Page 302 - Dynamics and Control of Nuclear Reactors

304 APPENDIX F State variable models and transient analysis

Therefore, for a MIMO system described by Eq. (F.7a), the solution X(t) is
written as.

ð t
xtðÞ ¼ exp AtðÞx 0ðÞ + exp At τފBf τðÞdτ (F.23)
½
ð
0
For the linear time-invariant system, the matrix φ(t) in Eq. (F.20) is the same as the
matrix function exp. (At). Thus, for this time-invariant linear system.

φ tðÞ ¼ exp AtðÞ (F.24)

Example F.2
A linear system is described by
dx 2 3 x 1
0
¼ + f
dt 1 4 x 2 1
y ¼ x 1 + x 2 :

(a) Determine the transfer function Y(s)/F(s).
(b) Determine the response y(t) for a unit step f(t). Assume zero initial conditions.

Solution

1
XsðÞ ¼ sI AÞ bF sðÞ
ð

s +2 3

j sI Aj ¼ ¼ s +2Þ s +4Þ 3 ¼ s +1Þ s +5Þ
ð
ð
ð
ð
1 s +4

1 s +4 3
1
ð sI AÞ ¼
ð s +1Þ s +5Þ 1 s +2
ð
½
YsðÞ ¼ 11ŠXsðÞ

1 0
½
¼ 11Š sI AÞ FsðÞ
ð
1
YsðÞ ð s 1Þ
¼
FsðÞ ð s +1Þ s +5Þ
ð
(a) For a unit step input, F(s)¼1/s s 1
YsðÞ ¼
ð
ð
ss +1Þ s +5Þ