Page 306 - Dynamics and Control of Nuclear Reactors

308 APPENDIX F State variable models and transient analysis

Using the linear system response given in Eq. (F.23), and iterative solution of X(t)
at time step {i} is given in terms of its value at time step {i – 1} by

xiðÞ ¼ e AΔt xi 1Þ +A 1 e AΔt I fiðÞ (F.34)
ð
where
Δt¼time step between solution evaluations.
I¼the identity matrix (value of 1 on the diagonal sand zeroes elsewhere).
The matrix exponential is defined as
3
2
e AΔt ¼ I+AΔt+ AΔtÞ =2! +AΔtÞ =3! + (F.35)
ð
ð
Note the similarity to the scalar exponential definition. Also note that it is necessary
to evaluate e AΔt only once for use in every calculation of X(i). For simplicity, define
C 1 ¼ e AΔt (F.36)
Now consider the other term in Eq. (F.34).
h i
3
2

A 1 e AΔt I ¼ A 1 I+AΔt+ AΔtÞ =2! +AΔtÞ =3! + … I
ð
ð
Simplifying gives
h 2 i

A 1 e AΔt I ¼ Δt I+ AΔtÞ=2! +AΔtÞ =3! + … (F.37)
ð
ð
It is also necessary to evaluate this quantity only once and apply it at every time step.
Define this constant matrix as
C 2 ¼ A 1 e AΔt I (F.38)
We now may write the recursive solution as
ð
xiðÞ ¼ C 1 xi 1Þ +C 2 fiðÞ (F.39)
Thus, the solution involves repeated matrix multiplications using the same values of
C 1 and C 2 at each time step. The matrix exponential method is exact for linear homo-
geneous models (initial value problems) or inhomogeneous models with a constant
forcing vector. The matrix exponential method can provide approximate solutions by
treating variable forcing vectors or nonlinear terms as piecewise constant terms. The
accuracy of these solutions increases as time steps get smaller.
Other numerical solution methods are considered in Sections F.6 and F.7.

F.5 Sensitivity analysis
Sensitivity analysis is sometimes used to determine the importance of specified sys-
tem parameters in determining the system response. Since sensitivity analysis is most
easily performed with the matrix exponential approach, it is considered here before
addressing other numerical methods later in this appendix [5].