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HW 7 Solutions
1a. The equation to find a transfer function from a state space representation: () ( [ )
[ [ [
[ ] () Or since the given A matrix is not in the controllable canonical form we must find a transformation matrix, P, to
bring the given A, B, C matrices into that form.
[
[ 2 Where a1 comes from the characteristic equation: s + a1s + a2.
2 The characteristic equation is: s – 5s + 3.
Then we can calculate the transformation matrix P.
[ Now we can obtain the controllable canonical form of our system.
[ [ So using the controllable canonical form we can get the transfer function, H(s)
() 1b. The given A matrix is in controllable canonical form so we don’t need to find a transformation matrix and we
can find the transfer function from the state space representation without any calculations. ()
1c. . The equation to find a transfer function from a state space representation:
() ( [ ) [
[ [ () Or since the given A matrix is not in the controllable canonical form we must find a transformation matrix, P, to
bring the given A, B, C matrices into that form.
[
[
3 2 Where a1 comes from the characteristic equation: s + a1s + a2s +a3.
3 2 The characteristic equation is: s – 3s – 5s + 7. Then we can calculate the transformation matrix P.
[ Now we can obtain the controllable canonical form of our system.
[ [ So using the controllable canonical form we can get the transfer function, H(s) ()
2a.
[ [ [ 2b. [ 2c. 3. Place closed loop poles at s = 5,3±2j.
Construct controllability matrix
[ [ Find the closed loop scalar polynomial
() ( )( )( ) Find the closed loop matrix polynomial
() [ Use Ackermann’s Formula to find K
[ () [ Place observer poles at s=10,1,1.
Construct Observability Matrix
[ [ Find the closed loop scalar polynomial
() ( )( )( ) Find the closed loop matrix polynomial
() [ Use Ackermann’s Formula to find L
() [ ...
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This note was uploaded on 09/05/2011 for the course EML 4312 taught by Professor Dixon during the Spring '07 term at University of Florida.
 Spring '07
 Dixon

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