Internal Stability#

An LTI 🇺🇸 State Space Model system is asymptotically stable if the solution to (free response) satisfies

Analyzing the behavior of 🇺🇸 State Space Model > Solutions#Free Response, we see that is asymptotically stable if and only if all eigenvalues of have negative real parts. Such a matrix with eigenvalues that have strictly negative real parts is called Hurwitz.

Alternatively, the system is marginally stable if:

1. All eigenvalues have non-positive real parts.
2. Those with real parts equal to have Jordan subblocks of size 1.

BIBO Stability#

We can define other type of stability regarding the input and output of the system. First, a signal is bounded if there exists constant such that

for all . Then, a system is BIBO stable if every bounded input produces a bounded output.

Note that asymptotic stability implies BIBO stability. Intuitively, this is because both states and inputs are bounded; more formally, we can check

Since is a combination of the eigenvalue terms with negative real parts, they are absolutely integrable and thus bounded.

However, BIBO stability does not imply asymptotic stability. Intuitively, this is because the output equation may "hide" some of the instability in the system, such that we still get bounded outputs even when states grow exponentially. For instance, with

we have , where grows exponentially and thus fails to be asymptotically stable. However, only checks the second state, and thus , making it BIBO stable.