2.10. EXERCISES AND PROJECTS 81
- Each of the following is a linear time-invariant system.
(i)
∑
x ̇ 1
x ̇ 2
∏
=
∑
10
22
∏∑
x 1
x 2
∏
+
∑
1
0
∏
uy=
£
21
§
∑
x 1
x 2
∏
(ii)
x ̇ 1
x ̇ 2
x ̇ 3
=
01 0
00 1
0 − 2 − 3
x 1
x 2
x 3
+
0
1
1
uy=
£
101
§
x 1
x 2
x 3
(iii)
x ̇ 1
x ̇ 2
x ̇ 3
=
10 0
01 1
0 − 2 − 1
x 1
x 2
x 3
+
1
0
1
uy=
£
011
§
x 1
x 2
x 3
(iv)
x ̇ 1
x ̇ 2
x ̇ 3
=
−10 0
0 − 10
0 − 2 − 2
x 1
x 2
x 3
+
0
1
1
uy=
£
110
§
x 1
x 2
x 3
(a) Explain why each system is or is not fully controllable.
(b) Explain why each system is or is not fully observable.
(c) Find the transfer function for each system.
(d) Explain why each system is or is not stable.
- For each of the open-loop transfer functions of linear time-invariant sys-
tems, specified below, we are required to obtain state-variable feedback
controllers. The corresponding desired roots (characteristic equation) are
specified for each system.
(i)G 1 (s)=
(s+2)
(s+3)(s+7)
; Desired roots:{− 5 ,− 8 }
(ii)G 2 (s)=
10
s(s+1)(s+5)
;Desiredroots:{− 0. 708 ±j 0. 706 ,− 50 }
(iii)G 3 (s)=
(s^2 +2)
(s+3)(s+4)(s+5)
; Desired characteristic equation:
s^3 +8s^2 +22s+24=0
(iv)G 4 (s)=
(s^2 +s+1)
(s^3 +8s^2 +22s+ 24)
; Desired characteristic equation:
s^3 +15s^2 +71s+ 105 = 0
(a) For each system obtain the step input open-loop response usingMat-
lab.
(b) For each system with state-variable feedback, obtain the step input
response usingMatlab.
- Explain whyC(sI−A)
− 1
B+E=
yà(s)
uà(s)
as claimed in Equations 2.39
and 2.40.
