Draw a block diagram with transfer functions and get the closed-loop transfer function with step-by-step reduction. Check stability with the Routh–Hurwitz criterion and the stable range of K, the steady-state error, the step response, the root locus with its rules, the Bode and Nyquist plots with gain and phase margins, and design controllers.
All closed-loop poles must have a negative real part. Routh's criterion checks it without computing the roots.
The paths of the closed-loop poles as a gain K goes from 0 to infinity.
A classic way is Ziegler–Nichols: find the ultimate gain Ku and period Pu, then Kp = 0.6Ku, Ti = Pu/2 and Td = Pu/8.
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