Fall 2001 16.3113-5 Ackermann’ s Formula The previous outlined a design procedure and showed how to do it by hand for second-order systems Extends to higher order(controllable)systems, but tedious Ackermann's Formula gives us a method of doing this entire design process is one easy step K=[0…01]Ma(A) nere M=[BAB…A-B] pa(s) is the characteristic equation for the closed-loop poles which we then evaluate for s= A It is explicit that the system must be controllable because we are inverting the controllability matrix · Revisit example#1:Φa(s)=s2+11+30 A=1:211]-1 11 11 +11 +301 12 4314 1457 Automated in Matlab: place. m &z acker. m(see polyvalm m tooFall 2001 16.31 13–5 Ackermann’s Formula • The previous outlined a design procedure and showed how to do it by hand for second-order systems. – Extends to higher order (controllable) systems, but tedious. • Ackermann’s Formula gives us a method of doing this entire design process is one easy step. K = 0 ... 0 1  M−1 c Φd(A) where – Mc = B AB . . . An−1B  – Φd(s) is the characteristic equation for the closed-loop poles, which we then evaluate for s = A. – It is explicit that the system must be controllable because we are inverting the controllability matrix. • Revisit example # 1: Φd(s) = s2 + 11s + 30 Mc = B AB  =

1 0
1 1 1 2
1 0 =
1 1 0 1 So K = 0 1 
1 1 0 1 −1 
1 1 1 2 2 + 11
1 1 1 2 + 30I  = 0 1  
43 14 14 57 = 14 57  • Automated in Matlab: place.m & acker.m (see polyvalm.m too)