GRED = bstmr(G) GRED = bstmr(G,order) [GRED,redinfo] = bstmr(G,key1,value1,...) [GRED,redinfo] = bstmr(G,order,key1,value1,...)
bstmr
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 phase
matrix of the original system [2].
The error bound is computed based on Hankel singular
values of the phase matrix 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 these values.
With only one input argument G
, the function
will show a Hankel singular value plot of the phase matrix of G
and
prompt for model order number to reduce.
This method guarantees an error bound on the infinity norm of
the multiplicative ∥ GRED
–1(GGRED)
∥
∞ or relative error ∥ G
^{}–1(GGRED)
∥
∞ for wellconditioned model reduction problems [1]:
$${\Vert {G}^{1}(GGred)\Vert}_{\infty}\le {\displaystyle \prod _{k+1}^{n}\left(1+2{\sigma}_{i}(\sqrt{1+{\sigma}_{i}^{2}}+{\sigma}_{i})\right)}1$$
This table describes input arguments for bstmr
.
Argument  Description 

 LTI model to be reduced (without any other inputs will plot its Hankel singular values and prompt for reduced order) 
 (Optional) an integer for the desired order of the reduced model, or a vector of 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 accumulated product
of Hankel singular values shown in the above equation reaches the 'MaxError'
.
Argument  Value  Description 

'MaxError'  Real number or vector of different errors  Reduce to achieve H_{∞} error. When
present, 
'Display' 
 Display Hankel singular plots (default 
'Order'  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. Become multidimension 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 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); G.D = zeros(5,4); [g1, redinfo1] = bstmr(G); % display Hankel SV plot % and prompt for order (try 15:20) [g2, redinfo2] = bstmr(G,20); [g3, redinfo3] = bstmr(G,[10:2:18]); [g4, redinfo4] = bstmr(G,'MaxError',[0.01, 0.05]); for i = 1:4 figure(i) eval(['sigma(G,g' num2str(i) ');']); end
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 controllability grammian P and observability grammian Q of the left spectral factor Φ = Γ(σ)Γ*(–σ) = Ω*(–σ)Ω(σ) by solving the following Lyapunov and Riccati equations
AP + PA^{T} + BB^{T} = 0
B_{W} = PC^{T} + BD^{T}
QA + A^{T} Q + (QB_{W} – C^{T}) (–DD^{T}) (QB_{W} – C^{T})^{T} = 0
Find the Schur decomposition for PQ in both ascending and descending order, respectively,
$$\begin{array}{l}{V}_{A}^{T}PQ{V}_{A}=\left[\begin{array}{ccc}{\lambda}_{1}& \cdots & \cdots \\ 0& \cdots & \cdots \\ 0& 0& {\lambda}_{n}\end{array}\right]\\ {V}_{D}^{T}PQ{V}_{D}=\left[\begin{array}{ccc}{\lambda}_{n}& \cdots & \cdots \\ 0& \cdots & \cdots \\ 0& 0& {\lambda}_{1}\end{array}\right]\end{array}$$
Find the left/right orthonormal eigenbases of PQ associated with the k^{th} big Hankel singular values of the allpass phase matrix (W^{*}(s))^{–1}G(s).
k
$$\begin{array}{l}{V}_{A}=[{V}_{R,SMALL},\stackrel{k}{\overbrace{{V}_{L,BIG}}}]\\ {V}_{D}=[\stackrel{}{\overbrace{{V}_{R,BIG}}},{V}_{L,SMALL}]\end{array}$$
Find the SVD of (V^{T }_{L,BIG}V_{R,BIG}) = U Σ ςΤ
Form the left/right transformation for the final k^{th} order reduced model
S_{L,BIG} = V_{L,BIG} U Σ(1:k,1:k)^{–½}
S_{R,BIG} = V_{R,BIG} V 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 Schur BST algorithm can be found in [1].
The BST model reduction theory requires that the original model D matrix
be full rank, for otherwise the Riccati solver fails. For any problem
with strictly proper model, you can shift the jωaxis
via bilin
such that BST/REM approximation can be
achieved up to a particular frequency range of interests. Alternatively,
you can attach a small but full rank D matrix to
the original problem but remove the D matrix of
the reduced order model afterwards. As long as the size of D matrix
is insignificant inside the control bandwidth, the reduced order model
should be fairly close to the true model. By default, the bstmr
program
will assign a full rank D matrix scaled by 0.001
of the minimum eigenvalue of the original model, if its D matrix
is not full rank to begin with. This serves the purpose for most problems
if user does not want to go through the trouble of model pretransformation.
[1] Zhou, K., “Frequencyweighted model reduction with L∞ error bounds,” Syst. Contr. Lett., Vol. 21, 115125, 1993.
[2] Safonov, M.G., and R.Y. Chiang, “Model Reduction for Robust Control: A Schur Relative Error Method,” International J. of Adaptive Control and Signal Processing, Vol. 2, p. 259272, 1988.