Hankel minimum degree approximation (MDA) without balancing
GRED = hankelmr(G) GRED = hankelmr(G,order) [GRED,redinfo] = hankelmr(G,key1,value1,...) [GRED,redinfo] = hankelmr(G,order,key1,value1,...)
hankelmr
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 Hankel singular
values indicate the respective state energy of the system. Hence,
reduced order can be directly determined by examining the system Hankel
SV's, 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]:
$${\Vert GGred\Vert}_{\infty}\le 2{\displaystyle \sum _{k+1}^{n}{\sigma}_{i}}$$
It seems this method is similar to the additive model reduction
routines balancmr
and schurmr
,
but actually it can produce more reliable reduced order model when
the desired reduced model has nearly controllable and/or observable
states (has Hankel singular values close to machine accuracy). hankelmr
will
then select an optimal reduced system to satisfy the error bound criterion
regardless the order one might naively select at the beginning.
This table describes input arguments for hankelmr
.
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) an 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 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 sv's reaches the 'MaxError
'.
Argument  Value  Description 

'MaxError'  Real number or vector of different errors  Reduce to achieve H_{∞} error. When
present, 
'Weights' 
 Optimal 1x2 cell array of LTI weights 
'Display' 
 Display Hankel singular plots (default 
'Order'  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. Become multidimensional array when input is a serial of different model order array. 
REDINFO  A STRUCT array with 4 fields:

G
can be stable or unstable, continuous or
discrete.
If size(GRED)
is not equal to the order you
specified. The optimal Hankel MDA algorithm has selected the best
Minimum Degree Approximate it can find within the allowable machine
accuracy.
Given a continuous or discrete, stable or unstable system, G
,
the following commands can get a set of reduced order models based
on your selections:
rng(1234,'twister'); G = rss(30,5,4); [g1, redinfo1] = hankelmr(G); % display Hankel SV plot % and prompt for order (try 15:20) [g2, redinfo2] = hankelmr(G,20); [g3, redinfo3] = hankelmr(G,[10:2:18]); [g4, redinfo4] = hankelmr(G,'MaxError',[0.01, 0.05]); for i = 1:4 figure(i); eval(['sigma(G,g' num2str(i) ');']); end
Singular Value Bode Plot of G (30state, 5 outputs, 4 inputs) shows
a singular value Bode plot of a random system G
with
20 states, 5 output and 4 inputs. The error system between G
and
its Zeroth order Hankel MDA has it infinity norm
equals to an all pass function, as shown in AllPass Error System Between G and Zeroth Order G Anticausal.
The Zeroth order Hankel MDA and its error system sigma plot are obtained via commands
[g0,redinfo0] = hankelmr(G,0); sigma(Gredinfo0.Ganticausal)
This interesting allpass property is unique in Hankel MDA model reduction.
Singular Value Bode Plot of G (30state, 5 outputs, 4 inputs)
AllPass Error System Between G and Zeroth Order G Anticausal
Given a statespace (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 controllability and observability grammians P and Q.
Form the descriptor
$$E=QP{\rho}^{2}I$$
where $${\sigma}_{k}>\rho \ge {\sigma}_{k+1}$$, and descriptor statespace
Take SVD of descriptor E and partition the result into k^{th} order truncation form
$$\begin{array}{l}\left[\begin{array}{cc}Es\overline{A}& \overline{B}\\ \overline{C}& \overline{D}\end{array}\right]=\left[\begin{array}{cc}{\rho}^{2}{A}^{T}+QAP& QB\\ CP& D\end{array}\right]\\ E=\left[{U}_{E1},{U}_{E2}\right]\left[\begin{array}{cc}{\Sigma}_{E}0& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}{V}_{E1}^{T}\\ {V}_{E2}^{T}\end{array}\right]\end{array}$$
Apply the transformation to the descriptor statespace system above we have
$$\begin{array}{l}\left[\begin{array}{cc}{A}_{11}& {A}_{12}\\ {A}_{21}& {A}_{22}\end{array}\right]=\left[\begin{array}{c}{U}_{E1}^{T}\\ {U}_{E2}^{T}\end{array}\right]({\rho}^{2}{A}^{T}+QAP)\left[\begin{array}{cc}{V}_{E1}& {V}_{E2}\end{array}\right]\\ \left[\begin{array}{c}{B}_{1}\\ {B}_{2}\end{array}\right]=\left[\begin{array}{c}{U}_{E1}^{T}\\ {U}_{E2}^{T}\end{array}\right]\left[\begin{array}{cc}QB& {C}^{T}\end{array}\right]\\ \left[\begin{array}{cc}{C}_{1}& {C}_{2}\end{array}\right]=\left[\begin{array}{c}CP\\ \rho {B}^{T}\end{array}\right]\left[\begin{array}{cc}{V}_{E1}& {V}_{E2}\end{array}\right]\\ {D}_{1}=D\end{array}$$
Form the equivalent statespace model.
$$\left[\begin{array}{cc}\tilde{A}& \tilde{B}\\ \tilde{C}& \tilde{D}\end{array}\right]=\left[\begin{array}{cc}{\sum}_{E}^{1}({A}_{11}{A}_{12}{A}_{22}{}^{\u2020}{A}_{21})& {\sum}_{E}^{1}({B}_{1}{A}_{12}{A}_{22}{}^{\u2020}{B}_{2})\\ {C}_{1}{C}_{2}{A}_{22}{}^{\u2020}{A}_{21}& {D}_{1}{C}_{2}{A}_{22}{}^{\u2020}{B}_{2}\end{array}\right]$$
The final k^{th} order
Hankel MDA is the stable part of the above statespace realization.
Its anticausal part is stored in redinfo.Ganticausal
.
The proof of the Hankel MDA algorithm can be found in [2]. The error system between the original system G and the Zeroth Order Hankel MDA G_{0} is an allpass function [1].
[1] Glover, K., “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] Safonov, M.G., 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.