1. The Value of Kc for which the control system is stable:

To determine the value of Kc for which the control system is stable, we need to find the range of values for Kc that will result in all the roots of the characteristic equation having negative real parts. This is because for a system to be stable, all the poles (roots of the characteristic equation) must have negative real parts.

The characteristic equation is given as:
S^3 + 6S^2 + 11S + 6(1 - Kc) = 0

To find the stability range, we can use the Routh-Hurwitz stability criterion. According to this criterion, for a polynomial equation of the form:
a_n S^n + a_(n-1) S^(n-1) + ... + a_1 S + a_0 = 0

The necessary and sufficient condition for stability is that all the coefficients a_i in the first column of the Routh array must be positive.

In our case, the characteristic equation is:
S^3 + 6S^2 + 11S + 6(1 - Kc) = 0

Using the Routh-Hurwitz stability criterion, we can construct the Routh array:

1 11
6(1 - Kc) 6

The first column of the Routh array is [1, 6(1 - Kc)]. For stability, both elements in this column must be positive. Therefore, we have:

1 > 0
6(1 - Kc) > 0

Simplifying the second inequality, we get:
1 - Kc > 0
Kc < />

Therefore, the control system is stable for all values of Kc such that Kc < />

2. The roots of the characteristic equation for the value of Kc for which the system is on the threshold of instability:

To determine the roots of the characteristic equation for the value of Kc at the threshold of instability, we need to find the value of Kc at the boundary between stability and instability. This occurs when the first coefficient in the first column of the Routh array becomes zero. In other words, when the first element in the first column changes sign.

In our case, the first element in the first column is Kc - 1. Therefore, the system is on the threshold of instability when Kc - 1 = 0, or Kc = 1.

For Kc = 1, the characteristic equation becomes:
S^3 + 6S^2 + 11S + 6(1 - 1) = 0
S^3 + 6S^2 + 11S = 0

To find the roots of this equation, we can use methods such as factoring, synthetic division, or numerical methods like Newton-Raphson.

The roots of the characteristic equation for Kc = 1 are:
S = 0
S = -3 ± √2i

Therefore, for Kc = 1, the system is on the threshold of instability, with two complex conjugate roots and one real root at the origin.

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