%% Approximate Plant Model by Multiplicative Error Method
% In most cases, the multiplicative error model reduction method bstmr tends to bound the relative error between the original and reduced-order models across the frequency range of interest, hence producing a more accurate reduced-order model than the additive error methods. This characteristic is obvious in system models with low damped poles.
% The following commands illustrate the significance of a multiplicative error model reduction method as compared to any additive error type. Clearly, the phase-matching algorithm using bstmr provides a better fit in the Bode plot.

rng(123456); 
G = rss(30,1,1);   % random 30-state model

[gr,infor] = reduce(G,'Algorithm','balance','order',7);
[gs,infos] = reduce(G,'Algorithm','bst','order',7);

figure(1)
bode(G,'b-',gr,'r--')
title('Additive Error Method')
legend('Original','Reduced')

figure(2)
bode(G,'b-',gs,'r--')
title('Relative Error Method')
legend('Original','Reduced')

% Therefore, for some systems with low damped poles or zeros, the balanced stochastic method (bstmr) produces a better reduced-order model fit in those frequency ranges to make multiplicative error small. Whereas additive error methods such as balancmr, schurmr, or hankelmr only care about minimizing the overall "absolute" peak error, they can produce a reduced-order model missing those low damped poles/zeros frequency regions.
% Copyright 2015 The MathWorks, Inc.