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Fall 2001 6.3117-6 Mechanics ● basics: y, y Gcs): ic= Acic+ C G(s) A Bu · Loop dynamics l=G(S)G(s)→y=L(s)e A +bc +A +B e ABC1「x 0 A B C O To "close the loop, note that e=r-y, then A BC 0 C 0 0 A B A BC Bc A B C 0 Ac is not exactly the same as on page 17-1 because we have re- arranged where the negative sign enters into the problem. Same result thoughFall 2001 16.31 17—6 Mechanics • Basics: e = r − y, u = Gce, y = Gu Gc(s) : x˙ c = Acxc + Bce, u = Ccxc G(s) : x˙ = Ax + Bu , y = Cx • Loop dynamics L = Gc(s)G(s) ⇒ y = L(s)e x˙ = Ax +BCc xc x˙ c = +Ac xc +Bce L(s) ∙ x˙ x˙ c ¸ = ∙ A BCc 0 Ac ¸ ∙ x xc ¸ + ∙ 0 Bc ¸ e y = £ C 0 ¤ ∙ x xc ¸ • To “close the loop”, note that e = r − y, then ∙ x˙ x˙ c ¸ = ∙ A BCc 0 Ac ¸ ∙ x xc ¸ + ∙ 0 Bc ¸ µr − £ C 0 ¤ ∙ x xc ¸¶ = ∙ A BCc −BcC Ac ¸ ∙ x xc ¸ + ∙ 0 Bc ¸ r y = £ C 0 ¤ ∙ x xc ¸ — Acl is not exactly the same as on page 17-1 because we have re￾arranged where the negative sign enters into the problem. Same result though
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