BIBO Stability

Deals with the zero-state response ($x(0)=0$). Does a bounded input lead to a bounded response?

An input is bounded if there exists a constant $u_m$ such that

$$|u(t)|\le u_m<\infty$$

An output is bounded if there exists a finite $M$ such that

$${\int }_0^{\infty }|g(\tau )|d\tau \le M$$

---

We care about the second part of the general solution

Link to original

SISO system

$$\begin{array}{rl}y(t)& ={\int }_0^t{e}^{A(t-\tau )}Bu(\tau )d\tau \\ |y(t)|& \le {\int }_0^t|g(\tau )|u(t-\tau )|d\tau \le {\int }_0^t|g(\tau )|d\tau \cdot u_m\\ |y(t)|& \le {\int }_0^t|g(\tau )|d\tau \le M\end{array}$$

Where $u_m$ is the largest input signal (a constant), such that we get the following inequality, and $M$ is a constant smaller than $\infty$.

A SISO system with proper rational transfer function $\hat{g}(s)$ is BIBO stable if and only if every pole of $\hat{g}(s)$ has a negative real part

• This means that a pole in zero is NOT BIBO STABLE