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]:
$${\Vert GGred\Vert}_{\infty}\le 2{\displaystyle \sum _{k+1}^{n}{\sigma}_{i}}$$
Note
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.
Note
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.