{"id":6966,"date":"2024-03-24T21:23:29","date_gmt":"2024-03-24T12:23:29","guid":{"rendered":"https:\/\/now0930.pe.kr\/wordpress\/?p=6966"},"modified":"2026-10-04T10:11:37","modified_gmt":"2026-10-04T01:11:37","slug":"matrix-exponential","status":"publish","type":"post","link":"https:\/\/now0930.pe.kr\/wordpress\/matrix-exponential\/","title":{"rendered":"Matrix Exponential"},"content":{"rendered":"\n

state state \ud574\uc11d\uc2dc \uae30\ubcf8\uc2dd \uc815\ub9ac. \uc5ec\uae30\uc5d0\uc11c \uc2dc\uc791\ud558\uc5ec controlibility,observability \uc720\ub3c4 \ud544\uc694<\/p>\n\n\n\n

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106\ud68c \uae30\ucd9c\ubb38\uc81c<\/p>\n\n\n\n

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Matrix Exponential<\/h1>\n\n
\n State Equation \u00b7 Matrix Exponential \u00b7 State Transition \u00b7 Controllability \u00b7 Observability\n <\/div>\n\n\n \n

1. \uc804\uccb4 \uad6c\uc131<\/h2>\n\n
\n\n

\n \ud544\uae30\uc5d0\uc11c\ub294 State Equation\uc744\n Homogeneous\uc640 Non-homogeneous \ud615\ud0dc\ub85c \ub098\ub204\uace0,\n \uac01\uac01 Scalar, Matrix, Laplace \ubc29\ubc95\uc73c\ub85c \ud574\ub97c \uad6c\ud55c\ub2e4.\n <\/p>\n\n 

\n      State Equation\n           \u2502\n           \u251c\u2500\u2500 Homogeneous\n           \u2502     \u251c\u2500\u2500 Scalar\n           \u2502     \u251c\u2500\u2500 Matrix\n           \u2502     \u2514\u2500\u2500 Laplace\n           \u2502\n           \u2514\u2500\u2500 Non-homogeneous\n                 \u251c\u2500\u2500 Matrix\n                 \u2514\u2500\u2500 Laplace\n    <\/pre>\n\n  <\/div>\n\n\n  \n  
2. Homogeneous Scalar Equation<\/h2>\n\n  
\n\n    
\n      \u1e8b(t) = a x(t)\n    <\/div>\n\n    
\n      Scalar \uc0c1\ud0dc\ubc29\uc815\uc2dd\uc758 \ud574\ub294\n      \ub2e4\uc74c\uacfc \uac19\uc774 \uae30\ub85d\ub418\uc5b4 \uc788\ub2e4.\n    <\/p>\n\n    
\n      x(t) = eat<\/sup>x(0)\n    <\/div>\n\n    
\n      \n        Scalar\uc758 eat<\/sup>\uac00\n        Matrix \ud615\ud0dc\uc5d0\uc11c\ub294 eAt<\/sup>\ub85c \ud655\uc7a5\ub41c\ub2e4.\n      <\/strong>\n    <\/div>\n\n  <\/div>\n\n\n  \n  
3. Homogeneous Matrix Equation<\/h2>\n\n  
\n\n    
\n      \u1e8b(t) = A x(t)\n    <\/div>\n\n    
\n      Matrix \ud615\ud0dc\uc758 \ud574\ub294\n      \ub2e4\uc74c\uacfc \uac19\uc774 \uae30\ub85d\ud55c\ub2e4.\n    <\/p>\n\n    
\n      x(t) = eAt<\/sup>x(0)\n    <\/div>\n\n    
\n      \n        Matrix Exponential eAt<\/sup>\uac00\n        \uc0c1\ud0dc x(0)\uc744 \uc2dc\uac04 t\uc758 \uc0c1\ud0dc x(t)\ub85c \uc774\ub3d9\uc2dc\ud0a8\ub2e4.\n      <\/strong>\n    <\/div>\n\n  <\/div>\n\n\n  \n  
4. Laplace Transform\uc5d0 \uc758\ud55c \ud574<\/h2>\n\n  
\n\n    
\n      \u1e8b(t) = A x(t)\n    <\/div>\n\n    
\n      Laplace Transform\uc744 \uc801\uc6a9\ud558\uba74\n    <\/p>\n\n    
\n      sX(s) – x(0) = A X(s)\n    <\/div>\n\n    
\n      (sI – A)X(s) = x(0)\n    <\/div>\n\n    
\n      X(s)\n      =\n      (sI-A)-1<\/sup>x(0)\n    <\/div>\n\n    
\n      \ub2e4\uc2dc Inverse Laplace Transform\uc744 \uc801\uc6a9\ud558\uba74\n    <\/p>\n\n    
\n      x(t)\n      =\n      L-1<\/sup>\n      {\n      (sI-A)-1<\/sup>\n      }\n      x(0)\n    <\/div>\n\n  <\/div>\n\n\n  \n  
5. Matrix Exponential\uacfc Laplace \uad00\uacc4<\/h2>\n\n  
\n\n    
\n      \ud544\uae30\uc5d0\uc11c\ub294 \ub2e4\uc74c \ub450 \ud45c\ud604\uc774\n      \ub3d9\uc77c\ud55c \uacb0\uacfc\ub97c \uac16\ub294\ub2e4\uace0 \uc815\ub9ac\ud55c\ub2e4.\n    <\/p>\n\n    
\n      L-1<\/sup>\n      {\n      (sI-A)-1<\/sup>\n      }\n      =\n      eAt<\/sup>\n    <\/div>\n\n    
\n      eAt<\/sup>\n      =\n      I + At + A\u00b2t\u00b2\/2! + A\u00b3t\u00b3\/3! + \u00b7\u00b7\u00b7\n    <\/div>\n\n    
\n      \n        eAt<\/sup>\ub294\n        Matrix Exponential\uc758 \uae09\uc218\uc804\uac1c\ub85c \uacc4\uc0b0\ud560 \uc218 \uc788\ub2e4.\n      <\/strong>\n    <\/div>\n\n  <\/div>\n\n\n  \n  
6. Matrix Exponential \uae09\uc218<\/h2>\n\n  
\n\n    
\n      eAt<\/sup>\n      =\n      \u03a3k=0<\/sub>\u221e<\/sup>\n      (Ak<\/sup>tk<\/sup> \/ k!)\n    <\/div>\n\n    
\n      e^(At)\n        =\n      I\n      + At\n      + A\u00b2t\u00b2\/2!\n      + A\u00b3t\u00b3\/3!\n      + ...\n    <\/pre>\n\n  <\/div>\n\n\n  \n  
7. Non-homogeneous State Equation<\/h2>\n\n  
\n\n    
\n      \u1e8b(t)\n      =\n      A x(t)\n      +\n      B u(t)\n    <\/div>\n\n    
\n      \uc785\ub825 u(t)\uac00 \ud3ec\ud568\ub41c\n      Non-homogeneous \uc0c1\ud0dc\ubc29\uc815\uc2dd\uc774\ub2e4.\n    <\/p>\n\n  <\/div>\n\n\n  \n  
8. \uc801\ubd84\uc778\uc790\ub97c \uc774\uc6a9\ud55c \uc720\ub3c4<\/h2>\n\n  
\n\n    
\n      \uc2dd\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\ub9ac\ud55c\ub2e4.\n    <\/p>\n\n    
\n      \u1e8b(t) – A x(t)\n      =\n      B u(t)\n    <\/div>\n\n    
\n      \uc591\ubcc0\uc5d0 e-At<\/sup>\ub97c \uacf1\ud55c\ub2e4.\n    <\/p>\n\n    
\n      e-At<\/sup>\n      {\n      \u1e8b(t) – A x(t)\n      }\n      =\n      e-At<\/sup>B u(t)\n    <\/div>\n\n    
\n      \uc88c\ubcc0\uc740 \ub2e4\uc74c \ubbf8\ubd84\ud615\ud0dc\ub85c \uc4f8 \uc218 \uc788\ub2e4.\n    <\/p>\n\n    
\n      d\/dt\n      {\n      e-At<\/sup>x(t)\n      }\n      =\n      e-At<\/sup>B u(t)\n    <\/div>\n\n  <\/div>\n\n\n  \n  
9. \uc801\ubd84<\/h2>\n\n  
\n\n    
\n      0\ubd80\ud130 t\uae4c\uc9c0 \uc801\ubd84\ud558\uba74\n    <\/p>\n\n    
\n      e-At<\/sup>x(t)\n      –\n      x(0)\n      =\n      \u222b0<\/sub>t<\/sup>\n      e-A\u03c4<\/sup>B u(\u03c4)d\u03c4\n    <\/div>\n\n    
\n      \ub530\ub77c\uc11c\n    <\/p>\n\n    
\n      x(t)\n      =\n      eAt<\/sup>x(0)\n      +\n      eAt<\/sup>\n      \u222b0<\/sub>t<\/sup>\n      e-A\u03c4<\/sup>B u(\u03c4)d\u03c4\n    <\/div>\n\n  <\/div>\n\n\n  \n  
10. Non-homogeneous State Solution<\/h2>\n\n  
\n\n    
\n      \uc704 \uc2dd\uc744 \uc815\ub9ac\ud558\uba74\n      \ub2e4\uc74c \ud615\ud0dc\uac00 \ub41c\ub2e4.\n    <\/p>\n\n    
\n      x(t)\n      =\n      eAt<\/sup>x(0)\n      +\n      \u222b0<\/sub>t<\/sup>\n      eA(t-\u03c4)<\/sup>B u(\u03c4)d\u03c4\n    <\/div>\n\n    
\n      \n        \uccab \ubc88\uc9f8 \ud56d\uc740 \ucd08\uae30\uc0c1\ud0dc\uc5d0 \uc758\ud55c \uc751\ub2f5,\n        \ub450 \ubc88\uc9f8 \ud56d\uc740 \uc785\ub825\uc5d0 \uc758\ud55c \uc751\ub2f5\uc774\ub2e4.\n      <\/strong>\n    <\/div>\n\n  <\/div>\n\n\n  \n  
11. Laplace Form<\/h2>\n\n  
\n\n    
\n      \u1e8b(t)\n      =\n      A x(t)\n      +\n      B u(t)\n    <\/div>\n\n    
\n      Laplace Transform\uc744 \uc801\uc6a9\ud558\uba74\n    <\/p>\n\n    
\n      sX(s) – x(0)\n      =\n      A X(s)\n      +\n      B U(s)\n    <\/div>\n\n    
\n      (sI-A)X(s)\n      =\n      x(0)\n      +\n      B U(s)\n    <\/div>\n\n    
\n      X(s)\n      =\n      (sI-A)-1<\/sup>x(0)\n      +\n      (sI-A)-1<\/sup>B U(s)\n    <\/div>\n\n  <\/div>\n\n\n  \n  
12. Laplace\uc640 Matrix Form\uc758 \ub3d9\uc77c\uc131<\/h2>\n\n  
\n\n    
\n      Inverse Laplace Transform\uc744 \uc801\uc6a9\ud558\uba74\n    <\/p>\n\n    
\n      x(t)\n      =\n      eAt<\/sup>x(0)\n      +\n      \u222b0<\/sub>t<\/sup>\n      eA(t-\u03c4)<\/sup>B u(\u03c4)d\u03c4\n    <\/div>\n\n    
\n      Matrix \ubc29\uc2dd\uacfc Laplace \ubc29\uc2dd\uc774\n      \ub3d9\uc77c\ud55c \ud574\ub97c \uc900\ub2e4\uace0 \ud544\uae30\uc5d0 \uc815\ub9ac\ub418\uc5b4 \uc788\ub2e4.\n    <\/p>\n\n  <\/div>\n\n\n  \n  
13. State Equation Solution \uc815\ub9ac<\/h2>\n\n  
\n\n    
\n      \u1e8b(t) = Ax(t) + Bu(t)\n               \u2502\n               \u25bc\n      x(t)\n        =\n      e^(At)x(0)\n        +\n      \u222b e^[A(t-\u03c4)] B u(\u03c4)d\u03c4\n    <\/pre>\n\n  <\/div>\n\n\n  \n  
14. Controllability \uc815\uc758<\/h2>\n\n  
\n\n    
\n      \ud544\uae30\uc5d0\uc11c\ub294 Controllability\ub97c\n      \ud604\uc7ac\uc758 \uc0c1\ud0dc x(t)\ub97c\n      \uc6d0\ud558\ub294 \uc0c1\ud0dc\ub85c \ub9cc\ub4e4 \uc218 \uc788\ub294\uc9c0\uc5d0 \ub300\ud55c \uac1c\ub150\uc73c\ub85c \uae30\ub85d\ud55c\ub2e4.\n    <\/p>\n\n    
\n      Input u(t)\n      \u2192\n      State x(t)\n    <\/div>\n\n    
\n      \n        \uc785\ub825\uc744 \uc801\uc808\ud558\uac8c \uc120\ud0dd\ud558\uc5ec\n        \uc0c1\ud0dc\ub97c \uc6d0\ud558\ub294 \uac12\uc73c\ub85c \uc774\ub3d9\uc2dc\ud0ac \uc218 \uc788\ub294\uac00?\n      <\/strong>\n    <\/div>\n\n  <\/div>\n\n\n  \n  
15. Controllability \uc720\ub3c4<\/h2>\n\n  
\n\n    
\n      \uc0c1\ud0dc\ubc29\uc815\uc2dd\uc758 \ud574\n    <\/p>\n\n    
\n      x(t)\n      =\n      eAt<\/sup>x(0)\n      +\n      \u222b0<\/sub>t<\/sup>\n      eA(t-\u03c4)<\/sup>B u(\u03c4)d\u03c4\n    <\/div>\n\n    
\n      \uc5d0\uc11c \ud2b9\uc815 \uc2dc\uac04 t\u2081\uc5d0\n      \uc6d0\ud558\ub294 \uc0c1\ud0dc\ub97c \ub9cc\ub4e0\ub2e4\uace0 \uac00\uc815\ud55c\ub2e4.\n    <\/p>\n\n    
\n      x(t\u2081) = 0\n    <\/div>\n\n    
\n      \uadf8\ub7ec\uba74 \ud544\uae30\uc5d0\uc11c\ub294\n      \ucd08\uae30\uc0c1\ud0dc x(0)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \ud45c\ud604\ud55c\ub2e4.\n    <\/p>\n\n    
\n      x(0)\n      =\n      –\n      \u222b0<\/sub>t\u2081<\/sup>\n      e-A\u03c4<\/sup>B u(\u03c4)d\u03c4\n    <\/div>\n\n  <\/div>\n\n\n  \n  
16. Matrix Exponential \uc804\uac1c<\/h2>\n\n  
\n\n    
\n      e-A\u03c4<\/sup>\ub97c A\uc758 \uac70\ub4ed\uc81c\uacf1\uc73c\ub85c \uc804\uac1c\ud55c\ub2e4.\n    <\/p>\n\n    
\n      e-A\u03c4<\/sup>\n      =\n      \u03a3\n      \u03b1k<\/sub>Ak<\/sup>\n    <\/div>\n\n    
\n      \uc774\ub97c \uc0c1\ud0dc\uc2dd\uc5d0 \ub300\uc785\ud558\uba74\n      Ak<\/sup>B \ud56d\ub4e4\uc774 \uc0dd\uc131\ub41c\ub2e4.\n    <\/p>\n\n    
\n      B,\n      AB,\n      A\u00b2B,\n      …\n    <\/div>\n\n  <\/div>\n\n\n  \n  
17. Controllability Matrix<\/h2>\n\n  
\n\n    
\n      C\n      =\n      [\n      B\n       \n      AB\n       \n      A\u00b2B\n       \n      \u00b7\u00b7\u00b7\n       \n      An-1<\/sup>B\n      ]\n    <\/div>\n\n    
\n      \ud544\uae30\uc5d0\uc11c\ub294 \uc774 Matrix\uac00\n      Full Rank\uc774\uba74 \uc6d0\ud558\ub294 \uc0c1\ud0dc\ub85c \uc81c\uc5b4 \uac00\ub2a5\ud558\ub2e4\uace0 \uc815\ub9ac\ud55c\ub2e4.\n    <\/p>\n\n    
\n      rank(C) = n\n      \u2192\n      Controllable\n    <\/div>\n\n    
\n      \n        [B AB A\u00b2B \u00b7\u00b7\u00b7 A\u207f\u207b\u00b9B]\uac00 Full Rank\uc774\uba74\n        \uc2dc\uc2a4\ud15c\uc740 Controllable\uc774\ub2e4.\n      <\/strong>\n    <\/div>\n\n  <\/div>\n\n\n  \n  
18. 106\ud68c \uae30\ucd9c\ubb38\uc81c<\/h2>\n\n  
\n\n    
\n      \ud398\uc774\uc9c0\uc5d0\ub294 \ub2e4\uc74c \uae30\ucd9c\ubb38\uc81c \uc774\ubbf8\uc9c0\uac00 \ud3ec\ud568\ub418\uc5b4 \uc788\ub2e4.\n    <\/p>\n\n    
\n      x[k+1]\n      =\n      Ax[k]\n      +\n      Bu[k]\n    <\/div>\n\n    
\n      y[k]\n      =\n      Cx[k]\n    <\/div>\n\n    
\n      \ubb38\uc81c\uc5d0\uc11c\ub294 \ud589\ub82c\n    <\/p>\n\n    
\n      [\n      B\n       \n      AB\n       \n      A\u00b2B\n       \n      \u00b7\u00b7\u00b7\n       \n      An-1<\/sup>B\n      ]\n    <\/div>\n\n    
\n      \uc758 \uc5ed\ud589\ub82c \ub610\ub294 Rank \uc870\uac74\uacfc\n      Controllability\uc758 \uad00\uacc4\ub97c \uc124\uba85\ud558\ub3c4\ub85d \ub418\uc5b4 \uc788\ub2e4.\n    <\/p>\n\n  <\/div>\n\n\n  \n  
19. Observability \uc815\uc758<\/h2>\n\n  
\n\n    
\n      \ud544\uae30\uc5d0\uc11c\ub294 Observability\ub97c\n      \uc2dc\uc2a4\ud15c \ucd9c\ub825 y(t)\ub97c \uc774\uc6a9\ud558\uc5ec\n      \ub0b4\ubd80 \uc0c1\ud0dc\ub97c \ud655\uc778\ud560 \uc218 \uc788\ub294\uac00\uc5d0 \ub300\ud55c \uac1c\ub150\uc73c\ub85c \uc815\ub9ac\ud55c\ub2e4.\n    <\/p>\n\n    
\n      State x(t)\n      \u2192\n      Output y(t)\n    <\/div>\n\n    
\n      \n        \ucd9c\ub825\uc73c\ub85c\ubd80\ud130\n        \ucd08\uae30\uc0c1\ud0dc \ub610\ub294 \ub0b4\ubd80 \uc0c1\ud0dc\ub97c \uc54c\uc544\ub0bc \uc218 \uc788\ub294\uac00?\n      <\/strong>\n    <\/div>\n\n  <\/div>\n\n\n  \n  
20. Observability \uc720\ub3c4<\/h2>\n\n  
\n\n    
\n      \u1e8b(t)\n      =\n      A x(t)\n      +\n      B u(t)\n    <\/div>\n\n    
\n      y(t)\n      =\n      Cx(t)\n    <\/div>\n\n    
\n      \uc0c1\ud0dc\ud574\ub97c \ucd9c\ub825\uc2dd\uc5d0 \ub300\uc785\ud558\uba74\n    <\/p>\n\n    
\n      y(t)\n      =\n      C eAt<\/sup>x(0)\n      +\n      C\n      \u222b0<\/sub>t<\/sup>\n      eA(t-\u03c4)<\/sup>B u(\u03c4)d\u03c4\n    <\/div>\n\n  <\/div>\n\n\n  \n  
21. Known Input \uc81c\uac70<\/h2>\n\n  
\n\n    
\n      \ud544\uae30\uc5d0\uc11c\ub294 A, B, u(t)\ub294\n      \uc54c\uace0 \uc788\ub294 \uac12\uc73c\ub85c \ubcf4\uace0,\n      x(0)\uc5d0 \uad00\ub828\ub41c \ud56d\uc744 \uc911\uc2ec\uc73c\ub85c \ubd84\uc11d\ud55c\ub2e4.\n    <\/p>\n\n    
\n      y(t)\n      ~\n      C eAt<\/sup>x(0)\n    <\/div>\n\n  <\/div>\n\n\n  \n  
22. Observability Matrix \uc720\ub3c4<\/h2>\n\n  
\n\n    
\n      Matrix Exponential\uc744 \uc804\uac1c\ud558\uba74\n    <\/p>\n\n    
\n      eAt<\/sup>\n      =\n      \u03a3\n      \u03b1k<\/sub>Ak<\/sup>\n    <\/div>\n\n    
\n      \ub530\ub77c\uc11c \ucd9c\ub825\uc740\n    <\/p>\n\n    
\n      y(t)\n      ~\n      \u03a3\n      \u03b1k<\/sub>\n      C Ak<\/sup>x(0)\n    <\/div>\n\n    
\n      \uc640 \uac19\uc774 \ud45c\ud604\ub41c\ub2e4.\n    <\/p>\n\n  <\/div>\n\n\n  \n  
23. Observability Matrix<\/h2>\n\n  
\n\n    
\n      O\n      =\n      [\n      C\n      ;\n      CA\n      ;\n      CA\u00b2\n      ;\n      \u00b7\u00b7\u00b7\n      ;\n      CAn-1<\/sup>\n      ]\n    <\/div>\n\n    
\n      \ud544\uae30\uc5d0\uc11c\ub294 \uc774 Matrix\uac00\n      Full Rank\uc774\uba74\n      \ucd9c\ub825\uc73c\ub85c\ubd80\ud130 \uc0c1\ud0dc\ub97c \ud655\uc778\ud560 \uc218 \uc788\ub2e4\uace0 \uc815\ub9ac\ud55c\ub2e4.\n    <\/p>\n\n    
\n      rank(O) = n\n      \u2192\n      Observable\n    <\/div>\n\n    
\n      \n        [C; CA; CA\u00b2; \u00b7\u00b7\u00b7; CA\u207f\u207b\u00b9]\uac00 Full Rank\uc774\uba74\n        \uc2dc\uc2a4\ud15c\uc740 Observable\uc774\ub2e4.\n      <\/strong>\n    <\/div>\n\n  <\/div>\n\n\n  \n  
24. Controllability\uc640 Observability \ube44\uad50<\/h2>\n\n  
\n\n    \n\n      \n        \n          \n\n        \n          \n          \n          \n          
\uad6c\ubd84<\/th>\n          Controllability<\/th>\n          Observability<\/th>\n        <\/tr>\n      <\/thead>\n\n      
\ubc29\ud5a5<\/strong><\/td>\n          Input \u2192 State<\/td>\n          State \u2192 Output<\/td>\n        <\/tr>\n\n        
\uc758\ubbf8<\/strong><\/td>\n          \uc785\ub825\uc73c\ub85c \uc0c1\ud0dc\ub97c \uc6d0\ud558\ub294 \uac12\uc73c\ub85c \uc774\ub3d9<\/td>\n          \ucd9c\ub825\uc73c\ub85c \ub0b4\ubd80 \uc0c1\ud0dc\ub97c \ucd94\uc815<\/td>\n        <\/tr>\n\n        
Matrix<\/strong><\/td>\n          [B AB A\u00b2B \u00b7\u00b7\u00b7 A\u207f\u207b\u00b9B]<\/td>\n          [C; CA; CA\u00b2; \u00b7\u00b7\u00b7; CA\u207f\u207b\u00b9]<\/td>\n        <\/tr>\n\n        
\uc870\uac74<\/strong><\/td>\n          Full Rank<\/td>\n          Full Rank<\/td>\n        <\/tr>\n\n      <\/tbody>\n\n    <\/table>\n\n  <\/div>\n\n\n  \n  
25. \uc804\uccb4 \uad00\uacc4<\/h2>\n\n  
\n\n    
\n      \u1e8b = Ax + Bu\n           \u2502\n           \u25bc\n      Matrix Exponential\n           \u2502\n           \u25bc\n      x(t)\n       =\n      e^(At)x(0)\n       +\n      \u222be^[A(t-\u03c4)]Bu(\u03c4)d\u03c4\n           \u2502\n           \u251c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2510\n           \u2502               \u2502\n           \u25bc               \u25bc\n      Controllability   Observability\n           \u2502               \u2502\n           \u25bc               \u25bc\n      [B AB ...]       [C; CA; ...]\n           \u2502               \u2502\n           \u25bc               \u25bc\n        Full Rank        Full Rank\n           \u2502               \u2502\n           \u25bc               \u25bc\n      \uc0c1\ud0dc \uc81c\uc5b4 \uac00\ub2a5    \uc0c1\ud0dc \uad00\uce21 \uac00\ub2a5\n    <\/pre>\n\n  <\/div>\n\n\n  \n  
26. \uc774\ubbf8\uc9c0 OCR \ud575\uc2ec \uc815\ub9ac<\/h2>\n\n  
\n\n    \n\n      \n        \n          \n\n        \n          \n          \n          \n          \n          \n          \n          \n          
\ud56d\ubaa9<\/th>\n          \ud544\uae30 \ub0b4\uc6a9<\/th>\n        <\/tr>\n      <\/thead>\n\n      
Homogeneous<\/strong><\/td>\n          \u1e8b = Ax<\/td>\n        <\/tr>\n\n        
Homogeneous Solution<\/strong><\/td>\n          x(t) = eAt<\/sup>x(0)<\/td>\n        <\/tr>\n\n        
Matrix Exponential<\/strong><\/td>\n          I + At + A\u00b2t\u00b2\/2! + \u00b7\u00b7\u00b7<\/td>\n        <\/tr>\n\n        
Laplace \uad00\uacc4<\/strong><\/td>\n          L\u207b\u00b9{(sI-A)\u207b\u00b9} = eAt<\/sup><\/td>\n        <\/tr>\n\n        
Non-homogeneous<\/strong><\/td>\n          \u1e8b = Ax + Bu<\/td>\n        <\/tr>\n\n        
State Solution<\/strong><\/td>\n          \n            eAt<\/sup>x(0)\n            +\n            \u222beA(t-\u03c4)<\/sup>Bu(\u03c4)d\u03c4\n          <\/td>\n        <\/tr>\n\n        
Controllability<\/strong><\/td>\n          [B AB \u00b7\u00b7\u00b7 A\u207f\u207b\u00b9B] Full Rank<\/td>\n        <\/tr>\n\n        
Observability<\/strong><\/td>\n          [C; CA; \u00b7\u00b7\u00b7; CA\u207f\u207b\u00b9] Full Rank<\/td>\n        <\/tr>\n\n      <\/tbody>\n\n    <\/table>\n\n  <\/div>\n\n\n  \n  
\n\n    \n      \uc774 HTML\uc740 \ud398\uc774\uc9c0 \ubcf8\ubb38 \uc124\uba85\uc744 \uc0ac\uc6a9\ud558\uc9c0 \uc54a\uace0,\n      \uc77c\ubc18 \uc774\ubbf8\uc9c0 6\uc7a5\ub9cc OCR\/\uc2dc\uac01 \ud310\ub3c5\ud558\uc5ec \uc791\uc131\ud588\ub2e4.\n\n      \uc774 \ud398\uc774\uc9c0\uc5d0\ub294 NeoStudio PDF\uac00 \uc5c6\ub2e4.\n\n      \uccab 4\uc7a5\uc758 \uc190\uae00\uc528 \uc774\ubbf8\uc9c0\ub294\n      Matrix Exponential \u2192 State Equation Solution \u2192\n      Controllability\uae4c\uc9c0 \uc5f0\uc18d\ub41c \uc720\ub3c4\uc774\uace0,\n      \ub9c8\uc9c0\ub9c9 \uc190\uae00\uc528 \uc774\ubbf8\uc9c0\ub294 Observability \uc720\ub3c4\uc774\ub2e4.\n\n      \uc77c\ubd80 \uacc4\uc218 \u03b1k, \u03b2k\uc758 \uc138\ubd80 \uc804\uac1c\ub294\n      \uc190\uae00\uc528\uac00 \ubd88\uba85\ud655\ud558\uc5ec \uc784\uc758\ub85c \ubcf5\uc6d0\ud558\uc9c0 \uc54a\uace0,\n      \ucd5c\uc885\uc801\uc73c\ub85c \uba85\ud655\ud558\uac8c \ud655\uc778\ub418\ub294\n      Controllability \/ Observability Matrix\uc640\n      Full Rank \uc870\uac74\uc744 \uc911\uc2ec\uc73c\ub85c \uc62e\uacbc\ub2e4.\n    <\/strong>\n\n  <\/div>\n\n\n  \n  
\n    Matrix Exponential,\n    State Space,\n    State Equation,\n    State Transition Matrix,\n    Homogeneous System,\n    Non-homogeneous System,\n    Laplace Transform,\n    eAt,\n    Controllability,\n    Observability,\n    Controllability Matrix,\n    Observability Matrix,\n    Full Rank\n  <\/div>\n\n<\/div>\n“`\n","protected":false},"excerpt":{"rendered":"
state state \ud574\uc11d\uc2dc \uae30\ubcf8\uc2dd \uc815\ub9ac. \uc5ec\uae30\uc5d0\uc11c \uc2dc\uc791\ud558\uc5ec controlibility,observability \uc720\ub3c4 \ud544\uc694 106\ud68c \uae30\ucd9c\ubb38\uc81c “`html id=”matrix-exp-ocr” Matrix Exponential State Equation \u00b7 Matrix Exponential \u00b7 State Transition \u00b7 Controllability \u00b7 Observability 1. \uc804\uccb4 \uad6c\uc131 \ud544\uae30\uc5d0\uc11c\ub294 State Equation\uc744 Homogeneous\uc640 Non-homogeneous \ud615\ud0dc\ub85c \ub098\ub204\uace0, \uac01\uac01 Scalar, Matrix, Laplace \ubc29\ubc95\uc73c\ub85c \ud574\ub97c \uad6c\ud55c\ub2e4. State Equation \u2502 \u251c\u2500\u2500 Homogeneous \u2502 \u251c\u2500\u2500 Scalar \u2502 \u251c\u2500\u2500 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_feature_clip_id":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_post_was_ever_published":false},"categories":[1185],"tags":[1539,1538,1540,1214],"class_list":["post-6966","post","type-post","status-publish","format-standard","hentry","category-1185","tag-controllability","tag-matrix-exponential","tag-observability","tag-state-space"],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/now0930.pe.kr\/wordpress\/wp-json\/wp\/v2\/posts\/6966","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/now0930.pe.kr\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/now0930.pe.kr\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/now0930.pe.kr\/wordpress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/now0930.pe.kr\/wordpress\/wp-json\/wp\/v2\/comments?post=6966"}],"version-history":[{"count":4,"href":"https:\/\/now0930.pe.kr\/wordpress\/wp-json\/wp\/v2\/posts\/6966\/revisions"}],"predecessor-version":[{"id":9615,"href":"https:\/\/now0930.pe.kr\/wordpress\/wp-json\/wp\/v2\/posts\/6966\/revisions\/9615"}],"wp:attachment":[{"href":"https:\/\/now0930.pe.kr\/wordpress\/wp-json\/wp\/v2\/media?parent=6966"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/now0930.pe.kr\/wordpress\/wp-json\/wp\/v2\/categories?post=6966"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/now0930.pe.kr\/wordpress\/wp-json\/wp\/v2\/tags?post=6966"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}