Stability Criterion

• An LTI system with impulse response function $g(t)$ is stable if and only if

$${\int }_0^{\infty }|g(t)|dt<\infty$$

• A signal is said to be absolutely integrable over an integral if the integral of the absolute value of the signal over the integral is infinite. A linear system is stable if its impulse response is absolutely integrable over $[0,\infty )$.

• A system with transfer function $G(s)$ is stable if and only if $G(s)$ is proper (degree of numerator < degree of denominator) and all poles of $G(s)$ have negative real parts.

Example

Determine the stability of $G(s)=\frac{1}{s+\gamma }$ by checking the sign of the real part of the pole.

The pole of $G(s)$ is $s_1=-\gamma$, and only has a real part.

• If $\gamma \le 0$, then $s_1\ge 0$, hence the system is unstable.
• If $\gamma >0$, then $s_1<0$. Stable!