2.10. EXERCISES AND PROJECTS 83
(b) If the inputu(t)is a unit step function, withx 1 (0) =x 2 (0) = 0,find
y(t),t> 0.
(c) If the inputu(t)is a unit step function, withx 1 (0) =x 2 (0) =− 1 ,
findy(t),t> 0.
(d) Obtain aMatlabsolution to verify your results.
- For each second-order system given below, obtain a complete analytical
solutiony(t),t> 0 whenx 1 (0) = 1andx 2 (0) = 0. Assume the input
u(t)=u 1 (t)=u 2 (t)is a unit step function. UsingMatlab,verifyyour
solutions in each case.
(a)
∑
x ̇ 1 (t)
x ̇ 2 (t)
∏
=
∑
− 32
− 2 − 3
∏∑
x 1 (t)
x 2 (t)
∏
+
∑
1
1
∏
u(t)
y(t)=
£
10
§
∑
x 1 (t)
x 2 (t)
∏
(b)
∑
x ̇ 1 (t)
x ̇ 2 (t)
∏
=
∑
03
− 5 − 8
∏∑
x 1 (t)
x 2 (t)
∏
+
∑
11
0 − 1
∏∑
u 1 (t)
u 2 (t)
∏
∑
y 1 (t)
y 2 (t)
∏
=
∑
10
12
∏∑
x 1 (t)
x 2 (t)
∏
(c)
∑
x ̇ 1 (t)
x ̇ 2 (t)
∏
=
∑
04
0 − 5
∏∑
x 1 (t)
x 2 (t)
∏
+
∑
0
1
∏
u(t)
∑
y 1 (t)
y 2 (t)
∏
=
∑
01
11
∏∑
x 1 (t)
x 2 (t)
∏
- Consider the following plant models:
(a)Gp 1 (s)=
10
(s+1)(s+ 10)
(b)Gp 2 (s)=
1
(s+1)(s−3)
(c)Gp 3 (s)=
1
(s^2 +s+1)
(d)Gp 4 (s)=
1
s(s+1)
Making reasonable assumptions for settling time and steady-state error
criteria, for each plant model derive a suitable set of PID control para-
meters. UseMatlabto simulate the behavior of the system. Assume a
tolerable overshoot of less than 5% in all cases.
- Synthesize a PI controller for the plant given by the transfer function
Gp(s)=
1
(s^2 +6s+9)
Simulate the dynamics of the plant usingMatlab. State all assumptions.
- Consider the plant model of a system as follows:
Gp(s)=
−αs+1
(s^2 +3s+2)
