Understanding System Stability Range for G(s)

The question asks for the range of the parameter k that ensures stability for a unity gain negative feedback control system. The open-loop transfer function (OLTF) is given as:

$$ \rm G(s) = \frac{k}{s^2 + 4s - 5} $$

For a unity gain negative feedback system, the closed-loop transfer function (CLTF) is given by:

$$ \rm CLTF(s) = \frac{G(s)}{1 + G(s)} $$

The stability of the system is determined by the roots of the characteristic equation, which is the denominator of the CLTF set to zero:

$$ 1 + G(s) = 0 $$

Substituting the given G(s):

$$ 1 + \frac{k}{s^2 + 4s - 5} = 0 $$

To find the characteristic equation, we clear the denominator:

$$ s^2 + 4s - 5 + k = 0 $$

Rearranging the terms to match the standard polynomial form:

$$ s^2 + 4s + (k-5) = 0 $$

Applying Routh-Hurwitz Criterion

The Routh-Hurwitz criterion is a method used to determine the stability of a linear time-invariant (LTI) system by examining the coefficients of its characteristic equation without explicitly calculating the roots. For a polynomial to have all its roots in the left-half of the s-plane (which is required for stability), all coefficients must be present and positive, and all elements in the first column of the Routh array must have the same sign (typically positive).

The characteristic polynomial is: $$ P(s) = 1 \cdot s^2 + 4 \cdot s^1 + (k-5) \cdot s^0 $$

We construct the Routh array using the coefficients of the characteristic polynomial:

Simplifying the term for the \( s^0 \) row:

$$ \text{Term for } s^0 = \frac{4(k-5)}{4} = k-5 $$

The Routh array is:

Determining the Range of k for Stability

For the system to be stable, all the elements in the first column of the Routh array must be positive.

• The coefficient of \( s^2 \) is 1, which is positive.
• The coefficient of \( s^1 \) is 4, which is positive.
• The coefficient of \( s^0 \) is \( k-5 \). For stability, this must also be positive.

Therefore, we require:

$$ k-5 > 0 $$

Solving for k:

$$ k > 5 $$

If \( k = 5 \), the \( s^0 \) row becomes zero, indicating the presence of roots on the imaginary axis (marginal stability). For strict stability, k must be greater than 5.

Thus, the range of k for which the system is stable is k > 5.