Fall 2001 16.3113-6 . Where did this formula come from? For simplicity, consider a third-order system(case #2), but this extends to any order 100B b1 b2b3 See key benefit of using control canonical state-space model This form is useful because the characteristic equation for the system is obvious => det(sI-A)=s+a1s+a2S+a3=0 ● Can show that Ac=A-BK 100 0k k2 ks 0 a1-k1-a2-k2-a3-k3 0 o that the characteristic equation for the system is still obvious 重a(s)=det(s-Aa) s3+(a1+k1)s2+(a2+k2)s+(a3+ka)=0Fall 2001 16.31 13–6 • Where did this formula come from? • For simplicity, consider a third-order system (case #2), but this extends to any order. A =   −a1 −a2 −a3 100 010   B =   1 0 0   C = b1 b2 b3  – See key benefit of using control canonical state-space model – This form is useful because the characteristic equation for the system is obvious ⇒ det(sI − A) = s3 + a1s2 + a2s + a3 = 0 • Can show that Acl = A − BK =   −a1 −a2 −a3 100 010   −   1 0 0   k1 k2 k3  =   −a1 − k1 −a2 − k2 −a3 − k3 100 010   so that the characteristic equation for the system is still obvious: Φcl(s) = det(sI − Acl) = s3 + (a1 + k1)s2 + (a2 + k2)s + (a3 + k3)=0