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

A PID Controller is given as follows:

\begin{gather} C(s)=K_p+K_i\frac{1}{s}+K_ds \end{gather} where $K_p=0.07847$, $K_i=0.03587$ and $K_d=0.04291$.

The transfer function of the controlled system is: \begin{gather} G(s) = \frac{1}{s^2+s+1} \end{gather}

Now as a feedback loop I have following:

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

$$ F(s) = \frac{0.04291s^2+0.07847s+0.03587}{s^3+1.043s^2+1.078s+0.03587} $$

The step response looks like:

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.

For example to solve for ISE for $W(s)$ as step response in frequency domain: \begin{gather} e(s)=w(s)-y(s) \\ \\ e(s) = w(s)-(F(s)w(s)) \\ \\ e(s) = \frac{1}{s}-(\frac{F(s)}{s}) \end{gather}

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 $K_p$, $K_i$ and $K_d$?

Matlab code is here.

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

A PID Controller is given as follows:

\begin{gather} C(s)=K_p+K_i\frac{1}{s}+K_ds \end{gather} where $K_p=0.07847$, $K_i=0.03587$ and $K_d=0.04291$.

The transfer function of the controlled system is: \begin{gather} G(s) = \frac{1}{s^2+s+1} \end{gather}

Now as a feedback loop I have following:

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

$$ F(s) = \frac{0.04291s^2+0.07847s+0.03587}{s^3+1.043s^2+1.078s+0.03587} $$

The step response looks like:

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.

For example to solve for ISE for $W(s)$ as step response in frequency domain: \begin{gather} e(s)=w(s)-y(s) \\ \\ e(s) = w(s)-(F(s)w(s)) \\ \\ e(s) = \frac{1}{s} - \left(\frac{F(s)}{s}\right) \end{gather}

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 $K_p$, $K_i$ and $K_d$?

Matlab code is here.

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:

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:

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.

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.

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

A PID Controller is given as follows:

\begin{gather} C(s)=K_p+K_i\frac{1}{s}+K_ds \end{gather} where $K_p=0.07847$, $K_i=0.03587$ and $K_d=0.04291$.

The transfer function of the controlled system is: \begin{gather} G(s) = \frac{1}{s^2+s+1} \end{gather}

Now as a feedback loop I have following:

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

$$ F(s) = \frac{0.04291s^2+0.07847s+0.03587}{s^3+1.043s^2+1.078s+0.03587} $$

The step response looks like:

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.

For example to solve for ISE for $W(s)$ as step response in frequency domain: \begin{gather} e(s)=w(s)-y(s) \\ \\ e(s) = w(s)-(F(s)w(s)) \\ \\ e(s) = \frac{1}{s}-(\frac{F(s)}{s}) \end{gather}

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 $K_p$, $K_i$ and $K_d$?

Matlab code is here.