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# Stability of systems

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1. A single-loop negative feedback system has a loop transfer function.

GH(s) = K(s+2)^2 / s(s^2+1)(s+8)

Determine the range of the gain K for which the system is stable.

Choose either
a. 23 < K < 367
b. K > 14
c. 0 < K < 5

2. A system has a characteristic equation q(s) = s^4 +9s^3 +45s^2 +87s +50 +0. Determine whether the system is stable using the Routh-Hurwitz criterion and also determine the roots of the characteristic equation.

Choose either
a. unstable and s1,2 = -3 ±j4. s3 = -2, s4 = -1
b. stable and s1,2 = -3 ± j4. s3 = -2, s4 = -1
c. marginally stable and s1,2 = -3 ± j4. s3 = -.2, s4 = -1

3. A system has a characteristic equation s3 + 9s2 + 26s + 24 = 0. Using the Routh-Hurwitz criterion, determine if the system is stable.

Choose either
a. Stable
b. Unstable

https://brainmass.com/math/linear-algebra/stability-systems-145128

#### Solution Preview

1. A single-loop negative feedback system has a loop transfer function.

GH(s) = K(s+2)^2 / s(s^2+1)(s+8)

Determine the range of the gain K for which the system is stable.

Choose either
a. 23 < K < 367
b. K > 14
c. 0 < K < 5

Solution
The characteristic equation of the system is

1
8 0
0
0

The system is stable if all of the signs of the first column are ...

#### Solution Summary

This is a set of questions regarding control systems, the Routh-Hurwitz criterion, and stability.

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