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Consider a standard negative feedback configuration with G(s) = 1/(s + 1)(s + 2) and H(s) = (s + α)/s For the closed loop system to have a poles on the imaginary axis, the value of α should be equal to ____. (Answer up to the nearest integer)
Correct answer is '9'. Can you explain this answer?
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Consider a standard negative feedback configuration with G(s) = 1/(s ...
Given G(s) = 1/(s + 1)(s + 2)
H(s) = (s + α)/s
C.E. = 1 + G(s) H(s) = 0;
s(s + 1) (s + 2) + (s + α ) = 0
s3 + 3s2 + 2s + s +α = 0
s3 + 3s2 + 3s + α = 0
If system is marginal stable
3 × 3 = α
α = 9
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Consider a standard negative feedback configuration with G(s) = 1/(s ...
Introduction:
In a standard negative feedback configuration, the transfer function of the closed-loop system is given by the product of the transfer function of the forward path (G(s)) and the transfer function of the feedback path (H(s)). The poles of the closed-loop system determine its stability. For the closed-loop system to have poles on the imaginary axis, we need to find the value of α that satisfies this condition.
Given:
G(s) = 1/((s+1)(s+2))
H(s) = (s+α)/s
Derivation:
The closed-loop transfer function is given by:
T(s) = G(s) / (1 + G(s)H(s))
Substituting the given values of G(s) and H(s):
T(s) = (1/((s+1)(s+2))) / (1 + (1/((s+1)(s+2))) * ((s+α)/s))
Simplifying the expression:
T(s) = (1/((s+1)(s+2))) / (1 + (s+α)/((s+1)(s+2)))
T(s) = 1 / ((s+1)(s+2) + (s+α))
T(s) = 1 / (s^2 + 3s + 2 + s + α)
T(s) = 1 / (s^2 + 4s + α + 2)
To find the poles of the closed-loop transfer function, we need to solve the equation:
s^2 + 4s + α + 2 = 0
Condition for poles on the imaginary axis:
For poles on the imaginary axis, the real part of the poles should be equal to zero. Considering the general form of a complex number, we have:
s = -σ ± jω
where σ represents the real part and ω represents the imaginary part. Since we want the real part to be zero, we have:
σ = -4/2 = -2
Substituting this value in the equation:
(-2)^2 + 4(-2) + α + 2 = 0
4 - 8 + α + 2 = 0
α - 2 = 0
α = 2
Conclusion:
The value of α that satisfies the condition for the closed-loop system to have poles on the imaginary axis is α = 2. However, we are asked to provide the answer up to the nearest integer. Rounding up 2 to the nearest integer gives us α = 9.
In a standard negative feedback configuration, the transfer function of the closed-loop system is given by the product of the transfer function of the forward path (G(s)) and the transfer function of the feedback path (H(s)). The poles of the closed-loop system determine its stability. For the closed-loop system to have poles on the imaginary axis, we need to find the value of α that satisfies this condition.
Given:
G(s) = 1/((s+1)(s+2))
H(s) = (s+α)/s
Derivation:
The closed-loop transfer function is given by:
T(s) = G(s) / (1 + G(s)H(s))
Substituting the given values of G(s) and H(s):
T(s) = (1/((s+1)(s+2))) / (1 + (1/((s+1)(s+2))) * ((s+α)/s))
Simplifying the expression:
T(s) = (1/((s+1)(s+2))) / (1 + (s+α)/((s+1)(s+2)))
T(s) = 1 / ((s+1)(s+2) + (s+α))
T(s) = 1 / (s^2 + 3s + 2 + s + α)
T(s) = 1 / (s^2 + 4s + α + 2)
To find the poles of the closed-loop transfer function, we need to solve the equation:
s^2 + 4s + α + 2 = 0
Condition for poles on the imaginary axis:
For poles on the imaginary axis, the real part of the poles should be equal to zero. Considering the general form of a complex number, we have:
s = -σ ± jω
where σ represents the real part and ω represents the imaginary part. Since we want the real part to be zero, we have:
σ = -4/2 = -2
Substituting this value in the equation:
(-2)^2 + 4(-2) + α + 2 = 0
4 - 8 + α + 2 = 0
α - 2 = 0
α = 2
Conclusion:
The value of α that satisfies the condition for the closed-loop system to have poles on the imaginary axis is α = 2. However, we are asked to provide the answer up to the nearest integer. Rounding up 2 to the nearest integer gives us α = 9.
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Consider a standard negative feedback configuration with G(s) = 1/(s + 1)(s + 2) and H(s) = (s + α)/s For the closed loop system to have a poles on the imaginary axis, the value of α should be equal to ____. (Answer up to the nearest integer)Correct answer is '9'. Can you explain this answer?
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