720 Chapter 9 / Control Systems Analysis in State SpaceProblems
B–9–1.Consider the following transfer-function system:Obtain the state-space representation of this system in (a)
controllable canonical form and (b) observable canonical
form.B–9–2.Consider the following system:Obtain a state-space representation of this system in a di-
agonal canonical form.B–9–3.Consider the system defined bywhereTransform the system equations into the controllable canon-
ical form.B–9–4.Consider the system defined bywhereObtain the transfer function Y(s)/U(s).B–9–5.Consider the following matrix A:Obtain the eigenvalues l 1 ,l 2 ,l 3 , and l 4 of the matrix A.
Then obtain a transformation matrix Psuch thatP-^1 AP=diagAl 1 , l 2 , l 3 , l 4 BA= D
0
0
0
1
1
0
0
0
0
1
0
0
0
0
1
0
T
A=C
- 1
1
0
0
- 2
0
1
0
- 3
S, B= C
0
0
1
S, C=[ 1 1 0 ]
y=Cxx#=Ax+BuA=B
1
- 4
2
- 3
R, B= B
1
2
R, C=[ 1 1 ]
y=Cxx#=Ax+Buy%+ 6 y$+ 11 y# + 6 y= 6 uY(s)
U(s)=
s+ 6
s^2 +5s+ 6B–9–6.Consider the following matrix A:ComputeeAtby three methods.B–9–7.Given the system equationfind the solution in terms of the initial conditions x 1 (0),
x 2 (0), and x 3 (0).B–9–8.Find x 1 (t)andx 2 (t)of the system described bywhere the initial conditions areB–9–9.Consider the following state equation and output
equation:Show that the state equation can be transformed into the
following form by use of a proper transformation matrix:Then obtain the output yin terms of z 1 , z 2 , and z 3.B–9–10.Obtain a state-space representation of the follow-
ing system with MATLAB:Y(s)
U(s)=
10.4s^2 +47s+ 160
s^3 +14s^2 +56s+ 160C
z# 1
z# 2
z# 3S = C
0
1
0
0
0
1
- 6
- 11
- 6
SC
z 1
z 2
z 3S + C
1
0
0
Su
y =[1 0 0]C
x 1
x 2
x 3S
C
x# 1
x# 2
x# 3S = C
- 6
- 11
- 6
1
0
0
0
1
0
SC
x 1
x 2
x 3S + C
2
6
2
Su
B
x 1 (0)
x 2 (0)R= B
1
- 1
R
B
x# 1
x# 2R= B
0
- 3
1
- 2
RB
x 1
x 2R
C
x# 1
x# 2
x# 3S =C
2
0
0
1
2
0
0
1
2
SC
x 1
x 2
x 3S
A= B
0
- 2
1
- 3
R
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