GATE 2021 EC – Question 58
A unity feedback system that uses proportional-integral (PI) control is shown in the figure.
The stability of the overall system is controlled by tuning the PI control parameters $K_P$ and $K_I$. The maximum value of $K_I$ that can be chosen so as to keep the overall system stable or, in the worst case, marginally stable (rounded off to three decimal places) is ________.
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Correct answer: 3.10 to 3.15
Explanation
The characteristic equation is $s(s^3+4s^2+5s+2)+2(K_Ps+K_I)=s^4+4s^3+5s^2+(2+2K_P)s+2K_I=0$. From the Routh array, the first-column entries are $1,\ 4,\ \frac{18-2K_P}{4},\ \ldots,\ 2K_I$, and stability requires $K_I<\frac{(9-K_P)(1+K_P)}{8}$ (with $K_P<9$, $K_I>0$). This bound is largest at $K_P=4$, where $K_I<\frac{5\times5}{8}=3.125$. The maximum $K_I$ is 3.125.