Determine the stability of systems that have the following characteristic equations:

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Accepted Answer

(a) [latex] 7s^{4} + 5s^{3} − 12s^{2} + 6s − 1 = 0 [/latex]. The coefficients in this characteristic equation have different signs and thus the set of necessary conditions for stability is not met. The system is unstable, and thus there is no need to look at the Hurwitz determinants.

(b) [latex] s^{3} + 6s^{2} + 11s + 6 = 0 [/latex]. The necessary condition is satisfied. Next, the Hurwitz determinants must be examined (a more stringent test). Because the necessary condition is satisfied, only [latex]D_{2}[/latex] needs to be checked in this case. It is

[latex] D_{2}=\left | \begin{matrix} 6 & 6 \ 1 & 11 \end{matrix} ight | =66-6=60\gt 0 [/latex].

Thus all determinants are positive and the system is stable.

(c) [latex] 2s^{4} + s^{3} + 3s^{2} + 5s + 10 = 0 [/latex]. The necessary condition is satisfied. The determinants [latex]D_{2}[/latex] and [latex]D_{3}[/latex] are

[latex]D_{2}= -7, D_{3}= -45[/latex]

The set of necessary and sufficient conditions is not satisfied because both [latex]D_{2}[/latex] and [latex]D_{3}[/latex] are negative; the system is unstable.

Question 13.1: Determine the stability of systems that have the following c...

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