{"id":6547,"date":"2025-11-27T22:05:44","date_gmt":"2025-11-27T14:05:44","guid":{"rendered":"https:\/\/nullthought.net\/?p=6547"},"modified":"2025-11-27T22:59:20","modified_gmt":"2025-11-27T14:59:20","slug":"%e5%87%b8%e4%bc%98%e5%8c%96%ef%bc%88convex-optimization%ef%bc%89%ef%bc%9a%e7%94%a8%e4%ba%8e%e7%81%ab%e7%ae%ad%e5%8a%a9%e6%8e%a8%e5%99%a8%e7%9a%84%e5%9b%9e%e6%94%b6%e9%99%8d%e8%90%bd%e6%8e%a7%e5%88%b6","status":"publish","type":"post","link":"https:\/\/nullthought.net\/?p=6547","title":{"rendered":"\u51f8\u4f18\u5316\uff08Convex Optimization\uff09\uff1a\u7528\u4e8e\u706b\u7bad\u52a9\u63a8\u5668\u7684\u56de\u6536\u964d\u843d\u63a7\u5236"},"content":{"rendered":"\n
\"\"\/
\u706b\u7bad\u7740\u9646\u51f8\u4f18\u5316\u7b97\u6cd5\u6a21\u62df\uff083D\uff09<\/figcaption><\/figure>\n\n\n\n

\u51f8\u4f18\u5316\u7528\u5728\u706b\u7bad\u52a9\u63a8\u5668\u56de\u6536\u964d\u843d<\/a>\u4e0a\uff0c\u672c\u8d28\u4e0a\u662f\uff1a\u628a\u4e00\u4e2a\u672c\u6765\u975e\u5e38\u590d\u6742\u3001\u975e\u7ebf\u6027\u7684\u201c\u600e\u4e48\u5b89\u5168\u3001\u8282\u7701\u71c3\u6599\u5730\u964d\u843d\u201d\u95ee\u9898\uff0c\u6539\u5199\u6210\u4e00\u7c7b\u8ba1\u7b97\u673a\u80fd\u53c8\u5feb\u53c8\u7a33\u5b9a\u6c42\u89e3\u7684\u51f8\u4f18\u5316\u95ee\u9898\uff0c\u518d\u5728\u98de\u884c\u7535\u8111\u4e0a\u5b9e\u65f6\u89e3\u51fa\u6765\uff0c\u6307\u5bfc\u63a8\u529b\u548c\u59ff\u6001\u63a7\u5236\u3002<\/p>\n\n\n\n

\u4e00\u3001\u706b\u7bad\u52a9\u63a8\u5668\u56de\u6536\u964d\u843d\u8981\u89e3\u51b3\u4ec0\u4e48\u95ee\u9898\uff1f<\/h4>\n\n\n\n

\u4ee5\u730e\u9e70 9 \u4e00\u5b50\u7ea7\u4e3a\u4f8b\uff0c\u4ece\u63a7\u5236\/\u4f18\u5316\u89d2\u5ea6\u770b\uff0c\u52a9\u63a8\u5668\u56de\u6536\u964d\u843d\u4e3b\u8981\u662f\u4e00\u4e2a\u6709\u7ea6\u675f\u7684\u6700\u4f18\u63a7\u5236\u95ee\u9898\uff1a<\/p>\n\n\n\n

\u00b7 \u72b6\u6001\u53d8\u91cf\uff1a\u4f4d\u7f6e\uff08x,y,z\uff09\u3001\u901f\u5ea6\uff08vx,vy,vz\uff09\u3001\u59ff\u6001\u3001\u89d2\u901f\u5ea6\u3001\u5269\u4f59\u63a8\u8fdb\u5242\u8d28\u91cf\u7b49\uff1b<\/p>\n\n\n\n

\u00b7 \u63a7\u5236\u53d8\u91cf\uff1a\u63a8\u529b\u5927\u5c0f T(t)\u3001\u63a8\u529b\u65b9\u5411\uff08\u53d1\u52a8\u673a\u77e2\u91cf\uff09\u3001\u6805\u683c\u7ffc\u504f\u89d2\u3001\u51b7\u6c14\u4f53\u59ff\u63a7\u7b49\uff1b<\/p>\n\n\n\n

\u00b7 \u76ee\u6807\u51fd\u6570\uff1a\u901a\u5e38\u662f\u6700\u5c0f\u71c3\u6599\u6d88\u8017\uff0c\u6216\u5728\u7ed9\u5b9a\u71c3\u6599\u4e0b\u6700\u5927\u5316\u7740\u9646\u5b89\u5168\u88d5\u5ea6\uff1b<\/p>\n\n\n\n

\u00b7 \u7ea6\u675f\uff1a\u5305\u62ec\u52a8\u529b\u5b66\u7ea6\u675f\uff08\u521a\u4f53\u52a8\u529b\u5b66\u548c\u8d28\u91cf\u53d8\u5316\u65b9\u7a0b\uff09\u3001\u63a8\u529b\u5927\u5c0f\u548c\u504f\u8f6c\u89d2\u7ea6\u675f\u3001\u7ed3\u6784\u8f7d\u8377\u7ea6\u675f\u3001\u7ec8\u7aef\u7740\u9646\u6761\u4ef6\u7ea6\u675f\u3001\u4ee5\u53ca\u98de\u884c\u8d70\u5eca\u7b49\u5b89\u5168\u533a\u57df\u7ea6\u675f\u3002<\/p>\n\n\n\n

\u628a\u8fd9\u4e9b\u4e1c\u897f\u5168\u653e\u4e00\u8d77\uff0c\u5c31\u662f\u4e00\u4e2a\u5927\u578b\u7684\u201c\u5e26\u7ea6\u675f\u975e\u7ebf\u6027\u6700\u4f18\u63a7\u5236\u95ee\u9898\u201d\u3002<\/p>\n\n\n\n

\u4e8c\u3001\u4e3a\u4ec0\u4e48\u8981\u7528\u51f8\u4f18\u5316\u800c\u4e0d\u662f\u968f\u4fbf\u4e00\u4e2a\u975e\u7ebf\u6027\u4f18\u5316\uff1f<\/h4>\n\n\n\n

\u4f20\u7edf\u975e\u7ebf\u6027\u89c4\u5212\uff08NLP\uff09\u6216\u542f\u53d1\u5f0f\u641c\u7d22\u6709\u51e0\u4e2a\u81f4\u547d\u95ee\u9898\uff1a<\/p>\n\n\n\n

1. \u975e\u51f8\u95ee\u9898\u4f1a\u6709\u5f88\u591a\u5c40\u90e8\u6700\u4f18\u89e3\uff0c\u5f80\u5f80\u4e0d\u77e5\u9053\u7b97\u51fa\u6765\u7684\u662f\u4e0d\u662f\u5168\u5c40\u6700\u4f18\uff1b<\/p>\n\n\n\n

2. \u6536\u655b\u65f6\u95f4\u548c\u7ed3\u679c\u4e0d\u53ef\u9884\u6d4b\uff0c\u8fed\u4ee3\u6b21\u6570\u3001\u662f\u5426\u6536\u655b\u90fd\u4e0d\u7a33\u5b9a\uff1b<\/p>\n\n\n\n

3. \u96be\u4ee5\u7ed9\u51fa\u201c\u80af\u5b9a\u80fd\u627e\u5230\u53ef\u884c\u89e3\/\u6700\u4f18\u89e3\u201d\u7684\u6570\u5b66\u4fdd\u8bc1\uff0c\u8fd9\u5bf9\u5b89\u5168\u5173\u952e\u7cfb\u7edf\u662f\u5927\u95ee\u9898\u3002<\/p>\n\n\n\n

\u800c\u5982\u679c\u80fd\u628a\u95ee\u9898\u53d8\u6210\u51f8\u4f18\u5316\u95ee\u9898\uff0c\u60c5\u51b5\u5c31\u5b8c\u5168\u4e0d\u540c\uff1a\u76ee\u6807\u548c\u7ea6\u675f\u90fd\u662f\u51f8\u7684\u65f6\uff0c\u4efb\u4f55\u5c40\u90e8\u6700\u4f18\u90fd\u662f\u5168\u5c40\u6700\u4f18\uff1b\u6709\u6210\u719f\u7684\u5185\u70b9\u6cd5\u3001\u4e8c\u9636\u9525\u89c4\u5212\uff08SOCP\uff09\u3001\u7ebf\u6027\u89c4\u5212\uff08LP\uff09\u6c42\u89e3\u5668\uff0c\u6536\u655b\u6027\u548c\u7cbe\u5ea6\u90fd\u6709\u5f3a\u7406\u8bba\u4fdd\u8bc1\uff0c\u53ef\u4ee5\u63a7\u5236\u8fd0\u7b97\u65f6\u95f4\uff0c\u673a\u8f7d\u5b9e\u65f6\u6c42\u89e3\u6210\u4e3a\u53ef\u80fd\u3002<\/p>\n\n\n\n

\u4e09\u3001\u539f\u59cb\u7684\u52a9\u63a8\u5668\u7740\u9646\u95ee\u9898\u6709\u54ea\u4e9b\u201c\u975e\u51f8\u6027\u201d\uff1f<\/h4>\n\n\n\n

\u5178\u578b\u975e\u51f8\u6765\u6e90\u5305\u62ec\uff1a<\/p>\n\n\n\n

1. \u63a8\u529b\u5927\u5c0f\u7684\u201c\u73af\u72b6\u201d\u7ea6\u675f\uff1a\u63a8\u529b\u5927\u5c0f\u5fc5\u987b\u5728 [T_min, T_max] \u4e4b\u95f4\uff0c\u662f\u7a7a\u5fc3\u7403\u58f3\/\u5706\u73af\uff0c\u96c6\u5408\u672c\u8eab\u662f\u975e\u51f8\u7684\uff1b<\/p>\n\n\n\n

2. \u63a8\u529b\u65b9\u5411\u9525\u5f62\u9650\u5236\uff1a\u63a8\u529b\u76f8\u5bf9\u706b\u7bad\u7eb5\u8f74\u504f\u8f6c\u89d2\u6709\u9650\u5236\uff0c\u6d89\u53ca\u5411\u91cf\u8303\u6570\u548c\u6bd4\u503c\uff0c\u7ea6\u675f\u975e\u7ebf\u6027\u3001\u975e\u51f8\uff1b<\/p>\n\n\n\n

3. \u516d\u81ea\u7531\u5ea6\u52a8\u529b\u5b66\u548c\u6c14\u52a8\u529b\u7684\u975e\u7ebf\u6027\uff1b<\/p>\n\n\n\n

4. \u53d1\u52a8\u673a\u5f00\u5173\u3001\u6a21\u5f0f\u5207\u6362\u7b49\u201c\u79bb\u6563\u201d\u51b3\u7b56\uff0c\u4f1a\u5f15\u5165\u6df7\u5408\u6574\u6570\u548c\u975e\u51f8\u7ed3\u6784\u3002<\/p>\n\n\n\n

\u76f4\u63a5\u628a\u8fd9\u4e9b\u975e\u51f8\u7ea6\u675f\u585e\u7ed9\u4e00\u4e2a\u901a\u7528\u975e\u7ebf\u6027\u89c4\u5212\u6c42\u89e3\u5668\uff0c\u5f88\u96be\u4fdd\u8bc1\u65e2\u5feb\u53c8\u7a33\u3002<\/p>\n\n\n\n

\u56db\u3001\u201c\u65e0\u635f\u51f8\u5316\u201d\u662f\u600e\u4e48\u628a\u5b83\u201c\u53d8\u6210\u51f8\u201d\u7684\uff1f<\/h4>\n\n\n\n

Acikmese\u3001Blackmore \u7b49\u4eba\u7684\u5173\u952e\u601d\u60f3\u662f\uff1a\u901a\u8fc7\u5f15\u5165\u65b0\u53d8\u91cf\u548c\u9002\u5f53\u7684\u677e\u5f1b\uff0c\u628a\u975e\u51f8\u7ea6\u675f\u53d8\u6210\u51f8\u7ea6\u675f\uff0c\u7136\u540e\u8bc1\u660e\u8fd9\u79cd\u677e\u5f1b\u662f\u201c\u65e0\u635f\u201d\u7684\u2014\u2014\u51f8\u95ee\u9898\u7684\u6700\u4f18\u89e3\u4e00\u5b9a\u6ee1\u8db3\u539f\u6765\u7684\u975e\u51f8\u7ea6\u675f\uff0c\u56e0\u6b64\u4e5f\u662f\u539f\u95ee\u9898\u7684\u5168\u5c40\u6700\u4f18\u89e3\u3002<\/p>\n\n\n\n

\u5178\u578b\u6b65\u9aa4\u5305\u62ec\uff1a<\/p>\n\n\n\n

1. \u65f6\u95f4\u79bb\u6563\u5316 + \u52a8\u529b\u5b66\u7ebf\u6027\u5316\uff1a\u628a\u8fde\u7eed\u6700\u4f18\u63a7\u5236\u95ee\u9898\u79bb\u6563\u6210\u6709\u9650\u7ef4\u7ebf\u6027\u7cfb\u7edf\u3002<\/p>\n\n\n\n

2. \u5c06\u63a8\u529b\u201c\u7403\u58f3\u201d\u7ea6\u675f [T_min, T_max] \u677e\u5f1b\u4e3a\u7403\u7ea6\u675f [0, T_max]\uff1a\u5148\u5141\u8bb8 0 \u5230 T_min \u8fd9\u6bb5\u672c\u6765\u4e0d\u5141\u8bb8\u7684\u533a\u57df\uff0c\u518d\u901a\u8fc7\u76ee\u6807\u51fd\u6570\u548c\u7ec8\u7aef\u7ea6\u675f\u8bbe\u8ba1\uff0c\u8bc1\u660e\u6700\u4f18\u89e3\u4e0d\u4f1a\u505c\u5728\u8fd9\u6bb5\u533a\u57df\uff0c\u4ece\u800c\u677e\u5f1b\u662f\u65e0\u635f\u7684\u3002<\/p>\n\n\n\n

3. \u5c06\u63a8\u529b\u65b9\u5411\u7b49\u9650\u5236\u5199\u6210\u4e8c\u9636\u9525\u7ea6\u675f\uff0c\u4f8b\u5982\u8303\u6570\u5f62\u5f0f \u2016a_k\u2016 \u2264 b_k\uff0c\u8fd9\u5c5e\u4e8e\u6807\u51c6 SOCP \u7ea6\u675f\uff0c\u5177\u6709\u51f8\u6027\u3002<\/p>\n\n\n\n

4. \u901a\u8fc7\u4e25\u683c\u8bc1\u660e\uff1a\u5982\u679c\u539f\u975e\u51f8\u95ee\u9898\u6709\u53ef\u884c\u89e3\uff0c\u677e\u5f1b\u540e\u7684\u51f8\u95ee\u9898\u4e5f\u6709\u53ef\u884c\u89e3\uff0c\u5e76\u4e14\u51f8\u95ee\u9898\u7684\u6700\u4f18\u89e3\u81ea\u52a8\u6ee1\u8db3\u539f\u6765\u7684\u975e\u51f8\u7ea6\u675f\uff0c\u56e0\u6b64\u540c\u65f6\u662f\u539f\u95ee\u9898\u7684\u5168\u5c40\u6700\u4f18\u89e3\u3002<\/p>\n\n\n\n

\u4e94\u3001\u5728\u706b\u7bad\u56de\u6536\u4e2d\u7684\u5b9e\u9645\u4f7f\u7528\u65b9\u5f0f\uff1a\u5206\u5c42\u63a7\u5236\u7ed3\u6784<\/h4>\n\n\n\n

\u5728\u5b9e\u9645\u5de5\u7a0b\u4e2d\uff0c\u51f8\u4f18\u5316\u901a\u5e38\u5904\u4e8e\u201c\u8f68\u8ff9\u89c4\u5212 \/ \u5236\u5bfc\u201d\u8fd9\u4e00\u5c42\uff1a<\/p>\n\n\n\n

1. \u6700\u4e0a\u5c42\uff08\u8f68\u8ff9\/\u5236\u5bfc\uff09\uff1a\u6bcf\u9694\u51e0\u5341\u6beb\u79d2\u5230\u51e0\u767e\u6beb\u79d2\uff0c\u57fa\u4e8e\u5f53\u524d\u72b6\u6001\uff0c\u6784\u5efa\u4e00\u4e2a\u6709\u9650\u9884\u6d4b\u65f6\u57df\u7684\u51f8\u4f18\u5316\u95ee\u9898\uff0c\u6c42\u51fa\u672a\u6765\u51e0\u79d2\u7684\u201c\u6700\u4f18\u8f68\u8ff9\u201d\u548c\u5bf9\u5e94\u63a8\u529b\u5e8f\u5217\uff1b<\/p>\n\n\n\n

2. \u4e2d\u95f4\u5c42\uff08\u59ff\u6001\u4e0e\u63a8\u529b\u63a7\u5236\uff09\uff1a\u63a5\u6536\u4e0a\u5c42\u7ed9\u51fa\u7684\u671f\u671b\u63a8\u529b\u77e2\u91cf\uff0c\u7528\u4f20\u7edf\u63a7\u5236\uff08\u5982 PID\/LQR\/\u975e\u7ebf\u6027\u63a7\u5236\uff09\u9a71\u52a8\u53d1\u52a8\u673a\u77e2\u91cf\u3001\u6805\u683c\u7ffc\u548c\u59ff\u63a7\u7cfb\u7edf\u53bb\u8ddf\u8e2a\uff1b<\/p>\n\n\n\n

3. \u5e95\u5c42\uff08\u6267\u884c\u673a\u6784\u4e0e\u6545\u969c\u7ba1\u7406\uff09\uff1a\u6267\u884c\u53d1\u52a8\u673a\u8282\u6d41\u3001\u70b9\u706b\u3001\u8235\u673a\u63a7\u5236\u3001\u843d\u811a\u67b6\u5c55\u5f00\uff0c\u4ee5\u53ca\u6545\u969c\u68c0\u6d4b\u4e0e\u5197\u4f59\u7ba1\u7406\u3002<\/p>\n\n\n\n

\u5728\u8fd9\u4e2a\u7ed3\u6784\u4e2d\uff0c\u51f8\u4f18\u5316\u8d1f\u8d23\u201c\u7b97\u6e05\u695a\u4e0b\u4e00\u6bb5\u65f6\u95f4\u5185\u6700\u4f73\u7684\u4e0b\u843d\u8def\u5f84\u548c\u63a8\u529b\u8ba1\u5212\u201d\uff0c\u4e0b\u9762\u7684\u63a7\u5236\u5668\u8d1f\u8d23\u8ba9\u706b\u7bad\u6cbf\u7740\u8fd9\u6761\u8def\u5f84\u98de\u3002<\/p>\n\n\n\n

\u516d\u3001\u5982\u4f55\u5904\u7406\u504f\u5dee\u548c\u4e0d\u786e\u5b9a\u6027\uff1f<\/h4>\n\n\n\n

\u5b9e\u9645\u98de\u884c\u4e2d\u6709\u98ce\u3001\u53d1\u52a8\u673a\u63a8\u529b\u504f\u5dee\u3001\u8d28\u91cf\u4f30\u8ba1\u8bef\u5dee\u7b49\u4e0d\u786e\u5b9a\u6027\uff0c\u8f68\u8ff9\u4f1a\u548c\u8ba1\u5212\u4ea7\u751f\u504f\u5dee\u3002\u5728\u51f8\u4f18\u5316\u6846\u67b6\u4e0b\uff0c\u53ef\u4ee5\u901a\u8fc7\u9c81\u68d2\u4f18\u5316\u6216\u673a\u4f1a\u7ea6\u675f\u7b49\u65b9\u5f0f\uff0c\u628a\u4e0d\u786e\u5b9a\u6027\u5bf9\u8f68\u8ff9\u504f\u5dee\u7684\u7edf\u8ba1\u8981\u6c42\u7eb3\u5165\u5230\u4f18\u5316\u95ee\u9898\u4e2d\uff0c\u6bd4\u5982\uff1a<\/p>\n\n\n\n

\u00b7 \u8981\u6c42\u201c\u8fdd\u53cd\u7ea6\u675f\u7684\u6982\u7387\u4e0d\u8d85\u8fc7 1%\u201d\uff1b<\/p>\n\n\n\n

\u00b7 \u5728\u6700\u574f\u60c5\u5f62\u7684\u6270\u52a8\u8303\u56f4\u5185\uff0c\u4ecd\u7136\u4fdd\u6301\u843d\u5728\u5b89\u5168\u7740\u9646\u533a\u3002<\/p>\n\n\n\n

\u4e03\u3001\u590d\u6742\u516d\u81ea\u7531\u5ea6\u95ee\u9898\u4e2d\u7684\u8fde\u7eed\u51f8\u5316<\/h4>\n\n\n\n

\u5f53\u52a0\u5165\u516d\u81ea\u7531\u5ea6\u521a\u4f53\u52a8\u529b\u5b66\u3001\u5f3a\u6c14\u52a8\u529b\u548c\u72b6\u6001\u89e6\u53d1\u7ea6\u675f\u7b49\u590d\u6742\u56e0\u7d20\u65f6\uff0c\u5355\u6b21\u7ebf\u6027\u5316\u7684\u51f8\u95ee\u9898\u53ef\u80fd\u4e0d\u591f\u7cbe\u786e\u3002\u8fd9\u65f6\u5e38\u7528\u201c\u8fde\u7eed\u51f8\u5316 \/ \u9012\u63a8\u51f8\u5316\u201d\uff08successive convexification\uff09\uff1a<\/p>\n\n\n\n

1. \u4ece\u4e00\u4e2a\u521d\u59cb\u8f68\u8ff9\u51fa\u53d1\uff0c\u5728\u5176\u9644\u8fd1\u7ebf\u6027\u5316\u52a8\u529b\u5b66\u548c\u7ea6\u675f\uff0c\u6784\u9020\u5e76\u6c42\u89e3\u4e00\u4e2a\u51f8\u4f18\u5316\u95ee\u9898\uff1b<\/p>\n\n\n\n

2. \u5f97\u5230\u65b0\u7684\u8f68\u8ff9\u540e\uff0c\u518d\u4ee5\u5b83\u4e3a\u4e2d\u5fc3\u91cd\u65b0\u7ebf\u6027\u5316\uff0c\u7ee7\u7eed\u6c42\u89e3\u65b0\u7684\u51f8\u95ee\u9898\uff1b<\/p>\n\n\n\n

3. \u7ecf\u8fc7\u82e5\u5e72\u6b21\u8fed\u4ee3\uff0c\u8f68\u8ff9\u6536\u655b\u5230\u6ee1\u8db3\u7ea6\u675f\u4e14\u6027\u80fd\u4f18\u826f\u7684\u89e3\u3002<\/p>\n\n\n\n

\u516b\u3001CVXGEN \u7b49\u4ee3\u7801\u751f\u6210\u5de5\u5177\u5982\u4f55\u4fdd\u8bc1\u5b9e\u65f6\u6027\uff1f<\/h4>\n\n\n\n

\u4e3a\u4e86\u5728\u673a\u8f7d\u8ba1\u7b97\u673a\u4e0a\u5b9e\u73b0\u6beb\u79d2\u7ea7\u7684\u6c42\u89e3\u65f6\u95f4\uff0c\u5de5\u7a0b\u4e0a\u901a\u5e38\u4f7f\u7528\u5982 CVXGEN \u4e4b\u7c7b\u7684\u51f8\u4f18\u5316\u4ee3\u7801\u751f\u6210\u5de5\u5177\uff1a<\/p>\n\n\n\n

1. \u9884\u5148\u5206\u6790\u95ee\u9898\u7ed3\u6784\uff08\u53d8\u91cf\u89c4\u6a21\u3001\u7a00\u758f\u7ed3\u6784\u3001\u7ea6\u675f\u7c7b\u578b\u7b49\uff09\uff1b<\/p>\n\n\n\n

2. \u81ea\u52a8\u751f\u6210\u9488\u5bf9\u8be5\u7ed3\u6784\u9ad8\u5ea6\u4f18\u5316\u7684 C \u4ee3\u7801\uff0c\u907f\u514d\u8fd0\u884c\u65f6\u5f00\u9500\uff1b<\/p>\n\n\n\n

3. \u5728\u5c0f\u89c4\u6a21\u95ee\u9898\u4e0a\u5b9e\u73b0\u6beb\u79d2\u7ea7\u6216\u66f4\u5feb\u7684\u6c42\u89e3\u6027\u80fd\u3002<\/p>\n\n\n\n

\u4e5d\u3001\u76f4\u89c2\u51e0\u4f55\u56fe\u50cf\u7684\u7406\u89e3<\/h4>\n\n\n\n

\u53ef\u4ee5\u628a\u6574\u4e2a\u95ee\u9898\u60f3\u8c61\u4e3a\uff1a\u5728\u72b6\u6001\u7a7a\u95f4\u91cc\u5b58\u5728\u4e00\u4e2a\u201c\u5b89\u5168\u8d70\u5eca\u201d\uff0c\u7531\u4f4d\u7f6e\u3001\u901f\u5ea6\u3001\u59ff\u6001\u3001\u8d28\u91cf\u7b49\u6784\u6210\uff0c\u63a8\u529b\u7ea6\u675f\u548c\u59ff\u6001\u7ea6\u675f\u9650\u5236\u4e86\u6bcf\u4e00\u6b65\u53ef\u884c\u7684\u201c\u65b9\u5411\u548c\u6b65\u957f\u201d\u3002\u51f8\u4f18\u5316\u8981\u505a\u7684\uff0c\u5c31\u662f\u5728\u8fd9\u4e2a\u8d70\u5eca\u91cc\uff0c\u627e\u5230\u4e00\u6761\u4ece\u201c\u5f53\u524d\u72b6\u6001\u201d\u8d70\u5230\u201c\u5b89\u5168\u7740\u9646\u72b6\u6001\u201d\u7684\u6700\u7701\u6cb9\u8def\u5f84\u3002<\/p>\n\n\n\n

\u65e0\u635f\u51f8\u5316\u4fdd\u8bc1\uff1a\u867d\u7136\u539f\u59cb\u95ee\u9898\u4e2d\u63a8\u529b\u6709\u201c\u6700\u5c0f\u503c\u201d\u7b49\u975e\u51f8\u7ea6\u675f\uff0c\u4f46\u901a\u8fc7\u9002\u5f53\u677e\u5f1b\u548c\u4ee3\u4ef7\u8bbe\u8ba1\uff0c\u6700\u4f18\u89e3\u81ea\u7136\u4f1a\u8d34\u5728\u539f\u6765\u7684\u7269\u7406\u8fb9\u754c\u4e0a\uff0c\u800c\u4e0d\u4f1a\u5229\u7528\u677e\u5f1b\u51fa\u6765\u7684\u201c\u975e\u6cd5\u533a\u57df\u201d\uff0c\u6240\u4ee5\u677e\u5f1b\u5e76\u4e0d\u4f1a\u5f15\u5165\u5047\u89e3\u3002<\/p>\n\n\n\n

\u5341\u3001\u4e00\u53e5\u8bdd\u538b\u7f29\u603b\u7ed3<\/h4>\n\n\n\n

\u706b\u7bad\u52a9\u63a8\u5668\u56de\u6536\u964d\u843d\uff0c\u4ece\u6570\u5b66\u4e0a\u770b\u662f\u4e00\u4e2a\u201c\u5728\u590d\u6742\u7ea6\u675f\u4e0b\u6c42\u71c3\u6599\u6700\u4f18\u3001\u5b89\u5168\u6700\u4f18\u8f68\u8ff9\u201d\u7684\u95ee\u9898\uff0c\u5176\u4e2d\u8bb8\u591a\u7ea6\u675f\u5929\u751f\u662f\u975e\u51f8\u7684\uff08\u6700\u5c0f\u63a8\u529b\u3001\u63a8\u529b\u65b9\u5411\u9650\u5236\u3001\u516d\u81ea\u7531\u5ea6\u975e\u7ebf\u6027\u52a8\u529b\u5b66\u7b49\uff09\u3002\u901a\u8fc7\u65e0\u635f\u51f8\u5316\u548c\u9012\u63a8\u51f8\u5316\u7b49\u65b9\u6cd5\uff0c\u53ef\u4ee5\u628a\u8fd9\u7c7b\u975e\u51f8\u6700\u4f18\u63a7\u5236\u95ee\u9898\u8f6c\u5316\u4e3a\u4e00\u7cfb\u5217\u51f8\u4f18\u5316\u95ee\u9898\uff0c\u518d\u7528\u4e13\u95e8\u5b9a\u5236\u7684\u51f8\u4f18\u5316\u6c42\u89e3\u5668\u5728\u673a\u8f7d\u8ba1\u7b97\u673a\u4e0a\u5b9e\u65f6\u6c42\u89e3\uff0c\u4e3a\u6bcf\u4e2a\u63a7\u5236\u5468\u671f\u751f\u6210\u201c\u6700\u4f73\u4e0b\u843d\u8def\u5f84\u201d\u548c\u63a8\u529b\u6307\u4ee4\uff0c\u8fd9\u6784\u6210\u4e86\u730e\u9e70 9 \u7b49\u73b0\u4ee3\u53ef\u56de\u6536\u706b\u7bad\u5b9e\u73b0\u9ad8\u7cbe\u5ea6\u81ea\u52a8\u7740\u9646\u7684\u91cd\u8981\u6570\u5b66\u57fa\u7840\u4e4b\u4e00\u3002<\/p>\n\n\n\n

\n\n\n\n

\u4e00\u3001\u603b\u4f53\u601d\u8def\uff1a\u6211\u4eec\u5230\u5e95\u60f3\u5e72\u4ec0\u4e48\uff1f<\/h4>\n\n\n\n

\u60f3\u8c61\u4e00\u4e2a\u6781\u7b80\u7248\u201c\u730e\u9e70 9\u201d\u7ad6\u76f4\u843d\u5730\u95ee\u9898\uff1a<\/p>\n\n\n\n

\u00b7 \u53ea\u6709\u4e00\u4e2a\u7ef4\u5ea6\uff1a\u7ad6\u76f4\u9ad8\u5ea6 h(t)\uff1b<\/p>\n\n\n\n

\u00b7 \u63a8\u529b\u53ea\u80fd\u671d\u4e0a\uff1b<\/p>\n\n\n\n

\u00b7 \u6211\u4eec\u5e0c\u671b\u4ece\u521d\u59cb\u9ad8\u5ea6 h\u2080\u3001\u5411\u4e0b\u7684\u901f\u5ea6 v\u2080<0 \u5f00\u59cb\uff0c\u5728\u7ed9\u5b9a\u65f6\u95f4 T_f \u65f6\u523b\u201c\u6b63\u597d\u5728\u5730\u9762\u3001\u9ad8\u5ea6\u4e3a 0\uff0c\u901f\u5ea6\u4e3a 0\uff08\u6216\u5f88\u5c0f\uff09\u201d\uff1b<\/p>\n\n\n\n

\u00b7 \u540c\u65f6\u5c3d\u53ef\u80fd\u5c11\u70e7\u71c3\u6599\u3002<\/p>\n\n\n\n

\u4ece\u63a7\u5236\u89d2\u5ea6\u770b\uff0c\u8fd9\u5c31\u662f\u4e00\u4e2a\u201c\u5982\u4f55\u5728\u6709\u9650\u65f6\u95f4\u5185\u5239\u4f4f\u5e76\u6070\u597d\u843d\u5730\u201d\u7684\u6700\u4f18\u63a7\u5236\u95ee\u9898\u3002\u4f46\u771f\u6b63\u9ebb\u70e6\u7684\u662f\u2014\u2014\u73b0\u5b9e\u4e2d\u7684\u53d1\u52a8\u673a\u6709\u5f88\u591a\u7ea6\u675f\uff0c\u6bd4\u5982\u6700\u5c0f\u63a8\u529b\u3001\u6700\u5927\u63a8\u529b\u3001\u63a8\u529b\u65b9\u5411\u9650\u5236\u7b49\u3002\u8fd9\u91cc\u6211\u4eec\u5148\u805a\u7126\u5728\u201c\u6700\u5c0f\u63a8\u529b\u201d\u8fd9\u4e00\u70b9\u3002<\/p>\n\n\n\n

\u73b0\u5b9e\u53d1\u52a8\u673a\u901a\u5e38\u662f\uff1a\u8981\u4e48\u5173\u6389\uff0c\u8981\u4e48\u63a8\u529b\u5728 [T_min, T_max] \u533a\u95f4\u5185\u5de5\u4f5c\u3002\u4e8e\u662f\uff0c\u63a8\u529b\u7684\u53ef\u884c\u96c6\u5408\u662f\uff1a<\/p>\n\n\n\n

    U = {0} \u222a [T_min, T_max],  0 < T_min < T_max<\/p>\n\n\n\n

\u4e5f\u5c31\u662f\u8bf4\uff0c(0, T_min) \u8fd9\u6bb5\u88ab\u201c\u6316\u7a7a\u201d\u4e86\uff0c\u8fd9\u4e2a\u96c6\u5408\u672c\u8eab\u662f\u975e\u51f8\u7684\uff0c\u8fd9\u4e5f\u662f\u6211\u4eec\u9700\u8981\u201c\u51f8\u5316\u201d\u7684\u6839\u672c\u539f\u56e0\u3002<\/p>\n\n\n\n

\u4e8c\u3001\u4e00\u7ef4\u5782\u76f4\u964d\u843d\u7684\u73a9\u5177\u52a8\u529b\u5b66\u6a21\u578b<\/h4>\n\n\n\n

\u5148\u5199\u4e00\u4e2a\u6781\u7b80\u52a8\u529b\u5b66\uff08\u6682\u4e0d\u8003\u8651\u8d28\u91cf\u53d8\u5316\uff09\uff1a<\/p>\n\n\n\n

\u00b7 \u9ad8\u5ea6 h(t)\uff1a\u5411\u4e0a\u4e3a\u6b63\uff1b<\/p>\n\n\n\n

\u00b7 \u901f\u5ea6 v(t)\uff1a\u5411\u4e0a\u4e3a\u6b63\uff1b<\/p>\n\n\n\n

\u00b7 \u63a7\u5236\u91cf\u4e3a\u63a8\u529b T(t)\uff0c\u5411\u4e0a\uff1b<\/p>\n\n\n\n

\u52a8\u529b\u5b66\u65b9\u7a0b\u53ef\u4ee5\u5199\u6210\uff1a<\/p>\n\n\n\n

    \u1e8b\u2081 = v(t)<\/p>\n\n\n\n

    \u1e8b\u2082 = T(t)\/m \u2212 g<\/p>\n\n\n\n

\u5176\u4e2d m \u6682\u65f6\u770b\u4f5c\u5e38\u6570\uff0cg \u4e3a\u91cd\u529b\u52a0\u901f\u5ea6\uff0c\u5411\u4e0b\uff08\u6240\u4ee5\u662f \u201c\u2212g\u201d\uff09\u3002<\/p>\n\n\n\n

\u8fb9\u754c\u6761\u4ef6\u4e3a\uff1a<\/p>\n\n\n\n

\u00b7 \u521d\u59cb\uff1ah(0) = h\u2080,  v(0) = v\u2080\uff1b<\/p>\n\n\n\n

\u00b7 \u7ec8\u7aef\uff1ah(T_f) = 0, v(T_f) = 0\uff08\u7406\u60f3\u8f6f\u7740\u9646\uff09\u3002<\/p>\n\n\n\n

\u53d1\u52a8\u673a\u63a8\u529b\u7ea6\u675f\u4e3a\uff1a<\/p>\n\n\n\n

\u00b7 \u8981\u4e48\u5173\u673a\uff1aT(t) = 0\uff1b<\/p>\n\n\n\n

\u00b7 \u8981\u4e48\u5f00\u673a\uff1aT(t) \u2208 [T_min, T_max]\u3002<\/p>\n\n\n\n

\u76ee\u6807\u51fd\u6570\u53ef\u7b80\u5355\u8bbe\u4e3a\uff1a<\/p>\n\n\n\n

    min \u222b\u2080^{T_f} T(t) dt<\/p>\n\n\n\n

\u53ef\u4ee5\u7406\u89e3\u4e3a\u603b\u63a8\u529b\u79ef\u5206\uff08\u603b\u51b2\u91cf\uff09\uff0c\u4e0e\u71c3\u6599\u6d88\u8017\u7ebf\u6027\u76f8\u5173\u3002<\/p>\n\n\n\n

\u4e09\u3001\u4e3a\u4ec0\u4e48\u8fd9\u4e2a\u95ee\u9898\u662f\u201c\u975e\u51f8\u201d\u7684\uff1f<\/h4>\n\n\n\n

\u4ece\u51f8\u4f18\u5316\u89c6\u89d2\u770b\uff0c\u8981\u6ee1\u8db3\u4e24\u70b9\uff1a<\/p>\n\n\n\n

1\uff09\u51b3\u7b56\u53d8\u91cf\u7684\u53ef\u884c\u96c6\u5408\u8981\u662f\u51f8\u96c6\uff1b<\/p>\n\n\n\n

2\uff09\u76ee\u6807\u51fd\u6570\u548c\u7ea6\u675f\u5728\u8be5\u96c6\u5408\u4e0a\u662f\u51f8\u7684\uff08\u76ee\u6807\u51f8\u3001\u7b49\u5f0f\u7ea6\u675f\u4eff\u5c04\u3001\u4e0d\u7b49\u5f0f\u7ea6\u675f\u5b9a\u4e49\u51f8\u96c6\uff09\u3002<\/p>\n\n\n\n

\u5728\u8fd9\u91cc\uff0c\u51b3\u7b56\u53d8\u91cf\u662f\u63a7\u5236\u51fd\u6570 T(t) \u7684\u8f68\u8ff9\uff1a<\/p>\n\n\n\n

\u00b7 \u76ee\u6807 \u222b T(t) dt \u662f\u7ebf\u6027\u7684\uff0c\u56e0\u6b64\u662f\u51f8\u7684\uff1b<\/p>\n\n\n\n

\u00b7 \u52a8\u529b\u5b66\u7ea6\u675f\u662f\u7ebf\u6027\u7684\u5fae\u5206\u65b9\u7a0b\uff08\u79bb\u6563\u540e\u662f\u7ebf\u6027\u7b49\u5f0f\uff09\uff1b<\/p>\n\n\n\n

\u00b7 T \u2264 T_max\u3001T \u2265 0 \u8fd9\u7c7b\u7ea6\u675f\u90fd\u662f\u51f8\u7684\uff08\u7ebf\u6027\u4e0d\u7b49\u5f0f\uff09\uff1b<\/p>\n\n\n\n

\u00b7 \u4f46\u662f\u6838\u5fc3\u95ee\u9898\u5728\u4e8e\uff1aT(t) \u2208 {0} \u222a [T_min, T_max]\uff0c\u8fd9\u662f\u201c\u4e00\u4e2a\u70b9 + \u4e00\u4e2a\u533a\u95f4\u201d\u7684\u5e76\u96c6\uff0c\u660e\u663e\u4e0d\u662f\u51f8\u96c6\u3002<\/p>\n\n\n\n

\u56e0\u6b64\uff0c\u5373\u4f7f\u52a8\u529b\u5b66\u662f\u7ebf\u6027\u7684\u3001\u76ee\u6807\u662f\u7ebf\u6027\u7684\uff0c\u53ea\u8981\u63a7\u5236\u57df\u662f\u8fd9\u6837\u7684\u5e76\u96c6\uff0c\u8fd9\u4e2a\u6700\u4f18\u63a7\u5236\u95ee\u9898\u4ecd\u7136\u662f\u975e\u51f8\u7684\u3002<\/p>\n\n\n\n

\u56db\u3001\u7b2c\u4e00\u6b65\uff1a\u65f6\u95f4\u79bb\u6563\u5316\uff0c\u628a\u6700\u4f18\u63a7\u5236\u53d8\u6210\u6709\u9650\u7ef4\u4f18\u5316<\/h4>\n\n\n\n

\u4e3a\u4e86\u7528\u901a\u7528\u4f18\u5316\u5668\uff08\u5305\u62ec\u51f8\u4f18\u5316\u5668\uff09\uff0c\u6211\u4eec\u901a\u5e38\u628a\u65f6\u95f4\u533a\u95f4 [0, T_f] \u79bb\u6563\u5316\uff1a<\/p>\n\n\n\n

\u00b7 \u628a [0, T_f] \u5207\u6210 N \u4e2a\u7b49\u957f\u65f6\u95f4\u6b65\uff0c\u6b65\u957f \u0394t = T_f \/ N\uff1b<\/p>\n\n\n\n

\u00b7 \u7528\u79bb\u6563\u65f6\u523b k = 0,1,\u2026,N \u8868\u793a\u3002<\/p>\n\n\n\n

\u5b9a\u4e49\u79bb\u6563\u53d8\u91cf\uff1a<\/p>\n\n\n\n

\u00b7 \u9ad8\u5ea6 h_k \u2248 h(k\u0394t)\uff1b<\/p>\n\n\n\n

\u00b7 \u901f\u5ea6 v_k \u2248 v(k\u0394t)\uff1b<\/p>\n\n\n\n

\u00b7 \u63a8\u529b T_k \u2248 T(k\u0394t)\u3002<\/p>\n\n\n\n

\u7528\u524d\u5411\u6b27\u62c9\u6cd5\u79bb\u6563\u52a8\u529b\u5b66\uff1a<\/p>\n\n\n\n

    h_{k+1} = h_k + \u0394t \u00b7 v_k<\/p>\n\n\n\n

    v_{k+1} = v_k + \u0394t \u00b7 (T_k\/m \u2212 g)<\/p>\n\n\n\n

\u8fb9\u754c\u6761\u4ef6\uff1a<\/p>\n\n\n\n

\u00b7 h_0 = h\u2080, v_0 = v\u2080\uff1b<\/p>\n\n\n\n

\u00b7 h_N = 0, v_N = 0\u3002<\/p>\n\n\n\n

\u79bb\u6563\u540e\u63a8\u529b\u7ea6\u675f\uff1a<\/p>\n\n\n\n

    T_k \u2208 {0} \u222a [T_min, T_max],  k = 0,\u2026,N\u22121<\/p>\n\n\n\n

\u76ee\u6807\u51fd\u6570\u79bb\u6563\u4e3a\uff1a<\/p>\n\n\n\n

    min \u03a3_{k=0}^{N\u22121} T_k \u0394t<\/p>\n\n\n\n

\u5982\u679c\u6211\u4eec\u6682\u65f6\u5ffd\u7565\u201c\u6700\u5c0f\u63a8\u529b\u201d\u548c\u201c\u5f00\u5173\u7ed3\u6784\u201d\uff0c\u53ea\u4fdd\u7559 0 \u2264 T_k \u2264 T_max\uff0c\u90a3\u4e48\u6574\u4e2a\u95ee\u9898\u5c31\u53d8\u6210\u4e86\u4e00\u4e2a\u7ebf\u6027\u89c4\u5212\uff08LP\uff09\uff1a<\/p>\n\n\n\n

\u00b7 \u76ee\u6807 \u03a3 T_k \u0394t \u7ebf\u6027\uff1b<\/p>\n\n\n\n

\u00b7 \u52a8\u529b\u5b66\u7ea6\u675f\u7ebf\u6027\u7b49\u5f0f\uff1b<\/p>\n\n\n\n

\u00b7 box \u7ea6\u675f\u7ebf\u6027\u4e0d\u7b49\u5f0f\uff1b<\/p>\n\n\n\n

\u00b7 \u7ec8\u7aef\u7ea6\u675f\u7ebf\u6027\u7b49\u5f0f\u3002<\/p>\n\n\n\n

\u8fd9\u662f\u5b8c\u5168\u51f8\u7684\u3002\u5173\u952e\u95ee\u9898\u662f\uff1a\u80fd\u4e0d\u80fd\u628a\u539f\u6765\u7684\u975e\u51f8\u96c6\u5408 {0} \u222a [T_min, T_max] \u677e\u5f1b\u6210 [0, T_max]\uff0c\u540c\u65f6\u4e0d\u4e22\u5931\u6700\u4f18\u89e3\uff1f<\/p>\n\n\n\n

\u4e94\u3001\u201c\u65e0\u635f\u51f8\u5316\u201d\uff1a\u628a\u975e\u51f8\u63a8\u529b\u96c6\u5408\u677e\u5f1b\u4e3a\u51f8\u96c6\u5408<\/h4>\n\n\n\n

\u6211\u4eec\u8003\u8651\u5982\u4e0b\u201c\u677e\u5f1b\u201d\uff1a<\/p>\n\n\n\n

\u00b7 \u539f\u7ea6\u675f\uff1aT_k \u2208 {0} \u222a [T_min, T_max]\uff1b<\/p>\n\n\n\n

\u00b7 \u677e\u5f1b\u7ea6\u675f\uff1a0 \u2264 T_k \u2264 T_max\u3002<\/p>\n\n\n\n

\u76f4\u89c9\u4e0a\uff0c\u677e\u5f1b\u540e\u53ef\u884c\u96c6\u5408\u66f4\u5927\uff0c\u6240\u4ee5\u677e\u5f1b\u95ee\u9898\u7684\u6700\u4f18\u503c\u4e0d\u5927\u4e8e\u539f\u95ee\u9898\u7684\u6700\u4f18\u503c\u3002\u5173\u952e\u662f\u8981\u8bc1\u660e\uff1a\u5728\u677e\u5f1b\u95ee\u9898\u7684\u6700\u4f18\u89e3\u4e2d\uff0c\u4e0d\u4f1a\u51fa\u73b0 0 < T_k < T_min \u7684\u60c5\u51b5\uff0c\u4e5f\u5c31\u662f\u8bf4\uff0c\u6700\u4f18\u89e3\u4e0d\u4f1a\u7528\u5230\u90a3\u4e00\u6bb5\u88ab\u201c\u65b0\u653e\u5f00\u7684\u533a\u57df\u201d\u3002<\/p>\n\n\n\n

\u7c7b\u6bd4\u73b0\u5b9e\u751f\u6d3b\uff1a\u5728\u201c\u6700\u5c0f\u5316\u71c3\u6599\u201d\u7684\u524d\u63d0\u4e0b\uff0c\u4e0e\u5176\u4e00\u76f4\u534a\u8e29\u6cb9\u95e8\uff0c\u4e0d\u5982\u5728\u8be5\u5239\u8f66\u7684\u65f6\u5019\u8e29\u4e00\u811a\u5b9e\u5728\u7684\u3002\u5728\u672c\u95ee\u9898\u7ed3\u6784\u4e0b\uff0c\u53ef\u4ee5\u6784\u9020\u8bc1\u660e\uff1a\u4efb\u4f55\u51fa\u73b0\u5728 (0, T_min) \u533a\u95f4\u5185\u7684\u63a8\u529b\u4f7f\u7528\uff0c\u90fd\u53ef\u4ee5\u7528\u201c\u5728\u67d0\u4e9b\u65f6\u523b\u5173\u673a + \u5728\u67d0\u4e9b\u65f6\u523b\u7528\u4e0d\u5c0f\u4e8e T_min \u7684\u63a8\u529b\u201d\u66ff\u4ee3\uff0c\u4e14\u4e0d\u589e\u52a0\u603b\u63a8\u529b\u79ef\u5206\u3001\u4e0d\u7834\u574f\u7ec8\u7aef\u6761\u4ef6\u3002<\/p>\n\n\n\n

\u66f4\u6b63\u5f0f\u5730\u8bf4\uff1a\u5728\u7ebf\u6027\u52a8\u529b\u5b66 + \u7ebf\u6027\u76ee\u6807\u4e0b\uff0c\u7cfb\u7edf\u5bf9\u63a8\u529b\u5e8f\u5217\u7684\u54cd\u5e94\u5177\u6709\u67d0\u79cd\u201c\u7ebf\u6027\u53ef\u52a0\u6027\u201d\uff0c\u4f7f\u5f97\u5c0f\u63a8\u529b\u957f\u65f6\u95f4\u4f5c\u7528\u53ef\u4ee5\u88ab\u7b49\u6548\u4e3a\u5927\u63a8\u529b\u77ed\u65f6\u95f4\u4f5c\u7528\u3002\u901a\u8fc7\u4e00\u7c7b\u201c\u6324\u538b\u5f0f\u201d\u7684\u6784\u9020\uff0c\u53ef\u4ee5\u8bc1\u660e\u677e\u5f1b\u95ee\u9898\u7684\u6700\u4f18\u89e3\u5b8c\u5168\u53ef\u4ee5\u9009\u62e9 T_k \u2208 {0} \u222a [T_min, T_max]\uff0c\u4ece\u800c\u677e\u5f1b\u662f\u65e0\u635f\u7684\uff08lossless\uff09\u3002<\/p>\n\n\n\n

\u516d\u3001\u5f15\u5165\u53d1\u52a8\u673a\u6a21\u5f0f\u53d8\u91cf\uff1a\u4ece\u201c\u534a\u8fde\u7eed\u8f93\u5165\u201d\u5230\u51f8\u4f18\u5316<\/h4>\n\n\n\n

\u5982\u679c\u8fdb\u4e00\u6b65\u5f62\u5f0f\u5316\uff0c\u53ef\u4ee5\u5728\u79bb\u6563\u95ee\u9898\u4e2d\u5f15\u5165\u4e00\u4e2a\u201c\u53d1\u52a8\u673a\u5f00\u5173\u53d8\u91cf\u201d \u03c3_k\uff1a<\/p>\n\n\n\n

\u00b7 \u03c3_k = 0\uff1a\u53d1\u52a8\u673a\u5173\u673a\uff1b<\/p>\n\n\n\n

\u00b7 \u03c3_k = 1\uff1a\u53d1\u52a8\u673a\u5f00\u673a\u3002<\/p>\n\n\n\n

\u539f\u6765\u7684\u63a8\u529b\u7ea6\u675f\u53ef\u4ee5\u7b49\u4ef7\u5199\u6210\uff1a<\/p>\n\n\n\n

    \u03c3_k \u2208 {0,1}<\/p>\n\n\n\n

    T_k \u2265 \u03c3_k T_min<\/p>\n\n\n\n

    T_k \u2264 \u03c3_k T_max<\/p>\n\n\n\n

\u8fd9\u65f6\u95ee\u9898\u53d8\u6210\u4e86\u5e26\u4e8c\u5143\u53d8\u91cf\u7684\u6df7\u5408\u6574\u6570\u89c4\u5212\uff08MILP\uff09\uff0c\u66f4\u52a0\u975e\u51f8\u3002<\/p>\n\n\n\n

\u51f8\u5316\u7684\u5173\u952e\u4e00\u6b65\u662f\uff1a\u628a \u03c3_k \u2208 {0,1} \u653e\u677e\u4e3a\u8fde\u7eed\u53d8\u91cf 0 \u2264 \u03c3_k \u2264 1\u3002\u8fd9\u6837\u914d\u5408 T_k \u2265 \u03c3_k T_min\u3001T_k \u2264 \u03c3_k T_max\uff0c\u6240\u6709\u7ea6\u675f\u90fd\u662f\u7ebf\u6027\u7684\uff0c\u95ee\u9898\u53d8\u6210\u7eaf\u51f8\uff08\u751a\u81f3\u662f LP \u6216 QP\uff09\u3002\u968f\u540e\uff0c\u518d\u901a\u8fc7\u4e0e\u524d\u4e00\u8282\u7c7b\u4f3c\u7684\u7ed3\u6784\u6027\u8bc1\u660e\uff0c\u8bf4\u660e\u5728\u6700\u4f18\u89e3\u4e2d \u03c3_k \u4f1a\u81ea\u52a8\u8dd1\u5230 0 \u6216 1\uff0c\u56e0\u6b64\u8fd9\u4e00\u653e\u677e\u4e5f\u662f\u65e0\u635f\u7684\u3002<\/p>\n\n\n\n

\u4e03\u3001\u628a\u5b83\u6b63\u5f0f\u5199\u6210\u4e00\u4e2a\u201c\u51f8\u4f18\u5316\u95ee\u9898\u201d\u7684\u6807\u51c6\u5f62\u5f0f<\/h4>\n\n\n\n

\u7ecf\u8fc7\u4e0a\u8ff0\u79bb\u6563\u5316\u548c\u677e\u5f1b\uff0c\u53ef\u4ee5\u5f97\u5230\u4e00\u4e2a\u5178\u578b\u7684\u51f8\u4f18\u5316\u95ee\u9898\uff0c\u53d8\u91cf\u5305\u62ec\uff1a<\/p>\n\n\n\n

\u00b7 \u72b6\u6001\uff1a{h_k, v_k}, k = 0,\u2026,N\uff1b<\/p>\n\n\n\n

\u00b7 \u63a7\u5236\uff1a{T_k, \u03c3_k}, k = 0,\u2026,N\u22121\u3002<\/p>\n\n\n\n

\u76ee\u6807\u51fd\u6570\uff1a<\/p>\n\n\n\n

    min \u03a3_{k=0}^{N\u22121} T_k \u0394t<\/p>\n\n\n\n

\u7ea6\u675f\u5305\u62ec\uff1a<\/p>\n\n\n\n

1\uff09\u521d\u503c\u7ea6\u675f\uff1ah_0 = h\u2080, v_0 = v\u2080\uff1b<\/p>\n\n\n\n

2\uff09\u7ebf\u6027\u52a8\u529b\u5b66\u7ea6\u675f\uff1a<\/p>\n\n\n\n

    h_{k+1} = h_k + \u0394t \u00b7 v_k<\/p>\n\n\n\n

    v_{k+1} = v_k + \u0394t \u00b7 (T_k\/m \u2212 g)<\/p>\n\n\n\n

3\uff09\u7ec8\u7aef\u7ea6\u675f\uff1ah_N = 0, v_N = 0\uff1b<\/p>\n\n\n\n

4\uff09\u63a8\u529b\u548c\u5f00\u5173\u53d8\u91cf\u7ea6\u675f\uff08\u677e\u5f1b\u540e\uff09\uff1a<\/p>\n\n\n\n

    0 \u2264 \u03c3_k \u2264 1<\/p>\n\n\n\n

    T_k \u2265 \u03c3_k T_min<\/p>\n\n\n\n

    T_k \u2264 \u03c3_k T_max<\/p>\n\n\n\n

    T_k \u2265 0<\/p>\n\n\n\n

\u8fd9\u662f\u4e00\u4e2a\u6807\u51c6\u7684\u51f8\u4f18\u5316\u95ee\u9898\uff08\u901a\u5e38\u662f LP \u6216 QP\uff09\u3002\u7ed3\u5408\u201c\u65e0\u635f\u51f8\u5316\u201d\u7684\u8bc1\u660e\uff0c\u6211\u4eec\u77e5\u9053\u6700\u4f18\u89e3\u81ea\u52a8\u6ee1\u8db3 \u03c3_k \u2208 {0,1}\uff0c\u4ece\u800c\u4e5f\u6ee1\u8db3\u539f\u59cb\u7269\u7406\u63a8\u529b\u7ea6\u675f\u3002<\/p>\n\n\n\n

\u516b\u3001\u56de\u5230\u201c\u706b\u7bad\u52a9\u63a8\u5668\u56de\u6536\u201d\u7684\u73b0\u5b9e\u610f\u4e49<\/h4>\n\n\n\n

\u4e0a\u9762\u8fd9\u4e00\u5927\u5708\uff0c\u662f\u5728\u4e00\u4e2a 1D \u73a9\u5177\u6a21\u578b\u91cc\u6f14\u793a\u4e86\u51e0\u4ef6\u4e8b\uff1a<\/p>\n\n\n\n

1\uff09\u539f\u59cb\u95ee\u9898\u56e0\u4e3a\u63a8\u529b\u96c6\u5408 {0} \u222a [T_min, T_max] \u7684\u7f18\u6545\u662f\u975e\u51f8\u7684\uff1b<\/p>\n\n\n\n

2\uff09\u901a\u8fc7\u5f15\u5165\u8f85\u52a9\u53d8\u91cf\u548c\u677e\u5f1b\uff0c\u53ef\u4ee5\u628a\u5b83\u6539\u5199\u6210\u4e00\u4e2a\u51f8\u4f18\u5316\u95ee\u9898\uff1b<\/p>\n\n\n\n

3\uff09\u5728\u9002\u5f53\u8bbe\u8ba1\u76ee\u6807\u4e0e\u7ea6\u675f\u7684\u60c5\u51b5\u4e0b\uff0c\u677e\u5f1b\u662f\u65e0\u635f\u7684\uff0c\u51f8\u95ee\u9898\u7684\u6700\u4f18\u89e3\u81ea\u7136\u6ee1\u8db3\u539f\u6765\u7684\u7269\u7406\u7ea6\u675f\uff1b<\/p>\n\n\n\n

4\uff09\u4e00\u65e6\u53d8\u6210\u51f8\u95ee\u9898\uff0c\u5c31\u53ef\u4ee5\u7528\u6210\u719f\u7684 convex solver\uff08LP\/QP\/SOCP\uff09\u505a\u5b9e\u65f6\u6c42\u89e3\uff1b<\/p>\n\n\n\n

5\uff09\u5728\u771f\u6b63\u7684\u706b\u7bad\u4e2d\uff0c\u8fd9\u4e00\u5957\u901a\u5e38\u88ab\u653e\u5728\u201c\u8f68\u8ff9\u89c4\u5212\/\u5236\u5bfc\u201d\u5c42\uff0c\u4e0b\u9762\u518d\u53e0\u52a0\u59ff\u6001\u63a7\u5236\u548c\u6267\u884c\u673a\u6784\u63a7\u5236\u7b49\u6a21\u5757\u3002<\/p>\n\n\n\n

\u5982\u679c\u628a\u6a21\u578b\u4ece 1D \u6269\u5c55\u5230 2D\/3D\uff0c\u63a8\u529b\u53d8\u6210\u5411\u91cf T\u20d7_k\uff0c\u63a8\u529b\u65b9\u5411\u3001\u504f\u8f6c\u89d2\u7b49\u7ea6\u675f\u53ef\u4ee5\u5199\u6210\u4e8c\u9636\u9525\u7ea6\u675f\uff08\u4f8b\u5982 \u2016T\u22a5\u2016 \u2264 tan(\u03b8_max)\u00b7T_z\uff09\uff0c\u6574\u4e2a\u95ee\u9898\u81ea\u7136\u5347\u7ea7\u4e3a\u4e8c\u9636\u9525\u89c4\u5212\uff08SOCP\uff09\u3002\u8fd9\u5c31\u662f\u5f88\u591a\u706b\u7bad\u56de\u6536\u8bba\u6587\u91cc\u63d0\u5230\u7684\u201c\u57fa\u4e8e SOCP \u7684\u8f68\u8ff9\u4f18\u5316\u201d\u7684\u6765\u5386\u3002<\/p>\n\n\n\n

\u5728\u5de5\u7a0b\u5b9e\u8df5\u4e2d\uff0c\u8fd8\u4f1a\u53e0\u52a0\u975e\u7ebf\u6027\u6c14\u52a8\u529b\u3001\u516d\u81ea\u7531\u5ea6\u521a\u4f53\u52a8\u529b\u5b66\u7b49\uff0c\u8fd9\u65f6\u901a\u5e38\u4f1a\u5728\u8fd9\u4e00\u5957\u201c\u51f8\u5316\u6846\u67b6\u201d\u4e0a\u4f7f\u7528\u8fde\u7eed\u51f8\u5316(successive convexification)\uff0c\u5373\uff1a\u9009\u4e00\u4e2a\u521d\u59cb\u8f68\u8ff9\uff0c\u7ebf\u6027\u5316\u5f97\u5230\u4e00\u4e2a\u51f8\u95ee\u9898\uff0c\u6c42\u89e3\u5f97\u5230\u65b0\u8f68\u8ff9\uff0c\u518d\u56f4\u7ed5\u65b0\u8f68\u8ff9\u7ee7\u7eed\u7ebf\u6027\u5316\u3001\u6c42\u89e3\uff0c\u8fed\u4ee3\u82e5\u5e72\u6b21\u76f4\u5230\u6536\u655b\u3002<\/p>\n\n\n\n

\u603b\u4e4b\uff0c\u51f8\u4f18\u5316\u5728\u706b\u7bad\u52a9\u63a8\u5668\u56de\u6536\u964d\u843d\u4e2d\u7684\u6838\u5fc3\u4f5c\u7528\uff0c\u662f\u628a\u4e00\u4e2a\u539f\u672c\u975e\u51f8\u3001\u96be\u4ee5\u76f4\u63a5\u6c42\u89e3\u7684\u6700\u4f18\u63a7\u5236\u95ee\u9898\uff0c\u901a\u8fc7\u7cbe\u5fc3\u8bbe\u8ba1\u7684\u677e\u5f1b\u548c\u91cd\u5199\uff0c\u53d8\u6210\u4e00\u7cfb\u5217\u53ef\u7531\u53ef\u9760\u3001\u5b9e\u65f6\u7684\u51f8\u4f18\u5316\u6c42\u89e3\u5668\u89e3\u51b3\u7684\u95ee\u9898\uff0c\u4ece\u800c\u5b9e\u73b0\u9ad8\u7cbe\u5ea6\u3001\u53ef\u8bc1\u660e\u3001\u53ef\u5de5\u7a0b\u843d\u5730\u7684\u81ea\u52a8\u7740\u9646\u63a7\u5236\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

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