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State Estimation and Tracking Control of Nonlinear Dynamical

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Page 1: State Estimation and Tracking Control of Nonlinear Dynamical

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Page 2: State Estimation and Tracking Control of Nonlinear Dynamical

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Page 16: State Estimation and Tracking Control of Nonlinear Dynamical

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Page 17: State Estimation and Tracking Control of Nonlinear Dynamical

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Desired Fully Linear Linear SE Gain Fully Nonlinear

Ê/Ë1Ì3Í�Î6Ï8Ð:Ñ<Ò>Ó3ÔAÕCÖDÎEË�×GÓ3ØIÓKÙ/ÚGÎEÖKÛ6Ü�ËQØ�Ì,Û�Ó3ØSÚGÎ6Ó3ÝQ×NÞ!×6Ú6ÖDÚGÏlÏV×GÚ6Ë�Ô,ÖDÚGÓKÎE×WÓ3Øpß�à�ÖKÔAÕCÝ1ÏAáDâ_ã`ËQÚ6äYØ�Ó3ËQ×GÏKåæ ç èWé�ê¿é é�ê¿é é�ê¿é é�ê¿é ëDé�ê¿é�ì"í,êî ä�ÏtÛ�Ó3×GÚ�Ù�Í_ØCÛ�Ú6Ë1Ó3Ø_ÖKÝyÚGÓ,ï{ÏtÔ,ËQØ_Ë�Ô,Ë1ð"ÏVñIËQ×ò[ó�ôyõDöG÷{ø[ç áëtùpúõ û ó�ü)ý�þ ø ímÿ ó�ü)ý�þ ø�� ÷ í�� ÷�����gêã`ä�ÏVØ�ËQØIÚGÎEÖKÛEÜ:Ë�Ø�Ì,Õ_Î6ÓKï�Ý1ÏVÔ Ù ÓKÎEÔYå î ä�Ï×GÚ6ÖDÚGÏ�!ÖDÎEË�ÖDïCÝ1ÏV×�ËQØcÚ6ä_ËQ×�Ô,Ó:ñ�ÏVÝ{ÎEÏ"Õ_Î6ÏV×GÏVØSÚWÚ6ä�Ï ��ËQÌ3ä Ú�Û�Ó3Ø_ñ_ËQÚ6Ë1Ó3Ø_×<ÓKÙÚ6ä�ÏAÖKË1ÎEÛ�ÎEÖDÙ�Ú"ÑWô��·ËQ×�Ú6ä�Ï)ñ_Ï�:ËQÖDÚ6ËQÓ3ØpÓKÙqÚ6ä�Ï�KÏVÝQÓ:Û"Ë1Ú��YÙ�Î6Ó3Ô Ú6ä�Ï)ÝQÏ�KÏVÝ��yË1Ì3ä Ú�ÚGÎEËQÔ�DÖKÝQÍ�Ï8ÓKÙ�á3ó á"éKéKÔ�Þ!×NøAó�Ì3Ë�KÏVØËQØ�Í_Ø_Ë1Ú6×·ÓKÙó á"éKéKÔ�Þ!×NøGøNâ/ô��)Ë�×Ú6ä�Ï,ñ�Ï��ËQÖDÚ6Ë1Ó3ØvÓKÙ>Ú6ä�ÏRÖKØ�Ì3ÝQÏAÓKÙ>ÖDÚGÚ6ÖKÛEÜ�Ù ÎEÓ3Ô Ú6ä�Ï,ÚGÎEËQÔ�DÖKÝQÍ�Ï ÓKÙ���êÅë�ó����:á��Dé øÎEÖKñ_ËQÖKØ_×VâCô��8ËQ×`Ú6ä_Ï8ÕCË1Ú6ÛNäpÎNÖDÚGÏ)ËQØYÎEÖKñ�Þ!×"âyô! )ËQ×WÚ6ä�Ï"��Ë1Ì3äSÚ`ÕCÖDÚ6ä�ÖKØ�Ì3Ý1Ï)ËQØYÎNÖKñ_ËQÖKØ_×"â�ÖKØ_ñpô�#8ËQ×`Ú6ä�Ïlñ_Ï�:ËQÖDÚ6ËQÓ3ØÓKÙ[Ú6ä�Ï Û"ÖKØ_ÖDÎEñ�ñ�Ï$�CÏVÛ�Ú6Ë1Ó3Ø�ÖKØ�Ì3Ý1Ï)ËQØpÎNÖKñ_ËQÖKØ_×`Ù Î6Ó3Ô§Ú6ä�ÏlÚGÎEËQÔ�!ÖKÝQÍ_ÏKâCã`ä_Ë�ÛEäpË�×�Ø�ÓKÚ�Ì3Ë�KÏVØ å î ä_ÏlÛ�Ó3ØSÚGÎ6Ó3Ým÷vË�×Ú6ä�ÏAËQØ�ÕCÍ�Ú�Û"ÖKØ_ÖDÎEñvñ�Ï$�CÏVÛ�Ú6ËQÓ3Ø�ËQØpÎNÖKñ_ËQÖKØ_×"å î ä�Ï Û"ÖKØ_ÖDÎEñC×�ÖDÎEÏ Û�Ó3ØSÚGÎ6Ó3Ý%��ÖDÕ�×�ã�ä_ËQÛEä�Û"ÖKØ�ñ�Ï$�CÏVÛ�Útñ_Ó ã`ØSã>ÖDÎNñï&��Í�Õ�ÚGÓ('Dé_ó����:á��Dé ø`ÎEÖKñCËQÖKØ_×"å�ó î ä�ÏlÔAÏVÖKØ_ËQØ�Ì ÓKÙ/Ú6ä�ÏtÔAÓ�ñ�ÏVÝxË�×�ñ�ÏV×6Û�ÎEË1ï{ÏVñIËQØ�ÔAÓKÎEÏ8ñ�Ï"Ú6ÖKËQÝ ËQØ*),+�á�-|åÉø.iØ Ú6ä_Ï0/CÎE×GÚAÚGÏV×GÚRãqÏpÛ�Ó3ØC×6ËQñ�Ï"Î,Ö�Û�Ó3ØSÚGÎ6Ó3ÝWÕCÎ6ÓKïCÝ1ÏVÔ ã`ËQÚ6ä ×GÚ6ÖDÚGÏYÏV×GÚ6ËQÔRÖDÚ6Ë1Ó3Ø ïCÍ�Ú,Ø_Ó�ÚGÎEÖKÛ6Ü�ËQØ�Ì�å î ä�ÏÙ Ï"ÏVñ�ï�ÖKÛ6ÜIÛ�Ó3Ø ÚGÎ6Ó3Ý{ã�ËQÝQÝyÖDÚGÚGÏVÔAÕ_Ú�ÚGÓ Ù ÓKÎNÛ�ÏÚ6ä�Ï×GÚ6ÖDÚGÏ1!ÖDÎNËQÖDïCÝ1ÏV×>ÚGÓ,é)Ù ÎEÓ3Ô ÖKØ�Ë�Ø_Ë1Ú6ËQÖKÝQÝ2� Ý�ÖDÎ6ÌKÏÖKØ�Ì3Ý1Ï·ÓKÙMÖDÚGÚ6ÖKÛEÜ�âÌ3Ë�KÏVØ�ï3��ÖKØ�Ë�Ø_Ë1Ú6ËQÖKÝ�Û�Ó3Ø_ñCË1Ú6Ë1Ó3Ø�ÓKÙ�ôyõRç óré�öVá�4�ó����:á��Dé øNö6é�ö6é�öEé ø í å65`Ó�ãqÏ�KÏ"Î â�Ú6ä�Ï�ËQØCË1Ú6ËQÖKÝ/ÏV×GÚ6ËQÔRÖDÚGÏVñ�×GÚ6ÖDÚGÏËQ×�ó�ô�7gøiõYç óré�öEëDé_ó����:á��Dé øNö6é�ö6é�ö6é ø í å î ä�ÏIÔAÏVÖK×6Í�Î6ÏVÔ,ÏVØ ÚAÙ�ÓKÎ Ú6ä�ÏIÏV×GÚ6ËQÔ,ÖDÚGÓKÎ,ËQ×)ÓKÙ�Ú6ä�Ï0KÏVÝ1Ó:Û"ËQÚ8��ÖKØCñ�Ú6ä_ÏÛ"ÖKØ_ÖDÎEñ�ñ_Ï$�CÏVÛ�Ú6Ë1Ó3Ø âm×GÓ�Ú6ä_ÖDÚ"9 ç;:Kó�ô{øç�) ô � öGô # - í å î ä�Ï,Û�Ó3×GÚ8ÙrÍ_Ø_Û�Ú6ËQÓ3Ø_ÖKÝ/ãqÏVËQÌ3ä Ú6×lÖDÎ6Ï,Ì3Ë�KÏVØ�ï3� ÿ ç=< #ÖKØ_ñ � ç á"éKé�âCÖKØ_ñIÚ6ä_ÏãqÏVË1Ì3äSÚ6×�Ù�ÓKÎ�Ú6ä�ÏÏV×GÚ6ËQÔRÖDÚGÓKÎ�Ì3ÖKËQØ�Õ_Î6ÓKï�Ý1ÏVÔ ÖDÎEÏ?> ç á"éKé@<�#ÖKØ_ñBA çC<���å�Ê/Ë1Ì3Í_Î6Ï"Dñ�Ï"ÕCËQÛ�Ú6×qÚ6ä�Ï�ÖKÛ�Ú6ÍCÖKÝyÖKØCñ,ÏV×GÚ6ËQÔRÖDÚGÏVñc×6Ú6ÖDÚGÏV×qÙ�ÓKÎ>Ú6ä_ËQ×[ÕCÎ6ÓKïCÝ1ÏVÔ Í_×6ËQØ�Ì8Ú6ä�Ï�ÝQË�Ø�ÏVÖDÎ�E�ß�Ì3ÖKËQØvó î ä_ÖKÍ�ø[Û�Ó3Ø ÚGÎEÓ3ÝyÖKØCñÚ6ä�ÏAØ�Ó3Ø_ÝQËQØ_ÏVÖDÎ E�ß Ì3ÖKËQØvÛ�Ó3ØSÚGÎ6Ó3Ý/Ù Ó3ÍCØ_ñpã�Ë1Ú6ä ÖKØ*EGF H�ß�×GÓ3ÝQÍ�Ú6Ë1Ó3Øxå Ê�ÓKÎÏVÖKÛEä�ÔAÏ"Ú6ä�Ó�ñ Ë1Ú�Ú6ÖDÜKÏV×t×GÓ3Ô,Ï)Ú6Ë�ÔAÏÙ ÓKÎ<Ú6ä�Ï�ÏV×GÚ6ËQÔ,ÖDÚGÏVñI×GÚ6ÖDÚGÏÚGÓ Û�Ó3Ø&KÏ"Î6ÌKÏÚGÓ Ú6ä�Ï�ÖKÛ�Ú6ÍCÖKÝ ×GÚ6ÖDÚGÏKâ_ÖKØ_ñ�Ù ÓKÎ<ï{ÓKÚ6äRÚGÓ ï{Ï`Ù�ÓKÎEÛ�ÏVñcÚGÓAé�â�ïCÍ�Ú¥Ú6ä_Ï��,ï{ÓKÚ6äñ�Ó ×GÓ ËQØ�ÖY×EÔAÓSÓKÚ6ä Ô,ÖKØ_Ø_Ï"ÎVå î ä�Ï"Î6ÏRË�×8ÖpÝ�ÖDÎ6ÌKÏ"Î8Ó3×6Û"Ë�ÝQÝQÖDÚ6Ë1Ó3Ø�Ø_ÓKÚ6ËQÛ�ÏVÖDïCÝ1ÏRË�Ø�Ú6ä�Ï�Ý�ËQØ�ÏVÖDÎIE�ß Ì3ÖKËQØ Û�Ó3Ø ÚGÎ6Ó3Ý|âáJD

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f Atta

ck x

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Actual State: Linear SE Gain Estimated State: Linear SE Gain Actual State: Nonlinear SE Gain Estimated State: Nonlinear SE Gain

KML�NOQPSR�T&UWVYX$ZSO\[@]^[@_\`0R�abZSL2cd[eZbR�`fabZS[eZbR�ahgji@PkglR�R�`Qm![@XonBX$i_pZbPSi]qasrtabZS[eZbRIR�abZSL2cd[eZbi@PoauL2_BvxwQ[@czy!]�R"{3|X�[@O\aSL2_QN0aS]�i~}�R�P?X$i_&�@R�PSN@R�_\X$R@|��1_fZS�\R�i@ZS�QR�P?�![@_\`���}uL�ZS�fZS�QRIglO\]q]��B]2L2_QR�[eP abZS[eZbRzR�abZSL2c�[eZSL�i_�[@_\`�X$i_pZbPSi][@]�N@i@PoL�ZS�\c�[ey\y!]2L�R�`��&ZS�QR�ab�3aSZbR�c�PSR�cd[@L2_!a�O\_\abZS[em!]2R?[@_\`B`\L��@R�PSN@R�auZbi�L2_G��_\L�Z8�@|�uR$wGZ�}�R�}uL2]2]�X$i_\aSL2`\R�PI[fy\PSi@m!]2R�c�}uL�ZS��ZbPo[@Xon3L2_\Nfm!O\Z�_Qi�abZS[eZbR0R�abZSL2cd[eZSL2i_�|��k�QR�i@mG�bR�X$ZSL��@RBL2a�ZbiZbPo[@XSn�[�`QR�aSL2PSR�`���L2N�pZ?y�[eZS��[@_QN]2R��!�e�xL2_\X$PoR�[@aSL2_QN�L2Z?N@Po[@`\O\[@]q]���gjPSic��fO\y�Zbi��p�Q�� �¡3¢�£e�p¤s���Qi]2`!L2_QNQ�^[@_!`ZS�QR�_�PSR�ZSOQPo_\Lq_QN¥Zbi�����[@azao�QiJ}u_�L2_�KML�NOQPSRB£¤s|¦�k�\R�L2_!L�ZSL2[@]§X$i_\`!L�ZSL�i_�L2a"ZS�\R�PSR�gji@PoR�[eZ�ZS�QR�i@PoL�NL2_��M[@_!`ZS�QRIX$iaSZY}¨R�L�N�&ZSa [ePSR�©«ª=¢�[@_!`�¬«ª=¢��G|=�k�QR"PoR�aSO\]�ZSah[ePSR�y!]�i@ZbZbR�`�L2_fKML�NO\PSRI£�gji@P ZS�QR�_Qi_\]qL2_QR�[ePh[@_!`]2L2_QR�[ePkZbPs[@XSnGL2_QN(X$i_&ZbPSi]2a�|®­hR�PSR�ZS�QR�_Qi_!]2L2_QR�[ePkX$i_&ZbPSi]�`\i&R�ah_Qi@Zuy�R�PSgli@Poc¯aSO\m!abZS[@_&ZSL2[@]2]��(m�R�ZbZbR�PhZS�\[@_¥ZS�\R]2L2_QR�[ePWX$i_&ZbPSi]±°dLq_zgl[@X$Z¨ZS�QRu]qL2_QR�[ePWX$i_&ZbPSi]��3L�R�]q`\axm�R�ZbZbR�PWPSR�aoO\]�ZSa§[eZ¨ZS�QRkZbi@y�i@g�ZS�QRuPo[@cdy(OQydZbi��p�Q�� �¡3¢�£e�p¤b¤ZS�QiOQN�¥L2_0ZS�\R"]2[eZbR?ZSL2cdR1y�R�PsL�i3`�ZS�QR�]2L2_QR�[eP�X$i_&ZbPSi]�PSR�ZSO\Po_\a�Zbi��dcdi@PSR"aS]�i~}u]��(ZS�\[@_BZS�QR"_\i_\]2L2_QR�[ePJ|²³_´X$i_\aSL2`QR�PsL2_QN*[*y!PSi@m!]�R�cµ}uL�ZS�¶m�i@ZS�·ZbPs[@XSnGL2_QN�[@_!`¸abZS[eZbR�R�abZSL2c�[eZSL�i_���}�R�O\abRfZS�QRfaS[@cdRf`QR�aSL2PSR�`ZbPo[~�¹R�X$Zbi@Po�d[@a§m�R�gji@PoR@�pgli@PoX�L2_QN ZS�QR���L�N�pZxy�[eZS��[@_QN]�R�����gjPoic«�?OQydZbi"�p�Q�� �¡3¢�£e�p¤W[@_!`zm![@XSn�|�v�abZSL2cd[eZSL�i_zLqa[@`\`QR�`¥O\aoL2_QN�[zczR�[@aSOQPSR�cdR�_pZki@g%ZS�QR1�@R�]�iGX�L�Z��@�![@_QN]�R1i@gM[eZbZS[@XSn���[@_!`BX�[@_\[ePo`¥`QR$��R�X$ZSL�i_�U§ºIª«» ��¼�½b��¾e½b��¿ÁÀ�§|�k�QR�}�R�L2N�pZSa?[ePoRd©Ãª¯¢e��¬;ª�¢e�^Ä ª�¢��@�@Å ¿ [@_\`�ƵªÇÅ�Èt|É�k�QR�[@X$ZSO\[@]§abZS[eZbRdabZS[ePoZSa?[eZ1ZS�QR�i@PoL�NL2_���[@aL2_�ZS�QRzy\PSR��GL�iO\a ZbPs[@XSnGL2_QNBy\PSi@m�]�R�cB��m!OQZ1ZS�\RzR�abZSL2cd[eZbR�`*aSZS[eZbR�abZS[ePSZSa�aS]2L�N�&ZS]��¥ieʶZS�QRd[@X$ZSO\[@]Ë�^[eZ(�l��ÌÁ¤³Ízª���G½S�G½S�G½o�Q�� �¡3¢�£e�p¤s½o�p¤  |ÎKML�NOQPSR Ï�`\R�y!L2X$ZSa¨ZS�QRYR�abZSL2cd[eZbR�`BabZS[eZbR1[@]2cdiabZ�X$i_p�@R�PSNLq_QN�ZbiIZS�\RY[@X$ZSO\[@]�aSZS[eZbRYm3�ZS�QR�ZSLqczR�i@gxZS�\RI`QR�aoL�PSR�`¥����L2_!X$PSR�[@abR@��[@_\`fPSR�cd[@L2_\Lq_QNdX�]�iabRIZbi0ZS�QRI[@X$ZSO![@]%aSZS[eZbRIgli@PYZS�QR�PoR�abZhi@gWZS�QR�ZSLqczRy�R�PsL�i3`�|в¹_fKML�NO\PSR�¢��(}¨R�y!]2i@ZhZS�QRI[@X$ZSO\[@]MabZS[eZbRz}u�QR�_fX$i_pZbPoi]2]�R�`�O\aSL2_\NdiOQPYglO\]q]���_\i_\]2L2_QR�[ePY[@]�N@i@PoL�ZS�!cB�[@a1}�R�]q]^[@a1}h�QR�_�O\aSL2_QN(ZS�QRz]2L2_QR�[eP?ÑGvCN[@Lq_�X$i_&ZbPSi]±�^[@_\`�ZS�QR�glO\]q]��¥]2L2_QR�[eP X$i_&ZbPSi]±|¯²³Z1X�[@_�m�RzabR�R�_�ZS�![eZZS�QR�]2L2_\R�[eP?X$i_pZbPSi]§iJ�@R�PoaS�\i&i@ZSaIaSL�N_\L���X�[@_pZS]2�f[eZ?ZS�QRzZbi@y�i@g�ZS�\R�[@aSX$R�_&Z"[@_\`*L2a?�@R�PS��ao]�iJ}«Zbi�PSR�ZSOQPo_�Zbif�G|�k�QRIi@ZS�QR�P Z8}¨i¥czR�ZS�QiG`\auy!PSi3`!O\X$R"�GL�PSZSO\[@]q]��BL2`QR�_&ZSL2X�[@]�PoR�aSO\]�ZSa���ZS�QR�`\LÒÊ�R�PSR�_\X$R�L2_\`\LqaSX$R�Po_\[em!]2R?L2_�ZS�\R�y!]2i@ZSaL2_BKML�NOQPSRd¢��G|K%Lq_\[@]2]��@�§}¨R¥[@]�ZbR�PdZS�\R�y!PSi@m!]�R�cÓO\abR�`¶Lq_¶K%L2NOQPSR�adÏ�[@_\`C¢��G��L2_&�@i]��3Lq_QNfm�i@ZS�¶abZS[eZbR�R�abZSL2cd[eZSL�i_¶[@_!`

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Desired Linear Nonlinear

ÔMÕ�Ö×QØSÙ"Ú3Û¨Ü�ÝÞzß!àeØoÕ2áSÝâ0Ý@ã^ãlÙ�Ù�äQå!à@æoçBèbØoà@æSçGÕ2âQÖdæ$Ýâ&èbØSÝé2áuÝâ¥êxëQà@Þzß!é�Ù"ì3í

0 5 10 15−0.2

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0.2

0.4

0.6

0.8

1

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Time (s)

Flig

ht P

ath

Ang

le x

4 (ra

d)

Desired State Actual State Estimated State

ÔMÕ�Ö×QØSÙ"î3ÛWïYæ$èS×\à@é^à@â\ä0Ù�ábèSÕ2ÞdàeèbÙ�äfábèSàeèbÙ�áhãjÝ@Øuâ\Ýâ\é2Õ2âQÙ�àeØ�èbØoà@æoç3ÕqâQÖ�æ$Ýâ&èbØSÝéjðtábèSàeèbÙ�Ù�ábèSÕ2ÞdàeèbÝ@ØYÕ2âBêxëQà@Þzß!é�Ù"ì3í

ñ î

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Desired Fully Linear Linear SE Gain Fully Nonlinear

òMó�ôõ\öS÷dø�ùGú¨û¨üýzþ�ÿeöoó��bü��0ü����böoÿ��3ó��\ô��$ü����böoü� ��������Sÿ��b÷�÷����Só2ýdÿ��bü@ö�uü������Qÿ@ýzþ� 2÷������uó��! Bó��!ÿ��"��õQöoÿ��b÷�#%$'&"(*)�+�böoÿ��!Gó��Qô,�.-0/�öS÷�ýzü213ó��\ô3�! Q÷54Q÷"1Gó2ÿ��Só�ü���ó��6�! Q÷dó7�\ó��Só2ÿ� %÷����Só2ý�ÿ��b÷�48���Sÿ��b÷�ÿ���4�ÿ�4�4\ó��Qô0öoÿ���4Qüý9�Qüó��b÷:�bü��! \÷þ\öSü�-� �÷�ý�+<;= Q÷�ó���4Q÷�þ�÷���4Q÷��0� õ��\ó��lü@öoý94\ó����bösó�-!õ,�Só�ü����?>�@2#%A�(?ÿ���4B>2C�#%A�(����uó��! ED >GF,DIHJ��KL��M,#ONQP3ø�Reù�(3#SM�T�ü���! Q÷Iý�ÿ��Gó2ýIõ\ýU4Q÷��Só�öo÷�43V�ó�ô� 0�uþ�ÿ��! �ÿ��\ô� �÷���ÿ���4fÿ���÷"1@÷��6 2ÿeöSô@÷�ö þ�÷�ö�$÷��0�Sÿeô@÷Iü��W�! Q÷�ÿ��X�Sõ\ÿ� ��!�Sÿ��b÷Y1eÿeöoó2ÿ�-' �÷��ÿ��uýzü����=�Sóqýz÷���(��!ÿeöS÷?ÿ�4�4Q÷�43�bü5�! Q÷Z4,/��!ÿ@ýdó��"��ÿ���43�! \÷?ýz÷�ÿ��SõQöS÷�ýd÷����köo÷��bþ�÷��X�Só�1@÷� �/�ú

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