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Frontiers in Adaptive Control Part 5

Tham khảo tài liệu 'frontiers in adaptive control part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | An Adaptive Controller Design for Flexible-joint Electrically-driven Robots With Consideration of Time-Varying Uncertainties 91 7. Appendix Lemma A.1 Let s G 9Ì n s G 9Ì n and K is the n X n positive definite matrix. Then - sTKs sTs 1 K si2 - Is J. 2L 11 2min K J A.1 Proof s s Ks s s s -Zmin K si 1 HI 1 s jh rmnwii si VA_ K 2 2 L n K si2 Lin K - 2 L n K si2 4.in K 2 2 Lemma A.2 Let wT wn wi 2 win G 9ilxn i 1 . m and W is a block diagonal defined as W diag w1 w2 L wm G mnXn . Then T - W W W w SI i 1 Q.E.D. matrix A.2 The notation Tr . denotes the trace operation. Proof The proof is straightforward as below 92 Frontiers in Adaptive Control WT W N w1n 0 0 0 0 1 0 0 w21 w2n 0 . 0 0 0 0 0 wm1 w.n J rW1 0 L 0 w1n 0 L 0 0 w21 L 0 0 W2n L 0 0 0 L wm1 _ 0 0 L w_ mn _ T rw1T 0 0 rw1 0 0 0 wT2 0 0 w2 0 . 0 0 m 0 0 w m 0 T W1W1 0 0 T w2 w 2 0 0 0 T ww m m INI2 0 0 llw 2II2 0 0 0 0 llw JI2 _ The last equality holds because by definition __T__ w w w ĩ I j w . w2 im hl2 Therefore we have Tr W W W ÊIWi II -. i 1 Q.E.D. J v L that are Lemma A.3 Suppose wT wii wi2 L win eĩV and VT v i 1 . m. Let W and V be block diagonal matrices W diag wi w 2 l w m gW 1 and V diag Vi V 2 L V m G respectively. Then Vn G 1Xn defined as mnXm An Adaptive Controller Design for Flexible-joint Electrically-driven Robots With Consideration of Time-Varying Uncertainties 93 Tr V V W ÊI v JI w 1 . A.3 Proof The proof is also straightforward v 0 L 0 wi 0 L 0 VT W 0 M vT v2 L 0 O M 0 M w2 L 0 M O M _ 0 0 L vT m _ 0 0 Lw m vT w 1 0 L 0 1 0 vTw2 L 0 _ 0 0L T vw m m _ Hence Tr V W vT w 1 vT w 2 . v vm w m hill Iwi l Iv dll Iw 2 II . 1 Iv mil Iw ml Q.E.D. Lemma A.4 m ỆI lv JI lw 1 Let W be defined as in Lemma A.2 and W is a matrix defined as W W - W where W is a matrix with proper dimension. Then Tr WT W 2Tr WT W - 2 Tr WT W . A.4 Proof