Page 299 - Dynamics and Control of Nuclear Reactors

APPENDIX F State variable models and transient analysis 301

where
x¼solution vector (state variables).
A¼system matrix of coefficients.
f ¼ vector of forcing functions.
g ¼ vector of nonlinear terms in the state variables.

F.3 General solution of the multiple-input multiple-output
(MISO) linear
State variable model
The application of the Laplace transform method for representing the system
transfer function and for obtaining the time-domain response to an external pertur-
bation can be easily extended to multivariate system [3]. The following topics are
discussed in this section:

(i) General representation of time-invariant systems in the state-space form.
(ii) Transfer function representation of multiple-input multiple output (MIMO)
systems.
(iii) Transient response of MIMO systems.
(iv) The state transition matrix.

F.3.1 Definition of multiple-input multiple output (MIMO) systems
We define a multivariate linear system in the following form.
dx
¼ Ax + Bf (F.7a)
dt

y ¼ Cx (F.7b)

x ¼ x 1 , x 2 , …, x n g is an n-dimensional state vector.
f

f ¼ f 1, f 2 , …, f p is an p-dimensional input vector.
y ¼ y 1 , y 2 , …, y m g is an m-dimensional output or measurement vector.
f
A is an (n x n) time-independent (or time-invariant) system matrix.
B is an (n x p) time-independent input matrix.
C is an (m x n) time-independent output matrix.
If C¼I, the identity matrix, then y ¼ x.
If p¼1, then matrix B becomes an (n x 1) vector and f is a scalar input. In general,
the number of measurements (m) is less than the number of state variables (n). In
actual systems, it is often not possible to measure all the state variables in the model.
If y is a scalar measurement, then matrix C is a row vector (1 x n).