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 RHPzeros all the RHPpoles of G(s). The coprimeness also implies that there should be no common RHPzeros in N_{r}(s) and M_{r}(s), i.e., when forming $$G={N}_{r}(s){M}_{r}^{1}(s)$$, there should be no polezero 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
antistable 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 jωaxis 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 multidimensional 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 statespace system to the k^{th} order reduced model.
Find the normalized coprime factors of G by solving Hamiltonian described in [1].
$$\begin{array}{l}{G}_{l}=\left[\begin{array}{cc}{N}_{l}& {M}_{l}\end{array}\right]\\ {G}_{r}=\left[\begin{array}{c}{N}_{r}\\ {M}_{r}\end{array}\right]\end{array}$$
Perform k^{th} order square root balanced model truncation on G_{l} (or G_{r}) [2].
The reduced model GRED
is:
$$\left[\begin{array}{cc}\widehat{A}& \widehat{B}\\ \widehat{C}& \widehat{D}\end{array}\right]=\left[\begin{array}{cc}{A}_{c}{B}_{m}{C}_{l}& {B}_{n}{B}_{m}{D}_{l}\\ {C}_{l}& {D}_{l}\end{array}\right]$$
where
N_{l}(:= A_{c}, B_{n}, C_{c}, D_{n})
M_{l} := (A_{c}, B_{m}, C_{c}, D_{m})
C_{l} = (D_{m})^{–1}C_{c}
D_{l} = (D_{m})^{–1}D_{n}
[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. AC2, no. 7, July 1989, pp. 729733.