?''
Since MPC optimizes over candidate decisions, a control-dependent predictor \eqref{eq:learned_dyn_clean} is necessary
to evaluate the predicted thickness/tension behavior under different candidate actuator trajectories.

\paragraph{Input/output dimensions.}
Let $d=3$ (state dimension), $|Z_i|$ be the number of neighbors of stand $i$ in \eqref{eq:Zi_clean}, and $p=6$ in \eqref{eq:Gamma_clean}.
Define the input vector
\begin{equation}
X_{i,\text{in}} \triangleq
\big[
\Delta x_i(t_n)^\top,,
\Delta x_{Z_i}(t_n)^\top,,
\Gamma_{i,n}^\top,,
\delta_n
\big]^\top
\in \mathbb{R}^{d(1+|Z_i|)+p+1}.
\label{eq:X_in_clean}
\end{equation}
The network mapping is
\begin{equation}
\mathcal{N}_i:\mathbb{R}^{d(1+|Z_i|)+p+1}\rightarrow\mathbb{R}^{d}.
\end{equation}

\paragraph{Residual (shortcut) structure.}
To improve training stability and long-horizon rollout robustness, we use a residual form.
Let $\hat{I}_i\in\mathbb{R}^{d\times(d(1+|Z_i|)+p+1)}$ be a selection matrix extracting the local state block:
\begin{equation}
\hat{I}i = [I_d,, 0{d\times(d|Z_i|+p+1)}].
\label{eq:Ihat_clean}
\end{equation}
Then the one-step predictor is written as
\begin{equation}
X_{i,\text{out}} = \hat{I}i X{i,\text{in}} + \mathcal{N}i(X{i,\text{in}}; \Theta_i),
\label{eq:res_predict_clean}
\end{equation}
where $X_{i,\text{out}}$ represents the predicted $\Delta x_i(t_{n+1})$.
This structure implements a baseline-plus-correction interpretation:
the shortcut propagates the current deviation state $\Delta x_i(t_n)$, while the network learns the correction capturing
unmodeled nonlinearities and inter-stand coupling (via $\Delta x_{Z_i}$) under varying operating conditions.

\paragraph{Auxiliary branch for variable interval length (avoid symbol conflict).}
To improve robustness when $\delta_n$ varies, we introduce an auxiliary branch inside $\mathcal{N}_i$:
\begin{equation}
\mathcal{N}i(X{i,\text{in}};\Theta_i)\triangleq
\psi_i(X_{i,\text{in}};\Theta_{\psi_i}) + \rho_i(X_{i,\text{in}};\theta_i),
\label{eq:aux_clean}
\end{equation}
where $\psi_i(\cdot)$ is a lightweight feedforward branch that captures low-frequency/scale effects strongly related to $\delta_n$,
and $\rho_i(\cdot)$ captures the remaining nonlinear coupling corrections.
When $\psi_i(\cdot)\equiv 0$, the model reduces to a standard residual network.
(We use $\psi_i$ and $\rho_i$ to avoid notation conflicts with the sampling sets $\mathcal{I}_x,\mathcal{I}_\Gamma$ and the optimizer learning rate.)

\paragraph{One-step supervised target.}
For the $j$-th sample in \eqref{eq:S_i_clean}, define
\begin{equation}
X_{i,\text{in}}^{(j)} =
\big[
\Delta x_i^{(j)}(t_n),\ \Delta x_{Z_i}^{(j)}(t_n),\
\Gamma_{i,n}^{(j)},\ \delta_n^{(j)}
\big]^{\top},
\end{equation}
and the supervised residual target
\begin{equation}
\Delta r_i^{(j)}=\Delta x_i^{(j)}(t_{n+1})-\Delta x_i^{(j)}(t_n).
\label{eq:target_clean}
\end{equation}

\subsection{Training, learned model, and system prediction (multi-step stability and control usage)}
\label{subsec:train_clean}

To suppress accumulation drift induced by long-horizon recursion and to improve long-term predictive stability,
we train the forward predictor jointly with an auxiliary backward residual model
and impose a multi-step reciprocal-consistency regularization over a $K$-step segment from $S_i^{(K)}$.

\paragraph{Backward residual model.}
Construct a backward residual network
\begin{equation}
\mathcal{B}i:\mathbb{R}^{d(1+|Z_i|)+p+1}\rightarrow\mathbb{R}^{d},
\end{equation}
parameterized by $\bar{\Theta}_i$. For the backward step associated with interval $[t_n,t_{n+1}]$, define
\begin{equation}
\begin{aligned}
X{i,\mathrm{in}}^{b}
&=
\big[
\Delta x_i(t_{n+1}),\ \Delta x_{Z_i}(t_{n+1}),
\Gamma_{i,n},\ \delta_n
\big]^{\top},\
X_{i,\mathrm{out}}^{b}
&=
\hat{I}i X{i,\mathrm{in}}^{b} + \mathcal{B}i(X{i,\mathrm{in}}^{b};\bar{\Theta}i),
\end{aligned}
\label{eq:back_clean}
\end{equation}
where $X_{i,\mathrm{out}}^{b}$ represents the backward estimate of $\Delta x_i(t_n)$.
The supervised backward residual target is
\begin{equation}
\Delta r_i^{b}=\Delta x_i(t_n)-\Delta x_i(t{n+1}).
\end{equation}

\paragraph{Forward rollout on a $K$-step segment.}
Given a segment sample $\mathcal{W}_{i,n}\in S_i^{(K)}$, initialize
\begin{equation}
\Delta \hat{x}i(t_n)=\Delta x_i(t_n),
\end{equation}
and recursively apply the forward predictor for $K$ steps:
\begin{equation}
\begin{aligned}
\Delta \hat{x}i(t{n+s+1})
&=
\Delta \hat{x}i(t{n+s})
+
\mathcal{N}i!\Big(
\Delta \hat{x}i(t{n+s}),,\Delta \hat{x}{Z_i}(t{n+s}),,
\Gamma_{i,n+s},,\delta_{n+s};,\Theta_i
\Big),\
&\qquad s=0,\ldots,K-1.
\end{aligned}
\label{eq:fwd_roll_clean}
\end{equation}

\paragraph{Backward rollout and reciprocal consistency.}
Set the terminal condition
\begin{equation}
\Delta \bar{x}i(t{n+K})=\Delta \hat{x}i(t{n+K}),
\end{equation}
and roll back using $\mathcal{B}_i$:
\begin{equation}
\begin{aligned}
\Delta \bar{x}i(t{n+s})
&=
\hat{I}i X{i,\mathrm{in}}^{b}(t_{n+s})
+
\mathcal{B}i!\Big(X{i,\mathrm{in}}^{b}(t_{n+s});,\bar{\Theta}i\Big),
\quad s=K-1,\ldots,0,
\end{aligned}
\label{eq:bwd_roll_clean}
\end{equation}
where
\begin{equation}
X{i,\mathrm{in}}^{b}(t_{n+s})=
\big[
\Delta \bar{x}i(t{n+s+1}),\ \Delta \hat{x}{Z_i}(t{n+s+1}),
\Gamma_{i,n+s},\ \delta_{n+s}
\big]^{\top}.
\end{equation}

Define the multi-step reciprocal prediction error
\begin{equation}
E_i(t_n)

\sum_{s=0}^{K}
\left|
\Delta \hat{x}i(t{n+s})-\Delta \bar{x}i(t{n+s})
\right|^2.
\end{equation}

\paragraph{Training objectives (meaning of each loss).}
We jointly minimize:
\begin{equation}
\begin{aligned}
L_{\mathrm{1step}}(\Theta_i)
&= \frac{1}{J_K}\sum_{j=1}^{J_K}\frac{1}{K}\sum_{s=0}^{K-1}
\Big|
\big(\Delta x_i^{(j)}(t_{n+s+1})-\Delta x_i^{(j)}(t_{n+s})\big)
-\mathcal{N}i!\left(
X{i,\mathrm{in}}^{(j)}(t_{n+s});\Theta_i
\right)
\Big|^2,\[2mm]
L_{\mathrm{bwd}}(\bar{\Theta}i)
&= \frac{1}{J_K}\sum{j=1}^{J_K}\frac{1}{K}\sum_{s=0}^{K-1}
\Big|
\big(\Delta x_i^{(j)}(t_{n+s})-\Delta x_i^{(j)}(t_{n+s+1})\big)
-\mathcal{B}i!\left(
X{i,\mathrm{in}}^{b,(j)}(t_{n+s});\bar{\Theta}i
\right)
\Big|^2,\[2mm]
L{\mathrm{msrp}}(\Theta_i,\bar{\Theta}i)
&= \frac{1}{J_K}\sum{j=1}^{J_K} E_i^{(j)}(t_n),\[2mm]
L_{\mathrm{roll}}(\Theta_i)
&= \frac{1}{J_K}\sum_{j=1}^{J_K}\sum_{s=1}^{K}
\Big|
\Delta x_i^{(j)}(t_{n+s})-\Delta \hat{x}i^{(j)}(t{n+s})
\Big|^2.
\end{aligned}
\label{eq:loss_clean}
\end{equation}
Here, $L_{\mathrm{1step}}$ enforces one-step accuracy; $L_{\mathrm{roll}}$ explicitly suppresses long-horizon drift under recursion;
$L_{\mathrm{msrp}}$ regularizes the learned dynamics by enforcing reciprocal consistency between forward and backward rollouts;
and $L_{\mathrm{bwd}}$ trains the backward model for the consistency regularization.
In implementation, these terms are combined as
\begin{equation}
L_{\mathrm{total}}=\lambda_1 L_{\mathrm{1step}}+\lambda_2 L_{\mathrm{roll}}+\lambda_3 L_{\mathrm{msrp}}+\lambda_4 L_{\mathrm{bwd}},
\end{equation}
where $\lambda_1,\lambda_2,\lambda_3,\lambda_4>0$ are tuned on a validation set.

\paragraph{Learned forward predictor used in control.}
After training, the forward predictor is
\begin{equation}
\Delta \hat{x}i(t{n+1})

\Delta x_i(t_n)
+
\mathcal{N}i!\Big(
\Delta x_i(t_n),,\Delta x{Z_i}(t_n),,
\Gamma_{i,n},,\delta_n;,\Theta_i^*
\Big),
\label{eq:pred_clean}
\end{equation}
and multi-step prediction is obtained by recursive rollout of \eqref{eq:pred_clean}.
This learned predictor is the internal model used by the MPC optimizer in the next section.

Finally, network parameters are optimized using Adam:
\begin{equation}
\Theta_{i,t+1} = \Theta_{i,t} - \alpha \frac{\hat{m}{i,t}}{\sqrt{\hat{v}{i,t}} + \varepsilon},
\end{equation}
where $\alpha$ is the learning rate (we use $\alpha$ to avoid conflict with other symbols),
$\hat{m}_{i,t}$ and $\hat{v}_{i,t}$ are bias-corrected moment estimates, and $\varepsilon>0$ is a small constant for numerical stability.
Figure~\ref{fig:rnn_logic} illustrates the overall structure.

\begin{figure}[htbp]
\centering
\includegraphics[scale=0.85]{picture/x6.pdf}
\caption{Logic diagram of the residual neural network.}
\label{fig:rnn_logic}
\end{figure}

%========================
\section{Nash Equilibrium-Based RNE-DMPC}
%========================

The five-stand tandem cold rolling system is strongly coupled through inter-stand tension propagation.
As a result, changes in control actions (roll gap and stand speed) at one stand can affect both upstream and downstream stands,
making centralized online optimization over all stands' decision variables computationally demanding.

To mitigate this issue, we decompose the global predictive-control problem into $N=5$ local subproblems associated with individual stands.
Each local controller optimizes its own decision variables while accounting for coupling via limited information exchange with neighboring controllers.
Motivated by game-theoretic coordination \citep{rawlings2008coordinating}, we formulate distributed coordination as a Nash-equilibrium-seeking iteration.
Based on the trained residual neural network surrogate model, we construct a Nash-equilibrium-based distributed MPC method (RNE-DMPC)
for coordinated thickness--tension regulation/tracking. The overall control structure is shown in Figure~\ref{4}.

\begin{figure*}[htbp]
\centering
\includegraphics[width=\linewidth]{picture/x2.pdf}
\caption{Schematic diagram of the control architecture for a tandem cold rolling mill.}\label{4}
\end{figure*}

\subsection{Prediction model used in MPC and the prediction--control interface}
\label{subsec:interface_clean}

\paragraph{Key idea: prediction serves control through model constraints.}
At each sampling time $t_n$, MPC evaluates candidate actuator trajectories (encoded by $\Gamma_{i,n+s}$) by rolling out predictions of
thickness/tension deviations using the learned surrogate \eqref{eq:pred_clean}.
Therefore, the learned predictor directly provides the multi-step forecasts required to compute the MPC objective and enforce constraints.

\paragraph{Local polynomial parameterization over the horizon.}
Over each interval $[t_{n+s},t_{n+s+1}]$ with length $\delta_{n+s}$, the control increment trajectory of stand $i$ is
\begin{equation}
\Delta u_{i,n+s}(\tau;\Gamma_{i,n+s})

\Gamma_{i,n+s,0}
+\Gamma_{i,n+s,1}\tau
+\Gamma_{i,n+s,2}\tau^2,\qquad \tau \in [0,\delta_{n+s}],
\label{eq:du_poly_mpc_clean}
\end{equation}
where $\Gamma_{i,n+s}\in\mathbb{R}^{p}$ with $p=6$.

\paragraph{Neural-network-based multi-step prediction inside MPC.}
Define the prediction horizon $N_p$ (number of future intervals predicted) and the control horizon $N_c$ (number of future intervals optimized),
with $N_c\le N_p$.
Given the current measured/estimated deviation states $\Delta x_i(t_n)$ and a candidate decision sequence
$\mathbf{\Gamma}_i(t_n)=\big[\Gamma_{i,n}^\top,\ldots,\Gamma_{i,n+N_c-1}^\top\big]^\top$,
stand $i$ predicts its deviation-state evolution by
\begin{equation}
\begin{aligned}
\Delta \hat{x}i(t{n+s+1})
&=
\Delta \hat{x}i(t{n+s})
+
\mathcal{N}i!\Big(
\Delta \hat{x}i(t{n+s}),,
\Delta \hat{x}{Z_i}(t_{n+s}),,
\Gamma_{i,n+s},,
\delta_{n+s};,
\Theta_i^*
\Big), \
&\qquad s=0,\ldots,N_p-1,
\end{aligned}
\label{eq:rollout_mpc_clean}
\end{equation}
with initialization $\Delta \hat{x}_i(t_n)=\Delta x_i(t_n)$.
Here $\Delta \hat{x}_{Z_i}(t_{n+s})$ is obtained from the latest communication with neighbors during the Nash iteration.
Equation \eqref{eq:rollout_mpc_clean} is the explicit mathematical interface: \textbf{control decisions} $(\Gamma_{i,n+s})$
$\rightarrow$ \textbf{predicted thickness/tension deviations} $(\Delta \hat{x}_i)$ $\rightarrow$ \textbf{objective/constraints evaluation}.

\subsection{Local optimization problem (objective, reference meaning, constraints, and numerical solution)}
\label{subsec:local_opt_clean}

\paragraph{Decision variables.}
At time $t_n$, the local decision vector for stand $i$ is
\begin{equation}
\mathbf{\Gamma}_i(t_n)

\big[
\Gamma_{i,n}^\top,,
\Gamma_{i,n+1}^\top,,
\ldots,,
\Gamma_{i,n+N_c-1}^\top
\big]^\top
\in \mathbb{R}^{pN_c}.
\label{eq:Gamma_seq_clean}
\end{equation}

\paragraph{Reference meaning (remove ambiguity of $\Delta x_{\mathrm{ref}}$).}
Because the deviation state is defined as $\Delta x_i(t)=x_i(t)-x_i^{\mathrm{ref}}(t)$ in \eqref{eq:dev_def} and \eqref{eq:xi_def_clean},
the desired regulation/tracking objective in deviation coordinates is always
\begin{equation}
\Delta x_i(t)\rightarrow 0.
\end{equation}
Therefore, the reference in deviation form is simply the zero vector, i.e.,
\begin{equation}
\Delta x_{i,\mathrm{ref}}(t_{n+s})\equiv 0\in\mathbb{R}^{d}.
\label{eq:dxref_zero_clean}
\end{equation}
Equivalently, one can view the cost as penalizing $(\hat x_i-x_i^{\mathrm{ref}})$ in absolute coordinates; in this paper we keep the deviation formulation.

\paragraph{Local objective (explicitly thickness+tension).}
Let $\Delta \hat{x}_i(t_{n+s})=[\Delta \hat h_i(t_{n+s}),\,\Delta \widehat T_{i-1}(t_{n+s}),\,\Delta \widehat T_i(t_{n+s})]^\top$.
The local cost is
\begin{equation}
\begin{aligned}
J_i
&=
\sum_{s=1}^{N_p}
\big|
\Delta \hat{x}i(t{n+s}) - \Delta x_{i,\mathrm{ref}}(t_{n+s})
\big|{Q_i}^2
+
\sum{s=0}^{N_c-1}
\big|
\Gamma_{i,n+s}
\big|_{R_i}^2,
\end{aligned}
\label{eq:Ji_clean}
\end{equation}
where $Q_i\in\mathbb{R}^{d\times d}$ weights thickness and tension deviations,
and $R_i\in\mathbb{R}^{p\times p}$ penalizes the polynomial-parameter magnitude to encourage smooth increments.
Using \eqref{eq:dxref_zero_clean}, the tracking term reduces to penalizing predicted deviation states directly.

\paragraph{Constraints.}
We enforce both absolute-input bounds and increment-trajectory bounds.

\emph{Absolute input bounds:}
\begin{equation}
u_{i,\min} \le u_i(t_{n+s}) \le u_{i,\max},
\qquad s=0,\ldots,N_p-1,
\label{eq:u_abs_clean}
\end{equation}
where $u_{i,\min},u_{i,\max}\in\mathbb{R}^{2}$ provide component-wise bounds for $(s_i,v_i)$.

\emph{Increment trajectory bounds for all $\tau$:}
\begin{equation}
\Delta u_{i,\min}
\le
\Delta u_{i,n+s}(\tau;\Gamma_{i,n+s})
\le
\Delta u_{i,\max},
\qquad \forall\tau\in[0,\delta_{n+s}],
\label{eq:du_traj_clean}
\end{equation}
where $\Delta u_{i,\min},\Delta u_{i,\max}\in\mathbb{R}^{2}$ are component-wise bounds for $(\Delta s,\Delta v)$.

\paragraph{Practical enforcement of \eqref{eq:du_traj_clean} (no nonstandard symbols).}
For a scalar quadratic $q(\tau)=a+b\tau+c\tau^2$ on $\tau\in[0,\delta]$, its extrema over the interval occur at
$\tau=0$, $\tau=\delta$, and possibly at the stationary point $\tau^\star=-b/(2c)$ if $c\neq 0$ and $\tau^\star\in[0,\delta]$.
Therefore, to enforce \eqref{eq:du_traj_clean} for the two-channel vector $\Delta u_{i,n+s}(\tau)=[\Delta s_{i,n+s}(\tau),\,\Delta v_{i,n+s}(\tau)]^\top$,
we check the above candidate points \emph{separately for each channel} using the corresponding coefficients in \eqref{eq:du_components_clean}.

\paragraph{Consistency between within-interval trajectory and discrete execution (interval average).}
To update the discrete absolute input and enforce \eqref{eq:u_abs_clean} consistently, we use the interval-averaged increment:
\begin{equation}
\Delta u_i(t_{n+s})

\frac{1}{\delta_{n+s}}\int_0^{\delta_{n+s}}\Delta u_{i,n+s}(\tau;\Gamma_{i,n+s}),d\tau

\Gamma_{i,n+s,0}
+\Gamma_{i,n+s,1}\frac{\delta_{n+s}}{2}
+\Gamma_{i,n+s,2}\frac{\delta_{n+s}^2}{3}.
\label{eq:du_avg_clean}
\end{equation}
Then propagate the absolute input sequence:
\begin{equation}
\begin{aligned}
u_i(t_n) &= u_i(t_{n-1}) + \Delta u_i(t_n), \
u_i(t_{n+s}) &= u_i(t_{n+s-1}) + \Delta u_i(t_{n+s}), \qquad s=1,\ldots,N_p-1.
\end{aligned}
\label{eq:u_prop_clean}
\end{equation}

\paragraph{Local optimization problem (solved at each Nash iteration).}
At Nash-iteration index $l$, subsystem $i$ solves
\begin{equation}
\mathbf{\Gamma}i^{(l)}=\arg\min{\mathbf{\Gamma}_i}; J_i
\quad \text{s.t.}\quad
\eqref{eq:rollout_mpc_clean},\ \eqref{eq:u_abs_clean},\ \eqref{eq:du_traj_clean},\ \eqref{eq:u_prop_clean}.
\label{eq:local_prob_clean}
\end{equation}
Because the learned surrogate $\mathcal{N}_i(\cdot)$ is differentiable, \eqref{eq:local_prob_clean} is a differentiable nonlinear program (NLP),
which can be solved by standard gradient-based NLP solvers (e.g., SQP or interior-point methods) using automatic differentiation to compute gradients.

\subsection{Nash equilibrium coordination (where conflict comes from, how it is resolved, and how the solution is obtained)}
\label{subsec:nash_clean}

\paragraph{Why a Nash iteration is needed (explicit conflict explanation).}
Inter-stand tensions are shared coupling variables: the tension $T_i$ is influenced by both stand $i$ and stand $i+1$ (notably through their speed actions),
and changes in roll gap can also indirectly affect tensions via strip deformation and transport.
Therefore, purely independent local optimization can lead to conflicting actions: improving local thickness may worsen neighbor tensions, and vice versa.
To resolve this coupling conflict with limited communication, we adopt a Nash-equilibrium-seeking distributed best-response iteration.

\paragraph{Distributed best-response iteration.}
At iteration $l$, each stand computes its best response to the latest neighbor strategies and predictions.
The procedure is summarized in Table~\ref{tab:nash_iter_en}.

\begin{table}[t]
\centering
\small
\renewcommand{\arraystretch}{1.12}
\setlength{\tabcolsep}{3.5pt}
\caption{Distributed Nash best-response iteration for RNE-DMPC (five-stand).}
\label{tab:nash_iter_en}
\begin{tabularx}{\linewidth}{>{\centering\arraybackslash}p{0.11\linewidth} X}
\toprule
\textbf{Step} & \textbf{Description} \
\midrule
A &
Initialize $l=1$ and initialize $\mathbf{\Gamma}_i^{(0)}$ for all subsystems (e.g., warm-start from previous time step). \

B &
Using \eqref{eq:rollout_mpc_clean}, compute $\Delta \hat{x}_i^{(l)}(t_{n+s})$ for $s=1,\ldots,N_p$ \
&
given $\mathbf{\Gamma}_i^{(l-1)}$ and the latest neighbor predictions
$\Delta \hat{x}_{Z_i}^{(l-1)}(t_{n+s})$. \

C & Solve the local NLP \eqref{eq:local_prob_clean} to update $\mathbf{\Gamma}_i^{(l)}$ (best response). \

D &
Broadcast $\mathbf{\Gamma}_i^{(l)}$ and predicted trajectories
$\Delta \hat{x}_i^{(l)}(t_{n+s})$ to the communication system. \

E &
Update neighbor predictions $\Delta \hat{x}_{Z_i}^{(l)}(t_{n+s})$ using received information; re-generate predictions if needed. \

F & Compute the maximum relative change $\varsigma^{(l)}$ as the convergence metric. \

G &
If $\varsigma^{(l)}\le \varsigma_{\mathrm{tol}}$, stop and set $\mathbf{\Gamma}_i^*=\mathbf{\Gamma}_i^{(l)}$; \
& otherwise set $l\leftarrow l+1$ and repeat Steps B--F. \
\bottomrule
\end{tabularx}
\end{table}

\paragraph{Convergence metric.}
Define
\begin{equation}
\varsigma^{(l)}

\max_i
\frac{\left|
\mathbf{\Gamma}_i^{(l)}-\mathbf{\Gamma}_i^{(l-1)}
\right|_2}{
\left|
\mathbf{\Gamma}_i^{(l-1)}
\right|_2+\epsilon},
\label{eq:nash_metric_clean}
\end{equation}
where $\epsilon>0$ is a small constant to avoid division by zero.

\paragraph{Receding-horizon implementation (how the final control is applied).}
Only the first-interval parameters $\Gamma_{i,n}^*$ are applied.
The increment trajectory over $[t_n,t_{n+1}]$ is
\begin{equation}
\Delta u_{i,n}(\tau)=\Delta u_{i,n}(\tau;\Gamma_{i,n}^*),
\quad \tau\in[0,\delta_n].
\end{equation}
The discrete input increment applied for updating the absolute input is the interval average:
\begin{equation}
\Delta u_i(t_n)

\frac{1}{\delta_n}\int_0^{\delta_n}\Delta u_{i,n}(\tau),d\tau

\Gamma_{i,n,0}^*
+
\Gamma_{i,n,1}^\frac{\delta_n}{2}
+
\Gamma_{i,n,2}^\frac{\delta_n^2}{3}.
\label{eq:apply_avg_clean}
\end{equation}
Then the absolute input is updated by
\begin{equation}
u_i(t_n)=u_i(t_{n-1})+\Delta u_i(t_n),
\label{eq:apply_u_clean}
\end{equation}
which ensures smooth evolution of both roll gap and stand speed and avoids abrupt actuator changes.

\paragraph{Closed-loop prediction--control connection (complete loop).}
At each sampling instant $t_n$:
(i) measure/estimate the current deviation states $\Delta x_i(t_n)$ (thickness and tensions) and initialize neighbor information;
(ii) perform Nash best-response iterations; in each iteration solve \eqref{eq:local_prob_clean} using the learned predictor \eqref{eq:rollout_mpc_clean};
(iii) after convergence, apply $\Gamma_{i,n}^*$ by generating the within-interval increment trajectory and updating $u_i(t_n)$ via \eqref{eq:apply_avg_clean}--\eqref{eq:apply_u_clean};
(iv) move to $t_{n+1}$ and repeat.
In this way, the learned neural predictor supplies the multi-step forecasts needed by MPC, and the distributed optimization computes coordinated
gap/speed increment trajectories that regulate thickness and tensions while resolving coupling conflicts via Nash equilibrium.

The overall control flow chart is shown in Fig.~\ref{liu}.

\begin{figure}[htbp]
\centering
\includegraphics[width=\linewidth]{picture/x5.pdf}
\caption{The overall system control flow chart}\label{liu}
\end{figure}

%========================
% Optional: a compact symbol paragraph you can keep or move to Appendix
%========================
\paragraph{Summary of key symbols (for reviewer clarity).}
$i\in\{1,\dots,5\}$: stand index; $t_n$: sampling instant; $\delta_n$: sampling interval; $\tau$: within-interval time.
$h_i,T_i$: thickness and inter-stand tension; $h_i^{\mathrm{ref}},T_i^{\mathrm{ref}}$: references.
$\Delta h_i,\Delta T_i$: deviations; $\Delta x_i$: local deviation state in \eqref{eq:xi_def_clean}.
$u_i=[s_i,v_i]^\top$: actuators (gap and speed); $\Delta u_i(t_n)=u_i(t_n)-u_i(t_{n-1})$: discrete input increment.
$\Delta u_{i,n}(\tau;\Gamma_{i,n})$: within-interval increment trajectory; $\Gamma_{i,n}\in\mathbb{R}^{p}$: polynomial parameters ($p=6$).
$Z_i$: neighbor set; $\Delta x_{Z_i}$: neighbor stack.
$\mathcal{N}_i(\cdot)$: learned forward residual model; $\mathcal{B}_i(\cdot)$: backward model.
$K$: segment length; $J,J_K$: numbers of one-step and segment samples.
$N_p,N_c$: prediction and control horizons; $Q_i,R_i$: weights; $\epsilon,\varepsilon$: small constants.
这是我的论文第二到第四部分。老师说优化目标函数看不出来多个机架博弈；建立了优化目标求解方案是不是得改变。

可以，帮我改吧

d为什么等于3

有没有用多项式参数化应用在五机架冷轧的论文呢

有没有用多项式参数化应用在五机架冷轧的论文呢

We keep the same constraints as in \eqref{eq:u_abs_clean}--\eqref{eq:u_prop_clean}.
Collect them into a compact feasible set:
\begin{equation}
\Omega_i \triangleq
\Big{\mathbf{\Gamma}_i\ \Big|\
\eqref{eq:rollout_mpc_game}\ \text{holds and}\
\eqref{eq:u_abs_clean},\eqref{eq:du_traj_clean},\eqref{eq:u_prop_clean}\ \text{are satisfied}
\Big}.
\end{equation}这部分是要什么公式，为什么显示出来都是问号

\eqref{eq:rollout_mpc_game}\ \text{holds and}\
\eqref{eq:u_abs_clean},\eqref{eq:du_traj_clean},\eqref{eq:u_prop_clean}\ \text{are satisfied}这几个公式标号前面没有么

那对应的，\eqref{eq:u_abs_clean},\eqref{eq:du_traj_clean},\eqref{eq:u_prop_clean}这三个公式在你现在新改的里面叫什么

%========================
\section{Nash Equilibrium-Based RNE-DMPC}
%========================

The five-stand tandem cold rolling system is strongly coupled through inter-stand tension propagation.
As a result, changes in control actions (roll gap and stand speed) at one stand can affect both upstream and downstream stands,
making centralized online optimization over all stands' decision variables computationally demanding.

To mitigate this issue, we decompose the global predictive-control problem into $N=5$ local subproblems associated with individual stands.
Each local controller optimizes its own decision variables while accounting for coupling via limited information exchange with neighboring controllers.
Motivated by game-theoretic coordination \citep{rawlings2008coordinating}, we formulate distributed coordination as a Nash-equilibrium-seeking iteration.
Based on the trained residual neural network surrogate model, we construct a Nash-equilibrium-based distributed MPC method (RNE-DMPC)
for coordinated thickness--tension regulation and tracking. The overall control structure is shown in Figure~\ref{4}.

\begin{figure*}[htbp]
\centering
\includegraphics[width=\linewidth]{picture/x2.pdf}
\caption{Schematic diagram of the control architecture for a tandem cold rolling mill.}\label{4}
\end{figure*}

At sampling time $t_n$, stand $i$ chooses the polynomial-parameter sequence
$\mathbf{\Gamma}_i(t_n)\in\mathbb{R}^{pN_c}$, where $p=6$.
Let $\mathbf{\Gamma}(t_n)\triangleq \mathrm{col}\{\mathbf{\Gamma}_1(t_n),\ldots,\mathbf{\Gamma}_5(t_n)\}$
denote the joint strategy profile, and let $\mathbf{\Gamma}_{-i}(t_n)$ denote the collection of all strategies except stand $i$.

Given the current measured/estimated deviation state $\Delta x_i(t_n)$ and the strategies
$(\mathbf{\Gamma}_i(t_n),\mathbf{\Gamma}_{Z_i}(t_n))$,
the multi-step prediction used by stand $i$ is written explicitly as
\begin{equation}
\begin{aligned}
\Delta \hat{x}i(t{n+s+1};\mathbf{\Gamma}i,\mathbf{\Gamma}{Z_i})
&=
\Delta \hat{x}i(t{n+s};\mathbf{\Gamma}i,\mathbf{\Gamma}{Z_i})
+
\mathcal{N}i!\Big(
\Delta \hat{x}i(t{n+s};\cdot),,
\Delta \hat{x}{Z_i}(t_{n+s};\mathbf{\Gamma}{Z_i}),\
&\qquad
\Gamma{i,n+s},,
\delta_{n+s};\Theta_i^*
\Big),
\end{aligned}
\label{eq:rollout_mpc_game}
\end{equation}
for $s=0,\ldots,N_p-1$, with initialization $\Delta \hat{x}_i(t_n;\cdot)=\Delta x_i(t_n)$.
Here the neighbor stack $\Delta \hat{x}_{Z_i}(t_{n+s};\mathbf{\Gamma}_{Z_i})$ is generated from neighbors' strategies via the same learned predictors.