Balanced model truncation via Schur method
GRED = schurmr(G) GRED = schurmr(G,order) [GRED,redinfo] = schurmr(G,key1,value1,...) [GRED,redinfo] = schurmr(G,order,key1,value1,...)
schurmr
returns a reduced order model GRED
of G and a struct array redinfo containing the error bound of the reduced model and Hankel singular values of the original system.
The error bound is computed based on Hankel singular values of G
. For a stable system Hankel singular values indicate the respective state energy of the system. Hence, reduced order can be directly determined by examining the system Hankel SV's, σι.
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.
This method guarantees an error bound on the infinity norm of the additive error ∥ GGRED
∥_{∞} for wellconditioned model reduced problems [1]:
$${\Vert GGred\Vert}_{\infty}\le 2{\displaystyle \sum _{k+1}^{n}{\sigma}_{i}}$$
This table describes input arguments for schurmr
.
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) an 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.
'
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 sv's reaches the '
MaxError
'
.
Argument  Value  Description 

'MaxError'  A real number or a vector of different errors  Reduce to achieve H_{∞} error. When present, 
'Weights' 
 Optimal 1x2 cell array of LTI weights 
'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. 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. G
and GRED
can be either 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); [g1, redinfo1] = schurmr(G); % display Hankel SV plot % and prompt for order (try 15:20) [g2, redinfo2] = schurmr(G,20); [g3, redinfo3] = schurmr(G,[10:2:18]); [g4, redinfo4] = schurmr(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 [16].
Find the controllability and observability grammians P and Q.
Find the Schur decomposition for PQ in both ascending and descending order, respectively,
$$\begin{array}{l}{V}_{A}^{T}PQ{V}_{A}=\left[\begin{array}{ccc}{\lambda}_{1}& \dots & \dots \\ 0& \dots & \dots \\ 0& 0& {\lambda}_{n}\end{array}\right]\\ {V}_{D}^{T}PQ{V}_{D}=\left[\begin{array}{ccc}\lambda n& \dots & \dots \\ 0& \dots & \dots \\ 0& 0& {\lambda}_{1}\end{array}\right]\end{array}$$
Find the left/right orthonormal eigenbases of PQ associated with the k^{th} big Hankel singular values.
$${V}_{A}=[{V}_{R,SMALL},\stackrel{}{\overbrace{{V}_{L,BIG}]}}$$
Find the SVD of (V^{T}_{L,BIG} V_{R,BIG}) = U Σ V^{T}
$${V}_{D}=[\stackrel{}{\overbrace{{V}_{R,BIG}}},{V}_{L,SMALL}]$$
Form the left/right transformation for the final k^{th} order reduced model
S_{L,BIG} = V_{ L,BIG} UΣ(1:k,1:k)^{–½}
S_{R,BIG} = V_{R,BIG}VΣ(1:k,1:k)^{–½}
Finally,
$$\left[\begin{array}{cc}\widehat{A}& \widehat{B}\\ \widehat{C}& \widehat{D}\end{array}\right]=\left[\begin{array}{cc}{S}_{L,BIG}^{T}A{S}_{R,BIG}& {S}_{L,BIG}^{T}B\\ C{S}_{R,BIG}& D\end{array}\right]$$
The proof of the Schur balance truncation algorithm can be found in [2].
[1] K. Glover, “All Optimal Hankel Norm Approximation of Linear Multivariable Systems, and Their L_{∝} error Bounds,” Int. J. Control, vol. 39, no. 6, pp. 11451193, 1984.
[2] M. G. Safonov and R. Y. Chiang, “A Schur Method for Balanced Model Reduction,” IEEE Trans. on Automat. Contr., vol. 34, no. 7, July 1989, pp. 729733.