5.6 KiB
这是一个非常棒的切入点！你提供的雷达图直观地展示了“省电模式”与“高性能模式”的硬性割裂（Trade-off）。要拿到MCM的O/F奖，仅仅展示这种割裂是不够的，核心在于打破这种二元对立，建立一个连续的、动态的控制模型。
你需要建立的是一个**“基于效用最大化的自适应动态控制策略” (Utility-Maximization Adaptive Control Strategy, UM-ACS)**。
我们可以利用你在 模型3.md 中建立的电池物理模型（SOC, 等），结合这张雷达图的概念，构建一个第四问的数学模型。
以下是为你设计的建模思路、数学公式和论文段落。
核心建模思路：从“二选一”到“最优控制”
- 量化雷达图： 将雷达图的五个维度定义为状态变量，它们是控制变量 的函数。
- 引入控制变量 ： 定义一个连续变量 ，代表“激进程度”。
- ：纯省电模式（Green Polygon）。
- ：纯高性能模式（Red Polygon）。
- ：中间混合状态。
- 建立目标函数（Utility Function）： 我们需要在每一时刻 寻找最优的 ，使得用户体验收益减去电量焦虑惩罚的值最大。
- 动态反馈： 随着电量（SOC）下降，电量焦虑惩罚权重增加，系统自动迫使 向 0 滑动，从而实现你所说的“自动动态调整”。
正式建模内容 (可直接用于论文第四部分)
4. Adaptive Power Management Strategy Based on Utility Optimization
Traditional power management forces users to choose between two static extremes: "Power Saver" and "High Performance" (as shown in Figure 4). This binary approach is inefficient because user needs and battery status fluctuate continuously. We propose a Continuous Adaptive Control Model that dynamically optimizes the trade-off between User Experience (UX) and Battery Sustainability.
4.1. Definition of Control Space and Metrics
Let be the Performance Aggressiveness Coefficient, which serves as the continuous control variable connecting the two modes in the radar chart.
We map the radar chart metrics to using linear interpolation (a valid simplification for control logic):
- Performance Index:
- Display Quality:
- Connectivity:
- User Experience (UX): Defined as the weighted sum of the above functional metrics:
where is strictly increasing with . 5. Power Consumption Cost: Conversely, higher performance implies higher power drain. Based on our Model 3 equation (), the power consumption is:
4.2. The "Battery Anxiety" Feedback Mechanism
To make the model compliant with reality, the system implies a "cost" for using power. This cost is not constant; it depends on the current State of Charge, (from Model 3).
We define the Marginal Cost of Energy (MCE), denoted as . This represents the user's "battery anxiety."
- When (100% battery), energy is "cheap," anxiety is low ( is small).
- When (Low battery), energy is "expensive," anxiety is infinite ( is large).
We model this as a convex barrier function:
Where is a scaling factor and controls how sharply anxiety rises as battery drops.
4.3. The Optimization Objective (The "Smart" Logic)
At any time instance , the operating system solves the following optimization problem to determine the optimal mode :
Interpretation:
- High SOC: is small. The gain dominates. The system chooses (High Performance Mode).
- Low SOC: becomes huge. The penalty term dominates. The system is forced to choose (Power Saver Mode) to minimize , regardless of the user's desire for performance.
4.4. Closed-Loop Solution
Solving , we can derive the dynamic switching logic. Since and are linear in , the derivative is constant for a given . However, to make it smoother (more realistic O-level modeling), we assume diminishing returns on User Experience (e.g., or ).
Assuming a logarithmic utility (diminishing returns for ultra-high performance), the optimal control law becomes:
This equation is the mathematical embodiment of your "Automatic Dynamic Adjustment." It continuously outputs the optimal mix of the Red and Green polygons based on the real-time battery level .
针对O奖的“点睛之笔” (Visual & Analysis)
为了拿到O奖，你需要在论文中加入对这个模型的可视化分析（利用你上传的图的概念）：
建议你在论文中加入这样一段描述（Visual Interpretation）：
"Figure 4 shows the static boundaries. Our model creates a dynamic breathing polygon that interpolates between the Green and Red zones.
- Phase 1 (SOC > 60%): The polygon adheres to the Red boundary (High Performance).
- Phase 2 (20% < SOC < 60%): The polygon shrinks smoothly. The system automatically dims the screen () and caps the CPU frequency () proportionally to .
- Phase 3 (SOC < 20%): The 'Battery Anxiety' factor diverges, forcing the polygon to collapse to the Green boundary (Power Saver), prioritizing survival over experience."
如何整合进你的现有回答 (p4_response)?
你需要替换或增强 p4_response.md 中的 "Recommendations" 部分。原稿主要是文字建议，现在你要把它升级为**"System Design"**。
操作步骤：
- 保留原稿中关于“屏幕亮度影响最大”的结论（这是物理事实）。
- 插入上述模型（4.1 - 4.4节）。
- 结论升级： 你的建议不再是简单的“用户应该降低亮度”，而是“手机厂商应该部署这套 -Adaptive Control Algorithm”。