Designing a PID controller for a pendulum

we have the following system:

• (4.545 s) / (s^3 + 0.1818 s^2 - 31.21 s - 4.459)

we have a upside down pendulum, and we need to design a PID controller which can hold it up straight. such that the following conditions occurs:

.

% setteling time: <5 Sec

% overshoot: 20 degree

% risetime: <0.5 Sec

.

any tips??

i, first, put Ti = Inf and Td = 0;

then found the value for Kp = Kcritical according to Routh Table: Kp = Kcr = 159.4546249

but when I want to calculate Natural Frequency (Omega), i get three answers which have imaginary parts:

.

 0.0000 + 7.3625i
 2.9609 - 3.5903i
-2.9609 - 3.5903i

.

I'm guessing I am not using the correct method

[Arkadiy Turevskiy]

You could tune the PID controller with PID Tuner app, unless you are trying to stcik with your method specifically.

>>s=tf('s');
>> sys=(4.545*s) / (s^3 + 0.1818*s^2 - 31.21*s - 4.459)

sys =

                 4.545 s
    ----------------------------------
    s^3 + 0.1818 s^2 - 31.21 s - 4.459

Continuous-time transfer function.

>> pidTuner(sys)

Then playing a little bit with sliders you could get to something like this:

Answer 1

[Arkadiy Turevskiy]

You could tune the PID controller with PID Tuner app, unless you are trying to stcik with your method specifically.

>>s=tf('s');
>> sys=(4.545*s) / (s^3 + 0.1818*s^2 - 31.21*s - 4.459)

sys =

                 4.545 s
    ----------------------------------
    s^3 + 0.1818 s^2 - 31.21 s - 4.459

Continuous-time transfer function.

>> pidTuner(sys)

Then playing a little bit with sliders you could get to something like this: