{"id":6070,"date":"2024-01-13T22:16:42","date_gmt":"2024-01-13T13:16:42","guid":{"rendered":"https:\/\/now0930.pe.kr\/wordpress\/?p=6070"},"modified":"2026-10-04T08:25:40","modified_gmt":"2026-10-03T23:25:40","slug":"modal-canonical-form","status":"publish","type":"post","link":"https:\/\/now0930.pe.kr\/wordpress\/modal-canonical-form\/","title":{"rendered":"Modal Canonical Form"},"content":{"rendered":"\n

\uac15\uc758 \uc601\uc0c1\uc744 \ubcf4\ub2e4 eigenvector\uac00 \uc774\uc0c1\ud558\uc5ec \ud655\uc778\uc744 python\uc73c\ub85c \ud574\ubcf4\uc558\ub2e4.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n
modal_canonical_form<\/a>\ub2e4\uc6b4\ub85c\ub4dc<\/a><\/div>\n\n\n\n“`html\n
\n\n

Modal Canonical Form<\/h1>\n\n
\n State Space \u00b7 Eigenvalue \u00b7 Eigenvector \u00b7 Similarity Transformation \u00b7 Modal Canonical Form\n <\/div>\n\n\n \n

1. \ub300\uc0c1 \uc2dc\uc2a4\ud15c<\/h2>\n\n
\n\n

\n \uc0c1\ud0dc\uacf5\uac04 \ubaa8\ub378\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc124\uc815\ud55c\ub2e4.\n <\/p>\n\n

\n A =\n [\n -7    -12\n
\n 1      0\n ]\n <\/div>\n\n
\n B =\n [\n 1\n
\n 0\n ]\n <\/div>\n\n
\n C =\n [\n 1    2\n ]\n <\/div>\n\n
\n D = 0\n <\/div>\n\n

\n Python control library\uc5d0\uc11c \uc0c1\ud0dc\uacf5\uac04 \uc2dc\uc2a4\ud15c\uc744 \uc0dd\uc131\ud55c\ub2e4.\n <\/p>\n\n

\n sys = ss(A, B, C, D)\n <\/div>\n\n <\/div>\n\n\n \n

2. State Space \u2192 Transfer Function<\/h2>\n\n
\n\n

\n \uc0c1\ud0dc\uacf5\uac04 \ubaa8\ub378\uc744 \uc804\ub2ec\ud568\uc218\ub85c \ubcc0\ud658\ud55c \uacb0\uacfc\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.\n <\/p>\n\n

\n G(s)\n =\n Y(s) \/ U(s)\n <\/div>\n\n
\n G(s)\n =\n (s + 2)\n \/\n (s2<\/sup> + 7s + 12)\n <\/div>\n\n

\n \ubd84\ubaa8\ub97c \uc778\uc218\ubd84\ud574\ud558\uba74\n <\/p>\n\n

\n s2<\/sup> + 7s + 12\n =\n (s+3)(s+4)\n <\/div>\n\n

\n \ub530\ub77c\uc11c pole\uc740\n <\/p>\n\n

\n s = -3, -4\n <\/div>\n\n <\/div>\n\n\n \n

3. Eigenvalue<\/h2>\n\n
\n\n

\n \uc0c1\ud0dc\ud589\ub82c A\uc758 eigenvalue\ub97c \uacc4\uc0b0\ud55c\ub2e4.\n <\/p>\n\n

\n det(\u03bbI – A) = 0\n <\/div>\n\n

\n Python \uacc4\uc0b0 \uacb0\uacfc\ub294\n <\/p>\n\n

\n \u03bb1<\/sub> = -4\n <\/div>\n\n
\n \u03bb2<\/sub> = -3\n <\/div>\n\n

\n \uc774\ub2e4.\n <\/p>\n\n

\n \n State Matrix A\uc758 eigenvalue\ub294\n \uc804\ub2ec\ud568\uc218\uc758 pole -4, -3\uacfc \uc77c\uce58\ud55c\ub2e4.\n <\/strong>\n <\/div>\n\n <\/div>\n\n\n \n

4. Eigenvector<\/h2>\n\n
\n\n

\n Python\uc758 numpy.linalg.eig()\ub97c \uc774\uc6a9\ud558\uc5ec\n eigenvector\ub97c \uacc4\uc0b0\ud55c \uacb0\uacfc\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.\n <\/p>\n\n

\n X =\n [\n -0.9701425     0.9486833\n
\n 0.24253563   -0.31622777\n ]\n <\/div>\n\n

\n \uac01 column\uc774 \uac01\uac01\uc758 eigenvalue\uc5d0 \ub300\uc751\ud558\ub294 eigenvector\uc774\ub2e4.\n <\/p>\n\n

\n\n \n\n \n \n \n\n \n \n
Eigenvalue<\/th>\n Eigenvector<\/th>\n <\/tr>\n <\/thead>\n\n
\u03bb = -4<\/td>\n \n [-0.9701425, 0.24253563]T<\/sup>\n <\/td>\n <\/tr>\n\n
\u03bb = -3<\/td>\n \n [0.9486833, -0.31622777]T<\/sup>\n <\/td>\n <\/tr>\n\n <\/tbody>\n\n <\/table>\n\n <\/div>\n\n <\/div>\n\n\n \n

5. Eigenvector Matrix\uc758 \uc5ed\ud589\ub82c \ud655\uc778<\/h2>\n\n
\n\n

\n Eigenvector matrix\ub97c X\ub77c\uace0 \ud558\uba74\n \uc5ed\ud589\ub82c X-1<\/sup>\uc774 \uc81c\ub300\ub85c \uacc4\uc0b0\ub418\uc5c8\ub294\uc9c0 \ud655\uc778\ud55c\ub2e4.\n <\/p>\n\n

\n X-1<\/sup>X = I\n <\/div>\n\n

\n Python \uacc4\uc0b0 \uacb0\uacfc\n <\/p>\n\n

\n X-1<\/sup>X\n \u2248\n [\n 1    0\n
\n 0    1\n ]\n <\/div>\n\n

\n \uc2e4\uc81c \uacc4\uc0b0\uc5d0\uc11c\ub294 \ubd80\ub3d9\uc18c\uc218\uc810 \uc624\ucc28\ub85c\n \uc57d 4.44\u00d710-16<\/sup> \uc218\uc900\uc758 \ub9e4\uc6b0 \uc791\uc740 \uac12\uc774 \ub098\ud0c0\ub09c\ub2e4.\n <\/p>\n\n <\/div>\n\n\n \n

6. Eigenvalue \/ Eigenvector \uad00\uacc4 \ud655\uc778<\/h2>\n\n
\n\n

\n Eigenvalue\uc640 eigenvector\uc758 \uae30\ubcf8 \uad00\uacc4\ub294\n <\/p>\n\n

\n Ax = \u03bbx\n <\/div>\n\n

\n \uc774\ub2e4.\n <\/p>\n\n

\n \uc5ec\ub7ec eigenvector\ub97c matrix X\ub85c \ubb36\uc73c\uba74\n <\/p>\n\n

\n AX = X\u039b\n <\/div>\n\n

\n \ub85c \ud45c\ud604\ud560 \uc218 \uc788\ub2e4.\n <\/p>\n\n

\n \u039b =\n [\n -4    0\n
\n 0    -3\n ]\n <\/div>\n\n

\n Python\uc5d0\uc11c \uacc4\uc0b0\ud55c\n <\/p>\n\n

\n AX\n <\/div>\n\n

\n \uacb0\uacfc\ub294\n <\/p>\n\n

\n [\n 3.88057    -2.84604989\n
\n -0.9701425    0.9486833\n ]\n <\/div>\n\n

\n \uc774\uba70,\n <\/p>\n\n

\n X\u039b\n <\/div>\n\n

\n \uacc4\uc0b0 \uacb0\uacfc\ub3c4 \ub3d9\uc77c\ud558\uac8c\n <\/p>\n\n

\n [\n 3.88057    -2.84604989\n
\n -0.9701425    0.9486833\n ]\n <\/div>\n\n

\n \uc774\ub2e4.\n <\/p>\n\n

\n \n AX = X\u039b \uad00\uacc4\uac00 \uc218\uce58\uc801\uc73c\ub85c \ud655\uc778\ub41c\ub2e4.\n <\/strong>\n <\/div>\n\n <\/div>\n\n\n \n

7. Modal Canonical Form<\/h2>\n\n
\n\n

\n \uc55e\uc758 \uad00\uacc4\n <\/p>\n\n

\n AX = X\u039b\n <\/div>\n\n

\n \uc758 \uc591\ubcc0 \uc67c\ucabd\uc5d0 X-1<\/sup>\uc744 \uacf1\ud558\uba74\n <\/p>\n\n

\n X-1<\/sup>AX = \u039b\n <\/div>\n\n

\n \uac00 \ub41c\ub2e4.\n <\/p>\n\n

\n Python \uacc4\uc0b0 \uacb0\uacfc\n <\/p>\n\n

\n X-1<\/sup>AX\n =\n [\n -4    1.776\u00d710-15<\/sup>\n
\n 0    -3\n ]\n <\/div>\n\n

\n \ub9e4\uc6b0 \uc791\uc740 \uc218\uce58 \uc624\ucc28\ub97c 0\uc73c\ub85c \ubcf4\uba74\n <\/p>\n\n

\n X-1<\/sup>AX\n \u2248\n [\n -4    0\n
\n 0    -3\n ]\n <\/div>\n\n
\n =\n \u039b\n <\/div>\n\n
\n \n Modal Canonical Form\uc5d0\uc11c\ub294\n State Matrix A\uac00 eigenvalue\ub97c \ub300\uac01\uc131\ubd84\uc73c\ub85c \uac16\ub294\n diagonal matrix \u039b\ub85c \ubcc0\ud658\ub41c\ub2e4.\n <\/strong>\n <\/div>\n\n <\/div>\n\n\n \n

8. Similarity Transformation<\/h2>\n\n
\n\n

\n \uc0c1\ud0dc\ubcc0\uc218\ub97c eigenvector matrix X\ub97c \uc774\uc6a9\ud558\uc5ec\n \uc0c8\ub85c\uc6b4 \uc88c\ud45c\uacc4\ub85c \ubcc0\ud658\ud55c\ub2e4.\n <\/p>\n\n

\n x = Xz\n <\/div>\n\n

\n \ub610\ub294\n <\/p>\n\n

\n z = X-1<\/sup>x\n <\/div>\n\n

\n \uc6d0\ub798 \uc0c1\ud0dc\ubc29\uc815\uc2dd\n <\/p>\n\n

\n \u1e8b = Ax + Bu\n <\/div>\n\n

\n \uc5d0 x = Xz\ub97c \ub300\uc785\ud558\uba74\n <\/p>\n\n

\n X\u017c = AXz + Bu\n <\/div>\n\n

\n \uc591\ubcc0\uc5d0 X-1<\/sup>\uc744 \uacf1\ud558\uba74\n <\/p>\n\n

\n \u017c\n =\n X-1<\/sup>AXz\n +\n X-1<\/sup>Bu\n <\/div>\n\n

\n \ub530\ub77c\uc11c\n <\/p>\n\n

\n \u017c\n =\n \u039bz\n +\n Bm<\/sub>u\n <\/div>\n\n
\n Bm<\/sub>\n =\n X-1<\/sup>B\n <\/div>\n\n

\n \ucd9c\ub825\ubc29\uc815\uc2dd\ub3c4\n <\/p>\n\n

\n y = Cx + Du\n <\/div>\n\n

\n \uc5d0\uc11c\n <\/p>\n\n

\n y = CXz + Du\n <\/div>\n\n

\n \ub85c \ubcc0\ud658\ub41c\ub2e4.\n <\/p>\n\n

\n Cm<\/sub> = CX\n <\/div>\n\n <\/div>\n\n\n \n

9. Modal State Matrix<\/h2>\n\n
\n\n

\n \ubcc0\ud658 \uc804 State Matrix\ub294\n <\/p>\n\n

\n A =\n [\n -7    -12\n
\n 1      0\n ]\n <\/div>\n\n

\n \uc774\ub2e4.\n <\/p>\n\n

\n Modal Transformation \ud6c4\uc5d0\ub294\n <\/p>\n\n

\n Am<\/sub>\n =\n X-1<\/sup>AX\n <\/div>\n\n
\n Am<\/sub>\n =\n [\n -4    0\n
\n 0    -3\n ]\n <\/div>\n\n
\n \n \uc6d0\ub798 \uc11c\ub85c \uacb0\ud569\ub418\uc5b4 \uc788\ub358 \uc0c1\ud0dc\ubc29\uc815\uc2dd\uc744\n eigenvector \uc88c\ud45c\uacc4\ub85c \ubcc0\ud658\ud558\uba74\n \uac01 mode\uac00 \ub300\uac01\uc131\ubd84\uc73c\ub85c \ubd84\ub9ac\ub41c\ub2e4.\n <\/strong>\n <\/div>\n\n <\/div>\n\n\n \n

10. Modal Equation<\/h2>\n\n
\n\n

\n Modal State Matrix\uac00 diagonal\uc774\ubbc0\ub85c\n \uc0c1\ud0dc\ubc29\uc815\uc2dd\uc740 \uac01 mode\ubcc4\ub85c \ubd84\ub9ac\ud560 \uc218 \uc788\ub2e4.\n <\/p>\n\n

\n \u017c\n =\n [\n -4    0\n
\n 0    -3\n ]\n z\n +\n Bm<\/sub>u\n <\/div>\n\n

\n \uc0c1\ud0dc\ubc29\uc815\uc2dd\uc744 \uac1c\ubcc4\uc801\uc73c\ub85c \uc4f0\uba74\n <\/p>\n\n

\n \u017c1<\/sub>\n =\n -4z1<\/sub>\n +\n bm1<\/sub>u\n <\/div>\n\n
\n \u017c2<\/sub>\n =\n -3z2<\/sub>\n +\n bm2<\/sub>u\n <\/div>\n\n
\n \n Modal Canonical Form\uc740\n \ub2e4\ubcc0\uc218 \uc0c1\ud0dc\ubc29\uc815\uc2dd\uc744 \uac01 eigenvalue\uc5d0 \ub300\uc751\ud558\ub294\n \ub3c5\ub9bd\uc801\uc778 mode \ud615\ud0dc\ub85c \ud45c\ud604\ud55c\ub2e4.\n <\/strong>\n <\/div>\n\n <\/div>\n\n\n \n

11. \uc774\ubbf8\uc9c0\uc640 Python \uacc4\uc0b0 \uacb0\uacfc<\/h2>\n\n
\n\n \n\n \n \n \n\n \n \n \n \n \n \n \n
\ud56d\ubaa9<\/th>\n \uacb0\uacfc<\/th>\n <\/tr>\n <\/thead>\n\n
Transfer Function<\/strong><\/td>\n \n (s+2)\/(s\u00b2+7s+12)\n <\/td>\n <\/tr>\n\n
State Matrix A<\/strong><\/td>\n \n [-7 -12; 1 0]\n <\/td>\n <\/tr>\n\n
Eigenvalue<\/strong><\/td>\n \n -4, -3\n <\/td>\n <\/tr>\n\n
Eigenvector Matrix X<\/strong><\/td>\n \n [-0.9701 0.9487; 0.2425 -0.3162]\n <\/td>\n <\/tr>\n\n
Eigenvector \uad00\uacc4<\/strong><\/td>\n \n AX = X\u039b\n <\/td>\n <\/tr>\n\n
Modal Transformation<\/strong><\/td>\n \n X\u207b\u00b9AX = \u039b\n <\/td>\n <\/tr>\n\n
Modal Matrix<\/strong><\/td>\n \n [-4 0; 0 -3]\n <\/td>\n <\/tr>\n\n <\/tbody>\n\n <\/table>\n\n <\/div>\n\n\n \n

12. \ud575\uc2ec \uae30\uc5b5 \ubb38\uc7a5<\/h2>\n\n
\n\n
    \n\n
  1. \n \n State Matrix A\uc758 eigenvalue\ub294\n \uc2dc\uc2a4\ud15c\uc758 pole\uacfc \uc5f0\uacb0\ub41c\ub2e4.\n <\/strong>\n <\/li>\n\n
  2. \n \n Eigenvalue\uc640 eigenvector\ub294\n Ax = \u03bbx \uad00\uacc4\ub97c \ub9cc\uc871\ud55c\ub2e4.\n <\/strong>\n <\/li>\n\n
  3. \n \n Eigenvector\ub97c column\uc73c\ub85c \ubaa8\uc73c\uba74\n AX = X\u039b\uac00 \ub41c\ub2e4.\n <\/strong>\n <\/li>\n\n
  4. \n \n X\uac00 \uac00\uc5ed\uc774\uba74\n X\u207b\u00b9AX = \u039b\ub85c \ubcc0\ud658\ud560 \uc218 \uc788\ub2e4.\n <\/strong>\n <\/li>\n\n
  5. \n \n Modal Canonical Form\uc5d0\uc11c\n \u039b\uc758 \ub300\uac01\uc131\ubd84\uc740 eigenvalue\uc774\ub2e4.\n <\/strong>\n <\/li>\n\n
  6. \n \n \uc0c1\ud0dc\uc88c\ud45c\ub97c x = Xz\ub85c \ubcc0\ud658\ud558\uba74\n \uc6d0\ub798 \uc0c1\ud0dc\ubcc0\uc218 \ub300\uc2e0 modal state z\ub97c \uc0ac\uc6a9\ud55c\ub2e4.\n <\/strong>\n <\/li>\n\n
  7. \n \n Modal State Matrix\uac00 diagonal\uc774\uba74\n \uac01 mode\uc758 \ub3d9\ud2b9\uc131\uc744 \uac1c\ubcc4\uc801\uc73c\ub85c \ud655\uc778\ud560 \uc218 \uc788\ub2e4.\n <\/strong>\n <\/li>\n\n <\/ol>\n\n <\/div>\n\n\n \n
    \n Modal Canonical Form, State Space, Eigenvalue, Eigenvector,\n Similarity Transformation, Diagonalization,\n State Matrix, Modal Matrix, AX=X\u039b, X^-1AX=\u039b\n <\/div>\n\n<\/div>\n“`\n","protected":false},"excerpt":{"rendered":"

    \uac15\uc758 \uc601\uc0c1\uc744 \ubcf4\ub2e4 eigenvector\uac00 \uc774\uc0c1\ud558\uc5ec \ud655\uc778\uc744 python\uc73c\ub85c \ud574\ubcf4\uc558\ub2e4. “`html Modal Canonical Form State Space \u00b7 Eigenvalue \u00b7 Eigenvector \u00b7 Similarity Transformation \u00b7 Modal Canonical Form 1. \ub300\uc0c1 \uc2dc\uc2a4\ud15c \uc0c1\ud0dc\uacf5\uac04 \ubaa8\ub378\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc124\uc815\ud55c\ub2e4. A = [ -7    -12 1      0 ] B = [ 1 0 ] C = [ 1    2 […]<\/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":[1211,1212],"class_list":["post-6070","post","type-post","status-publish","format-standard","hentry","category-1185","tag-eigenvalue","tag-eigenvector"],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/now0930.pe.kr\/wordpress\/wp-json\/wp\/v2\/posts\/6070","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=6070"}],"version-history":[{"count":13,"href":"https:\/\/now0930.pe.kr\/wordpress\/wp-json\/wp\/v2\/posts\/6070\/revisions"}],"predecessor-version":[{"id":9545,"href":"https:\/\/now0930.pe.kr\/wordpress\/wp-json\/wp\/v2\/posts\/6070\/revisions\/9545"}],"wp:attachment":[{"href":"https:\/\/now0930.pe.kr\/wordpress\/wp-json\/wp\/v2\/media?parent=6070"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/now0930.pe.kr\/wordpress\/wp-json\/wp\/v2\/categories?post=6070"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/now0930.pe.kr\/wordpress\/wp-json\/wp\/v2\/tags?post=6070"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}