balancmr
(Not recommended) Balanced model truncation via square root method
balancmr
is not recommended. Use
reducespec
instead. (since R2023b) For more information on updating your code, see Version History.
Syntax
GRED = balancmr(G) GRED = balancmr(G,order) [GRED,redinfo] = balancmr(G,key1,value1,...) [GRED,redinfo] = balancmr(G,order,key1,value1,...)
Description
balancmr
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 these values
indicate the respective state energy of the system. Hence, reduced order can be directly
determined by examining the system Hankel singular values,
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 balancmr
.
Argument  Description 

G  LTI model to be reduced. Without any other inputs,

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 positive 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 singular values
reaches the 'MaxError'
.
This table lists the input arguments 'key'
and its
'value'
.
Argument  Value  Description 

 Real number or vector of different errors  Reduce to achieve H_{∞}
error. When present,


 Optional 1by2 cell array of LTI weights $${W}_{out}^{1}\left(G{G}_{red}\right){W}_{in}^{1}.$$ You can use weighting functions to make the model reduction algorithm focus on frequency bands of interest. As an alternative, you
can use Default weights are both identity. 

 Display Hankel singular plots (default

 Integer, vector or cell array  Order of reduced model. Use only if not specified as 2nd argument. 
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 three fields:

G
can be stable or unstable, continuous or discrete.
Examples
Algorithms
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 SVD of the controllability and observability Gramians
P = U_{p} Σ_{p} V_{p}^{T}
Q = U_{q}Σ_{q} V_{q}^{T}
Find the square root of the Gramians (left/right eigenvectors)
L_{p} = U_{p} Σ_{p}^{½}
L_{o} = U_{q} Σ_{q}^{½}
Find the SVD of (L_{o}^{T}L_{p})
L_{o}^{T} L_{p} = U Σ V^{T}
Then the left and right transformation for the final k^{th} order reduced model is
S_{L,BIG} = L_{o} U(:,1:k) Σ(1;k,1:k))^{–½}
S_{R,BIG} = L_{p} V(:,1:k) Σ(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 square root balance truncation algorithm can be found in [2].
References
[1] Glover, K., “All Optimal Hankel Norm Approximation of Linear Multivariable Systems, and Their Lµerror Bounds,“ Int. J. Control, Vol. 39, No. 6, 1984, p. 11451193
[2] Safonov, M.G., and R.Y. Chiang, “A Schur Method for Balanced Model Reduction,” IEEE Trans. on Automat. Contr., Vol. 34, No. 7, July 1989, p. 729733