Page 314 - Dynamics and Control of Nuclear Reactors

316 APPENDIX F State variable models and transient analysis

F.3 Determine the transfer function vector for a system with an additional
delay terms.
dx
¼ Ax + d + bf
dt

ð
x 1 12 x 1 t 1ð Þ +2x 2 t 3Þ 1
x ¼ A ¼ d ¼ b ¼
x 2 3 4 4x 1 t 2ð Þ 3
F.4 Consider the second order system with two inputs.
dx 1
¼ x 1 +2x 2 + f 1 +2f 2
dt
dx 2
¼ 3x 1 4x 2 + f 2
dt
(a) Compute the transfer function matrix

G 11 sðÞ G 12 sðÞ
GsðÞ ¼
G 21 sðÞ G 22 sðÞ
where
G ij sðÞ ¼ X i sðÞ=f j sðÞ

(b) Compute the response of x1 and x2 due to an impulse in f1 for the above sys-
tem. Assume that f2¼0.

F.5 Verify Eq. (F.37).

At
F.6 Calculate e for the following matrix. Truncate the solution after a few terms.

12
A ¼
3 4
F.7 What is the maximum number of sensitivities to matrix elements for a
dynamic system with a (n x n) coefficient matrix? Explain.

F.8 Consider the following first-order differential equation.
dx
¼ 3x +5
dt

a. Calculate the solution at t¼1witha Δt¼0.5 using the Euler method and the
Runge-Kutta2 method.
b. Repeat the calculations for a Δt¼0.2.
c. Discuss your results.

F.9 Calculate the eigenvalues of the matrix used in Exercise F.6.