Talk:Minimal realization

Given any irreducible rational transfer function, $H(s)={\frac {b(s)}{a(s)}}$, any state model of order n

${\frac {d}{dt}}x(t)=Ax(t)+bu(t)$,

$y(t)=c\,x(t)$

that has the same input-output behaviour as the transfer function is said to be a minimal realization of the transfer function if this transfer function cannot be realized with a fewer number of states.

Properties

1. A minimal realization of H(s) is both reachable (the notion controllable is also in use), and observable. Conversely, any reachable and observable state space realization of H(s) is minimal.

2. The realization polynomials $:b(s)=c\;Adj(sI-A)b$, and $a(s)=det(sI-A)$, associated with a minimal realization $(A,b,c)$ are relatively prime (or coprime).

3. If $(A,b,c)$ and $(F,g,h)$ are two minimal realizations of the same transfer function H(s), then the realizations are related by a similarity transformation.

For discrete time systems, the definition and properties are identical.

References

• Kailath, Thomas, Linear Systems, Prentice-Hall, (1980). ISBN 0-13-536961-4. This textbook focuses on the structure of finite dimensional linear systems and treats the multi-input multi-output (or multivariable) systems in great detail, emphasizing that transfer function descriptions and state space descriptions are two extremes of a whole spectrum of possible descriptions of finite dimensional systems.

Mastlab 19:01, 10 September 2006 (UTC)

Merger proposal

I propose that Minimal realization be merged into Realization (systems). I think that the content in the former article can easily be explained in the context of the latter, and the latter article is of a reasonable size that the merging of this article will not cause any problems as far as article size is concerned. Saung Tadashi (talk) 02:54, 6 November 2018 (UTC)