Fa2004 16.3339-8 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重 [BAB…A-B] dpa(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:重(Ss)=s2+115+30 11 BlAB 0 So 11 11 01 +301 01 4314 =1457 Automated in Matlab: place. m acker m(see polyvalmm too� � � � � � � � �� �� Fall 2004 16.333 9–8 Ackermann’s Formula • The previous outlined a design procedure and showed how to do it by hand for second­order 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 Φd(A) c – Mc = B AB . . . An−1B – Φd(s) is the characteristic equation for the closed­loop 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 � � � � � � � 1 1 1 � 1 1 1 Mc = B AB = = 0 1 2 0 0 1 So ⎞ � � 1 1 �−1 ⎛� �2 � � � 1 1 1 1 K = 0 1 ⎝ + 11 + 30I⎠ 0 1 1 2 1 2 � � 43 14 � � = 0 1 = 14 57 14 57 • Automated in Matlab: place.m & acker.m (see polyvalm.m too)