By Goro Obinata PhD, Brian D. O. Anderson PhD (auth.)
Modern equipment of clear out layout and controller layout frequently yield structures of very excessive order, posing an issue for his or her implementation. during the last twenty years or so, subtle equipment were built to accomplish simplification of filters and controllers. Such equipment frequently include easy-to-use mistakes bounds, and on the subject of controller simplification tools, such mistakes bounds will often be regarding closed-loop properties.
This ebook is the 1st finished remedy of approximation equipment for filters and controllers. it really is absolutely brand new, and it truly is authored through prime researchers who've in my opinion contributed to the advance of a few of the tools. Balanced truncation, Hankel norm aid, multiplicative relief, weighted tools and coprime factorization equipment are all discussed.
The booklet is abundantly illustrated with examples, and should equip practicing keep watch over engineers and graduates for clever use of industrial software program modules for version and controller reduction.
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Extra resources for Model Reduction for Control System Design
Example text
Let G, E RHoo have degree less than or equal to r < n, and let (Vi, Wi) for i = 1, ... , n be the Schmidt pairs of G. 16) v = Laivi. i=1 Since the Vi for i = 1, ... , r + 1 span an r + 1 dimensional space and the Hankel operator of G, has rank:::: r, we can select the ai, which are not all zero, such that (r,v)(t) = O. Using such a V, we obtain r, lI(r - r,)vlI~ '+1 12 ,+1 1=1 2 1=1 = IIrvlI~ = 11 ~aiaiWi = ~afa/ ,+1 ~ a;+1 Laf = a,2+ 1 IIvll~. 17). If a, > a,+I, then ai > a,+1 for i = 1, ... , r and we conclude that ai = 0 for i = 1, ...
Liu and Anderson (1989) discusses these issues in some detail. Approximation by finite impulse response transfer functions Finite impulse response (FIR) transfer functions are widely used because they have good numerical properties and are easy to implement. Approximating infinite impulse response (UR) transfer functions by FIR transfer functions is a kind of model reduction. ) We can apply ideas similar to some introduced for the continuous time case to this problem. For example, if we apply a Markov parameter matching technique to a discrete time UR transfer function, then the simple truncation retaining lower order coefficients is obtained as the approximation.
3) where nr and mr are the order of the denominator and numerator of Gr(s), respectively. It can be seen from this relation that the obtained model approximates the original impulse response in the low frequency range. It is also possible to expand G (s) in negative powers of s instead of (1. 1): 00 G(s) = Lh;s-;. 4) ;=0 The coefficients are called Markov parameters, and they also characterize the system response. The matching of these parameters leads to better approximation in the high frequency range.