This is a basic control theory question, since Control Theory is a part of applied mathematics but also of engineering I was unsure whether to ask this here.
The question says:
Given the transfer function of a system is $G(s)=1/(s^2+3s+2)$, consider the design of a PI closed loop control system with unit feedback using proportional gain $k_p$ and integral gain $k_i$, both of which are positive. Determine the range of gain for which the closed loop system is stable. What I did was this, I went the Routh-Hurwitz way:
After completing the Routh table, I went ahead and reasoned a little about what conditions need to be met in order to avoid sign changes on the main column, however it seems that these conditions are never met! I get that $k_i$ should be less than zero when the problem clearly specifies it will always be positive. Have I done something wrong? Is my reasoning right? Is the answer "The controller is never stable"?