Contact US

Log In

Come Join Us!

Are you an
Engineering professional?
Join Eng-Tips Forums!
  • Talk With Other Members
  • Be Notified Of Responses
    To Your Posts
  • Keyword Search
  • One-Click Access To Your
    Favorite Forums
  • Automated Signatures
    On Your Posts
  • Best Of All, It's Free!

*Eng-Tips's functionality depends on members receiving e-mail. By joining you are opting in to receive e-mail.

Posting Guidelines

Promoting, selling, recruiting, coursework and thesis posting is forbidden.

Students Click Here

Minimal Realization

Minimal Realization

Minimal Realization

One of the ways to obtain a minimal realization of a system model transfer function is to create a balanced system first from its corresponding state space equation, and eliminate states with small eigenvalues. To go back to a transfer function, it seems like we'd have to know what the new states are. How do states map from the original (higher-order) model to the reduced balanced system? For example, say a state space model's full state vector is [x1 x2 ... xn] before reducing, and [x1'...xm'] (m < n) after reducing. How do x1 = x1' relate?

RE: Minimal Realization

I don't think there'd be any definite relationship - all the states would adjust as necessary to accommodate the space. Is it usual to think of a polynomial fit analogy: suppose you have a signal y(t) that varied with time. Some knowledge of the physics of the signal lead you to believe it can be modelled with a 6th degree polynomial, and via least squares reduction you arrive at a model: y(x) = ax^6 + bx^5 + cx^4 + dx^3 + ex^2 + fx + e. The vector [a, b, c, d, e, f, e] adequately represents your system.

But then you decide that there's very little 4th order behaviour, so you re-model as y(x) = a'x^6 + b'x^5 + d'x^3 + e'x^2 + f'x + e'. You get a fit and now your system is represented by [a', b', d', e', f', e']. Depending on how much c really contributed to model, a' through e' will adjust to accommodate in an unpredictable way.

RE: Minimal Realization

That's what I was thinking for a SISO transfer function, however, what if it's a state space system that is being balanced and reduced? Say the state vector is [position, acceleration, another position, another acceleration].
Say you first you balance: mathworks.com/help/control/ref/balreal.html
Then you create the minimal realization: mathworks.com/help/control/ref/modred.html
How to know which states were eliminated in the reduction? Was it position 2, or ...?
The reduction happened in a balanced state which is a mix of various states, so after reducing, presumably there is also a way to transform back to the original state space form...and thus we'd have to know which of the positions or accelerations were kept.

RE: Minimal Realization

Ah I see. Good question. That's definitely something I don't know off the top of my head and would require some study of the balancing algorithm. I'm intrigued, because as you say the reverse transform seems achievable, but not intrigued enough to warrant the time to study it. Please let us know if you discover a solution.

Red Flag This Post

Please let us know here why this post is inappropriate. Reasons such as off-topic, duplicates, flames, illegal, vulgar, or students posting their homework.

Red Flag Submitted

Thank you for helping keep Eng-Tips Forums free from inappropriate posts.
The Eng-Tips staff will check this out and take appropriate action.

Reply To This Thread

Posting in the Eng-Tips forums is a member-only feature.

Click Here to join Eng-Tips and talk with other members! Already a Member? Login


Close Box

Join Eng-Tips® Today!

Join your peers on the Internet's largest technical engineering professional community.
It's easy to join and it's free.

Here's Why Members Love Eng-Tips Forums:

Register now while it's still free!

Already a member? Close this window and log in.

Join Us             Close