Continuous-time Stochastic Grey-box Model of the Nonlinear Feedback System based on Residual Analysis Rune Grønborg Junker Workshop on Nonlinear Benchmarks TU Eindhoven, April, 10, 2019 Rishi Relan 28 March 2019 DTU Compute 1
Modelling Philosophies 28 March 2019 DTU Compute 2
<latexit sha1_base64="CPNKUJxJDu7WBncys+oG4e96xk=">ACHXicbVDLSgMxFM3UV62vqks3wSJUlDLTFnQjFN24rGAf0I5DJpO2oZkHyR1pGfojbvwVNy4UceFG/BvTB6itB8I9OfeR3ONGgiswzS8jtbS8srqWXs9sbG5t72R39+oqjCVlNRqKUDZdopjgAasB8GakWTEdwVruP2rcb5xz6TiYXALw4jZPukGvMpAS052bLf9rwQksEoD8cn2Gv/XPqONZjG0uCupBm+wLEOpxknmzML5gR4kVgzkMzVJ3sh5LY58FQAVRqmWZEdgJkcCpYKNMO1YsIrRPuqylaUB8puxkst0IH2nFw51Q6hMAnqi/OxLiKzX0XV3pE+ip+dxY/C/XiqFzbic8iGJgAZ0+1IkFhCPrcIel4yCGpCqOT6r5j2iCQUtKFjE6z5lRdJvViwSoXiTlXuZzZkUYH6BDlkYXOUAVdoyqIYoe0BN6Qa/Go/FsvBnv09KUMevZR39gfH4D1kefLQ=</latexit> Silverbox Benchmark x ( t ) + k 1 x ( t ) + k 3 x 3 ( t ) = u ( t ) , m ¨ x ( t ) + d ˙ 28 March 2019 DTU Compute 3
<latexit sha1_base64="w3z4tdLo21OJR6jWrFZL8fgwky8=">ACF3icbVC7TgMxEPTxDOEVoKSxEiHREN2FAsoIGgqKgMhLSTj5HCex4vOd7D2k6HR/QcNfUNQgICBiQK/gbnUDCSJbGM7va3fFCwTXY9rc1N7+wuLScWkmvrq1vbGa2tis6iBRlZRqIQNU8opngkpWBg2C1UDHie4JVvf7p0K/eMKV5IK9gELKWT7qSdzglYCQ3k6+7Mbj9BDe5xE2fQI8SEdfNX0eZjDWPC+TK7j8TN5Oy8PQKeJc6E5IrZ9f7g/pXyc18NtsBjXwmgQqidcOxQ2jFRAGngiXpZqRZSGifdFnDUEl8plvx6K4E7xmljTuBMk8CHqm/O2Liaz3wPVM53FJPe0PxP68RQe4FXMZRsAkHQ/qRAJDgIch4TZXjIYGEKo4mZXTHtEQomyrQJwZk+eZUCnMF+4MGmcoDFSaBdl0T5y0BEqojNUQmVE0S16QE/o2bqzHq0X621cOmdNenbQH1gfPxy4pEc=</latexit> <latexit sha1_base64="avq70iNCGhQ9n6XLG2uAeWxiCA=">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</latexit> <latexit sha1_base64="zHs9+HV3fyzf5H18X2hPCzWoX+E=">ACI3icbVDLSgMxFM34rPVdekmWIQWSplRQXElunElFW0VOmXIpLdtaCYzJHeEMvRf3PgrblwoxY0L/8X0sfB1IbmHc+4lOSdMpDouh/O3PzC4tJybiW/ura+sVnY2m6YONUc6jyWsb4PmQEpFNRoIT7RAOLQgl3Yf9irN89gDYiVrc4SKAVsa4SHcEZWionEKQYdAfUl8o6kcMe5zJ7GpYciv0ZiqV6tNeofauUB97gKxcpkGh6FbdSdG/wJuBIplVLSiM/HbM0wgUcsmMaXpugq2MaRcwjDvpwYSxvusC0LFYvAtLKJxyHdt0ybdmJtj0I6Yb9vZCwyZhCFdnLswvzWxuR/WjPFzkrEypJERSfPtRJcWYjgOjbaGBoxYwLgW9q+U95hmHG2seRuC9vyX9A4qHqH1YPro+LZ+SyOHNkle6REPHJMzsglqZE64eSRPJNX8uY8OS/OyHmfjs45s50d8qOczy8BHaM0</latexit> <latexit sha1_base64="hb2nARE+qc3xor3yPM/EA+axkQ=">ACEXicbVDLSgMxFM3UV62vqks3wSJ0VWaqoMuiG5dV7AM6tSRp2oZmMkNyRyhDf8GNv+LGhSJu3bnzb8y0s9DWA4HDOeSew+NpDgut9ObmV1bX0jv1nY2t7Z3SvuHzRNGvGyUoW5TYrgUijdAgOTtSHMSUMlbdHyV+q0Hro0I1R1MIt4NyFCJgWAErNQrlts9wL5Q2A8IjBiRSXuKfRNTw2GuUZrcTu9tORW3BnwMvEyUkIZ6r3il98PWRxwBUwSYzqeG0E3IRoEk3xa8GPDI8LGZMg7lioScNZhdN8YlV+ngQavsU4Jn6eyIhgTGTgNpkuqNZ9FLxP68Tw+CimwgVxcAVm380iCWGEKf14L7QnIGcWEKYFnZXzEZEwa2xItwVs8eZk0qxXvtFK9OSvVLrM68ugIHaMy8tA5qFrVEcNxNAjekav6M15cl6cd+djHs052cwh+gPn8weAjp1</latexit> <latexit sha1_base64="2KlTFU3zWPGmceOLaGKSMVW5+c=">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</latexit> <latexit sha1_base64="Vt9lcBLYl4FcUmrzeRGe5IQ7RQ=">ACD3icbZC7SgNBFIZnvcZ4i1raDAbFKuxGQcugjWU3CAbw+zkJBkyO7vMnBXCkjew8VsLBSxtbXzbZxcCk38YeDjP+cw5/xBLIVB1/12lpZXVtfWMxvZza3tnd3c3n7NRInmUOWRjHQjYAakUFBFgRIasQYWBhLqweB6XK8/gDYiUhUcxtAKWU+JruAMrdXOnfjYB2TUF4r6lSmaJDCA1A8Z9oMgvRvdx+1c3i24E9F8GaQJzOV27kvxPxJASFXDJjmp4bYytlGgWXMr6iYGY8QHrQdOiYiGYVjq5Z0SPrdOh3Ujbp5BO3N8TKQuNGYaB7RzvaOZrY/O/WjPB7mUrFSpOEBSftRNJMWIjsOhHaGBoxaYFwLuyvlfaYZRxth1obgzZ+8CLViwTsrFG/P86WrWRwZckiOyCnxyAUpkRtSJlXCySN5Jq/kzXlyXpx352PauTMZg7IHzmfP7OtnHY=</latexit> <latexit sha1_base64="nJFXjlrheozQMxlw+mgKD3y5MFc=">ACHicbVDLSgMxFM3UV62vqks3wSIoSJlpBd0IRTcuq7SdQqcdMmnahmYyQ3JHKEM/xI2/4saFIm5cCP6N6WOh1gMJ+fcS+49QSy4Btv+sjJLyura9n13Mbm1vZOfnevoaNEUVankYhUMyCaCS5ZHTgI1owVI2EgmBsMrye+e8+U5pGswShm7ZD0Je9xSsBIfr7s+oAv8bHbcXw4dTslc3u0G4E2j9CHk04Ne1xiLyQwCIL0bmzUfMEu2lPgReLMSQHNUfXzH143oknIJFBtG45dgztlCjgVLBxzks0iwkdkj5rGSpJyHQ7nS43xkdG6eJepMyRgKfqz46UhFqPwsBUTmbUf72J+J/XSqB30U65jBNgks4+6iUCQ4QnSeEuV4yCGBlCqOJmVkwHRBEKJs+cCcH5u/IiaZSKTrlYuj0rVK7mcWTRATpEx8hB56iCblAV1RFD+gJvaBX69F6t6s91lpxpr37KNfsD6/Ad0AoA4=</latexit> Stochastic Differential Equations d X t = f t ( X t , U t , t, θ )d t + σ t ( X t , U t , t, θ )d W t , | {z } | {z } Drift Di ff usion X t ∈ X ⊂ R n Y t k = g t k ( X t k , U t k , t k , θ ) + e t k , |{z} Measurement noise Y t k ∈ Y ⊂ R L t ) T ∈ R m W t = ( W 1 t , W 2 t , · · · , W m e t k ∈ N (0 , S t k ( U t k , t k , θ )) θ ∈ Θ ⊂ R p 28 March 2019 DTU Compute 4
<latexit sha1_base64="w3z4tdLo21OJR6jWrFZL8fgwky8=">ACF3icbVC7TgMxEPTxDOEVoKSxEiHREN2FAsoIGgqKgMhLSTj5HCex4vOd7D2k6HR/QcNfUNQgICBiQK/gbnUDCSJbGM7va3fFCwTXY9rc1N7+wuLScWkmvrq1vbGa2tis6iBRlZRqIQNU8opngkpWBg2C1UDHie4JVvf7p0K/eMKV5IK9gELKWT7qSdzglYCQ3k6+7Mbj9BDe5xE2fQI8SEdfNX0eZjDWPC+TK7j8TN5Oy8PQKeJc6E5IrZ9f7g/pXyc18NtsBjXwmgQqidcOxQ2jFRAGngiXpZqRZSGifdFnDUEl8plvx6K4E7xmljTuBMk8CHqm/O2Liaz3wPVM53FJPe0PxP68RQe4FXMZRsAkHQ/qRAJDgIch4TZXjIYGEKo4mZXTHtEQomyrQJwZk+eZUCnMF+4MGmcoDFSaBdl0T5y0BEqojNUQmVE0S16QE/o2bqzHq0X621cOmdNenbQH1gfPxy4pEc=</latexit> <latexit sha1_base64="hb2nARE+qc3xor3yPM/EA+axkQ=">ACEXicbVDLSgMxFM3UV62vqks3wSJ0VWaqoMuiG5dV7AM6tSRp2oZmMkNyRyhDf8GNv+LGhSJu3bnzb8y0s9DWA4HDOeSew+NpDgut9ObmV1bX0jv1nY2t7Z3SvuHzRNGvGyUoW5TYrgUijdAgOTtSHMSUMlbdHyV+q0Hro0I1R1MIt4NyFCJgWAErNQrlts9wL5Q2A8IjBiRSXuKfRNTw2GuUZrcTu9tORW3BnwMvEyUkIZ6r3il98PWRxwBUwSYzqeG0E3IRoEk3xa8GPDI8LGZMg7lioScNZhdN8YlV+ngQavsU4Jn6eyIhgTGTgNpkuqNZ9FLxP68Tw+CimwgVxcAVm380iCWGEKf14L7QnIGcWEKYFnZXzEZEwa2xItwVs8eZk0qxXvtFK9OSvVLrM68ugIHaMy8tA5qFrVEcNxNAjekav6M15cl6cd+djHs052cwh+gPn8weAjp1</latexit> <latexit sha1_base64="nJFXjlrheozQMxlw+mgKD3y5MFc=">ACHicbVDLSgMxFM3UV62vqks3wSIoSJlpBd0IRTcuq7SdQqcdMmnahmYyQ3JHKEM/xI2/4saFIm5cCP6N6WOh1gMJ+fcS+49QSy4Btv+sjJLyura9n13Mbm1vZOfnevoaNEUVankYhUMyCaCS5ZHTgI1owVI2EgmBsMrye+e8+U5pGswShm7ZD0Je9xSsBIfr7s+oAv8bHbcXw4dTslc3u0G4E2j9CHk04Ne1xiLyQwCIL0bmzUfMEu2lPgReLMSQHNUfXzH143oknIJFBtG45dgztlCjgVLBxzks0iwkdkj5rGSpJyHQ7nS43xkdG6eJepMyRgKfqz46UhFqPwsBUTmbUf72J+J/XSqB30U65jBNgks4+6iUCQ4QnSeEuV4yCGBlCqOJmVkwHRBEKJs+cCcH5u/IiaZSKTrlYuj0rVK7mcWTRATpEx8hB56iCblAV1RFD+gJvaBX69F6t6s91lpxpr37KNfsD6/Ad0AoA4=</latexit> <latexit sha1_base64="avq70iNCGhQ9n6XLG2uAeWxiCA=">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</latexit> <latexit sha1_base64="f2sgEWK6mex8SehMqiohe5oYHI=">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</latexit> <latexit sha1_base64="Vt9lcBLYl4FcUmrzeRGe5IQ7RQ=">ACD3icbZC7SgNBFIZnvcZ4i1raDAbFKuxGQcugjWU3CAbw+zkJBkyO7vMnBXCkjew8VsLBSxtbXzbZxcCk38YeDjP+cw5/xBLIVB1/12lpZXVtfWMxvZza3tnd3c3n7NRInmUOWRjHQjYAakUFBFgRIasQYWBhLqweB6XK8/gDYiUhUcxtAKWU+JruAMrdXOnfjYB2TUF4r6lSmaJDCA1A8Z9oMgvRvdx+1c3i24E9F8GaQJzOV27kvxPxJASFXDJjmp4bYytlGgWXMr6iYGY8QHrQdOiYiGYVjq5Z0SPrdOh3Ujbp5BO3N8TKQuNGYaB7RzvaOZrY/O/WjPB7mUrFSpOEBSftRNJMWIjsOhHaGBoxaYFwLuyvlfaYZRxth1obgzZ+8CLViwTsrFG/P86WrWRwZckiOyCnxyAUpkRtSJlXCySN5Jq/kzXlyXpx352PauTMZg7IHzmfP7OtnHY=</latexit> <latexit sha1_base64="zHs9+HV3fyzf5H18X2hPCzWoX+E=">ACI3icbVDLSgMxFM34rPVdekmWIQWSplRQXElunElFW0VOmXIpLdtaCYzJHeEMvRf3PgrblwoxY0L/8X0sfB1IbmHc+4lOSdMpDouh/O3PzC4tJybiW/ura+sVnY2m6YONUc6jyWsb4PmQEpFNRoIT7RAOLQgl3Yf9irN89gDYiVrc4SKAVsa4SHcEZWionEKQYdAfUl8o6kcMe5zJ7GpYciv0ZiqV6tNeofauUB97gKxcpkGh6FbdSdG/wJuBIplVLSiM/HbM0wgUcsmMaXpugq2MaRcwjDvpwYSxvusC0LFYvAtLKJxyHdt0ybdmJtj0I6Yb9vZCwyZhCFdnLswvzWxuR/WjPFzkrEypJERSfPtRJcWYjgOjbaGBoxYwLgW9q+U95hmHG2seRuC9vyX9A4qHqH1YPro+LZ+SyOHNkle6REPHJMzsglqZE64eSRPJNX8uY8OS/OyHmfjs45s50d8qOczy8BHaM0</latexit> Stochastic Differential Equations d X t = f t ( X t , U t , t, θ )d t + σ t ( U t , t, θ )d W t , | {z } | {z } Drift Di ff usion X t ∈ X ⊂ R n Y t k = g t k ( X t k , U t k , t k , θ ) + e t k , |{z} Measurement noise Y t k ∈ Y ⊂ R L t ) T ∈ R m W t = ( W 1 t , W 2 t , · · · , W m e t k ∈ N (0 , S t k ( U t k , t k , θ )) θ ∈ Θ ⊂ R p 28 March 2019 DTU Compute 5
Recommend
More recommend
