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The Integral Absolute value Error (IAE) and the Integral Squared Error (ISE) is to be analysed.

A PID Controller is given as follows:

C(s)=  Kp + Ki * (1/s) + Kd * s
Where kp = 0.07847, ki = 0.03587, kd = 0.04291

and a controlled system is given as

G(s) = 1/(s^2 + s + 1)

Now as a feedback loop I have following:

enter image description here

Calculating the overall Transfer function F(s) we get:

             0.04291 s^2 + 0.07847 s + 0.03587
   F(s) =   -----------------------------------
            s^3 + 1.043 s^2 + 1.078 s + 0.03587

The step response looks like:

enter image description here

Now I wanted to find the Integral Absolute value Error (IAE) and the Integral Squared Error (ISE). By researching google I came to know the principle behind it and I also saw the formulas but how really to mathmetically solve it.

enter image description here

For example to solve for ISE for W(s) as step response in frequency domain:

e(s) = w(s) - y(s)
e(s) = w(s) - { F(s) * w(s) }
e(s) = (1/s) - { F(s) * (1/s)}

Now I replace F(s) and solve for e(s) ??

After that integrate from 0 to infinity to the e(s)^2?? I don't get it. How am I supposed to get the tuned value for kp, ki and kd?

Matlab code is here.

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$e(s)=w(s)(1-F(s))$

You know both F and w (w is the reference). Then calculate e(s), later e(t) (use the Laplace anti-transform) and integrate.

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