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Minimal Realization

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Minimal Realization

(OP)
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?
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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

(OP)
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.

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