Electrical – step response data extraction from response data of a model(simulink)

A not-at-all elegant way to do it is:

zeta=[...]; %your zeta values
wn = ... % calculate your wn values according to your zeta values
figure;
hold('on');
for idx = 1:length(zeta)
    % sys = tf([wn(idx)],[1 2*wn(idx)*zeta(idx) wn(idx)^2]); %system's transfer function
    % EDIT : numerator corrected
    sys = tf([wn(idx)^2],[1 2*wn(idx)*zeta(idx) wn(idx)^2]); %system's transfer function
    step(alpharef*sys);
end

Integrator wind up. That's what seems likely after a 10 second review of your plots. Do you restrict the maximum and minimum values of the integrator sum? You should. The purpose of the integrator is to correct for the loop error left over by the proportional part of the loop, due to the proportional part's finite gain. The integrator sum should not be allowed to be much greater than is required to correct the proportional error.

At steady state the integrator will have reached some value that offsets the proportional error. Then there is the step change, and the integrator value is now badly wrong. It corrects, but it takes a lot of time. Meanwhile the loop is railed out as the integrator slowly increments (or decrements) its way back to balance the proportional contribution.

Here are 2 ways to vastly improve integrator performance:

• Restrict the size of the integrator sum. Calculate, worst case, the value it takes to balance the error leftover from the proportional loop, and then use 1.5 to 2 times that to limit the value extremes of the integrator sum. This will also prevent rollover.

• Phase reset of the integrator sum. Compare the sign of the error with the sign of the integrator sum. If the signs are the same, then set the integrator sum to zero. This will give your integrator a leg up, getting rid of a wrong value that it will have to work off over time, and allow it to contribute to minimizing the error right away. This is hard to do with an analog loop, but for an algorithm in a DSP or micro-controller is easy.

Anyway, check out the possibility of integrator wind up as cause.