MATLAB: Does relative error model reduction using the BSTSCHMR and BSTSCHML functions in the Robust Control Toolbox 2.0.10 (R14) produce wrong results

Question

I am using the Robust Control Toolbox to reduce my system model using BST relative error model reduction. However, I find that the resulting reduced order system found with the BSTSCHMR exhibit significant errors when compared with the original system.

Answer 1

This phenomenon is a result of ill-conditioned system which causes numerical inaccuracies in the model reduction algorithm.

This is often a result of a nearly singular D matrix. In this case, adjustments to this D matrix may result in a better solution. However, this may distort the systems high frequency dynamics. This procedure is shown in the following code (variables "sys" and num_reduced_states must be defined in the workspace):

% With a system with a nearly singular D matrix
% Assume sys.d = 1e-5
sys1 = sys;
sys1.d = 1e-2; %this will disregard the higher frequency dynamics
[sysr1,aug,hsv] = bstschml(sys1,1,num_reduced_states);
sysr1a = sysr1;
sysr1a.d = sys.d; % put D back to original value
% Now compare the frequency response

bode(sys, sysr1, sysr1a);

As an alternative, if you have the Mu-Analysis and Synthesis Toolbox, you can first try rebalancing the system using the SRELBAL function before performing this model reduction. This would be done as follows:

[a,b,c,d] = ssdata(sys);
sysmat = pck(a,b,c,d);
[sysmatrb,sigrel] = srelbal(sysmat);
[a2,b2,c2,d2] = unpck(sysmatrb);
sys2 = ss(a2,b2,c2,d2); 
[sysr2,aug,hsv] = bstschml(sys,1,num_reduced_states);
% Now compare the frequency response
bode(sys,sysr2);