Compensator design

I got a plant $$G(s)=\left(0.13s+1\right)/s^2$$ to design a compensator which provides below demands:

• Settling time : max 2s
• Overshoot : max %35
• Gain margin : min 10 dB
• Phase margin : min 30 deg
• Controller effort (r to u) : max 0.9
• Bandwith : min 10 rad/s

The best architecture so far was the one below but I couldn't reach the demands.

assigning $$C1=0.03\cdot\left(\left(s+80\right)\left(s+10\right)\right)/\left(\left(s+0.12\right)\left(s+1\right)\right)$$, $$C2=17.5\text{m}$$ and $$H=1$$ results as below:

Can anyone explain or guide a design approach on how to handle $$\left(s+a\right)/s^2$$ type plants when designing compensators or mention some tips/shortcuts for architecture selection? How do we select the order of the controller?

• Is this a homework question? Or like a job? Could you show what your compensator could accomplish in a plot? What it does manage, and also doesn‘t manage? What don‘t you understand when it comes to design and choosing an order? – morbo Oct 23 '19 at 8:07
• it is a job like question and I editted the question to show how my compensator affects. As to order, I know that we should select lower order compensator as much as possible and there is no exact prescription for the type of compensator selection. what I want to learn is that is there a relation or at least an approach between plant order with compensator order when we start to design a compensator. for example my plant is second order and so should I start with 2th order compensator (like lead-lag) in order to save time by skipping working on 1th order compensators. – lsn Oct 23 '19 at 9:09
• Is there a reason why you choose that specific architecture, instead of for example a single feedback loop? – fibonatic Oct 23 '19 at 9:20
• this architecture performed better than the others. I tried the others also. – lsn Oct 23 '19 at 9:40
• Maybe this will help, $-\frac{8.20546 \left(1. s^2-3.84499 s+118.343\right)}{s}$ – morbo Oct 23 '19 at 21:50

If you set $$C=(2\zeta\omega_ns+\omega_n^2)/(0.13s+1)$$, then you'll get the following closed-loop system transfer function: $$T=\frac{GC}{1+GC}=\frac{2\zeta\omega_ns+\omega_n^2}{s^2+2\zeta\omega_ns+\omega_n^2}.$$
This can be achieved through zero-pole cancelation, which is doable since $$G$$ has a zero in LHP.
Finally, you can easily attain your design requirements with $$\omega_n \approx 10\,\text{rad/s}$$ and e.g. $$\zeta=\sqrt{2}/2$$.