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# Stability control with the App:  Polynomial Root Finder

## Explanation of the method

With the application software program Polysmlt -Poly Root Finder, you can easily calculate the poles of a closed loop system. If there are poles with a positive real part, then the system in not stable. This method is simpler than the Routh-Hurwitz criteria.

As an example, we choose s closed loop system with the transfer function :

H=Kc(s+1)/(10s^3+17s^2+8s+1+Kc):

According to the control system theory, it yields that :

A linear system is stable if and only if all roots (poles) of the denominator in the transfer function are negative or have negative real parts. Otherwise, the system is unstable.

With the Poly root finder, we can show that the system has positive real parts for Kc>12.6 and therefore unstable.

For Kc<12.6 the system is stable and for Kc=12.6 it's in practice not stable. We will demonstrate the App. The denominator = 10s^3+17s^2+8s+1+Kc.

We take Kc as 5, 12.6 and 20respectively .

See the results here below:

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