MEM255Su06-07.12b.ssRealizatn_studs

MEM255Su06-07.12b.ssRealizatn_studs - Dr. Ajmal Yousuff...

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Unformatted text preview: Dr. Ajmal Yousuff Dept. MEM Drexel University & set u = 0 and consider x = Ax + Bu; z = Mx z (t ) = Mx(t ) = Me At x0 = M [ 0 (t ) I + 1 (t ) A + 2 (t ) A2 + .. + n -1 (t ) An -1 ]x0 M MA = [ 0 I 1 I L n -1 I ] x0 M n -1 MA The system (M,A) is observable if and only if any x0 can be determined from the output {z( ); 0 < < t < }. collect {z ( ), 0 z (t ) = Mx(t ) & & z (t ) = Mx(t ) = MAx(t ) M z ( n -1) = MAn -1 x(t ) t; t > 0}. Differentiate z0 M z MA & 0 0 = x M M ( n -1) n -1 z MA 0 z = O x0 & x 1 0 0 & x = 2 3 a0 & x - 1 0 -a1 0 0 1 + x 0 u -a2 1 1 -a2 rank (C ) = 3 # C= B AB 0 0 A2 B 1 = 0 - a2 1 & x - 1 a2 a & x = - 2 1 3 a0 & x - M 1 O = a2 MA = - 2 MA # 1 0 0 1 ; z = [ 1 0 0 ] x x 0 0 0 1 -a2 0 0 rank (O ) = 3 1 3 - 1 1 & x= x+ u - 2 2 0 y = [ 3 2] x ( A) = {1, 2} y(s) 1 = C ( sI - A) -1 B = u (s) s -1 poles of y(s) = {1} u ( s) What happened to the eigenvalue {2} ? Yousuff MEM 255 Controls 7 1 1 0 & x= x + 0 u 1 -1 y = [ 1 -2] x ( A) = {1, -1} y(s) 1 = C ( sI - A) -1 B = u (s) s +1 poles of y( s) = {-1} u (s) What happened to the eigenvalue {1} ? Yousuff MEM 255 Controls 8 #2 #3 for 1 = 1 and 2 = 2, i are : - 1 1 1 = 2 = , 2 - 1 Transform : x = Tq; T = [1 2 ] for 1 = 1 and 2 = -1, i are : 2 0 1 = 2 = , 1 1 Transform : x = Tq; T = [1 2 ] 1 1 0 & q = + q u 0 0 2 y = [ 1 1] q u(t) does not affect q2(t) 1 1/ 0 2 & q= q+ -1/ 2 u 0 -1 y = [ 0 -2 ] q q2(t) has no effect on y(t) Yousuff MEM 255 Controls 9 ...
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