t04 control system

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(c) Find the control canonical form of the system. 
 
 
(a) Find the poles of the system.
(b) Obtain the transfer function of the system.
(c) Find the similarity transformation which converts the original model to the control canon-
ical form.
Answer:
Solving the characteristic equation of the system
det(sI − A) = s2 + 5s+ 7 = 0
yields the poles
p =
−5±√3j
2
The transfer function is given by
H(s) = C(sI − A)−1B = s+ 7
s2 + 5s+ 7
For part (c), note that
P =
[
B AB
]
=
[
1 −5
1 2
]
Q =
[
1 5
0 1
]
T = PQ =
[
1 0
1 7
]
So under the similarity transformation
x = Tz =
[
1 0
1 7
]
z
or
z = T−1x =
[
1 0
−1/7 1/7
]
x
the given model is converted into the control canonical realization
z˙ =
[−5 −7
1 0
]
z +
[
1
0
]
u
y =
[
1 7
]
z
3. Consider the system shown in Figure 1. Determine the value of k such that the damping ratio
ζ is 0.4. Then obtain the rise time tr, peak time tp, maximum overshoot Mp, and 2% settling
time ts in the unit-step response.
Σ
+
−
Σ
+
−
16
s+1.2
k
1
s
R Y
Figure 1. Feedback system.
Answer:
The closed-loop transfer function is
Y (s)
R(s)
=
16
s2 + (1.2 + 16k)s+ 16
It follows that
2ζωn = 1.2 + 16k and ω
2
n = 16
Thus, we obtain
k =
1
8
and ωn = 4
In addition, the rise time, the peak time, the settling time, and the overshoot can be determined
as follows:
tr =
1
ωd
tan−1
(
ωd
−σ
)
= 0.32 sec.
tp =
pi
ωd
= 0.86 sec.
ts ≈ 4/σ = 2.5 sec. (2% criterion)
Mp = e
−piζ/
√
1−ζ2 = 25.4%