Page 303 - Dynamics and Control of Nuclear Reactors

APPENDIX F State variable models and transient analysis 305

Using partial fractions (or the method of residues) gives.
1 1 3 1
YsðÞ ¼ +
5s 2 s +1Þ 10 2+ 5Þ
ð
ð
1 t
1
3
Time response ytðÞ ¼ + e e 5t
5 2 10
F.3.4 The state transition matrix
The matrix φ(t) in Eq. (F.20) is often called the State Transition Matrix (STM)
because it provides the solution x(t) at time t from an initial time t¼0 (or t¼t 0 ).
In general Eq. (F.20) may be written in the form.
ð t
xtðÞ ¼ φ t, t 0 Þxt 0 + φ t, τÞBf τðÞdτ (F.25)
ðÞ
ð
ð
t 0
We showed that for the special case of time-invariant system.
h 1 i
φ tðÞ ¼ L 1 ð sI AÞ (F.26a)
and
φ tðÞ ¼ exp AtðÞ (F.26b)
For time-invariant systems the state transition matrix φ(t) satisfies the following
properties [3].
dφ
¼ Aφ tðÞ (F.27)
dt
φ t 0 , t 0 Þ ¼ I,φ 0 ðÞ ¼ I (F.28)
ð
ð
ð
φ t, t 0 Þ ¼ φ t t 0 Þ (F.29)
These properties satisfy the solution.
ð t
xtðÞ ¼ φ tðÞx 0ðÞ + φ t τÞBf τðÞdτ (F.30)
ð
0
Differentiate Eq. (F.30) with respect to t.
ð t
dx dφ tðÞ
ð
¼ x 0ðÞ + φ 0ðÞBftðÞ + Aφ t τÞBf τðÞdτ
dt dt
0
Using Eqs. (F.27) and (F.28) in the above gives.
ð t
dx
¼ Aφ tðÞx 0ðÞ + BftðÞ + A φ t τð ÞBf τðÞdτ
dt
0
ð t
2 3
¼ A φ tðÞx 0ðÞ + φ t τð ÞBf τðÞdτ + BftðÞ
4
5
0