Final answer:
The range of K for stability is determined using the Routh–Hurwitz criterion, and for marginal stability, we look for a critical gain that introduces a zero in the first column of the Routh array, allowing for calculation of the frequency of oscillation.
Step-by-step explanation:
To determine the range of K for stability in the unity feedback system with G(s) = K(s+6)/s(s+1)(s+4), we use the Routh–Hurwitz criterion. This criterion provides a systematic method for investigating the stability of the closed-loop system by arranging the coefficients of the characteristic polynomial into an array and evaluating the number of sign changes in the first column. For a stable system, there should be no sign changes.
For marginal stability, the system has a pair of purely imaginary roots, and a sign change occurs in the Routh array. Setting the K value such that the first column of the array has a zero and solving for K yields the critical gain for marginal stability. To find the frequency of oscillation under marginal stability, we substitute the critical gain back into the characteristic equation and solve for the imaginary roots, which will give us the frequency of sustained oscillations.