I am trying to design a PID controller, to get the response I want to, so I am beginning to think that my approach with tuning the values might be incorrect.
I have to Design a controller for the system G(S) such, that the step response to close loop transfer function has an overshoot of 0% and a settling time of less than 1 sec.
$$G(s) = \frac{1}{10s^2+5s+10}$$
the overshoot has to be 0% which means that we interested in a critical damped system ζ = 1 since the settling time has to be lower than 1 sec, I can deduce that ωn has to be less than 4. which means that, my poles shall be on the real axis and less than -4.
I chose I want my poles to be at -5 and -7, and using solve I am able to see that a PD controller consisting of d = 115 and p = 340 would do the job. But the step response shows me that there is about 12.5 percent overshoot, but the settling time matches. The close loop transfer function I end up with
What mathematically do: I write up the mathematical equation for the close loop transfer each each consisting of P,PI,PD,and PID, and solve the denominator for it's pole.
So if I want the poles to lie at s = -5 and s = -7 I would for P controller solve this equation
s1:= -5
s2:= -7
Solve [10s1^2 +5s1 + c+kp ==0 && 10s2^2 +5s2 + c+kp ==0 , p] for a PI controller.
Solve [s1 (a*s1^2 + b*s1 + c) + i + p*s1 == 0 &&
s2 (a*s2^2 + b*s2 + c) + i + p*s2 == 0 , {p, i} ]
And for a PID controller:
Solve[s1 (a*s1^2 + b*s1 + c) d*s1^2 + p*s1 + i == 0 &&
s2 (a*s2^2 + b*s2 + c) d*s2^2 + p*s2 + i == 0, {p, i, d}]
Where a, b, c is parameter corresponding a second order equation $$as^2 + bs + c$$
$$G(s)=\frac{115s+340}{10s^2+120s+350}$$
