# For what step disturbance value the system will become unstable?

I am new to control theory and have difficulties about this question R is the input, Td is the disturbance, C is the output Given that G1=K/(s^2+A*s), where K is the gain and A is a parameter greater than zero. The question is, for what value of the step disturbance the system can become unstable? How do I solve this?

## 1 Answer

The stability is a property of the linear systems themselves, hence there is no meaning in considering stability as regarded with the input disturbance $$T_d$$.

To verify if the closed-loop system $$C/T_d$$ is stable/unstable, you ought to compute the roots of the characteristic polynomial.

Given that $$\frac{C}{T_d} = - \frac{s^2+As}{s^2+As+K},$$ the roots of $$s^2+As+K$$ − poles of the closed-loop system − are: $$s_{1,2} = \frac{-A \pm \sqrt{A^2-4K}}{2}.$$ For $$C/T_d$$ to be unstable, at least one pole needs to have a positive real part, thus $$K<0$$.
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