\begin{equation}
\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}),,
\Gamma{i,n+s},,
\delta_{n+s};\Theta_i^*
\Big),
\label{eq:rollout_mpc_game}
\end{equation}这个公式太长了需要换行。

\begin{equation}
J_i^{\mathrm{cpl}}

\mu_i\sum_{s=1}^{N_p}
\Big(
\mathbb{I}_{{i\ge 2}}
\big|
C^-\Delta \hat{x}i(t{n+s};\mathbf{\Gamma})

C^+\Delta \hat{x}{i-1}(t{n+s};\mathbf{\Gamma})
\big|^2
+
\mathbb{I}_{{i\le 4}}
\big|
C^+\Delta \hat{x}i(t{n+s};\mathbf{\Gamma})

C^-\Delta \hat{x}{i+1}(t{n+s};\mathbf{\Gamma})
\big|^2
\Big),
\label{eq:coupling_cost}
\end{equation}这个公式太长了需要换行

%========================
\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.

Over $[t_{n+s},t_{n+s+1}]$ with length $\delta_{n+s}$,
\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}],
\end{equation}
and the interval-averaged increment is
\begin{equation}
\Delta u_i(t_{n+s})

\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}

\begin{remark}
Because inter-stand tension $T_i$ is jointly affected by the adjacent stands $i$ and $i+1$,
the predicted evolution of $\Delta x_i$ depends on neighbors' future actions,
hence the MPC problems are not independent but form a coupled dynamic game.
\end{remark}

At time $t_n$, the local strategy of stand $i$ is
\begin{equation}
\mathbf{\Gamma}_i(t_n)

\mathrm{col}{\Gamma_{i,n},\Gamma_{i,n+1},\ldots,\Gamma_{i,n+N_c-1}}
\in \mathbb{R}^{pN_c}.
\end{equation}

In deviation coordinates, the regulation/tracking objective is $\Delta x_i(t)\rightarrow 0$, i.e.
\begin{equation}
\Delta x_{i,\mathrm{ref}}(t_{n+s})\equiv 0\in\mathbb{R}^{d},\qquad d=3.
\end{equation}

Recall $\Delta x_i=[\Delta h_i,\Delta T_{i-1},\Delta T_i]^\top$.
Define the row selectors
\begin{equation}
C^- \triangleq [0\ \ 1\ \ 0]\in\mathbb{R}^{1\times 3},\qquad
C^+ \triangleq [0\ \ 0\ \ 1]\in\mathbb{R}^{1\times 3},
\end{equation}
so that $C^- \Delta x_i=\Delta T_{i-1}$ (upstream interface) and $C^+\Delta x_i=\Delta T_i$ (downstream interface).

For the interface between stands $i$ and $i+1$,
stand $i$'s prediction provides $C^+\Delta \hat{x}_i$, while stand $i+1$'s prediction provides $C^- \Delta \hat{x}_{i+1}$.
Their mismatch measures coupling inconsistency:
\begin{equation}
e_{i}^{\mathrm{sh}}(t_{n+s};\mathbf{\Gamma})
\triangleq
C^+\Delta \hat{x}i(t{n+s};\mathbf{\Gamma})

C^-\Delta \hat{x}{i+1}(t{n+s};\mathbf{\Gamma}),\qquad i=1,\ldots,4.
\label{eq:shared_tension_mismatch}
\end{equation}

We define the stage cost of stand $i$ as a function of all players' strategies:
\begin{equation}
J_i(\mathbf{\Gamma}i;\mathbf{\Gamma}{-i})

\sum_{s=1}^{N_p}
\left|
\Delta \hat{x}i(t{n+s};\mathbf{\Gamma}i,\mathbf{\Gamma}{Z_i})
\right|{Q_i}^{2}
+
\sum{s=0}^{N_c-1}
\left|\Gamma_{i,n+s}\right|_{R_i}^{2}
+
J_i^{\mathrm{cpl}}(\mathbf{\Gamma}i;\mathbf{\Gamma}{-i})
\label{eq:Ji_game}
\end{equation}
where $Q_i\succeq 0$ weights thickness and tension deviations, and $R_i\succeq 0$ penalizes actuation magnitudes.

The coupling term $J_i^{\mathrm{cpl}}$ explicitly reflects the game/coordination requirement on shared tensions.
A simple and effective choice is to penalize the interface mismatches adjacent to stand $i$:
\begin{equation}
\begin{aligned}
J_i^{\mathrm{cpl}}
&=
\mu_i\sum_{s=1}^{N_p}
\Big(
\mathbb{I}_{{i\ge 2}}
\big|
C^-\Delta \hat{x}i(t{n+s};\mathbf{\Gamma})

C^+\Delta \hat{x}{i-1}(t{n+s};\mathbf{\Gamma})
\big|^2 \
&\qquad\quad
+
\mathbb{I}_{{i\le 4}}
\big|
C^+\Delta \hat{x}i(t{n+s};\mathbf{\Gamma})

C^-\Delta \hat{x}{i+1}(t{n+s};\mathbf{\Gamma})
\big|^2
\Big),
\end{aligned}
\label{eq:coupling_cost}
\end{equation}
with $\mu_i>0$ and indicator $\mathbb{I}_{\{\cdot\}}$.
This term makes the coupling conflict explicit: unilateral actions that locally reduce thickness error may worsen shared-tension
compatibility and thus increase $J_i$, and also affect neighbors' objectives.

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}

\paragraph{Local best-response problem (player $i$).}
Given neighbors' current strategies (or the communicated predicted trajectories induced by them),
stand $i$ solves the differentiable NLP:
\begin{equation}
\mathbf{\Gamma}_i^{\mathrm{BR}}

\arg\min_{\mathbf{\Gamma}_i\in\Omega_i}\
J_i(\mathbf{\Gamma}i;\mathbf{\Gamma}{-i}).
\label{eq:local_BR}
\end{equation}
Because the learned surrogate is differentiable, \eqref{eq:local_BR} can be solved by standard gradient-based NLP solvers
(e.g., SQP or interior-point methods with automatic differentiation).

%========================
\subsection{Nash equilibrium coordination and relaxed best-response solution}
\label{subsec:nash_clean}
%========================

\paragraph{Dynamic game definition (five-stand).}
At each sampling time $t_n$, the distributed MPC coordination induces a finite-horizon dynamic game:
players are stands $i=1,\ldots,5$; strategy sets are $\Omega_i$; and payoff (cost) functions are $J_i(\mathbf{\Gamma}_i;\mathbf{\Gamma}_{-i})$
defined in \eqref{eq:Ji_game}--\eqref{eq:coupling_cost}.

\paragraph{Nash equilibrium (NE).}
A joint strategy profile $\mathbf{\Gamma}^*=\mathrm{col}\{\mathbf{\Gamma}_1^*,\ldots,\mathbf{\Gamma}_5^*\}$
is a Nash equilibrium if
\begin{equation}
\forall i\in{1,\ldots,5},\qquad
\mathbf{\Gamma}i^*\in
\arg\min{\mathbf{\Gamma}_i\in\Omega_i}
J_i(\mathbf{\Gamma}i;\mathbf{\Gamma}{-i}^*).
\label{eq:NE_def}
\end{equation}
This definition explicitly characterizes the strategic coupling:
each player's optimal decision depends on neighbors' decisions through the shared-tension dynamics and the coupling term.

\paragraph{Relaxed distributed best-response iteration (improved robustness).}
To compute an NE online with limited communication, we employ a relaxed best-response iteration.
Let $l$ denote the Nash-iteration index.
Given $\mathbf{\Gamma}^{(l-1)}$, each stand computes a best response $\mathbf{\Gamma}_i^{\mathrm{BR},(l)}$
by solving \eqref{eq:local_BR}, and then updates with relaxation:
\begin{equation}
\mathbf{\Gamma}_i^{(l)}

(1-\omega)\mathbf{\Gamma}_i^{(l-1)}
+
\omega,\mathbf{\Gamma}_i^{\mathrm{BR},(l)},
\qquad \omega\in(0,1].
\label{eq:relaxed_BR}
\end{equation}
The relaxation factor $\omega$ mitigates oscillations caused by strong coupling and improves practical convergence.

\paragraph{Convergence metric (unchanged).}
\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},
\end{equation}
with $\epsilon>0$ small.

\begin{table}[t]
\centering
\small
\renewcommand{\arraystretch}{1.12}
\setlength{\tabcolsep}{3.5pt}
\caption{Relaxed 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 warm-start $\mathbf{\Gamma}_i^{(0)}$ (e.g., from the previous sampling time). \

B &
Communicate $\mathbf{\Gamma}_i^{(l-1)}$ (or the induced predicted trajectories) among neighbors; form $\mathbf{\Gamma}_{Z_i}^{(l-1)}$. \

C &
Given $\mathbf{\Gamma}_{-i}^{(l-1)}$, solve the best-response NLP \eqref{eq:local_BR} to obtain $\mathbf{\Gamma}_i^{\mathrm{BR},(l)}$. \

D &
Update the relaxed strategy using \eqref{eq:relaxed_BR} and compute the induced predictions
$\Delta \hat{x}_i^{(l)}(\cdot;\mathbf{\Gamma}^{(l)})$ via \eqref{eq:rollout_mpc_game}. \

E &
Broadcast $\mathbf{\Gamma}_i^{(l)}$ and the predicted interface quantities needed in \eqref{eq:coupling_cost}
(e.g., $C^\pm\Delta \hat{x}_i^{(l)}$) to neighbors. \

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

\paragraph{Receding-horizon implementation (unchanged).}
After convergence, only the first-interval parameters $\Gamma_{i,n}^*$ are applied.
The within-interval increment is $\Delta u_{i,n}(\tau;\Gamma_{i,n}^*)$ and the discrete applied increment uses the interval average:
\begin{equation}
\Delta u_i(t_n)

\Gamma_{i,n,0}^*
+
\Gamma_{i,n,1}^\frac{\delta_n}{2}
+
\Gamma_{i,n,2}^\frac{\delta_n^2}{3},
\end{equation}
followed by $u_i(t_n)=u_i(t_{n-1})+\Delta u_i(t_n)$.

\paragraph{Remark (what makes the ``game'' visible).}
Equations \eqref{eq:Ji_game}--\eqref{eq:coupling_cost} and the NE definition \eqref{eq:NE_def} explicitly show that each player's objective
depends on others' strategies through (i) the coupled prediction dynamics and (ii) the shared-tension interface mismatch penalty,
thus making the multi-stand strategic interaction and coordination mechanism transparent.这里面有的公式标号没有这个公式，怎么解决