Measurement & Control (Summer 2021): Control part

Comment on the stability of a control system whose characteristic equation is given b:

\({s^5} + 24{s^4} + 24{s^3} + 48{s^2} - 25s - 50 = 0\)

(AMIE Summer 2021)

Solution

s⁵	1	24	-25
s⁴	2	48	-50
s³	a = 0	b = 0
s²
s¹
s⁰

\(\begin{array}{l}a = \frac{{1x48 - 24x2}}{2}\\b = \frac{{1x( - 50) - ( - 25x2)}}{2}\end{array}\)

If all elements in a row are zero, then differentiate the coefficients of the above row.

\(\begin{array}{l}\frac{{d(2{s^4} + 48{s^2} - 50)}}{{ds}}\\ = 8{s^3} + 96s\end{array}\)

s⁵	1	24	-25
s⁴	2	48	-50
s³	a = 0 (8)	b = 0 (96)
s²	-24	50
s¹	-112.66
s⁰	-50

There is a sign change in the first column of the Routh table, hence the system is unstable.

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