Balanced model truncation via square root method
GRED = balancmr(G) GRED = balancmr(G,order) [GRED,redinfo] = balancmr(G,key1,value1,...) [GRED,redinfo] = balancmr(G,order,key1,value1,...)
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 $${W}_{out}^{1}\left(G{G}_{red}\right){W}_{in}^{1}.$$ 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.
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 grammians
P = 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 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].
[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