reduce
(Not recommended) Simplified access to Hankel singular value based model reduction functions
reduce
is not recommended. Use
reducespec
instead. (since R2023b) For more information on updating your
code, see Version History.
Syntax
GRED = reduce(G) GRED = reduce(G,order) [GRED,redinfo] = reduce(G,'key1','value1',...) [GRED,redinfo] = reduce(G,order,'key1','value1',...)
Description
reduce
returns a reduced order model
GRED
of G
and a struct array
redinfo
containing the error bound of the reduced model, Hankel
singular values of the original system and some other relevant model reduction
information.
An error bound is a measure of how close GRED
is to
G
and is computed based on either additive
error, ∥ GGRED
∥_{∞},
multiplicative error,
∥G
^{–1}(GGRED)
∥_{∞}, or nugap error (ref.:
ncfmr
) [1],[4],[5].
Hankel singular values of 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. Model reduction routines, which based on Hankel singular values are grouped by
their error bound types. In many cases, the additive error method
GRED=reduce(G,ORDER)
is adequate to provide a good reduced order
model. But for systems with lightly damped poles and/or zeros, a multiplicative error
method (namely, GRED=reduce(G,ORDER,'ErrorType','mult')
) that
minimizes the relative error between G
and GRED
tends to produce a better fit.
This table describes input arguments for reduce
.
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 physical 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
'
after an
'
ErrorType
'
is
selected. 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 


 Default for Option for
Option for Default for
Default for 

 Additive error (default) Multiplicative error at model output NCF nugap error 
 A real number or a vector of different errors  Reduce to achieve H_{∞} error. When present,


 Optimal 1x2 cell array of LTI weights 

 Display Hankel singular plots (default

 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.
A successful model reduction with a wellconditioned original model
G
will ensure that the reduced model GRED
satisfies the infinity norm error bound.
Examples
References
[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. AC2, no. 7, July 1989, pp. 729733.
[3] M. G. Safonov, R. Y. Chiang and D. J. N. Limebeer, “Optimal Hankel Model Reduction for Nonminimal Systems,” IEEE Trans. on Automat. Contr., vol. 35, No. 4, April, 1990, pp. 496502.
[4] M. G. Safonov and R. Y. Chiang, “Model Reduction for Robust Control: A Schur RelativeError Method,” International Journal of Adaptive Control and Signal Processing, vol. 2, pp. 259272, 1988.
[5] K. Zhou, “Frequency weighted L[[BULLET]] error bounds,” Syst. Contr. Lett., Vol. 21, 115125, 1993.