How to decide the optimal order of reduction except the knowledge of Hankel singular Values??
This is always a subjective question. Normally you look for a significant drop in the singular values. You could divide the singular values by the sum of singular values and then work with explained variance, for instance, suppose the first 10 singular values gives you a ratio of 0.90, than they explain 90% of the variance (after scaling each by the sum of the singular values). There are connections to AIC and entropy measures. If the Hankel singular values are canonical correlations, then an AIC based measure or entropy measure can be applied, search the literature for that. I might have a reference for that, I'll try to find them.
The references are
1. W. J. Glynn and J. R. Muirhead, "Inference in canonical correlation analysis," Journal of multivariate analysis, vol. 8, pp. 468-478.
2. I. M. Gelfand and A. M. Yaglom, "Calculation of the amount of information about a random function contained in another such function." Amer. Math. Soc. Transl., vol. 2, no. 12, pp. 199-246.
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