balancmr - Balanced model truncation via square root method

Syntax

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

Description

balancmr 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 these values indicate the respective state energy of the system. Hence, reduced order can be directly determined by examining the system Hankel singular values, 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]:

This table describes input arguments for balancmr.

A batch run of a serial of different reduced order models can be generated by specifying order = x:y, or a vector of positive 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 singular values reaches the 'MaxError'.

This table lists the input arguments 'key' and its 'value'.

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.

G can be stable or unstable, continuous or discrete.

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.

1. Find the SVD of the controllability and observability grammians

   P = U_p Σ_p V_p^T

   Q = U_qΣ_q V_q^T

2. Find the square root of the grammians (left/right eigenvectors)

   L_p = U_p Σ_p^½

   L_o = U_q Σ_q^½

3. Find the SVD of (L_o^TL_p)

   L_o^T L_p = U Σ V^T

4. Then the left and right transformation for the final k^th order reduced model is

   S_L,BIG = L_o U(:,1:k) Σ(1;k,1:k))^–½

   S_R,BIG = L_p V(:,1:k) Σ(1;k,1:k))^–½

5. Finally,

   

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

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:

rand('state',1234); randn('state',5678);G = rss(30,5,4);
[g1, redinfo1] = balancmr(G); % display Hankel SV plot
                              % and prompt for order (try 15:20)
[g2, redinfo2] = balancmr(G,20); 
[g3, redinfo3] = balancmr(G,[10:2:18]);
[g4, redinfo4] = balancmr(G,'MaxError',[0.01, 0.05]);
rand('state',12345); randn('state',6789);
wt1 = rss(6,5,5); wt1.d = eye(5)*2;
wt2 = rss(6,4,4); wt2.d = 2*eye(4);
[g5, redinfo5] = balancmr(G, [10:2:18], 'weight',{wt1,wt2});
for i = 1:5
    figure(i); eval(['sigma(G,g' num2str(i) ');']);
end

References

[1] Glover, K., "All Optimal Hankel Norm Approximation of Linear Multivariable Systems, and Their Lµ-error Bounds," Int. J. Control, Vol. 39, No. 6, 1984, p. 1145-1193

[2] Safonov, M.G., and R.Y. Chiang, "A Schur Method for Balanced Model Reduction," IEEE Trans. on Automat. Contr., Vol. 34, No. 7, July 1989, p.729-733

See Also

bstmr | hankelmr | hankelsv | ncfmr | reduce | schurmr

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