Stability Analysis for G(s) and P(s)

[Logarythmic]

Homework Statement

I have a transfer function $G(s) = \frac{1}{s^2}$ and a PI controller $P(s) = 6 \left( 1 + \frac{1}{s} \right)$.

How do I check for stability? Just use 1 + P(s)G(s) = 0 and check the roots?

 

[CEL]

Yes, if all the roots have a negative real part the system is stable.

 

[piercas]

I would recommend using the Routh-Hurwitz stability criterion to check for stability. This method involves constructing a Routh array using the coefficients of the characteristic equation 1 + P(s)G(s) = 0 and analyzing the signs of the first column. If all the signs are positive, then the system is stable. If there is a sign change, then the number of sign changes in the first column represents the number of poles in the right-half plane, indicating instability. Additionally, you can also use the Nyquist stability criterion to plot the Nyquist diagram and determine stability based on the number of encirclements of the critical point (-1,0). Both of these methods are more reliable and comprehensive than simply checking the roots of the characteristic equation.