schurmr

Balanced model truncation via Schur method

Syntax

GRED = schurmr(G)
GRED = schurmr(G,order)
[GRED,redinfo] = schurmr(G,key1,value1,...)
[GRED,redinfo] = schurmr(G,order,key1,value1,...)

Description

schurmr 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 ∥ G-GRED ∥_∞ for well-conditioned model reduced problems [1]:

${‖ G - G r e d ‖}_{\infty} \leq 2 \sum_{k + 1}^{n} σ_{i}$

This table describes input arguments for schurmr.

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 anti-stable 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'`	A real number or a vector of different errors	Reduce to achieve H_∞ error. When present, `'``MaxError``'` overrides `ORDER` input.
`'Weights'`	`{Wout,Win}` cell array	Optimal 1x2 cell array of LTI weights `Wout` (output) and `Win` (input); default is both identity; Weights must be invertible.
`'Display'`	`'``on``'` or `'``off``'`	Display Hankel singular plots (default `'off'`).
`'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. Becomes multi-dimensional array when input is a serial of different model order array.
`REDINFO`	A STRUCT array with 3 fields: `REDINFO.ErrorBound` `REDINFO.StabSV` `REDINFO.UnstabSV`

G can be stable or unstable. G and GRED can be either continuous or discrete.

Examples

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] = schurmr(G); % display Hankel SV plot
                             % and prompt for order (try 15:20)
[g2, redinfo2] = schurmr(G,20); 
[g3, redinfo3] = schurmr(G,[10:2:18]);
[g4, redinfo4] = schurmr(G,'MaxError',[0.01, 0.05]);
for i = 1:4
     figure(i); eval(['sigma(G,g' num2str(i) ');']);
end

Algorithms

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 state-space system to the k^th order reduced model [16].

Find the controllability and observability Gramians P and Q.
Find the Schur decomposition for PQ in both ascending and descending order, respectively,
$\begin{array}{l} V_{A}^{T} P Q V_{A} = [\begin{matrix} λ_{1} & \dots & \dots \\ 0 & \dots & \dots \\ 0 & 0 & λ_{n} \end{matrix}] \\ V_{D}^{T} P Q V_{D} = [\begin{matrix} λ n & \dots & \dots \\ 0 & \dots & \dots \\ 0 & 0 & λ_{1} \end{matrix}] \end{array}$
Find the left/right orthonormal eigen-bases of PQ associated with the k^th big Hankel singular values.
$V_{A} = [V_{R, S M A L L}, \overset{}{\overset{︷}{V_{L, B I G}]}}$
Find the SVD of (V^T_L,BIG V_R,BIG) = U Σ V^T
$V_{D} = [\overset{}{\overset{︷}{V_{R, B I G}}}, V_{L, S M A L L}]$
Form the left/right transformation for the final k^th order reduced model
S_L,BIG = V_L,BIG UΣ(1:k,1:k)^–½
S_R,BIG = V_R,BIGVΣ(1:k,1:k)^–½
Finally,
$[\begin{matrix} \hat{A} & \hat{B} \\ \hat{C} & \hat{D} \end{matrix}] = [\begin{matrix} S_{L, B I G}^{T} A S_{R, B I G} & S_{L, B I G}^{T} B \\ C S_{R, B I G} & D \end{matrix}]$

The proof of the Schur balance truncation algorithm can be found in [2].

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. 1145-1193, 1984.

[2] M. G. Safonov and R. Y. Chiang, “A Schur Method for Balanced Model Reduction,” IEEE Trans. on Automat. Contr., vol. 34, no. 7, July 1989, pp. 729-733.

Version History

Introduced before R2006a