Upgrade to Pro — share decks privately, control downloads, hide ads and more …

数理最適化に基づく制御

 数理最適化に基づく制御

数理最適化に基づく制御ーモデル予測制御を中心にー

Avatar for MIKIO KUBO

MIKIO KUBO

May 25, 2025
Tweet

More Decks by MIKIO KUBO

Other Decks in Research

Transcript

  1. 2 MIT/Illinois U. Tokyo 2006-2010 2010-2012 2012 2013-2015 2015 2016-

    ERATO U. Canterbury U. Magdeburg NII/Sokendai U. Illinois CREST JST
  2. § § 5 : § : § : § :

    : § : § : PID § :
  3. 11

  4. § § 12 i vs R C L vc +

    _ + _ https://jp.mathworks.com/academia/courseware/mass-spring-damper-systems.html <latexit sha1_base64="fNVoAq7bnV7VURo3ZBbHY5ydDtM=">AAACGXicbZDLSgMxFIYzXmu9jboSN8EiCEKZEa1uhIIgLivYC7RDyWTSNjQzGZIz0jIUH8S1W30Gd+LWlY/gW5i2s9DWHwIf/zknyfn9WHANjvNlLSwuLa+s5tby6xubW9v2zm5Ny0RRVqVSSNXwiWaCR6wKHARrxIqR0Bes7vevx/X6A1Oay+gehjHzQtKNeIdTAsZq2/thKwgkpIPRCW1l0B/gK3zTtgtO0ZkIz4ObQQFlqrTtb3MBTUIWARVE66brxOClRAGngo3yrUSzmNA+6bKmwYiETHvpZIURPjJOgDtSmRMBnri/J1ISaj0MfdMZEujp2drY/Lc2dkBKoWc+AJ1LL+VRnACL6PT9TiIwSDyOCQdcMQpiaIBQxc0KmPaIIhRMmHmTjTubxDzUTotuqXh+d1Yol7KUcugAHaJj5KILVEa3qIKqiKJH9Ixe0Kv1ZL1Z79bHtHXBymb20B9Znz8R7aDV</latexit> m¨ x + c ˙ x + kx = F
  5. 15 m q f b k § : f §

    : q x B u A C x
  6. u § § 22 <latexit sha1_base64="f4jX17UXSt6gAHisS04Eb4EgOp4=">AAACHnicbZDLSgMxFIYz9VbrbdSlIMEitFDKjNTLRii4cVnBXqAtJZNm2tDMZEjOSMvQnQ/i2q0+gztxq4/gW5heFtr6Q+DnO+ckOb8XCa7Bcb6s1Mrq2vpGejOztb2zu2fvH9S0jBVlVSqFVA2PaCZ4yKrAQbBGpBgJPMHq3uBmUq8/MKW5DO9hFLF2QHoh9zklYFDHPm51JSTDMb7Gfm5YiPMF3MLDnJM3YNhxOnbWKTpT4WXjzk0WzVXp2N/mQhoHLAQqiNZN14mgnRAFnAo2zrRizSJCB6THmsaGJGC6nUz3GONTQ7rYl8qcEPCU/p5ISKD1KPBMZ0CgrxdrE/hvbUJASqEXPgD+VTvhYRQDC+nsfT8WGCSeZIW7XDEKYmQMoYqbFTDtE0UomEQzJht3MYllUzsruhfF87tStlyap5RGR+gE5ZCLLlEZ3aIKqiKKHtEzekGv1pP1Zr1bH7PWlDWfOUR/ZH3+AMJUoFQ=</latexit> ˙ x = f(x,

    u), x(0) = x0 <latexit sha1_base64="irofdI0xx9d3x9UAY7QOQIfbkJM=">AAACIXicbVDLSgMxFM3UV62vqks30SJUkDIj9bERCm5EXFSwKtg6ZNJMG5pJhuSOtAxd+yGu3eo3uBN34hf4F6aPhVYPXO7hnHtJ7gliwQ247oeTmZqemZ3LzucWFpeWV/Kra1dGJZqyGlVC6ZuAGCa4ZDXgINhNrBmJAsGug87JwL++Z9pwJS+hF7NGRFqSh5wSsJKf3zwrdn13Bx/jOpfgu3ep7SH0+ufF7m6yg5vg5wtuyR0C/yXemBTQGFU//1VvKppETAIVxJhbz42hkRINnArWz9UTw2JCO6TFbi2VJGKmkQ5P6eNtqzRxqLQtCXio/txISWRMLwrsZESgbSa9gfivN1BAKWEmPgDhUSPlMk6ASTp6P0wEBoUHceEm14yC6FlCqOb2BEzbRBMKNtSczcabTOIvudoreQel/YtyoVIep5RFG2gLFZGHDlEFnaIqqiGKHtATekYvzqPz6rw576PRjDPeWUe/4Hx+A+AUoqw=</latexit> J(x0) = Z 1 0 L(x, u)dt
  7. § A T B § 23 <latexit sha1_base64="TlQ++rRdK9rnxNFCKIjkcssLLCs=">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</latexit> J(x0) =

    Z T 0 kuk2dt, x(T) = xgoal <latexit sha1_base64="S4M4KDQK5QwGzIVe1MJc4O8DQQc=">AAACGHicbVDJSgNBEO1xjXGLehG8NAYhXsKMxOUY8OIxglkgE0JPpydp0tM9dtcIYRI/xLNX/QZv4tWbn+Bf2FkOmvig4PFeFVX1glhwA6775Swtr6yurWc2sptb2zu7ub39mlGJpqxKlVC6ERDDBJesChwEa8SakSgQrB70r8d+/YFpw5W8g0HMWhHpSh5ySsBK7dyhP0wKcOoP26nPZQiDEfYFu8deO5d3i+4EeJF4M5JHM1TauW+/o2gSMQlUEGOanhtDKyUaOBVslPUTw2JC+6TLmpZKEjHTSicfjPCJVTo4VNqWBDxRf0+kJDJmEAW2MyLQM/PeWPzXGyuglDBzB0B41Uq5jBNgkk73h4nAoPA4JdzhmlEQA0sI1dy+gGmPaELBZpm12XjzSSyS2lnRuyie35by5dIspQw6QseogDx0icroBlVQFVH0iJ7RC3p1npw35935mLYuObOZA/QHzucPDsGgSQ==</latexit> ku(t)k1  1 <latexit sha1_base64="1yVbZegkjjEuXl3/OOsvBmpWrOU=">AAACMnicbVDLSsNAFJ3Ud31VXboZLEIFKYn42giCG3GlYFVoaphMJnZwMhNnbsQS+x9+iGu3+gu6E1eCH+G0ZqGtBy4czrmXe+8JU8ENuO6rUxoZHRufmJwqT8/Mzs1XFhbPjMo0ZQ2qhNIXITFMcMkawEGwi1QzkoSCnYfXBz3//JZpw5U8hU7KWgm5kjzmlICVgsrGUe0ucNfwHva5hMC9zE+7OIJ17GP/PqvBmn8f5NaKodPFvmA32AsqVbfu9oGHiVeQKipwHFQ+/UjRLGESqCDGND03hVZONHAqWLfsZ4alhF6TK9a0VJKEmVbe/62LV60S4VhpWxJwX/09kZPEmE4S2s6EQNsMej3xX6+ngFLCDBwA8W4r5zLNgEn6sz/OBAaFe/nhiGtGQXQsIVRz+wKmbaIJBZty2WbjDSYxTM426t52fetks7q/WaQ0iZbRCqohD+2gfXSIjlEDUfSAntAzenEenTfn3fn4aS05xcwS+gPn6xtlAqkM</latexit> J(x0) = Z T 0 dt, ku(t)k1  1 A B <latexit sha1_base64="xh4DQY8SILI18Bt6tNDzsjSfO9E=">AAACFXicbVDLSsNAFJ34rPUVFVduBotQNyWRUt0IBTcuK/QFbQiT6aQdOsmEmRtpCf0O1271G9yJW9d+gn/h9LHQ1gMXDufcy733BIngGhzny1pb39jc2s7t5Hf39g8O7aPjppapoqxBpZCqHRDNBI9ZAzgI1k4UI1EgWCsY3k391iNTmsu4DuOEeRHpxzzklICRfPt0VKxf3uKRn3WBjSDrSyImE98uOCVnBrxK3AUpoAVqvv3d7UmaRiwGKojWHddJwMuIAk4Fm+S7qWYJoUPSZx1DYxIx7WWz8yf4wig9HEplKgY8U39PZCTSehwFpjMiMNDL3lT815sqIKXQSwdAeONlPE5SYDGd7w9TgUHiaUS4xxWjIMaGEKq4eQHTAVGEggkyb7Jxl5NYJc2rklspVR7KhWp5kVIOnaFzVEQuukZVdI9qqIEoytAzekGv1pP1Zr1bH/PWNWsxc4L+wPr8AS4wn1s=</latexit> x(T) = xgoal
  8. u = −Kx K § § 25 <latexit sha1_base64="rVbhlpJGGEpOJPafNOIyL4cPzhI=">AAACHHicbVDLSgMxFM3UV62vqktdBItQUcqMFHVTqLpxWcE+oC0lk6Y1NDMZkjsyZejGD3HtVr/BnbgV/AT/wrSdhbYeuHA4597k3uMGgmuw7S8rtbC4tLySXs2srW9sbmW3d2pahoqyKpVCqoZLNBPcZ1XgIFgjUIx4rmB1d3A99usPTGku/TsYBqztkb7Pe5wSMFInu9/qSoijUQlfRsdX4Qlu4ShvH+ESjjp2J5uzC/YEeJ44CcmhBJVO9ts8R0OP+UAF0brp2AG0Y6KAU8FGmVaoWUDogPRZ01CfeEy348kVI3xolC7uSWXKBzxRf0/ExNN66Lmm0yNwr2e9sfivN1ZASqFnFoDeRTvmfhAC8+n0/14oMEg8Tgp3uWIUxNAQQhU3J2B6TxShYPLMmGyc2STmSe204JwVirfFXLmYpJRGe+gA5ZGDzlEZ3aAKqiKKHtEzekGv1pP1Zr1bH9PWlJXM7KI/sD5/AODin+o=</latexit> ˙

    x = Ax + Bu, x(0) = x0 <latexit sha1_base64="LRQ01vCSX018FAxITnvEQ3ykT6U=">AAACHXicbZDLSgMxFIYz9VbrbdSlCMEiuJAyI/WyLLhx2Yq9QGcomTTThmaSMckIZejKB3HtVp/BnbgVH8G3MNPOQlt/CPx85xxOzh/EjCrtOF9WYWl5ZXWtuF7a2Nza3rF391pKJBKTJhZMyE6AFGGUk6ammpFOLAmKAkbaweg6q7cfiFRU8Ds9jokfoQGnIcVIG9SzDxvQUwnGOJFsTO6hc+rB2xmCDuzZZafiTAUXjZubMshV79nfXl/gJCJcY4aU6rpOrP0USU0xI5OSlygSIzxCA9I1lqOIKD+dnjGBx4b0YSikeVzDKf09kaJIqXEUmM4I6aGar2Xw31pGtBBMzX1Ah1d+SnmcaMLxbH+YMKgFzKKCfSoJ1mxsDMKSmhMgHiKJsDaBlkw27nwSi6Z1VnEvKueNarlWzVMqggNwBE6ACy5BDdyAOmgCDB7BM3gBr9aT9Wa9Wx+z1oKVz+yDP7I+fwDRa6EJ</latexit> Q < 0, R 0 <latexit sha1_base64="vylmWdfSGF33Hz+S/wtQdZdQLtc=">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</latexit> J(x0) = Z 1 0 x>Qx + u>Rudt
  9. Hamiltonian λ 26 <latexit sha1_base64="PFgDpe/+lWggQIJ2IfFL6t2cLUc=">AAACMHicbZDLTgIxFIY7eEO8oS7dNBITDAmZMXjZmKBuWIKRSwJIOqUDDZ3ppD1jIITX8EFcu9Vn0JVxqU9huSwU/JMmX/5zTtvzu6HgGmz73YotLa+srsXXExubW9s7yd29ipaRoqxMpZCq5hLNBA9YGTgIVgsVI74rWNXt3Yzr1QemNJfBHQxC1vRJJ+AepwSM1UraBXyJ+/cNkCEu9XEGR1O+xRHONIS5qE2mTvqqn7mOjlvJlJ21J8KL4MwghWYqtpJfjbakkc8CoIJoXXfsEJpDooBTwUaJRqRZSGiPdFjdYEB8ppvDyWYjfGScNvakMicAPHF/TwyJr/XAd02nT6Cr52tj89/a2AEphZ77AHgXzSEPwghYQKfve5HAIPE4PdzmilEQAwOEKm5WwLRLFKFgMk6YbJz5JBahcpJ1zrKnpVwqn5ulFEcH6BClkYPOUR4VUBGVEUWP6Bm9oFfryXqzPqzPaWvMms3soz+yvn8A4zqnqw==</latexit> H = x > Qx

    + u > Ru + >(Ax + Bu) <latexit sha1_base64="rXG2ER3ndaUYkW2LnY2o+CfRJ/w=">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</latexit> ˙ x = ✓ @H @ ◆> = Ax + Bu, x(0) = x0 <latexit sha1_base64="jZjVesepblvHkIDK3GK+dLJvAsE=">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</latexit> ˙ = ✓ @H @x ◆> = Qx + A > <latexit sha1_base64="/sxurwIfzcoPE9zPI7LtvWZBYrk=">AAACGHicbVDLSgMxFM34rPU16kZwEyyCm5YZqY+NUHTjsop9QKctmTRtQzOTIbkjlKF+iGu3+g3uxK07P8G/MNN2oa0HAodz7uHmHj8SXIPjfFkLi0vLK6uZtez6xubWtr2zW9UyVpRVqBRS1X2imeAhqwAHweqRYiTwBav5g+vUrz0wpbkM72EYsWZAeiHvckrASG17P77E+btWkndHVy0PZIQ9YdId0rZzTsEZA88Td0pyaIpy2/72OpLGAQuBCqJ1w3UiaCZEAaeCjbJerFlE6ID0WMPQkARMN5PxBSN8ZJQO7kplXgh4rP5OJCTQehj4ZjIg0NezXir+66UKSCn0zAege9FMeBjFwEI62d+NBQaJ05ZwhytGQQwNIVRxcwKmfaIIBdNl1nTjzjYxT6onBfescHpbzJWK05Yy6AAdomPkonNUQjeojCqIokf0jF7Qq/VkvVnv1sdkdMGaZvbQH1ifP/Rzn6Q=</latexit> u = R 1B> <latexit sha1_base64="Wm98pOwu6NexLjDTcC7R21V17D4=">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</latexit> 0 = ✓ @H @u ◆> = Ru + B >
  10. x Algebraic Riccati Equation 27 <latexit sha1_base64="NoR2vwyjcjR4VV8/BsIDfTkqNmQ=">AAACCXicbVDLSgMxFM3UV62vqks3wSK4KjNSHxuh4MZlBfuAdiiZTKYNzSRDckcspV/g2q1+gztx61f4Cf6FmXYW2nogcDjnXO7NCRLBDbjul1NYWV1b3yhulra2d3b3yvsHLaNSTVmTKqF0JyCGCS5ZEzgI1kk0I3EgWDsY3WR++4Fpw5W8h3HC/JgMJI84JWClbk/YaEiuceOxX664VXcGvEy8nFRQjka//N0LFU1jJoEKYkzXcxPwJ0QDp4JNS73UsITQERmwrqWSxMz4k9nJU3xilRBHStsnAc/U3xMTEhszjgObjAkMzaKXif96mQJKCbNwAERX/oTLJAUm6Xx/lAoMCme14JBrRkGMLSFUc/sFTIdEEwq2vJLtxltsYpm0zqreRfX8rlap1/KWiugIHaNT5KFLVEe3qIGaiCKFntELenWenDfn3fmYRwtOPnOI/sD5/AFnkpqi</latexit> = Px <latexit

    sha1_base64="T9Ra5EpWXv2krLD5+N48qRdHLvA=">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</latexit> ˙ = P ˙ x = P(Ax + Bu) = P(A BR 1B>P)x <latexit sha1_base64="jZjVesepblvHkIDK3GK+dLJvAsE=">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</latexit> ˙ = ✓ @H @x ◆> = Qx + A > <latexit sha1_base64="wJHzXQ551Tn/IdVfagn60fiHncc=">AAACKHicbVDLTgIxFO3gG1+jLt00EhOMkcwYXxsTwY1LMPJIAEmnFGjsTCftHQOZ8At+iGu3+g3uDFt3/oUFZqHoSW5yes696b3HCwXX4DgjKzU3v7C4tLySXl1b39i0t7YrWkaKsjKVQqqaRzQTPGBl4CBYLVSM+J5gVe/heuxXH5nSXAZ3MAhZ0yfdgHc4JWCklp3FxWz+qHB7Hx+5w8J9A2SIiwf9w1IfH+J88u7jS6dlZ5ycMwH+S9yEZFCCYsv+arQljXwWABVE67rrhNCMiQJOBRumG5FmIaEPpMvqhgbEZ7oZTy4a4n2jtHFHKlMB4In6cyImvtYD3zOdPoGenvXG4r/eWAEphZ5ZADoXzZgHYQQsoNP/O5HAIPE4NdzmilEQA0MIVdycgGmPKELBZJs22bizSfwlleOce5Y7LZ1krk6SlJbRLtpDWeSic3SFblARlRFFT+gFvaI369l6tz6s0bQ1ZSUzO+gXrM9vpm6j4w==</latexit> P(A BR 1B>P)x + Qx + A>Px = 0 <latexit sha1_base64="nuaI6+XOkogO060NlQ4CUnjMXqs=">AAACI3icbVDLSgMxFL3j2/qqunQTLIIglhmpj41g68blVKwK9kEmTW1oZjIkd4Qy9AP8ENdu9RvciRsXfoB/YfoQtHogcDjnXG7uCWIpDLruuzMxOTU9Mzs3n1lYXFpeya6uXRqVaMYrTEmlrwNquBQRr6BAya9jzWkYSH4VdE77/tUd10ao6AK7Ma+F9DYSLcEoWqmRzRXrVVQx8XeIX9z1S+f1dNfrlb7FMiHHrk25eXcA8pd4I5KDEfxG9rPaVCwJeYRMUmNuPDfGWko1CiZ5L1NNDI8p69BbfmNpRENuaungmB7ZskqTtJS2L0IyUH9OpDQ0phsGNhlSbJtxry/+6/UVVEqasQ9g66iWiihOkEdsuL+VSIKK9AsjTaE5Q9m1hDIt7AmEtammDG2tGduNN97EX3K5l/cO8vvlQu6kMGppDjZgE7bBg0M4gTPwoQIM7uERnuDZeXBenFfnbRidcEYz6/ALzscXR7qiKA==</latexit> A>P + PA PBR 1B>P + Q = 0 <latexit sha1_base64="kiImzKzNqas4M/LSCaA4sngC4Nw=">AAACE3icbVDLTgJBEJz1ifhCOXqZSEy8QHYNPi4mRC8e0cgjgYXMDgNMmN3ZzPQaNxs+w7NX/QZvxqsf4Cf4Fw6wBwUr6aRS1Z3uLi8UXINtf1lLyyura+uZjezm1vbObm5vv65lpCirUSmkanpEM8EDVgMOgjVDxYjvCdbwRtcTv/HAlOYyuIc4ZK5PBgHvc0rASN1cPrrExbtOUnTGV502yBBXH7u5gl2yp8CLxElJAaWodnPf7Z6kkc8CoIJo3XLsENyEKOBUsHG2HWkWEjoiA9YyNCA+024yPX6Mj4zSw32pTAWAp+rviYT4Wse+Zzp9AkM9703Ef72JAlIKPXcA9C/chAdhBCygs/39SGCQeBIQ7nHFKIjYEEIVNy9gOiSKUDAxZk02znwSi6R+UnLOSqe35UKlnKaUQQfoEB0jB52jCrpBVVRDFMXoGb2gV+vJerPerY9Z65KVzuTRH1ifP/JrnX0=</latexit> u = R 1B>Px K
  11. Lyapunov K 28 <latexit sha1_base64="Ayr7p2n2TMS1XGqxTH64dOc7OfA=">AAACKHicbZDLSgMxFIYzXmu9VV26CRahMlBmpF42QtWN0E0r9gJtLZk0bUMzkyE5I5Shr+CDuHarz+BOunXnW5heFtr6Q+DnO+dwcn4vFFyD44yspeWV1bX1xEZyc2t7Zze1t1/RMlKUlakUUtU8opngASsDB8FqoWLE9wSrev3bcb36xJTmMniAQciaPukGvMMpAYNaqUzm2r4pnDw2QIa4iG1cnAK7ZOPClN4X8BV2Wqm0k3UmwovGnZk0mqnYSn032pJGPguACqJ13XVCaMZEAaeCDZONSLOQ0D7psrqxAfGZbsaTi4b42JA27khlXgB4Qn9PxMTXeuB7ptMn0NPztTH8tzYmIKXQcx+AzmUz5kEYAQvodH8nEhgkHqeG21wxCmJgDKGKmxMw7RFFKJhskyYbdz6JRVM5zbrn2bNSLp3PzVJKoEN0hDLIRRcoj+5QEZURRc/oFb2hd+vF+rA+rdG0dcmazRygP7K+fgAspaJo</latexit> (A + BK)>P + P(A

    + BK) + Q + K>RK = 0 <latexit sha1_base64="q83zwfvim9E6+voUPL9NkUqcGwA=">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</latexit> P = Z 1 0 e(A+BK)>t(Q + K>RK)e(A+BK)tdt <latexit sha1_base64="TCtdMoOftKHuZF+sLvyzA7m+n5Q=">AAACDHicbZDLSgMxFIYz9VbrrerSTbAIrsqMeFsW3LisYC/QTksmzbShmWRIzohl6Cu4dqvP4E7c+g4+gm9hpp2Ftv4Q+PjPOZyTP4gFN+C6X05hZXVtfaO4Wdra3tndK+8fNI1KNGUNqoTS7YAYJrhkDeAgWDvWjESBYK1gfJPVWw9MG67kPUxi5kdkKHnIKQFr9R77bq8LKsZ1bLFfrrhVdya8DF4OFZSr3i9/dweKJhGTQAUxpuO5Mfgp0cCpYNNSNzEsJnRMhqxjUZKIGT+dXT3FJ9YZ4FBp+yTgmft7IiWRMZMosJ0RgZFZrGXmv7XMAaWEWTgAwms/5TJOgEk63x8mAoPCWTJ4wDWjICYWCNXcfgHTEdGEgs2vZLPxFpNYhuZZ1busXtydV2rneUpFdISO0Sny0BWqoVtURw1EkUbP6AW9Ok/Om/PufMxbC04+c4j+yPn8ASWAm4k=</latexit> x> 0 Px0 <latexit sha1_base64="yO2Bi3EgaBV0xeh9gxcLdueDBN4=">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</latexit> A>P + PA PBR 1B>P + Q + (R 1B>P + K)>R(R 1B>P + K) = 0 <latexit sha1_base64="kiImzKzNqas4M/LSCaA4sngC4Nw=">AAACE3icbVDLTgJBEJz1ifhCOXqZSEy8QHYNPi4mRC8e0cgjgYXMDgNMmN3ZzPQaNxs+w7NX/QZvxqsf4Cf4Fw6wBwUr6aRS1Z3uLi8UXINtf1lLyyura+uZjezm1vbObm5vv65lpCirUSmkanpEM8EDVgMOgjVDxYjvCdbwRtcTv/HAlOYyuIc4ZK5PBgHvc0rASN1cPrrExbtOUnTGV502yBBXH7u5gl2yp8CLxElJAaWodnPf7Z6kkc8CoIJo3XLsENyEKOBUsHG2HWkWEjoiA9YyNCA+024yPX6Mj4zSw32pTAWAp+rviYT4Wse+Zzp9AkM9703Ef72JAlIKPXcA9C/chAdhBCygs/39SGCQeBIQ7nHFKIjYEEIVNy9gOiSKUDAxZk02znwSi6R+UnLOSqe35UKlnKaUQQfoEB0jB52jCrpBVVRDFMXoGb2gV+vJerPerY9Z65KVzuTRH1ifP/JrnX0=</latexit> u = R 1B>Px 0 K K K
  12. 31 <latexit sha1_base64="SbkwophhDdRZdoTSi+PTBsuyVWg=">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</latexit> xt+1 = f(xt, ut), t = 0,

    · · · , p 1 <latexit sha1_base64="4XkrTQgnLggvIWr9g9yBHaZIXP4=">AAACCHicbVDLSgNBEJyNrxhfUY9eBoPgKexKfFyEgBePEcwDkyXMTmaTIbMzy0yvJIT8gGev+g3exKt/4Sf4F84me9DEgoaiqpvuriAW3IDrfjm5ldW19Y38ZmFre2d3r7h/0DAq0ZTVqRJKtwJimOCS1YGDYK1YMxIFgjWD4U3qNx+ZNlzJexjHzI9IX/KQUwJWehh1XXyNR+2h3y2W3LI7A14mXkZKKEOtW/zu9BRNIiaBCmJM23Nj8CdEA6eCTQudxLCY0CHps7alkkTM+JPZxVN8YpUeDpW2JQHP1N8TExIZM44C2xkRGJhFLxX/9VIFlBJm4QAIr/wJl3ECTNL5/jARGBROU8E9rhkFMbaEUM3tC5gOiCYUbHYFm423mMQyaZyVvYvy+V2lVK1kKeXRETpGp8hDl6iKblEN1RFFEj2jF/TqPDlvzrvzMW/NOdnMIfoD5/MHAm2Z1Q==</latexit> x0 = x[k] k subject to <latexit sha1_base64="fKQy4fvbzO1+4xkl3S9SkmdRlnM=">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</latexit> min u0,··· ,up 1 p 1 X t=0 `t(xt, ut) + `p(xp) <latexit sha1_base64="sTGHTCL47RYQM0eoV6yVI1Z8FDs=">AAACVXicbVHLSgMxFM2MWnV8tOrSTbAoLmqZEV8boeDGZQVHC51SMmmqoZlkSO6IZeiX+SHi0qX6DYKZWkFbD4Qczj2X3HsSp4Ib8P0Xx52bXygtLi17K6tr6+XKxuaNUZmmLKRKKN2KiWGCSxYCB8FaqWYkiQW7jQcXRf32gWnDlbyGYco6CbmTvM8pASt1K+FjF/BexCWOEgL3lIi8NarhCMM5DuxNewpMDaeWRl424w1/vP4v70HQrVT9uj8GniXBhFTRBM1u5S3qKZolTAIVxJh24KfQyYkGTgUbeVFmWErogNyxtqWSJMx08vH6I7xrlR7uK22PBDxWf3fkJDFmmMTWWcxtpmuF+G+tUEApYaYGgP5ZJ+cyzYBJ+v1+PxMYFC4ixj2uGQUxtIRQze0KmN4TTSjYj/BsNsF0ErPk5rAenNSPr46qjaNJSktoG+2gfRSgU9RAl6iJQkTRE3pF7+jDeXY+3Xm39G11nUnPFvoDt/wFsLexpw==</latexit> xt 2 X, t = 1, · · · , p, ut 2 U, t = 0, · · · , p 1 [ ] subscript
  13. MPC § 1963 Propoi Application of linear programming methods for

    the synthesis of automatic sampled-data systems § 1978 Richalet et al. 33 § §
  14. § § quadratic program : 34 <latexit sha1_base64="Oz2UEjC9bIugHY8y9Nu0ZRL5FNs=">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</latexit> min u0,···

    ,up 1 p 1 X t=0 x> t Qxt + u> t Rut + x> p Qf xp <latexit sha1_base64="yiEw6RggZAsYi1UZg/GCYoCkC2I=">AAACKHicbZDLSgMxFIYz9VbrrerSTbAIhdYyI/WyEapuXFawF2jLkEnTNjQzGZIz0jL0FXwQ1271GdxJt+58C9PLQlt/CPx85xxOzu+Fgmuw7bGVWFldW99Ibqa2tnd299L7B1UtI0VZhUohVd0jmgkesApwEKweKkZ8T7Ca17+b1GtPTGkug0cYhqzlk27AO5wSMMhNZwduDDlnhK/xzcCF3G3kQh43MRhgG0PbEnQ+PHXcdMYu2FPhZePMTQbNVXbT3822pJHPAqCCaN1w7BBaMVHAqWCjVDPSLCS0T7qsYWxAfKZb8fSiET4xpI07UpkXAJ7S3xMx8bUe+p7p9An09GJtAv+tTQhIKfTCB6Bz1Yp5EEbAAjrb34kEBoknqeE2V4yCGBpDqOLmBEx7RBEKJtuUycZZTGLZVM8KzkXh/KGYKRXnKSXRETpGWeSgS1RC96iMKoiiZ/SK3tC79WJ9WJ/WeNaasOYzh+iPrK8fCfmkGA==</latexit> xt+1 = Axt + But, t = 0, · · · , p 1 <latexit sha1_base64="8Y8vH0VuwhjjI5U3nKgDqvwrUlw=">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</latexit> u = 2 6 4 u0 . . . up 1 3 7 5 x0 , Q, R, Qf <latexit sha1_base64="1irHdw9QUP/XzlFvEnHABXwSR0I=">AAACHnicbZDLSgMxFIYz9VbrbdSlIMEiCEKZkXpZFtx0WcFeoB1LJs20oZlkTM4IpXTng7h2q8/gTtzqI/gWppeFtv4Q+PjPOZycP0wEN+B5X05maXlldS27ntvY3NrecXf3akalmrIqVULpRkgME1yyKnAQrJFoRuJQsHrYvx7X6w9MG67kLQwSFsSkK3nEKQFrtd3DVsxlO8XpXQtUgsspPr2foiWs227eK3gT4UXwZ5BHM1Xa7nero2gaMwlUEGOavpdAMCQaOBVslGulhiWE9kmXNS1KEjMTDCd3jPCxdTo4Uto+CXji/p4YktiYQRzazphAz8zXxua/tbEDSgkz9wGIroIhl0kKTNLp/igVGBQeZ4U7XDMKYmCBUM3tCZj2iCYUbKI5m40/n8Qi1M4K/kXh/KaYLxVnKWXRATpCJ8hHl6iEyqiCqoiiR/SMXtCr8+S8Oe/Ox7Q148xm9tEfOZ8/StSh3Q==</latexit> min u u > Hu + q > u + r
  15. § § § 35 <latexit sha1_base64="zHaWoOaSOlSa6PHwzfU/JVyLSaY=">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</latexit> min u0,··· ,up 1

    E " p 1 X t=0 `t(xt, ut) + `p(xp) # <latexit sha1_base64="4WUkiEyhiEZdop9oKezwbKlPWQQ=">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</latexit> xt+1 = f(xt, ut, wt), t = 0, · · · , p 1 <latexit sha1_base64="XNEChlX59Pw3e8tqKgQI19Shq0U=">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</latexit> P[xt / 2 X]  , t = 1, · · · , p ut 2 U, t = 0, · · · , p 1 δ
  16. § § § 36 <latexit sha1_base64="15SHQ1W2yoS+AQG+8yViNBM2WOY=">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</latexit> xt+1 = f(xt, ut,

    dt), t = 0, · · · , p 1 <latexit sha1_base64="cSK7IBGNtFERol5QdtFCAX4YmyE=">AAACkXicbZHLThsxFIY9U64pl7QsEdJRI9IuQjSD6G2BFAELJDZUaiASE408Hg9YeOyRfaYiGuUh+ng8Qp+gWzwhIEg4G//6z2f7XJJCCotBcO/57xYWl5ZXVhvv19Y3NpsfPl5YXRrG+0xLbQYJtVwKxfsoUPJBYTjNE8kvk9vjOn/5hxsrtPqNo4IPc3qtRCYYRWfFzb8R8jusMm2ASgljgHYaYyRUlFO8YVRWJ+MORICHELiTpRptB4q9EKKocRcjtB0Lz/DgCQ5fwJ2aLefY/tsPx81W0A0mAfMinIoWmcZ53PwXpZqVOVfIJLX2KgwKHFbUoGCSjxtRaXlB2S295ldOKppzO6wmoxvDrnNSqNvPtEKYuC9vVDS3dpQnjqzrtrO52nwzVzuotbQzBWD2Y1gJVZTIFXv8PysloIZ6PZAKwxnKkROUGeFaAHZDDWXolthwswlnJzEvLva74bfu118Hrd7BdEorZJt8Il9ISL6THjkl56RPGPnv7Xht77O/5f/0e/7RI+p70ztb5FX4Zw9QzcKp</latexit> for all dt 2 D, t = 0, · · · , p 1 xt 2 X, t = 1, · · · , p, ut 2 U, t = 0, · · · , p 1 <latexit sha1_base64="rMtRH3zW08HTFInouS4fD2pxhb4=">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</latexit> min u0,··· ,up 1 max dt,··· ,dp 1 p 1 X t=0 `t(xt, ut) + `p(xp)
  17. § MPC 1999 § § 1980 § {ui , yi

    }i=1 N A, B, C, D α § § non-iterative MIMO SISO § horizon 39 MPC <latexit sha1_base64="Sc8gKuXW7+dyhsoRtwppkHqpg4U=">AAACJ3icbZBPSwJBGMZn7Z/ZP6tjlyGJAkF2Q6xLYNmho0FqoLLMjqMOzu4sM++GsvgR+iCdu9Zn6BZ17Ni3aFY9lPbAwMPvfV/eeR8vFFyDbX9aqaXlldW19HpmY3Nreye7u1fXMlKU1agUUt17RDPBA1YDDoLdh4oR3xOs4Q0qSb3xwJTmMriDUcjaPukFvMspAYPc7PHQjSHvjPEFvhy6kL+KXGi1MiMXDKkYgvP42jA3m7ML9kR40Tgzk0MzVd3sd6sjaeSzAKggWjcdO4R2TBRwKtg404o0CwkdkB5rGhsQn+l2PDlojI8M6eCuVOYFgCf090RMfK1Hvmc6fQJ9PV9L4L+1hICUQs99ALrn7ZgHYQQsoNP93UhgkDgJDXe4YhTEyBhCFTcnYNonilAw0WZMNs58EoumflpwSoXSbTFXLs5SSqMDdIhOkIPOUBndoCqqIYoe0TN6Qa/Wk/VmvVsf09aUNZvZR39kff0AisSkdw==</latexit> xt+1 = Axt + But yt = Cxt + Dut
  18. Favoreel et. al § § +QR MPC § § SVD

    40 <latexit sha1_base64="HwrZGe5BeobXbyjoSEGfwBy/gPg=">AAACC3icbVDLSgNBEJyNrxhfUY9eBoMgHsKuhOhFCHjxIkR0k0CyhtnJbDJkHuvMrBCWfIJnr/oN3sSrH+En+BfOJjloYkFDUdVNd1cYM6qN6345uaXlldW1/HphY3Nre6e4u9fQMlGY+FgyqVoh0oRRQXxDDSOtWBHEQ0aa4fAy85uPRGkqxZ0ZxSTgqC9oRDEyVgquL/zOLe1zBBv3J91iyS27E8BF4s1ICcxQ7xa/Oz2JE06EwQxp3fbc2AQpUoZiRsaFTqJJjPAQ9UnbUoE40UE6OXoMj6zSg5FUtoSBE/X3RIq41iMe2k6OzEDPe5n4r5cpRkqm5w4w0XmQUhEnhgg83R8lDBoJs2BgjyqCDRtZgrCi9gWIB0ghbGx8BZuNN5/EImmclr1quXpTKdUqs5Ty4AAcgmPggTNQA1egDnyAwQN4Bi/g1Xly3px352PamnNmM/vgD5zPHyrymv8=</latexit> M = U⌃V ⇤ m×n r m×m ( ) n×n ( ) <latexit sha1_base64="6+EgYY+hjj8BvIhfbIMW1Fm0LyI=">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</latexit> ⌃ = 8 > > > > < > > > > : h O i , (m  n) , (m = n) " O # , (m n) = diag( 1, . . . , q), q = min(m, n), r+1 = · · · = q = 0
  19. Favoreel et. al § QR § QR 41 <latexit sha1_base64="RhQAWvqgvnJUENNtlJy6OOLgTo8=">AAACA3icbVBNS8NAEJ3Ur1q/qh69LBbBU0mkVC9CxYvHVkxbaEPZbDft0k027G6EEnr07FV/gzfx6g/xJ/gv3LQ5aOuDgcd7M8zM82POlLbtL6uwtr6xuVXcLu3s7u0flA+P2kokklCXCC5k18eKchZRVzPNaTeWFIc+px1/cpv5nUcqFRPRg57G1AvxKGIBI1gbyb25Rq37QbliV+050CpxclKBHM1B+bs/FCQJaaQJx0r1HDvWXoqlZoTTWamfKBpjMsEj2jM0wiFVXjo/dobOjDJEgZCmIo3m6u+JFIdKTUPfdIZYj9Wyl4n/epmiheBq6QAdXHkpi+JE04gs9gcJR1qgLBA0ZJISzaeGYCKZeQGRMZaYaBNbyWTjLCexStoXVaderbdqlUYtT6kIJ3AK5+DAJTTgDprgAgEGz/ACr9aT9Wa9Wx+L1oKVzxzDH1ifPz7Il8Y=</latexit>

    A = QR m×n m×m ( ) m×n <latexit sha1_base64="jjCU+qTS5iurm7+L3b4oi375TmE=">AAACLXicbZDLSgMxFIYz9VbrrerSTbAIddEyI6W6KVTcuGyLvYCtJZOmbWhmMiRnhDL0KXwQ1271GVwI4lZ8C9NpBW09EPjy/+eQnN8NBNdg229WYmV1bX0juZna2t7Z3UvvHzS0DBVldSqFVC2XaCa4z+rAQbBWoBjxXMGa7uhq6jfvmdJc+jcwDljHIwOf9zklYKRuOnd5F+WcSQlnq7XTGHEJ12KozpyfSxtk0E1n7LwdF14GZw4ZNK9KN/3V7kkaeswHKojWt44dQCciCjgVbJJqh5oFhI7IgN0a9InHdCeK15rgE6P0cF8qc3zAsfp7IiKe1mPPNZ0egaFe9Kbiv95UASmFXvgA9C86EfeDEJhPZ+/3Q4FB4ml0uMcVoyDGBghV3KyA6ZAoQsEEnDLZOItJLEPjLO8U88VqIVMuzFNKoiN0jLLIQeeojK5RBdURRQ/oCT2jF+vRerXerY9Za8KazxyiP2V9fgNdZqZp</latexit> A 1 = (QR) 1 = R 1Q 1 = R 1Q> <latexit sha1_base64="CG7O2cG0g9CQio5dmK6D1uwMxDA=">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</latexit> det(A) = det(QR) = det(Q) det(R) = det(R) = Y Rii https://dsc-spidal.github.io/harp/docs/harpdaal/QR/
  20. Favoreel et. al § § j >> M, N Yp

    , Yf 42 <latexit sha1_base64="pfcmm1fNJPPlC1dQyN4DKc2Kzes=">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</latexit> min u0,··· ,up 1 p 1 X t=0 (ˆ yt rt)>Qt(ˆ yt rt) + u> t Rtut <latexit sha1_base64="LVg3ScB2RVMd3ID6TNHIQrC51gU=">AAACG3icbVDLSsNAFJ3UV62vqMtuBosgVEoipboRCm7cKBXsA5oYJtNpO3aSCTMToYQs/BDXbvUb3IlbF36Cf+Gk7UJbD1w4nHMv997jR4xKZVlfRm5peWV1Lb9e2Njc2t4xd/dakscCkybmjIuOjyRhNCRNRRUjnUgQFPiMtP3RRea3H4iQlIe3ahwRN0CDkPYpRkpLnll0ktijx3DsUSf1Enpup3fJVfm6fJ9CzyxZFWsCuEjsGSmBGRqe+e30OI4DEirMkJRd24qUmyChKGYkLTixJBHCIzQgXU1DFBDpJpMnUniolR7sc6ErVHCi/p5IUCDlOPB1Z4DUUM57mfivlymKcybnDlD9MzehYRQrEuLp/n7MoOIwCwr2qCBYsbEmCAuqX4B4iATCSsdZ0NnY80ksktZJxa5VajfVUr06SykPiuAAHAEbnII6uAQN0AQYPIJn8AJejSfjzXg3PqatOWM2sw/+wPj8AZxfoQs=</latexit> {ui, yi }M+N+j i=1 <latexit sha1_base64="LVg3ScB2RVMd3ID6TNHIQrC51gU=">AAACG3icbVDLSsNAFJ3UV62vqMtuBosgVEoipboRCm7cKBXsA5oYJtNpO3aSCTMToYQs/BDXbvUb3IlbF36Cf+Gk7UJbD1w4nHMv997jR4xKZVlfRm5peWV1Lb9e2Njc2t4xd/dakscCkybmjIuOjyRhNCRNRRUjnUgQFPiMtP3RRea3H4iQlIe3ahwRN0CDkPYpRkpLnll0ktijx3DsUSf1Enpup3fJVfm6fJ9CzyxZFWsCuEjsGSmBGRqe+e30OI4DEirMkJRd24qUmyChKGYkLTixJBHCIzQgXU1DFBDpJpMnUniolR7sc6ErVHCi/p5IUCDlOPB1Z4DUUM57mfivlymKcybnDlD9MzehYRQrEuLp/n7MoOIwCwr2qCBYsbEmCAuqX4B4iATCSsdZ0NnY80ksktZJxa5VajfVUr06SykPiuAAHAEbnII6uAQN0AQYPIJn8AJejSfjzXg3PqatOWM2sw/+wPj8AZxfoQs=</latexit> {ui, yi }M+N+j i=1 <latexit sha1_base64="0UnrwlH0qWoz8OaRTYINHYO1z2U=">AAACG3icbVDLSsNAFJ3UV62vqMtuBosgqCWRUt0IBTduhAr2AW0Nk+mkHTvJhJmJEEIWfohrt/oN7sStCz/Bv3DSZqHVAxcO59zLvfe4IaNSWdanUVhYXFpeKa6W1tY3NrfM7Z225JHApIU546LrIkkYDUhLUcVINxQE+S4jHXdykfmdeyIk5cGNikMy8NEooB7FSGnJMcv9JHLoEYwd2k+dhJ7b6W1ydWwf3qXQMStW1ZoC/iV2TiogR9Mxv/pDjiOfBAozJGXPtkI1SJBQFDOSlvqRJCHCEzQiPU0D5BM5SKZPpHBfK0PocaErUHCq/pxIkC9l7Lu600dqLOe9TPzXyxTFOZNzByjvbJDQIIwUCfBsvxcxqDjMgoJDKghWLNYEYUH1CxCPkUBY6ThLOht7Pom/pH1StevV+nWt0qjlKRVBGeyBA2CDU9AAl6AJWgCDB/AEnsGL8Wi8Gm/G+6y1YOQzu+AXjI9vcJyg8A==</latexit> {ui, yi }M 1+j i=1 <latexit sha1_base64="OA/6mxPGGNbo7sS24AQS/HEjO7w=">AAACHXicbVDLSsNAFJ34rPVVdSnCYBGESkmkVDdCwY2bSgX7gCaGyXTajp1kwsxECCErP8S1W/0Gd+JW/AT/wmmbhbYeuHA4517uvccLGZXKNL+MhcWl5ZXV3Fp+fWNza7uws9uSPBKYNDFnXHQ8JAmjAWkqqhjphIIg32Ok7Y0ux377gQhJeXCr4pA4PhoEtE8xUlpyCwd2Ern0BMYutVM3oRf1kpXeJfXSdek+hW6haJbNCeA8sTJSBBkabuHb7nEc+SRQmCEpu5YZKidBQlHMSJq3I0lChEdoQLqaBsgn0kkmb6TwSCs92OdCV6DgRP09kSBfytj3dKeP1FDOemPxX2+sKM6ZnDlA9c+dhAZhpEiAp/v7EYOKw3FUsEcFwYrFmiAsqH4B4iESCCsdaF5nY80mMU9ap2WrWq7eVIq1SpZSDuyDQ3AMLHAGauAKNEATYPAInsELeDWejDfj3fiYti4Y2cwe+APj8we5HaGX</latexit> {ui, yi }M+N+j i=M+1 <latexit sha1_base64="cKcIGnx9UucgRXK5waGX4hQxYCw=">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</latexit> Uf = 2 6 6 6 4 uM+1 uM+2 · · · uM+j uM+2 uM+3 · · · uM+j+1 . . . . . . ... . . . uM+N uM+N+1 · · · uM+N+j 3 7 7 7 5 <latexit sha1_base64="uQ4IBzjxtri56IlKbV7qu6v0nV0=">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</latexit> Up = 2 6 6 6 4 u1 u2 · · · uj u2 u3 · · · uj+1 . . . . . . ... . . . uM uM+1 · · · uM 1+j 3 7 7 7 5
  21. Favoreel et. al § 43 <latexit sha1_base64="oF4zvr5o9TzYKmva5Z9Ek3XxdmM=">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</latexit> r = 2

    6 6 6 4 C CA . . . CAr 1 3 7 7 7 5 <latexit sha1_base64="0oygu0xbuvgwp8RArNguoJ2obOw=">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</latexit> Hr = 2 6 6 6 4 D 0 · · · 0 CB D · · · 0 . . . . . . ... . . . CAr 2 B CAr 3 B · · · D 3 7 7 7 5 <latexit sha1_base64="bd26n9gTLGQJcJEi3Y+bAijYz00=">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</latexit> Yp = M Xp + HM Up Yf = N Xf + HN Uf <latexit sha1_base64="FF5qDrvoILI6sUAAgPeI//UDz/o=">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</latexit> Xp = [x1, · · · , xj] Xf = [xM+1, · · · , xM+j] <latexit sha1_base64="r7355TlX6DPlxSKwnxx1XWRxXvg=">AAACLHicbZDLSgMxFIYz9VbrrerSTbAoQqXMSKluhGpduKxgL9CWIZOmbWhmMiRnpGXoS/ggrt3qM7gRcSv4FqaXhbb+EPjznXNIzu+Fgmuw7XcrsbS8srqWXE9tbG5t76R396paRoqyCpVCqrpHNBM8YBXgIFg9VIz4nmA1r18a12sPTGkug3sYhqzlk27AO5wSMMhNnw7cGLLYGeHjS3w1cAFn8XXkQrOZGpqLgaUpvDHQTWfsnD0RXjTOzGTQTGU3/d1sSxr5LAAqiNYNxw6hFRMFnAo2SjUjzUJC+6TLGsYGxGe6FU+2GuEjQ9q4I5U5AeAJ/T0RE1/roe+ZTp9AT8/XxvDf2piAlELPfQA6F62YB2EELKDT9zuRwCDxODnc5opREENjCFXcrIBpjyhCweSbMtk480ksmupZzinkCnf5TDE/SymJDtAhOkEOOkdFdIvKqIIoekTP6AW9Wk/Wm/VhfU5bE9ZsZh/9kfX1A4MIpVU=</latexit> xt+1 = Axt + But yt = Cxt + Dut § §
  22. Favoreel et. al § § § SVD 44 <latexit sha1_base64="6FdxKx9r/jLKkGtx6Mv4l2MxXN8=">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</latexit>

    min Lw1,Lw2,Lu kYf (Lw1Yp + Lw2Up + LuUf )k2 F <latexit sha1_base64="2tvNL+ZAYJy04NHp+aYDkpVMLpI=">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</latexit> ⇥ Lw1 Lw2 ⇤ = ⇥ U1 U2 ⇤  ⌃1 0 0 ⌃2  V > 1 V > 2 ⇡ U1⌃1V > 1 <latexit sha1_base64="9STlJfff/clzGec/jHpf0HrMnEs=">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</latexit> N = U1⌃1/2 1 , ˆ Xf = ⌃1/2 1 V > 1  Yp Up <latexit sha1_base64="bd26n9gTLGQJcJEi3Y+bAijYz00=">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</latexit> Yp = M Xp + HM Up Yf = N Xf + HN Uf <latexit sha1_base64="TTArrxXMAQHwZN1/epenJLR0C+s=">AAACKXicbZDLSsNAFIYn3q23qEs3g0VQhJIUUTeC4MaFiwrGC20Ik+mkHTrJhJkTpYQ8gw/i2q0+gzt168q3cNJ2oa0/DHz85xzOnD9MBdfgOB/W1PTM7Nz8wmJlaXlldc1e37jWMlOUeVQKqW5DopngCfOAg2C3qWIkDgW7CXtnZf3mninNZXIF/ZT5MekkPOKUgLECe6/VJZDfFUGET/BFkD+4xV2Q4v0S64UXpIYy7AVRJbCrTs0ZCE+CO4IqGqkR2N+ttqRZzBKggmjddJ0U/Jwo4FSwotLKNEsJ7ZEOaxpMSMy0nw9OKvCOcdo4ksq8BPDA/T2Rk1jrfhyazphAV4/XSvPfWumAlEKPfQCiYz/nSZoBS+hwf5QJDBKXseE2V4yC6BsgVHFzAqZdoggFE26ZjTuexCRc12vuYe3w8qB6ejBKaQFtoW20i1x0hE7ROWogD1H0iJ7RC3q1nqw36936HLZOWaOZTfRH1tcPaLGmBw==</latexit> ˆ Yf = Lw1Yp + Lw2Up + LuUf
  23. Favoreel et. al § k 45 <latexit sha1_base64="BoELxHYxtniJF8Nstss1r9dUmKk=">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</latexit> min uf

    (ˆ yf rf )>Q(ˆ yf rf ) + u> f Ruf <latexit sha1_base64="pfcmm1fNJPPlC1dQyN4DKc2Kzes=">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</latexit> min u0,··· ,up 1 p 1 X t=0 (ˆ yt rt)>Qt(ˆ yt rt) + u> t Rtut N=p <latexit sha1_base64="UfD+/hl521PV0Ux60lIyeW/g+hA=">AAACKXicbZDLSsNAFIYnXmu9RV26GSyCIpSklOpGKLhx4aKCvYAtYTKd2MFJJsycKCHkGXwQ1271GdypW1e+hZO2C63+MPDxn3M4c34/FlyD47xbc/MLi0vLpZXy6tr6xqa9td3RMlGUtakUUvV8opngEWsDB8F6sWIk9AXr+rdnRb17x5TmMrqCNGaDkNxEPOCUgLE8+7A/IpCluRfgU3zhZfdunnoxPiqwlidebCjBiReUPbviVJ2x8F9wp1BBU7U8+6s/lDQJWQRUEK2vXSeGQUYUcCpYXu4nmsWE3pIbdm0wIiHTg2x8Uo73jTPEgVTmRYDH7s+JjIRap6FvOkMCIz1bK8x/a4UDUgo98wEITgYZj+IEWEQn+4NEYJC4iA0PuWIURGqAUMXNCZiOiCIUTLhFNu5sEn+hU6u6jWrjsl5p1qcpldAu2kMHyEXHqInOUQu1EUUP6Ak9oxfr0Xq13qyPSeucNZ3ZQb9kfX4DPyCmhw==</latexit> ˆ yf = Lw1yp + Lw2up + Luuf <latexit sha1_base64="nJlJ0Sparfql84J5D35tf1JfEqg=">AAACP3icbZDNSgMxFIUz/tb6V3XpJliEilhmRNSNWHDjwoWKVcHWkEkzNpiZhOSOUoZ5HB/EtVtd+ATiRty6M60FtXog8HHuvdzcE2opLPj+szc0PDI6Nl6YKE5OTc/MlubmT61KDeN1pqQy5yG1XIqE10GA5OfacBqHkp+F13vd+tkNN1ao5AQ6mjdjepWISDAKziKl3ZREeAdXjlcPSHrZAKXxkaOVy2wtyL+tiiHR2gHJboO8Q3SP1vOU6BVSKvtVvyf8F4I+lFFfh6T02mgplsY8ASaptReBr6GZUQOCSZ4XG6nlmrJresUvHCY05raZ9Q7N8bJzWjhSxr0EcM/9OZHR2NpOHLrOmELbDta65r+1rgNKSTvwAYi2m5lIdAo8YV/7o1RiULgbJm4JwxnIjgPKjHAnYNamhjJwkRddNsFgEn/hdL0abFY3jzbKtY1+SgW0iJZQBQVoC9XQPjpEdcTQHXpAj+jJu/devDfv/at1yOvPLKBf8j4+AV6mrpQ=</latexit> uf = (R + L> u QLu) 1L> u Q(rf Lw1yp Lw2up) <latexit sha1_base64="Ssvcf/2O6dMgWJNgMz/v7ReTcgg=">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</latexit> uf = 2 6 6 6 4 u0 u2 . . . uN 1 3 7 7 7 5 , ˆ yf = 2 6 6 6 4 ˆ y0 ˆ y2 . . . ˆ yN 1 3 7 7 7 5 , rf = 2 6 6 6 4 r0 r2 . . . rN 1 3 7 7 7 5 , yp = 2 6 6 6 4 y[k M + 1] . . . y[k 1] y[k] 3 7 7 7 5 , up = 2 6 6 6 4 u[k M + 1] . . . u[k 1] u[k] 3 7 7 7 5 ,
  24. Iplikci § : nonlinear autoregressive exogeneous model (NARX) MPC 49

    t nu ny <latexit sha1_base64="udUyzllvGZeBO3hLkwUh8Qazu+E=">AAACIHicbVDLSsNAFJ34rPFVdelmsAguakmkVJcFN+Kqgn1AE8JkOm2HTmbCzI1QQrd+iGu3+g3uxKX+gX9h0nahrQcuHM65l3vvCWPBDTjOp7Wyura+sVnYsrd3dvf2iweHLaMSTVmTKqF0JySGCS5ZEzgI1ok1I1EoWDscXed++4Fpw5W8h3HM/IgMJO9zSiCTgiK2vYjLIE0Cp4w92lNgyjgJ0vjcnUzwrW0HxZJTcabAy8SdkxKaoxEUv72eoknEJFBBjOm6Tgx+SjRwKtjE9hLDYkJHZMC6GZUkYsZPp59M8Gmm9HBf6awk4Kn6eyIlkTHjKMw6IwJDs+jl4r9eroBSwiwcAP0rP+UyToBJOtvfTwQGhfO0cI9rRkGMM0Ko5tkLmA6JJhSyTPNs3MUklknrouLWKrW7aqlenadUQMfoBJ0hF12iOrpBDdREFD2iZ/SCXq0n6816tz5mrSvWfOYI/YH19QOgC6Hy</latexit> min u0,··· ,up 1 J f <latexit sha1_base64="dvn9iqzOCVqwOzeQq450TD6twDo=">AAACRHicbZDNSsNAFIUn/tb6V3XpZrAILdSSSFE3QkEXLitYFWwJk+mkHZzMhJkbMYS8kA/i2m19A8GduBWTtqC2Xhj4OOde7tzjhYIbsO2hNTe/sLi0XFgprq6tb2yWtravjYo0ZW2qhNK3HjFMcMnawEGw21AzEniC3Xj3Z7l/88C04UpeQRyybkD6kvucEsgkt3Qeu4BPsV+JXKjhDu0pMDUcuQkcSDdKazjO0Ul/vHjsxWn11K88ulAtFt1S2a7bo8Kz4EygjCbVcktvnZ6iUcAkUEGMuXPsELoJ0cCpYGmxExkWEnpP+uwuQ0kCZrrJ6NoU72dKD/tKZ08CHqm/JxISGBMHXtYZEBiYaS8X//VyBZQSZuoD4J90Ey7DCJik4/1+JDAonCeKe1wzCiLOgFDNsxMwHRBNKGS559k400nMwvVh3TmqH102ys3GJKUC2kV7qIIcdIya6AK1UBtR9IRe0BC9Ws/Wu/VhfY5b56zJzA76U9bXNyMjr70=</latexit> yt = f(ut, · · · , ut nu , yt 1, · · · , yt ny ) = f(xt) nu ny <latexit sha1_base64="70RphR5tJREZ+/PcGq/u0JOZRfc=">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</latexit> J = p 1 X t=0 (ˆ yt rt)2 + t(ut ut 1)2 + 0 @ µ ⇢ 2 ✓ + ut + µ ⇢ 2 + ✓ ut 4 ⇢ 1 A
  25. Iplikci § 1. {xi, yi}i=1 N 2. SVM dual αi

    b 50 <latexit sha1_base64="IE2DEecFHdKEhwOBWnoXWTqhB+Y=">AAACU3icbZFBaxNBFMcna6tttJrq0ctgaElpCbsSopdAwYsgSETTFDrp8nYyyQ6dnVlm3taEJR/MD+LBk1f7GXpxNslBkz4Y+PP7v8fM+0+SK+kwDH/Vgkc7u4+f7O3Xnz47eP6icfjywpnCcjHgRhl7mYATSmoxQIlKXOZWQJYoMUxuPlT+8FZYJ43+hvNcjDKYajmRHNCjuPGVpYDlfNGandDjHlOgp0rQ72esn8qKMbsipwllrH5Me5S5IotL2YsW158pA5WnEEtKP7VmZ7NreeIb40YzbIfLotsiWosmWVc/btyxseFFJjRyBc5dRWGOoxIsSq7Eos4KJ3LgNzAVV15qyIQblcvlF/TIkzGdGOuPRrqk/06UkDk3zxLfmQGmbtOr4INeRdAY5TYegJP3o1LqvECh+er+SaEoGloFTMfSCo5q7gVwK/0KlKdggaP/hrrPJtpMYltcvG1H3Xb3S6d53lmntEdekzekRSLyjpyTj6RPBoSTH+Q3+UPuaj9r90EQ7Kxag9p65hX5r4KDv+pbsVs=</latexit> ˆ y(x) = hw, (x)i + b = N X i=1 ↵iK(x, xi) + b <latexit sha1_base64="2U83NyY0vaT7mGTQk64XbYVFj2Y=">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</latexit> exp ✓ kx xik2 2 2 ◆ <latexit sha1_base64="tN3/rH7JZiA1yKEXBsScEZJpphE=">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</latexit> min w,b,⇠,⇠⇤ 1 2 kwk2 + c N X i=1 (⇠i + ⇠⇤ i ) subject to hw, (xi)i b + yi  " + ⇠i hw, (xi)i + b yi  " + ⇠⇤ i ⇠i, ⇠⇤ i 0
  26. Iplikci § 3. 51 <latexit sha1_base64="8T1WWE4kOP40YXnV/vhzrePdap4=">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</latexit> ck = [u[k], ·

    · · , u[k nu], y[k 1], · · · , y[k ny]]> <latexit sha1_base64="wMAEwPFX1I8RGPRwMDzzRXeV1Yk=">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</latexit> ck+1 = [u0, u[k], · · · , u[k nu 1], ˆ y0, y[k 1], · · · , y[k ny 1]]> <latexit sha1_base64="HnNL/pkk6j87ns6nPfwyGFh6Lbg=">AAACCHicbVDLSsNAFJ3UV62vqks3g0VwVRKR6rLgxmUF+8A2lMl00g6dzISZGyGE/oBrt/oN7sStf+En+BdO2iy09cCFwzn3cu89QSy4Adf9ckpr6xubW+Xtys7u3v5B9fCoY1SiKWtTJZTuBcQwwSVrAwfBerFmJAoE6wbTm9zvPjJtuJL3kMbMj8hY8pBTAlZ6GEwIZOls6OFhtebW3TnwKvEKUkMFWsPq92CkaBIxCVQQY/qeG4OfEQ2cCjarDBLDYkKnZMz6lkoSMeNn84tn+MwqIxwqbUsCnqu/JzISGZNGge2MCEzMspeL/3q5AkoJs3QAhNd+xmWcAJN0sT9MBAaF81TwiGtGQaSWEKq5fQHTCdGEgs2uYrPxlpNYJZ2LuteoN+4ua83LIqUyOkGn6Bx56Ao10S1qoTaiSKJn9IJenSfnzXl3PhatJaeYOUZ/4Hz+AAC5mnE=</latexit> ˆ y1 <latexit sha1_base64="KbXQ4sIqHUswluVwLwbVaI37CC8=">AAACDHicbVDLSsNAFJ3UV62vqks3g0VwY0mkVJcFNy4r2Ae0sUymk3boJBNmboQQ8guu3eo3uBO3/oOf4F84abPQ1gMXDufcy733eJHgGmz7yyqtrW9sbpW3Kzu7e/sH1cOjrpaxoqxDpZCq7xHNBA9ZBzgI1o8UI4EnWM+b3eR+75EpzWV4D0nE3IBMQu5zSsBID8MpgTTJRml04WR4VK3ZdXsOvEqcgtRQgfao+j0cSxoHLAQqiNYDx47ATYkCTgXLKsNYs4jQGZmwgaEhCZh20/nVGT4zyhj7UpkKAc/V3xMpCbROAs90BgSmetnLxX+9XAEphV46APxrN+VhFAML6WK/HwsMEufJ4DFXjIJIDCFUcfMCplOiCAWTX8Vk4ywnsUq6l3WnWW/eNWqtRpFSGZ2gU3SOHHSFWugWtVEHUaTQM3pBr9aT9Wa9Wx+L1pJVzByjP7A+fwAyXZwu</latexit> ˆ yp 1 <latexit sha1_base64="OJx/2bbwCUmpso6zd2bP+FToML4=">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</latexit> ˆ y0 = N X i=1 ↵i exp ✓ kck xik2 2 2 ◆ + b
  27. Iplikci § 5. 6. u ← u + sp a)

    p i. ii. b) s 52 <latexit sha1_base64="86rKDpv7EaLbrhvv3NPSOtzSZL0=">AAACL3icbZBNS8MwGMfT+Tbr29Sjl+AQPOhsZUwvwsCLeJrgXmAtI03TLSxtSpIKo/Rj+EE8e9XPIF7E676F6VZQp38I/Pg/z5Pk+Xsxo1JZ1rtRWlpeWV0rr5sbm1vbO5XdvY7kicCkjTnjouchSRiNSFtRxUgvFgSFHiNdb3yd17sPREjKo3s1iYkbomFEA4qR0tagchbDK3g6PIEOHGpy/EAgnDoxEooiBm+zb04y0zQHlapVs2aCf8EuoAoKtQaVqeNznIQkUpghKfu2FSs3ze/EjGSmk0gSIzxGQ9LXGKGQSDedLZbBI+34MOBCn0jBmftzIkWhlJPQ050hUiO5WMvNf2u5ozhncuEDKrh0UxrFiSIRnr8fJAwqDvPwoE8FwYpNNCAsqF4B4hHSiSkdcZ6NvZjEX+ic1+xGrXFXrzbrRUplcAAOwTGwwQVoghvQAm2AwSN4Bi/g1Xgy3owP43PeWjKKmX3wS8b0C3a/p4I=</latexit> p = g, g = @J @u <latexit sha1_base64="g1KEUdh5mAO6kMrLpKqXO4EKq3U=">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</latexit> p = H 1 g, G = @2 J @u2 <latexit sha1_base64="DIY5R1yfsAy1vLx6duOO5HUJVOA=">AAACI3icbVDLSgMxFM3UVx1fVZdugkVwUcuMlOpGKLhxWcE+oB1LJs20oZnJkNwRytAP8ENcu9VvcCduXPgB/oWZtgttPRA4nHMuN/f4seAaHOfTyq2srq1v5Dftre2d3b3C/kFTy0RR1qBSSNX2iWaCR6wBHARrx4qR0Bes5Y+uM7/1wJTmMrqDccy8kAwiHnBKwEi9QjHBV7iT9JwSxl3al6BLOOml8Zk78e67IGPbtk3KKTtT4GXizkkRzVHvFb67fUmTkEVABdG64zoxeClRwKlgE7ubaBYTOiID1jE0IiHTXjo9ZoJPjNLHgVTmRYCn6u+JlIRaj0PfJEMCQ73oZeK/XqaAlEIvfACCSy/lUZwAi+hsf5AIDBJnheE+V4yCGBtCqOLmBEyHRBEKptasG3exiWXSPC+71XL1tlKsVeYt5dEROkanyEUXqIZuUB01EEWP6Bm9oFfryXqz3q2PWTRnzWcO0R9YXz/bgqKG</latexit> u = [u0, · · · , up 1]> <latexit sha1_base64="3wvy4p/rWCAzKfzzcbwA58Y6W24=">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</latexit> min u0,··· ,up 1 p 1 X t=0 (ˆ yt rt)2 + t(ut ut 1)2 + 0 @ µ ⇢ 2 ✓ + nt + µ ⇢ 2 + ✓ nt 4 ⇢ 1 A
  28. T. Parisini and R. Zoppoli § (FNN) § § x[k]

    § FNN( ) 55 https://fr.wikipedia.org/wiki/R%C3%A9seau_de_neurones_%C3%A0_action_directe
  29. <latexit sha1_base64="fKQy4fvbzO1+4xkl3S9SkmdRlnM=">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</latexit> min u0,··· ,up 1 p 1 X t=0

    `t(xt, ut) + `p(xp) T. Parisini and R. Zoppoli § k § k u0 § k curse of dimensionality FNN 56 <latexit sha1_base64="SbkwophhDdRZdoTSi+PTBsuyVWg=">AAACK3icbZDLSgMxFIYz9VbrrerSTbAIldYyI/WyEQpuXFawF2jLkEkzNTQzGZIz0jL0IXwQ1271GVwpbl34FqaXhbb+EPj5zjmcnN+LBNdg2+9Waml5ZXUtvZ7Z2Nza3snu7tW1jBVlNSqFVE2PaCZ4yGrAQbBmpBgJPMEaXv96XG88MKW5DO9gGLFOQHoh9zklYJCbLQzcBArOCF9hPz9woYhjF46LuI3BINsY2pWgizg6cdxszi7ZE+FF48xMDs1UdbPf7a6kccBCoIJo3XLsCDoJUcCpYKNMO9YsIrRPeqxlbEgCpjvJ5KgRPjKki32pzAsBT+jviYQEWg8Dz3QGBO71fG0M/62NCUgp9NwHwL/sJDyMYmAhne73Y4FB4nFwuMsVoyCGxhCquDkB03uiCAUTb8Zk48wnsWjqpyXnvHR2W85VyrOU0ugAHaI8ctAFqqAbVEU1RNEjekYv6NV6st6sD+tz2pqyZjP76I+srx9VE6Sr</latexit> xt+1 = f(xt, ut), t = 0, · · · , p 1 <latexit sha1_base64="4XkrTQgnLggvIWr9g9yBHaZIXP4=">AAACCHicbVDLSgNBEJyNrxhfUY9eBoPgKexKfFyEgBePEcwDkyXMTmaTIbMzy0yvJIT8gGev+g3exKt/4Sf4F84me9DEgoaiqpvuriAW3IDrfjm5ldW19Y38ZmFre2d3r7h/0DAq0ZTVqRJKtwJimOCS1YGDYK1YMxIFgjWD4U3qNx+ZNlzJexjHzI9IX/KQUwJWehh1XXyNR+2h3y2W3LI7A14mXkZKKEOtW/zu9BRNIiaBCmJM23Nj8CdEA6eCTQudxLCY0CHps7alkkTM+JPZxVN8YpUeDpW2JQHP1N8TExIZM44C2xkRGJhFLxX/9VIFlBJm4QAIr/wJl3ECTNL5/jARGBROU8E9rhkFMbaEUM3tC5gOiCYUbHYFm423mMQyaZyVvYvy+V2lVK1kKeXRETpGp8hDl6iKblEN1RFFEj2jF/TqPDlvzrvzMW/NOdnMIfoD5/MHAm2Z1Q==</latexit> x0 = x[k] subject to <latexit sha1_base64="sTGHTCL47RYQM0eoV6yVI1Z8FDs=">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</latexit> xt 2 X, t = 1, · · · , p, ut 2 U, t = 0, · · · , p 1 <latexit sha1_base64="+wohoKj6jxuGfCZO6wzWbnKr/uM=">AAACJXicbVDLSgMxFM34rPVVdekmWAoVapkRUZcFN+Kqgn1AOwyZNG1DM8mQ3BHL0C/wQ1y71W9wJ4Ir1/6F6WOhrQcuHM65N7n3hLHgBlz301laXlldW89sZDe3tnd2c3v7daMSTVmNKqF0MySGCS5ZDTgI1ow1I1EoWCMcXI39xj3Thit5B8OY+RHpSd7llICVglyhTXSvHXEZpEngltodBaaEkyCNT7zRCN8UH1oD/zjI5d2yOwFeJN6M5NEM1SD3bV+iScQkUEGMaXluDH5KNHAq2CjbTgyLCR2QHmtZKknEjJ9OzhnhglU6uKu0LQl4ov6eSElkzDAKbWdEoG/mvbH4rzdWQClh5haA7qWfchknwCSd/t9NBAaFx5HhDteMghhaQqjm9gRM+0QTCjbYrM3Gm09ikdRPy955+fz2LF85m6WUQYfoCBWRhy5QBV2jKqohih7RM3pBr86T8+a8Ox/T1iVnNnOA/sD5+gFJxKUF</latexit> arg min u0,...,up 1 J(x[k]) <latexit sha1_base64="MssXBBfILdaQ3P+7T8qeRcg8ASI=">AAACEnicbZDLSgMxFIYz9VbrbaxLN8Ei1E2ZkVLdCAU3LivYC7RDyaRpG5rLkGSkZehbuHarz+BO3PoCPoJvYaadhbb+EPj4zzmckz+MGNXG876c3Mbm1vZOfrewt39weOQeF1taxgqTJpZMqk6INGFUkKahhpFOpAjiISPtcHKb1tuPRGkqxYOZRSTgaCTokGJkrNV3i3F3Etz0RohzVJ5avoB9t+RVvIXgOvgZlECmRt/97g0kjjkRBjOkddf3IhMkSBmKGZkXerEmEcITNCJdiwJxooNkcfscnltnAIdS2ScMXLi/JxLEtZ7x0HZyZMZ6tZaa/9ZSx0jJ9MoBZngdJFREsSECL/cPYwaNhGk+cEAVwYbNLCCsqP0CxGOkEDY2xYLNxl9NYh1alxW/VqndV0v1apZSHpyCM1AGPrgCdXAHGqAJMJiCZ/ACXp0n5815dz6WrTknmzkBf+R8/gDYQZ14</latexit> u[k] = (x[k]) <latexit sha1_base64="pBd+HyYWmRnJrF/YcgSuohtg/kY=">AAACBXicbVDLSsNAFJ3UV62vqks3wSLUTUmkVJcFN+Kqgn1AG8pkOmmHTGbCzI1YQteu3eo3uBO3foef4F84abPQ1gMXDufcy733+DFnGhznyyqsrW9sbhW3Szu7e/sH5cOjjpaJIrRNJJeq52NNORO0DQw47cWK4sjntOuH15nffaBKMynuYRpTL8JjwQJGMBipe1t97Ife+bBccWrOHPYqcXNSQTlaw/L3YCRJElEBhGOt+64Tg5diBYxwOisNEk1jTEI8pn1DBY6o9tL5uTP7zCgjO5DKlAB7rv6eSHGk9TTyTWeEYaKXvUz818sUkJLrpQMguPJSJuIEqCCL/UHCbZB2Fok9YooS4FNDMFHMvGCTCVaYgAmuZLJxl5NYJZ2LmtuoNe7qlWY9T6mITtApqiIXXaImukEt1EYEhegZvaBX68l6s96tj0VrwcpnjtEfWJ8/FqGYzw==</latexit> J(x[k])
  30. T. Parisini and R. Zoppoli § FNN § x §

    q 57 <latexit sha1_base64="june7amgC4W11d8iIkptqDv3znM=">AAACEnicbVDLSgNBEJz1GeMrxqOXwSAoSNiVoB4DXjxGMFEwS+idTJLBmZ1lplcNS/7Cs1f9Bm/i1R/wE/wLJ8keNFrQUFR1090VJVJY9P1Pb25+YXFpubBSXF1b39gsbZVbVqeG8SbTUpvrCCyXIuZNFCj5dWI4qEjyq+j2bOxf3XFjhY4vcZjwUEE/Fj3BAJ3UKZXbA8Cs3QelYLT/cHh/QDulil/1J6B/SZCTCsnR6JS+2l3NUsVjZBKsvQn8BMMMDAom+ajYTi1PgN1Cn984GoPiNswmt4/onlO6tKeNqxjpRP05kYGydqgi16kAB3bWG4v/emMFtZZ25gDsnYaZiJMUecym+3uppKjpOB/aFYYzlENHgBnhXqBsAAYYuhSLLptgNom/pHVUDY6rxxe1Sr2Wp1QgO2SX7JOAnJA6OScN0iSMPJAn8kxevEfv1Xvz3qetc14+s01+wfv4Bjv2nbQ=</latexit> ˆ(x, w) <latexit sha1_base64="dKRsw3Ldsw8WSAlkB4f1tiOkQjc=">AAACB3icbVDLTgJBEOzFF+IL9ehlIjHBC9k1BD2SePGIiTwMbMjsMAsTZnY2M7NGQvgAz171G7wZr36Gn+BfOAt7ULCSTipV3enuCmLOtHHdLye3tr6xuZXfLuzs7u0fFA+PWlomitAmkVyqToA15SyiTcMMp51YUSwCTtvB+Dr12w9UaSajOzOJqS/wMGIhI9hY6b43xELg8uN5v1hyK+4caJV4GSlBhka/+N0bSJIIGhnCsdZdz42NP8XKMMLprNBLNI0xGeMh7VoaYUG1P50fPENnVhmgUCpbkUFz9ffEFAutJyKwnQKbkV72UvFfL1WMlFwvHWDCK3/KojgxNCKL/WHCkZEoDQUNmKLE8IklmChmX0BkhBUmxkZXsNl4y0msktZFxatVarfVUr2apZSHEziFMnhwCXW4gQY0gYCAZ3iBV+fJeXPenY9Fa87JZo7hD5zPH+xVmdU=</latexit> (x) <latexit sha1_base64="dKRsw3Ldsw8WSAlkB4f1tiOkQjc=">AAACB3icbVDLTgJBEOzFF+IL9ehlIjHBC9k1BD2SePGIiTwMbMjsMAsTZnY2M7NGQvgAz171G7wZr36Gn+BfOAt7ULCSTipV3enuCmLOtHHdLye3tr6xuZXfLuzs7u0fFA+PWlomitAmkVyqToA15SyiTcMMp51YUSwCTtvB+Dr12w9UaSajOzOJqS/wMGIhI9hY6b43xELg8uN5v1hyK+4caJV4GSlBhka/+N0bSJIIGhnCsdZdz42NP8XKMMLprNBLNI0xGeMh7VoaYUG1P50fPENnVhmgUCpbkUFz9ffEFAutJyKwnQKbkV72UvFfL1WMlFwvHWDCK3/KojgxNCKL/WHCkZEoDQUNmKLE8IklmChmX0BkhBUmxkZXsNl4y0msktZFxatVarfVUr2apZSHEziFMnhwCXW4gQY0gYCAZ3iBV+fJeXPenY9Fa87JZo7hD5zPH+xVmdU=</latexit> (x) <latexit sha1_base64="june7amgC4W11d8iIkptqDv3znM=">AAACEnicbVDLSgNBEJz1GeMrxqOXwSAoSNiVoB4DXjxGMFEwS+idTJLBmZ1lplcNS/7Cs1f9Bm/i1R/wE/wLJ8keNFrQUFR1090VJVJY9P1Pb25+YXFpubBSXF1b39gsbZVbVqeG8SbTUpvrCCyXIuZNFCj5dWI4qEjyq+j2bOxf3XFjhY4vcZjwUEE/Fj3BAJ3UKZXbA8Cs3QelYLT/cHh/QDulil/1J6B/SZCTCsnR6JS+2l3NUsVjZBKsvQn8BMMMDAom+ajYTi1PgN1Cn984GoPiNswmt4/onlO6tKeNqxjpRP05kYGydqgi16kAB3bWG4v/emMFtZZ25gDsnYaZiJMUecym+3uppKjpOB/aFYYzlENHgBnhXqBsAAYYuhSLLptgNom/pHVUDY6rxxe1Sr2Wp1QgO2SX7JOAnJA6OScN0iSMPJAn8kxevEfv1Xvz3qetc14+s01+wfv4Bjv2nbQ=</latexit> ˆ(x, w) <latexit sha1_base64="azBQDh3VXnyJ1JfTO/Udfn/hWzM=">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</latexit> min w Z X k (x) ˆ(x, w)k2dx <latexit sha1_base64="HZhYCpPbMQDBP1mN0yicimPdpT8=">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</latexit> yq(s) = g " ns 1 X i=1 wiq(s)yi(s 1) + w0q(s) # , s = 1, . . . , L, q = 1, . . . , ns <latexit sha1_base64="32ODzTC8F2ac3KVDnsfpQm6StSg=">AAACDHicbVDLSgNBEJyNrxhfUY9eBoMQL2FXQvQiBLx4jGAekKxhdjKbDJmdWWZ6JWHJL3j2qt/gTbz6D36Cf+HkcdDEgoaiqpvuriAW3IDrfjmZtfWNza3sdm5nd2//IH941DAq0ZTVqRJKtwJimOCS1YGDYK1YMxIFgjWD4c3Ubz4ybbiS9zCOmR+RvuQhpwSs9NBvj3x83QEiB8XReTdfcEvuDHiVeAtSQAvUuvnvTk/RJGISqCDGtD03Bj8lGjgVbJLrJIbFhA5Jn7UtlSRixk9nV0/wmVV6OFTalgQ8U39PpCQyZhwFtjMiMDDL3lT815sqoJQwSwdAeOWnXMYJMEnn+8NEYFB4mgzucc0oiLElhGpuX8B0QDShYPPL2Wy85SRWSeOi5FVKlbtyoVpepJRFJ+gUFZGHLlEV3aIaqiOKNHpGL+jVeXLenHfnY96acRYzx+gPnM8fTcObow==</latexit> g[x] = tanh(x)
  31. T. Parisini and R. Zoppoli § § § ⇨ x

    § § x(i) § 58 <latexit sha1_base64="T1bV6h7c8sABx6tFcD9yHtsOnlg=">AAACKHicbVDLSgNBEJz1GeMr6tHLYhAiaNgVUS+CoAePEUwiuDH0TibJ4MzOMtOrCWt+wQ/x7FW/wZvk6s2/cPI4aLSgoaaqm+muMBbcoOf1nanpmdm5+cxCdnFpeWU1t7ZeMSrRlJWpEkpfh2CY4BErI0fBrmPNQIaCVcO7s4FfvWfacBVdYTdmNQmtiDc5BbRSPVc4L3R2H3ZOgsegBVJCobOzF7QB09GzN3SDx9v9ei7vFb0h3L/EH5M8GaNUz30FDUUTySKkAoy58b0Yaylo5FSwXjZIDIuB3kGL3VgagWSmlg4v6rnbVmm4TaVtRegO1Z8TKUhjujK0nRKwbSa9gfivN1BQKWEmFsDmcS3lUZwgi+jo/2YiXFTuIDW3wTWjKLqWANXcnuDSNmigaLPN2mz8yST+ksp+0T8sHl4e5E8PxillyCbZIgXikyNySi5IiZQJJU/khbySN+fZeXc+nP6odcoZz2yQX3A+vwEYdKXy</latexit> D(x, w) = k (x) ˆ(x, w)k2 <latexit sha1_base64="+EedBidCz6KvU5aYxoC+IaSLIzE=">AAACC3icbVDLSgNBEJyNrxhfUY9eBoMQL2FXQvQYEMFjBPOAZAmzk9lkyDzWmVlDWPIJnr3qN3gTr36En+BfOJvsQRMLGoqqbrq7gohRbVz3y8mtrW9sbuW3Czu7e/sHxcOjlpaxwqSJJZOqEyBNGBWkaahhpBMpgnjASDsYX6d++5EoTaW4N9OI+BwNBQ0pRsZKPuxxKvoTCG/Kk/N+seRW3DngKvEyUgIZGv3id28gccyJMJghrbueGxk/QcpQzMis0Is1iRAeoyHpWioQJ9pP5kfP4JlVBjCUypYwcK7+nkgQ13rKA9vJkRnpZS8V//VSxUjJ9NIBJrzyEyqi2BCBF/vDmEEjYRoMHFBFsGFTSxBW1L4A8QgphI2Nr2Cz8ZaTWCWti4pXq9TuqqV6NUspD07AKSgDD1yCOrgFDdAEGDyAZ/ACXp0n5815dz4WrTknmzkGf+B8/gC4HZq4</latexit> min w E(w) <latexit sha1_base64="IBFA7mX+j+XCOYWoLr2vhqIie/8=">AAACIHicbVBNSwMxEM3Wr1q/qh69BItQQcquFPUiCCp4rGBtwZYlm03b0GyyJLNqWXr1h3j2qr/Bm3jUf+C/MFt70NYHA4/3ZpiZF8SCG3DdDyc3Mzs3v5BfLCwtr6yuFdc3ro1KNGV1qoTSzYAYJrhkdeAgWDPWjESBYI2gf5r5jVumDVfyCgYxa0ekK3mHUwJW8ov4vHy3e9ziEvy0FRHoUSLS5nB4Vr7fu9vF4b1fLLkVdwQ8TbwxKaExan7xqxUqmkRMAhXEmBvPjaGdEg2cCjYstBLDYkL7pMtuLJUkYqadjj4Z4h2rhLijtC0JeKT+nkhJZMwgCmxndqyZ9DLxXy9TQClhJg6AzlE75TJOgEn6s7+TCAwKZ2nhkGtGQQwsIVRz+wKmPaIJBZtpwWbjTSYxTa73K95B5eCyWjqpjlPKoy20jcrIQ4foBF2gGqojih7QE3pGL86j8+q8Oe8/rTlnPLOJ/sD5/Aa/c6M+</latexit> E(w) = Z X D(x, w)dx <latexit sha1_base64="UQ76fSg7ttzFwtzgO6JKUtzU5eQ=">AAACCXicbVDLSgNBEJyNrxhfUY9eBoMQL2FXQvQYEMFjBPOAZAmzk0kyZHZmmelVwpIv8OxVv8GbePUr/AT/wtlkD5pY0FBUddPdFUSCG3DdLye3tr6xuZXfLuzs7u0fFA+PWkbFmrImVULpTkAME1yyJnAQrBNpRsJAsHYwuU799gPThit5D9OI+SEZST7klICVuj1JAkHwTfnxvF8suRV3DrxKvIyUUIZGv/jdGygah0wCFcSYrudG4CdEA6eCzQq92LCI0AkZsa6lkoTM+Mn85Bk+s8oAD5W2JQHP1d8TCQmNmYaB7QwJjM2yl4r/eqkCSgmzdAAMr/yEyygGJuli/zAWGBROY8EDrhkFMbWEUM3tC5iOiSYUbHgFm423nMQqaV1UvFqldlct1atZSnl0gk5RGXnoEtXRLWqgJqJIoWf0gl6dJ+fNeXc+Fq05J5s5Rn/gfP4A1PmaSA==</latexit> rE(w) <latexit sha1_base64="vt6bJfINRZpSWZV1Ncj79OBdnPI=">AAACA3icbVBNS8NAEN3Ur1q/qh69LBahXkoipXosiOCxgmkLbSib7aZdusmG3YlSQo+evepv8CZe/SH+BP+FmzYHbX0w8Hhvhpl5fiy4Btv+sgpr6xubW8Xt0s7u3v5B+fCorWWiKHOpFFJ1faKZ4BFzgYNg3VgxEvqCdfzJdeZ3HpjSXEb3MI2ZF5JRxANOCRjJxTfVx/NBuWLX7DnwKnFyUkE5WoPyd38oaRKyCKggWvccOwYvJQo4FWxW6ieaxYROyIj1DI1IyLSXzo+d4TOjDHEglakI8Fz9PZGSUOtp6JvOkMBYL3uZ+K+XKSCl0EsHQHDlpTyKE2ARXewPEoFB4iwQPOSKURBTQwhV3LyA6ZgoQsHEVjLZOMtJrJL2Rc1p1Bp39UqznqdURCfoFFWRgy5RE92iFnIRRRw9oxf0aj1Zb9a79bFoLVj5zDH6A+vzBx52l7I=</latexit> E(w) <latexit sha1_base64="HbARAQpI3rshpNx9jp/fZwtfi/0=">AAACBHicbVDLSsNAFJ3UV62vqks3g0WoICWRUl0WdOGygn1AG8pkOmnHTjJh5kYtoVvXbvUb3Ilb/8NP8C+ctFlo9cCFwzn3cu89XiS4Btv+tHJLyyura/n1wsbm1vZOcXevpWWsKGtSKaTqeEQzwUPWBA6CdSLFSOAJ1vbGF6nfvmNKcxnewCRibkCGIfc5JWCk1mX54eT+uF8s2RV7BvyXOBkpoQyNfvGrN5A0DlgIVBCtu44dgZsQBZwKNi30Ys0iQsdkyLqGhiRg2k1m107xkVEG2JfKVAh4pv6cSEig9STwTGdAYKQXvVT810sVkFLohQPAP3cTHkYxsJDO9/uxwCBxmggecMUoiIkhhCpuXsB0RBShYHIrmGycxST+ktZpxalVatfVUr2apZRHB+gQlZGDzlAdXaEGaiKKbtETekYv1qP1ar1Z7/PWnJXN7KNfsD6+ARfumD8=</latexit> D(x, w) <latexit sha1_base64="RcE9DrxaCRHiJXnJNjZJWm1mk+E=">AAACCXicbVDLSgMxFL1TX7W+qi7dBIvgqsxIqS4LblxWsA9oh5JJ0zY0MxmSO0IZ+gWu3eo3uBO3foWf4F+YaWehrQcCh3Pu5Z6cIJbCoOt+OYWNza3tneJuaW//4PCofHzSNirRjLeYkkp3A2q4FBFvoUDJu7HmNAwk7wTT28zvPHJthIoecBZzP6TjSIwEo2ilXj+kOGFUpt35oFxxq+4CZJ14OalAjuag/N0fKpaEPEImqTE9z43RT6lGwSSfl/qJ4TFlUzrmPUsjGnLjp4vIc3JhlSEZKW1fhGSh/t5IaWjMLAzsZBbRrHqZ+K+XKaiUNCsBcHTjpyKKE+QRW94fJZKgIlktZCg0ZyhnllCmhf0CYROqKUNbXsl24602sU7aV1WvXq3f1yqNWt5SEc7gHC7Bg2towB00oQUMFDzDC7w6T86b8+58LEcLTr5zCn/gfP4AcOObRw==</latexit> X <latexit sha1_base64="7EHnAMBEwy0r2c/vobdjRPKWn+s=">AAACC3icbVDLSgNBEOyNrxhfUY9eBoMQL2FXQvQYEMFjBPOAZAmzk9lkyOzMOjNrCEs+wbNX/QZv4tWP8BP8CyePgyYWNBRV3XR3BTFn2rjul5NZW9/Y3Mpu53Z29/YP8odHDS0TRWidSC5VK8CaciZo3TDDaStWFEcBp81geD31m49UaSbFvRnH1I9wX7CQEWys5HcEDjjujtBNcXTezRfckjsDWiXeghRggVo3/93pSZJEVBjCsdZtz42Nn2JlGOF0kuskmsaYDHGfti0VOKLaT2dHT9CZVXoolMqWMGim/p5IcaT1OApsZ4TNQC97U/Ffb6oYKbleOsCEV37KRJwYKsh8f5hwZCSaBoN6TFFi+NgSTBSzLyAywAoTY+PL2Wy85SRWSeOi5FVKlbtyoVpepJSFEziFInhwCVW4hRrUgcADPMMLvDpPzpvz7nzMWzPOYuYY/sD5/AGACpsy</latexit> rwE(w) <latexit sha1_base64="mPD5my4HgAQzLog8hqR5Y/L8LxU=">AAACD3icbVDLSgMxFM3UV62Pjrp0EyxCBSkzUqrLgi5cVrAPaIchk6ZtaCYZkoy1DP0I1271G9yJWz/BT/AvzLSz0NYDFw7n3Mu99wQRo0o7zpeVW1vf2NzKbxd2dvf2i/bBYUuJWGLSxIIJ2QmQIoxy0tRUM9KJJEFhwEg7GF+nfvuBSEUFv9fTiHghGnI6oBhpI/l2scdRwJA/gTflx/PJmW+XnIozB1wlbkZKIEPDt797fYHjkHCNGVKq6zqR9hIkNcWMzAq9WJEI4TEakq6hHIVEecn88Bk8NUofDoQ0xTWcq78nEhQqNQ0D0xkiPVLLXir+66WKFoKppQP04MpLKI9iTThe7B/EDGoB03Bgn0qCNZsagrCk5gWIR0girE2EBZONu5zEKmldVNxapXZXLdWrWUp5cAxOQBm44BLUwS1ogCbAIAbP4AW8Wk/Wm/VufSxac1Y2cwT+wPr8AVO0nBo=</latexit> rwD(x, w) <latexit sha1_base64="rCdGskCXp0tmmIuctSrYG+0/+uM=">AAACPXicbZDNSgMxFIUz/tb6V3XpJliEFmuZEVE3gqALlwpWBaeUO2lqQzPJkNyxluLT+CCu3erKB3AhiFu3ZmoXWr0Q+HLOvST3RIkUFn3/xRsbn5icms7N5Gfn5hcWC0vL51anhvEa01Kbywgsl0LxGgqU/DIxHOJI8ouoc5j5FzfcWKHVGfYSXo/hWomWYIBOahT2uyWxEZTpPnVQ3gxBJm1wFCqIJDS69Kh0666VzC1XQipcp18JKmFTo6WNQtGv+oOifyEYQpEM66RReHODLI25QibB2qvAT7DeB4OCSX6XD1PLE2AduOZXDhXE3Nb7gzXv6LpTmrSljTsK6UD9OdGH2NpeHLnOGLBtR71M/NfLFNRa2pEPYGuv3hcqSZEr9v1+K5UUNc2ipE1hOEPZcwDMCLcCZW0wwNAFnnfZBKNJ/IXzrWqwU9053S4ebA9TypFVskZKJCC75IAckxNSI4zck0fyRJ69B+/Ve/c+vlvHvOHMCvlV3ucX97aqWw==</latexit> w(i + 1) = w(i) ↵(i)rwD(x(i), w(i)), i = 0, 1, . . . x
  32. 59 § DNN § ReLU MPC § § § §

    § § MPC § LSTM § §