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A题/成文/2问题重述与建模目标.md
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A题/成文/2问题重述与建模目标.md
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\section{Problem Restatement \& Objectives}
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\label{sec:problem}
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\subsection{Restatement of the Problem}
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\label{subsec:restatement}
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The 2026 MCM Problem A concerns continuous-time prediction of smartphone battery drain under time-varying usage.
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A smartphone is subject to multiple power-consuming components---most prominently the display, CPU workload, and cellular/Wi-Fi communication---whose intensities evolve over time according to user behavior and network conditions.
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Meanwhile, the battery exhibits coupled electro-thermal dynamics and gradual health degradation.
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The task is to construct a mechanism-driven, continuous-time model that maps future usage profiles to battery states and terminal voltage, and then to estimate the remaining operating time before the device shuts down.
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In particular, given (measured, prescribed, or scenario-generated) time series describing user/device usage and ambient conditions, we aim to:
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(i) predict the trajectories of key battery states (e.g., state-of-charge and temperature) and the terminal voltage;
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(ii) compute the time-to-empty (TTE) defined by physically meaningful shutdown criteria; and
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(iii) interpret sudden shutdown phenomena through explicit feasibility mechanisms rather than black-box regression.
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\subsection{Inputs, Outputs, and Prediction Tasks}
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\label{subsec:io_tasks}
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We represent the battery-in-phone system by a state vector
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\[
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\mathbf{x}(t)=[z(t),\,v_p(t),\,T_b(t),\,S(t),\,w(t)]^\top,
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\]
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where \(z\) is state-of-charge (SOC), \(v_p\) is polarization voltage (memory effect of the ECM),
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\(T_b\) is battery temperature, \(S\) is state-of-health (SOH, capacity fraction), and \(w\) is the radio tail state.
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The exogenous inputs are collected as
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\[
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\mathbf{u}(t)=[L(t),\,C(t),\,N(t),\,\Psi(t),\,T_a(t)]^\top,
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\]
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where \(L\in[0,1]\) denotes normalized screen brightness,
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\(C\in[0,1]\) the normalized CPU load,
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\(N\in[0,1]\) the normalized network activity level (throughput/airtime proxy),
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\(\Psi>0\) a signal-quality indicator (larger is better),
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and \(T_a\) the ambient temperature.
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\paragraph{Primary predicted outputs.}
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The model produces the battery terminal voltage and SOC trajectories,
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\[
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V_{\mathrm{term}}(t), \qquad z(t),
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\]
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as well as the time-to-empty (TTE), defined as the first time the device becomes inoperable under the specified shutdown criteria:
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\[
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\mathrm{TTE}=\inf\Big\{t>0:\ V_{\mathrm{term}}(t)\le V_{\mathrm{cut}}\ \text{or}\ z(t)\le 0\Big\}.
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\]
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Here \(V_{\mathrm{cut}}\) is the cutoff voltage dictated by system protection (BMS/PMIC).
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\paragraph{Prediction tasks.}
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Given \(\mathbf{x}(0)\) and a future input profile \(\mathbf{u}(t)\) on a horizon \([0,T]\), the prediction tasks are:
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\begin{enumerate}
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\item \textbf{State/voltage forecasting:} compute \(\mathbf{x}(t)\) and \(V_{\mathrm{term}}(t)\) for \(t\in[0,T]\);
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\item \textbf{Runtime estimation:} compute \(\mathrm{TTE}\) from the stopping rule above;
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\item \textbf{Mechanistic interpretation:} attribute shutdown to depletion (\(z\to 0\)) or voltage protection (\(V_{\mathrm{term}}\le V_{\mathrm{cut}}\)), and quantify risk of power infeasibility (Section~\ref{subsec:metrics_scenarios}).
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\end{enumerate}
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\subsection{Performance Metrics and Usage-Scenario Description}
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\label{subsec:metrics_scenarios}
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\paragraph{Operational termination and reliability-oriented metrics.}
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The principal performance metric is the operating time before shutdown, \(\mathrm{TTE}\).
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For evaluation and comparison across scenarios, we also report:
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\begin{itemize}
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\item \textbf{Terminal-voltage margin:} \(\min_{t\in[0,\mathrm{TTE}]}(V_{\mathrm{term}}(t)-V_{\mathrm{cut}})\), which indicates how close the device operates to the cutoff boundary;
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\item \textbf{Delivered-energy proxy:} \(E_{\mathrm{del}}=\int_{0}^{\mathrm{TTE}} V_{\mathrm{term}}(t)I(t)\,dt\) (when current \(I(t)\) is available from the closure), which supports sanity checks against SOC depletion;
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\item \textbf{Thermal exposure:} \(\max_{t\in[0,\mathrm{TTE}]} T_b(t)\), reflecting potential thermal throttling or safety constraints.
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\end{itemize}
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\paragraph{Risk event: CPL feasibility (voltage-collapse risk).}
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Because the load is modeled as a constant-power demand (CPL) coupled to the electrochemical model, a feasibility condition naturally arises.
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Let
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\[
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\Delta(t)=\big(V_{\mathrm{oc}}(z(t)) - v_p(t)\big)^2 - 4R_0(T_b(t),S(t))\,P_{\mathrm{tot}}(t),
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\]
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where \(P_{\mathrm{tot}}(t)\) is the demanded total power and \(R_0\) the ohmic resistance.
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When \(\Delta(t)<0\), the CPL algebraic closure admits no real current solution, indicating that the demanded power is infeasible given the instantaneous battery capability and may lead to abrupt voltage collapse.
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We therefore introduce the \emph{first risk time}
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\[
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t_{\Delta}=\inf\{t>0:\ \Delta(t)\le 0\},
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\]
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as an auxiliary diagnostic.
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In later sections, we use \(t_\Delta\) to distinguish \emph{infeasibility-driven} shutdown risk from ordinary energy depletion.
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\paragraph{Representative usage scenarios.}
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To ensure that conclusions are interpretable and reproducible, we evaluate the model under a small set of canonical usage scenarios, each defined by a characteristic input pattern \(\mathbf{u}(t)\).
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Table~\ref{tab:scenarios} summarizes the scenarios used throughout the paper.
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\begin{table}[t]
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\centering
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\caption{Representative usage scenarios and their qualitative input characteristics.}
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\label{tab:scenarios}
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\begin{tabular}{p{2.4cm}p{10.6cm}}
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\hline
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Scenario & Input characteristics \(\mathbf{u}(t)=[L(t),C(t),N(t),\Psi(t),T_a(t)]^\top\) \\
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\hline
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Standby/Idle &
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Low brightness \(L\approx 0\) (screen off), low CPU \(C\ll 1\), sporadic network \(N\approx 0\) with residual tail \(w\), typical \(\Psi\), moderate \(T_a\). \\
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Browsing/Social &
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Moderate \(L\), moderate CPU \(C\), intermittent network bursts \(N(t)\) with tail effects, typical-to-good \(\Psi\), moderate \(T_a\). \\
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Video Streaming &
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High \(L\), sustained moderate CPU \(C\), sustained network activity \(N\) (downlink), sensitivity to \(\Psi\); moderate \(T_a\). \\
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Gaming/High Compute &
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High \(L\), high CPU \(C\) (near saturation), moderate network \(N\), typical \(\Psi\); emphasizes thermal rise and possible throttling. \\
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Weak Signal (Stress) &
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Moderate-to-high \(L\), moderate CPU \(C\), nontrivial \(N\) under poor signal \(\Psi\downarrow\); stresses the signal-quality penalty in \(P_{\mathrm{net}}(N,\Psi,w)\) and increases collapse risk. \\
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Cold Ambient (Stress) &
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Any of the above with low \(T_a\); highlights increased \(R_0\) and reduced \(Q_{\mathrm{eff}}\), potentially shortening TTE and increasing \(t_\Delta\) likelihood. \\
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\hline
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\end{tabular}
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\end{table}
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The above scenarios are not tied to a specific dataset; they can be instantiated using recorded traces or generated synthetically (e.g., piecewise-smooth profiles or stochastic processes) while keeping the same physical meaning of each input channel.
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This design supports both deterministic simulations and uncertainty quantification (Monte Carlo) in later sections.
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%========================================================
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