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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 U_{r}(s), V_{r}(s), U_{l}(s) and V_{l}(s) such that
$$\begin{array}{l}{U}_{r}{N}_{r}+{V}_{r}{M}_{r}=I\\ {N}_{l}{U}_{l}+{M}_{l}{V}_{l}=I\end{array}$$
The left/right coprime factors are stable, hence implies M_{r}(s) should contain as RHP-zeros all the RHP-poles of G(s). The comprimeness also implies that there should be no common RHP-zeros in N_{r}(s) and M_{r}(s), i.e., when forming $$G={N}_{r}(s){M}_{r}^{-1}(s)$$, there should be no pole-zero cancellations.
This table describes input arguments for ncmfr.
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, 'MaxError'overides ORDER input. |
'Display' | 'on' or 'off' | Display Hankel singular plots (default 'off'). |
'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.
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