Balanced model truncation for normalized coprime factors
GRED = ncfmr(G) GRED = ncfmr(G,order) [GRED,redinfo] = ncfmr(G,key1,value1,...) [GRED,redinfo] = ncfmr(G,order,key1,value1,...)
ncfmr returns a reduced order model GRED
formed by a set of balanced normalized coprime factors and a struct array redinfo
containing the left and right coprime factors of G and their coprime Hankel singular
values.
Hankel singular values of coprime factors of such a stable system indicate the respective “state energy” of the system. Hence, reduced order can be directly determined by examining the system Hankel SV's.
With only one input argument G, the function will show a Hankel
singular value plot of the original model and prompt for model order number to
reduce.
The left and right normalized coprime factors are defined as [1]
where there exist stable Ur(s), Vr(s), Ul(s) and Vl(s) such that
The left/right coprime factors are stable, hence implies Mr(s) should contain as RHP-zeros all the RHP-poles of G(s). The coprimeness also implies that there should be no common RHP-zeros in Nr(s) and Mr(s), i.e., when forming , there should be no pole-zero cancellations.
This table describes input arguments for ncmfr.
Argument | Description |
|---|---|
G | LTI model to be reduced (without any other inputs will plot its Hankel singular values and prompt for reduced order) |
ORDER | (Optional) Integer for the desired order of the reduced model, or optionally a vector packed with desired orders for batch runs |
A batch run of a serial of different reduced order models can be generated by
specifying order = x:y, or a vector of integers. By default, all the
anti-stable part of a system is kept, because from control stability point of view,
getting rid of unstable state(s) is dangerous to model a system. The
ncfmr method allows
the original model to have jω-axis singularities.
'MaxError' can be
specified in the same fashion as an alternative for
'ORDER'.
In this case, reduced order will be determined when the sum of the tails of the Hankel
singular values reaches the
'MaxError'.
Argument | Value | Description |
|---|---|---|
'MaxError' | A real number or a vector of different errors | Reduce to achieve H∞ error. When present,
|
'Display' |
| Display Hankel singular plots (default
|
'Order' | integer, vector or cell array | Order of reduced model. Use only if not specified as 2nd argument. |
Weights on the original model input and/or output can make the model reduction algorithm focus on some frequency range of interests. But weights have to be stable, minimum phase, and invertible.
This table describes output arguments.
Argument | Description |
|---|---|
GRED | LTI reduced order model, that becomes multi-dimensional array when input is a serial of different model order array. |
REDINFO | A STRUCT array with 3 fields:
|
G can be stable or unstable, continuous or discrete.
Given a continuous or discrete, stable or unstable system, G, the following commands can get a set of reduced order models based on your selections:
rng(1234,'twister');
G = rss(30,5,4);
G.D = zeros(5,4);
[g1, redinfo1] = ncfmr(G); % display Hankel SV plot
% and prompt for order (try 15:20)
[g2, redinfo2] = ncfmr(G,20);
[g3, redinfo3] = ncfmr(G,[10:2:18]);
[g4, redinfo4] = ncfmr(G,'MaxError',[0.01, 0.05]);
for i = 1:4
figure(i)
eval(['sigma(G,g' num2str(i) ');']);
end
Given a state space (A,B,C,D) of a system and k, the desired reduced order, the following steps will produce a similarity transformation to truncate the original state-space system to the kth order reduced model.
where
Nl(:= Ac, Bn, Cc, Dn)
Ml := (Ac, Bm, Cc, Dm)
Cl = (Dm)–1Cc
Dl = (Dm)–1Dn
[1] M. Vidyasagar. Control System Synthesis - A Factorization Approach. London: The MIT Press, 1985.
[2] M. G. Safonov and R. Y. Chiang, “A Schur Method for Balanced Model Reduction,” IEEE Trans. on Automat. Contr., vol. AC-2, no. 7, July 1989, pp. 729-733.