Page 301 - Dynamics and Control of Nuclear Reactors
P. 301
APPENDIX F State variable models and transient analysis 303
Example F.1
Consider a system with two inputs (f) and two outputs (y). Then Eq. (F.15) may be written as
y 1 s ðÞ G 11 G 12 f 1 s ðÞ
¼ (F.16)
y 2 sðÞ G 21 G 22 f 2 sðÞ
Thus, the transfer function between output y i and input f j is given by the matrix element.
y i s ðÞ
¼ G ij sðÞ; i ¼ 1,2; j ¼ 1,2 (F.17)
f j s ðÞ
F.3.3 Transient response of MIMO systems
The response x(t) for a given initial condition x(0) and forcing term f(t) is determined
by taking the inverse Laplace transform of Eq. (F.9). Thus,
h i h i
1
1
xtðÞ ¼ L 1 ð sI AÞ x 0ðÞ + L 1 ð sI AÞ BFsðÞ (F.18)
h i
Define ϕ tðÞ L 1 ð sI AÞ 1 (F.19)
The inverse transform in Eq. (F.18) can be accomplished by using the convolution
integral (see Appendix D). If Y(s) ¼G(s)X(s), the inverse transform is achieved by
the following convolution integral.
ð ∞
ytðÞ ¼ L 1 ½ GsðÞXsðÞ ¼ gt τÞx τðÞ dτ
ð
0
Using the convolution integral and the definition (F.19), x(t) becomes.
ð t
xtðÞ ¼ φ tðÞx 0ðÞ + φ t τÞBf τðÞdτ (F.20)
ð
0
(The convolution between two time functions p(t) and q(t) is defined by the following
integral: t
Ð
ð
ptðÞ∗qtðÞ ¼ pt τÞ q τðÞ dτ, p(t) and q(t) are causal functions).
0
Eq. (F.20) is similar the solution of a linear first order system with initial condi-
tion. Furthermore, the solution of a first order linear time invariant system.
dx
¼ ax + ftðÞ (F.21)
dt
is given by
ð t
at
ð
xtðÞ ¼ e x 0ðÞ + e at τÞ f τðÞdτ (F.22)
0