In systems theory, a realization of a state space model is an implementation of a given input-output behavior. That is, given an input-output relationship, a realization is a quadruple of (time-varying) matrices [ A ( t ) , B ( t ) , C ( t ) , D ( t ) ] {\displaystyle [A(t),B(t),C(t),D(t)]} such that
x ˙ ( t ) = A ( t ) x ( t ) + B ( t ) u ( t ) {\displaystyle {\dot {\mathbf {x} }}(t)=A(t)\mathbf {x} (t)+B(t)\mathbf {u} (t)}
y ( t ) = C ( t ) x ( t ) + D ( t ) u ( t ) {\displaystyle \mathbf {y} (t)=C(t)\mathbf {x} (t)+D(t)\mathbf {u} (t)}
with ( u ( t ) , y ( t ) ) {\displaystyle (u(t),y(t))} describing the input and output of the system at time t {\displaystyle t} .
LTI System For a linear time-invariant system specified by a transfer matrix, H ( s ) {\displaystyle H(s)} , a realization is any quadruple of matrices ( A , B , C , D ) {\displaystyle (A,B,C,D)} such that H ( s ) = C ( s I − A ) − 1 B + D {\displaystyle H(s)=C(sI-A)^{-1}B+D} .
Canonical realizations Any given transfer function which is strictly proper can easily be transferred into state-space by the following approach (this example is for a 4-dimensional, single-input, single-output system)): Given a transfer function, expand it to reveal all coefficients in both the numerator and denominator. This should result in the following form:
H ( s ) = n 3 s 3 + n 2 s 2 + n 1 s + n 0 s 4 + d 3 s 3 + d 2 s 2 + d 1 s + d 0 {\displaystyle H(s)={\frac {n_{3}s^{3}+n_{2}s^{2}+n_{1}s+n_{0}}{s^{4}+d_{3}s^{3}+d_{2}s^{2}+d_{1}s+d_{0}}}} . The coefficients can now be inserted directly into the state-space model by the following approach:
… excerpt ends here. Continue reading the full article.
