29
Blind subspace-based eigenstructure identification under nonstationary excitation using moving sensors Laurent Mevel, Albert Benveniste, Mich` ele Basseville, Maurice Goursat To cite this version: Laurent Mevel, Albert Benveniste, Mich` ele Basseville, Maurice Goursat. Blind subspace-based eigenstructure identification under nonstationary excitation using moving sensors. [Research Report] RR-4024, INRIA. 2000. <inria-00072616> HAL Id: inria-00072616 https://hal.inria.fr/inria-00072616 Submitted on 24 May 2006 HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destin´ ee au d´ epˆ ot et ` a la diffusion de documents scientifiques de niveau recherche, publi´ es ou non, ´ emanant des ´ etablissements d’enseignement et de recherche fran¸cais ou ´ etrangers, des laboratoires publics ou priv´ es.

Blind subspace-based eigenstructure identification under … · 2016-12-27 · Blind subspace-based eigenstructure identi cation under nonstationary excitation using moving sensors

  • Upload
    others

  • View
    2

  • Download
    0

Embed Size (px)

Citation preview

Page 1: Blind subspace-based eigenstructure identification under … · 2016-12-27 · Blind subspace-based eigenstructure identi cation under nonstationary excitation using moving sensors

Blind subspace-based eigenstructure identification under

nonstationary excitation using moving sensors

Laurent Mevel, Albert Benveniste, Michele Basseville, Maurice Goursat

To cite this version:

Laurent Mevel, Albert Benveniste, Michele Basseville, Maurice Goursat. Blind subspace-basedeigenstructure identification under nonstationary excitation using moving sensors. [ResearchReport] RR-4024, INRIA. 2000. <inria-00072616>

HAL Id: inria-00072616

https://hal.inria.fr/inria-00072616

Submitted on 24 May 2006

HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, estdestinee au depot et a la diffusion de documentsscientifiques de niveau recherche, publies ou non,emanant des etablissements d’enseignement et derecherche francais ou etrangers, des laboratoirespublics ou prives.

Page 2: Blind subspace-based eigenstructure identification under … · 2016-12-27 · Blind subspace-based eigenstructure identi cation under nonstationary excitation using moving sensors

ISS

N 0

249-

6399

ISR

N IN

RIA

/RR

--40

24--

FR

+E

NG

ap por t de r ech er ch e

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Blind subspace-based eigenstructureidentification under nonstationary excitation

using moving sensors

Laurent Mevel, Albert Benveniste, Michèle Basseville, Maurice Goursat

N˚4024

Octobre 2000

THÈME 4

Page 3: Blind subspace-based eigenstructure identification under … · 2016-12-27 · Blind subspace-based eigenstructure identi cation under nonstationary excitation using moving sensors
Page 4: Blind subspace-based eigenstructure identification under … · 2016-12-27 · Blind subspace-based eigenstructure identi cation under nonstationary excitation using moving sensors

������������ ������������ �������������������������� �!���"�#�������$�!�&%����'�!��(���������)���(�����!�'�!��(����'�"*+�-,.�-�/�!�0�"��(��1��!�����324(657�����8!����!('��

9;:=<">@?BA�C�DE?GFH?BIKJML�ION�?P>QCSRT?PA�F=?BA�UOVQC@?XW�JYDEU[Z]\"^BI[?_R`:aV@V@?GF�U[I[IO?XbcJYDE:=<�>@UOZB?edgfH<�>@V@:aC]h

i0j�kclnm.o�p qarslSt�uwvBx&rzyX{|mcx$yX}�x&rslgrz~�vBx&rzyX{� mS~&�=~�x�kclnm]~��]yXlg}�uzm@�Hm]~

�;��yB�/mcxTqars��l_v��

�$vB}�}�yX�&x � m`��m]��j�mc����j�m�{���o����Bo|p��T�cx�yX����m���������p����g}�vB��m]~

�e�-�]���P������� qat���~&}�vX�]m�r � mc{ax&r�����vBx&rzyX{�vBus��yX�&rsx&j�ln~�j�v] �m$}���y� �mc{¡m@¢_�crzmc{�x�£¤yX�;}�mc�/£¥yX�&lgrs{��yXt�x&}�t�x/¦§yX{�us��r � mc{ax&r�����vBx&rzyX{¨yB£)x&j�m7mcrs��mc{�~&x&�&t��cx&t���mSyB£;v�usrs{�m�vB�ª©¨«K©¬�­~&�H~&x�mcl®~&t��H�&m]�cx�x�yt�{��]yX{ax&��yXusuzm �M¯ t�{�lnm�vX~&t���m �M¯ vB{ � {�yX{�~&x�vBx&rzyX{�vB�&�°m@���crsx�vBx&rzyX{Y±¡qat���j²v¡}���yX��uzmcl³vB�&rz~�m]~Srs{lnm]��j�vB{�rz��vBu-mc{���rs{�m]mc�&rs{�� ¯ £¤yX�ªlny � vBu�vB{�vBus�H~&rz~$yB£- =rs����vBx&rs{��e~&x&�&t��cx&t���m]~TvB{ � l_vX��j�rs{�m]~G±�´�]yXlglnyX{µ}���vX�cx&rz�]m�x&j�mc��mgrz~`x�y¶�]yXusuzm]�cx � vBx�v¡£¤��yXl· �vB�&�=rs{��¸~�mc{�~�yX�7uzy=��vBx&rzyX{�~ ¯ t�~&rs{��¶��yXx&j����m � vB{ � lny� =rs{���~�mc{�~�yX��~ ¯ rs{eyX� � mc��x�y7lgrslgrz�$x&j�mªv] �vBrsuwvB��rsusrsx§�_yB£ v.uwvB�&��mc��~�mcx6yB£Y~�mc{�~�yX��~�±i0j�m�}�t��&}�y�~�m�yB£$x&j�rz~�}�vB}�mc�grz~�x�y¹rs{� �m]~�x&rs��vBx�m¶j�y�º»~&t���~�}�vX�]m@¦¼��vX~�m � r � mc{�x&r�����vBx&rzyX{½��vB{��m.v � vB}�x�m � x�y�j�vB{ � uzm.~�t���j|v�~&rsx&t�vBx&rzyX{ ¯ x�y�}���y� �m7rsx�~��]yX{�~�rz~&x�mc{��c��t�{ � mc�'{�yX{�~&x�vBx&rzyX{�vB�&�m@���crsx�vBx&rzyX{ ¯ vB{ � x�yn��mc}"yX�&x�yX{|v���m�vBu-vB}�}�usrz��vBx&rzyX{|m@��vBlg}�uzm�±¾¸¿�À�Á�Â�à �PÄ-��� Å usrs{ � mcrs��mc{�~&x&�&t��cx&t���m¡r � mc{�x&r�����vBx&rzyX{ ¯ {�yX{�~&x�vBx&rzyX{�vB�&�µm@���crsx�vBx&rzyX{ ¯ ~&t��H¦~�}�vX�]m.vBus��yX�&rsx&j�ln~ ¯ lnyG ars{��_~�mc{�~�yX��~ ¯  =rs����vBx&rzyX{¨vB{�vBus�H~&rz~�±

Æ�Ç ÈKɼÊ�Ë�È�Ì ÍsɼΥÏ]Ð

Ñ�ÒBÓÕÔ�Ö�×�ؼÙ�ÒBÚQÔ�Û�Ü/Ü/Ý0Þ/Ú�إؼÓÕÜ&ßà×QáPâ�ÖYÓãâ¼ÒBÓÕÝàâ¼ÒBÜ�äåاÚQæ'ÜçÖ"×QؼÙ6×Qä�è�áPؼÜçÙ@Ú-éGؼ×�ê[Ü/ÞçâYë�ì�ícî6ï�ðòñ�ó=ôGõçöaõ|÷sæ'×�ßPÜ/øÛXÚQÔ¥Ü/ß¡ù�â¥Ø¼áBÞçâ¼áGاÚQø�æ'×QÝBÓãâ¼×QؼÓÕÝBú.áPÔ¥ÓûÝPú�üKý$þ¤ÿ��)ÜçاÚ�â¼ÓÕ×@Ý�ù��`ù�â¼Ü/æ ÓûßPÜ/Ý]â¼Ó��XÞ&Ú�â¼ÓÕ×@Ý���!Þ/×]×�اßPÓÕÝXÚ�â¼Ü&ßgÛ� ��������aÜ/á��cÜ/Ý����"ÜçøûúQÓÕáBæ��

����� ðòõ��� � ñ � ð!��"$#�%'&)()*'+�,.-0/�*21)*)3�4657*'+�18*'+:98;',8*8<�9'(792;2%=-?>8(@�A ��� ðòõ��� :B�ñ � õC"�57%�;2;D*217923232*8<�9'(�98;D%=-?>2(:�E � ñ � ðF��"�/�%'&�(792GD*=-IH)JK&�(7;D%2,8<�9K+�(79'%=-?>8(:�

Unité de recherche INRIA RennesIRISA, Campus universitaire de Beaulieu, 35042 RENNES Cedex (France)

Téléphone : 02 99 84 71 00 - International : +33 2 99 84 71 00Télécopie : 02 99 84 71 71 - International : +33 2 99 84 71 71

Page 5: Blind subspace-based eigenstructure identification under … · 2016-12-27 · Blind subspace-based eigenstructure identi cation under nonstationary excitation using moving sensors

���-����(7��� !(���������������� �e(ª�S� ���¼�������$�!�&%����0�"��(���;5����������®��� ����������� �!���"�#���"(������#���§��� �*7����;24� ����������&���®�(ª�� �-,.�-�/�!�0�!��(����(��.����!�'�!��(������ª�&�"�®�����.�!��������ª�ª� ���� �-�'���!�����"­24(' ��������

�� ����� ��� m]~TvBus��yX�&rsx&j�lnm]~ ��� r � mc{�x&r�����vBx&rzyX{¬}�vB�T~�yXt�~/¦§m]~&}�vX�]m]~TyX{�xª£¥vBrsx$uzmct���~ª}���mct� �m]~}�yXt��Su � r � mc{ax&r�����vBx&rzyX{ v] �mct���uzm � meuwv ~�x&�&t��cx&t���m_}���yX}���m ��� t�{²~&�=~�x�kclnmnusrs{���vBrs��m�~�yXt�lgrz~��t�{�mTm@���crsx�vBx&rzyX{¡{�yX{��]yX{�x&���Xu��]m ¯ {�yX{¡lnm]~&t����]m`mcxà{�yX{H¦§~&x�vBx&rzyX{�{�vBrs��m�±��àm$x§�=}"m � mª}���yX��uzkclnmrs{ax�mc�& arzmc{ax|mc{ ���c{�rzm¬l��]��vB{�r��at�m ¯ }�yXt��¶u � vB{�vBus�=~�m¹lny � vBuzm � m¬~&x&�&t��cx&t���m]~|yXt l_vX��j�rs{�m]~mc{  ars����vBx&rzyX{�~G± «§u$m]~&x ��� t�~�vB��m|�]yXt���vB{�x ¯$� vB{�~e�]m � yXl_vBrs{�m ¯�� m �]yXusuzm]�cx�mc� � m]~ � yX{�{��]m]~mc{²}���yG �mc{�vB{��]m � m¡��vB}�x�mct���~! ��]mc�&x�vBrs{�~�����m]~ ¯���� vBt�x&��m]~�lnyX��rsuzm]~ ¬}�uwvX�"�]~�� � r$#��c��mc{�x�~mc{ � ��yXrsx�~ ¯ vP��{ ��� rslgrsx�mc��t�{�m�~&rsx&t�vBx&rzyX{¡y&%nt�{n{�yXlS����m � m���vB}�x�mct���~6�"m�vBt��]yXt�}e}�ust�~;rslg}�yX�/¦x�vB{ax.~�mc��vBrsx � rz~&}�yX{�rs��uzm�±('7vB{�~7�]mcx7vB�&x&rz�cuzm ¯ yX{)�cx&t � rzm��]yXlglnmc{ax�v � vB}�x�mc�7u � r � mc{�x&r�����vBx&rzyX{}�vB��~�yXt�~/¦§m]~&}�vX�]m]~��t�{�mgx�mcusuzm_~�rsx&t�vBx&rzyX{ ¯�� m � �clnyX{ax&��mc��~Qv¶�]yX{�~&rz~&x�vB{��]mn~�yXt�~Sm@���crsx�vBx&rzyX{{�yX{H¦§~�x�vBx&rzyX{�{�vBrs��m ¯ mcx � mn}����]~�mc{�x�mc� � m]~S���]~&t�usx�vBx�~�m@�H}*�c�&rslnmc{ax�vBtH�EyX��x�mc{�t�~�~&t�� � yX{�{��]m]~���]mcusuzm]~G±

�àm�x&��v� XvBrsu�v+�cx��0m,#Mm]�cx&t�� � vB{�~)uzmà��v � ��m � t�}���yB�&mcx.-�t���m0/Xv${�y!1��32��54*687:9�;=<0>?<A@¤©|y � mcu��vX~�m �CB x&�&t��cx&t���vBu6lnyX{�rsx�yX�&rs{�� t�~&rs{��EDGFYÁ�Ã=H-mc��vBx&rzyX{ � À � x�mcl3r � mc{ax&r�����vBx&rzyX{�I ¯ ��t�r;m]~�x.�]yB¦yX� � yX{�{��.}�vB��uwv�~�y=�cr��cx��5JLKM< ¯N� mct� �mc{ ¯!Å mcus��r���t�m�±O­Ã ��� Á �QP � « � mc{ax&r�����vBx&rzyX{|v] �mct���uzm � m`~&x&�&t��cx&t���m`}���yX}���m ¯ m@���crsx�vBx&rzyX{¶{�yX{H¦§~&x�vBx&rzyX{�{�vBrs��m ¯vBus��yX�&rsx&j�lnm]~ � m7x§�=}�m7~�yXt�~/¦§m]~&}�vX�]m ¯ ��vB}�x�mct���~ªlnyX��rsuzm]~ ¯ vB{�vBus�H~�m � m]~' =rs����vBx&rzyX{�~�±

Page 6: Blind subspace-based eigenstructure identification under … · 2016-12-27 · Blind subspace-based eigenstructure identi cation under nonstationary excitation using moving sensors

����������� �������������������������������� ��� �!���#"$�%�&���'�&�)(*"$+,�'�&�*�������"$��� -

. / �$����(7����� �!��(��

qat���~�}�vX�]m@¦¼��vX~�m � usrs{�m�vB�'~&�=~�x�mcl�r � mc{�x&r�����vBx&rzyX{ lnmcx&j�y � ~1032 ¯ 1�� ¯ 1,-54"j�v� �mT}���y� �mc{¸m@¢_�crzmc{�x£¥yX�_x&j�m|r � mc{�x&r�����vBx&rzyX{ yB£.x&j�m|mcrs��mc{�~&x&�&t��cx&t���m¬yB£�vµusrs{�m�vB��lSt�usx&r�¦¼ �vB�&rwvB��uzm¬~&�H~&x�mcl¸± ´${rslg}�yX�&x�vB{�x�rs{�~&x�vB{��]meyB£àx&j�rz~�}���yX��uzmcl rz~�~&x&�&t��cx&t���vBu6 =rs����vBx&rzyX{ vB{�vBus�H~&rz~ ¯ ��vX~�m � yX{µlnm�vP¦~�t���mclnmc{�x�~�£[��yXl+vX�]�]mcuzmc��yXlnmcx�mc��~gyX��~�x&��vBrs{µ��vBt���m]~ ¯ º'j�mc��mex&j�m�/�mc�¬rz~�~�t�mnrz~�x�y¸r � mc{�x&r�£¤� =rs����vBx&rs{�����j�vB��vX�cx�mc�&rz~&x&rz�]~ @[lny � m]~àvB{ � lny � vBu!~&j�vB}�m]~ I-yB£Mlnm]��j�vB{�rz��vBuM~&x&�&t��cx&t���m]~à~&t��H�&m]�cxx�y.t�{��]yX{�x&��yXusuzm �M¯ ���6(7��5����� ��`vB{ � ��"$�6���8�!���#"$���$��90m@�H�crsx�vBx&rzyX{Y±;i6�a}�rz��vBu"m@��vBlg}�uzm]~àvB��mªy #"¦~�j�yX��m�~�x&�&t��cx&t���m]~7~&t��H�/m]�cx`x�y�~&ºàmcusu ¯ ��t�rsu � rs{���~7~&t��H�/m]�cx`x�y¡º'rs{ � yX�7m�vB�&x&j���t�v /�m ¯ ���&r � ��m]~ ¯� vBln~ ¯ º'rs{���~�~&t��H�/m]�cxSx�y:�t�x&x�mc�Srs{;:�rs��j�x ¯ vB{ � x&t��&��rs{�m]~�~&t��H�/m]�cxSx�y¸~&x�m�vBl x&t��&��t�uzmc{��]m ¯£¤�&rz�cx&rzyX{�rs{��"m�vB�&rs{���~ ¯ vB{ � rslg}�mc�/£¤m]�cx)��vBuwvB{��crs{���±�´ ��mcuzmc �vB{�x;vB}�}���y�vX��jnx�y$~�x&�&t��cx&t���vBu�vB{�vBu�¦�H~&rz~Sj�vX~.��m]mc{E~&j�yGº'{Ex�y ��mcus�°yX{µx&j�mnlny � mcusrs{��¹yB£�lny � m]~�vB{ � lny � vBu�~&j�vB}�m]~�x&j���yXt���j~�x�vBx�m¡~�}�vX�]m���mc}���m]~�mc{�x�vBx&rzyX{�~%0 1��<4 ¯ vB{ � x&j�m¡t�~�m�yB£�"$�=�?>��=��@A"$���?9¸vB{ � � "$+5�$���#�$�����@A�$���'+5��mcrs��mc{�~�x&�&t��cx&t���m.r � mc{�x&r�����vBx&rzyX{¨lnmcx&j�y � ~B0 2 ¯ 1�4§±

´ �]yXlglnyX{E}���vX�cx&rz�]mnrs{° =rs����vBx&rzyX{ vB{�vBus�H~&rz~7rz~7x�y �]yXusuzm]�cx � vBx�v¡£¤��yXl3 �vB�&�ars{�� ~�mc{�~�yX�uzy=��vBx&rzyX{�~ ¯ t�~�rs{��T��yXx&j�����m � vB{ � lnyG ars{���~�mc{�~�yX��~ ¯ rs{nyX� � mc��x�yTlgrslgrz�'x&j�m'v] �vBrsuwvB��rsusrsx§�nyB£Mv@[l�t���j�I�uwvB�&��mc�$~�mcx$yB£�~�mc{�~�yX��~�±'q=mc �mc��vBu-~&t��]�]m]~�~&rs �m � vBx�vn~�mcx�~$vB��m7��m]�]yX� � m �M¯ º'rsx&j|~�mc{�~�yX��~vBx � r$#�mc��mc{axeuzy=��vBx&rzyX{�~¡yX{�x&j�m ~�x&�&t��cx&t���m�± q=yXlnm|yB£7x&j�m¨~�mc{�~�yX��~ ¯ ��vBusuzm � x&j�mC� 'D,�������E~�mc{�~�yX��~ ¯ vB��m /�mc}�xg����m �M¯ º'j�rsuzm¶x&j�m yXx&j�mc��~�vB��m¶lnyG �m � ±½i0j�rz~¡~�mcx&t�} ¯ t�~&t�vBusus�²��m@£¥mc�&��m �x�y�vX~�>="$��95� 'D,�� ����E.~�mcx&t�}F0Õ�<4;vB{ � x§�=}�rz��vBusus� ��vX~�m � yX{¹vB��yXt�xTx�mc{¨~�mc{�~�yX��~ ¯ ��vB{¨lgrslgrz�gv~�rsx&t�vBx&rzyX{¸rs{¶º'j�rz��j|j�t�{ � ��m � ~TyB£)~�mc{�~�yX��~ªvB��m.v] �vBrsuwvB��uzm�±

����ya�]m]~�~&rs{��|}�yXus�=��m@£¤mc��mc{��]m � vBx�v¸£¤yX��~&x&�&t��cx&t���vBu�vB{�vBus�=~�rz~Srz~gvX��j�rzmc �m � x�y � v]�E���°}�mc�/¦£¥yX�&lgrs{��¬mcrs��mc{�~&x&�&t��cx&t���m�r � mc{�x&r�����vBx&rzyX{�£¤yX��m�vX��j ��m]�]yX� � ~�mc}�vB��vBx�mcus� ¯ vB{ � x&j�mc{ lnmc�&��rs{��x&j�me��m]~&t�usx�~�yX��x�vBrs{�m � £¤yX����m]�]yX� � ~g�]yX�&��m]~&}�yX{ � rs{��¹x�y � r$#Mmc��mc{�xg~�mc{�~�yX��}�yayXuz~�±HGTy�º�mc �mc� ¯}�yXuzm�l_vBx���j�rs{�� l_v]�¹�"m�{�yXx.m�vX~&�¨rs{¬}���vX�cx&rz�]m ¯ vB{ � x&jat�~7x&j�mg��m]~&t�usx.yB£0mcrs��mc{� �m]�cx�yX����ustH¦rs{��¡l_v]�|{�yXx`�"m��]yX{�~&rz~&x�mc{ax�±.i0j�mc��m@£¤yX��m�x&j�m!�at�m]~&x&rzyX{°vB�&rz~�m]~Tx�y_}�mc�/£¤yX�&l8mcrs��mc{�~&x&�&t��cx&t���mr � mc{ax&r�����vBx&rzyX{��a�µlnmc�&��rs{��°x&j�m � vBx�v¹yB£Tx&j�m¶~&t��]�]m]~�~&rs �m¡��m]�]yX� � ~ ¯ vB{ � }���ya�]m]~�~�rs{��¬x&j�mcl��uzyX��vBusus� ¯ rs{�~&x�m�v � yB£)lnmc�&��rs{��_x&j�m`r � mc{�x&r�����vBx&rzyX{¨��m]~&t�usx�~�±

i0j�m }�t��&}�y�~�m¨yB£.x&j�rz~e}�vB}�mc�¡rz~_x�yErs{� �m]~&x&rs��vBx�m¹j�yGº·~�t���~&}�vX�]m@¦¼��vX~�m � r � mc{�x&r�����vBx&rzyX{lnmcx&j�y � ~`��vB{ ��m�v � vB}�x�m � x�yej�vB{ � uzm�vn}"yXus�=��m@£¥mc��mc{��]m�~�mcx&t�} ¯ rs{|x&j�mS}���m]~�mc{��]m�yB£àvB{|t�{H¦lnm�vX~�t���m � vB{ � {�yX{�~&x�vBx&rzyX{�vB�&� m@���crsx�vBx&rzyX{Y±ª«K{|~�m]�cx&rzyX{¬� ¯ º�m����&rzm�:��¸��mc arzmcº­yXt�x&}�t�x/¦§yX{�us��]yG XvB�&rwvB{��]m@¦ � �&rs �mc{ ~&t���~�}�vX�]mnr � mc{�x&r�����vBx&rzyX{²lnmcx&j�y � ~�±¶i0j�mnl_vBx&j�mcl_vBx&rz��vBu0lny � mcusrs{��|yB£x&j�mS}"yXus�=��m@£¥mc��mc{��]mS~�mcx&t�}¸rz~$~&x�vBx�m � rs{¨~�m]�cx&rzyX{H-H±0´ ����~&x�lnmc�&��m�yB£�x&j�m � vBx�vgrz~ª}���yX}"y�~�m �£¥yX��x&j�m�~&x�vBx&rzyX{�vB�&�gm@���crsx�vBx&rzyX{e��vX~�m ¯ vB{ � ~&j�y�º'{ex�yT£¥vBrsu�rs{nx&j�m0{�yX{�~&x�vBx&rzyX{�vB�&�nyX{�m�±�«K{n~�m]�@¦x&rzyX{�o ¯ v$�]y� �vB�&rwvB{��]m�{�yX�&l_vBusr IGvBx&rzyX{g}��&rzyX�)x�y$lnmc�&��màrz~ }���yX}�y�~�m � £¤yX�-x&j�m�~�rslg}�usr���m � ��vX~�m�yB£��m]�]yX� � ¦ � mc}�mc{ � mc{�x`m@���crsx�vBx&rzyX{ ¯ vB{ � rsx�~�rslg}�uzmclnmc{ax�vBx&rzyX{|rz~���rs �mc{Y±;i0j�m`£¤t�usus�e{�yX{�~&x�vBx&rzyX{H¦vB�&�e��vX~�m ¯ rs{eº'j�rz��j¶x&j�m$m@���crsx�vBx&rzyX{¶rz~6{�yX{�~�x�vBx&rzyX{�vB�&�eº'rsx&j�rs{�x&j�mT��m]�]yX� � ~ ¯ rz~'v ��� ��m]~�~�m � rs{~�m]�cx&rzyX{¶� ¯ º'j�mc��m.v��]yX{�~&rz~&x�mc{��c�e��m]~&t�usx0rz~à}���y� �mc{Y±KJ$t�lnmc�&rz��vBuM��m]~&t�usx�~0yX��x�vBrs{�m � yX{¸v���m�vBu

LML ÝON�P�Qcî P

Page 7: Blind subspace-based eigenstructure identification under … · 2016-12-27 · Blind subspace-based eigenstructure identi cation under nonstationary excitation using moving sensors

o � �!�&� ������ �+5������ ��� ���� � ��6+5��6����� �� �#�� ��� � �5�����+ ���'� ���H�$�������E���"$�������!�

vB}�}�usrz��vBx&rzyX{¬m@��vBlg}�uzmgvB��mS��mc}�yX�&x�m � rs{¨~�m]�cx&rzyX{ 2H±��-rs{�vBusus� ¯ ~�yXlnm��]yX{��cust�~&rzyX{�~.vB��m � ��v]º'{rs{¸~�m]�cx&rzyX{H2=±

� � �.�����S����(�����*³�-(;5à�'�"�������-�����S���&5���� !�S �������-� �������$�!�&%����'�!��(���°me�]yX{�~&r � mc�Susrs{�m�vB��lSt�usx&r�¦¼ �vB�&rwvB��uzm�~&�H~&x�mcln~ � m]~��c�&rs��m � �a�µv � rz~��c��mcx�m@¦¼x&rslnm�~&x�vBx�mn~�}�vX�]mlny � mcu ����������� !���#"�$%�&���' � � ()��� @ 1 I

º'j�mc��me~&x�vBx�m�

vB{ � yX��~�mc�& �m � yXt�x&}�t�x*'·j�v] �m � rslnmc{�~&rzyX{�~*+�vB{ �-,�¯ ��m]~&}�m]�cx&rs �mcus� ¯ vB{ �º'j�mc��mTx&j�m`~&x�vBx�mT{�yXrz~�m$}���ya�]m]~�~/. $%�102� rz~'vB{ �&�6(7��$��������437vBt�~�~&rwvB{¡º'j�rsx�m`{�yXrz~�mT~�m"�at�mc{��]mº'rsx&j I�mc��yglnm�vB{|vB{ �M¯ rs{¸x&j�m.~�x�vBx&rzyX{�vB�&����vX~�m ¯ �]yX{�~&x�vB{�xT�]y� �vB�&rwvB{��]m.l_vBx&�&r��65879 $%�:$<;�=?>�@BA� 5#CD.BE8FGEDH 0º'j�mc��m 9 .2I 0 � mc{�yXx�m]~�x&j�m0m@�H}"m]�cx�vBx&rzyX{nyX}�mc��vBx�yX��±�i0j�màº'j�rsx�mc{�m]~�~6vX~�~�t�lg}�x&rzyX{nyX{gx&j�m'~&x�vBx�m{�yXrz~�m$vB{ � x&j�mTvB��~�mc{��]mTyB£ lnm�vX~&t���mclnmc{ax0{�yXrz~�mªrs{ @ 1 I;vB��mª£¤t��&x&j�mc� � rz~��ct�~�~�m � rs{¡~�m]�cx&rzyX{ 2H±� mcxYx&j�m;}�vBrs��~J.LKNMPORQ 0 �"m;x&j�m;mcrs��mc{� �vBust�m]~-vB{ � mcrs��mc{� �m]�cx�yX��~)yB£�x&j�m;~&x�vBx�m�x&��vB{�~�rsx&rzyX{Sl_vBx&�&r�� ¯ vB{ ��� m@��{�m:STQ >�@BA�U( ORQ�±��°m7vX~�~&t�lnm$x&j�vBx'x&j�m`~&�=~�x�mcl j�vX~à{�y�lSt�usx&rs}�uzm7mcrs��mc{a XvBust�m]~ ¯

vB{ � x&j�t�~ex&j�vBx_x&j�mVK � ~evB{ � STQ � ~¡vB��m¶}�vBrs�&º'rz~�m|�]yXlg}�uzm@���]yX{X�çt���vBx�m�±½«§{�}�vB�&x&rz�ct�uwvB� ¯XWrz~ ��"!��vB{�mcrs��mc{a XvBust�m|yB£`~&x�vBx�m¸x&��vB{�~&rsx&rzyX{�l_vBx&�&r��

± i0j�m �]yXusuzm]�cx&rzyX{ yB£`}�vBrs��~V.LKYMZS�Q 0

£¥yX�&l·vn��vB{�yX{�rz��vBu-}�vB��vBlnmcx�mc�&r IGvBx&rzyX{¬yB£;x&j�mS}"yXuzmS}�vB�&x$yB£;~&�H~&x�mcl @ 1 I ¯ ��m@£¤mc�&��m � x�yevX~ªx&j�m~��=~&x�mcl»mcrs��mc{�~�x&�&t��cx&t���m�±'´ ��vB{�yX{�rz��vBu-}�vB��vBlnmcx�mc�&r IGvBx&rzyX{|rz~Tv�}�vB��vBlnmcx�mc�&r IGvBx&rzyX{¹º'j�rz��j|rz~rs{a XvB�&rwvB{axªº.± ��± x�±;��j�vB{���m]~ªrs{¶x&j�mS~&x�vBx�m`��vX~&rz~�±

����ya�]m]~�~&rs{���yXt�x&}�t�x'�]yG XvB�&rwvB{��]m`l_vBx&�&rz�]m]~0rz~0yB£Yrs{�x�mc��m]~�x0£¤yX�� �mc�&�_uwvB�&��m � vBx�v�~�mcx�~�±)i0j�m� r$#Mmc��mc{��]m��"mcxKº�m]mc{¬x&j�m��]y� �vB�&rwvB{��]m@¦ � �&rs �mc{¬£¥yX�&l8yB£�~&t���~&}�vX�]mgvBus��yX�&rsx&j�ln~`º'j�rz��j¬ºàm � m@¦~��c�&rs�"m${�y�º vB{ � x&j�mTt�~&t�vBu � vBx�vP¦ � �&rs �mc{¸£¤yX�&l rz~àlgrs{�yX� ¯ vBx�uzm�vX~&xࣥyX�àmcrs��mc{�~�x&�&t��cx&t���m`r � mc{H¦x&r�����vBx&rzyX{ 0 1�� ¯\[ 4§± �àyG XvB�&rwvB{��]m@¦ � �&rs �mc{­~&t���~&}�vX�]m¹r � mc{�x&r�����vBx&rzyX{­yB£�x&j�m¬mcrs��mc{�~&x&�&t��cx&t���m.LKYMZSTQ 0 � ~�rz~���vX~�m � yX{¸x&j�mT£¥yXusuzy�º'rs{��e~�x�mc}�~�± � mcx]_^ >�@BA� 9 ' � ' ;�a` ^vB{ �

b*c ���ed f >�@BA� ghhhhhhi]<j ] � ±±± ] fP`N�] � ]�k ±±± ] f±±± ±±± ±±± ±±±] c ] c ��� ±±±l] c �mfP`N�

nZoooooop>�@BA�rq�s1tvu .w] ^ 0 @¥��I

xzy L{xw|

Page 8: Blind subspace-based eigenstructure identification under … · 2016-12-27 · Blind subspace-based eigenstructure identi cation under nonstationary excitation using moving sensors

����������� �������������������������������� ��� �!���#"$�%�&���'�&�)(*"$+,�'�&�*�������"$��� �

��m`x&j�m7x&j�m]yX��mcx&rz��vBu-yXt�x&}�t�x/¦§�]y� �vB�&rwvB{��]m�vB{ � G`vB{N/�mcuYl_vBx&�&rz�]m]~ ¯ ��m]~&}�m]�cx&rs �mcus��±;«§{ax&��y � t��crs{��x&j�m7�c��y�~�~/¦§�]y� �vB�&rwvB{��]m.��mcx§ºàm]mc{ x&j�m.~&x�vBx�m�vB{ � x&j�m.yX��~�mc�& �m � yXt�x&}�t�x�~ 7

� >�@BA� 9 ��� ' ;�� rs��m]�cx$�]yXlg}�t�x�vBx&rzyX{�~$yB£)x&j�m/]�^ � ~'£[��yXl®x&j�m7lny � mcuYm"�at�vBx&rzyX{�~�uzm�v � x�y]�^ � ( ^ � @#-�IvB{ � x�y�x&j�m7º�mcusu:/={�y�º'{ 0 131�4M£¥vX�cx�yX�&r IGvBx&rzyX{b c ���ed f � � c ��� . ( M *0 � f . M � 0 @[o I

º'j�mc��m

� c . ( M *0 >�@BA� ghhhhi((V ±±±(V c `N�

nZoooopvB{ � � f . M � 0 >�@BA� . � ������� fP`N� � 0vB��m x&j�m yX��~�mc�& �vB��rsusrsx§��vB{ � �]yX{�x&��yXusuwvB��rsusrsxK� l_vBx&�&rz�]m]~ ¯ ��m]~�}"m]�cx&rs �mcus��± «K{�x&j�m|~�m"��t�mcu ¯ ºàmyB£¤x�mc{ � ��yX}¸x&j�m.yX� � mc��~$yB£-x&j�m7~��=~&x�mcl®l_vBx&�&rz�]m]~G±

i0j�m'yX��~�mc�& XvBx&rzyX{gl_vBx&�&r��(rz~;x&j�mc{�£¤yXt�{ � rs{gx&j�mà����~&x���uzy=��/a¦¼��y�º�yB£"x&j�m'yX��~�mc�& �vB��rsusrsx§�

l_vBx&�&r�� � ¯ º'j�mc��m�vX~0m@�=}�uzyXrsx&rs{���x&j�mª~&j�r�£¤x6rs{a XvB�&rwvB{��]m$}���yX}"mc�&xK�nyB£ � }���y� =r � m]~àt�~;º'rsx&jex&j�m~�x�vBx�m@¦¼x&��vB{�~&rsx&rzyX{ l_vBx&�&r��

¯ £¤��yXl+º'j�rz��j²x&j�m�mcrs��mc{�~&x&�&t��cx&t���m-.LKNMZSTQ 0 ��m]~&t�usx�~G±¨i0j�m�/�mc�

£¥m�vBx&t���m�rs{_£¥vX�cx�yX�&r IGvBx&rzyX{ @[o I)º'j�rz��jeº'rsusu"��m�mcuwvB�"yX��vBx�m � yX{ ¯ rz~;x&j�vBx6x&j�mªuzm@£[x6£¥vX�cx�yX� � yX{�us�� mc}�mc{ � ~-yX{�x&j�m6}�vBrs�R. ( M *0 ¯ vB{ � x&j�t�~-yX{�x&j�m�mcrs��mc{�~&x&�&t��cx&t���m0yB£�~&�=~�x�mcl @ 1 I ¯ º'j�mc��m�vX~-x&j�mm@���crsx�vBx&rzyX{

$ �yX{�us�¶v�#Mm]�cx�~�x&j�m`�&rs��j�xª£¥vX�cx�yX� � x&j���yXt���j¸x&j�mS�c��y�~�~/¦§�]y� �vB�&rwvB{��]m.l_vBx&�&r�� � ±

i0j�m�vX�cx&t�vBu)rslg}�uzmclnmc{�x�vBx&rzyX{EyB£;x&j�rz~T~�t���~&}�vX�]m�vBus��yX�&rsx&j�l8j�vX~$x&j�m�mclg}�rs�&rz��vBu;�]y� �vB�&r�¦vB{��]m]~

�] ^ � �

���� ' � ' ;�a` ^ @¥��I

~�t���~&x&rsx&t�x�m � £¤yX�<] ^ rs{ b*c ���ed f ¯ �=rzmcu � rs{��¶x&j�m�mclg}�rs�&rz��vBuOGTvB{N/�mcu)l_vBx&�&r�� �b*c ���ed f ±BG$yGº x�y~�mcuzm]�cx6x&j�m�{at�lS��mc�'yB£YuwvB��~�.� "�� 0 vB{ � x&j�t�~�x&j�m$~&r I�m$yB£ �b c ���ed f rz~ � rz~��ct�~�~�m � rs{e~�m]�cx&rzyX{ 2H±i0j�m_q�� ' yB£ �b c ���ed f vB{ � rsx�~`x&�&t�{���vBx&rzyX{EvBx7x&j�m � m]~&rs��m � lny � mcu�yX� � mc�.�arzmcu �M¯ rs{¬x&j�m� 'D��D�� ���8"!� ¯ vB{¶m]~&x&rsl_vBx�m �� £¥yX��x&j�mTyX��~�mc�& �vB��rsusrsx§�_l_vBx&�&r�� � ¯ £¤��yXl º'j�rz��j . �( M � 0 vB{ � . � KYM �STQ 0vB��m`��m]�]yG �mc��m � vX~ª~ /�mcx���j�m � vB��y� �m�±

LML ÝON�P�Qcî P

Page 9: Blind subspace-based eigenstructure identification under … · 2016-12-27 · Blind subspace-based eigenstructure identi cation under nonstationary excitation using moving sensors

2 � �!�&� ������ �+5������ ��� ���� � ��6+5��6����� �� �#�� ��� � �5�����+ ���'� ���H�$�������E���"$�������!�

� � (��&*`�����]�)�"�����-�®��� �������-�#�������ª�"�&%��-�0�!��(�������!�'�!��(����'�"*³�-,.�-�ç��!�0�!��(��

«K{�~&x�m�v � yB£;v�~&rs{���uzm7��m]�]yX� � £¥yX�'x&j�m.yXt�x&}�t�x*.w' �10 yB£)~��=~&x�mcl @ 1 I ¯ ºàm7{�y�º j�v] �m��¬��m]�]yX� � ~� ' j d ����' ���� � ��� ��ªm]�]yX� � �� ' j d k ��' k �� � ��� ��ªm]�]yX� ���

I&I&I � '� j d ����' ���� � ��� ��$m]�]yX� � � @ 2�I

�]yXusuzm]�cx�m � ~�t��]�]m]~�~&rs �mcus��± -�vX��j ��m]�]yX� ��� �]yX{ax�vBrs{�~ � vBx�vG'� j d ����£[��yXl vµ���Hm � � 'D,�������E

~�mc{�~�yX�¶}"y=yXu ¯ vB{ � � vBx�v ' ���� £¤��yXl v (*"$+,�'�&�¬~�mc{�~�yX�¶}�y=yXu¼±®i0j�m¬{�t�l��"mc�¨yB£�~�mc{�~�yX��~l_v�� ��m � r$#�mc��mc{ax7rs{¨x&j�m�����m � vB{ � x&j�m�lnyG ars{��¶}�y=yXuz~ ¯ vB{ � x&j�t�~`rs{¬m�vX��j¬��m]�]yX� ����¯ x&j�mlnm�vX~�t���mclnmc{�x` �m]�cx�yX��~_' j d ����

vB{ � ' ���� l_v��¶j�v] �m � r$#Mmc��mc{�x � rslnmc{�~&rzyX{�~�±ªi0j�rz~�rz~M/a{�yGº'{t�{ � mc�ªx&j�m7{�vBlnm.yB£)}"yXus�=��m@£¤mc��mc{��]mS~�mcx&t�}C0Õ�<4§±

i ynm�vX��j¸��m]�]yX� ��� @ � � � � ��I0�]yX�&��m]~&}�yX{ � ~ªvg~&x�vBx�m@¦§~&}�vX�]m.��m�vBusr IGvBx&rzyX{ rs{¶x&j�m`£¥yX�&l!""# ""$� �������� � !� ���� "?$ ��������'� j d ���� � ( j � ���� .[��m@£¥mc��mc{��]m7}�yayXu 0' ���� � ( � � ���� .¤~�mc{�~�yX�'}"y=yXuM{&% � 0 @823I

º'rsx&j v¶~&rs{���uzm_~�x�vBx�mgx&��vB{�~&rsx&rzyX{µl_vBx&�&r�� ¦$~&rs{��]mnx&j�m_~�vBlnmn~&�H~&x�mcl3rz~7��mcrs{��|yX��~�mc�& �m �M¯

ve����m � yX��~�mc�& �vBx&rzyX{°l_vBx&�&r��( j�£¥yX�.x&j�m����Hm � ~�mc{�~�yX�7}�yayXu ¯ vB{ � v�~&}�m]�cr����nyX��~�mc�& �vBx&rzyX{

l_vBx&�&r��( �

�]yX�&��m]~&}"yX{ � rs{��_x�yguzya��vBx&rzyX{ � yB£)x&j�m`lny� =rs{��e~�mc{�~�yX�'}"y=yXu¼±i0j�m$}���yX��uzmcl®rz~àx�y���{ � j�y�º x�ygv � vB}�x0x&j�mTyXt�x&}�t�x/¦§yX{�us���]y� �vB�&rwvB{��]m@¦ � �&rs �mc{ ~&t���~�}�vX�]m

vBus��yX�&rsx&j�l1yB£`~�m]�cx&rzyX{½�¨rs{�yX� � mc�ex�y¬r � mc{�x&r�£¤�²x&j�m¸mcrs��mc{�~&x&�&t��cx&t���m¨yB£ rs{ @823I ¯ yX{�x&j�m

��vX~�rz~gyB£Tx&j�m¶lnm�vX~�t���mclnmc{�x�~E@ 2�I�º'j�rz��j ºàm¸~&j�yXt�u � lnmc�&��m¸~�yXlnmcj�yGº»£¥yX��x&j�rz~g}�t��&}"y�~�m�±´0xSx&j�rz~7}"yXrs{ax ¯ xKº�y }"yXrs{ax�~S~&j�yXt�u � ��mg~&x&��m]~�~�m � ±��-rs��~&x ¯ x&j�mn�]yX�&{�mc��~�x�yX{�mnrs{°~�m]�cx&rzyX{E�¶rz~£¼vX�cx�yX�&r IGvBx&rzyX{ @#-�I)º'j�rz��j ¯ yB£ �]yXt���~�m ¯ j�yXu � ~;£¥yX�6m�vX��j�yB£Mx&j�m � ��m]�]yX� � ~�º'rsx&j¡v7�$� ' ��������uzm@£¤x£¼vX�cx�yX�

(vB{ � v7�$� ' ���������&rs��jax;£¼vX�cx�yX� � ±-«K{n}�vB�&x&rz�ct�uwvB� ¯ mc �mc{_x&j�yXt���j_ºàm � y7j�v� �mTvX�]�]m]~�~

x�y mclg}�rs�&rz��vBuàm]~&x&rsl_vBx�m]~�yB£'x&j�m_�]yG XvB�&rwvB{��]m]~ 9 ' ���� ' ����a` ^ ; yB£'x&j�mnlnyG ars{��¹~�mc{�~�yX�S}"y=yXu ¯x&j�mc�°vB��meyB£0}�y=yX�.j�mcus}µ��m]��vBt�~�mnx&j�mcrs�S£¥vX�cx�yX�&r IGvBx&rzyX{�~ @#-�I.~&j�y�º t�}Ev¶��m]�]yX� � ¦ � mc}�mc{ � mc{�x~�x�vBx�m@¦§yXt�x&}�t�x��]yX�&��mcuwvBx&rzyX{ � ~�rs{��]m_x&j�m�~�mc{�~�yX��~�uzy=��vBx&rzyX{²��j�vB{���m�±Eq=m]�]yX{ �M¯ ºàmej�v� �m¡{�yvX�]�]m]~�~;x�y.�c��y�~�~&¦¼��m]�]yX� � �]y� �vB�&rwvB{��]m]~ 9 ' � = �� ' ���� ` ^ ; º'rsx&j �)(� � H ¯ ~&rs{��]m�~�mc{�~�yX�6}"y=yXuz~ � vB{ �� H � y�{�yXx7��m]�]yX� �¬� vBx�v�~�rslSt�usx�vB{�m]yXt�~&us��±gi0jat�~ ¯ º�mg��vB{�{�yXx�~&x�vX��/¹vBusu�x&j�m � vBx�v�~�mcx�~`rs{µvt�{�r��at�m7 �m]�cx�yX� ¯ yXx&j�mc�&º'rz~�m7º�m7ºàyXt�u � j�v� �m7x�ygj�vB{ � uzm7rs{��]yXlg}�uzmcx�m��]y� �vB�&rwvB{��]m.l_vBx&�&rz�]m]~�±

´ �]yX{�~�m"�at�mc{��]m7yB£ x&j�y�~�m$x§º�y�£¼vX�cx�~ ¯ rs{�x&j�mT}"mc��~&}�m]�cx&rs �m`yB£ ��t�rsu � rs{��_v GTvB{N/�mcu�l_vBx&�&r��mc{X�&y��=rs{��¡v�£¥vX�cx�yX�&r IGvBx&rzyX{|}���yX}"mc�&xK� ¯ rz~ªx�yn�]yX{��]mc{�x&��vBx�m�yX{¸x&j�m.�]yG XvB�&rwvB{��]m]~Trs{a �yXus ars{��ex&j�m

xzy L{xw|

Page 10: Blind subspace-based eigenstructure identification under … · 2016-12-27 · Blind subspace-based eigenstructure identi cation under nonstationary excitation using moving sensors

����������� �������������������������������� ��� �!���#"$�%�&���'�&�)(*"$+,�'�&�*�������"$��� 2

� 'D,�� ����E ����6��"$���@±!����yXl{�yGº3yX{ ¯ º�m¨x&jat�~¡£¤y=�ct�~eyX{ x&j�m¨x§ºày £¤yXusuzyGº'rs{�� £¥vBlgrsusrzm]~¶yB£�]yG XvB�&rwvB{��]m]~ ] j d �^ >�@BA� 9 ' j d ���� ' j d ���� ` ^ ; M ] � ^ >�@BA� 9 ' ���� ' j d ����a` ^ ; @ [ IyB£)º'j�rz��j º�m.��vB{ �]yXlg}�t�x�m�mclg}�rs�&rz��vBu�m]~&x&rsl_vBx�m]~ ¯ £¤yX��uwvB��~���� W ±

´ ����~&x)lnmc�&��m'yB£!x&j�m � vBx�vTrz~){�yGº }���yX}"y�~�m � £¤yX�)x&j�m'~&x�vBx&rzyX{�vB�&��rs{�}�t�x;m@�H�crsx�vBx&rzyX{e��vX~�m�±«Kx'rz~'~&j�yGº'{ x�y�£¼vBrsuMrs{¶x&j�m7{�yX{�~&x�vBx&rzyX{�vB�&�¸��vX~�m ¯ mc �mc{¸t�{ � mc�ªx&j�m7~&rslg}�usr�£[�=rs{��evX~�~�t�lg}�x&rzyX{yB£�~&x�vBx&rzyX{�vB�&�¶m@���crsx�vBx&rzyX{¸º'rsx&j�rs{¸x&j�m7��m]�]yX� � ~�±

� Ã*F ����� F � ¿�� � D ���H� DOÃ*F � Ã�� ��� D � F � ¿ �µ�H��� D� «K{|x&j�m�~&x�vBx&rzyX{�vB�&�|��vX~�m ¯ x&j�m�m@���crsx�vBx&rzyX{�]yG XvB�&rwvB{��]mSl_vBx&�&r�� � yam]~0{�yXx � mc}�mc{ � yX{¸��m]�]yX� ��� 79 $ ���� $ ���� = ; � 5#CD.BE8FlEDH 0vB{ � x&j�m7�c��y�~�~/¦§�]y� �vB�&rwvB{��]m.��mcx§ºàm]mc{ x&j�m.~�x�vBx�m.vB{ � x&j�m`����m � ~�mc{�~�yX��~�yXt�x&}�t�x

� >�@BA� 9 � ���� ' j d ���� ;� y=m]~O��"!� � mc}�mc{ � yX{ � mcrsx&j�mc��±)i0j�mc��m@£¤yX��m ¯ £¥yX�-vBusu=x&j�m���m]�]yX� � ~ � � � M&I&I&IaM �¡vB{ � uwvB��~��� W�¯x&j�m7�]yG XvB�&rwvB{��]m]~$rs{ @ [ I6£¼vX�cx�yX�&r I�m ¯ º'rsx&j|vg�]yX{�~�x�vB{�xB��� � �&��£¼vX�cx�yX� ¯ vX~] j d �^ � ( j ^ � >�@BA� ] j^ @���I] � ^ � ( � ^ ��àyX{�~�m"��t�mc{ax&us� ¯ £¥yX��m�vX��j¸uwvB� ��� W�¯ º�mS��vB{ ~&x�vX��/�x&j�m:] � ^ � ~0rs{�x�yev���uzy=��/a¦§�]yXust�lg{  �m]�cx�yX�

]��^ >�@BA� ghhhhi] j^] �^±±±] �^n oooop @ 1���I

º'j�rz��j¶£¼vX�cx�yX�&r I�m]~$vX~ ] �^ � ( ^ � @ 131 Iº'j�mc��m

(rz~'x&j�m.��uzya� /�¦§�]yXust�lg{¨yB£)yX��~�mc�& �vBx&rzyX{¸l_vBx&�&rz�]m]~

( >�@BA� ghhhhi( j( �±±±( �nZoooop @ 1���I

LML ÝON�P�Qcî P

Page 11: Blind subspace-based eigenstructure identification under … · 2016-12-27 · Blind subspace-based eigenstructure identi cation under nonstationary excitation using moving sensors

[ � �!�&� ������ �+5������ ��� ���� � ��6+5��6����� �� �#�� ��� � �5�����+ ���'� ���H�$�������E���"$�������!�

i0j�mc��m@£¥yX��m ¯ x&j�m GTvB{N/�mcuªl_vBx&�&r��E��usuzm � º'rsx&j�x&j�y�~�m¸�]yXust�lg{H¦§~&x�vX� /�m � �]y� �vB�&rwvB{��]m]~ ¯ º'j�rz��jj�vX~*.�� " � 0 x&rslnm]~ªvX~'l_vB{a����y�º�~$vX~ª�]yXust�lg{�~ ¯ £¼vX�cx�yX�&r I�m]~$vX~b � >�@BA� q�s1tvu .w]��^ 0 � � . ( M *0 � . M � 0 @ 1,-�IvB{ � x&j�m¸vBus��yX�&rsx&j�l � m]~��c�&rs��m � rs{ ~�m]�cx&rzyX{��|£¤yX�gx&j�m¶~�x�vB{ � vB� � ��vX~�m¶��vB{ ��m¸vB}�}�usrzm � x�yb � ±¡«§{EyXx&j�mc�Sº�yX� � ~ ¯ º'j�mc{Ex&j�mgrs{�}�t�x�m@���crsx�vBx&rzyX{µrz~�~&x�vBx&rzyX{�vB�&� ¯ º�me��vB{°}���ya�]m]m � vX~.r�£ºàm�j�v � x&j�m�£¤t�usu�~�mcx`yB£�~�mc{�~�yX� � vBx�v�v] �vBrsuwvB��uzmgvBusx�yX��mcx&j�mc� ¯ rs{�~&x�m�v � yB£�~�mc{�~�yX�T}"y=yXuz~$º'rsx&j �vB�&�=rs{��¡uzy=��vBx&rzyX{�~�±�«§{¹x&j�rz~.��vX~�m ¯M� rs��m]�cxSvB}�}�usrz��vBx&rzyX{µyB£�x&j�mg~&t���~�}�vX�]mnvBus��yX�&rsx&j�l3x�y¡x&j�m�]yXust�lg{H¦§~�x�vX��/�m � �]yG XvB�&rwvB{��]m]~+@ 1���IYrz~;vT~&rslg}�uzm0~�yXust�x&rzyX{gx�yTx&j�m0}�yXus�a��m@£¥mc��mc{��]m0yXt�x&}�t�x/¦§yX{�us�mcrs��mc{�~�x&�&t��cx&t���m.r � mc{�x&r�����vBx&rzyX{¨}���yX��uzmcl¸±

�¡¿ � à �BÄ Á Ä ¿QH;¿ F Ä ¿ F � ¿�� � D ���H� DOÃ*F � Ã�� ��� D � F � ¿ �µ�H��� D� «§£�ºàm$vX~�~�t�lnm'x&j�vBx6x&j�mªrs{�}�t�xm@���crsx�vBx&rzyX{ �]yG XvB�&rwvB{��]mSl_vBx&�&r�� � mc}�mc{ � ~�yX{¸x&j�m.��m]�]yX� � rs{ � m@� � 79 $ ���� $ ���� = ; � 5 � CD.BE FlE H 0x&j�m7�c��y�~�~/¦§�]y� �vB�&rwvB{��]m.l_vBx&�&r��

� � >�@BA� 9 � � ���� ' j d ���� ;vBuz~�y �& >�����$�)"$� � � GTmc{��]m ¯ £¤yX� � � � M&I&I&I M � vB{ � ��� W�¯ £¼vX�cx�yX�&r IGvBx&rzyX{�~ @���I${�y�º�º'�&rsx�mº'rsx&j¨v���m]�]yX� � ¦ � mc}"mc{ � mc{ax � ] j d �^ � ( j ^ � � @ 1co I] � ^ � ( � ^ � � @ 1���IvB{ �  �m]�cx�yX�_] �^ yB£�~�x�vX��/�m � �]yG XvB�&rwvB{��]m]~ � m@��{�m � rs{ @ 1���I0{�yguzyX{���mc�ª£¥vX�cx�yX�&r I�m]~`vX~�rs{ @ 131 I@±´T~6v`�]yX{�~�m"�at�mc{��]m ¯ ���&t�x�m0£¤yX���]m$vB}�}�usrz��vBx&rzyX{eyB£M~&t���~&}�vX�]mªvBus��yX�&rsx&j�l ���g~&x�vX��/=rs{��.x&j�m\] � ^ � ~rz~){�yXx;vB{_vB}�}���yX}��&rwvBx�m�~�yXust�x&rzyX{gx�y$x&j�m'mcrs��mc{�~�x&�&t��cx&t���m'r � mc{�x&r�����vBx&rzyX{e}���yX��uzmcl rs{�x&j�m'��vX~�myB£-��m]�]yX� � ¦ � mc}"mc{ � mc{ax$rs{�}�t�x$m@�H�crsx�vBx&rzyX{¸{�yXrz~�m�±�i yn�crs���ct�l� �mc{�xªx&j�rz~ � r�¢_�ct�usx§� ¯ x&j�m.r � m�v�rz~x�y)��"$��(*�$��� � �x&j�mT�]yG XvB�&rwvB{��]m`l_vBx&�&rz�]m]~(@ 1co I§¦ @ 1���I�x�y�l_v /�mTx&j�mcl#uzyay&/=rs{��nvX~àr�£ x&j�mc�_ºàmc��myX��x�vBrs{�m � º'rsx&j¸x&j�m.~�vBlnm.m@���crsx�vBx&rzyX{Y±

� � (ª����!�0�!��(ª���'�"*+�-,7�-�/�!�'�!��(�� � ���0�!�8��('��24�ª�����-�0�!��(����yX����{ � rs{��¡x&j�m�vB}�}���yX}��&rwvBx�m�{�yX�&l_vBusr IGvBx&rzyX{ ¯ º�m�}"mc�/£¥yX�&l � r$#Mmc��mc{�x`�]y� �vB�&rwvB{��]m�~�x�vX��/=rs{��yX}�mc��vBx&rzyX{�~�£¤yX��x&j�m � vBx�v.£[��yXl x&j�mª��m@£¥mc��mc{��]m`vB{ � lny� =rs{���~�mc{�~�yX��~ ¯ ��m]~&}�m]�cx&rs �mcus��±�«§{eº'j�vBx£¥yXusuzy�º�~ ¯ º�m7t�~�m`x&j�m7��mc{�mc�&rz�7{�yXx�vBx&rzyX{�~�rs{ax&��y � t��]m � rs{ ~�m]�cx&rzyX{|�H±

xzy L{xw|

Page 12: Blind subspace-based eigenstructure identification under … · 2016-12-27 · Blind subspace-based eigenstructure identi cation under nonstationary excitation using moving sensors

����������� �������������������������������� ��� �!���#"$�%�&���'�&�)(*"$+,�'�&�*�������"$��� �

�¡¿��K¿ � ¿ F � ¿ � ¿ F � à �B� i0j�vB{N/=~¡x�y @ 1co I ¯ vBusuTx&j�m%G`vB{N/�mcu`l_vBx&�&rz�]m]~¡��t�rsusx�yX{½x&j�m � vBx�v� mcusrs �mc��m � �a�nx&j�mª��m@£¥mc��mc{��]m`~�mc{�~�yX��~àyB£Yx&j�m � ��m]�]yX� � ~ࣥvX�cx�yX�&r I�m$º'rsx&j�x&j�m$~QvBlnm � 'D���£¼vX�cx�yX�vX~ b�j d � >@LA� q�s1tvu .w] j d ���^ 0?� � . ( j M *0 � . M � � 0 @ 1 2�I�°m7��vB{ x&j�t�~�}���y=�]m]m � vX~0£¥yXusuzy�º�~G± � mcx] j^ >�@BA��� ] j d �^ ] j d k^ I&I&I_] j d �^����m.x&j�mS��uzya� /�¦¼��yGº  �m]�cx�yX�TyB£;x&j�m�mc{�x&rs��m�~�mcx$yB£;��m@£¥mc��mc{��]m � vBx�v_�]yG XvB�&rwvB{��]m]~�±$«§xª£¥vX�cx�yX�&r I�m]~vX~ ] j^ � ( j ^ � M º'j�mc��m � >�@BA� . � � � klI&I&I � � 0 @ 1 23Ii0jat�~6x&j�m GTvB{N/�mcu!l_vBx&�&r�����usuzm � º'rsx&j_x&j�y�~�m���y�º0¦§~&x�vX� /�m � �]yG XvB�&rwvB{��]m]~ ¯ º'j�rz��j_j�vX~ �¸x&rslnm]~vX~�l_vB{��¶�]yXust�lg{�~$vX~'��yGº�~ ¯ vBuz~�y�£¼vX�cx�yX�&r I�m]~$vX~b j >@LA� q�s1tvu .w] j^ 0?� � . ( j M *0 � . M � 0 @ 1 [ IJTyXx�m¡x&j�vBx�l_vBx&�&r��Gb je��vB{²��m¡yX��x�vBrs{�m � �a�Ers{ax�mc�&uzm�v] =rs{��°x&j�m¡��uzya� /�¦§�]yXust�lg{�~eyB£ªx&j�m �G`vB{N/�mcu l_vBx&�&rz�]m]~_b�j d � rs{ @ 1 2�I@±'´T~�~&t�lnm7x&j�vBx$}�vBrs�4. ( j M *0 rz~$yX��~�mc�& XvB��uzm�±0i0j�mc{ ¯ }�vB�&x&r�¦x&rzyX{�rs{��¶x&j�mn�&rs��j�x`£¼vX�cx�yX� � yB£ @ 1 [ ITrs{¹x&j�mg~�vBlnm�l_vB{�{�mc��vX~ � rs{ @ 1 23I ¯ º�m���m]�]y� �mc�Sx&j�m�&rs��jax0£¥vX�cx�yX��~ � . M � � 0 yB£�@ 1 2�I;£¤yX��vBusu � � � M&I&I&I M �;± JTyXx�m$x&j�vBx'}���y=�]m]~�~&rs{���x&j�mT��m@£¥mc��mc{��]m� vBx�v�' j d ����

�$���8"E����z�=�� ¯ yX{¸x&j�m7��vX~&rz~�yB£)x&j�m������&�!� �GTvB{N/�mcuYl_vBx&�&r���£¼vX�cx�yX�&r IGvBx&rzyX{C@ 1 [ I ¯ rz~l_vB{ � vBx�yX�&�¡£¥yX�'l_v /ars{��e~&t���m7x&j�vBx�ºàm � m�vBuMº'rsx&j¸x&j�m����$(70~&x�vBx�m7��vX~�m]~�rs{ @823I@±

O­Ã�� DGF�� � ¿ F � à �P� i0j�vB{N/H~�x�y @ 1���I ¯ x&j�m7G`vB{N/�mcu6l_vBx&�&r��°��t�rsusx�yX{µx&j�m_�]yG XvB�&rwvB{��]m]~�yB£x&j�m � vBx�v ' ���� £¤��yXl+x&j�m¡lny� =rs{��¹~�mc{�~�yX��}�y=yXu�rs{²��m]�]yX� � � mc{X�&y��H~�v|��m]�]yX� � ¦ � mc}�mc{ � mc{�x£¼vX�cx�yX�&r IGvBx&rzyX{ b � >�@BA� q_s1t%u .w] � ^ 0?� � . ( � M 40 � . M � � 0 @ 1 ��I´T~�~&t�lnm$x&j�vBx ¯ £¥yX�0vBusu ��¯ x&j�mT}�vBrs� . M � � 0 rz~'�]yX{ax&��yXusuwvB��uzm�±6i0j�rz~0rz~�v���m�vX~�yX{�vB��uzm.vX~�~�t�lg}H¦x&rzyX{ ¯ rsx;�/t�~&x$x�mcusuz~ªx&j�vBx ¯ º'j�rsuzmSx&j�m�m@�H�crsx�vBx&rzyX{¨l_v]�|��j�vB{���m ¯ º�m�{�m]m � rsx$x�yem@���crsx�m�vBusu�x&j�mlny � m]~`yB£6x&j�m�~&�H~&x�mcl ¯ £¤yX�`m�vX��j � vBx�v¡~�mcx�±7´T~�~&t�lnm�vBuz~�y¡x&j�vBx ¯ t�}¹x�y¶v_}"mc�&l�t�x�vBx&rzyX{°yX{x&j�m0��m]�]yX� � rs{ � m@�Hm]~ ¯ � . M � � 0 rz~)x&j�m'��m]~&x;�]yX{ � rsx&rzyX{�m � �]yX{�x&��yXusuwvB��rsusrsx§�_l_vBx&�&r��nvBlnyX{��Sx&j�m� . M � � 0 � ~�± ��yX�TvBusu ��¯ � � � M&I&I&I M � ¯ ºàm � m@��{�m.x&j�m`{�yX�&l_vBusr I�m � l_vBx&�&rz�]m]~

b � >@LA� b �� � ; . M � � 0 � � . M � � 0 � ; . M � � 0 � `N� � . M � � 0� @¥����I

LML ÝON�P�Qcî P

Page 13: Blind subspace-based eigenstructure identification under … · 2016-12-27 · Blind subspace-based eigenstructure identi cation under nonstationary excitation using moving sensors

1�� � �!�&� ������ �+5������ ��� ���� � ��6+5��6����� �� �#�� ��� � �5�����+ ���'� ���H�$�������E���"$�������!�

�°m0{�yXx�m0x&j�vBx b � �]yXrs{��cr � m]~6º'rsx&jgx&j�m GTvB{N/�mcu�l_vBx&�&r��q�s1tvu . ] � ^ 0 ��t�rsusx;yX{n�]y� �vB�&rwvB{��]m]~ ] � ^

º'j�rz��j¶£¼vX�cx�yX�&r I�m�vX~ ] � ^ � ( � ^ � � @¥� 1 Ii0j�mc��m@£¥yX��m ¯ º�m6��vB{S{�y�º°}���y=�]m]m � vX~Mrs{Sx&j�m;��vX~�m�yB£��]yX{�~&x�vB{�x-m@�H�crsx�vBx&rzyX{��]y� �vB�&rwvB{��]m;l_vBx&�&r��M±��yX�ªm�vX��j¸uwvB� ��� W�¯ ºàm � m@��{�m7x&j�m7��uzy=��/a¦§�]yXust�lg{|~&x�vX� /�m � �]y� �vB�&rwvB{��]m]~

] ^ >�@BA� ghhhhhi] j^] �^±±±] �^nZooooop M º'j�mc��m ] j^ >�@BA� ] j d �^ � ( j ^ � � @¥����I

i0j�vB{N/H~�x�y @¥� 1 I ¯ x&j�y�~�m�~&x�vX� /�m � �]y� �vB�&rwvB{��]m]~ ]�^ª��mcj�v] �m¸vX~8] �^ � m@��{�m � rs{ @ 1���I7¡x&j�mc�£¼vX�cx�yX�&r I�mSvX~ ]<^ � (V ^ � �º'rsx&j

(� m@��{�m � rs{ @ 1���I@± i0j�mc��m@£¥yX��m ¯ x&j�mHGTvB{N/�mcuTl_vBx&�&r�����usuzm � º'rsx&j½x&j�y�~�m|�]yXust�lg{H¦

~�x�vX��/�m � �]y� �vB�&rwvB{��]m]~ ¯ º'j�rz��j²j�vX~ .�� " � 0 x&rslnm]~�vX~Sl_vB{a�°��y�º�~gvX~��]yXust�lg{�~ ¯ £¥vX�cx�yX�&r I�m]~vX~ b >�@BA� q�s1tvu . ]<^ 0 � � . ( M 40 � . M � � 0 @¥�!-�IJTyXx�m`x&j�vBx�l_vBx&�&r�� b4��vB{¶�"m7yX��x�vBrs{�m � ����rs{�x�mc�&uzm�v� ars{��ex&j�m7��uzya� /�¦¼��yGº�~$yB£b j >@LA� q�s1tvu . ] j^ 0�� b j d �vB{ � yB£'x&j�m ��l_vBx&�&rz�]m]~ b � � m@��{�m � rs{ @¥����I@±�i0j�vB{N/H~Sx�y @¥�!-�I ¯ x&j�m¡vBus��yX�&rsx&j�l ��rs �mc{ rs{~�m]�cx&rzyX{|����vB{¸��m.vB}�}�usrzm � x�y b|±��P ��à � D ���?� ��yX�'rslg}�uzmclnmc{�x&rs{��_x&j�m.vB�"yG �m$��m�vX~�yX{�rs{�� ¯ º�mT}���yX}�y�~�mTx&j�mT£¤yXusuzyGº'rs{��_vBus��yB¦�&rsx&j�l ¯ º'j�mc��m �½rz~Sx&j�me{�t�l��"mc�nyB£���m]�]yX� � ~nvB{ ���� � mc{�yXx�m]~gvB{ mclg}�rs�&rz��vBu'm]~&x&rsl_vBx�m�yB£l_vBx&�&r���� ±

� Å t�rsu � x&j�m � G`vB{N/�mcu'l_vBx&�&rz�]m]~ �b j d � � q�s1tvu . �] j d ���^ 0£¤��yXl x&j�m���m@£¤mc��mc{��]m¸~�mc{�~�yX�

� vBx�vH±

� Å t�rsu � �b�j¨���½rs{�x�mc�&uzm�v� ars{��½x&j�m¹��uzya� /�¦§�]yXust�lg{�~|yB£�x&j�m � l_vBx&�&rz�]m]~ �b j d � � ~ @ � �� M&I&I&I M ��I@±

� q��+'$¦ � m]�]yXlg}�y�~�m �b jB±��7m]mc}nyX{�us��x&j�m'�&rs��j�x;£¥vX�cx�yX� � . M � 0 rs{ @ 1 [ I ¯ vB{ � ��vBusu�rsx��/t�~&x� ±

xzy L{xw|

Page 14: Blind subspace-based eigenstructure identification under … · 2016-12-27 · Blind subspace-based eigenstructure identi cation under nonstationary excitation using moving sensors

����������� �������������������������������� ��� �!���#"$�%�&���'�&�)(*"$+,�'�&�*�������"$��� 131

� �-vB�&x&rsx&rzyX{ � rs{�x�y � >�@BA� . � � � k I&I&I � � 0 vX~ � rs{ @ 1 23I@± ´$~�~�t�lnm � � rz~_x&j�m¶��m]~&x�]yX{ � rsx&rzyX{�m � ±

� �àyXlg}�t�x�m.x&j�m`{�yX�&l_vBusr I]rs{��e£¥vX�cx�yX��~� � ;� . � � � ;� 0 `N� � � � ±

� Å t�rsu � x&j�m��CGTvB{N/�mcu l_vBx&�&rz�]m]~ �b � � q�s1tvu . �] � ^ 0 £[��yXl®x&j�m.lny� =rs{��_~�mc{�~�yX� � vBx�vH±

� �àyXlg}�t�x�m¸x&j�m � {�yX�&l_vBusr I�m � l_vBx&�&rz�]m]~ �b � >�@BA� �b � � � ;� . � � � ;� 0 `N� � � � ± '`m@��{�m�b j >�@BA� �b�j d � ±

� Å t�rsu � �b ��� rs{�x�mc�&uzm�v] =rs{���x&j�m¨��uzy=��/a¦¼��y�º�~|yB£Sx&j�m?.�� " � 0 l_vBx&�&rz�]m]~ �b � � ~ @ � �W M&I&I&I M ��I@±� ´ª}�}�us�¡x&j�mSvBus��yX�&rsx&j�l·yB£)~�m]�cx&rzyX{¨��x�y �b¨±i0j�mࣤyXusuzyGº'rs{��.�]yXlglnmc{�x�~�vB��m0rs{gyX� � mc��±��-rs��~&x ¯ x&j�m'�]yXlg}�t�x�vBx&rzyX{�vBu��]yXlg}�uzm@�Hrsx§�gyB£"x&j�rz~

vBus��yX�&rsx&j�l � y=m]~ ��" � � mc}�mc{ � yX{ex&j�m�{�t�l��"mc� �|yB£Y��m]�]yX� � ~ ¯ ~&rs{��]m$x&j�mª~�l_vBusuzm]~&x � rslnmc{�~�rzyX{yB£"x&j�m0x§ºàyBGTvB{N/�mcu�l_vBx&�&rz�]m]~ �b j vB{ � �b�x�y`��m � m]�]yXlg}"y�~�m �n� mc}"mc{ � ~�yX{�us��yX{gx&j�m0{�t�l��"mc�yB£0��m@£¥mc��mc{��]mn~�mc{�~�yX��~�±_q=m]�]yX{ �M¯ x&j�m�rs{�x�mc�&uzm�v� ars{��¹yB£0x&j�mg��uzya� /�¦¼��yGº�~SyB£0x&j�m �b � � ~.��vB{°��m� yX{�m|mc �mc{�º'j�mc{ x&j�m¸{�t�l��"mc��yB£.~�mc{�~�yX��~_rs{ x&j�m¸lnyG ars{�� }"y=yXuz~grz~ ��"!���]yX{�~&x�vB{ax 7¨x&j�mlnm�vX~�t���mclnmc{�x� �m]�cx�yX��~ ' ���� vB{ � ' � = ��

l_v��°j�v] �m � r$#Mmc��mc{�x � rslnmc{�~&rzyX{�~�±|i0j�rs� �M¯ º�m � y{�yXxà{�m]m � x�yS�]yXlg}�t�x�m`m@�=}�usrz�crsx&us�_x&j�m � � l_vBx&�&rz�]m]~ ¯ ~&rs{��]mTyX{�us�nx&j�mª�&rs��j�xࣥvX�cx�yX��~ � . M � � 0rs{ @ 1 ��I�vB��m7{�m]m � m � £¥yX���]yXlg}�t�x&rs{�� �b � rs{ @¥����I@±� � (ª����!�0�!��(ª���'�"*+�-,7�-�/�!�'�!��(���� �/������������� ������(��"��

«K{e~�m]�cx&rzyX{eo ¯ º�m$j�v] �m7vX~�~�t�lnm � ~&x�vBx&rzyX{�vB�&�¡m@�H�crsx�vBx&rzyX{¡º'rsx&j�rs{¶m�vX��j���m]�]yX� �M¯ º'rsx&j���m]�]yX� � ¦� mc}�mc{ � mc{�x'�]yG XvB�&rwvB{��]mTl_vBx&�&r��M±�i0j�rz~6rz~'v7��m�vX~�yX{�vB��uzm`vB}�}���y��=rsl_vBx&rzyX{�º'j�mc{�x&j�mªmc{a ars��yX{H¦lnmc{ax�vBu;�]yX{ � rsx&rzyX{�~.vB��m�~&uzy�º'us�¸x&rslnm@¦¼ �vB�&�=rs{���± Å t�x`v_lnyX��m���m�vBusrz~&x&rz�gvX~�~&t�lg}�x&rzyX{¹rz~Tx&j�vBxx&j�m7m@���crsx�vBx&rzyX{|�]yG XvB�&rwvB{��]mSl_vBx&�&r��m79 $ ���� $ ���� = ; � 5 � CD.BE8FlE H 0rz~ ���'(*�@ +<�$��9$������� �'�z�&��� �� ��� � �� "$����±OG$mc{��]mªx&j�m$yXt�x&}�t�x � vBx�v�yB£Yx&j�m�}�yXus�a��m@£¥mc��mc{��]m$~�mcx&t�}@823I ��vB{�{�yXx)�"m0vX~�~&t�lnm � ~&x�vBx&rzyX{�vB�&��º'rsx&j�rs{gm�vX��j�rs{ � rs ar � t�vBuH��m]�]yX� �M¯ vB{ � º�m0��vB{�{�yXx;vB}�}�us�x&j�m7vB�&��t�lnmc{ax�yB£�~�m]�cx&rzyX{�o�± J$mc �mc�&x&j�mcuzm]~�~ ¯ x&j�m7vBus��yX�&rsx&j�l � m]~��c�&rs�"m � rs{¡x&j�vBx'~�m]�cx&rzyX{¶��vB{~�x&rsusu;�"mevB}�}�usrzm � º'rsx&jEusrsx&x&uzmnlny � r�����vBx&rzyX{ ¯ vB{ � ��vB{°��mn}���yG �mc{ �]yX{�~&rz~�x�mc{�x ¯ vX~.ºàm_~&j�yGº{�yGº t�~&rs{��nx&j�m`��m]~�t�usx�~ªyB£ 0 o$4§±

LML ÝON�P�Qcî P

Page 15: Blind subspace-based eigenstructure identification under … · 2016-12-27 · Blind subspace-based eigenstructure identi cation under nonstationary excitation using moving sensors

1�� � �!�&� ������ �+5������ ��� ���� � ��6+5��6����� �� �#�� ��� � �5�����+ ���'� ���H�$�������E���"$�������!�

��yX� � � � M&I&I&IaM � ¯ uzmcx�� � ��m6x&j�m���{�rsx�m0~�mcx)yB£�~�vBlg}�uzmà}"yXrs{ax�~)rs{���m]�]yX� � ��¯ uzmcx�� � >�@BA� � �� �

��mTx&j�mT{�t�lS��mc�ªyB£�~QvBlg}�uzm]~ ¯ vB{ �¶� m@��{�m`x&j�m$£¤yXusuzyGº'rs{��n~�m]�]yX{ � yX� � mc��mclg}�rs�&rz��vBuMlnyXlnmc{�x�~

�� .�� � 0 >@LA� �� � � ��

���� � ' j d ���� � k "�� ' �� � k �

� � .�� � 0 >@LA� ��� .�� � 0 ��

���� ���� ' j d ���� ;

] j d �^ .�� � 0 >@LA� ��� .�� � 0 ��

���' j d ���� ' j d ���� ` ^ ; @¥�Bo I

] � ^ .�� � 0 >@LA� ��� .�� � 0 ��

���' �� ' j d ���� ` ^ ;

�àyX{�~�r � mc��x&j�m6£¥yXusuzy�º'rs{��SvX~�~&t�lg}�x&rzyX{�~ ¯ r � mc{ax&rz��vBu�x�y7�]yX{ � rsx&rzyX{�~L�+1'vB{ � �0�TyB£�~�m]�cx&rzyX{g«�«Q±ã´yB£ 0 o$4§±�e��� �?� H � D[Ã�F ���

� ����������� ���� �

� .�� � 0 � "� � ���z� >�� " ��� ���'�?����9 � �

��� �"! �= $#����'�8� ����"$� � "$+<�!���#�$���E (*�!�����%# 5 � ��� ���6� D�"$��(��?9#��"$������ �CD�� "!(� ��"$+$P�O�'� E �

JTyXx�màx&j�vBx;�]yX{ � rsx&rzyX{ � � rz~6v$ �mc�&��lgrsu � ��m"��t�m]~�x)~�x�vBx&rs{��7x&j�vBx ¯ £¤yX�;m�vX��jgrs{ � rs ar � t�vBu���m]�]yX� ���¯ x&j�m�~&�=~�x�mcl rz~`~&tH¢_�crzmc{ax&us�¨m@���crsx�m � ±.i0j�m�£¤yXusuzyGº'rs{��¶��m]~&t�usx`rz~T}���y� �m � rs{¹x&j�m�����~&x7~&x�mc}yB£)x&j�m`}���y=yB£)yB£)x&j�m]yX��mcl 1`rs{;0 o$4 0ûq=m]��±ã«�«�«Q±ã´ 4 7& � ¿aà � ¿ �'�)( � ���� � "$� �$�'���#"$��� � � Á ��� � �O:���$+5] j d �^ .�� � 0 � ( j ^ � � .�� � 0N" * .�� � 0] � ^ .�� � 0 � ( � ^ � � .�� � 0Y" * .�� � 0������� (*�!�����%# * .�� 0 ��" �� �8" � ���" ���=���� � " �� �8"*�'�,� ���'��9 �i0j�m]yX��mcl 1Tm@�=}���m]~�~�m]~6x&j�vBx @ 1co I§¦ @ 1���I�j�yXu � vX~&�=lg}�x�yXx&rz��vBusus��±L�àyX{�~&r � mc�à{�y�º x&j�m'£¥yXusuzy�º'rs{��v ��� rsx&rzyX{�vBu)vX~�~&t�lg}�x&rzyX{�~�±�e��� �?� H � D[Ã�F �+�

�-,.�0/ "$� �+5���9 � �$� �1��

� �$�#��P� �z�� + @A�z�32 ��� ���������$���'�&� "$������54 ���'�&�$��@� �!� +<�$���� " D �z��%�E"$������"$��� � ���'���'��9 ( � �����6# � � .�� � 0 >�@BA� � f . M � � .�� � 0e0 ����5��9$( > �8"!���#� �$�'��9*�&��� D�"$��( ��9 ��"!�&���&�� D���"$( �E�� " � � ���z� >�� " ��� ���'�?����9 � �

xzy L{xw|

Page 16: Blind subspace-based eigenstructure identification under … · 2016-12-27 · Blind subspace-based eigenstructure identi cation under nonstationary excitation using moving sensors

����������� �������������������������������� ��� �!���#"$�%�&���'�&�)(*"$+,�'�&�*�������"$��� 1,-

��� �0/ "$� �+5���9 � � �z�= >=�!��� . ( � M *0 ��� " �������+<� ��� � �z���!�� � ���

s1tvu � c . ( � M *0� + ��àyX{ � rsx&rzyX{ �-, ¯ º'j�rz��j �]yXrs{��cr � m]~�º'rsx&j²�]yX{ � rsx&rzyX{ �-, yB£ 0 o$4 0ûq=m]��±ã«�«@±ã´ 4 ¯ lnm�vB{�~�x&j�vBx�x&j�m}�vBrs�4. M � � .�� � 0e0 rz~$vX~&�alg}�x�yXx&rz��vBusus��t�{�r�£¥yX�&lgus�¶�]yX{�x&��yXusuwvB��uzmS£¤yX��mc �mc�&� � vB{ � � � uwvB�&��m�±

«§{ax&��y � t��]m7x&j�m GTvB{N/�mcuYl_vBx&�&r��b � >�@BA� q�s1tvu .w] � ^ .�� � 0e0vB{ � x&j�m`{�yX�&l_vBusr I�m � l_vBx&�&r�� b � >@LA� b � � � ;� . � � � ;� 0 `N� � � � @¥����I

º'j�mc��m � >�@BA� . � � � k I&I&I � � 0 rz~ªx&j�m.�&rs��jax$£¼vX�cx�yX�$yB£ b j ¯ vB{ � b j rz~$��t�rsusxª�a�¶rs{�x�mc�&uzm�v� ars{��x&j�m`��uzy=��/a¦§�]yXust�lg{�~TyB£-x&j�m��¬l_vBx&�&rz�]m]~b j d � >�@BA� q�s1tvu .w] j d �^ .�� � 0e0JTyXx�m'x&j�vBx ¯ vX~;rs{)@¥����I ¯ b � rz~àvB{)G`vB{N/�mcu�l_vBx&�&r��n��t�rsusx�yX{¡�]y� �vB�&rwvB{��]m]~�º'j�rz��j_ºàmª{�yXx�m ] � ^ ±i0j�m7£¤yXusuzyGº'rs{��nx&j�m]yX��mcl j�yXu � ~�±

& � ¿aà � ¿ � � ( � ������ "$���!�'���#"$��� � � Á ��� � �z�="5�� � "$+<�!���#�$���E�� �����"$9B�z�=���> >���" #$�'( � � D�� ��@�8"!��� � �!���#"$� � ] � ^ � ( � ^ � � .�� � 0N" * .�� � 0 @¥�32�Iº'j�mc��m.l_vBx&�&r�� * .�� � 0 ��yam]~'x�y*I�mc��ygº'j�mc{ �

���y=m]~'x�ygrs{H��{�rsxK��±

� � ÃMà ����yX�$vBusuYx&j�m � � uwvB�&��m ¯�� m@��{�m] � ^ >@LA�U( � ^ � � .�� � 0b � >�@BA� q�s1tvu .w] � ^ 0vB{ � b � >�@BA� q�s1tvu . ( � ^ � � .�� � 0e0�½rsx&j¶x&j�m]~�m7{�yXx�vBx&rzyX{�~ ¯ x&j�m]yX��mcl 1`º'�&rsx�m]~] � ^ .�� � 0 � ] � ^ " * .�� � 0

LML ÝON�P�Qcî P

Page 17: Blind subspace-based eigenstructure identification under … · 2016-12-27 · Blind subspace-based eigenstructure identi cation under nonstationary excitation using moving sensors

1co � �!�&� ������ �+5������ ��� ���� � ��6+5��6����� �� �#�� ��� � �5�����+ ���'� ���H�$�������E���"$�������!�

vB{ � º�m7j�v� �m7x&j�m`£¥yXusuzy�º'rs{��n��mcuwvBx&rzyX{ £¤yX�'x&j�mBG`vB{N/�mcuYl_vBx&�&rz�]m]~b � � b � � � ;� .�� � 0 . � � .�� � 0 � ;� .�� � 0e0 `N� � � .�� � 0 � @¥� 23I��yX�;}���y� =rs{�� @¥�32�I ¯ rsx6rz~;~&tH¢_�crzmc{�x�x�y`}���y� �mªx&j�vBx ¯ £¥yX�6vBusu � vB{ � £¥yX� � � uwvB�&��m ¯ x&j�mࣥyXusuzy�º'rs{��j�yXu � ~ 7 b � � b � " * . �+� t� �

� 0@¥� [ I

i�yg}���y� �m @¥� [ I ¯ º�m7{�yXx�m.x&j�vBx5@¥����I'vB{ � @¥� 23Ià��m]~�t�usx'rs{b � F b � � b � � � ;� . � � � ;� 0 `N� � � �� ��� ��

@¥� ��I

F b � � � ;� .�� � 0 . � � .�� � 0 � ;� .�� � 0e0 `N� � � .�� � 0 �� ��� ��

i0j�vB{N/H~`x�y��]yX{ � rsx&rzyX{ �-,Y¯ �]yX{ax&��yXusuwvB��rsusrsx§�¬l_vBx&�&rz�]m]~ � � vB{ � � � vB��m�t�{�r�£¤yX�&lgus�¨��yXt�{ � m �£¤��yXl ��mcuzy�º ¯ £¤yX�0�

�vB{ � �

�uwvB�&��m ¯ vB{ � l_vBx&�&r��l. � � � ;� 0 `N� rz~�t�{�r�£¥yX�&lgus�¶��yXt�{ � m � £¤��yXl

vB��y� �m�±6i0j�rz~ªvB{ � x&j�m]yX��mcl 1`x�yX��mcx&j�mc�$rslg}�us�ex&j�vBx ¯ £¤yX� ��uwvB�&��m ¯b � � b � " * .�� � 0 M�� � � � � " * . � � t� �

� 0@#-���I

- ��t�vBx&rzyX{�~ @¥� ��IMvB{ � @#-���IMx�yX��mcx&j�mc� }���yG �mM@¥� [ I@±-i0j�rz~��]yX{��cust � m]~�x&j�m�}���y=yB£�yB£�x&j�m�x&j�m]yX��mcl¸±i0j�m]yX��mcln~(17vB{ � ��mc{�~�t���m$x&j�vBx'x&j�mTlnmcx&j�y � yB£�~�m]�cx&rzyX{¶onvB}�}�usrzm]~ ¯ }���yG ar � m � x&j�vBx'ºàm

t�~�mgx&j�m�] j d �^ .�� � 0 � ~�vB{ � ] � ^ .�� � 0 � ~�rs{ @¥�Bo I�vX~Smclg}�rs�&rz��vBu0m]~&x&rsl_vBx�m]~�yB£'x&j�m_�]yG XvB�&rwvB{��]m]~�±��t��&x&j�mc�&lnyX��m ¯ x&j�m`£¥yXusuzy�º'rs{��¡�]yX{�~&rz~&x�mc{��c����vB{¸�"m`}���yG �mc{ ¯ vX~'rs{ 0 o$4§±

& � ¿aà � ¿ � , ( � ������E"$���$������"!��� � � Á ��� � �z�= �$� � "$���'�z�&( "�D �� ������"!��� ����� "$����� ��� ���� � �z�=��� #$� ���'�F� ��������� �E "�DC(*�!�����#�E����/� � ���z�� �$� ��

`N����6� D�"$��(��?9 ��"$�������� � ������ �z�=�!�

`N�� � �

M �(�(� ���z� >�� " ��� ���'�?����9 � �

qars{��]mTºàmªj�v� �m.vX~�~&t�lnm � x&j�vBxàx&j�mT~&�H~&x�mcl j�vX~�{�y�lSt�usx&rs}�uzm7mcrs��mc{a XvBust�m]~�vB{ � ��m]��vBt�~�m$x&j�mmcrs��mc{�~�x&�&t��cx&t���m7rz~�v��]yX{�x&rs{at�yXt�~'£¤t�{��cx&rzyX{¸yB£-x&j�mT}�vBrs� . ( M *0 ¯ x&j�rz~0x&j�m]yX��mcl vBuz~�ygmc{�~�t���m]~0 o$4Yx&j�mS�]yX{�~&rz~&x�mc{��c�¶yB£-x&j�m7mcrs��mc{�~�x&�&t��cx&t���mSm]~&x&rsl_vBx�m . � KNM �STQ 0 ±� � �!�"���Y�"�S�����¸57�& ����0�!��(��1�������&*7��� �-,.�ª2·�����

´T~0lnmc{�x&rzyX{�m � rs{¸x&j�m`rs{�x&��y � t��cx&rzyX{ ¯ ~�x&�&t��cx&t���vBuY ars����vBx&rzyX{¹vB{�vBus�=~�rz~'yB£)�]yXlg}�uzm@�¶lnm]��j�vB{H¦rz��vBu�~&x&�&t��cx&t���m]~$vB{ � ��yXx�vBx&rs{��el_vX��j�rs{�m]~T~&t��H�&m]�cx'x�y_vBlS��rzmc{axTm@���crsx�vBx&rzyX{¸rz~ªvB{¸rslg}�yX�&x�vB{�x

xzy L{xw|

Page 18: Blind subspace-based eigenstructure identification under … · 2016-12-27 · Blind subspace-based eigenstructure identi cation under nonstationary excitation using moving sensors

����������� �������������������������������� ��� �!���#"$�%�&���'�&�)(*"$+,�'�&�*�������"$��� 1��

rs{�~�x�vB{��]m�yB£�mcrs��mc{�~�x&�&t��cx&t���m�r � mc{ax&r�����vBx&rzyX{Y±7i0j�m]~�m�~&�=~�x�mcln~TvB��m�~&t��H�&m]�cx$x�y_£¥vX~�xTvB{ � t�{H¦lnm�vX~�t���m �  XvB�&rwvBx&rzyX{�~ªrs{¸x&j�mcrs�ªmc{a ars��yX{�lnmc{ax ¯ vB{ � x&j�t�~�x&j�m.rs{�}�t�xªm@�H�crsx�vBx&rzyX{|rz~'x§�=}�rz��vBusus�x&t��&��t�uzmc{axªrs{¸{�vBx&t���m�vB{ � {�yX{�~&x�vBx&rzyX{�vB�&��±J��yX�ªm@��vBlg}�uzm ¯ y #M~&j�yX��m7~&x&�&t��cx&t���m]~`vB��m.~&t��H�&m]�cxx�y�x&j�m�x&t��&��t�uzmc{ax�vB{ � j�rs��j�us�¨x&rslnm@¦¼ �vB�&�ars{��|vX�cx&rzyX{EyB£àx&j�mg~&ºàmcusu |~&j�y=��/¹m,#�m]�cx�~ � t�mgx�y:�t�r ��� ~�x&�&t��cx&t���mgrs{�x�mc��vX�cx&rzyX{) º'j�rz��j°��vB{�{�yXx.��m��]yX{�~�r � mc��m � vX~`lnm�vX~&t���vB��uzm�±�i0j�mg~�vBlnmj�yXu � ~Tx&�&t�mS£¤yX�Tº'rs{ � vB{ � x&��vP¢_��yX{¨���&r � ��m]~ ¯ º'j�rz��j¹��m]~&t�usxTrs{¨t�{�lnm�vX~&t���m � vB{ � {�yX{�~�x�vP¦x&rzyX{�vB�&� m@���crsx�vBx&rzyX{Y± ´ � r$#Mmc��mc{�x¡m@��vBlg}�uzm|�]yX{��]mc�&{�~_��yXx�vBx&rs{��El_vX��j�rs{�m]~¡~&t���j vX~njat���mvBusx�mc�&{�vBx�yX��~ªrs{ mcuzm]�cx&�&rz�crsx§�¶}�y�ºàmc��}�uwvB{�x�~ 0 1��<4§±

«§{¬x&j�rz~�~�m]�cx&rzyX{ ¯ ºàm � m]~��c�&rs��mgx&j�m_vB}�}�usrz��vBx&rzyX{²yB£0x&j�mg}"yXus�=��m@£¥mc��mc{��]m_~&t���~&}�vX�]mnr � mc{H¦x&r�����vBx&rzyX{ lnmcx&j�y � }���yX}�y�~�m � rs{ ~�m]�cx&rzyX{ o x�y ~�x&�&t��cx&t���vBu� =rs����vBx&rzyX{²vB{�vBus�H~&rz~�± �-rs��~&x ¯ ºàmrs{ax&��y � t��]mnx&j�m�lny � mcusrs{��¸rz~�~&t�m]~ ¯ vB{ � x&j�mc{°ºàm���mc}"yX�&xS~�yXlnm�{at�lnmc�&rz��vBu6��m]~&t�usx�~.º�mgj�v� �myX��x�vBrs{�m � yX{|v���m�vBu m@��vBlg}�uzm�±

����� ��� ����������°m'vX~�~�t�lnm6x&j�vBx)x&j�mà�"mcj�v� arzyX��yB£�x&j�màlnm]��j�vB{�rz��vBu�~&�H~&x�mcl­��vB{���m � m]~��c�&rs�"m � ����vª~�x�vBx&rzyX{H¦vB�&�¡usrs{�m�vB� � �={�vBlgrz��vBu�~��=~&x�mcl ¯ vB{ � x&j�vBx ¯ rs{¸x&j�m`£¤��m"�at�mc{��c�¡��vB{���mSyB£)rs{ax�mc��m]~&x ¯ x&j�m.rs{�}�t�x£¥yX���]m]~T��vB{|��m7lny � mcuzm � vX~`vn{�yX{�~�x�vBx&rzyX{�vB�&�¸º'j�rsx�mS{�yXrz~�m�±ªi0j�rz~$��m]~&t�usx�~ªrs{¨x&j�m7£¥yXusuzy�º'rs{��l_vBx&�&r�� � r$#Mmc��mc{�x&rwvBu-m"��t�vBx&rzyX{ 7�

� �� .�� 0N"����� .�� 0{" ����.�� 0 �

� .�� 0' .�� 0 � ���.�� 0 @#- 1 I

º'j�mc��m�� � mc{�yXx�m]~`�]yX{�x&rs{at�yXt�~7x&rslnm ¯ � M � M �vB��m.x&j�m�l_vX~�~ ¯"� vBlg}�rs{��¸vB{ � ~&x&r$#�{�m]~�~ªl_vP¦x&�&rz�]m]~6��m]~&}�m]�cx&rs �mcus� ¯ @[j�rs��j � rslnmc{�~�rzyX{�vBu I� �m]�cx�yX� �­�]yXusuzm]�cx�~�x&j�m � rz~&}�uwvX�]mclnmc{ax�~�yX�0vX�]�]mcuzmc�/¦vBx&rzyX{�~$yB£-x&j�m � mc����m]m]~$yB£-£[��m]m � yXl yB£)x&j�mS~&x&�&t��cx&t���m"!�x&j�m.m@�Hx�mc�&{�vBuL@[{�yX{|lnm�vX~&t���m � IࣥyX���]m� rz~nlny � mcuzm � vX~ev¨{�yX{�~&x�vBx&rzyX{�vB�&�²º'j�rsx�m¶{�yXrz~�m¶º'rsx&j x&rslnm@¦¼ �vB�&�ars{��E�]y� �vB�&rwvB{��]m¸l_vBx&�&r��5$#D.�� 0 ¯ lnm�vX~&t���mclnmc{ax�~_vB��m��]yXusuzm]�cx�m � rs{Ex&j�m @[uzyGº � rslnmc{�~&rzyX{�vBu I� �m]�cx�yX�8' ¯ vB{ � l_vBx&�&r���rs{ � rz��vBx�m]~$º'j�rz��j|�]yXlg}�yX{�mc{�x�~`yB£�x&j�mS~�x�vBx�m7 �m]�cx�yX�TvB��m�vX�cx&t�vBusus�¶lnm�vX~&t���m � @[º'j�mc��mSx&j�m

~�mc{�~�yX��~�vB��meuzy=��vBx�m � I@±V�½rsx&j�yXt�x�uzy�~�~�yB£���mc{�mc��vBusrsx§� ¯ ºàm � y {�yXx��]yX{�~&r � mc��lnm�vX~&t���mclnmc{�x{�yXrz~�m ¯ º'j�rz��j¨��vB{¨�"mSmc{��]yXlg}�vX~�~�m � rs{¨~&t���~�}�vX�]m�vBus��yX�&rsx&j�ln~T��� ~�j�r�£[x&rs{��ex&j�m.uwvB��~`yB£�x&j�m�]yG XvB�&rwvB{��]m]~_t�~�m � £¥yX����ususrs{��°x&j�m G`vB{N/�mcu�l_vBx&�&rz�]m]~ 0 1�� ¯ �<4§± qars{�t�~�yXr � vBuTyX�n�]yXuzyX��m � m@�=¦�crsx�vBx&rzyX{ {�yXrz~�m ��vB{��"m¸mc{��]yXlg}�vX~�~�m � vX~_º�mcusu 0 1��$4§± GTy�º x&j�m¶}���m]~�mc{��]m|vB{ � {�vBx&t���m yB£m@���crsx�vBx&rzyX{¡vB{ � lnm�vX~&t���mclnmc{�xà{�yXrz~�m]~àvB��m'j�vB{ � uzm � rs{_}���vX�cx&rz�]m ¯ º'j�mc{et�~&rs{���vB{evBus��yX�&rsx&j�l{�yX{|mc{��]yXlg}�vX~�~&rs{��nx&j�mcl ¯ rz~$v ��� ��m]~�~�m � rs{¸~&t���~�m]�cx&rzyX{E2H±Õ�H±

i0j�mS}�vB��vBlnmcx�mc��~8. � M � M � 0 ��vB{�{�yXx`��m.��m]�]yG �mc��m � £¤��yXl·lnm�vX~&t���m � yXt�x&}�t�x�~G± GTmc{��]m ¯r � mc{ax&r��!vB��uzmS}�vB��vBlnmcx�mc��~`vB��m7rs{ax&��y � t��]m � ±0i0j�m]~�mSvB��m.x&j�m7lny � m]~$yX�ªmcrs��mc{H¦¥£¤��m"��t�mc{��crzm]~&% ¯vB{ � x&j�m`lny � vBu ~�j�vB}"m]~�yX��mcrs��mc{a �m]�cx�yX��~('�) ¯ ~�yXust�x&rzyX{�~�yB£e7. �*% k "�� % " �

0�+)� W M,'�)

�-�.+) @#-���I

LML ÝON�P�Qcî P

Page 19: Blind subspace-based eigenstructure identification under … · 2016-12-27 · Blind subspace-based eigenstructure identi cation under nonstationary excitation using moving sensors

1 2 � �!�&� ������ �+5������ ��� ���� � ��6+5��6����� �� �#�� ��� � �5�����+ ���'� ���H�$�������E���"$�������!�

qHvBlg}�usrs{��`lny � mcu:@#- 1 I�vBx-��vBx�m ����� �arzmcu � ~-x&j�m � rz~��c��mcx�m�x&rslnmà~�x�vBx�mà~�}�vX�]m6lny � mcu:@ 1 I ¯ º'j�mc��mx&j�m7~�x�vBx�mSvB{ � x&j�m7yXt�x&}�t�xTvB��m87���:� � � .BE � 0�

� .BE � 0 M ' �:� ' .BE � 0 M @#-!-�I

x&j�m7~�x�vBx�m7x&��vB{�~&rsx&rzyX{|vB{ � yX��~�mc�& XvBx&rzyX{¸l_vBx&�&rz�]m]~TvB��m�7 � ����� M � � W �F � `N�� F � `N� � M ( � . � W 0 M

vB{ � º'j�mc��m.~&x�vBx�m`{�yXrz~�m$ �&���

rz~ I�mc��yB¦¼lnm�vB{ ¯ º'j�rsx�m ¯ º'rsx&j|�]y� �vB�&rwvB{��]m.l_vBx&�&r��-75 �&��� � � �&����� ��

� � �����5�.�� 0 � ������� �º'j�mc��m

�5�.�� 0 � � W WW �

`N� 5$#D.�� 0 � ` ; i0j�m�lny � vBu;}�vB��vBlnmcx�mc��~Srs{ @#-���I7vB��mnm"�at�rs XvBuzmc{ax&us�|£¥yXt�{ � rs{¬x&j�mnmcrs��mc{�~&x&�&t��cx&t���m .LKNMPOXQ 0yB£ 7

� � ) � KYM '�) � STQ >�@BA�U( ORQÅ m]��vBt�~�m�yB£�x&j�m�~�x&�&t��cx&t���m�yB£;x&j�m�~&x�vBx�m.rs{ @#-!-�I ¯ x&j�m�mcrs��mc{a �m]�cx�yX��~.vB��mS}�vBrs�&º'rz~�m��]yXlg}�uzm@��]yX{X�/t���vBx�m�±-i0j�mc�7vB��m;��m�vBu�r�£�}���yX}"yX�&x&rzyX{�vBu � vBlg}�rs{��ªrz~ vX~�~�t�lnm �M¯ x&j�vBxYrz~ r�£

� ����

"���²±

i0j�mc��m@£¤yX��m ¯  ars����vBx&rzyX{ vB{�vBus�H~&rz~���vB{ ��m¨~&x�vBx�m � vX~ex&j�m|}���yX��uzmcl yB£Sr � mc{ax&r�£[�=rs{��²x&j�mmcrs��mc{�~�x&�&t��cx&t���mªyB£�x&j�m�~&x�vBx�m'x&��vB{�~&rsx&rzyX{_l_vBx&�&r��gyB£Yv`usrs{�m�vB� � �={�vBlgrz��~&�H~&x�mcl º'rsx&jn{�yX{�~�x�vP¦x&rzyX{�vB�&�¹~&x�vBx�m�{�yXrz~�m�±gqax�vBx�m

�vB{ � yX��~�mc�& �m � yXt�x&}�t�x/' j�v] �m � rslnmc{�~�rzyX{�~:+ � ��� ��� �

vB{ � , ��m]~&}�m]�cx&rs �mcus� ¯ º'rsx&j , @[l�t���j�I'~�l_vBusuzmc��x&j�vB{ +¬rs{¶}���vX�cx&rz�]m�±

����� "!$# �&% �('�) �*% �,+�! ��-.+�°m|{�y�º8��mc}�yX�&x¶~�yXlnm|{�t�lnmc�&rz��vBu.��m]~&t�usx�~�yX��x�vBrs{�m � º'rsx&j x&j�m|}"yXus�=��m@£¤mc��mc{��]m¹~&t���~�}�vX�]mvBus��yX�&rsx&j�l }���yX}�y�~�m � rs{ ~�m]�cx&rzyX{ o�± Å m@£¥yX��m¡}���ya�]m]m � rs{�� ¯ ~�yXlnm��]yXlglnmc{�x�~¡vB��m�rs{ yX� � mc�yX{ x&j�m¸}���vX�cx&rz��vBu�t�~�m¶yB£`x&j�m¶�cuwvX~�~&rz��vBu5@[lnyX{�yX��m@£¥mc��mc{��]m|vB{ � £¤t�usu��]y� �vB�&rwvB{��]m Ig~&t���~�}�vX�]mr � mc{ax&r�����vBx&rzyX{¹vBus��yX�&rsx&j�l·yB£)~�m]�cx&rzyX{ �H±

xzy L{xw|

Page 20: Blind subspace-based eigenstructure identification under … · 2016-12-27 · Blind subspace-based eigenstructure identi cation under nonstationary excitation using moving sensors

����������� �������������������������������� ��� �!���#"$�%�&���'�&�)(*"$+,�'�&�*�������"$��� 1 2

B à � ¿ H �B����� D ����P D ��� � ¿ � DGF � ����� H ��� ¿ ¿ D ��¿ F �]���&����� � � ¿ D Ä ¿ F � D�� ���H� DOÃ*F i0j�m7~�mcuzm]�@¦x&rzyX{EyB£0x&j�mnlny � mcu6yX� � mc� ¯ vB{ � x&jat�~�yB£0x&j�mn~&r I�m_yB£0x&j�m7GTvB{N/�mcu�l_vBx&�&rz�]m]~SyX{EyX{�mnj�vB{ �M¯vB{ � x&j�m�j�vB{ � usrs{��¸yB£6x&j�m�}���m]~�mc{��]mnvB{ � {�vBx&t���mgyB£�m@���crsx�vBx&rzyX{µvB{ � lnm�vX~&t���mclnmc{�x.{�yXrz~�m]~yX{¶x&j�mSyXx&j�mc��j�vB{ �M¯ vB��m7xKº�ynl_vP�&yX�'}���vX�cx&rz��vBu rz~�~&t�m]~'º'j�rz��j º�mSv ��� ��m]~�~ª{�y�º.±

´$~¡~&x&��m]~�~�m � rs{ 0 1�� ¯ - ¯R[ 4 ¯ º'j�mc{ £¤mcº4~�mc{�~�yX��~�vB��m¬v] �vBrsuwvB��uzm ¯ vµ{�t�lS��mc�¶yB£Slny � m]~º'j�rz��j°rz~7l�t���jµ~&l_vBusuzmc�Sx&j�vB{¬x&j�mg{�t�l��"mc�SyB£'~�yXust�x&rzyX{�~.yB£(@#-���I ¯ ��t�x.l�t���j°uwvB�&��mc�Sx&j�vB{x&j�me{�t�l��"mc�nyB£T~�mc{�~�yX��~ ¯ ��vB{²�"mer � mc{�x&r���m � ±¬«K{²yX� � mc��x�y¨m@�Hx&��vX�cx�x&j�melny � m]~S£¤��yXl³x&j�m� vBx�v ¯ ºàm¸j�v] �m x�yµvB}�}�us�²x&j�m ~&t���~�}�vX�]m¶lnmcx&j�y � º'rsx&j½vB{ rs{��c��m�vX~&rs{�� q��+'4x&�&t�{���vBx&rzyX{yX� � mc�gvB{ � x�y|uzy=y&/|£¥yX�Sx&j�me��mcuzmc �vB{�x�lny � m]~�rs{Ex&j�mn£[��m"�at�mc{��c�¬��vB{ � yB£�rs{�x�mc��m]~&x�±¹qars{��]mx&j�m$~&t���~&}�vX�]m$lnmcx&j�y � �arzmcu � ~0v.~�mcx�yB£Ylny � m]~;º'rsx&j¡�"yXx&j¡~&x&�&t��cx&t���vBu� �£[��yXl x&j�m$~�x&�&t��cx&t���m�!���0x&j�mªm@���crsx�vBx&rzyX{� gvB{ � ~&}�t��&rzyXt�~;l_vBx&j�mcl_vBx&rz��vBu"lny � m]~ ¯ ºàm�j�v] �mªx�y � mc�&rs �mªv`}���vX�cx&rz��vBuº0v]�ex�y � rz~�x&rs{���t�rz~&j¶�"mcxKº�m]mc{¶x&j�mTx§ºày�x§�=}"m]~0yB£-lny � m]~�± GTyX}�m@£[t�usus� ¯ ~&}�t��&rzyXt�~0lny � m]~àx�mc{ �x�y� �vB�&�¸£¤��yXl3yX{�m�lny � mcu;yX� � mc�7x�y�x&j�m�{�m@�=x�±�i0j�vBx.rz~`º'j��¨t�~�vB��mg~&t�����m]~�x�~`x�y�}�uzyXx7x&j�m£¤��m"�at�mc{��crzm]~�vB��vBrs{�~�xnq�� '»x&�&t�{���vBx&rzyX{ yX� � mc� ¯ rs{ v¨~�yB¦§��vBusuzm � ~&x�vB��rsusr IGvBx&rzyX{ � rwvB����vBl 0 [ 4§±J$t�lnmc��yXt�~ªm@�H}�mc�&rslnmc{�x�~�º'rsx&j x&j�m7�cuwvX~�~&rz��vBu ~&t���~�}�vX�]m7lnmcx&j�y � j�v] �m.~&j�yGº'{¸x&j�vBx 7� «§x�rz~0yB£Y{�yS}���vX�cx&rz��vBu�j�mcus}¡x�y�~�mcuzm]�cx�£¥yX��b c ���ed f rs{)@¥��I6yXx&j�mc�� �vBust�m]~àx&j�vB{ �/� " � ±i0j�m�{at�lS��mc�0yB£YuwvB��~ � ¡£¤yX�;x&j�m � vBx�v.�]yG XvB�&rwvB{��]m]~0~&j�yXt�u � ��m'uwvB�&��m@[xK�a}�rz��vBusus�_lnyX��mx&j�vB{¸x§ºàynjat�{ � ��m � ~�rs{¸x&j�m.m@�H}�mc�&rslnmc{�x�~���mcuzy�ºMI ¯

� i0j�m�{�t�l��"mc�-yB£�lny � m]~Y~�j�yXt�u � �"m6y� �mc��m]~&x&rsl_vBx�m �M¯ ��m]��vBt�~�m�lny � m]~Yx�mc{ � x�y�~�x�vB��rsusr I�m ¯vB{ � mc �mc{�x�y7~&j�y�º²t�} ¯ £¥yX�)j�rs��jnyX� � mc��~�±�«K{gyXx&j�mc��º�yX� � ~ ¯ x&j�mªq��+'�rs{gx&j�m0vBus��yX�&rsx&j�lyB£"~�m]�cx&rzyX{_�`~�j�yXt�u � �"màx&�&t�{���vBx�m � vBx6vB{nyX� � mc��l�t���jn����m�vBx�mc�;x&j�vB{gx&j�m��ãx&j�m]yX��mcx&rz��vBu �yX� � mc� ¯ x&j�vBx�rz~ ��� � , F � ¯ º'j�mc��m � rz~'x&j�m.{�t�lS��mc�$yB£ � m]~&rs��m � lny � m]~ªvB{ � , rz~�x&j�m{�t�l��"mc�TyB£)~�mc{�~�yX��~�±

´ª{¸m@���crsx�vBx&rzyX{¸{�yXrz~�m.vBx$v�~�rs{���uzm7�]yX{�~&x�vB{�x'£¤��m"�at�mc{��c�¡rz~0��m]�]yX��{�r I�m � vX~ªv�}�yXuzm`���¡x&j�mvBus��yX�&rsx&j�l³yB£�~�m]�cx&rzyX{E� ¯ � m]~�rs��{�m � t�{ � mc�Sx&j�mevX~�~&t�lg}�x&rzyX{µyB£ªv�º'j�rsx�mnm@���crsx�vBx&rzyX{µ{�yXrz~�m�±qat���jnv�}�yXuzm6��vB{���m6mcusrslgrs{�vBx�m � }���y� =r � m � x&j�vBx)vB{ �O>����#"$����rs{H£¤yX�&l_vBx&rzyX{�rz~)v] �vBrsuwvB��uzm�±�i0j�rz~rz~.x&j�mn��vX~�mnyB£'x&j�mgj�vB�&lnyX{�rz�]~�yB£'x&j�mg��yXx�vBx&rzyX{E~&}�m]m � £¥yX�7��yXx�vBx&rs{�� l_vX��j�rs{�m]~�º'rsx&jµuzy�v �t�{a��vBuwvB{��crs{���±ªqarslgrsuwvB�&us� ¯ v_�]yXuzyX��m � m@���crsx�vBx&rzyX{ {�yXrz~�m���vB{ m�vX~&rsus����m.mcusrslgrs{�vBx�m � }���yG ar � m �x&j�vBx'rsx�~�}"yXuzm]~�vB��m @¤~�rs��{�r�����vB{�x&us��IàlnyX��m � vBlg}�m � x&j�vB{¸x&j�y�~�m7yB£�x&j�mS~&x&�&t��cx&t���m�±;«Kxàrz~$vBuz~�yvn�]yXlglnyX{¸��{ � rs{��_x&j�vBxªx&j�m.}�yXuzm]~�yB£�x&j�m.~&x&�&t��cx&t���m.x�mc{ � x�y_~&x�vB��rsusr I�m7º'j�mc{|x&j�mBG`vB{N/�mcul_vBx&�&r��²yX� � mc��rz~�rs{��c��m�vX~�m �M¯ º'j�mc��m�vX~�x&j�y�~�m�yB£ªx&j�m�m@�H�crsx�vBx&rzyX{²lny�~&x&us� � y|{�yXx�±¬´»{�yX{H¦~�x�vBx&rzyX{�vB�&�_m@�H�crsx�vBx&rzyX{�rz~0vX�cx&t�vBusus�¡v.£¥v� �yX��vB��uzm`~&rsx&t�vBx&rzyX{ ¯ ~&rs{��]mªx&j�m`v] �mc��vB��rs{��g}�mc�/£¥yX�&lnm �rs{ @¥�Bo IT£¥yX�S�]yXlg}�t�x&rs{�� � x�mc{ � x�y º'j�rsx�mc{µx&j�mn{�yXrz~�m�±ei0j�rz~�rz~.x&j�mn��vX~�m_yB£�v¸��j�rs�&}H¦¼usr /�m{�yX{�~�x�vBx&rzyX{�vB�&��m@���crsx�vBx&rzyX{Y±

LML ÝON�P�Qcî P

Page 21: Blind subspace-based eigenstructure identification under … · 2016-12-27 · Blind subspace-based eigenstructure identi cation under nonstationary excitation using moving sensors

1 [ � �!�&� ������ �+5������ ��� ���� � ��6+5��6����� �� �#�� ��� � �5�����+ ���'� ���H�$�������E���"$�������!�

��rs��t���m 117;©|m�vB{�rs{��eyB£-x&j�m7~&�=lS��yXuz~'rs{6�-rs��±���x�y�o�±�¡¿ ��Pà� H?H P D ���H� D[Ã�F ¿ � ��� H P ¿ i0j�m7}���yX}�y�~�m � lnmcx&j�y � j�vX~'��m]mc{|vB}�}�usrzm � x�ygx&j�m�qaº'rz~�~���Bog���&r � ��m ¯ v��"mc{���j�l_vB� /¶yB£-x&j�m Å �ª«çi - � -��T�ª´T©3}���yB�&m]�cxTq=«ç©)��-àq�yX{¶r � mc{�x&r�����vBx&rzyX{vB{ � lnyX{�rsx�yX�&rs{�� yB£��crs arsu`mc{���rs{�m]mc�&rs{���~&x&�&t��cx&t���m]~ ¯ £¤yX��º'j�rz��j -;©|�"´ @[x&j�m°qaº'rz~�~ ��m � ¦mc��vBu � vB��yX��vBx�yX�&�¸£¥yX�$©¹vBx�mc�&rwvBuz~$i�m]~&x&rs{���vB{ � �ªm]~�m�vB����j�I�j�vX~$��vB�&�&rzm � yXt�x$x�m]~�x�~$vB{ �¨� vBx�v��m]�]yX� � rs{���±6i0j�mª��m]~&}�yX{�~�m$yB£Yx&j�mª���&r � ��mTx�y.x&��vP¢_�`m@���crsx�vBx&rzyX{et�{ � mc�0x&j�m$���&r � ��mªj�vX~à��m]mc{lnm�vX~�t���m � rs{ 1,- �$}"yXrs{ax�~ ¯ l_vBrs{�us�.rs{�x&j�m� �mc�&x&rz��vBu�vB{ � x&��vB{�~& �mc��~�m � rs��m]�cx&rzyX{�~ ¯ vB{ � ~QvBlg}�uzm �vBx 1����$G I�± Å m]��vBt�~�mªvBx;lny�~&x 1 �.~�mc{�~�yX��~�ºàmc��mªv] �vBrsuwvB��uzm ¯ � � vBx�v`~�mcx�~�j�v] �mª�"m]mc{g��m]�]yX� � m �M¯m�vX��j �]yX{�x�vBrs{�rs{��¬x&j�m¡lnm�vX~&t���mclnmc{�x�~�£¤��yXl³o|����m � vB{ � 1�� lny� =rs{��°~�mc{�~�yX��~�±HGTy�º�mc �mc� ¯rsxªrz~�{�yXxª��m]�]yXlglnmc{ � m � x�ynlnmc�&��m � vBx�vg��m]�]yX� � m � t�{ � mc�ªx�y=y � r$#Mmc��mc{�xªx�mclg}�mc��vBx&t���m�vB{ �x&��vP¢_�ª�]yX{ � rsx&rzyX{�~ 0 �54 ¯ vB{ � x�y`lgr�� � r$#Mmc��mc{�x � rs��m]�cx&rzyX{�~ 0 1��<4§±)i0j�mc��m@£¥yX��m ¯ £¥yX�;rs{� �m]~&x&rs��vBx&rs{��x&j�m�m@�H}"mc�&rslnmc{ax�vBu;}���yX}�mc�&x&rzm]~.yB£�x&j�m�}���yX}�y�~�m � }"yXus�=��m@£¥mc��mc{��]mg~&t���~&}�vX�]mnvBus��yX�&rsx&j�l¸±*�°mj�v� �m$~�mcuzm]�cx�m � � � � ��m]�]yX� � ~�£[��yXl#~�mc{�~�yX��~�rs{_x&j�m$ �mc�&x&rz��vBu � rs��m]�cx&rzyX{ ¯ º'rsx&j¡yX{�m$��m@£¥mc��mc{��]m~�mc{�~�yX� ¯ vB{ � xKº�y.lnyG ars{���~�mc{�~�yX��~�rs{_x&j�m0����~&x���m]�]yX� �M¯ vB{ � yX{�us�gyX{�m'lny� =rs{��S~�mc{�~�yX�6rs{gx&j�m~�m]�]yX{ � ��m]�]yX� � ±.-�vX��j¶~&rs��{�vBu��]yX{�x�vBrs{�~�����������~�vBlg}�uzm]~�±)´T~6lnmc{ax&rzyX{�m � vB�"yG �m ¯ }���ya�]m]~�~&rs{��lnyX��m ��m]�]yX� � ~_ºàyXt�u � j�v� �m¨v�#Mm]�cx�m � x&j�m ��m"��t�rs��m � lnmclnyX�&� ~&r I�m ¯ ��t�x � "!��x&j�m|�]yXlg}�tH¦x�vBx&rzyX{�vBu0�]yXlg}�uzm@�Hrsx§��± ´$uz~�y ¯ rsxSrz~�yB£�rs{�x�mc��m]~&x�x�y|rs{� �m]~&x&rs��vBx�mex&j�m_}���vX�cx&rz��vBuà��mcuzmc �vB{��]meyB£x&j�m`�]yX{��]mc}�x&t�vBuY��vB}�vB��rsusrsxK�¡yB£-x&j�mª}���yX}�y�~�m � vBus��yX�&rsx&j�l»x�y�j�vB{ � uzm`��m]�]yX� � ~0º'rsx&j � r$#Mmc��mc{�x{at�lS��mc��~ªyB£)lny� =rs{��e~�mc{�~�yX��~�±

i0j�mT�cuwvX~�~&rz��vBuM~&t���~&}�vX�]m`vBus��yX�&rsx&j�l»yB£-~�m]�cx&rzyX{¸� ¯ ��vX~�m � yX{e£¤t�usu � vBx�v��]yG XvB�&rwvB{��]m]~$vB{ �yX{

� � � ���H¯ j�vX~¡�"m]mc{ vB}�}�usrzm � ~�mc}�vB��vBx�mcus� x�y²m�vX��j½��m]�]yX� �M¯ º'j�rz��j �]yX{�x�vBrs{ - vB{ � �~�mc{�~�yX��~ ¯ ��m]~&}"m]�cx&rs �mcus��±|i0j�m_��m]~�t�usx�~�vB��m � rz~&}�uwv]��m � rs{G��rs��±-�|vB{ � - ¯ º'j�mc��mn£¤��m"��t�mc{��crzm]~@¤yX� � rs{�vBx�m I`vB��m � rz~&}�uwv���m � £¤yX�`rs{��c��m�vX~&rs{��|q�� ' x&�&t�{���vBx&rzyX{EyX� � mc��@¥vB��~��crz~�~�v�I ¯ rs{°v_~�x�vB��r�¦usr IGvBx&rzyX{ � rwvB����vBl»º'j�rz��j ~��al��"yXuz~�vB��m.m@�H}�uwvBrs{�m � rs{ �-rs��±�1�±;i;�=}�rz��vBuY �vBust�m]~'yB£)m]~�x&rsl_vBx�m �lny � m]~'º'j�rz��j|��vB{¶��m7m@�=x&��vX�cx�m � £[��yXl®x&j�m]~�m � rwvB����vBln~$vB��m � rz~&}�uwv���m � rs{ i-vB��uzm�1�±

xzy L{xw|

Page 22: Blind subspace-based eigenstructure identification under … · 2016-12-27 · Blind subspace-based eigenstructure identi cation under nonstationary excitation using moving sensors

����������� �������������������������������� ��� �!���#"$�%�&���'�&�)(*"$+,�'�&�*�������"$��� 1 �

i�vB��uzm 117 -;~&x&rsl_vBx�m � lny � m]~0¦���uwvX~�~�rz��vBu ~&t���~&}�vX�]m7r � mc{ax&r�����vBx&rzyX{Y±©¨y � m � � � � �����m"��t�mc{��c� @�G I�I � �DI � �DI � � W I � � �

2 8 14 20 26 32

0

4

8

12

16

20

+ + + + + + +

+

+ + + + + + +

+

+

+

+

+

+

+

+

+

+

+

+ +

+

++ + ++ +

+

+

+

++ ++ ++ ++ ++

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

++

+

++

+

+

++

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

++

+

+

+

+

+

+

+

+

++

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

++

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

+

++

+

+

++

+

+

2 8 14 20 26 32

0

4

8

12

16

20

××

×

××

×

××

××

×

××

××

××

××

×

××

××

×××

××

×

××

××

××

×

×

××

×

×

×

××

×

×

×××

××

×

××

××

×

×

×××

××

×

×××

×

×

×

××

×

×××

×××

×

×

×

×

×

××

×

×

×

×

××××

×××

×

××

×

×

×××××

×××

×

××

×

×

×

××××

××

××

××

×

×

×

××××

×

×

××

×

×

×

×

××

×

×

×

××××

×

×

××

×

×

×

×

×

×

×

×

×

××

×

×

×

×××

×

×

×

×

××

××

×

×

×

×

×

×

×

×

×

××

×

×

×

××

×

××

×

××

×××

×

×

×

×

×

×

×

×

×

×

×

×

×××××

×

××

×

××

×

×

×

×

×

×

×

×

×

×

×

×

××

×

××××××

××

××

×

×

×

×

×

×

×

×

×

×

×

×

×××××

×

×

××

××

×

×

×

×

×

×

××××

×

×

××

×

×

×

×

×

×

×

×

×

××××

××

×

×

××

×

×

×

×

×

×

××××××

×

×

×××

××

×

×

×

×

×

×

×

×

×

×

×××××××

××

××

×

×

×

×

××

×

×

×

×

×

××××××

×

××

×

×

×

×

×

××

×

×

×

×

×××××××××

××

×

×

××

××

×

×

×

×××××

×××

×××

×

×

×

×

×

×××

×

×

××××××

×××

××

×

×

×

×

×

××

×

××

×

××××××

×××

×

×

×××

×

×

×

×

×

××

×

×

×

×

××××××

×××

××

×

××

×

×

×

××

×

×

×

×

×××××

××

×

×

××

×

×

×

×

××

×

××

×

×××××

×××

××××

×

×

××

×

××××

×

×

×××××

××

××××

×

×

××

×

×

×

×

×

×

××××××

××

××××

×

×

××

×

×

×

×

×

××

××××××

××

××××

×

××

×

×

×

×

×

×

××××××

××

×××××

×

×

×

×

×

×

×

××

×

××

××××××

××

×××××

×

×

×

×

××

×

×

×

×

×

××

×

××××

××

×

×

×××

×

×

××

×

×

×

×

×

××

××

××××

××

×

×

×××

×

×

×

×

××

×

×

××

×

××

××××××

××

×

×

×

×

×

×

×

×

×

××

×

××

××

××

×××××××

××

××

×

×

×

××

×

×××

×

×

×××××××

××

×

××

×

×

×

××

×

×××

×

×

×

×××××××

××

×

×××

×

×

×

×

×

××

×

×××

×

×

××××××××

××

×

×××

×

×

×

××

×

×××

×

×

×

×××××××

××

××××××

×

×

××

×

×××

×

××

×

×××××

××

××××××

×

×××××

×

××

×

×××××

×××

×××××××

×

×

×

×

××

×××

××

×

×

×

×××××××

×××

××

×

×××

×

×

×

×

×

××

××

×××

××

×××

×

××××××

××

×××××

×

×

×

××

××

×××

××

×××

××

××××××

××

×

××××

×

×

×

×

×

×

××

×

×××

×

×

×

×

××××××××

×××

×

××××

×

×

×

×

×

××

×

×××

×

×

×

×

××××××××

×××

×

××

×

×

×

×

×

×

×

×

×

×

×

××

×××

×

×

×

×

××××××××

×××

×

××××

×

×

×

×

×

×

×

×

×

×

×

×

×××

×

××

×

×

××××××××

×××

×

××××

×

×

×

×

×

×

×

×

×

×

×

×

×

×××

×

××

××

×

×××××××

×××

×

××××

×

×

×

×

×

××

×

××

×××

×

×

×

×

××

×××××××

×××

×

××××

×

×

×

×

×

×

×

×

×

×

×

×

××

×

××

×

×

×

×××××××

×××

×

××××

×

×

×

×

×

×

×

××

×

×

×

×

××

×××

×

×

×

××

×

×××××××

×××

×

××××

×

×

×

×

×

×

×

×

×

×

×

××

×××

×

××

×

×

××××××××

×××

×

××××

×

×

×

×

×

×

×

×

×

×

×

×××

××

××

×

×

×××××××

×××

×

×××

×

×

×

×

×

××

×

×

×

×

×××

×

××

×

××

××××××××

×××

××

×

××

×

×

×

×

×

×

××

×

×

×

×

×××

×

××

×

××

××××××××

×××

×

×××

×

×

×

××

×

×

×

×

×××

×

××

×

××

××××××××

×××

××

×

×

××

2 8 14 20 26 32

0

4

8

12

16

20

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

2 8 14 20 26 32

0

4

8

12

16

20

♦♦

♦ ♦ ♦ ♦ ♦

♦♦

♦♦ ♦

♦♦♦

♦♦

♦ ♦

♦ ♦

♦♦

♦♦

♦♦

♦♦

2 8 14 20 26 32

0

4

8

12

16

20

◊◊

◊ ◊

◊◊

◊ ◊

◊◊

◊ ◊

◊◊

2 8 14 20 26 32

0

4

8

12

16

20

∆ ∆

∆ ∆∆

∆∆

∆ ∆

∆ ∆

∆ ∆∆ ∆∆

∆∆∆

∆∆∆ ∆∆∆ ∆∆

∆∆∆ ∆∆

∆∆ ∆∆

∆∆

∆∆∆∆ ∆∆∆

∆∆∆ ∆∆∆∆

∆∆∆

∆∆

∆∆

∆∆∆∆∆∆

∆∆

∆ ∆

∆∆∆

∆∆∆∆∆

∆∆∆∆∆

∆∆

∆∆ ∆∆

∆∆

∆∆∆∆∆∆

∆∆

∆∆∆∆∆∆∆

∆∆

∆∆∆

∆ ∆

∆∆∆∆∆

∆∆

∆ ∆

∆∆∆ ∆∆∆

∆∆

∆∆

∆∆∆

∆∆∆∆

∆∆ ∆

∆∆∆∆∆∆∆∆

∆∆

∆∆∆∆∆ ∆∆∆∆∆∆∆∆∆ ∆∆∆

∆∆∆∆∆ ∆∆∆∆

∆∆∆∆∆∆

∆∆∆∆

∆∆∆∆∆

∆∆

∆∆∆∆∆∆

∆∆

∆∆∆∆ ∆

∆∆∆∆∆

∆∆

∆∆∆

∆∆∆ ∆

∆∆∆∆∆∆∆∆

∆∆

∆∆

∆∆∆∆∆

�-rs��t���mS� 7 ��uwvX~�~&rz��vBu ~&t���~&}�vX�]m.r � mc{�x&r�����vBx&rzyX{Y76����~&x'��m]�]yX� � @#-g~�mc{�~�yX��~ I@±

LML ÝON�P�Qcî P

Page 23: Blind subspace-based eigenstructure identification under … · 2016-12-27 · Blind subspace-based eigenstructure identi cation under nonstationary excitation using moving sensors

��� � �!�&� ������ �+5������ ��� ���� � ��6+5��6����� �� �#�� ��� � �5�����+ ���'� ���H�$�������E���"$�������!�

2 8 14 20 26 32

0

4

8

12

16

20

+ +

+ + + + + + + + +

+

+ + + +

+

+

+

+

+

+

+

+

+

+

+ + +

+

+

+

+

+ +

+

+

+

+

+

+

+

+

+

+ +

+

+

+

+

+

+ ++ +

+

+ ++ ++ ++ ++

+

++

+

+

+

++

+

+

++

+

++

+

+

++

+

+

+++ ++

+

+

+

+

+

+

+

+

++ ++

+

+

++

+

+

+

+

+

+

+

+

+

++

+

+

+

+

+

+

+

+

+

+

+

+

+

2 8 14 20 26 32

0

4

8

12

16

20

×

×

×

×

××

×

×

×

×

××

×

×

×

×

×

×

×

×

××

×

××××××

×

××××××

×

×

×

×

×

×

×××××

×

×××××

×

×××××

×

×××××

×

××

××××

××

××××

××

××××

××

×××××

××

××××

××

××××

×

××

××××

×

××

×

××××

×

××

××××

×

××

××××

×

×

×

×

×

×

×

×

×××

×

×

×

×

×

×

×××

×

×

×

×

×

×

×

×××

×

×

×

×

×

×

×

××

×

×

×

×

×

×

××××

×

×

×

×

×

×

×

×

××

×

×

×

×

×

×

×

×

×

×

×××

×

××

××××

×

×

×

×××

×

×

×

×

×

×

×

×

×

×

×

×××

×

×

×

×

××××

×

×

×

×××

×

×

×

×

×

××××

×

×

×

×××

×

×

×

×

××××

×

×

×

×××

×

×

×

×

××××

×

×

×

×

×××

×

×

×

××××

×

×

×

×××

×

×

××××

×

×

×××

×

×

×

×

××××

×

×

×××

×

×

×

×

××××

×

×

×××

×

××

×

×

×

××××

×

×

×

×××

×

×

××

×

×

×

××××

×

×

×××

×

×

×

××

×

×

××××

×

×

×

×××××

×

×

×

×

×

××××

×

×

×

×

×××××××

×

×

××××

×

×

×

×××××

×

×××

××

×××

××××

×

×

×

×××××

×

×

×

×

××

×

××

×

×

×

××

×

×

×

×

×

×

×××

×

××

×

××

××

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×××

××

×

×

×

×

×

×

×

×

×

×

××××

×

××××

××

×

×

×

×

×

×

×

×

×

××

×

×

×××

×

×××××

××

×

××

×

××

××

×

×

×

××

×××

×

×

×

×

××

×

×

××

×

×

×

×

××

×

×

×

×

××××

×

×

××

×

×

×

×

×

×

××

×

×

×

××

×

×

×

××

×

×

××

×

××

×

×

×

××

×

×

×

×

×

×

×××

×

×××

×

×

×

×××

×

××

×

××

×

×

×

××

×

×

××××××

×

××

×

×

××

×

×

××

×

×

×

××

×××××××

×

××

×

×

×

×

×

×

×

×

××

×

×

×

×

×

×

×

××××××××

×

×

×

××

×

×××

×

××

×

×

×

×

×

×

××

×

×××××××

×

×

×

×

×

××

×

×

×

×

×

×

×

××

×××

×

×

××××××

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

×

××

×

×

×××××××

×

×

×

×

×

×

×

×

×

×

×

×

×

××

×××

××

×

××××××

×

×

×

×

×

×

×

×

×

×

×

×

××

×××

×

×

×××××××

×

×

×

×

×

×

×

×

×

×

×

×

×

×

××

××

×

×

×

×××

×

××

2 8 14 20 26 32

0

4

8

12

16

20

⊕ ⊕

⊕ ⊕

⊕ ⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

⊕⊕

2 8 14 20 26 32

0

4

8

12

16

20

♦ ♦ ♦ ♦

♦♦

♦♦

♦♦

♦♦

♦♦

♦♦

♦♦

♦♦

♦♦

♦♦

♦♦

♦♦

♦♦

♦♦

♦♦

♦♦

♦♦

♦♦

♦♦

♦♦

♦♦

♦♦

♦♦

♦♦

♦♦

♦♦

♦♦

♦♦

♦♦

2 8 14 20 26 32

0

4

8

12

16

20

◊ ◊◊

◊ ◊ ◊ ◊

◊ ◊ ◊

◊ ◊

◊ ◊◊

◊◊

◊ ◊

2 8 14 20 26 32

0

4

8

12

16

20

∆ ∆ ∆ ∆

∆ ∆ ∆ ∆

∆∆

∆∆ ∆∆∆∆

∆∆

∆∆ ∆

∆∆ ∆∆

∆∆

∆ ∆

∆∆∆ ∆∆

∆ ∆

∆∆

∆∆

∆∆∆∆

∆∆∆∆ ∆∆∆∆ ∆

∆∆

∆ ∆∆∆∆

∆ ∆∆∆∆

∆∆ ∆∆

∆∆∆ ∆

∆ ∆∆∆∆∆∆∆

∆∆∆ ∆∆∆ ∆∆

∆∆ ∆

∆∆ ∆

∆∆∆ ∆

∆∆

∆ ∆

∆∆

∆ ∆

∆∆

∆∆ ∆∆

∆∆ ∆∆

∆∆∆

∆∆

∆∆∆

∆∆∆

∆ ∆∆

∆ ∆

∆ ∆

∆∆∆ ∆

∆∆∆∆ ∆

∆∆∆∆ ∆

∆∆

∆∆∆∆

�-rs��t���m - 7L��uwvX~�~&rz��vBuY~�t���~&}�vX�]m7r � mc{�x&r�����vBx&rzyX{Y7à~�m]�]yX{ � ��m]�]yX� � @¥�g~�mc{�~�yX��~ I@±

2 8 14 20 26 32

0

4

8

12

16

20

+

+

+

+

+

+

+

+

+ + + + + + + + + + + +

2 8 14 20 26 32

0

4

8

12

16

20

× × ×

××

×

×

×

×

×

×

××

×

×

×

×

××

×

××

×

×

×

××

×

×

×

×

×

×××

××

×

×

×

××

××

××

×

×

×

×

×

××

×

×

××

××××

×

××

×

×

×

××××

×

××

×

×

××

××

×

××

×

×

×

××

×

××

×

×

×

××

×

××

×

×

×

××

××

×

××

×

×

×

×

×

××

×

××

×

×

×

×

××××

×

××

×

×

×

×

×

××

×

××

×

×

×

×

×

××

×

××

×

×

×

×

×

××

×

××

×

×

×

×

×

××

×

××

×

×

×

×

×

××

×

××

×

×

×

×

×

××

×

××

×

×

×

×

×

××

×

××

×

×

×

×

×

××

×

××

×

×

×

×

×

××

×

××

×

×

×

×

×

××

×

××

×

×

×

×

×

××

×

××

×

×

×

2 8 14 20 26 32

0

4

8

12

16

20

⊕ ⊕ ⊕ ⊕⊕

2 8 14 20 26 32

0

4

8

12

16

20

♦ ♦ ♦

2 8 14 20 26 32

0

4

8

12

16

20

◊ ◊

◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊

2 8 14 20 26 32

0

4

8

12

16

20

∆ ∆

�-rs��t���m7o%7���yXus�=��m@£¥mc��mc{��]mS~&t���~&}�vX�]m.r � mc{�x&r�����vBx&rzyX{¨º'rsx&j¸x&j�m`x§ºàyn��m]�]yX� � ~�±xzy L{xw|

Page 24: Blind subspace-based eigenstructure identification under … · 2016-12-27 · Blind subspace-based eigenstructure identi cation under nonstationary excitation using moving sensors

����������� �������������������������������� ��� �!���#"$�%�&���'�&�)(*"$+,�'�&�*�������"$��� � 1

�Tà P À � ¿��K¿ � ¿ F � ¿ ��� �-� H ��� ¿S¿ D �!¿ F �]���&� ��� ��� ¿D Ä ¿ F � D�� ���H� DOÃ*F i0j�m6}�yXus�a��m@£¥mc��mc{��]mà~&t��H¦~�}�vX�]m7r � mc{�x&r�����vBx&rzyX{¬vBus��yX�&rsx&j�l·yB£�~�m]�cx&rzyX{¸o ¯ ��vX~�m � yX{¸}�vB�&x&rwvBu��]y� �vB�&rwvB{��]m]~��!@ [ I'vB{ � yX{�/� � ���H¯ j�vX~)�"m]mc{gvB}�}�usrzm � x�yTx&j�m � vBx�vª£¤��yXl x&j�m�xKº�y`��m]�]yX� � ~�±)i0j�mà��m]~&t�usx�~�vB��m � rz~&}�uwv]��m �rs{ �-rs��±"o�±�qars{��]m�x&j�m G`vB{N/�mcu)l_vBx&�&r�� b3rz~T��t�rsusx7yX{¬~&x�vX��/�m � �]yG XvB�&rwvB{��]m]~ @¥����I ¯ rsx�~7q�� 'x&�&t�{���vBx&rzyX{°yX� � mc�7~&j�yXt�u � ��mSx�v /�mc{¹{�ye����m�vBx�mc�7x&j�vB{ � ����� ��� . � " � " � 0\�

� � ±:�-rs��±"o~�j�y�º�~�t�}|~�mc �mc��vBuYrslg}���y� �mclnmc{ax�~`y� �mc�_�-rs��±��nvB{ � ��rs��±�- ¯ yX{¸~�mc �mc��vBu�vX~&}�m]�cx�~�±;´'�cx&t�vBusus� ¯£¥yX�'uwvB�&��m.mc{�yXt���j¸x&�&t�{���vBx&rzyX{¨yX� � mc� 7� qa}�t��&rzyXt�~�lny � m]~ � rz~�vB}�}�m�vB� 76vB��yXt�{ � � G I ¯ £¥yX��m@��vBlg}�uzm"!� iY�&t�m¹lny � m]~¸vB}�}"m�vB�¸vB{ � ~&x�vB��rsusr I�m¨£¼vX~&x�mc�¸vB{ � :�t��cx&t�vBx�m¹uzm]~�~¡rs{½x&j�m � rwvB����vBln~ 7vB��yXt�{ � �DI �*G ISvB{ � �DI � G I ¯ £¥yX��m@��vBlg}�uzm"!

� i0j�m � vBlg}�rs{��|yB£àx&j�mgx&�&t�m�lny � m]~.rz~7lnyX��mnvX�ct���vBx�mcus�¬m]~&x&rsl_vBx�m �M¯ vBx.uzyGº�mc�SyX� � mc��~x&j�vB{ ��m@£¤yX��m ¯ vBusuzy�º'rs{���x�yn~�mc}�vB��vBx�m�x&�&t�m�vB{ � º'��yX{��elny � m]~$vBlnyX{��¡x&j�mS~�x�vB��rsusr I�m �yX{�m]~ 7`v_º'��yX{��¶lny � mgvB}�}"m�vB��~�vBx��)G I�º'rsx&jµv_j�rs��j � vBlg}�rs{�� ¯ º'j�mc��m�vX~7x&j�m�x&�&t�mlny � m�vBx �7G ISvB}�}�m�vB��~'º'rsx&j|v�uzyGº � vBlg}�rs{�� !

� q=yXlnm`lny � m]~ ¯ {�yXxàr � mc{ax&r���m � yX{ �-rs��±���vB{ � - ¯ {�y�º ~&j�yGº t�}¶rs{¶v�~�x�vB��uzm � rwvB����vBl 7vB��yXt�{ � � � G I ¯ vBlnyX{��_yXx&j�mc��~�±

i0j�mn}"yXus�=��m@£¤mc��mc{��]m�lnmcx&j�y � m@�Hx&��vX�cx�~ 2elny � m]~7rs{°x&j�m���vB{���m W F � � G I�±8�-rs �mnyB£�x&j�m]~�mlny � m]~��]yXlnmn£¤��yXl³x&j�m¡~�x&�&t��cx&t���m¡vB{ � x�mc{ � x�y¨vB}�}�m�vB��rs{El_vB{a� � rwvB����vBln~ � t��&rs{��¹x&j�mº'j�yXuzme��m�vB��±¶i0j�m �DI � G Inlny � m_rz~S �mc�&� � r�¢_�ct�usx�x�y|m@�=x&��vX�cx ¯�� t�m_l_vBrs{�us�°x�y¸uzyGº m@���cr�¦x�vBx&rzyX{²£¥yX�gx&j�rz~�£¤��m"��t�mc{��c��±²«K{ �"yXx&j �cuwvX~�~&rz��vBu'r � mc{�x&r�����vBx&rzyX{�~ ¯ rsx_vB}�}�m�vB��~g �mc�&� uwvBx�m¶rs{x&j�m � rwvB����vBln~�vB{ � j�vX~7usrsx&x&uzmn~�x�vB��rsusrsx§� ¯ º'j�mc��m�vX~Sx&j�y�~�mgxKº�y¶rz~�~�t�m]~SvB��mgy� �mc���]yXlnm_rs{¬x&j�m}�yXus�=��m@£¤mc��mc{��]m�x&�&rwvBu¼±_�-rs��±!oerz~`v_��mcl_vB� /XvB��us� �cuzm�vB� � rwvB����vBl4º'j�rz��j ¯ vB��y� �m�vnx&�&t�{���vBx&rzyX{yX� � mc�TyB£ � � ¯ rz~ª�cuzm�vB{�m � £[��yXl·vBusu�~&}�t��&rzyXt�~ª}"yXuzm]~ ¯ mc �mc{¸x&j�y�~�mSº'j�rz��j x�mc{ � x�y_~&x�vB��rsusr I�mSrs{�-rs��±��nvB{ � -H±� j�vBx�x&j�mª}���yX}�y�~�m � vBus��yX�&rsx&j�l#rz~�{�yXxà��vB}�vB��uzmTyB£Yrz~�x�yS~�mc}�vB��vBx�m$lny � m]~àyB£Yx&j�mª~�x&�&t��@¦x&t���m¡£¤��yXl+x&j�m�lny � m]~g�]yX�&��m]~�}"yX{ � rs{��¬x�y¹x&j�m¡{�yX{ �º'j�rsx�m¸m@���crsx�vBx&rzyX{ ¯ ��t�xgx&j�m��cuwvX~�~�rz��vBu~�t���~&}�vX�]m`r � mc{�x&r�����vBx&rzyX{¨}���ya�]m � t���mSrz~'t�{�vB��uzm7x�ynvX��j�rzmc �m.x&j�vBxªmcrsx&j�mc��± JTmc �mc�&x&j�mcuzm]~�~ ¯ x&j�m{�mcº®vBus��yX�&rsx&j�l vB}�}"m�vB��~�x�y|}"mc�/£¥yX�&l³º�mcusu�vBx�m@�Hx&��vX�cx&rs{��¹lny � m]~�º'j�rz��j vB}�}�m�vB�gyX{�us�°rs{~�yXlnm � rwvB����vBln~ ¯ £¥yX�em@��vBlg}�uzm¸lny � m]~eº'j�rz��j vB��m {�yXxe �mc�&� º�mcusu`m@�H�crsx�m � vBusu$x&j�m x&rslnm�±i0j�rz~$£¥m�vBx&t���mSrz~Trslg}�yX�&x�vB{�x`rs{¨}���vX�cx&rz�]m ¯ vB{ � ~&j�yXt�u � j�yX}�m@£[t�usus�¸��m��]yX{H���&lnm � rs{|£¤t��&x&j�mc�{at�lnmc�&rz��vBu rs{a �m]~&x&rs��vBx&rzyX{�~�±� y-×�â¼ÜSâ¼ÒBÚ�âªâ¼ÒBÜ/Ô¥ÜSÞ/× �QÚ�ؼÓûÚQÝPÞ/Ü�æ�Ú�â¥Ø¼ÓÕÞ/Ü/Ô`Ú�ؼÜ�Þç×@øÕáBæ'Ý �cÜçÞçâ¼×QؼÔ���Ô¥ÓÕÝBÞ/ÜS×@ÝPø �×@ÝBÜ7ؼÜçäòÜçؼÜ/ÝPÞ/Ü�Ô¥Ü/ÝPÔ¥×QØTÓûÔ

Ú �QÚQÓÕøûÚQÛPøÕÜ�ÓûÝTÜ&ÚQÞ§Ò`×Qä=â¼ÒBÜ-âOÖ�×0ؼÜ/Þ/×�اßPÔ��

LML ÝON�P�Qcî P

Page 25: Blind subspace-based eigenstructure identification under … · 2016-12-27 · Blind subspace-based eigenstructure identi cation under nonstationary excitation using moving sensors

��� � �!�&� ������ �+5������ ��� ���� � ��6+5��6����� �� �#�� ��� � �5�����+ ���'� ���H�$�������E���"$�������!�

� � (������������(���°m j�v] �m � rz~��ct�~�~�m � x&j�m }�yXus�=��m@£¤mc��mc{��]m|~�mcx&t�}�£¥yX�n��usrs{ � mcrs��mc{�~&x&�&t��cx&t���m¨r � mc{�x&r�����vBx&rzyX{Y±i0j�rz~¡xK�a}�m¹yB£�~�mcx&t�}½rz~e}�yX}�t�uwvB�¡£¤yX��rs{�~&x�vB{��]m¨rs{½ ars����vBx&rzyX{ lnm]��j�vB{�rz�]~ ¯ º'j�mc��m¨~�mc �mc��vBulnyG ars{��²~�mc{�~�yX��~¡lgrslgrz�|x&j�m¨v� XvBrsuwvB��rsusrsxK�½yB£�v°lSt���j uwvB�&��mc�¡~�mcx�yB£S~�mc{�~�yX��~G± i0j�m|}���yB¦}�y�~�m � r � mc{�x&r�����vBx&rzyX{°lnmcx&j�y � �]yX{�~&rz~&x�~Trs{¹lnmc�&��rs{���x&j�m � vBx�v_����~&x.vB{ � }���ya�]m]~�~&rs{���x&j�mcl��uzyX��vBusus� t�~&rs{���vB{¹yXt�x&}�t�x/¦§yX{�us�|�]yG XvB�&rwvB{��]m@¦¼��vX~�m � ~�t���~&}�vX�]m�vBus��yX�&rsx&j�l¸±`i0j�rz~Tj�vX~$x�y_��m�]yX{ax&��vX~&x�m � º'rsx&j �cuwvX~�~&rz��vBu�}"yXus�=��m@£¤mc��mc{��]m.r � mc{�x&r�����vBx&rzyX{ ¯ º'j�rz��j¨rz~�t�~&t�vBusus��}�mc�/£¤yX�&lnm � ���lnmc�&��rs{��¡r � mc{�x&r�����vBx&rzyX{¬��m]~�t�usx�~$yX��x�vBrs{�m � yX{¨x&j�m � r$#Mmc��mc{�x`��m]�]yX� � ~�±ªi0j�mS}���m]~�mc{��]m�yB£6vB{t�{�lnm�vX~�t���m � vB{ � {�yX{�~&x�vBx&rzyX{�vB�&�µm@���crsx�vBx&rzyX{El_v /�m]~�x&j�m_rz~�~&t�m_yB£ªyXt�x&}�t�x � vBx�v¶lnmc�&��rs{��{�yX{ax&�&rs arwvBu¼±Si0j�mA/�mc�¸r � m�v¡yB£6x&j�m�}���yX}�y�~�m � lnmcx&j�y � rz~7v_~�t�rsx�vB��uzm�{�yX�&l_vBusr IGvBx&rzyX{°yB£�x&j�myXt�x&}�t�xT�]y� �vB�&rwvB{��]m]~�±�i0j�rz~$lnmcx&j�y � j�vX~ª��m]mc{ }���yG �m � x�y_}���y� =r � m.t�~$º'rsx&j|�]yX{�~&rz~�x�mc{�x`m]~/¦x&rsl_vBx�m]~ªyB£-x&j�m7mcrs��mc{�~&x&�&t��cx&t���m ¯ mc �mc{¶t�{ � mc��{�yX{�~&x�vBx&rzyX{�vB�&�¶m@���crsx�vBx&rzyX{Y±�q=yXlnm7{�t�lnmc�&rz��vBu��m]~�t�usx�~�yX{¸yX{�m7��m�vBu m@��vBlg}�uzm7j�v] �m.�"m]mc{¶��mc}�yX�&x�m �M¯ º'j�rz��j¸j�rs��j�usrs��j�x$x&j�m7}"yXx�mc{ax&rwvBuY��mc{�m@¦��x�~àyB£Yx&j�m$}���yX}"y�~�m � lnmcx&j�y � ± ��rs��~�x ¯ x&j�m$lnmcx&j�y � m@¢_�crzmc{�x&us�elnmc�&��m]~ � vBx�v.�a�_~&lny=yXx&j�rs{��yXt�x$x&j�m � rwvB����vBln~ ¯ ��mç�/m]�cx&rs{���~&}�t��&rzyXt�~`vB{ � rs{�~&x�vB��uzmS£[��m"�at�mc{��crzm]~�±�q=m]�]yX{ �M¯ vB}�}�us�ars{��ex&j�m}�yXus�=��m@£¤mc��mc{��]m`lnmcx&j�y � y� �mc�ªx&rslnm`rz~'m@�=}�m]�cx�m � x�yg~&lny=yXx&j¸yXt�x0£[t��&x&j�mc�'x&j�m7m,#Mm]�cxªyB£-{�yX{H¦~�x�vBx&rzyX{�vB�&� m@���crsx�vBx&rzyX{Y± �-rs{�vBusus� ¯ x&j�m¹vX�cx&t�vBu.~�x�vB��rsusrsx§� yB£�x&j�m¨}"yXus�=��m@£¤mc��mc{��]m � rwvB����vBln~l_v /�m]~ªx&j�m7}���yX}�y�~�m � }�yXus�a��m@£¥mc��mc{��]m7lnmcx&j�y � v���y=y � ��vB{ � r � vBx�m.£¤yX�ªv�£[t�x&t���mSvBt�x�yXl_vBx�m �r � mc{ax&r�����vBx&rzyX{|}���y=�]m � t���m�±

xzy L{xw|

Page 26: Blind subspace-based eigenstructure identification under … · 2016-12-27 · Blind subspace-based eigenstructure identi cation under nonstationary excitation using moving sensors

����������� �������������������������������� ��� �!���#"$�%�&���'�&�)(*"$+,�'�&�*�������"$��� �!-

�­���]�)�������-��0 1�4�©E±-´ª� � mcus��j�vB{�r ¯ © ±T3`yXt���~QvBx ¯ i7± Å rzyXuz��j�rs{�r ¯ � ± G$mc�&l_vB{�~nvB{ � G�± �6vB{ � mc��´ªtaº0¦

mc��vXmc� ¯�� �-mc�/£¤yX�&l_vB{��]m�yB£)yXt�x&}�t�x/¦§yX{�us� r � mc{�x&r�����vBx&rzyX{¬vBus��yX�&rsx&j�ln~�£¥yX��lny � vBu�vB{�vBus�=~�rz~yB£YvBrs���c��vP£¤x�~�x&�&t��cx&t���m]~�� ¯�� � ��� @�� ��� � � ��� � �H",���$� � ���!�?95����� � "$�,D � ¯ �`rz~�~&rslglnm]m ¯ � �6¯��mc�Y± 1 � � �H±

0Õ�<4�©E± Å vX~�~�mc arsusuzm ¯ © ±!´$� � mcus��j�vB{�r�vB{ � ´�± Å mc{� �mc{�rz~&x�m ¯�� qat���~&}�vX�]m@¦¼��vX~�m � £¼vBt�usx � mcx�m]�@¦x&rzyX{_vBus��yX�&rsx&j�ln~�£¥yX�� ars����vBx&rzyX{nlnyX{�rsx�yX�&rs{���� ¯ � �=�8"$(*�!�����E� �� �yXu¼± -32 ¯ {�y�1 ¯ }�}Y± 1�� 1@¦ 1�� � ¯�������H±

0 -<4�©E± Å vX~�~�mc =rsusuzm ¯ ´�± Å mc{a �mc{�rz~&x�m ¯�Å ± 37vX��jH¦ '`mc �vBt���j�mcusuzm ¯ ©E± 37yXt���~�vBx ¯ '�± Å yX{�{�m]��vX~�m ¯��± '`yX��mc� ¯ ©E±"����mc �y�~&x�y¶vB{ � ©E±M�$uwvB��{�yX{ ¯�� '7vBl_vB��m�lnyX{�rsx�yX�&rs{��¶rs{| =rs����vBx&rzyX{¹lnm@¦��j�vB{�rz�]~ 7_rz~�~&t�m]~�rs{ � rwvB��{�y�~&x&rz�]~nvB{ � }���m � rz�cx&rs �m¡l_vBrs{�x�mc{�vB{��]m�� ¯ � ���� ��� 9<��� ��� � �$� ���� " � � ¯  �yXu¼±32 ¯ {Y±Õ� ¯ }�}Y± o�� 1@¦¼o��!- ¯ 1 � �!-H±

0 o$4�´�± Å mc{� �mc{�rz~&x�m�vB{ ��� ¦ � ± ��t���j�~ ¯�� qars{���uzm7~�vBlg}�uzm.lny � vBuMr � mc{ax&r�����vBx&rzyX{|yB£)v�{�yX{�~�x�vP¦x&rzyX{�vB�&�.~&x�ya��j�vX~&x&rz��}���ya�]m]~�~�� ¯�������� ! ���$��� � � �=�8"$(*�!� � � "$����� � ¯  �yXu¼± -�� ¯ {Y± 1 ¯ }�}Y± 232P¦A2Po ¯1 � [ �H±

0Õ�<44�à±�'`mc��uwvBt�º�m ¯ ��± � ±�´$usuzmcl_vB{���vB{ � '�± � ± Å ��yGº'{ ¯�� i0j�m�}"yXus�=��m@£¥mc��mc{��]m.x&rslnm � yXl_vBrs{x�m]��j�{�r��at�m�� ¯�� � ��� @�m� � ��� � �H",� �!� � ���$��95����� � "$� D � ¯ }�}Y± [ -��P¦ [ o�� ¯ 1 � [ 2=±

0 2<4 � ±�GTmc�&l_vB{�~àvB{ � G�±� XvB{ � mc�à´$t�ºàmc��vXmc� ¯�� ©¨y � vBu�x�m]~&x&rs{���vB{ � vB{�vBus�H~&rz~�yB£Y~&x&�&t��cx&t���m]~t�{ � mc��yX}"mc��vBx&rzyX{�vBu �]yX{ � rsx&rzyX{�~ 7�rs{ � t�~&x&�&rwvBu�vB}�}�usrz��vBx&rzyX{�~�� ¯ � ��� �!� 95��� �"� � �$� � ��� " � � ¯ �yXu¼± 1,- ¯ {Y±Õ� ¯ }�}Y± 1 �!-P¦K� 1 2 ¯ 1 � � �H±

032 4 Å ±��$x&x�mc��~&x�mc{ ¯ © ± �Trs�"mc�&��vB{ � i7± �.vBrsuwvBx&j ¯�� ´ª{�vBus�H~&rz~TyB£6~&t���~&}�vX�]mS��x&x&rs{��¶vB{ � © �x�m]��j�{�r��at�m]~0£¤yX�0}�vB��vBlnmcx�mc�ªm]~�x&rsl_vBx&rzyX{#� ¯�������� ! ���$��� �"� � �!� � � � ",� � ¯  �yXu¼± o�� ¯ }�}Y±Õ� ���P¦� � � ¯ 1 � ���H±

0 [ 4 Å ±���m]mcx�mc��~'vB{ � 3g± � mª�$yam]� / ¯�� �ªm@£¥mc��mc{��]m@¦¼��vX~�m � ~&x�y=��j�vX~�x&rz�ª~&t���~&}�vX�]mªr � mc{�x&r�����vBx&rzyX{£¤yX��yXt�x&}�t�x/¦§yX{�us�°lny � vBu'vB{�vBus�H~&rz~�� ¯ � ��� ��� 95��� ��� � �!� � ����",� � ¯  �yXu¼± 1,- ¯ {Y± 2 ¯ }�}Y± [ ���P¦[ 2 [ ±

0 �<4 Å ±!��m]mcx�mc��~7vB{ � 3�± � m��ªy=m]��/ ¯�� �${�m���m�vB�ªlnyX{�rsx�yX�&rs{��¶yB£�x&j�m ���BoX¦¼���&r � ��m17'mc{a ars��yX{H¦lnmc{�x�vBuMm,#�m]�cx�~� �mc��~&t�~ � vBl_vB��m`mc �mc{�x�~�� ¯$� � ��� @%�� ����� � � ��� � �H",���$� � ���$��9<��� � � "!�,D � ¯qHvB{ ´ª{ax�yX{�rzy ¯ i�& ¯ ��mc�Y±Õ�������H±

0 1��<4�©E±�����mc �y�~�x�y ¯ © ±a�$uwvB��{�yX{ ¯ ´�± Å mc{� �mc{�rz~�x�m ¯ © ± Å vX~�~�mc =rsusuzm ¯ 3�± � m �;mc� ¯�� qax�vBx�m@¦§~&}�vX�]m£¤yX�&l�t�uwvBx&rzyX{ ¯ v¡~�yXust�x&rzyX{¹x�y¡lny � vBu�}�vB��vBlnmcx�mc�.m]~&x&rsl_vBx&rzyX{#� ¯(' �$� � "!�&��� �$���) �B��� � ¯ �yXu¼± 1co [H¯ {Y±Õ� ¯ }�}Y± -�� �P¦ -Bo�� ¯ 1 � � 1�±

LML ÝON�P�Qcî P

Page 27: Blind subspace-based eigenstructure identification under … · 2016-12-27 · Blind subspace-based eigenstructure identi cation under nonstationary excitation using moving sensors

�Bo � �!�&� ������ �+5������ ��� ���� � ��6+5��6����� �� �#�� ��� � �5�����+ ���'� ���H�$�������E���"$�������!�

0 131�4���±�qax�y��¼��v.vB{ � ��± � ±�©|y�~�m]~ ¯ � ������",�$��������"!��8" � >���,��� �$�v� ���$��95����� ¯ ����mc{ax&rz�]m�GTvBusu ¯ 1 � � 2=±0 1��<4���± �6vB{¹�$ �mc��~���j�m]m�vB{ �¸Å ±N'Tm7©¨y=yX� ¯ � �v���8>=� �� � �������� �O� � ����"$� D�"$�_� �'� ��$� � 95��� �( � �

! �=�"$��9 � � ( > � �(7����8�!����"!�v�T�O> > �?�#� �!���#"$��� ¯ �`ust�º�mc� ¯ 1 � �32H±0 1,-<4�©E±��Trs��mc�&�TvB{ �.Å ±B�$x&x�mc��~�x�mc{ ¯ � q=mc{�~�yX�-vB�&��v]�S}���ya�]m]~�~�rs{�����vX~�m � yX{�~&t���~&}�vX�]m)��x&x&rs{���� ¯

������� ! ���$��� �#� � �!� � ����",� � ¯  �yXu¼± - � ¯ {�y_� ¯ }�}Y± 13131��P¦ 131�� 1 ¯ 1 � � 1�±

xzy L{xw|

Page 28: Blind subspace-based eigenstructure identification under … · 2016-12-27 · Blind subspace-based eigenstructure identi cation under nonstationary excitation using moving sensors

����������� �������������������������������� ��� �!���#"$�%�&���'�&�)(*"$+,�'�&�*�������"$��� ���

� (ª�ª�"���ª�"

� � F ��� à Ä?����� D[Ã�F ,� ��� � H � � Á�Ã�F P À � à � ��� D � F � ¿�Á Ä�� D�!¿ F � ����� H ��� ¿ D Ä ¿ F � D � ���H� D[Ã�F �

, �Tà P À � ¿��K¿ � ¿ F � ¿ � ���-� H ��� ¿ED Ä ¿ F � D�� ���H� D[Ã*F �.�]���H� D[Ã*F ��� À�¿ � � D ���H� D[Ã*F �

� � Ã�F �]���H� D[Ã�F ��� À ¿�� � D ���H� DOÃ*F ��Ä��H��� F�à �&�°��P D�� �H� D[Ã�F �

� � Ã�F �]���H� D[Ã�F ��� À ¿�� � D ���H� DOÃ*F  D ��� DGF ��� ¿ � ¿ � à �PÄ-� � �� B����&� ��� ���B��P � D �-�P�H� D[Ã�F � F ��P À � D � ¿�� ��� H P ¿ � �

2H± 1 ©¨y � mcusrs{�� ±7±.±S±7±.±7±.±7±.±.±7±S±7±.±7±.±.±7±.±7±S±.±7±.±7±.±7±.±S±7±.±7± 1��2H±Õ� J$t�lnmc�&rz��vBu ��m]~&t�usx�~E±.±7±.±7±.±.±7±S±7±.±7±.±.±7±.±7±S±.±7±.±7±.±7±.±S±7±.±7± 1 2

� � Ã*F � P8��� D[Ã�F � ��¡¿��K¿ � ¿ F � ¿ � � ,

LML ÝON�P�Qcî P

Page 29: Blind subspace-based eigenstructure identification under … · 2016-12-27 · Blind subspace-based eigenstructure identi cation under nonstationary excitation using moving sensors

Unité de recherche INRIA Lorraine, Technopôle de Nancy-Brabois, Campus scientifique,615 rue du Jardin Botanique, BP 101, 54600 VILLERS LÈS NANCY

Unité de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES CedexUnité de recherche INRIA Rhône-Alpes, 655, avenue de l’Europe, 38330 MONTBONNOT ST MARTIN

Unité de recherche INRIA Rocquencourt, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY CedexUnité de recherche INRIA Sophia-Antipolis, 2004 route des Lucioles, BP 93, 06902 SOPHIA-ANTIPOLIS Cedex

ÉditeurINRIA, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France)��������� ���� ���������� ��� ���

ISSN 0249-6399