Fa2004 16.3339-5 Find the closed-loop dynamics Ax+B(r-Ka =(A- BK)a+Br ld.C+B 4 C Objective: Pick K so that Ac has the desired properties, e. g A unstable, want A stable Put2 poles at-2±2 o Note that there are n parameters in k and n eigenvalues in a, so it looks promising, but what can we achieve Example #1: Consider 0 Then det(s-A)=(s-1)(s-2) 3s+1=0 so the system is unstable -Define u=-[ki k2]=-Ka,then 1-k11-k2 Ad=a-BK= [k1k2] So then we have that det(I-Aa)=s2+(k1-3)s+(1-2k1+k2)=0� � � � � � � � Fall 2004 16.333 9–5 • Find the closed­loop dynamics: x˙ = Ax + B(r − Kx) = (A − BK)x + Br = Aclx + Br y = Cx • Objective: Pick K so that Acl has the desired properties, e.g., – A unstable, want Acl stable – Put 2 poles at −2 ± 2j • Note that there are n parameters in K and n eigenvalues in A, so it looks promising, but what can we achieve? • Example #1: Consider: 1 1 x˙ = 1 2 1 x + u 0 – Then det(sI − A) = (s − 1)(s − 2) − 1 = s2 − 3s + 1 = 0 so the system is unstable. – Define u = − k1 k2 x = −Kx, then � � � � 1 1 1 � � 1 − k1 1 − k2 Acl = A−BK = k1 k2 = 1 2 − 0 1 2 – So then we have that det(sI − Acl) = s2 + (k1 − 3)s + (1 − 2k1 + k2) = 0