Rahul Sharma (Editor)

Realization (systems)

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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 ) ] such that

Contents

x ˙ ( t ) = A ( t ) x ( t ) + B ( t ) u ( t ) y ( t ) = C ( t ) x ( t ) + D ( t ) u ( t )

with ( u ( t ) , y ( t ) ) describing the input and output of the system at time t .

LTI System

For a linear time-invariant system specified by a transfer matrix, H ( s ) , a realization is any quadruple of matrices ( A , B , C , D ) such that H ( s ) = C ( s I − 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 1 s 3 + n 2 s 2 + n 3 s + n 4 s 4 + d 1 s 3 + d 2 s 2 + d 3 s + d 4 .

The coefficients can now be inserted directly into the state-space model by the following approach:

x ˙ ( t ) = [ − d 1 − d 2 − d 3 − d 4 1 0 0 0 0 1 0 0 0 0 1 0 ] x ( t ) + [ 1 0 0 0 ] u ( t ) y ( t ) = [ n 1 n 2 n 3 n 4 ] x ( t ) .

This state-space realization is called controllable canonical form (also known as phase variable canonical form) because the resulting model is guaranteed to be controllable (i.e., because the control enters a chain of integrators, it has the ability to move every state).

The transfer function coefficients can also be used to construct another type of canonical form

x ˙ ( t ) = [ − d 1 1 0 0 − d 2 0 1 0 − d 3 0 0 1 − d 4 0 0 0 ] x ( t ) + [ n 1 n 2 n 3 n 4 ] u ( t ) y ( t ) = [ 1 0 0 0 ] x ( t ) .

This state-space realization is called observable canonical form because the resulting model is guaranteed to be observable (i.e., because the output exits from a chain of integrators, every state has an effect on the output).

D = 0

If we have an input u ( t ) , an output y ( t ) , and a weighting pattern T ( t , σ ) then a realization is any triple of matrices [ A ( t ) , B ( t ) , C ( t ) ] such that T ( t , σ ) = C ( t ) ϕ ( t , σ ) B ( σ ) where ϕ is the state-transition matrix associated with the realization.

System identification

System identification techniques take the experimental data from a system and output a realization. Such techniques can utilize both input and output data (e.g. eigensystem realization algorithm) or can only include the output data (e.g. frequency domain decomposition). Typically an input-output technique would be more accurate, but the input data is not always available.

References

Realization (systems) Wikipedia
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