Fall 2001 16.311724 Second: this generalization does not change the closed-loop poles of the system, regardless of the selection of G and N, since 汇=Ax+Bu C AcIc+Ly+ gr Kr+ Nr A-BK BN LC A c So the closed-loop poles are the eigenvalues of/A BK LC A regardless of the choice of G and N G and N impact the forward path, not the feedback path Third: given this extra freedom, what is the best way to use it? One good objective is to select G and N so that the state esti- mation error is independent of r With this choice, changes in r do not tend to cause such large transients in a Note that for this analysis, take t=a-c since c= =i-i= Ar+ Bu-(Acc+Ly+ gr) A c+B(KIc+ Nr-A-BK-LCc+ LCa+ gr)Fall 2001 16.31 17–24 • Second: this generalization does not change the closed-loop poles of the system, regardless of the selection of G and N¯ , si nce x˙ = Ax + Bu , y = Cx x˙ c = Acxc + Ly + Gr u = −Kxc + Nr ¯ ⇒
x˙ x˙ c =
A −BK LC Ac
x xc +
BN¯ G r y = C 0 
x xc – So the closed-loop poles are the eigenvalues of
A −BK LC Ac regardless of the choice of G and N¯ – G and N¯ impact the forward path, not the feedback path • Third: given this extra freedom, what is the best way to use it? – One good objective is to select G and N¯ so that the state esti￾mation error is independent of r. – With this choice, changes in r do not tend to cause such large transients in ˜x – Note that for this analysis, take ˜x = x − xc since xc ≡ xˆ x˜˙ = ˙x − x˙ c = Ax + Bu − (Acxc + Ly + Gr) = Ax + B(−Kxc + Nr ¯ ) − ({A − BK − LC}xc + LCx + Gr)