模型复查
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ChuXun
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A题/分析/框架1/memory.md
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A题/分析/框架1/memory.md
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# 1) 必须文档 ①：Project Memory（核心模型备忘录）
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> **用途**：下个对话里快速恢复我们已完成的“假设 + 模型建立 + 求解框架”。
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> **你要做的**：原样粘贴到新对话开头（Prompt A 会包含它）。
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## A. Problem & Scope
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* Contest: **2026 MCM Problem A (continuous-time smartphone battery drain)**
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* Completed sections: **Assumptions + Model Formulation and Solution (Q1 core)**
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* Constraints: **mechanism-driven, no black-box regression**, continuous-time ODE/DAE, include numerical method + stability/convergence statements.
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## B. State, Inputs, Outputs
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* **State**: (\mathbf{x}(t)=[z(t),v_p(t),T_b(t),S(t),w(t)]^\top)
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* (z): SOC, (v_p): polarization voltage, (T_b): battery temperature, (S): SOH (capacity fraction), (w): radio tail state
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* **Inputs**: (\mathbf{u}(t)=[L(t),C(t),N(t),\Psi(t),T_a(t)]^\top)
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* (L): brightness, (C): CPU load, (N): network activity, (\Psi): signal quality (higher better), (T_a): ambient temp
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* **Outputs**: (V_{\text{term}}(t)), SOC (z(t)), **TTE**
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## C. Power mapping (component-level, explicit (\Psi) effect)
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[
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P_{\mathrm{tot}}(t)=P_{\mathrm{bg}}+P_{\mathrm{scr}}(L)+P_{\mathrm{cpu}}(C)+P_{\mathrm{net}}(N,\Psi,w)
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]
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[
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P_{\mathrm{scr}}(L)=P_{\mathrm{scr},0}+k_L L^\gamma,;\gamma>1
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]
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[
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P_{\mathrm{cpu}}(C)=P_{\mathrm{cpu},0}+k_C C^\eta,;\eta>1
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]
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[
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P_{\mathrm{net}}(N,\Psi,w)=P_{\mathrm{net},0}+k_N\frac{N}{(\Psi+\varepsilon)^\kappa}+k_{\mathrm{tail}}w,;\kappa>0
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]
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Tail dynamics (continuous, avoids discrete FSM):
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[
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\dot w=\frac{\sigma(N)-w}{\tau(N)},\quad
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\tau(N)=\begin{cases}\tau_\uparrow,&\sigma(N)\ge w\ \tau_\downarrow,&\sigma(N)<w\end{cases},;
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\tau_\uparrow\ll\tau_\downarrow,;
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\sigma(N)=\min(1,N)
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]
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## D. ECM + CPL current closure (nonlinear feedback source)
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Terminal voltage:
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[
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V_{\mathrm{term}}=V_{\mathrm{oc}}(z)-v_p-I R_0(T_b,S)
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]
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CPL constraint:
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[
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P_{\mathrm{tot}}=V_{\mathrm{term}}I=\big(V_{\mathrm{oc}}-v_p-IR_0\big)I
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]
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Quadratic current:
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[
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I=\frac{V_{\mathrm{oc}}-v_p-\sqrt{\Delta}}{2R_0},\quad
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\Delta=(V_{\mathrm{oc}}-v_p)^2-4R_0P_{\mathrm{tot}}
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]
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Shutdown/feasibility:
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* Require (\Delta\ge0); if (\Delta\le0) ⇒ power infeasible ⇒ voltage collapse/shutdown.
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## E. Coupled ODEs (SOC–polarization–thermal–SOH)
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[
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\dot z=-\frac{I}{3600,Q_{\mathrm{eff}}(T_b,S)}
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]
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[
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\dot v_p=\frac{I}{C_1}-\frac{v_p}{R_1C_1}
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]
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[
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\dot T_b=\frac{1}{C_{\mathrm{th}}}\Big(I^2R_0+Iv_p-hA(T_b-T_a)\Big)
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]
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SOH (Option A compact, used for Q1):
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[
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\dot S=-\lambda_{\mathrm{sei}}|I|^{m}\exp!\left(-\frac{E_{\mathrm{sei}}}{R_gT_b}\right),;0\le m\le1
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]
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(Option B SEI thickness (\delta) exists as upgrade path if needed.)
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## F. Constitutive relations
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Modified Shepherd OCV:
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[
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V_{\mathrm{oc}}(z)=E_0-K\Big(\frac{1}{z}-1\Big)+A e^{-B(1-z)}
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]
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Arrhenius resistance + SOH correction:
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[
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R_0(T_b,S)=R_{\mathrm{ref}}\exp!\Big[\frac{E_a}{R_g}\Big(\frac{1}{T_b}-\frac{1}{T_{\mathrm{ref}}}\Big)\Big],(1+\eta_R(1-S))
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]
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Effective capacity:
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[
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Q_{\mathrm{eff}}(T_b,S)=Q_{\mathrm{nom}}S\Big[1-\alpha_Q(T_{\mathrm{ref}}-T_b)\Big]_+
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]
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## G. Initial conditions & TTE
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[
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z(0)=z_0,;v_p(0)=0,;T_b(0)=T_a(0),;S(0)=S_0,;w(0)=0
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]
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[
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\mathrm{TTE}=\inf{t>0:;V_{\mathrm{term}}(t)\le V_{\mathrm{cut}}\ \text{or}\ z(t)\le0\ \text{or}\ \Delta(t)\le0}
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]
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## H. Numerical solution standard
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* Use RK4 (or ode45) with **nested algebraic solve** for (I) at each substep.
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* Step size: (\Delta t\le0.05,\tau_p) where (\tau_p=R_1C_1).
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* Convergence: step-halving until (|z_{\Delta t}-z_{\Delta t/2}|_\infty<10^{-4}); TTE change <1%.
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## I. Parameter estimation (hybrid, reproducible)
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* OCV params ((E_0,K,A,B)): least squares to OCV–SOC curve.
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* (R_0): pulse instantaneous drop (\Delta V(0^+)/\Delta I).
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* (R_1,C_1): pulse relaxation exponential fit.
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* (\kappa): fit (\ln P_{\mathrm{net}}) vs (-\ln(\Psi)) at fixed throughput.
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## J. References (BibTeX you already used)
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* Shepherd (1965), Tremblay & Dessaint (2009), Plett (2004) + smartphone energy paper as needed.
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---
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# 2) 必须文档 ②：“不可预测机制叙事”一句话模板
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> **用途**：下次写 Introduction/Modeling/Results 时保持口径一致
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> Battery-life variability arises from (i) time-varying usage inputs ((L,C,N,\Psi,T_a)), (ii) nonlinear CPL closure (P=VI) that amplifies current when voltage drops, and (iii) state memory through polarization (v_p) and thermal inertia (T_b), producing history-dependent discharge trajectories.
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---
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# 3) 必须文档 ③：你下次对话开场的 Prompt（复制即用）
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## Prompt A（必用：恢复上下文 + 锁定写作风格与约束）
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把下面整段复制到新对话的第一条消息：
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```markdown
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You are my MCM/ICM continuous-modeling O-award mentor and paper lead writer.
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We have already completed Assumptions + full Model Formulation and Solution (Q1 core).
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Do NOT reinvent the model; strictly continue from the finalized framework below, keeping all symbols consistent and mechanism-driven (no black-box regression).
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Write in academic English (SIAM/IEEE), equations in LaTeX, and ensure solution logic matches paper narrative.
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## Project Memory (do not alter)
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[PASTE THE ENTIRE "Project Memory" SECTION HERE]
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```
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> 你只需要把上面那个 `[PASTE ... HERE]` 换成我给你的 **Project Memory** 全文即可。
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---
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## Prompt B（如果你下一步要做 Q2/Q3：不确定性、策略、灵敏度）
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```markdown
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Continue with the same model. Now do: (1) uncertainty modeling for future usage inputs using a continuous-time stochastic process (e.g., OU / regime switching), (2) Monte Carlo to obtain a TTE distribution, (3) global sensitivity (Sobol or variance-based) on key parameters (k_L, gamma, k_N, kappa, T_a, etc.), and (4) produce figure descriptions that match the simulations. Keep all derivations and algorithmic steps explicit.
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```
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---
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## Prompt C（如果你下一步要做“Parameter Estimation”章节写作）
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```markdown
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Write a complete "Parameter Estimation" section for the existing model:
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- specify which parameters come from literature/datasheets vs which are fitted;
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- provide objective functions and constraints for fitting (OCV curve, pulse response for R0/R1/C1, signal exponent kappa);
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- include identifiability discussion and practical calibration workflow.
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No new model components unless strictly necessary.
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```
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