The two types of stability and "Why exponential"

When I was learning about LTI systems, I noticed that LTI system is said to be BIBO stable if and only if its square sum of impulse response is finite. This expression is found on many textbook of signal processing.

I continued study and found this expression LTI system is stable if and only if its impulse response is exponentially bounded. I could not understand this at first, but by further research, I elaborately acquire this interpretation. However, I could not find explicit description about this. So my question is "is these things true ?",

1. There are two types of stability (of the system), internal stability and external stability.
2. BIBO stability is equivalent to external stability.
3. If the system is controllable and observable, internal and external stability is equivalent.
4. If the system is linear, "exponentially bounded" is equivalent to internal stability.

and if 4. is true, why the function must be "exponential decaying" ? How is this restricted to be "exponential" ?

• whether they are continuous-time LTI (which are built out of adders, scalers, and integrator elements) or discrete-time LTI (built out of adders, scalers, and delay elements), the exponential function is what we call an "eigenfunction". exponential going in means an exponential comes out. Commented Mar 24, 2015 at 3:51

Let's consider discrete-time systems. The generalization to continuous-time systems is straightforward.

First note that BIBO stability does not require that the impulse response be square summable. It is necessary and sufficient that the impulse response be absolutely summable:

$$\sum_{n=-\infty}^{\infty}\left|h[n]\right|<\infty\tag{1}$$

In the context of stability of a causal LTI system, an exponentially bounded impulse response must satisfy

$$|h[n]|\le a\rho^{n},\quad n\ge 0\tag{2}$$

for some positive constants $a$ and $\rho$, with $0<\rho<1$.

If a causal LTI system is described by a rational transfer function, it can be shown that condition (1) for BIBO stability is equivalent to the condition that all poles of the transfer function lie inside the unit circle of the $z$-plane. This is equivalent with the system's impulse response being a weighted sum of decaying exponentials, and, consequently, condition (2) is always met by choosing $\rho$ equal to the largest pole radius, and by selecting an appropriate value for $a$.

In sum, condition (2) is a special case of general BIBO stability for LTI systems with rational transfer functions.

Also take a look at this answer, which also mentions non-linear systems.

• Thank you. Do you mean there are also systems which satisfies (1) and not satisfies (2)? and such system does not have rational transfer function?
– user15151
Commented Mar 23, 2015 at 11:46
• @ShuS: Yes, even though I don't know how to realize such a system. How about $h[n]=1/n^2$, $n\ge 1$ (and zero otherwise)? It describes a BIBO system but it's not exponentially bounded. Commented Mar 23, 2015 at 20:29