P+D Control, Finding parameters Wn and Zeta

Question

I have the system shown here;

I now need to introduce some P+D control. Assuming K(s) =kp

The specification I need to meet is overshoot = 30% and rise time = 0.5s

Having read online I am aware that I need to generate some simultaneous equations to find Wn and $\zeta$. This will allow me to generate an equation of this form to prove them;

$\frac{kp}{s^2+s(-kd+1)+kp}$

However I am unsure how to go about this. Can anyone give me a point in the right direction?

Thanks.

Answer 1

Well, the total transfer function looks like:

$$\mathscr{H}_{\space\text{T}}\left(\text{s}\right):=\frac{\theta_{\space\text{o}}\left(\text{s}\right)}{\theta_{\space\text{i}}\left(\text{s}\right)}:=\frac{\text{K}\cdot\frac{\text{a}}{\text{s}\cdot\left(\text{s}+\text{b}\right)}}{1+\text{K}\cdot\frac{\text{a}}{\text{s}\cdot\left(\text{s}+\text{b}\right)}}:=\frac{1}{\frac{1}{\omega_0^2}\cdot\text{s}^2+\frac{2\beta}{\omega_0}\cdot\text{s}+1}\tag1$$

Where:

• $$\omega_0=\sqrt{\text{a}\cdot\text{k}}\tag2$$
• $$\beta=\frac{\text{b}}{2\cdot\sqrt{\text{a}\cdot\text{k}}}\tag3$$

We have overshoot so we know that $0<\beta<1$ and we know the formula:

$$0.3=\exp\left(-\pi\cdot\frac{\beta}{\sqrt{1-\beta^2}}\right)\space\Longleftrightarrow\space\beta=\pm\frac{\ln\left(0.3\right)}{\sqrt{\pi^2+\ln^2\left(0.3\right)}}\tag4$$

Now, because we know that $0<\beta<1$ we know that:

$$\beta=-\frac{\ln\left(0.3\right)}{\sqrt{\pi^2+\ln^2\left(0.3\right)}}\approx0.357857\tag5$$

And for the rise time:

$$0.5=\frac{\pi}{\omega}\space\Longleftrightarrow\space\omega=\frac{\pi}{0.5}\tag6$$

The relation between $\omega$, $\omega_0$ and $\beta$ is given by:

$$\omega=\omega_0\cdot\sqrt{1-\beta^2}\tag7$$

So, we get:

$$\frac{\pi}{0.5}=\sqrt{0.36\cdot2.51\cdot\text{k}}\cdot\sqrt{1-\left(-\frac{\ln\left(0.3\right)}{\sqrt{\pi^2+\ln^2\left(0.3\right)}}\right)^2}\tag8$$