Fal!2001 16.3122-8 As noted, the closed-loop dynamics couple a(t) and A(t) and are given (t) A T BR-lB 入(t) -CTR,C 入(t) with the appropriate boundary conditions e OK, so where are the closed-loop poles of the system? They must be the eigenvalues of -Dufu CTRZZC2 -AT When we analyzed this before for a siso system, we found that the closed-loop poles could be related to a srl for the transfer function Guu(s=C2(sI-A"Bu b( and, in fact, the closed-loop poles were given by the Lhp roots of s)+ bs(-s)=0 R uu where we previously had Rz/ru=1/r . We now know enough to show that this is trueFall 2001 16.31 22—8 • As noted, the closed-loop dynamics couple x(t) and λ(t) and are given by ∙ x˙(t) λ˙(t) ¸ = " A −BuR−1 uu Bu T −Cz TRzzCz −AT # ∙ x(t) λ(t) ¸ with the appropriate boundary conditions. • OK, so where are the closed-loop poles of the system? — They must be the eigenvalues of " # H , A −BuR−1 uu Bu T −Cz TRzzCz −AT • When we analyzed this before for a SISO system, we found that the closed-loop poles could be related to a SRL for the transfer function Gzu(s) = Cz(sI − A) −1 Bu = b(s) a(s) and, in fact, the closed-loop poles were given by the LHP roots of Rzz a(s)a(−s) + b(s)b(−s)=0 Ruu where we previously had Rzz/Ruu ≡ 1/r • We now know enough to show that this is true