Page 300 - Dynamics and Control of Nuclear Reactors

302 APPENDIX F State variable models and transient analysis

F.3.2 Transfer function representation of MIMO systems
Assume the initial conditions are known and are given by X (0). Taking the Laplace
transform of Eq. (F.7a) gives.
sXsðÞ x 0ðÞ ¼ AXsðÞ + BFsðÞ
Simplification yields.

ð sI AÞXs ðÞ ¼ X 0 ðÞ + BFs ðÞ (F.8)
Solve for X(s) to give.

1 1
ð
ð
XsðÞ ¼ sI AÞ x 0ðÞ + sI AÞ BFsðÞ (F.9)
Eq. (F.9) may be used to solve for X(t). Assuming x(0)¼0, we can express X(s) using
the form.
XsðÞ ¼ G fX sðÞFsðÞ (F.10)
where
1
G fX ¼ sI AÞ B (F.11)
ð
G fX is an (nxp) matrix and is the transfer function matrix between f (input variables)
and X (state variables).
To derive the transfer function between Y(s) and f(s), take the Laplace transform
of Eq. (F.7b).

YsðÞ ¼ CXsðÞ (F.12)
Substituting Eq. (F.10) in Eq. (F.12) gives.
1
YsðÞ ¼ CsI AÞ BFsðÞ (F.13)
ð
From Eq. (F.13) define the transfer function matrix between input f and output Y as
1
G fY sðÞ ¼ CsI A B (F.14)
Using Eq. (F.14) in Eq. (F.13) gives

YsðÞ ¼ G fY sðÞFsðÞ (F.15)

Remark
Note that the roots of the denominator of the transfer functions in Eq. (F.14) are the roots of the
polynomial in the Laplace variable, s, and is given by the determinant of (sI – A). These roots
are also the poles of the transfer functions. For an n-th order matrix, there are n-poles. These poles
are also the eigenvalues of system matrix, A. Thus, the system stability can be easily checked by
calculating the eigenvalues of matrix, A.