Content of article
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 grammiansP = U_{p} Σ_{p}V_{p}^{T}
Q = U_{q}Σ_{q}V_{q}^{T}

Find the square root of the grammians
(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 isS_{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,
The proof of the square root balance truncation algorithm can
be found in [2].
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]:
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 
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 You can use weighting functions to make the model reduction algorithm 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 
This table describes output arguments.
Argument 
Description 

GRED 
LTI reduced order model. Becomes multidimensional array 
REDINFO 
A STRUCT array with three fields:

G
can be stable or unstable, continuous or
discrete.
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