if we know the breakaway point how to determine the range of K for unstable system

If we know the breakaway point, we can determine the range of K for an unstable system by performing a Routh-Hurwitz stability analysis.

The Routh-Hurwitz stability criterion is a mathematical tool used to determine the stability of a system based on the coefficients of its characteristic equation. By applying this criterion, we can determine the conditions that must be met for the system to be stable or unstable.

To apply the Routh-Hurwitz criterion, we first need to find the characteristic equation of the system. The characteristic equation is obtained by setting the denominator of the transfer function equal to zero:

1 + K G(s) = 0

where G(s) is the open-loop transfer function of the system.

Next, we write the coefficients of the characteristic equation in the form of a Routh array. The first row of the array contains the coefficients of the even powers of s, while the second row contains the coefficients of the odd powers of s. The subsequent rows are obtained by using the following formula:

ai = -1/ai-2 * | ai-2 ai-4 | | ai-1 ai-3 |

where ai-2 and ai-1 are the coefficients of the previous row.

Once we have constructed the Routh array, we can determine the range of K for an unstable system. If any of the coefficients in the first column of the array are negative, the system is unstable. The range of K for an unstable system is the interval between the breakaway point and the value of K at which the first coefficient in the first column of the array becomes negative