2173 lines
154 KiB
Python
2173 lines
154 KiB
Python
"""
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知识卡片内容 - 包含时域、频域、根轨迹、控制器设计、模型蒸馏、能量管理分析的公式和方法
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使用纯HTML格式，无需LaTeX渲染库
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"""
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# 时域分析知识卡片
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TIME_DOMAIN_KNOWLEDGE = """
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<div class='knowledge-card'>
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<h3 style='color: #667eea; margin-top: 0; display: flex; align-items: center; gap: 10px;'>
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<span style='font-size: 1.3em;'>📚</span> 时域分析常用公式
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</h3>
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<details open>
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<summary style='background: linear-gradient(135deg, #e3f2fd 0%, #f3e5f5 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
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📐 二阶系统标准形式
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</summary>
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<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #e0e7ff;'>
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<p style='margin: 10px 0;'><strong>标准传递函数：</strong></p>
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<div style='text-align: center; background: white; padding: 20px; border-radius: 8px; margin: 10px 0; box-shadow: 0 2px 6px rgba(0,0,0,0.05);'>
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<div style='font-size: 1.3em; display: inline-block;'>
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<i>G</i>(<i>s</i>) =
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<div style='display: inline-block; vertical-align: middle; margin: 0 8px;'>
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<div style='text-align: center; border-bottom: 2px solid #333; padding: 0 10px 5px;'>ω<sub>n</sub><sup>2</sup></div>
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<div style='text-align: center; padding: 5px 10px 0;'><i>s</i><sup>2</sup> + 2ζω<sub>n</sub><i>s</i> + ω<sub>n</sub><sup>2</sup></div>
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</div>
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</div>
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</div>
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<h4 style='color: #667eea; margin-top: 20px; font-size: 1.1em;'>📊 关键参数</h4>
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<table style='width: 100%; border-collapse: collapse; margin: 15px 0; font-size: 0.95em; box-shadow: 0 2px 10px rgba(0,0,0,0.08); border-radius: 8px; overflow: hidden;'>
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<tr style='background: linear-gradient(135deg, #667eea 0%, #764ba2 100%); color: white;'>
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<th style='padding: 12px; text-align: left; font-weight: 700;'>参数</th>
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<th style='padding: 12px; text-align: left; font-weight: 700;'>含义</th>
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<th style='padding: 12px; text-align: left; font-weight: 700;'>影响</th>
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</tr>
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<tr style='background: white;'>
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<td style='padding: 10px; border: 1px solid #e0e7ff;'><strong>ζ (zeta)</strong></td>
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<td style='padding: 10px; border: 1px solid #e0e7ff;'>阻尼比</td>
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<td style='padding: 10px; border: 1px solid #e0e7ff;'>控制超调量和振荡</td>
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</tr>
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<tr style='background: #f8f9ff;'>
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<td style='padding: 10px; border: 1px solid #e0e7ff;'><strong>ω<sub>n</sub></strong></td>
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<td style='padding: 10px; border: 1px solid #e0e7ff;'>无阻尼自然频率 (rad/s)</td>
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<td style='padding: 10px; border: 1px solid #e0e7ff;'>决定响应速度</td>
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</tr>
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</table>
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<h4 style='color: #764ba2; margin-top: 20px; font-size: 1.1em;'>🎯 阻尼比分类</h4>
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<ul style='line-height: 2; margin-left: 20px;'>
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<li><strong style='color: #f5576c;'>ζ < 0:</strong> 不稳定系统</li>
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<li><strong style='color: #ffa726;'>ζ = 0:</strong> 无阻尼振荡</li>
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<li><strong style='color: #4caf50;'>0 < ζ < 1:</strong> 欠阻尼（有振荡）⭐ 最常见</li>
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<li><strong style='color: #2196f3;'>ζ = 1:</strong> 临界阻尼（无超调）</li>
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<li><strong style='color: #9c27b0;'>ζ > 1:</strong> 过阻尼（响应慢）</li>
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</ul>
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</div>
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</details>
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<details open>
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<summary style='background: linear-gradient(135deg, #fff0f8 0%, #fce4ec 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
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📊 时域性能指标（欠阻尼系统）
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</summary>
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<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #e0e7ff;'>
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||
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<div style='background: linear-gradient(135deg, #e3f2fd 0%, #f3e5f5 100%); padding: 15px; border-radius: 10px; margin: 12px 0; border-left: 4px solid #667eea;'>
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<strong style='color: #667eea; font-size: 1.1em;'>1️⃣ 上升时间 (Rise Time, t<sub>r</sub>)</strong>
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<p style='margin: 8px 0;'>响应从 10% 上升到 90% 终值所需时间</p>
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<div style='text-align: center; background: white; padding: 15px; border-radius: 6px; margin: 8px 0;'>
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<div style='font-size: 1.2em; display: inline-block;'>
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<i>t</i><sub>r</sub> ≈
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<div style='display: inline-block; vertical-align: middle; margin: 0 5px;'>
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<div style='text-align: center; border-bottom: 2px solid #333; padding: 0 8px 3px;'>1.8</div>
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<div style='text-align: center; padding: 3px 8px 0;'>ω<sub>n</sub></div>
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</div>
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</div>
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</div>
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</div>
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<div style='background: linear-gradient(135deg, #fff4f0 0%, #ffe0b2 100%); padding: 15px; border-radius: 10px; margin: 12px 0; border-left: 4px solid #ff9800;'>
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<strong style='color: #f57c00; font-size: 1.1em;'>2️⃣ 峰值时间 (Peak Time, t<sub>p</sub>)</strong>
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<p style='margin: 8px 0;'>响应达到第一个峰值的时间</p>
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<div style='text-align: center; background: white; padding: 15px; border-radius: 6px; margin: 8px 0;'>
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<div style='font-size: 1.2em; display: inline-block;'>
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<i>t</i><sub>p</sub> =
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<div style='display: inline-block; vertical-align: middle; margin: 0 5px;'>
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<div style='text-align: center; border-bottom: 2px solid #333; padding: 0 8px 3px;'>π</div>
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<div style='text-align: center; padding: 3px 8px 0;'>ω<sub>n</sub>√(1−ζ²)</div>
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</div>
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=
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<div style='display: inline-block; vertical-align: middle; margin: 0 5px;'>
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<div style='text-align: center; border-bottom: 2px solid #333; padding: 0 8px 3px;'>π</div>
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<div style='text-align: center; padding: 3px 8px 0;'>ω<sub>d</sub></div>
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</div>
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</div>
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</div>
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<p style='margin: 8px 0; color: #666; font-size: 0.95em;'>其中 ω<sub>d</sub> = ω<sub>n</sub>√(1−ζ²) 为阻尼振荡频率</p>
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</div>
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<div style='background: linear-gradient(135deg, #f0fff4 0%, #c8e6c9 100%); padding: 15px; border-radius: 10px; margin: 12px 0; border-left: 4px solid #4caf50;'>
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<strong style='color: #388e3c; font-size: 1.1em;'>3️⃣ 超调量 (Overshoot, σ%)</strong>
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<p style='margin: 8px 0;'>响应超过稳态值的最大百分比</p>
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<div style='text-align: center; background: white; padding: 15px; border-radius: 6px; margin: 8px 0; font-size: 1.2em;'>
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σ% = <i>e</i><sup>−πζ/√(1−ζ²)</sup> × 100%
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</div>
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<p style='margin: 8px 0; color: #388e3c; font-weight: 600;'>💡 仅与 ζ 有关！</p>
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<table style='width: 100%; margin: 10px 0; font-size: 0.9em;'>
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<tr style='background: #e8f5e9;'>
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<td style='padding: 6px; border: 1px solid #c8e6c9;'><strong>ζ = 0.5</strong></td>
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<td style='padding: 6px; border: 1px solid #c8e6c9;'>σ% ≈ 16%</td>
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</tr>
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<tr style='background: white;'>
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<td style='padding: 6px; border: 1px solid #c8e6c9;'><strong>ζ = 0.707</strong></td>
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<td style='padding: 6px; border: 1px solid #c8e6c9;'>σ% ≈ 4.3% ⭐ 最佳</td>
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</tr>
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</table>
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</div>
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<div style='background: linear-gradient(135deg, #fff0f8 0%, #f3e5f5 100%); padding: 15px; border-radius: 10px; margin: 12px 0; border-left: 4px solid #9c27b0;'>
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<strong style='color: #7b1fa2; font-size: 1.1em;'>4️⃣ 调节时间 (Settling Time, t<sub>s</sub>)</strong>
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<p style='margin: 8px 0;'>响应达到并保持在稳态值 ±2%（或±5%）范围内的时间</p>
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<div style='text-align: center; background: white; padding: 15px; border-radius: 6px; margin: 8px 0;'>
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<div style='font-size: 1.2em;'>
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<i>t</i><sub>s</sub> ≈
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<div style='display: inline-block; vertical-align: middle; margin: 0 5px;'>
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<div style='text-align: center; border-bottom: 2px solid #333; padding: 0 8px 3px;'>4</div>
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<div style='text-align: center; padding: 3px 8px 0;'>ζω<sub>n</sub></div>
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</div>
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(2% 误差带)
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</div>
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<div style='font-size: 1.2em; margin-top: 10px;'>
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<i>t</i><sub>s</sub> ≈
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<div style='display: inline-block; vertical-align: middle; margin: 0 5px;'>
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<div style='text-align: center; border-bottom: 2px solid #333; padding: 0 8px 3px;'>3</div>
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<div style='text-align: center; padding: 3px 8px 0;'>ζω<sub>n</sub></div>
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</div>
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(5% 误差带)
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</div>
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</div>
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</div>
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</div>
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</details>
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||
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||
<details>
|
||
<summary style='background: linear-gradient(135deg, #e8f5e9 0%, #c8e6c9 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
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🎯 一阶系统特性
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</summary>
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<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #e0e7ff;'>
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<p><strong>标准传递函数：</strong></p>
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<div style='text-align: center; background: white; padding: 20px; border-radius: 8px; margin: 10px 0;'>
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<div style='font-size: 1.3em; display: inline-block;'>
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<i>G</i>(<i>s</i>) =
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<div style='display: inline-block; vertical-align: middle; margin: 0 8px;'>
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<div style='text-align: center; border-bottom: 2px solid #333; padding: 0 10px 5px;'><i>K</i></div>
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<div style='text-align: center; padding: 5px 10px 0;'>τ<i>s</i> + 1</div>
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</div>
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</div>
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</div>
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<p><strong>阶跃响应：</strong></p>
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<div style='text-align: center; background: white; padding: 20px; border-radius: 8px; margin: 10px 0; font-size: 1.2em;'>
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<i>y</i>(<i>t</i>) = <i>K</i>(1 − <i>e</i><sup>−<i>t</i>/τ</sup>)
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</div>
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<div style='background: #e3f2fd; padding: 12px; border-radius: 8px; margin: 10px 0;'>
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<p><strong>💡 关键时间点：</strong></p>
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<ul style='line-height: 1.8;'>
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<li><strong><i>t</i> = τ:</strong> 达到稳态值的 63.2%</li>
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<li><strong><i>t</i> = 3τ:</strong> 达到稳态值的 95%</li>
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<li><strong><i>t</i> = 4τ:</strong> 达到稳态值的 98.2%</li>
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<li><strong><i>t</i> = 5τ:</strong> 达到稳态值的 99.3%</li>
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</ul>
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</div>
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</div>
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</details>
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</div>
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"""
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# 频域分析知识卡片
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FREQUENCY_DOMAIN_KNOWLEDGE = """
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<div class='knowledge-card'>
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||
<h3 style='color: #667eea; margin-top: 0; display: flex; align-items: center; gap: 10px;'>
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<span style='font-size: 1.3em;'>📚</span> 频域分析常用方法
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</h3>
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||
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||
<details open>
|
||
<summary style='background: linear-gradient(135deg, #e3f2fd 0%, #bbdefb 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
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🎯 增益裕度 (Gain Margin, GM)
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</summary>
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||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #e0e7ff;'>
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<p><strong>定义：</strong>在相角为 -180° 时，系统增益可增加的最大倍数</p>
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<div style='text-align: center; background: white; padding: 20px; border-radius: 8px; margin: 10px 0;'>
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<div style='font-size: 1.3em;'>
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GM<sub>dB</sub> = −20log<sub>10</sub>|<i>G</i>(<i>j</i>ω<sub>pc</sub>)|
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</div>
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</div>
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<p style='margin: 8px 0;'><strong>ω<sub>pc</sub></strong> - 相角交越频率（相角 = -180°）</p>
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||
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<h4 style='color: #667eea; margin-top: 15px;'>判断准则</h4>
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||
<table style='width: 100%; border-collapse: collapse; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08); border-radius: 8px; overflow: hidden;'>
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||
<tr style='background: #e8f5e9;'>
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||
<td style='padding: 10px; border: 1px solid #c8e6c9; font-weight: 600;'>GM > 0 dB</td>
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<td style='padding: 10px; border: 1px solid #c8e6c9;'>✅ 系统稳定</td>
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</tr>
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||
<tr style='background: #fff3e0;'>
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||
<td style='padding: 10px; border: 1px solid #ffe0b2; font-weight: 600;'>GM = 0 dB</td>
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<td style='padding: 10px; border: 1px solid #ffe0b2;'>⚠️ 临界稳定</td>
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</tr>
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||
<tr style='background: #ffebee;'>
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||
<td style='padding: 10px; border: 1px solid #ffcdd2; font-weight: 600;'>GM < 0 dB</td>
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<td style='padding: 10px; border: 1px solid #ffcdd2;'>❌ 系统不稳定</td>
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||
</tr>
|
||
</table>
|
||
|
||
<div style='background: linear-gradient(135deg, #e3f2fd 0%, #f3e5f5 100%); padding: 12px; border-radius: 8px; margin: 10px 0;'>
|
||
<p><strong>💡 工程要求：</strong>通常要求 <strong>GM ≥ 6 dB</strong>（约2倍增益余量）</p>
|
||
</div>
|
||
</div>
|
||
</details>
|
||
|
||
<details open>
|
||
<summary style='background: linear-gradient(135deg, #f3e5f5 0%, #e1bee7 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
🎯 相角裕度 (Phase Margin, PM)
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #e0e7ff;'>
|
||
<p><strong>定义：</strong>在增益为 1 (0dB) 时，系统相角与 -180° 的差值</p>
|
||
<div style='text-align: center; background: white; padding: 20px; border-radius: 8px; margin: 10px 0; font-size: 1.3em;'>
|
||
PM = 180° + ∠<i>G</i>(<i>j</i>ω<sub>gc</sub>)
|
||
</div>
|
||
<p style='margin: 8px 0;'><strong>ω<sub>gc</sub></strong> - 增益交越频率（幅值 = 1 或 0dB）</p>
|
||
|
||
<h4 style='color: #764ba2; margin-top: 15px;'>判断准则</h4>
|
||
<table style='width: 100%; border-collapse: collapse; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08); border-radius: 8px; overflow: hidden;'>
|
||
<tr style='background: #e8f5e9;'>
|
||
<td style='padding: 10px; border: 1px solid #c8e6c9; font-weight: 600;'>PM > 0°</td>
|
||
<td style='padding: 10px; border: 1px solid #c8e6c9;'>✅ 系统稳定</td>
|
||
</tr>
|
||
<tr style='background: #fff3e0;'>
|
||
<td style='padding: 10px; border: 1px solid #ffe0b2; font-weight: 600;'>PM = 0°</td>
|
||
<td style='padding: 10px; border: 1px solid #ffe0b2;'>⚠️ 临界稳定</td>
|
||
</tr>
|
||
<tr style='background: #ffebee;'>
|
||
<td style='padding: 10px; border: 1px solid #ffcdd2; font-weight: 600;'>PM < 0°</td>
|
||
<td style='padding: 10px; border: 1px solid #ffcdd2;'>❌ 系统不稳定</td>
|
||
</tr>
|
||
</table>
|
||
|
||
<div style='background: linear-gradient(135deg, #fff3e0 0%, #ffe0b2 100%); padding: 12px; border-radius: 8px; margin: 10px 0;'>
|
||
<p><strong>💡 工程要求：</strong>通常要求 <strong>PM ∈ [30°, 60°]</strong></p>
|
||
<p style='margin: 8px 0;'>• PM ≈ 45° ~ 60°: 良好阻尼特性</p>
|
||
<p style='margin: 8px 0;'>• PM 越大，超调量越小</p>
|
||
</div>
|
||
|
||
<h4 style='color: #f57c00; margin-top: 15px;'>与时域性能的关系</h4>
|
||
<div style='text-align: center; background: white; padding: 15px; border-radius: 8px; margin: 10px 0; font-size: 1.2em;'>
|
||
ζ ≈ <div style='display: inline-block; vertical-align: middle; margin: 0 5px;'>
|
||
<div style='text-align: center; border-bottom: 2px solid #333; padding: 0 8px 3px;'>PM</div>
|
||
<div style='text-align: center; padding: 3px 8px 0;'>100</div>
|
||
</div> (PM 以度为单位)
|
||
</div>
|
||
<table style='width: 100%; margin: 10px 0; font-size: 0.9em; box-shadow: 0 2px 6px rgba(0,0,0,0.06); border-radius: 6px; overflow: hidden;'>
|
||
<tr style='background: #f5f5f5; font-weight: 600;'>
|
||
<td style='padding: 8px; border: 1px solid #ddd;'>PM</td>
|
||
<td style='padding: 8px; border: 1px solid #ddd;'>ζ</td>
|
||
<td style='padding: 8px; border: 1px solid #ddd;'>超调量</td>
|
||
</tr>
|
||
<tr><td style='padding: 8px; border: 1px solid #ddd;'>30°</td><td style='padding: 8px; border: 1px solid #ddd;'>≈ 0.3</td><td style='padding: 8px; border: 1px solid #ddd;'>≈ 37%</td></tr>
|
||
<tr><td style='padding: 8px; border: 1px solid #ddd;'>45°</td><td style='padding: 8px; border: 1px solid #ddd;'>≈ 0.45</td><td style='padding: 8px; border: 1px solid #ddd;'>≈ 20%</td></tr>
|
||
<tr><td style='padding: 8px; border: 1px solid #ddd;'>60°</td><td style='padding: 8px; border: 1px solid #ddd;'>≈ 0.6</td><td style='padding: 8px; border: 1px solid #ddd;'>≈ 10% ⭐</td></tr>
|
||
</table>
|
||
</div>
|
||
</details>
|
||
|
||
<details>
|
||
<summary style='background: linear-gradient(135deg, #fff0f8 0%, #fce4ec 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
📊 Bode 图与 Nyquist 图
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #e0e7ff;'>
|
||
<h4 style='color: #667eea;'>Bode 图（伯德图）</h4>
|
||
<p><strong>组成：</strong>幅频特性图 + 相频特性图</p>
|
||
<ul style='line-height: 1.8;'>
|
||
<li><strong>横轴：</strong>频率 ω (对数刻度)</li>
|
||
<li><strong>纵轴（幅频）：</strong>20log|<i>G</i>(<i>j</i>ω)| (dB)</li>
|
||
<li><strong>纵轴（相频）：</strong>∠<i>G</i>(<i>j</i>ω) (度)</li>
|
||
</ul>
|
||
|
||
<div style='background: #e3f2fd; padding: 12px; border-radius: 8px; margin: 10px 0;'>
|
||
<p><strong>💡 优点：</strong></p>
|
||
<ul style='line-height: 1.6;'>
|
||
<li>便于绘制（渐近线近似）</li>
|
||
<li>直观读取稳定裕度</li>
|
||
<li>串联系统可图形叠加</li>
|
||
</ul>
|
||
</div>
|
||
|
||
<h4 style='color: #764ba2; margin-top: 20px;'>Nyquist 图（奈奎斯特图）</h4>
|
||
<p><strong>定义：</strong>开环频率特性在复平面上的轨迹</p>
|
||
|
||
<div style='background: #fff3e0; padding: 12px; border-radius: 8px; margin: 10px 0;'>
|
||
<p><strong>🎯 奈奎斯特稳定判据：</strong></p>
|
||
<p style='text-align: center; font-weight: 600; margin: 10px 0; font-size: 1.2em;'><i>Z</i> = <i>P</i> − <i>N</i></p>
|
||
<p style='font-size: 0.95em;'>• <i>Z</i>: 闭环右半平面极点数<br>
|
||
• <i>P</i>: 开环右半平面极点数<br>
|
||
• <i>N</i>: 曲线逆时针包围 (−1, <i>j</i>0) 的圈数</p>
|
||
<p style='margin-top: 10px; color: #f57c00; font-weight: 600;'>稳定条件：<i>Z</i> = 0</p>
|
||
</div>
|
||
</div>
|
||
</details>
|
||
|
||
<details>
|
||
<summary style='background: linear-gradient(135deg, #e8f5e9 0%, #c8e6c9 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
🔧 典型环节频率特性
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #e0e7ff;'>
|
||
<div style='margin-bottom: 15px; padding: 12px; background: white; border-radius: 8px; border-left: 4px solid #667eea;'>
|
||
<h4 style='margin-top: 0; color: #667eea;'>1️⃣ 比例环节 <i>K</i></h4>
|
||
<p>• 幅频：20log<i>K</i> (dB) - 水平线<br>• 相频：0° - 水平线</p>
|
||
</div>
|
||
|
||
<div style='margin-bottom: 15px; padding: 12px; background: white; border-radius: 8px; border-left: 4px solid #f57c00;'>
|
||
<h4 style='margin-top: 0; color: #f57c00;'>2️⃣ 积分环节 1/<i>s</i></h4>
|
||
<p>• 幅频：-20dB/dec 斜率<br>• 相频：-90° - 水平线</p>
|
||
</div>
|
||
|
||
<div style='margin-bottom: 15px; padding: 12px; background: white; border-radius: 8px; border-left: 4px solid #4caf50;'>
|
||
<h4 style='margin-top: 0; color: #4caf50;'>3️⃣ 惯性环节 1/(<i>Ts</i>+1)</h4>
|
||
<p>• 转折频率：ω = 1/<i>T</i><br>• 幅频：低频 0dB，高频 -20dB/dec<br>• 相频：0° → -90°</p>
|
||
</div>
|
||
</div>
|
||
</details>
|
||
</div>
|
||
"""
|
||
|
||
# 根轨迹分析知识卡片
|
||
ROOT_LOCUS_KNOWLEDGE = """
|
||
<div class='knowledge-card'>
|
||
<h3 style='color: #667eea; margin-top: 0; display: flex; align-items: center; gap: 10px;'>
|
||
<span style='font-size: 1.3em;'>📚</span> 根轨迹分析要点
|
||
</h3>
|
||
|
||
<details open>
|
||
<summary style='background: linear-gradient(135deg, #e3f2fd 0%, #bbdefb 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
🎯 根轨迹基本概念
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #e0e7ff;'>
|
||
<p><strong>定义：</strong>当开环增益 <i>K</i> 从 0 → ∞ 变化时，闭环特征方程根（极点）在 <i>s</i> 平面上的运动轨迹</p>
|
||
|
||
<h4 style='color: #667eea; margin-top: 15px;'>特征方程</h4>
|
||
<div style='text-align: center; background: white; padding: 15px; border-radius: 8px; margin: 10px 0; font-size: 1.3em;'>
|
||
1 + <i>KG</i>(<i>s</i>)<i>H</i>(<i>s</i>) = 0
|
||
</div>
|
||
|
||
<h4 style='color: #764ba2; margin-top: 15px;'>主要作用</h4>
|
||
<ul style='line-height: 1.8;'>
|
||
<li>✅ 直观显示增益对极点位置的影响</li>
|
||
<li>✅ 判断系统稳定性</li>
|
||
<li>✅ 选择合适的增益值</li>
|
||
<li>✅ 设计控制器参数</li>
|
||
</ul>
|
||
</div>
|
||
</details>
|
||
|
||
<details open>
|
||
<summary style='background: linear-gradient(135deg, #f3e5f5 0%, #e1bee7 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
📐 根轨迹绘制基本条件
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #e0e7ff;'>
|
||
|
||
<div style='background: linear-gradient(135deg, #e8f5e9 0%, #c8e6c9 100%); padding: 15px; border-radius: 10px; margin: 12px 0; border-left: 4px solid #4caf50;'>
|
||
<h4 style='margin-top: 0; color: #388e3c;'>1️⃣ 幅值条件（充要条件）</h4>
|
||
<div style='text-align: center; background: white; padding: 15px; border-radius: 6px; margin: 8px 0; font-size: 1.2em;'>
|
||
|<i>KG</i>(<i>s</i><sub>0</sub>)<i>H</i>(<i>s</i><sub>0</sub>)| = 1
|
||
</div>
|
||
<p style='margin: 8px 0;'><strong>💡 用途：</strong>确定增益 <i>K</i> 的值</p>
|
||
</div>
|
||
|
||
<div style='background: linear-gradient(135deg, #fff3e0 0%, #ffe0b2 100%); padding: 15px; border-radius: 10px; margin: 12px 0; border-left: 4px solid #ff9800;'>
|
||
<h4 style='margin-top: 0; color: #f57c00;'>2️⃣ 相角条件（充要条件）</h4>
|
||
<div style='text-align: center; background: white; padding: 15px; border-radius: 6px; margin: 8px 0; font-size: 1.2em;'>
|
||
∠<i>G</i>(<i>s</i><sub>0</sub>)<i>H</i>(<i>s</i><sub>0</sub>) = (2<i>k</i>+1) × 180°
|
||
</div>
|
||
<p style='margin: 8px 0; font-size: 0.95em;'>其中 <i>k</i> = 0, ±1, ±2, ...</p>
|
||
<p style='margin: 8px 0;'><strong>💡 用途：</strong>判断 <i>s</i> 平面某点是否在根轨迹上</p>
|
||
</div>
|
||
</div>
|
||
</details>
|
||
|
||
<details open>
|
||
<summary style='background: linear-gradient(135deg, #fff0f8 0%, #fce4ec 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
🌟 根轨迹基本性质
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #e0e7ff;'>
|
||
|
||
<table style='width: 100%; border-collapse: collapse; margin: 15px 0; box-shadow: 0 2px 10px rgba(0,0,0,0.08); border-radius: 8px; overflow: hidden;'>
|
||
<tr style='background: linear-gradient(135deg, #667eea 0%, #764ba2 100%); color: white;'>
|
||
<th style='padding: 12px; text-align: left; font-weight: 700;'>规则</th>
|
||
<th style='padding: 12px; text-align: left; font-weight: 700;'>内容</th>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 10px; border: 1px solid #e0e7ff; font-weight: 600;'>起点</td>
|
||
<td style='padding: 10px; border: 1px solid #e0e7ff;'><i>K</i>=0，位于<strong>开环极点</strong></td>
|
||
</tr>
|
||
<tr style='background: #f8f9ff;'>
|
||
<td style='padding: 10px; border: 1px solid #e0e7ff; font-weight: 600;'>终点</td>
|
||
<td style='padding: 10px; border: 1px solid #e0e7ff;'><i>K</i>→∞，位于<strong>开环零点</strong>或无穷远</td>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 10px; border: 1px solid #e0e7ff; font-weight: 600;'>分支数</td>
|
||
<td style='padding: 10px; border: 1px solid #e0e7ff;'>max(<i>n</i>, <i>m</i>)，<i>n</i>=极点数，<i>m</i>=零点数</td>
|
||
</tr>
|
||
<tr style='background: #f8f9ff;'>
|
||
<td style='padding: 10px; border: 1px solid #e0e7ff; font-weight: 600;'>实轴段</td>
|
||
<td style='padding: 10px; border: 1px solid #e0e7ff;'>右侧零极点总数为<strong>奇数</strong>的区段</td>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 10px; border: 1px solid #e0e7ff; font-weight: 600;'>渐近线数</td>
|
||
<td style='padding: 10px; border: 1px solid #e0e7ff;'><strong><i>n</i> − <i>m</i></strong> 条</td>
|
||
</tr>
|
||
<tr style='background: #f8f9ff;'>
|
||
<td style='padding: 10px; border: 1px solid #e0e7ff; font-weight: 600;'>渐近线角度</td>
|
||
<td style='padding: 10px; border: 1px solid #e0e7ff;'>φ<sub>a</sub> = (2<i>k</i>+1)×180°/(<i>n</i>−<i>m</i>)</td>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 10px; border: 1px solid #e0e7ff; font-weight: 600;'>渐近线交点</td>
|
||
<td style='padding: 10px; border: 1px solid #e0e7ff;'>σ<sub>a</sub> = (Σ<i>p</i> − Σ<i>z</i>)/(<i>n</i>−<i>m</i>)</td>
|
||
</tr>
|
||
</table>
|
||
</div>
|
||
</details>
|
||
|
||
<details>
|
||
<summary style='background: linear-gradient(135deg, #e8f5e9 0%, #c8e6c9 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
🛡️ <i>s</i> 平面稳定性区域
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #e0e7ff;'>
|
||
<table style='width: 100%; border-collapse: collapse; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08); border-radius: 8px; overflow: hidden;'>
|
||
<tr style='background: #f5f5f5; font-weight: 600;'>
|
||
<th style='padding: 10px; border: 1px solid #ddd;'>区域</th>
|
||
<th style='padding: 10px; border: 1px solid #ddd;'>条件</th>
|
||
<th style='padding: 10px; border: 1px solid #ddd;'>稳定性</th>
|
||
</tr>
|
||
<tr>
|
||
<td style='padding: 10px; border: 1px solid #ddd;'>左半平面</td>
|
||
<td style='padding: 10px; border: 1px solid #ddd;'>Re(<i>s</i>) < 0</td>
|
||
<td style='padding: 10px; border: 1px solid #ddd; background: #c8e6c9;'><strong>✅ 稳定</strong></td>
|
||
</tr>
|
||
<tr>
|
||
<td style='padding: 10px; border: 1px solid #ddd;'>虚轴</td>
|
||
<td style='padding: 10px; border: 1px solid #ddd;'>Re(<i>s</i>) = 0</td>
|
||
<td style='padding: 10px; border: 1px solid #ddd; background: #fff9c4;'><strong>⚠️ 临界稳定</strong></td>
|
||
</tr>
|
||
<tr>
|
||
<td style='padding: 10px; border: 1px solid #ddd;'>右半平面</td>
|
||
<td style='padding: 10px; border: 1px solid #ddd;'>Re(<i>s</i>) > 0</td>
|
||
<td style='padding: 10px; border: 1px solid #ddd; background: #ffcdd2;'><strong>❌ 不稳定</strong></td>
|
||
</tr>
|
||
</table>
|
||
|
||
<div style='background: #e3f2fd; padding: 12px; border-radius: 8px; margin: 10px 0;'>
|
||
<p><strong>💡 虚轴（<i>x</i>=0）</strong>即为<strong>稳定性边界</strong></p>
|
||
<p style='margin: 8px 0;'>根轨迹穿越虚轴点对应<strong>临界增益 <i>K</i><sub>c</sub></strong></p>
|
||
</div>
|
||
</div>
|
||
</details>
|
||
|
||
<details>
|
||
<summary style='background: linear-gradient(135deg, #fff3e0 0%, #ffe0b2 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
📏 阻尼比等值线
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #e0e7ff;'>
|
||
<p>从原点出发的射线代表恒定阻尼比 ζ 的轨迹：</p>
|
||
<div style='text-align: center; background: white; padding: 15px; border-radius: 8px; margin: 10px 0; font-size: 1.2em;'>
|
||
θ = arccos(ζ)
|
||
</div>
|
||
|
||
<table style='width: 100%; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.06); border-radius: 6px; overflow: hidden;'>
|
||
<tr style='background: #f5f5f5; font-weight: 600;'>
|
||
<td style='padding: 8px; border: 1px solid #ddd;'>阻尼比 ζ</td>
|
||
<td style='padding: 8px; border: 1px solid #ddd;'>角度 θ</td>
|
||
<td style='padding: 8px; border: 1px solid #ddd;'>系统特性</td>
|
||
</tr>
|
||
<tr>
|
||
<td style='padding: 8px; border: 1px solid #ddd;'>0.5</td>
|
||
<td style='padding: 8px; border: 1px solid #ddd;'>60°</td>
|
||
<td style='padding: 8px; border: 1px solid #ddd;'>欠阻尼，较大超调</td>
|
||
</tr>
|
||
<tr>
|
||
<td style='padding: 8px; border: 1px solid #ddd;'><strong>0.707</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #ddd;'><strong>45°</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #ddd;'><strong>⭐ 最佳阻尼</strong></td>
|
||
</tr>
|
||
</table>
|
||
|
||
<div style='background: linear-gradient(135deg, #e8f5e9 0%, #c8e6c9 100%); padding: 12px; border-radius: 8px; margin: 10px 0;'>
|
||
<p><strong>💡 工程应用：</strong>通过根轨迹与阻尼比等值线的交点，选择满足性能要求的增益 <i>K</i></p>
|
||
</div>
|
||
</div>
|
||
</details>
|
||
</div>
|
||
"""
|
||
|
||
|
||
# ============================================================
|
||
# 发动机控制器设计知识卡片 (PID + MPC) — 中文
|
||
# ============================================================
|
||
ENGINE_CONTROL_KNOWLEDGE = """
|
||
<div class='knowledge-card'>
|
||
<h3 style='color: #e65100; margin-top: 0; display: flex; align-items: center; gap: 10px;'>
|
||
<span style='font-size: 1.3em;'>🔧</span> 发动机控制器设计理论
|
||
</h3>
|
||
|
||
<details open>
|
||
<summary style='background: linear-gradient(135deg, #fff3e0 0%, #ffe0b2 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
🎛️ PID 控制器 — 增量式形式
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #ffe0b2;'>
|
||
<p style='margin: 10px 0;'><strong>标准PID传递函数：</strong></p>
|
||
<div style='text-align: center; background: white; padding: 18px; border-radius: 8px; margin: 10px 0; box-shadow: 0 2px 6px rgba(0,0,0,0.05);'>
|
||
<div style='font-size: 1.2em;'>
|
||
<i>u</i>(<i>t</i>) = <i>K</i><sub>p</sub> <i>e</i>(<i>t</i>)
|
||
+ <i>K</i><sub>i</sub> ∫<i>e</i>(τ)dτ
|
||
+ <i>K</i><sub>d</sub>
|
||
<span style='display: inline-block; vertical-align: middle;'>
|
||
<span style='display: block; text-align: center; border-bottom: 2px solid #333; padding: 0 6px 3px;'>d<i>e</i>(<i>t</i>)</span>
|
||
<span style='display: block; text-align: center; padding: 3px 6px 0;'>d<i>t</i></span>
|
||
</span>
|
||
</div>
|
||
</div>
|
||
<p style='margin: 10px 0;'><strong>增量式（速度型）PID：</strong></p>
|
||
<div style='text-align: center; background: white; padding: 18px; border-radius: 8px; margin: 10px 0; box-shadow: 0 2px 6px rgba(0,0,0,0.05);'>
|
||
<div style='font-size: 1.1em;'>
|
||
Δ<i>u</i>(<i>k</i>) = <i>K</i><sub>p</sub>[<i>e</i>(<i>k</i>) − <i>e</i>(<i>k</i>−1)]
|
||
+ <i>K</i><sub>i</sub> <i>e</i>(<i>k</i>)
|
||
+ <i>K</i><sub>d</sub>[<i>e</i>(<i>k</i>) − 2<i>e</i>(<i>k</i>−1) + <i>e</i>(<i>k</i>−2)]
|
||
</div>
|
||
</div>
|
||
<div style='background: linear-gradient(135deg, #e3f2fd 0%, #f3e5f5 100%); padding: 12px; border-radius: 8px; margin: 10px 0;'>
|
||
<p><strong>💡 增量式优势：</strong></p>
|
||
<ul style='line-height: 1.8; margin-left: 15px;'>
|
||
<li>输出为控制量的增量 Δu，而非绝对值，避免积分饱和（Windup）</li>
|
||
<li>手动/自动切换时输出无跳变，适合工程实际</li>
|
||
<li>计算量恒定，不随运行时间增加</li>
|
||
<li>即使计算失误，影响仅限于当前步的增量</li>
|
||
</ul>
|
||
</div>
|
||
|
||
<h4 style='color: #e65100; margin-top: 18px;'>📊 各参数作用与调节规律</h4>
|
||
<table style='width: 100%; border-collapse: collapse; margin: 12px 0; font-size: 0.93em; box-shadow: 0 2px 10px rgba(0,0,0,0.08); border-radius: 8px; overflow: hidden;'>
|
||
<tr style='background: linear-gradient(135deg, #e65100 0%, #bf360c 100%); color: white;'>
|
||
<th style='padding: 10px; text-align: left;'>参数</th>
|
||
<th style='padding: 10px; text-align: left;'>增大效果</th>
|
||
<th style='padding: 10px; text-align: left;'>过大时</th>
|
||
<th style='padding: 10px; text-align: left;'>调节建议</th>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'><strong>K<sub>p</sub></strong> (比例)</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>加快响应速度，减小稳态误差</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>产生振荡甚至不稳定</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>先调大至出现振荡，再回退70%</td>
|
||
</tr>
|
||
<tr style='background: #fff8f0;'>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'><strong>K<sub>i</sub></strong> (积分)</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>彻底消除稳态误差</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>超调加重，积分饱和</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>从小值开始缓慢增大</td>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'><strong>K<sub>d</sub></strong> (微分)</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>抑制超调和振荡，改善动态品质</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>放大噪声，响应迟钝</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>通常取较小值，有噪声时可设为0</td>
|
||
</tr>
|
||
</table>
|
||
|
||
<div style='background: linear-gradient(135deg, #e8f5e9 0%, #c8e6c9 100%); padding: 12px; border-radius: 8px; margin: 10px 0;'>
|
||
<p><strong>🎯 经典调参步骤（Ziegler-Nichols 启发式）：</strong></p>
|
||
<ol style='line-height: 2; margin-left: 15px;'>
|
||
<li>设 K<sub>i</sub>=0, K<sub>d</sub>=0，逐渐增大 K<sub>p</sub> 直到系统出现等幅振荡</li>
|
||
<li>记录此时的临界增益 K<sub>u</sub> 和振荡周期 T<sub>u</sub></li>
|
||
<li>按经验公式设定：K<sub>p</sub>=0.6K<sub>u</sub>, K<sub>i</sub>=2K<sub>p</sub>/T<sub>u</sub>, K<sub>d</sub>=K<sub>p</sub>T<sub>u</sub>/8</li>
|
||
<li>根据实际响应微调各参数</li>
|
||
</ol>
|
||
</div>
|
||
</div>
|
||
</details>
|
||
|
||
<details open>
|
||
<summary style='background: linear-gradient(135deg, #e3f2fd 0%, #bbdefb 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
🧠 MPC — 模型预测控制
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #bbdefb;'>
|
||
<p style='margin: 10px 0;'>MPC（Model Predictive Control）利用系统的<strong>内部动态模型</strong>预测未来行为，
|
||
在<strong>滚动时域</strong>上优化控制序列，仅执行第一步，然后重新优化——这就是"滚动优化"原理。</p>
|
||
|
||
<h4 style='color: #1565c0; margin-top: 12px;'>🔑 优化目标函数</h4>
|
||
<div style='text-align: center; background: white; padding: 18px; border-radius: 8px; margin: 10px 0; box-shadow: 0 2px 6px rgba(0,0,0,0.05);'>
|
||
<div style='font-size: 1.1em;'>
|
||
min<sub><i>U</i></sub> <i>J</i> = Σ<sub>k=1</sub><sup>H</sup>
|
||
[ <i>W</i><sub>p</sub> (<i>P</i><sub>k</sub> − <i>P</i><sub>ref</sub>)²
|
||
+ <i>W</i><sub>Δu</sub> (Δ<i>u</i><sub>k</sub>)² ]
|
||
</div>
|
||
</div>
|
||
<p style='margin: 8px 0; color: #555; font-size: 0.93em;'>
|
||
其中 H 为预测时域长度，W<sub>p</sub> 为功率跟踪权重，W<sub>Δu</sub> 为控制增量惩罚权重。
|
||
</p>
|
||
|
||
<h4 style='color: #1565c0; margin-top: 18px;'>🔑 MPC 核心概念</h4>
|
||
<table style='width: 100%; border-collapse: collapse; margin: 12px 0; font-size: 0.93em; box-shadow: 0 2px 10px rgba(0,0,0,0.08); border-radius: 8px; overflow: hidden;'>
|
||
<tr style='background: linear-gradient(135deg, #1565c0 0%, #0d47a1 100%); color: white;'>
|
||
<th style='padding: 10px; text-align: left;'>概念</th>
|
||
<th style='padding: 10px; text-align: left;'>说明</th>
|
||
<th style='padding: 10px; text-align: left;'>调节效果</th>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 8px; border: 1px solid #bbdefb;'><strong>预测时域 (H)</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #bbdefb;'>优化器向前看的步数</td>
|
||
<td style='padding: 8px; border: 1px solid #bbdefb;'>H越大前瞻性越强，但计算量增加</td>
|
||
</tr>
|
||
<tr style='background: #f0f7ff;'>
|
||
<td style='padding: 8px; border: 1px solid #bbdefb;'><strong>W<sub>power</sub></strong></td>
|
||
<td style='padding: 8px; border: 1px solid #bbdefb;'>功率跟踪权重</td>
|
||
<td style='padding: 8px; border: 1px solid #bbdefb;'>越大跟踪越紧密，但控制信号波动可能增大</td>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 8px; border: 1px solid #bbdefb;'><strong>W<sub>Δcost</sub></strong></td>
|
||
<td style='padding: 8px; border: 1px solid #bbdefb;'>控制平滑度权重</td>
|
||
<td style='padding: 8px; border: 1px solid #bbdefb;'>越大控制越平滑，减少执行器磨损</td>
|
||
</tr>
|
||
<tr style='background: #f0f7ff;'>
|
||
<td style='padding: 8px; border: 1px solid #bbdefb;'><strong>超调限制</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #bbdefb;'>输出偏差的硬约束（如 ≤5%）</td>
|
||
<td style='padding: 8px; border: 1px solid #bbdefb;'>越小越安全但可能减缓响应速度</td>
|
||
</tr>
|
||
</table>
|
||
|
||
<div style='background: linear-gradient(135deg, #ede7f6 0%, #d1c4e9 100%); padding: 12px; border-radius: 8px; margin: 10px 0;'>
|
||
<p><strong>🔬 内部线性化模型：</strong>本仿真中，涡轴发动机被线性化为以下离散状态方程：</p>
|
||
<div style='text-align: center; background: white; padding: 12px; border-radius: 6px; margin: 8px 0;'>
|
||
<i>N</i><sub>k+1</sub> = <i>N</i><sub>k</sub> + Δt · <i>K</i><sub>inertia</sub> · (<i>W</i><sub>f,act</sub> − <i>W</i><sub>f,req</sub>)<br>
|
||
<i>W</i><sub>f,act,k+1</sub> = (1 − Δt/τ) · <i>W</i><sub>f,act,k</sub> + (Δt/τ) · <i>u</i><sub>k</sub>
|
||
</div>
|
||
<p style='margin: 8px 0; font-size: 0.93em; color: #555;'>
|
||
τ 为燃油执行机构时间常数，K<sub>inertia</sub> 为转子惯性增益。W<sub>f,req</sub> 和功率 P 由 NN 代理模型实时提供。
|
||
</p>
|
||
</div>
|
||
|
||
<div style='background: linear-gradient(135deg, #fff3e0 0%, #ffe0b2 100%); padding: 12px; border-radius: 8px; margin: 10px 0;'>
|
||
<p><strong>⚖️ PID vs MPC 对比：</strong></p>
|
||
<table style='width: 100%; border-collapse: collapse; margin: 8px 0; font-size: 0.91em;'>
|
||
<tr style='background: #f5f5f5; font-weight: 600;'>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>特性</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>PID</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>MPC</td>
|
||
</tr>
|
||
<tr>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>原理</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>基于误差的反馈校正（事后调节）</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>基于模型的前馈预测（提前规划）</td>
|
||
</tr>
|
||
<tr style='background: #fff8f0;'>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>约束处理</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>需额外加限幅/抗饱和机制</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>在优化中自然处理约束</td>
|
||
</tr>
|
||
<tr>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>计算量</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>极低，微秒级</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>较高，需在线优化</td>
|
||
</tr>
|
||
<tr style='background: #fff8f0;'>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>模型依赖</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>无需模型</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>需要准确的系统模型</td>
|
||
</tr>
|
||
<tr>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>适用场景</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>稳定对象，简单跟踪</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>多变量、有约束的复杂系统</td>
|
||
</tr>
|
||
</table>
|
||
</div>
|
||
</div>
|
||
</details>
|
||
|
||
<details>
|
||
<summary style='background: linear-gradient(135deg, #fff0f8 0%, #fce4ec 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
✈️ 涡轴发动机模型
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #fce4ec;'>
|
||
<p>涡轴发动机由三个耦合子系统建模：</p>
|
||
<table style='width: 100%; border-collapse: collapse; margin: 12px 0; font-size: 0.93em; box-shadow: 0 2px 10px rgba(0,0,0,0.08); border-radius: 8px; overflow: hidden;'>
|
||
<tr style='background: linear-gradient(135deg, #ad1457 0%, #880e4f 100%); color: white;'>
|
||
<th style='padding: 10px;'>子系统</th>
|
||
<th style='padding: 10px;'>动态特性</th>
|
||
<th style='padding: 10px;'>关键参数</th>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 8px; border: 1px solid #fce4ec;'><strong>NN 代理模型</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #fce4ec;'>映射 (高度, 马赫数, 转速) → (燃油流量, 功率)；基于 GPR/数据训练</td>
|
||
<td style='padding: 8px; border: 1px solid #fce4ec;'>输入3维, 输出2维</td>
|
||
</tr>
|
||
<tr style='background: #fff5f8;'>
|
||
<td style='padding: 8px; border: 1px solid #fce4ec;'><strong>燃油执行机构</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #fce4ec;'>一阶惯性环节: dW<sub>f</sub>/dt = (u − W<sub>f</sub>) / τ<sub>fuel</sub></td>
|
||
<td style='padding: 8px; border: 1px solid #fce4ec;'>τ 越大响应越慢</td>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 8px; border: 1px solid #fce4ec;'><strong>转子动力学</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #fce4ec;'>dN/dt = K<sub>inertia</sub> · (W<sub>f,actual</sub> − W<sub>f,required</sub>)</td>
|
||
<td style='padding: 8px; border: 1px solid #fce4ec;'>K 越大加速越快</td>
|
||
</tr>
|
||
</table>
|
||
|
||
<div style='background: linear-gradient(135deg, #e3f2fd 0%, #f3e5f5 100%); padding: 12px; border-radius: 8px; margin: 10px 0;'>
|
||
<p><strong>💡 仿真流程：</strong></p>
|
||
<ol style='line-height: 2; margin-left: 15px;'>
|
||
<li>给定当前转速 N，NN模型预测所需燃油流量 W<sub>f,req</sub> 和当前功率 P</li>
|
||
<li>控制器（PID或MPC）根据功率误差计算燃油命令 W<sub>f,cmd</sub></li>
|
||
<li>执行机构将命令滤波为实际燃油流量 W<sub>f,act</sub></li>
|
||
<li>W<sub>f,act</sub> − W<sub>f,req</sub> 产生净力矩驱动转子加速或减速</li>
|
||
<li>转速变化又改变NN模型的输入，形成闭环</li>
|
||
</ol>
|
||
</div>
|
||
|
||
<div style='background: linear-gradient(135deg, #fff3e0 0%, #ffe0b2 100%); padding: 12px; border-radius: 8px; margin: 10px 0;'>
|
||
<p><strong>📏 性能评价指标：</strong></p>
|
||
<ul style='line-height: 1.8; margin-left: 15px;'>
|
||
<li><strong>稳态误差：</strong>功率达到稳态后与目标的平均偏差</li>
|
||
<li><strong>超调量 (σ%)：</strong>输出超过目标值的最大百分比</li>
|
||
<li><strong>上升时间 (t<sub>r</sub>)：</strong>功率从10%变化到90%所需时间</li>
|
||
<li><strong>调节时间 (t<sub>s</sub>)：</strong>功率进入并持续保持在 ±2% 误差带内的时间</li>
|
||
</ul>
|
||
</div>
|
||
</div>
|
||
</details>
|
||
|
||
<details>
|
||
<summary style='background: linear-gradient(135deg, #e0f7fa 0%, #b2ebf2 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
📐 PID 控制器 — Laplace 域传递函数
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #b2ebf2;'>
|
||
<p><strong>连续域 PID 传递函数：</strong></p>
|
||
<div style='text-align: center; background: white; padding: 18px; border-radius: 8px; margin: 10px 0;'>
|
||
<div style='font-size: 1.15em;'>
|
||
<i>C</i>(<i>s</i>) = <i>K</i><sub>p</sub>
|
||
<span style='font-size: 1.2em;'>(</span>
|
||
1 +
|
||
<span style='display: inline-block; vertical-align: middle;'>
|
||
<span style='display: block; text-align: center; border-bottom: 2px solid #333; padding: 0 8px 3px;'>1</span>
|
||
<span style='display: block; text-align: center; padding: 3px 8px 0;'>T<sub>i</sub> s</span>
|
||
</span>
|
||
+ T<sub>d</sub> s
|
||
<span style='font-size: 1.2em;'>)</span>
|
||
</div>
|
||
</div>
|
||
<p style='color: #555; font-size: 0.93em;'>其中 T<sub>i</sub> = K<sub>p</sub>/K<sub>i</sub> 为积分时间常数，
|
||
T<sub>d</sub> = K<sub>d</sub>/K<sub>p</sub> 为微分时间常数。</p>
|
||
|
||
<h4 style='color: #00838f; margin-top: 15px;'>Ziegler-Nichols 整定公式表</h4>
|
||
<p>基于极限灵敏度(临界增益)法——设 K<sub>u</sub> 为临界增益，T<sub>u</sub> 为临界周期：</p>
|
||
<table style='width: 100%; border-collapse: collapse; margin: 12px 0; font-size: 0.93em; box-shadow: 0 2px 10px rgba(0,0,0,0.08); border-radius: 8px; overflow: hidden;'>
|
||
<tr style='background: linear-gradient(135deg, #00838f 0%, #006064 100%); color: white;'>
|
||
<th style='padding: 10px;'>控制器类型</th>
|
||
<th style='padding: 10px;'>K<sub>p</sub></th>
|
||
<th style='padding: 10px;'>T<sub>i</sub></th>
|
||
<th style='padding: 10px;'>T<sub>d</sub></th>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 8px; border: 1px solid #b2ebf2;'><strong>P</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #b2ebf2;'>0.5 K<sub>u</sub></td>
|
||
<td style='padding: 8px; border: 1px solid #b2ebf2;'>—</td>
|
||
<td style='padding: 8px; border: 1px solid #b2ebf2;'>—</td>
|
||
</tr>
|
||
<tr style='background: #f0fafa;'>
|
||
<td style='padding: 8px; border: 1px solid #b2ebf2;'><strong>PI</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #b2ebf2;'>0.45 K<sub>u</sub></td>
|
||
<td style='padding: 8px; border: 1px solid #b2ebf2;'>T<sub>u</sub> / 1.2</td>
|
||
<td style='padding: 8px; border: 1px solid #b2ebf2;'>—</td>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 8px; border: 1px solid #b2ebf2;'><strong>PID</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #b2ebf2;'>0.6 K<sub>u</sub></td>
|
||
<td style='padding: 8px; border: 1px solid #b2ebf2;'>T<sub>u</sub> / 2</td>
|
||
<td style='padding: 8px; border: 1px solid #b2ebf2;'>T<sub>u</sub> / 8</td>
|
||
</tr>
|
||
</table>
|
||
<p style='color: #555; font-size: 0.93em;'>
|
||
等价转换：K<sub>i</sub> = K<sub>p</sub>/T<sub>i</sub>，K<sub>d</sub> = K<sub>p</sub>·T<sub>d</sub>。
|
||
Z-N 公式通常产生约 25% 超调，实际应用中需进一步微调。
|
||
</p>
|
||
</div>
|
||
</details>
|
||
|
||
<details>
|
||
<summary style='background: linear-gradient(135deg, #f1f8e9 0%, #dcedc8 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
🧮 MPC 矩阵化优化问题
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #dcedc8;'>
|
||
<p>MPC 将有限时域优化问题转化为<strong>二次规划 (QP)</strong> 或非线性规划问题：</p>
|
||
|
||
<h4 style='color: #558b2f;'>离散状态空间模型</h4>
|
||
<div style='text-align: center; background: white; padding: 15px; border-radius: 8px; margin: 10px 0;'>
|
||
<div style='font-size: 1.1em;'>
|
||
<b>x</b><sub>k+1</sub> = A<b>x</b><sub>k</sub> + B<b>u</b><sub>k</sub><br>
|
||
<b>y</b><sub>k</sub> = C<b>x</b><sub>k</sub>
|
||
</div>
|
||
</div>
|
||
<p style='color: #555; font-size: 0.93em;'>
|
||
本系统中：<b>x</b> = [N, W<sub>f,act</sub>]<sup>T</sup>（转速和实际燃油流量），
|
||
<b>u</b> = W<sub>f,cmd</sub>（燃油命令），<b>y</b> = P（功率输出）。
|
||
</p>
|
||
|
||
<h4 style='color: #558b2f;'>预测矩阵展开</h4>
|
||
<p>利用递推关系，将 H 步预测写成矩阵形式：</p>
|
||
<div style='text-align: center; background: white; padding: 15px; border-radius: 8px; margin: 10px 0;'>
|
||
<div style='font-size: 1.05em;'>
|
||
<b>Y</b> = Ψ<b>x</b><sub>0</sub> + Θ<b>U</b>
|
||
</div>
|
||
</div>
|
||
<ul style='line-height: 1.8; margin-left: 15px; font-size: 0.95em;'>
|
||
<li><b>Y</b> = [y<sub>1</sub>, ..., y<sub>H</sub>]<sup>T</sup>：预测输出序列</li>
|
||
<li><b>U</b> = [u<sub>0</sub>, ..., u<sub>H−1</sub>]<sup>T</sup>：控制输入序列</li>
|
||
<li>Ψ = [CA, CA², ..., CA<sup>H</sup>]<sup>T</sup>：自由响应矩阵</li>
|
||
<li>Θ：Toeplitz 控制响应矩阵</li>
|
||
</ul>
|
||
|
||
<h4 style='color: #558b2f;'>带约束 QP 问题</h4>
|
||
<div style='text-align: center; background: white; padding: 15px; border-radius: 8px; margin: 10px 0;'>
|
||
<div style='font-size: 1.05em;'>
|
||
min<sub><b>U</b></sub> (<b>Y</b> − <b>R</b>)<sup>T</sup> Q (<b>Y</b> − <b>R</b>) + Δ<b>U</b><sup>T</sup> R<sub>Δ</sub> Δ<b>U</b><br><br>
|
||
s.t. u<sub>min</sub> ≤ u<sub>k</sub> ≤ u<sub>max</sub>,
|
||
|y<sub>k</sub> − r<sub>k</sub>| ≤ ε<sub>overshoot</sub>
|
||
</div>
|
||
</div>
|
||
<p style='color: #555; font-size: 0.93em;'>
|
||
Q 和 R<sub>Δ</sub> 分别对应界面中的 W<sub>power</sub> 和 W<sub>Δcost</sub> 权重滑块。
|
||
本系统使用 SciPy SLSQP 求解器进行约束非线性优化。
|
||
</p>
|
||
</div>
|
||
</details>
|
||
</div>
|
||
"""
|
||
MOTOR_CONTROL_KNOWLEDGE = """
|
||
<div class='knowledge-card'>
|
||
<h3 style='color: #1565c0; margin-top: 0; display: flex; align-items: center; gap: 10px;'>
|
||
<span style='font-size: 1.3em;'>⚡</span> 电机控制器设计理论
|
||
</h3>
|
||
|
||
<details open>
|
||
<summary style='background: linear-gradient(135deg, #e3f2fd 0%, #bbdefb 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
⚙️ PMSM 永磁同步电机模型
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #bbdefb;'>
|
||
<p><strong>永磁同步电机 (PMSM)</strong> 是电动推进系统的核心驱动组件，具有功率密度高、效率高、控制精度好等优点。</p>
|
||
|
||
<h4 style='color: #1565c0;'>机械动力学方程</h4>
|
||
<div style='text-align: center; background: white; padding: 15px; border-radius: 8px; margin: 10px 0; box-shadow: 0 2px 6px rgba(0,0,0,0.05);'>
|
||
<div style='font-size: 1.15em;'>
|
||
<i>J</i>
|
||
<span style='display: inline-block; vertical-align: middle;'>
|
||
<span style='display: block; text-align: center; border-bottom: 2px solid #333; padding: 0 6px 3px;'>dω</span>
|
||
<span style='display: block; text-align: center; padding: 3px 6px 0;'>dt</span>
|
||
</span>
|
||
= −(<i>T</i><sub>motor</sub> + <i>T</i><sub>load</sub> + <i>T</i><sub>ext</sub> + <i>B</i>ω)
|
||
</div>
|
||
</div>
|
||
<p style='margin: 8px 0; color: #555; font-size: 0.93em;'>
|
||
其中 T<sub>motor</sub> 为电磁转矩（电机产生），T<sub>load</sub> 为负载转矩，
|
||
T<sub>ext</sub> 为外部扰动，Bω 为粘性摩擦力矩。
|
||
</p>
|
||
|
||
<h4 style='color: #1565c0;'>电磁转矩表达式</h4>
|
||
<div style='text-align: center; background: white; padding: 15px; border-radius: 8px; margin: 10px 0; box-shadow: 0 2px 6px rgba(0,0,0,0.05);'>
|
||
<div style='font-size: 1.15em;'>
|
||
<i>T</i><sub>e</sub> = 1.5 · <i>n</i><sub>p</sub> · ψ<sub>f</sub> · <i>i</i><sub>q</sub>
|
||
</div>
|
||
</div>
|
||
<p style='margin: 8px 0; color: #555; font-size: 0.93em;'>
|
||
对表贴式PMSM (L<sub>d</sub> = L<sub>q</sub>)，转矩仅与q轴电流 i<sub>q</sub> 成正比。
|
||
</p>
|
||
|
||
<table style='width: 100%; border-collapse: collapse; margin: 12px 0; font-size: 0.93em; box-shadow: 0 2px 10px rgba(0,0,0,0.08); border-radius: 8px; overflow: hidden;'>
|
||
<tr style='background: linear-gradient(135deg, #1565c0 0%, #0d47a1 100%); color: white;'>
|
||
<th style='padding: 10px;'>符号</th>
|
||
<th style='padding: 10px;'>含义</th>
|
||
<th style='padding: 10px;'>典型值</th>
|
||
<th style='padding: 10px;'>对系统的影响</th>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 8px; border: 1px solid #bbdefb;'><strong>J</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #bbdefb;'>转动惯量 (kg·m²)</td>
|
||
<td style='padding: 8px; border: 1px solid #bbdefb;'>0.5 – 5.0</td>
|
||
<td style='padding: 8px; border: 1px solid #bbdefb;'>越大响应越慢但越平稳</td>
|
||
</tr>
|
||
<tr style='background: #f0f7ff;'>
|
||
<td style='padding: 8px; border: 1px solid #bbdefb;'><strong>B</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #bbdefb;'>粘性摩擦系数 (N·m·s)</td>
|
||
<td style='padding: 8px; border: 1px solid #bbdefb;'>3×10⁻⁴</td>
|
||
<td style='padding: 8px; border: 1px solid #bbdefb;'>提供自然阻尼</td>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 8px; border: 1px solid #bbdefb;'><strong>n<sub>p</sub></strong></td>
|
||
<td style='padding: 8px; border: 1px solid #bbdefb;'>极对数</td>
|
||
<td style='padding: 8px; border: 1px solid #bbdefb;'>3</td>
|
||
<td style='padding: 8px; border: 1px solid #bbdefb;'>影响电气频率和转矩常数</td>
|
||
</tr>
|
||
<tr style='background: #f0f7ff;'>
|
||
<td style='padding: 8px; border: 1px solid #bbdefb;'><strong>ψ<sub>f</sub></strong></td>
|
||
<td style='padding: 8px; border: 1px solid #bbdefb;'>永磁体磁链 (Wb)</td>
|
||
<td style='padding: 8px; border: 1px solid #bbdefb;'>0.15</td>
|
||
<td style='padding: 8px; border: 1px solid #bbdefb;'>决定转矩能力和反电动势</td>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 8px; border: 1px solid #bbdefb;'><strong>P<sub>rate</sub></strong></td>
|
||
<td style='padding: 8px; border: 1px solid #bbdefb;'>额定功率 (kW)</td>
|
||
<td style='padding: 8px; border: 1px solid #bbdefb;'>300</td>
|
||
<td style='padding: 8px; border: 1px solid #bbdefb;'>确定最大可用转矩</td>
|
||
</tr>
|
||
</table>
|
||
|
||
<div style='background: linear-gradient(135deg, #e8f5e9 0%, #c8e6c9 100%); padding: 12px; border-radius: 8px; margin: 10px 0;'>
|
||
<p><strong>💡 电压约束与弱磁：</strong>
|
||
当转速升高时，反电动势 e = n<sub>p</sub>·ψ<sub>f</sub>·ω 增大。
|
||
当反电动势接近母线电压 V<sub>dc</sub> 时，电机进入弱磁区，
|
||
需注入负的 i<sub>d</sub> 电流来削弱磁链，此时可用转矩下降。
|
||
</p>
|
||
</div>
|
||
</div>
|
||
</details>
|
||
|
||
<details open>
|
||
<summary style='background: linear-gradient(135deg, #e8f5e9 0%, #c8e6c9 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
🎯 转速控制环路
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #c8e6c9;'>
|
||
<p>典型电机控制采用<strong>双环级联</strong>结构：外环为转速环，内环为电流环。</p>
|
||
<div style='text-align: center; background: white; padding: 15px; border-radius: 8px; margin: 10px 0;'>
|
||
<span style='font-size: 1em; color: #555;'>
|
||
ω<sub>ref</sub> → [转速控制器] → <i>T</i><sub>cmd</sub> → [电流环 + SVPWM] → PMSM → ω<sub>actual</sub>
|
||
</span>
|
||
</div>
|
||
|
||
<div style='background: linear-gradient(135deg, #fff3e0 0%, #ffe0b2 100%); padding: 12px; border-radius: 8px; margin: 10px 0;'>
|
||
<p><strong>PID 模式：</strong></p>
|
||
<ul style='line-height: 1.8; margin-left: 15px;'>
|
||
<li>根据转速误差 e = ω<sub>ref</sub> − ω 计算转矩命令 T = K<sub>p</sub>·e + K<sub>i</sub>·∫e + K<sub>d</sub>·ė</li>
|
||
<li>转矩命令转换为功率请求 P = T·ω 送入逆变器</li>
|
||
<li>简单可靠，但无法提前预知负载变化</li>
|
||
</ul>
|
||
</div>
|
||
<div style='background: linear-gradient(135deg, #e3f2fd 0%, #bbdefb 100%); padding: 12px; border-radius: 8px; margin: 10px 0;'>
|
||
<p><strong>MPC 模式：</strong></p>
|
||
<ul style='line-height: 1.8; margin-left: 15px;'>
|
||
<li>利用电机离散动力学模型预测 H 步后的转速轨迹</li>
|
||
<li>优化 H 步的转矩序列，使转速跟踪误差和控制增量同时最小化</li>
|
||
<li>支持超调硬约束：转速偏差不得超过设定百分比</li>
|
||
</ul>
|
||
<div style='text-align: center; background: white; padding: 10px; border-radius: 6px; margin: 8px 0;'>
|
||
min<sub>T<sub>0..H</sub></sub> Σ [ W<sub>ω</sub>(ω<sub>k</sub> − ω<sub>ref</sub>)² + W<sub>Δ</sub>(T<sub>k</sub> − T<sub>k−1</sub>)² ]
|
||
</div>
|
||
<p style='margin: 8px 0; font-size: 0.93em; color: #555;'>
|
||
离散状态方程：ω<sub>k+1</sub> = (1−dt·B/J)·ω<sub>k</sub> − (dt/J)·T<sub>cmd</sub> − (dt/J)·(T<sub>ext</sub>+T<sub>load</sub>)
|
||
</p>
|
||
</div>
|
||
</div>
|
||
</details>
|
||
|
||
<details>
|
||
<summary style='background: linear-gradient(135deg, #fce4ec 0%, #f8bbd0 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
🔍 负载扰动测试设计
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #f8bbd0;'>
|
||
<p style='margin: 10px 0;'>本仿真在运行至 <strong>60% 时刻</strong>时自动施加 <strong>50% 负载阶跃扰动</strong>，
|
||
用于检验控制器的抗扰性能。</p>
|
||
|
||
<h4 style='color: #c62828; margin-top: 12px;'>📊 抗扰性能指标</h4>
|
||
<table style='width: 100%; border-collapse: collapse; margin: 12px 0; font-size: 0.93em; box-shadow: 0 2px 10px rgba(0,0,0,0.08); border-radius: 8px; overflow: hidden;'>
|
||
<tr style='background: linear-gradient(135deg, #c62828 0%, #b71c1c 100%); color: white;'>
|
||
<th style='padding: 10px;'>指标</th>
|
||
<th style='padding: 10px;'>定义</th>
|
||
<th style='padding: 10px;'>理想范围</th>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 8px; border: 1px solid #ffcdd2;'><strong>转速跌落 (Speed Dip)</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #ffcdd2;'>扰动后转速最大瞬时下降量</td>
|
||
<td style='padding: 8px; border: 1px solid #ffcdd2;'>< 5% 额定转速</td>
|
||
</tr>
|
||
<tr style='background: #fff5f5;'>
|
||
<td style='padding: 8px; border: 1px solid #ffcdd2;'><strong>恢复时间</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #ffcdd2;'>转速回到 ±2% 设定值的时间</td>
|
||
<td style='padding: 8px; border: 1px solid #ffcdd2;'>< 3 秒</td>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 8px; border: 1px solid #ffcdd2;'><strong>稳态误差</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #ffcdd2;'>扰动后新稳态与目标的偏差</td>
|
||
<td style='padding: 8px; border: 1px solid #ffcdd2;'>< 1% 或指定精度</td>
|
||
</tr>
|
||
</table>
|
||
|
||
<div style='background: linear-gradient(135deg, #fff3e0 0%, #ffe0b2 100%); padding: 12px; border-radius: 8px; margin: 10px 0;'>
|
||
<p><strong>⚙️ 转动惯量 J 的影响：</strong></p>
|
||
<ul style='line-height: 1.8; margin-left: 15px;'>
|
||
<li><strong>J 较大：</strong>响应较慢但转速跌落小，系统惯性大不易被扰动"推走"</li>
|
||
<li><strong>J 较小：</strong>响应快但对扰动更敏感，转速波动大</li>
|
||
<li>实际工程中，可通过飞轮或增大转子质量来调整等效惯量</li>
|
||
</ul>
|
||
</div>
|
||
|
||
<div style='background: linear-gradient(135deg, #e3f2fd 0%, #bbdefb 100%); padding: 12px; border-radius: 8px; margin: 10px 0;'>
|
||
<p><strong>💡 MPC 调参建议：</strong></p>
|
||
<ul style='line-height: 1.8; margin-left: 15px;'>
|
||
<li>如果扰动后转速恢复慢→增大 W<sub>speed</sub> 或减小 W<sub>Δcost</sub></li>
|
||
<li>如果转速振荡→增大 W<sub>Δcost</sub> 或减小 W<sub>speed</sub></li>
|
||
<li>如果超调过大→降低超调限制百分比</li>
|
||
<li>负载很大时，确认电机额定功率是否足以提供所需转矩</li>
|
||
</ul>
|
||
</div>
|
||
</div>
|
||
</details>
|
||
|
||
<details>
|
||
<summary style='background: linear-gradient(135deg, #f3e5f5 0%, #e1bee7 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
📐 PMSM dq 坐标系电压方程
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #e1bee7;'>
|
||
<p>通过 Park 变换将三相 ABC 坐标系转换为旋转 dq 坐标系，PMSM 的电压方程简化为：</p>
|
||
<div style='text-align: center; background: white; padding: 18px; border-radius: 8px; margin: 10px 0;'>
|
||
<div style='font-size: 1.1em; line-height: 2.0;'>
|
||
u<sub>d</sub> = R<sub>s</sub> i<sub>d</sub> + L<sub>d</sub>
|
||
<span style='display: inline-block; vertical-align: middle;'>
|
||
<span style='display: block; text-align: center; border-bottom: 2px solid #333; padding: 0 6px 3px;'>di<sub>d</sub></span>
|
||
<span style='display: block; text-align: center; padding: 3px 6px 0;'>dt</span>
|
||
</span>
|
||
− ω<sub>e</sub> L<sub>q</sub> i<sub>q</sub><br>
|
||
u<sub>q</sub> = R<sub>s</sub> i<sub>q</sub> + L<sub>q</sub>
|
||
<span style='display: inline-block; vertical-align: middle;'>
|
||
<span style='display: block; text-align: center; border-bottom: 2px solid #333; padding: 0 6px 3px;'>di<sub>q</sub></span>
|
||
<span style='display: block; text-align: center; padding: 3px 6px 0;'>dt</span>
|
||
</span>
|
||
+ ω<sub>e</sub> (L<sub>d</sub> i<sub>d</sub> + ψ<sub>f</sub>)
|
||
</div>
|
||
</div>
|
||
<ul style='line-height: 1.8; margin-left: 15px;'>
|
||
<li><strong>ω<sub>e</sub></strong> = n<sub>p</sub> · ω<sub>m</sub>：电气角速度</li>
|
||
<li><strong>−ω<sub>e</sub>L<sub>q</sub>i<sub>q</sub></strong>：d 轴耦合项（q 轴电流产生的交叉耦合）</li>
|
||
<li><strong>+ω<sub>e</sub>ψ<sub>f</sub></strong>：永磁体反电动势（ωψ<sub>f</sub> = E<sub>back-EMF</sub>）</li>
|
||
<li>表贴式 PMSM 中 L<sub>d</sub> = L<sub>q</sub>，无磁阻转矩分量</li>
|
||
</ul>
|
||
|
||
<h4 style='color: #7b1fa2; margin-top: 15px;'>磁场定向控制 (FOC) 原理</h4>
|
||
<div style='background: linear-gradient(135deg, #e3f2fd 0%, #bbdefb 100%); padding: 12px; border-radius: 8px; margin: 10px 0;'>
|
||
<p><strong>FOC 核心思想：</strong>通过控制 i<sub>d</sub> = 0（表贴式），使电磁转矩与 i<sub>q</sub> 线性正比：</p>
|
||
<div style='text-align: center; background: white; padding: 12px; border-radius: 6px; margin: 8px 0;'>
|
||
T<sub>e</sub> = 1.5 · n<sub>p</sub> · ψ<sub>f</sub> · i<sub>q</sub>
|
||
</div>
|
||
<ul style='line-height: 1.8; margin-left: 15px;'>
|
||
<li>外环（转速环）→ 给出转矩参考 → 换算为 i<sub>q</sub> 参考</li>
|
||
<li>内环（电流环）→ PI 控制 i<sub>d</sub>=0, i<sub>q</sub>=目标值</li>
|
||
<li>逆 Park 变换 → SVPWM 产生三相驱动电压</li>
|
||
</ul>
|
||
</div>
|
||
|
||
<h4 style='color: #7b1fa2; margin-top: 15px;'>坐标变换</h4>
|
||
<table style='width: 100%; border-collapse: collapse; margin: 12px 0; font-size: 0.93em;'>
|
||
<tr style='background: #f3e5f5; font-weight: 600;'>
|
||
<td style='padding: 8px; border: 1px solid #e1bee7;'>变换</td>
|
||
<td style='padding: 8px; border: 1px solid #e1bee7;'>作用</td>
|
||
<td style='padding: 8px; border: 1px solid #e1bee7;'>公式要点</td>
|
||
</tr>
|
||
<tr>
|
||
<td style='padding: 8px; border: 1px solid #e1bee7;'><strong>Clarke (ABC→αβ)</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #e1bee7;'>三相→两相静止坐标</td>
|
||
<td style='padding: 8px; border: 1px solid #e1bee7;'>i<sub>α</sub> = i<sub>a</sub>，i<sub>β</sub> = (i<sub>a</sub> + 2i<sub>b</sub>)/√3</td>
|
||
</tr>
|
||
<tr style='background: #faf5ff;'>
|
||
<td style='padding: 8px; border: 1px solid #e1bee7;'><strong>Park (αβ→dq)</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #e1bee7;'>静止→旋转坐标</td>
|
||
<td style='padding: 8px; border: 1px solid #e1bee7;'>i<sub>d</sub> = i<sub>α</sub>cosθ + i<sub>β</sub>sinθ</td>
|
||
</tr>
|
||
</table>
|
||
</div>
|
||
</details>
|
||
</div>
|
||
"""
|
||
GPR_KNOWLEDGE = """
|
||
<div class='knowledge-card'>
|
||
<h3 style='color: #4a148c; margin-top: 0; display: flex; align-items: center; gap: 10px;'>
|
||
<span style='font-size: 1.3em;'>📈</span> GPR 高斯过程回归 — 理论与公式详解
|
||
</h3>
|
||
|
||
<details open>
|
||
<summary style='background: linear-gradient(135deg, #f3e5f5 0%, #e1bee7 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
📖 什么是高斯过程回归 (Gaussian Process Regression)?
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #e1bee7;'>
|
||
<p style='margin: 10px 0;'><strong>高斯过程 (Gaussian Process, GP)</strong> 是一种定义在函数空间上的随机过程。
|
||
直觉上，GP 将"先验分布"从有限维的参数空间推广到无穷维的函数空间——它不是对参数建模，
|
||
而是直接对函数本身施加概率分布。</p>
|
||
|
||
<div style='background: linear-gradient(135deg, #ede7f6 0%, #d1c4e9 100%); padding: 15px; border-radius: 8px; margin: 12px 0; border-left: 4px solid #7c4dff;'>
|
||
<p style='margin: 0; font-weight: 700; color: #4a148c;'>📌 核心定义</p>
|
||
<p style='margin: 8px 0;'>一个高斯过程是指：对于任意有限输入点集合 {<b>x</b><sub>1</sub>, ..., <b>x</b><sub>n</sub>}，
|
||
其对应的函数值 [f(<b>x</b><sub>1</sub>), ..., f(<b>x</b><sub>n</sub>)] 服从联合多元高斯分布。</p>
|
||
<div style='text-align: center; background: white; padding: 15px; border-radius: 8px; margin: 10px 0;'>
|
||
<div style='font-size: 1.15em;'>
|
||
<i>f</i>(<b>x</b>) ~ GP( <i>m</i>(<b>x</b>), <i>k</i>(<b>x</b>, <b>x</b>') )
|
||
</div>
|
||
</div>
|
||
<ul style='line-height: 1.8; margin-left: 15px;'>
|
||
<li><strong>均值函数</strong> <i>m</i>(<b>x</b>) = E[<i>f</i>(<b>x</b>)]：描述函数的先验趋势</li>
|
||
<li><strong>协方差函数（核函数）</strong> <i>k</i>(<b>x</b>, <b>x</b>') = Cov[<i>f</i>(<b>x</b>), <i>f</i>(<b>x</b>')]：
|
||
刻画不同输入点处函数值之间的相关性</li>
|
||
</ul>
|
||
</div>
|
||
|
||
<p><strong>与参数化回归的本质区别：</strong></p>
|
||
<table style='width: 100%; border-collapse: collapse; margin: 12px 0; font-size: 0.93em; box-shadow: 0 2px 10px rgba(0,0,0,0.08); border-radius: 8px; overflow: hidden;'>
|
||
<tr style='background: linear-gradient(135deg, #6a1b9a 0%, #4a148c 100%); color: white;'>
|
||
<th style='padding: 10px;'>方面</th>
|
||
<th style='padding: 10px;'>参数化方法（如线性回归）</th>
|
||
<th style='padding: 10px;'>GPR（非参数贝叶斯）</th>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 8px; border: 1px solid #e1bee7;'>模型形式</td>
|
||
<td style='padding: 8px; border: 1px solid #e1bee7;'>f(x) = w<sup>T</sup>φ(x)，参数 w 有限</td>
|
||
<td style='padding: 8px; border: 1px solid #e1bee7;'>直接在函数空间上建模</td>
|
||
</tr>
|
||
<tr style='background: #faf5ff;'>
|
||
<td style='padding: 8px; border: 1px solid #e1bee7;'>复杂度</td>
|
||
<td style='padding: 8px; border: 1px solid #e1bee7;'>由模型结构（特征维度）固定</td>
|
||
<td style='padding: 8px; border: 1px solid #e1bee7;'>随数据量自适应增长</td>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 8px; border: 1px solid #e1bee7;'>不确定性</td>
|
||
<td style='padding: 8px; border: 1px solid #e1bee7;'>通常只提供点估计</td>
|
||
<td style='padding: 8px; border: 1px solid #e1bee7;'>天然提供预测方差（置信区间）</td>
|
||
</tr>
|
||
<tr style='background: #faf5ff;'>
|
||
<td style='padding: 8px; border: 1px solid #e1bee7;'>小样本</td>
|
||
<td style='padding: 8px; border: 1px solid #e1bee7;'>容易过拟合或欠拟合</td>
|
||
<td style='padding: 8px; border: 1px solid #e1bee7;'>先验正则化，小样本表现好</td>
|
||
</tr>
|
||
</table>
|
||
</div>
|
||
</details>
|
||
|
||
<details open>
|
||
<summary style='background: linear-gradient(135deg, #e8eaf6 0%, #c5cae9 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
🔬 核函数 (Kernel / Covariance Function) 详解
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #c5cae9;'>
|
||
<p>核函数是 GPR 的"灵魂"——它完全决定了 GP 的先验性质（平滑度、周期性、长程相关性等）。
|
||
核函数 <i>k</i>(<b>x</b>, <b>x</b>') 衡量两个输入点对应函数值的相关程度。</p>
|
||
|
||
<div style='background: linear-gradient(135deg, #e3f2fd 0%, #f3e5f5 100%); padding: 15px; border-radius: 10px; margin: 12px 0; border-left: 4px solid #1565c0;'>
|
||
<strong style='color: #1565c0; font-size: 1.05em;'>1️⃣ RBF 核 (径向基函数核 / 平方指数核)</strong>
|
||
<div style='text-align: center; background: white; padding: 15px; border-radius: 8px; margin: 10px 0;'>
|
||
<div style='font-size: 1.15em;'>
|
||
<i>k</i><sub>RBF</sub>(<b>x</b>, <b>x</b>') = σ<sub>f</sub><sup>2</sup> · exp
|
||
<span style='font-size: 1.2em;'>(</span>
|
||
−
|
||
<span style='display: inline-block; vertical-align: middle;'>
|
||
<span style='display: block; text-align: center; border-bottom: 2px solid #333; padding: 0 8px 3px;'>‖<b>x</b> − <b>x</b>'‖²</span>
|
||
<span style='display: block; text-align: center; padding: 3px 8px 0;'>2ℓ²</span>
|
||
</span>
|
||
<span style='font-size: 1.2em;'>)</span>
|
||
</div>
|
||
</div>
|
||
<ul style='line-height: 1.8; margin-left: 15px;'>
|
||
<li><strong>σ<sub>f</sub><sup>2</sup></strong> (输出尺度)：控制函数值的整体变化幅度</li>
|
||
<li><strong>ℓ</strong> (长度尺度)：控制平滑度——ℓ 越大，函数变化越缓慢；ℓ 越小，函数变化越剧烈</li>
|
||
<li>RBF 核假设函数<b>无限可微</b>，适合建模光滑连续的物理过程</li>
|
||
<li>当 ‖<b>x</b> − <b>x</b>'‖ ≫ ℓ 时，<i>k</i> → 0（远距离点不相关）</li>
|
||
</ul>
|
||
</div>
|
||
|
||
<div style='background: linear-gradient(135deg, #fff4f0 0%, #ffe0b2 100%); padding: 15px; border-radius: 10px; margin: 12px 0; border-left: 4px solid #ff9800;'>
|
||
<strong style='color: #e65100; font-size: 1.05em;'>2️⃣ Matérn 核（本项目使用 Matérn-5/2）</strong>
|
||
<div style='text-align: center; background: white; padding: 15px; border-radius: 8px; margin: 10px 0;'>
|
||
<div style='font-size: 1.1em;'>
|
||
<i>k</i><sub>5/2</sub>(<i>r</i>) = σ<sub>f</sub><sup>2</sup>
|
||
<span style='font-size: 1.1em;'>(</span>1 + √5 ·
|
||
<span style='display: inline-block; vertical-align: middle;'>
|
||
<span style='display: block; text-align: center; border-bottom: 2px solid #333; padding: 0 6px 3px;'><i>r</i></span>
|
||
<span style='display: block; text-align: center; padding: 3px 6px 0;'>ℓ</span>
|
||
</span>
|
||
+ <span style='display: inline-block; vertical-align: middle;'>
|
||
<span style='display: block; text-align: center; border-bottom: 2px solid #333; padding: 0 6px 3px;'>5<i>r</i>²</span>
|
||
<span style='display: block; text-align: center; padding: 3px 6px 0;'>3ℓ²</span>
|
||
</span>
|
||
<span style='font-size: 1.1em;'>)</span>
|
||
exp<span style='font-size: 1.1em;'>(</span>−√5 ·
|
||
<span style='display: inline-block; vertical-align: middle;'>
|
||
<span style='display: block; text-align: center; border-bottom: 2px solid #333; padding: 0 6px 3px;'><i>r</i></span>
|
||
<span style='display: block; text-align: center; padding: 3px 6px 0;'>ℓ</span>
|
||
</span>
|
||
<span style='font-size: 1.1em;'>)</span>
|
||
</div>
|
||
</div>
|
||
<p style='margin: 8px 0; color: #555;'>其中 <i>r</i> = ‖<b>x</b> − <b>x</b>'‖ 为欧氏距离</p>
|
||
<ul style='line-height: 1.8; margin-left: 15px;'>
|
||
<li>Matérn 核是 RBF 的推广，通过参数 ν 控制可微性：ν=1/2 → 不可微; ν=∞ → RBF</li>
|
||
<li><strong>ν=5/2</strong>：函数<b>两次可微</b>，比 RBF 更灵活，是工程建模的常用选择</li>
|
||
<li>相比 RBF 能更好地捕捉工程数据中的局部不规则性</li>
|
||
</ul>
|
||
</div>
|
||
|
||
<div style='background: linear-gradient(135deg, #f0fff4 0%, #c8e6c9 100%); padding: 15px; border-radius: 10px; margin: 12px 0; border-left: 4px solid #4caf50;'>
|
||
<strong style='color: #2e7d32; font-size: 1.05em;'>3️⃣ ARD (自动相关性判定) 机制</strong>
|
||
<p style='margin: 8px 0;'>对于多维输入 <b>x</b> = [x<sub>1</sub>, ..., x<sub>d</sub>]，ARD 核为每个输入维度分配独立的长度尺度：</p>
|
||
<div style='text-align: center; background: white; padding: 15px; border-radius: 8px; margin: 10px 0;'>
|
||
<div style='font-size: 1.1em;'>
|
||
<i>r</i><sub>ARD</sub><sup>2</sup> = Σ<sub>j=1</sub><sup>d</sup>
|
||
<span style='display: inline-block; vertical-align: middle;'>
|
||
<span style='display: block; text-align: center; border-bottom: 2px solid #333; padding: 0 8px 3px;'>(x<sub>j</sub> − x'<sub>j</sub>)²</span>
|
||
<span style='display: block; text-align: center; padding: 3px 8px 0;'>ℓ<sub>j</sub>²</span>
|
||
</span>
|
||
</div>
|
||
</div>
|
||
<ul style='line-height: 1.8; margin-left: 15px;'>
|
||
<li>若 ℓ<sub>j</sub> 很大 → 函数对第 j 维输入不敏感（自动"忽略"该维度）</li>
|
||
<li>若 ℓ<sub>j</sub> 很小 → 函数在第 j 维变化剧烈（高度相关）</li>
|
||
<li>ARD 通过数据自动学习每维的重要性，实现<b>隐式特征选择</b></li>
|
||
<li>本项目：3 个输入 [高度, 马赫数, RPM] → 3 个独立的 ℓ<sub>j</sub></li>
|
||
</ul>
|
||
</div>
|
||
</div>
|
||
</details>
|
||
|
||
<details open>
|
||
<summary style='background: linear-gradient(135deg, #e0f2f1 0%, #b2dfdb 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
📐 GPR 后验推断公式（预测均值与方差）
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #b2dfdb;'>
|
||
<p><strong>训练数据：</strong>给定 N 个观测 D = {(<b>x</b><sub>i</sub>, y<sub>i</sub>)}<sub>i=1</sub><sup>N</sup>，
|
||
假设 y = f(<b>x</b>) + ε，其中 ε ~ N(0, σ<sub>n</sub>²) 为观测噪声。</p>
|
||
|
||
<p><strong>联合先验分布：</strong></p>
|
||
<div style='text-align: center; background: white; padding: 15px; border-radius: 8px; margin: 10px 0;'>
|
||
<div style='font-size: 1.1em;'>
|
||
<span style='font-size: 1.3em;'>[</span>
|
||
<span style='display: inline-block; vertical-align: middle;'>
|
||
<span style='display: block; padding: 2px;'><b>y</b></span>
|
||
<span style='display: block; padding: 2px;'><b>f</b><sub>*</sub></span>
|
||
</span>
|
||
<span style='font-size: 1.3em;'>]</span>
|
||
~ N
|
||
<span style='font-size: 1.3em;'>(</span>
|
||
<span style='display: inline-block; vertical-align: middle;'>
|
||
<span style='display: block; padding: 2px;'><b>m</b></span>
|
||
<span style='display: block; padding: 2px;'><b>m</b><sub>*</sub></span>
|
||
</span>
|
||
,
|
||
<span style='display: inline-block; vertical-align: middle;'>
|
||
<span style='display: block; padding: 2px;'>[K + σ<sub>n</sub>²I K<sub>*</sub>]</span>
|
||
<span style='display: block; padding: 2px;'>[K<sub>*</sub><sup>T</sup> K<sub>**</sub>]</span>
|
||
</span>
|
||
<span style='font-size: 1.3em;'>)</span>
|
||
</div>
|
||
</div>
|
||
|
||
<div style='background: linear-gradient(135deg, #e8f5e9 0%, #c8e6c9 100%); padding: 15px; border-radius: 8px; margin: 12px 0; border-left: 4px solid #4caf50;'>
|
||
<p style='font-weight: 700; color: #2e7d32;'>🎯 后验预测分布（关键公式）</p>
|
||
<p>对新输入 <b>x</b><sub>*</sub>，后验 f<sub>*</sub> | D, <b>x</b><sub>*</sub> 仍为高斯分布：</p>
|
||
<div style='text-align: center; background: white; padding: 15px; border-radius: 8px; margin: 10px 0;'>
|
||
<div style='font-size: 1.1em; margin: 8px 0;'>
|
||
<strong>预测均值：</strong>
|
||
μ<sub>*</sub> = m(<b>x</b><sub>*</sub>) + <b>k</b><sub>*</sub><sup>T</sup> (K + σ<sub>n</sub>²I)<sup>−1</sup> (<b>y</b> − <b>m</b>)
|
||
</div>
|
||
<div style='font-size: 1.1em; margin: 8px 0;'>
|
||
<strong>预测方差：</strong>
|
||
σ<sub>*</sub>² = k<sub>**</sub> − <b>k</b><sub>*</sub><sup>T</sup> (K + σ<sub>n</sub>²I)<sup>−1</sup> <b>k</b><sub>*</sub>
|
||
</div>
|
||
</div>
|
||
<p style='margin: 8px 0; color: #333; font-size: 0.95em;'>其中：</p>
|
||
<ul style='line-height: 1.8; margin-left: 15px; font-size: 0.95em;'>
|
||
<li><strong>K</strong> ∈ ℝ<sup>N×N</sup>：训练数据之间的核矩阵，K<sub>ij</sub> = k(<b>x</b><sub>i</sub>, <b>x</b><sub>j</sub>)</li>
|
||
<li><strong>k<sub>*</sub></strong> ∈ ℝ<sup>N</sup>：测试点与所有训练点之间的核向量</li>
|
||
<li><strong>k<sub>**</sub></strong>：测试点自身的先验方差 k(<b>x</b><sub>*</sub>, <b>x</b><sub>*</sub>)</li>
|
||
<li><strong>(K + σ<sub>n</sub>²I)<sup>−1</sup></strong>：核心计算瓶颈，复杂度 O(N³)</li>
|
||
</ul>
|
||
</div>
|
||
|
||
<div style='background: linear-gradient(135deg, #fff3e0 0%, #ffe0b2 100%); padding: 12px; border-radius: 8px; margin: 10px 0;'>
|
||
<p><strong>📊 方差的物理含义：</strong></p>
|
||
<ul style='line-height: 1.8; margin-left: 15px;'>
|
||
<li>σ<sub>*</sub>² <strong>小</strong>（接近训练数据区域）→ 预测可信度高，GPR "有信心"</li>
|
||
<li>σ<sub>*</sub>² <strong>大</strong>（远离训练数据区域）→ 预测不确定性高，是外推区域</li>
|
||
<li>方差热力图可以直观展示模型"知道什么"和"不知道什么"</li>
|
||
<li>这是 GPR 优于 NN 的核心优势——自动量化认知不确定性</li>
|
||
</ul>
|
||
</div>
|
||
</div>
|
||
</details>
|
||
|
||
<details open>
|
||
<summary style='background: linear-gradient(135deg, #fce4ec 0%, #f8bbd0 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
⚙️ 超参数优化 — 最大化边际似然
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #f8bbd0;'>
|
||
<p>GPR 的超参数 θ = {σ<sub>f</sub>, ℓ<sub>1</sub>, ..., ℓ<sub>d</sub>, σ<sub>n</sub>} 通过
|
||
<strong>最大化对数边际似然 (Log Marginal Likelihood)</strong> 自动确定：</p>
|
||
|
||
<div style='text-align: center; background: white; padding: 18px; border-radius: 8px; margin: 10px 0; box-shadow: 0 2px 6px rgba(0,0,0,0.05);'>
|
||
<div style='font-size: 1.1em;'>
|
||
log <i>p</i>(<b>y</b>|X, θ) = −
|
||
<span style='display: inline-block; vertical-align: middle;'>
|
||
<span style='display: block; text-align: center; border-bottom: 2px solid #333; padding: 0 6px 3px;'>1</span>
|
||
<span style='display: block; text-align: center; padding: 3px 6px 0;'>2</span>
|
||
</span>
|
||
<b>y</b><sup>T</sup>K<sub>y</sub><sup>−1</sup><b>y</b>
|
||
−
|
||
<span style='display: inline-block; vertical-align: middle;'>
|
||
<span style='display: block; text-align: center; border-bottom: 2px solid #333; padding: 0 6px 3px;'>1</span>
|
||
<span style='display: block; text-align: center; padding: 3px 6px 0;'>2</span>
|
||
</span>
|
||
log|K<sub>y</sub>|
|
||
−
|
||
<span style='display: inline-block; vertical-align: middle;'>
|
||
<span style='display: block; text-align: center; border-bottom: 2px solid #333; padding: 0 6px 3px;'>N</span>
|
||
<span style='display: block; text-align: center; padding: 3px 6px 0;'>2</span>
|
||
</span>
|
||
log(2π)
|
||
</div>
|
||
</div>
|
||
<p style='margin: 8px 0; color: #555;'>其中 K<sub>y</sub> = K + σ<sub>n</sub>²I</p>
|
||
|
||
<div style='background: linear-gradient(135deg, #ede7f6 0%, #d1c4e9 100%); padding: 12px; border-radius: 8px; margin: 12px 0;'>
|
||
<p><strong>📐 三项分解的物理意义：</strong></p>
|
||
<table style='width: 100%; border-collapse: collapse; margin: 8px 0; font-size: 0.93em;'>
|
||
<tr style='background: #f5f5f5; font-weight: 600;'>
|
||
<td style='padding: 8px; border: 1px solid #d1c4e9;'>项</td>
|
||
<td style='padding: 8px; border: 1px solid #d1c4e9;'>数学形式</td>
|
||
<td style='padding: 8px; border: 1px solid #d1c4e9;'>含义</td>
|
||
</tr>
|
||
<tr>
|
||
<td style='padding: 8px; border: 1px solid #d1c4e9;'>数据拟合项</td>
|
||
<td style='padding: 8px; border: 1px solid #d1c4e9;'>−½ <b>y</b><sup>T</sup>K<sub>y</sub><sup>−1</sup><b>y</b></td>
|
||
<td style='padding: 8px; border: 1px solid #d1c4e9;'>模型对数据的拟合程度（越大越好拟合）</td>
|
||
</tr>
|
||
<tr style='background: #faf5ff;'>
|
||
<td style='padding: 8px; border: 1px solid #d1c4e9;'>复杂度惩罚项</td>
|
||
<td style='padding: 8px; border: 1px solid #d1c4e9;'>−½ log|K<sub>y</sub>|</td>
|
||
<td style='padding: 8px; border: 1px solid #d1c4e9;'>Occam 剃刀：自动惩罚过于复杂的模型</td>
|
||
</tr>
|
||
<tr>
|
||
<td style='padding: 8px; border: 1px solid #d1c4e9;'>归一化常数</td>
|
||
<td style='padding: 8px; border: 1px solid #d1c4e9;'>−(N/2) log(2π)</td>
|
||
<td style='padding: 8px; border: 1px solid #d1c4e9;'>与超参数无关，优化时可忽略</td>
|
||
</tr>
|
||
</table>
|
||
<p style='margin: 8px 0; color: #555; font-size: 0.93em;'>
|
||
优化过程在"拟合数据"和"模型简单性"之间自动取得平衡，天然防止过拟合。
|
||
本项目使用 BoTorch/GPyTorch 框架，通过 L-BFGS 优化器求解。
|
||
</p>
|
||
</div>
|
||
</div>
|
||
</details>
|
||
|
||
<details>
|
||
<summary style='background: linear-gradient(135deg, #e3f2fd 0%, #bbdefb 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
🔄 数据预处理流水线
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #bbdefb;'>
|
||
<p>发动机数据（燃油流量、功率）的数值范围跨越多个数量级，直接建模会导致核函数
|
||
无法有效捕捉小值区域的变化。采用两步预处理：</p>
|
||
|
||
<div style='background: white; padding: 18px; border-radius: 8px; margin: 12px 0; border: 1px dashed #90caf9;'>
|
||
<ol style='line-height: 2.5; margin-left: 15px;'>
|
||
<li>
|
||
<strong>Log1p 变换：</strong> y<sub>log</sub> = log(1 + y)<br>
|
||
<span style='color: #555; font-size: 0.93em;'>→ 压缩大值，拉伸小值，使数据分布更均匀。log1p(0)=0，避免 log(0) 的数值问题。</span>
|
||
</li>
|
||
<li>
|
||
<strong>Z-Score 标准化：</strong> ŷ = (y<sub>log</sub> − μ) / σ<br>
|
||
<span style='color: #555; font-size: 0.93em;'>→ 标准化为均值 0、标准差 1，使核函数的超参数在同一尺度上可比。</span>
|
||
</li>
|
||
<li>
|
||
<strong>反变换（预测时）：</strong><br>
|
||
y<sub>log</sub> = ŷ · σ + μ → y = expm1(y<sub>log</sub>) = e<sup>y<sub>log</sub></sup> − 1<br>
|
||
<span style='color: #555; font-size: 0.93em;'>→ 从标准化空间恢复到物理空间</span>
|
||
</li>
|
||
</ol>
|
||
</div>
|
||
|
||
<div style='background: linear-gradient(135deg, #e8f5e9 0%, #c8e6c9 100%); padding: 12px; border-radius: 8px; margin: 10px 0;'>
|
||
<p><strong>💡 相同的预处理同时应用于 GPR 和 NN：</strong></p>
|
||
<ul style='line-height: 1.8; margin-left: 15px;'>
|
||
<li>GPR 在 Z-Score 空间中训练和预测</li>
|
||
<li>NN（蒸馏学生模型）使用 GPR 相同的 scaler 参数</li>
|
||
<li>这确保了两个模型的输入输出在相同数值空间中进行比较</li>
|
||
</ul>
|
||
</div>
|
||
</div>
|
||
</details>
|
||
|
||
<details>
|
||
<summary style='background: linear-gradient(135deg, #fff3e0 0%, #ffe0b2 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
📊 模型评价指标与计算复杂度
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #ffe0b2;'>
|
||
<h4 style='color: #e65100;'>评价指标公式</h4>
|
||
<div style='background: white; padding: 15px; border-radius: 8px; margin: 12px 0;'>
|
||
<p><strong>MAPE (Mean Absolute Percentage Error)：</strong></p>
|
||
<div style='text-align: center; padding: 8px; font-size: 1.1em;'>
|
||
MAPE =
|
||
<span style='display: inline-block; vertical-align: middle;'>
|
||
<span style='display: block; text-align: center; border-bottom: 2px solid #333; padding: 0 8px 3px;'>1</span>
|
||
<span style='display: block; text-align: center; padding: 3px 8px 0;'>N</span>
|
||
</span>
|
||
Σ<sub>i=1</sub><sup>N</sup>
|
||
<span style='display: inline-block; vertical-align: middle;'>
|
||
<span style='display: block; text-align: center; border-bottom: 2px solid #333; padding: 0 8px 3px;'>|y<sub>i</sub> − ŷ<sub>i</sub>|</span>
|
||
<span style='display: block; text-align: center; padding: 3px 8px 0;'>y<sub>i</sub></span>
|
||
</span>
|
||
× 100%
|
||
</div>
|
||
<p style='color: #555; font-size: 0.93em;'>注意：MAPE 计算排除 y ≈ 0 的数据点（避免除零导致的虚高误差）</p>
|
||
</div>
|
||
<div style='background: white; padding: 15px; border-radius: 8px; margin: 12px 0;'>
|
||
<p><strong>R² (决定系数)：</strong></p>
|
||
<div style='text-align: center; padding: 8px; font-size: 1.1em;'>
|
||
R² = 1 −
|
||
<span style='display: inline-block; vertical-align: middle;'>
|
||
<span style='display: block; text-align: center; border-bottom: 2px solid #333; padding: 0 8px 3px;'>Σ(y<sub>i</sub> − ŷ<sub>i</sub>)²</span>
|
||
<span style='display: block; text-align: center; padding: 3px 8px 0;'>Σ(y<sub>i</sub> − ȳ)²</span>
|
||
</span>
|
||
</div>
|
||
<p style='color: #555; font-size: 0.93em;'>R² = 1 → 完美预测; R² = 0 → 与均值预测一样差; R² < 0 → 更差</p>
|
||
</div>
|
||
|
||
<h4 style='color: #e65100; margin-top: 15px;'>计算复杂度分析</h4>
|
||
<table style='width: 100%; border-collapse: collapse; margin: 12px 0; font-size: 0.93em; box-shadow: 0 2px 10px rgba(0,0,0,0.08); border-radius: 8px; overflow: hidden;'>
|
||
<tr style='background: linear-gradient(135deg, #e65100 0%, #bf360c 100%); color: white;'>
|
||
<th style='padding: 10px;'>操作</th>
|
||
<th style='padding: 10px;'>时间复杂度</th>
|
||
<th style='padding: 10px;'>空间复杂度</th>
|
||
<th style='padding: 10px;'>瓶颈</th>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>训练（Cholesky 分解）</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>O(N³)</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>O(N²)</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>核矩阵 K 的 Cholesky 分解</td>
|
||
</tr>
|
||
<tr style='background: #fff8f0;'>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>单点预测</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>O(N)</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>O(N)</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>需与所有训练点计算核值</td>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>批量预测 (M 点)</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>O(MN + N²)</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>O(MN)</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>大 M 时成为瓶颈</td>
|
||
</tr>
|
||
<tr style='background: #fff8f0;'>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>超参数优化</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>O(N³) × 迭代次数</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>O(N²)</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>每次迭代都需 Cholesky</td>
|
||
</tr>
|
||
</table>
|
||
<p style='margin: 8px 0; color: #c62828; font-weight: 600;'>
|
||
⚠️ 正是 O(N³) 的训练复杂度和 O(N) 的单点预测成本，使得 GPR 不适合 MPC 实时控制
|
||
（每个控制步需评估模型数十次），因此需要将 GPR 的知识蒸馏到 O(1) 推理的 NN 中。
|
||
</p>
|
||
</div>
|
||
</details>
|
||
|
||
<details>
|
||
<summary style='background: linear-gradient(135deg, #e8f5e9 0%, #c8e6c9 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
✈️ GPR 在发动机代理建模中的应用
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #c8e6c9;'>
|
||
<p>在本项目中，GPR 被用作涡轴发动机的<strong>高精度代理模型 (Surrogate Model)</strong>，
|
||
替代复杂的热力学仿真代码。</p>
|
||
|
||
<div style='background: white; padding: 15px; border-radius: 8px; margin: 12px 0; border: 1px solid #c8e6c9;'>
|
||
<p><strong>建模映射关系：</strong></p>
|
||
<div style='text-align: center; padding: 10px; font-size: 1.05em;'>
|
||
[高度(m), 马赫数, RPM] → GPR → [燃油流量(kg/h), 功率(kW)]
|
||
</div>
|
||
<table style='width: 100%; border-collapse: collapse; margin: 12px 0; font-size: 0.93em;'>
|
||
<tr style='background: #e8f5e9; font-weight: 600;'>
|
||
<td style='padding: 8px; border: 1px solid #c8e6c9;'>变量</td>
|
||
<td style='padding: 8px; border: 1px solid #c8e6c9;'>含义</td>
|
||
<td style='padding: 8px; border: 1px solid #c8e6c9;'>典型范围</td>
|
||
</tr>
|
||
<tr>
|
||
<td style='padding: 8px; border: 1px solid #c8e6c9;'>Altitude_m</td>
|
||
<td style='padding: 8px; border: 1px solid #c8e6c9;'>飞行高度</td>
|
||
<td style='padding: 8px; border: 1px solid #c8e6c9;'>0 ~ 10000 m</td>
|
||
</tr>
|
||
<tr style='background: #f8fdf8;'>
|
||
<td style='padding: 8px; border: 1px solid #c8e6c9;'>Mach</td>
|
||
<td style='padding: 8px; border: 1px solid #c8e6c9;'>飞行马赫数</td>
|
||
<td style='padding: 8px; border: 1px solid #c8e6c9;'>0 ~ 0.8</td>
|
||
</tr>
|
||
<tr>
|
||
<td style='padding: 8px; border: 1px solid #c8e6c9;'>RPM</td>
|
||
<td style='padding: 8px; border: 1px solid #c8e6c9;'>燃气涡轮转速</td>
|
||
<td style='padding: 8px; border: 1px solid #c8e6c9;'>1000 ~ 30000+</td>
|
||
</tr>
|
||
</table>
|
||
</div>
|
||
|
||
<div style='background: linear-gradient(135deg, #e3f2fd 0%, #bbdefb 100%); padding: 12px; border-radius: 8px; margin: 10px 0;'>
|
||
<p><strong>💡 为什么选择 GPR 作为代理模型？</strong></p>
|
||
<ul style='line-height: 1.8; margin-left: 15px;'>
|
||
<li>发动机热力学仿真每次调用耗时秒级，而 GPR 训练后毫秒级预测</li>
|
||
<li>发动机测试数据成本高昂，GPR 在小样本下表现优异</li>
|
||
<li>GPR 的方差输出可识别数据稀疏区域，指导后续试验设计</li>
|
||
<li>多输出 GPR (Multi-output GP) 可同时建模燃油流量和功率</li>
|
||
</ul>
|
||
</div>
|
||
</div>
|
||
</details>
|
||
</div>
|
||
"""
|
||
|
||
|
||
# ============================================================
|
||
# NN 神经网络知识蒸馏知识卡片 — 中文
|
||
# ============================================================
|
||
NN_KNOWLEDGE = """
|
||
<div class='knowledge-card'>
|
||
<h3 style='color: #00695c; margin-top: 0; display: flex; align-items: center; gap: 10px;'>
|
||
<span style='font-size: 1.3em;'>🧠</span> NN 神经网络 — 知识蒸馏与 MLP 详解
|
||
</h3>
|
||
|
||
<details open>
|
||
<summary style='background: linear-gradient(135deg, #e0f2f1 0%, #b2dfdb 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
📖 知识蒸馏 (Knowledge Distillation) 原理
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #b2dfdb;'>
|
||
<p style='margin: 10px 0;'><strong>知识蒸馏</strong>是一种模型压缩技术，
|
||
将大型"教师"模型（如 GPR）的知识迁移到轻量级"学生"模型（如 MLP 神经网络）中，
|
||
在几乎不损失精度的前提下大幅提升推理速度。</p>
|
||
|
||
<div style='text-align: center; background: white; padding: 18px; border-radius: 8px; margin: 10px 0; box-shadow: 0 2px 6px rgba(0,0,0,0.05);'>
|
||
<div style='font-size: 1.05em; color: #333;'>
|
||
<span style='background: #f3e5f5; padding: 6px 12px; border-radius: 6px;'>[GPR 教师模型]</span>
|
||
→ 密集网格预测 →
|
||
<span style='background: #e0f2f1; padding: 6px 12px; border-radius: 6px;'>[NN 学生模型]</span>
|
||
</div>
|
||
</div>
|
||
|
||
<h4 style='color: #00695c; margin-top: 15px;'>蒸馏流程</h4>
|
||
<div style='background: white; padding: 15px; border-radius: 8px; margin: 10px 0; border: 1px dashed #80cbc4;'>
|
||
<ol style='line-height: 2.2; margin-left: 15px;'>
|
||
<li><strong>训练 GPR 教师：</strong>用少量真实发动机数据训练 GPR 模型</li>
|
||
<li><strong>生成伪数据：</strong>在输入空间的密集网格上用 GPR 预测，生成大量"虚拟标签"</li>
|
||
<li><strong>训练 NN 学生：</strong>用 GPR 的预测结果作为训练标签，训练 MLP 网络</li>
|
||
<li><strong>部署学生网络：</strong>在 MPC 控制器中使用轻量 NN 替代 GPR</li>
|
||
</ol>
|
||
</div>
|
||
|
||
<h4 style='color: #00695c; margin-top: 15px;'>教师 vs 学生对比</h4>
|
||
<table style='width: 100%; border-collapse: collapse; margin: 12px 0; font-size: 0.93em; box-shadow: 0 2px 10px rgba(0,0,0,0.08); border-radius: 8px; overflow: hidden;'>
|
||
<tr style='background: linear-gradient(135deg, #00695c 0%, #004d40 100%); color: white;'>
|
||
<th style='padding: 10px;'>指标</th>
|
||
<th style='padding: 10px;'>GPR (教师)</th>
|
||
<th style='padding: 10px;'>NN (学生)</th>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 8px; border: 1px solid #b2dfdb;'><strong>推理速度</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #b2dfdb;'>~10 ms（核矩阵运算）</td>
|
||
<td style='padding: 8px; border: 1px solid #b2dfdb;'>~0.1 ms（矩阵乘法）</td>
|
||
</tr>
|
||
<tr style='background: #f0faf8;'>
|
||
<td style='padding: 8px; border: 1px solid #b2dfdb;'><strong>内存占用</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #b2dfdb;'>O(N²) — 存储全部训练数据</td>
|
||
<td style='padding: 8px; border: 1px solid #b2dfdb;'>O(W) — 约 4.5K 参数</td>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 8px; border: 1px solid #b2dfdb;'><strong>不确定性</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #b2dfdb;'>预测均值 + 后验方差</td>
|
||
<td style='padding: 8px; border: 1px solid #b2dfdb;'>仅点估计</td>
|
||
</tr>
|
||
<tr style='background: #f0faf8;'>
|
||
<td style='padding: 8px; border: 1px solid #b2dfdb;'><strong>MPC 适用性</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #b2dfdb;'>推理太慢，不适合实时</td>
|
||
<td style='padding: 8px; border: 1px solid #b2dfdb;'>✅ 满足实时控制需求</td>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 8px; border: 1px solid #b2dfdb;'><strong>批量预测</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #b2dfdb;'>大批量很慢</td>
|
||
<td style='padding: 8px; border: 1px solid #b2dfdb;'>天然并行，GPU 加速</td>
|
||
</tr>
|
||
</table>
|
||
</div>
|
||
</details>
|
||
|
||
<details open>
|
||
<summary style='background: linear-gradient(135deg, #e8eaf6 0%, #c5cae9 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
🏗️ MLP 多层感知机架构详解
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #c5cae9;'>
|
||
<p>学生网络采用 <strong>3 层全连接 MLP (Multi-Layer Perceptron)</strong> 架构：</p>
|
||
<div style='text-align: center; background: white; padding: 18px; border-radius: 8px; margin: 10px 0; box-shadow: 0 2px 6px rgba(0,0,0,0.05);'>
|
||
<div style='font-size: 1.05em; color: #333;'>
|
||
<span style='background: #e3f2fd; padding: 5px 10px; border-radius: 6px;'>输入 (3)</span>
|
||
→ <span style='background: #f3e5f5; padding: 5px 10px; border-radius: 6px;'>Linear(64) + Tanh</span>
|
||
→ <span style='background: #f3e5f5; padding: 5px 10px; border-radius: 6px;'>Linear(64) + Tanh</span>
|
||
→ <span style='background: #e8f5e9; padding: 5px 10px; border-radius: 6px;'>Linear(2)</span>
|
||
→ <span style='background: #fff3e0; padding: 5px 10px; border-radius: 6px;'>输出 (2)</span>
|
||
</div>
|
||
</div>
|
||
|
||
<h4 style='color: #303f9f; margin-top: 18px;'>前向传播数学公式</h4>
|
||
<p>每一层的计算可以表示为仿射变换 + 非线性激活：</p>
|
||
<div style='text-align: center; background: white; padding: 18px; border-radius: 8px; margin: 10px 0;'>
|
||
<div style='font-size: 1.1em; margin: 6px 0;'>
|
||
<strong>第 l 层：</strong>
|
||
<b>z</b><sup>(l)</sup> = W<sup>(l)</sup> <b>a</b><sup>(l−1)</sup> + <b>b</b><sup>(l)</sup>
|
||
</div>
|
||
<div style='font-size: 1.1em; margin: 6px 0;'>
|
||
<strong>激活后：</strong>
|
||
<b>a</b><sup>(l)</sup> = σ(<b>z</b><sup>(l)</sup>)
|
||
</div>
|
||
</div>
|
||
<ul style='line-height: 1.8; margin-left: 15px; font-size: 0.95em;'>
|
||
<li><strong>W<sup>(l)</sup></strong> ∈ ℝ<sup>d<sub>l</sub>×d<sub>l−1</sub></sup>：第 l 层的权重矩阵</li>
|
||
<li><strong>b<sup>(l)</sup></strong> ∈ ℝ<sup>d<sub>l</sub></sup>：第 l 层的偏置向量</li>
|
||
<li><strong>σ(·)</strong>：激活函数（本项目使用 Tanh）</li>
|
||
<li><strong>a<sup>(0)</sup></strong> = <b>x</b>（输入向量）</li>
|
||
</ul>
|
||
|
||
<h4 style='color: #303f9f; margin-top: 18px;'>本网络的完整前向计算</h4>
|
||
<div style='background: white; padding: 15px; border-radius: 8px; margin: 12px 0; border: 1px solid #c5cae9;'>
|
||
<div style='font-size: 1.05em; line-height: 2.5;'>
|
||
<b>h</b><sub>1</sub> = tanh(W<sub>1</sub> · <b>x̂</b> + <b>b</b><sub>1</sub>) <span style='color: #888;'>// 隐藏层 1: ℝ³ → ℝ⁶⁴</span><br>
|
||
<b>h</b><sub>2</sub> = tanh(W<sub>2</sub> · <b>h</b><sub>1</sub> + <b>b</b><sub>2</sub>) <span style='color: #888;'>// 隐藏层 2: ℝ⁶⁴ → ℝ⁶⁴</span><br>
|
||
<b>ŷ</b> = W<sub>3</sub> · <b>h</b><sub>2</sub> + <b>b</b><sub>3</sub> <span style='color: #888;'>// 输出层: ℝ⁶⁴ → ℝ² (无激活)</span>
|
||
</div>
|
||
</div>
|
||
|
||
<h4 style='color: #303f9f; margin-top: 18px;'>参数量计算</h4>
|
||
<table style='width: 100%; border-collapse: collapse; margin: 12px 0; font-size: 0.93em;'>
|
||
<tr style='background: #e8eaf6; font-weight: 600;'>
|
||
<td style='padding: 8px; border: 1px solid #c5cae9;'>层</td>
|
||
<td style='padding: 8px; border: 1px solid #c5cae9;'>形状</td>
|
||
<td style='padding: 8px; border: 1px solid #c5cae9;'>权重数</td>
|
||
<td style='padding: 8px; border: 1px solid #c5cae9;'>偏置数</td>
|
||
<td style='padding: 8px; border: 1px solid #c5cae9;'>合计</td>
|
||
</tr>
|
||
<tr>
|
||
<td style='padding: 8px; border: 1px solid #c5cae9;'>Linear-1</td>
|
||
<td style='padding: 8px; border: 1px solid #c5cae9;'>3 → 64</td>
|
||
<td style='padding: 8px; border: 1px solid #c5cae9;'>192</td>
|
||
<td style='padding: 8px; border: 1px solid #c5cae9;'>64</td>
|
||
<td style='padding: 8px; border: 1px solid #c5cae9;'>256</td>
|
||
</tr>
|
||
<tr style='background: #f5f5ff;'>
|
||
<td style='padding: 8px; border: 1px solid #c5cae9;'>Linear-2</td>
|
||
<td style='padding: 8px; border: 1px solid #c5cae9;'>64 → 64</td>
|
||
<td style='padding: 8px; border: 1px solid #c5cae9;'>4096</td>
|
||
<td style='padding: 8px; border: 1px solid #c5cae9;'>64</td>
|
||
<td style='padding: 8px; border: 1px solid #c5cae9;'>4160</td>
|
||
</tr>
|
||
<tr>
|
||
<td style='padding: 8px; border: 1px solid #c5cae9;'>Linear-3</td>
|
||
<td style='padding: 8px; border: 1px solid #c5cae9;'>64 → 2</td>
|
||
<td style='padding: 8px; border: 1px solid #c5cae9;'>128</td>
|
||
<td style='padding: 8px; border: 1px solid #c5cae9;'>2</td>
|
||
<td style='padding: 8px; border: 1px solid #c5cae9;'>130</td>
|
||
</tr>
|
||
<tr style='background: #e8eaf6; font-weight: 700;'>
|
||
<td style='padding: 8px; border: 1px solid #c5cae9;' colspan='4'>总参数量</td>
|
||
<td style='padding: 8px; border: 1px solid #c5cae9;'>4,546</td>
|
||
</tr>
|
||
</table>
|
||
</div>
|
||
</details>
|
||
|
||
<details open>
|
||
<summary style='background: linear-gradient(135deg, #fff3e0 0%, #ffe0b2 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
📐 激活函数详解
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #ffe0b2;'>
|
||
<p>激活函数为神经网络引入<strong>非线性</strong>——没有激活函数，无论多少层的网络都等价于单层线性变换。</p>
|
||
|
||
<div style='background: linear-gradient(135deg, #e8f5e9 0%, #c8e6c9 100%); padding: 15px; border-radius: 10px; margin: 12px 0; border-left: 4px solid #43a047;'>
|
||
<strong style='color: #2e7d32; font-size: 1.05em;'>Tanh (双曲正切) — 本项目所用</strong>
|
||
<div style='text-align: center; background: white; padding: 15px; border-radius: 8px; margin: 10px 0;'>
|
||
<div style='font-size: 1.15em;'>
|
||
tanh(x) =
|
||
<span style='display: inline-block; vertical-align: middle;'>
|
||
<span style='display: block; text-align: center; border-bottom: 2px solid #333; padding: 0 8px 3px;'>e<sup>x</sup> − e<sup>−x</sup></span>
|
||
<span style='display: block; text-align: center; padding: 3px 8px 0;'>e<sup>x</sup> + e<sup>−x</sup></span>
|
||
</span>
|
||
</div>
|
||
</div>
|
||
<ul style='line-height: 1.8; margin-left: 15px;'>
|
||
<li><strong>输出范围：</strong>(−1, +1)，以零为中心</li>
|
||
<li><strong>导数：</strong>tanh'(x) = 1 − tanh²(x)，最大值为 1（x=0 处）</li>
|
||
<li><strong>优点：</strong>输出零中心，有助于加速收敛</li>
|
||
<li><strong>缺点：</strong>|x| 较大时梯度接近 0（梯度饱和），deep网络可能出现梯度消失</li>
|
||
</ul>
|
||
</div>
|
||
|
||
<h4 style='color: #e65100; margin-top: 15px;'>常用激活函数对比</h4>
|
||
<table style='width: 100%; border-collapse: collapse; margin: 12px 0; font-size: 0.93em; box-shadow: 0 2px 10px rgba(0,0,0,0.08); border-radius: 8px; overflow: hidden;'>
|
||
<tr style='background: linear-gradient(135deg, #e65100 0%, #bf360c 100%); color: white;'>
|
||
<th style='padding: 10px;'>函数</th>
|
||
<th style='padding: 10px;'>公式</th>
|
||
<th style='padding: 10px;'>范围</th>
|
||
<th style='padding: 10px;'>特点</th>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'><strong>Sigmoid</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>1 / (1 + e<sup>−x</sup>)</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>(0, 1)</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>输出非零中心，双端饱和</td>
|
||
</tr>
|
||
<tr style='background: #fff8f0;'>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'><strong>Tanh</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>(e<sup>x</sup>−e<sup>−x</sup>)/(e<sup>x</sup>+e<sup>−x</sup>)</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>(−1, 1)</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>零中心，双端饱和</td>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'><strong>ReLU</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>max(0, x)</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>[0, ∞)</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>计算高效，但有"死神经元"问题</td>
|
||
</tr>
|
||
<tr style='background: #fff8f0;'>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'><strong>GELU</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>x · Φ(x)</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>连续</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>Transformer 常用，性能好</td>
|
||
</tr>
|
||
</table>
|
||
<p style='color: #555; font-size: 0.93em;'>对于本项目的回归任务（光滑连续的发动机特性映射），Tanh 是良好的选择，
|
||
因为数据经过 Z-Score 标准化后值域在 [−3, 3] 左右，刚好在 Tanh 的线性区间内。</p>
|
||
</div>
|
||
</details>
|
||
|
||
<details open>
|
||
<summary style='background: linear-gradient(135deg, #f3e5f5 0%, #e1bee7 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
📉 损失函数与反向传播
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #e1bee7;'>
|
||
<h4 style='color: #7b1fa2;'>损失函数 (Loss Function)</h4>
|
||
<p>本项目在<strong>标准化 log 空间</strong>中使用 MSE (Mean Squared Error) 损失：</p>
|
||
<div style='text-align: center; background: white; padding: 18px; border-radius: 8px; margin: 10px 0;'>
|
||
<div style='font-size: 1.1em;'>
|
||
L =
|
||
<span style='display: inline-block; vertical-align: middle;'>
|
||
<span style='display: block; text-align: center; border-bottom: 2px solid #333; padding: 0 8px 3px;'>1</span>
|
||
<span style='display: block; text-align: center; padding: 3px 8px 0;'>N</span>
|
||
</span>
|
||
Σ<sub>i=1</sub><sup>N</sup> ‖ŷ<sub>i</sub> − y<sub>i</sub>‖²
|
||
</div>
|
||
</div>
|
||
<p style='color: #555; font-size: 0.93em;'>
|
||
其中 ŷ 和 y 均为标准化 log 空间中的值（先 log1p 变换，再 Z-Score）。
|
||
在标准化空间中训练使得燃油流量和功率两个输出的损失贡献相当。
|
||
</p>
|
||
|
||
<h4 style='color: #7b1fa2; margin-top: 18px;'>反向传播算法 (Backpropagation)</h4>
|
||
<p>反向传播基于<strong>链式法则 (Chain Rule)</strong>，从输出层到输入层逐层计算梯度：</p>
|
||
<div style='background: white; padding: 15px; border-radius: 8px; margin: 12px 0; border: 1px solid #e1bee7;'>
|
||
<div style='font-size: 1.05em; line-height: 2.5;'>
|
||
<strong>输出层梯度：</strong>
|
||
<span style='display: inline-block; vertical-align: middle;'>
|
||
<span style='display: block; text-align: center; border-bottom: 2px solid #333; padding: 0 8px 3px;'>∂L</span>
|
||
<span style='display: block; text-align: center; padding: 3px 8px 0;'>∂W<sub>3</sub></span>
|
||
</span>
|
||
=
|
||
<span style='display: inline-block; vertical-align: middle;'>
|
||
<span style='display: block; text-align: center; border-bottom: 2px solid #333; padding: 0 8px 3px;'>∂L</span>
|
||
<span style='display: block; text-align: center; padding: 3px 8px 0;'>∂ŷ</span>
|
||
</span>
|
||
· <b>h</b><sub>2</sub><sup>T</sup>
|
||
<br>
|
||
<strong>隐藏层梯度：</strong>
|
||
<span style='display: inline-block; vertical-align: middle;'>
|
||
<span style='display: block; text-align: center; border-bottom: 2px solid #333; padding: 0 8px 3px;'>∂L</span>
|
||
<span style='display: block; text-align: center; padding: 3px 8px 0;'>∂W<sub>l</sub></span>
|
||
</span>
|
||
= δ<sup>(l)</sup> · <b>a</b><sup>(l−1)T</sup>
|
||
其中 δ<sup>(l)</sup> = (W<sub>l+1</sub><sup>T</sup> δ<sup>(l+1)</sup>) ⊙ σ'(<b>z</b><sup>(l)</sup>)
|
||
</div>
|
||
</div>
|
||
<ul style='line-height: 1.8; margin-left: 15px;'>
|
||
<li><strong>⊙</strong> 表示逐元素乘法 (Hadamard product)</li>
|
||
<li><strong>σ'</strong> 为激活函数的导数：tanh'(x) = 1 − tanh²(x)</li>
|
||
<li>反向传播的梯度从后向前"流动"，自动计算所有参数的偏导数</li>
|
||
<li>PyTorch 的 <code>autograd</code> 引擎自动构建计算图并执行反向传播</li>
|
||
</ul>
|
||
</div>
|
||
</details>
|
||
|
||
<details>
|
||
<summary style='background: linear-gradient(135deg, #e3f2fd 0%, #bbdefb 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
🔧 优化器与学习率调度
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #bbdefb;'>
|
||
<h4 style='color: #1565c0;'>Adam 优化器</h4>
|
||
<p>Adam (Adaptive Moment Estimation) 结合了动量 (Momentum) 和 RMSProp 的优点：</p>
|
||
<div style='background: white; padding: 15px; border-radius: 8px; margin: 12px 0; border: 1px solid #bbdefb;'>
|
||
<div style='font-size: 1.05em; line-height: 2.5;'>
|
||
<b>m</b><sub>t</sub> = β<sub>1</sub> <b>m</b><sub>t−1</sub> + (1−β<sub>1</sub>) <b>g</b><sub>t</sub>
|
||
<span style='color: #888;'>// 一阶矩（梯度均值）</span><br>
|
||
<b>v</b><sub>t</sub> = β<sub>2</sub> <b>v</b><sub>t−1</sub> + (1−β<sub>2</sub>) <b>g</b><sub>t</sub>²
|
||
<span style='color: #888;'>// 二阶矩（梯度方差）</span><br>
|
||
<b>m̂</b><sub>t</sub> = <b>m</b><sub>t</sub> / (1−β<sub>1</sub><sup>t</sup>)
|
||
|
||
<span style='color: #888;'>// 偏差修正</span><br>
|
||
<b>v̂</b><sub>t</sub> = <b>v</b><sub>t</sub> / (1−β<sub>2</sub><sup>t</sup>)<br>
|
||
θ<sub>t+1</sub> = θ<sub>t</sub> − η · <b>m̂</b><sub>t</sub> / (√<b>v̂</b><sub>t</sub> + ε)
|
||
</div>
|
||
</div>
|
||
<ul style='line-height: 1.8; margin-left: 15px;'>
|
||
<li><strong>β<sub>1</sub>=0.9, β<sub>2</sub>=0.999</strong>：动量和二阶矩的衰减率（默认值）</li>
|
||
<li><strong>ε=10⁻⁸</strong>：防止除零的小常数</li>
|
||
<li><strong>η</strong>：初始学习率（本项目约 1e-3 ~ 5e-3）</li>
|
||
<li>Adam 为每个参数维护独立的自适应学习率</li>
|
||
</ul>
|
||
|
||
<h4 style='color: #1565c0; margin-top: 18px;'>余弦退火学习率调度 (Cosine Annealing)</h4>
|
||
<div style='text-align: center; background: white; padding: 15px; border-radius: 8px; margin: 10px 0;'>
|
||
<div style='font-size: 1.15em;'>
|
||
η<sub>t</sub> = η<sub>min</sub> +
|
||
<span style='display: inline-block; vertical-align: middle;'>
|
||
<span style='display: block; text-align: center; border-bottom: 2px solid #333; padding: 0 8px 3px;'>η<sub>max</sub> − η<sub>min</sub></span>
|
||
<span style='display: block; text-align: center; padding: 3px 8px 0;'>2</span>
|
||
</span>
|
||
<span style='font-size: 1.2em;'>(</span>
|
||
1 + cos
|
||
<span style='font-size: 1.2em;'>(</span>
|
||
<span style='display: inline-block; vertical-align: middle;'>
|
||
<span style='display: block; text-align: center; border-bottom: 2px solid #333; padding: 0 6px 3px;'>πt</span>
|
||
<span style='display: block; text-align: center; padding: 3px 6px 0;'>T</span>
|
||
</span>
|
||
<span style='font-size: 1.2em;'>)</span>
|
||
<span style='font-size: 1.2em;'>)</span>
|
||
</div>
|
||
</div>
|
||
<ul style='line-height: 1.8; margin-left: 15px;'>
|
||
<li>学习率从 η<sub>max</sub> 平滑下降到 η<sub>min</sub>，呈余弦形状</li>
|
||
<li>训练初期学习率大 → 快速探索参数空间</li>
|
||
<li>训练后期学习率小 → 精细化调整，不会跳出最优解</li>
|
||
<li>比阶梯式衰减更平滑，收敛更稳定</li>
|
||
</ul>
|
||
</div>
|
||
</details>
|
||
|
||
<details>
|
||
<summary style='background: linear-gradient(135deg, #fce4ec 0%, #f8bbd0 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
🔄 数据处理流水线与超参数建议
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #f8bbd0;'>
|
||
<h4 style='color: #c62828;'>完整数据流水线</h4>
|
||
<div style='background: white; padding: 15px; border-radius: 8px; margin: 12px 0; border: 1px dashed #f48fb1;'>
|
||
<ol style='line-height: 2.3; margin-left: 15px;'>
|
||
<li><strong>原始输入：</strong>[高度(m), 马赫数, 转速(RPM)]</li>
|
||
<li><strong>输入 Z-Score：</strong>x̂ = (x − μ<sub>x</sub>) / σ<sub>x</sub></li>
|
||
<li><strong>NN 前向传播：</strong>ŷ<sub>norm</sub> = MLP(x̂)</li>
|
||
<li><strong>反 Z-Score：</strong>ŷ<sub>log</sub> = ŷ<sub>norm</sub> · σ<sub>y</sub> + μ<sub>y</sub></li>
|
||
<li><strong>反 Log1p：</strong>ŷ = expm1(ŷ<sub>log</sub>) = e<sup>ŷ<sub>log</sub></sup> − 1</li>
|
||
<li><strong>物理输出：</strong>[燃油流量(kg/h), 功率(kW)]</li>
|
||
</ol>
|
||
</div>
|
||
|
||
<h4 style='color: #c62828; margin-top: 15px;'>超参数调优建议</h4>
|
||
<table style='width: 100%; border-collapse: collapse; margin: 12px 0; font-size: 0.93em; box-shadow: 0 2px 10px rgba(0,0,0,0.08); border-radius: 8px; overflow: hidden;'>
|
||
<tr style='background: linear-gradient(135deg, #c62828 0%, #b71c1c 100%); color: white;'>
|
||
<th style='padding: 10px;'>超参数</th>
|
||
<th style='padding: 10px;'>推荐范围</th>
|
||
<th style='padding: 10px;'>增大效果</th>
|
||
<th style='padding: 10px;'>减小效果</th>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 8px; border: 1px solid #f8bbd0;'><strong>隐藏层宽度</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #f8bbd0;'>32 ~ 128</td>
|
||
<td style='padding: 8px; border: 1px solid #f8bbd0;'>拟合能力增强，速度稍降</td>
|
||
<td style='padding: 8px; border: 1px solid #f8bbd0;'>推理更快，但可能欠拟合</td>
|
||
</tr>
|
||
<tr style='background: #fff5f8;'>
|
||
<td style='padding: 8px; border: 1px solid #f8bbd0;'><strong>训练轮数</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #f8bbd0;'>2000 ~ 5000</td>
|
||
<td style='padding: 8px; border: 1px solid #f8bbd0;'>精度更高，但耗时增加</td>
|
||
<td style='padding: 8px; border: 1px solid #f8bbd0;'>训练快，但可能未收敛</td>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 8px; border: 1px solid #f8bbd0;'><strong>学习率</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #f8bbd0;'>1e-3 ~ 5e-3</td>
|
||
<td style='padding: 8px; border: 1px solid #f8bbd0;'>收敛更快，但容易振荡</td>
|
||
<td style='padding: 8px; border: 1px solid #f8bbd0;'>收敛更稳，但可能太慢</td>
|
||
</tr>
|
||
</table>
|
||
|
||
<div style='background: linear-gradient(135deg, #e8f5e9 0%, #c8e6c9 100%); padding: 12px; border-radius: 8px; margin: 10px 0;'>
|
||
<p><strong>🎯 观察训练 Loss 曲线的注意事项：</strong></p>
|
||
<ul style='line-height: 1.8; margin-left: 15px;'>
|
||
<li><strong>Loss 持续下降</strong>：正常训练中，一切良好</li>
|
||
<li><strong>Loss 振荡不降</strong>：学习率可能过大，尝试降低</li>
|
||
<li><strong>Loss 很快收敛到平台</strong>：模型容量不足（增大隐藏层宽度）或学习率太小</li>
|
||
<li><strong>Parity Plot 偏离对角线</strong>：模型预测有系统偏差，可能需要更多训练轮数</li>
|
||
<li>典型目标：MAPE < 1%，即 NN 与 GPR 的预测差异小于 1%</li>
|
||
</ul>
|
||
</div>
|
||
</div>
|
||
</details>
|
||
</div>
|
||
"""
|
||
|
||
|
||
# ============================================================
|
||
# 能量管理策略知识卡片 — 中文
|
||
# ============================================================
|
||
EMS_KNOWLEDGE = """
|
||
<div class='knowledge-card'>
|
||
<h3 style='color: #2e7d32; margin-top: 0; display: flex; align-items: center; gap: 10px;'>
|
||
<span style='font-size: 1.3em;'>🔋</span> 串联混动能量管理策略 (EMS)
|
||
</h3>
|
||
|
||
<details open>
|
||
<summary style='background: linear-gradient(135deg, #e8f5e9 0%, #c8e6c9 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
📊 SOC 迟滞控制 (Hysteresis Control)
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #c8e6c9;'>
|
||
<p>EMS 采用<strong>迟滞（滞回）控制策略</strong>，在"充电模式"与"功率跟随模式"
|
||
之间引入死区，避免因 SOC 在阈值附近波动而导致频繁切换：</p>
|
||
|
||
<div style='text-align: center; background: white; padding: 18px; border-radius: 8px; margin: 10px 0; box-shadow: 0 2px 6px rgba(0,0,0,0.05);'>
|
||
<div style='font-size: 1.05em; color: #333;'>
|
||
SOC < SOC<sub>low</sub> → <span style='color: #c62828; font-weight: bold;'>充电模式</span>（发动机以固定高功率运行）<br>
|
||
SOC > SOC<sub>high</sub> → <span style='color: #2e7d32; font-weight: bold;'>功率跟随模式</span>（发动机跟踪需求功率）<br>
|
||
SOC<sub>low</sub> ≤ SOC ≤ SOC<sub>high</sub> → <span style='color: #1565c0; font-weight: bold;'>保持当前模式</span>（迟滞死区）
|
||
</div>
|
||
</div>
|
||
|
||
<h4 style='color: #2e7d32; margin-top: 18px;'>运行模式详解</h4>
|
||
<table style='width: 100%; border-collapse: collapse; margin: 12px 0; font-size: 0.93em; box-shadow: 0 2px 10px rgba(0,0,0,0.08); border-radius: 8px; overflow: hidden;'>
|
||
<tr style='background: linear-gradient(135deg, #2e7d32 0%, #1b5e20 100%); color: white;'>
|
||
<th style='padding: 10px;'>模式</th>
|
||
<th style='padding: 10px;'>发动机功率</th>
|
||
<th style='padding: 10px;'>触发条件</th>
|
||
<th style='padding: 10px;'>设计目的</th>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 8px; border: 1px solid #c8e6c9;'><strong style='color:#c62828;'>紧急充电</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #c8e6c9;'>最大功率 P<sub>eng,max</sub></td>
|
||
<td style='padding: 8px; border: 1px solid #c8e6c9;'>SOC < 10%</td>
|
||
<td style='padding: 8px; border: 1px solid #c8e6c9;'>防止电池深放电损坏</td>
|
||
</tr>
|
||
<tr style='background: #f0fff0;'>
|
||
<td style='padding: 8px; border: 1px solid #c8e6c9;'><strong style='color:#e65100;'>充电模式</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #c8e6c9;'>恒定充电功率 P<sub>charge</sub></td>
|
||
<td style='padding: 8px; border: 1px solid #c8e6c9;'>SOC < SOC<sub>low</sub>（进入）</td>
|
||
<td style='padding: 8px; border: 1px solid #c8e6c9;'>高效率恒定工况点运行</td>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 8px; border: 1px solid #c8e6c9;'><strong style='color:#2e7d32;'>功率跟随</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #c8e6c9;'>P<sub>demand</sub> + Reserve + K<sub>SOC</sub>·ΔSOC</td>
|
||
<td style='padding: 8px; border: 1px solid #c8e6c9;'>SOC > SOC<sub>high</sub>（退出充电）</td>
|
||
<td style='padding: 8px; border: 1px solid #c8e6c9;'>实时匹配负载需求</td>
|
||
</tr>
|
||
<tr style='background: #f0fff0;'>
|
||
<td style='padding: 8px; border: 1px solid #c8e6c9;'><strong style='color:#1565c0;'>过充保护</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #c8e6c9;'>最低功率 P<sub>eng,min</sub></td>
|
||
<td style='padding: 8px; border: 1px solid #c8e6c9;'>SOC > 95%</td>
|
||
<td style='padding: 8px; border: 1px solid #c8e6c9;'>防止电池过充损坏</td>
|
||
</tr>
|
||
</table>
|
||
|
||
<div style='background: linear-gradient(135deg, #e3f2fd 0%, #bbdefb 100%); padding: 12px; border-radius: 8px; margin: 10px 0;'>
|
||
<p><strong>💡 迟滞控制原理：</strong></p>
|
||
<ul style='line-height: 1.8; margin-left: 15px;'>
|
||
<li><strong>上阈值 SOC<sub>high</sub></strong>（如 0.7）：SOC 超过此值时从充电切换到功率跟随</li>
|
||
<li><strong>下阈值 SOC<sub>low</sub></strong>（如 0.3）：SOC 降到此值时从功率跟随切换到充电</li>
|
||
<li>死区宽度 = SOC<sub>high</sub> − SOC<sub>low</sub>，越宽则切换频率越低，但 SOC 波动范围越大</li>
|
||
<li>死区内保持上一步的运行模式不变 → 有效抑制"抖动"(chattering)</li>
|
||
</ul>
|
||
</div>
|
||
</div>
|
||
</details>
|
||
|
||
<details open>
|
||
<summary style='background: linear-gradient(135deg, #fff3e0 0%, #ffe0b2 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
🔌 串联混合动力架构
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #ffe0b2;'>
|
||
<div style='text-align: center; background: white; padding: 18px; border-radius: 8px; margin: 10px 0;'>
|
||
<span style='font-size: 1em; color: #555;'>
|
||
[涡轴发动机] → 发电机 → 直流母线 ← → [锂电池组]<br>
|
||
直流母线 → 逆变器 → [PMSM 驱动电机] → 螺旋桨
|
||
</span>
|
||
</div>
|
||
<p>发动机与推进轴<strong>机械解耦</strong>：所有动力通过电气母线传输，
|
||
允许发动机始终运行在最优效率点，不受负载功率变化的直接影响。</p>
|
||
|
||
<h4 style='color: #e65100; margin-top: 15px;'>架构特点</h4>
|
||
<table style='width: 100%; border-collapse: collapse; margin: 12px 0; font-size: 0.93em; box-shadow: 0 2px 10px rgba(0,0,0,0.08); border-radius: 8px; overflow: hidden;'>
|
||
<tr style='background: linear-gradient(135deg, #e65100 0%, #bf360c 100%); color: white;'>
|
||
<th style='padding: 10px;'>特性</th>
|
||
<th style='padding: 10px;'>串联构型</th>
|
||
<th style='padding: 10px;'>并联构型（对比）</th>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>发动机-负载耦合</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>完全解耦</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>机械直连</td>
|
||
</tr>
|
||
<tr style='background: #fff8f0;'>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>发动机工况</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>可固定在最优效率点</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>随负载波动</td>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>EMS 复杂度</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>较低（功率分配）</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>较高（扭矩耦合）</td>
|
||
</tr>
|
||
<tr style='background: #fff8f0;'>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>能量传递效率</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>经两次电能转换（略低）</td>
|
||
<td style='padding: 8px; border: 1px solid #ffe0b2;'>机械直驱（略高）</td>
|
||
</tr>
|
||
</table>
|
||
</div>
|
||
</details>
|
||
|
||
<details>
|
||
<summary style='background: linear-gradient(135deg, #fce4ec 0%, #f8bbd0 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
⚙️ 关键参数与调节指南
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #f8bbd0;'>
|
||
<table style='width: 100%; border-collapse: collapse; margin: 12px 0; font-size: 0.93em; box-shadow: 0 2px 10px rgba(0,0,0,0.08); border-radius: 8px; overflow: hidden;'>
|
||
<tr style='background: linear-gradient(135deg, #880e4f 0%, #ad1457 100%); color: white;'>
|
||
<th style='padding: 10px;'>参数</th>
|
||
<th style='padding: 10px;'>作用</th>
|
||
<th style='padding: 10px;'>调大效果</th>
|
||
<th style='padding: 10px;'>调小效果</th>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 8px; border: 1px solid #f8bbd0;'><strong>P<sub>charge</sub></strong></td>
|
||
<td style='padding: 8px; border: 1px solid #f8bbd0;'>充电模式发动机功率</td>
|
||
<td style='padding: 8px; border: 1px solid #f8bbd0;'>SOC 恢复更快</td>
|
||
<td style='padding: 8px; border: 1px solid #f8bbd0;'>充电更温和，效率稍高</td>
|
||
</tr>
|
||
<tr style='background: #fff0f5;'>
|
||
<td style='padding: 8px; border: 1px solid #f8bbd0;'><strong>K<sub>SOC</sub></strong></td>
|
||
<td style='padding: 8px; border: 1px solid #f8bbd0;'>功率跟随模式 SOC 反馈增益</td>
|
||
<td style='padding: 8px; border: 1px solid #f8bbd0;'>SOC 偏差纠正更强</td>
|
||
<td style='padding: 8px; border: 1px solid #f8bbd0;'>发动机功率更平稳</td>
|
||
</tr>
|
||
<tr style='background: white;'>
|
||
<td style='padding: 8px; border: 1px solid #f8bbd0;'><strong>功率裕度 %</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #f8bbd0;'>发动机额外功率余量</td>
|
||
<td style='padding: 8px; border: 1px solid #f8bbd0;'>抗瞬态能力增强</td>
|
||
<td style='padding: 8px; border: 1px solid #f8bbd0;'>油耗更低</td>
|
||
</tr>
|
||
<tr style='background: #fff0f5;'>
|
||
<td style='padding: 8px; border: 1px solid #f8bbd0;'><strong>电池容量</strong></td>
|
||
<td style='padding: 8px; border: 1px solid #f8bbd0;'>能量缓冲区大小</td>
|
||
<td style='padding: 8px; border: 1px solid #f8bbd0;'>SOC 波动更小</td>
|
||
<td style='padding: 8px; border: 1px solid #f8bbd0;'>系统更轻、响应更快</td>
|
||
</tr>
|
||
</table>
|
||
|
||
<div style='background: linear-gradient(135deg, #e8f5e9 0%, #c8e6c9 100%); padding: 12px; border-radius: 8px; margin: 10px 0;'>
|
||
<p><strong>🎯 仿真观察要点：</strong></p>
|
||
<ul style='line-height: 1.8; margin-left: 15px;'>
|
||
<li>SOC 是否维持在安全范围（10% ~ 95%）内</li>
|
||
<li>充电-跟随模式切换是否过于频繁（理想为低频切换）</li>
|
||
<li>发动机功率曲线是否平稳（频繁剧烈波动 → 油耗升高、寿命下降）</li>
|
||
<li>最终 SOC 与初始 SOC 之差（SOC 平衡性）</li>
|
||
<li>电池充放电电流是否超过安全限值</li>
|
||
</ul>
|
||
</div>
|
||
|
||
<div style='background: linear-gradient(135deg, #fff3e0 0%, #ffe0b2 100%); padding: 12px; border-radius: 8px; margin: 10px 0;'>
|
||
<p><strong>💡 调参建议：</strong></p>
|
||
<ul style='line-height: 1.8; margin-left: 15px;'>
|
||
<li>先用默认参数跑一次，观察 SOC 整体走势</li>
|
||
<li>若 SOC 持续下降 → 增大 P<sub>charge</sub> 或 K<sub>SOC</sub></li>
|
||
<li>若模式切换过于频繁 → 加大 SOC<sub>high</sub> − SOC<sub>low</sub> 死区宽度</li>
|
||
<li>若发动机功率波动大 → 降低 K<sub>SOC</sub>，增大功率裕度</li>
|
||
</ul>
|
||
</div>
|
||
</div>
|
||
</details>
|
||
|
||
<details>
|
||
<summary style='background: linear-gradient(135deg, #e3f2fd 0%, #bbdefb 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
🔋 电池 SOC 动态方程与功率平衡
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #bbdefb;'>
|
||
<h4 style='color: #1565c0;'>SOC 状态方程（库仑计数法）</h4>
|
||
<div style='text-align: center; background: white; padding: 18px; border-radius: 8px; margin: 10px 0;'>
|
||
<div style='font-size: 1.15em;'>
|
||
<span style='display: inline-block; vertical-align: middle;'>
|
||
<span style='display: block; text-align: center; border-bottom: 2px solid #333; padding: 0 6px 3px;'>dSOC</span>
|
||
<span style='display: block; text-align: center; padding: 3px 6px 0;'>dt</span>
|
||
</span>
|
||
= −
|
||
<span style='display: inline-block; vertical-align: middle;'>
|
||
<span style='display: block; text-align: center; border-bottom: 2px solid #333; padding: 0 8px 3px;'>P<sub>bat</sub></span>
|
||
<span style='display: block; text-align: center; padding: 3px 8px 0;'>Q<sub>bat</sub></span>
|
||
</span>
|
||
</div>
|
||
</div>
|
||
<p style='color: #555; font-size: 0.93em;'>
|
||
P<sub>bat</sub> 为电池充放电功率（放电为正），Q<sub>bat</sub> 为电池总容量 (kWh)。
|
||
离散化：SOC<sub>k+1</sub> = SOC<sub>k</sub> − (P<sub>bat,k</sub> · Δt) / Q<sub>bat</sub>
|
||
</p>
|
||
|
||
<h4 style='color: #1565c0; margin-top: 15px;'>系统功率平衡方程</h4>
|
||
<div style='text-align: center; background: white; padding: 15px; border-radius: 8px; margin: 10px 0;'>
|
||
<div style='font-size: 1.1em;'>
|
||
P<sub>demand</sub>(t) = P<sub>engine</sub>(t) + P<sub>battery</sub>(t)
|
||
</div>
|
||
</div>
|
||
<ul style='line-height: 1.8; margin-left: 15px;'>
|
||
<li><strong>P<sub>demand</sub> > P<sub>engine</sub></strong>：电池补充差额（放电），SOC 下降</li>
|
||
<li><strong>P<sub>demand</sub> < P<sub>engine</sub></strong>：多余功率给电池充电，SOC 上升</li>
|
||
<li>EMS 的核心任务就是合理分配 P<sub>engine</sub>，使 SOC 维持在安全范围</li>
|
||
</ul>
|
||
|
||
<h4 style='color: #1565c0; margin-top: 15px;'>功率跟随模式公式</h4>
|
||
<div style='background: white; padding: 15px; border-radius: 8px; margin: 10px 0; border: 1px solid #bbdefb;'>
|
||
<div style='font-size: 1.05em; line-height: 2.0;'>
|
||
P<sub>eng,cmd</sub> = P<sub>demand</sub> × (1 + Reserve%) + K<sub>SOC</sub> × (SOC<sub>target</sub> − SOC)<br>
|
||
P<sub>eng,cmd</sub> = clamp(P<sub>eng,cmd</sub>, P<sub>eng,min</sub>, P<sub>eng,max</sub>)
|
||
</div>
|
||
</div>
|
||
<p style='color: #555; font-size: 0.93em;'>
|
||
第二项为 SOC 反馈补偿：当 SOC 低于目标时增加发动机功率以充电；
|
||
当 SOC 高于目标时减少发动机功率以节油。
|
||
</p>
|
||
</div>
|
||
</details>
|
||
|
||
<details>
|
||
<summary style='background: linear-gradient(135deg, #ede7f6 0%, #d1c4e9 100%); padding: 12px 16px; border-radius: 10px; cursor: pointer; font-weight: 700; color: #333; font-size: 1.05em; margin: 10px 0; box-shadow: 0 2px 8px rgba(0,0,0,0.08);'>
|
||
📚 进阶：ECMS 等效消耗最小化策略
|
||
</summary>
|
||
<div style='padding: 15px; background: #fafbff; margin-top: 5px; border-radius: 10px; border: 1px solid #d1c4e9;'>
|
||
<p>ECMS (Equivalent Consumption Minimization Strategy) 是一种更先进的实时优化 EMS。
|
||
它将电池充放电等效为燃油消耗，从而将双能源分配问题转化为单目标瞬时优化：</p>
|
||
|
||
<div style='text-align: center; background: white; padding: 18px; border-radius: 8px; margin: 10px 0;'>
|
||
<div style='font-size: 1.1em;'>
|
||
min<sub>P<sub>eng</sub></sub>
|
||
J = ṁ<sub>fuel</sub>(P<sub>eng</sub>) + s(t) ·
|
||
<span style='display: inline-block; vertical-align: middle;'>
|
||
<span style='display: block; text-align: center; border-bottom: 2px solid #333; padding: 0 8px 3px;'>P<sub>bat</sub></span>
|
||
<span style='display: block; text-align: center; padding: 3px 8px 0;'>η<sub>bat</sub> · Q<sub>LHV</sub></span>
|
||
</span>
|
||
</div>
|
||
</div>
|
||
<ul style='line-height: 1.8; margin-left: 15px;'>
|
||
<li><strong>s(t)</strong>：等效因子，将电能消耗折算为燃油消耗</li>
|
||
<li><strong>Q<sub>LHV</sub></strong>：燃料低热值 (MJ/kg)</li>
|
||
<li><strong>η<sub>bat</sub></strong>：电池充放电效率</li>
|
||
<li>当 s(t) 自适应调整以维持 SOC 时，ECMS 近似全局最优</li>
|
||
</ul>
|
||
|
||
<div style='background: linear-gradient(135deg, #fff3e0 0%, #ffe0b2 100%); padding: 12px; border-radius: 8px; margin: 10px 0;'>
|
||
<p><strong>💡 本平台使用的是基于规则的滞环策略（简单可靠），
|
||
ECMS 和 DP (动态规划) 等高级方法可作为后续扩展方向。</strong></p>
|
||
</div>
|
||
</div>
|
||
</details>
|
||
</div>
|
||
"""
|