Simplified access to Hankel singular value based model reduction functions
GRED = reduce(G) GRED = reduce(G,order) [GRED,redinfo] = reduce(G,'key1','value1',...) [GRED,redinfo] = reduce(G,order,'key1','value1',...)
returns a reduced order model
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
G and is computed based on either additive
G-GRED ∥∞, multiplicative
or nugap error (ref.:
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
adequate to provide a good reduced order model. But for systems with
lightly damped poles and/or zeros, a multiplicative error method (namely,
that minimizes the relative error between
to produce a better fit.
This table describes input arguments for
LTI model to be reduced (without any other inputs will plot its Hankel singular values and prompt for reduced 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 anti-stable 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.
be specified in the same fashion as an alternative for
selected. In this case, reduced order will be determined when the
sum of the tails of the Hankel SV's reaches the
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.
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.
LTI reduced order model. Becomes multi-dimensional array when input is a serial of different model order array.
A STRUCT array with 3 fields:
G can be stable or unstable.
be either continuous or discrete.
A successful model reduction with a well-conditioned original
G will ensure that the reduced model
the infinity norm error bound.
Reduce Model Order
Given a continuous or discrete, stable or unstable system,
G, create a set of reduced-order models based on your selections.
rng(1234,'twister'); % For reproducibility G = rss(30,5,4);
If you call
reduce without specifying an order for the reduced model, the software displays a Hankel singular-value plot and prompts you to select an order.
If you specify a reduced-model order,
reduce defaults to the
balancmr algorithm for model reduction.
[g1,redinfo1] = reduce(G,20);
Specify other algorithms using the
Algorithm argument. Use the
ErrorType argument to specify whether the algorithm uses multiplicative or additive error, and the maximum permissible error in the reduced model.
[g2,redinfo2] = reduce(G,[10:2:18],'Algorithm','schur'); [g3,redinfo3] = reduce(G,'ErrorType','mult','MaxError',[0.01 0.05]); [g4,redinfo4] = reduce(G,'ErrorType','add','Algorithm','hankel','MaxError',[0.01]); for i = 1:4 figure(i); eval(['sigma(G,g' num2str(i) ');']); end
 K. Glover, “All Optimal Hankel Norm Approximation of Linear Multivariable Systems, and Their L∝- error Bounds,” Int. J. Control, vol. 39, no. 6, pp. 1145-1193, 1984.
 M. G. Safonov and R. Y. Chiang, “A Schur Method for Balanced Model Reduction,” IEEE Trans. on Automat. Contr., vol. AC-2, no. 7, July 1989, pp. 729-733.
 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. 496-502.
 M. G. Safonov and R. Y. Chiang, “Model Reduction for Robust Control: A Schur Relative-Error Method,” International Journal of Adaptive Control and Signal Processing, vol. 2, pp. 259-272, 1988.
 K. Zhou, “Frequency weighted L[[BULLET]] error bounds,” Syst. Contr. Lett., Vol. 21, 115-125, 1993.
Introduced before R2006a