MATLAB: Diffrent results using tf(ss(A,B,C,D)) or Cinv(sI-A)*B+D

controlMATLABstatespace

Why do I obtain diffrent results?

A = rand(3,3);
B = rand(3,1);
C = rand(4,3);
D = rand(4,1);
% using matlab function
tf1 = tf(ss(A,B,C,D));
% calculated by hand
s = zpk('s');
tf2 = D + C*inv(s*eye(3)-A)*B;

Does anyone have an idea why tf1-tf2 is not zero?

Thank you

Best Answer

The results are not different. They are same. Minor differences can be due to numerical/computational tolerances and accuracy. At any given natural frequency, magnitude of (tf1-tf2) is indeed very close to zero. You can check the similarity yourself by comparing their bode plots like this:

bode(tf1)
hold on
bode(tf2)

Related Solutions

MATLAB: How do you extract the state space model from ssest()

sys1 is a discrete-time state space model.

If you want to compare the results with sys2, replace your code with:

%% Extract the A, B, C, and D matricies from sys1 to to put the state space model in a ss class
sys2 = ss(sys1.A,sys1.B,sys1.C,sys1.D,sys1.Ts);

You should obtain similar plots for both sys1 and sys2.

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

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);

Best Answer

Related Solutions

MATLAB: How do you extract the state space model from ssest()

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

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