# Documentation

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# bstmr

Balanced stochastic model truncation (BST) via Schur method

## Syntax

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

## Description

`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`(G-GRED)` ∥ ∞ or relative error`G`-–1`(G-GRED)` ∥ ∞ for well-conditioned model reduction problems [1]:

`${‖{G}^{-1}\left(G-Gred\right)‖}_{\infty }\le \prod _{k+1}^{n}\left(1+2{\sigma }_{i}\left(\sqrt{1+{\sigma }_{i}^{2}}+{\sigma }_{i}\right)\right)-1$`

This table describes input arguments for `bstmr`.

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 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 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 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, `'``MaxError``'`overides `ORDER` input.

`'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.

This table describes output arguments.

Argument

Description

`GRED`

LTI reduced order model. Become multi-dimension array when input is a serial of different model order array.

`REDINFO`

A STRUCT array with three fields:

• `REDINFO.ErrorBound` (bound on ∥G–1(G-GRED) ∥∞)

• `REDINFO.StabSV` (Hankel SV of stable part of G)

• `REDINFO.UnstabSV` (Hankel SV of unstable part of G)

`G` can be stable or unstable, 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); 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 ```

## 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 kth order reduced model.

1. 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 + PAT + BBT = 0

BW = PCT + BDT

QA + AT Q + (QBW – CT) (–DDT) (QBW – CT)T = 0

2. Find the Schur decomposition for PQ in both ascending and descending order, respectively,

`$\begin{array}{l}{V}_{A}^{T}PQ{V}_{A}=\left[\begin{array}{ccc}{\lambda }_{1}& \cdots & \cdots \\ 0& \cdots & \cdots \\ 0& 0& {\lambda }_{n}\end{array}\right]\\ {V}_{D}^{T}PQ{V}_{D}=\left[\begin{array}{ccc}{\lambda }_{n}& \cdots & \cdots \\ 0& \cdots & \cdots \\ 0& 0& {\lambda }_{1}\end{array}\right]\end{array}$`
3. Find the left/right orthonormal eigen-bases of PQ associated with the kth big Hankel singular values of the all-pass phase matrix (W*(s))–1G(s).

k

`$\begin{array}{l}{V}_{A}=\left[{V}_{R,SMALL},\stackrel{k}{\overbrace{{V}_{L,BIG}}}\right]\\ {V}_{D}=\left[\stackrel{}{\overbrace{{V}_{R,BIG}}},{V}_{L,SMALL}\right]\end{array}$`
4. Find the SVD of (VT L,BIGVR,BIG) = U Σ ςΤ

5. Form the left/right transformation for the final kth order reduced model

SL,BIG = VL,BIG U Σ(1:k,1:k)–½

SR,BIG = VR,BIG V Σ(1:k,1:k)–½

6. Finally,

`$\left[\begin{array}{cc}\stackrel{^}{A}& \stackrel{^}{B}\\ \stackrel{^}{C}& \stackrel{^}{D}\end{array}\right]=\left[\begin{array}{cc}{S}_{L,BIG}^{T}A{S}_{R,BIG}& {S}_{L,BIG}^{T}B\\ C{S}_{R,BIG}& D\end{array}\right]$`

The proof of the Schur BST algorithm can be found in [1].

 Note   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 jω-axis 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.

## References

[1] Zhou, K., "Frequency-weighted model reduction with L∞ error bounds," Syst. Contr. Lett., Vol. 21, 115-125, 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. 259-272, 1988.