Stability

In system theory, stability is defined as the defining characteristic of a system with a bounded system response. Stable systems are important in engineering. This often means that we can control how they behave.¹ Unstable systems are also dangerous — and can lead to component damage or failure.

One important definition of stability is that of bounded-input bounded-output (BIBO) stability. A linear CT system $T$ is said to be BIBO stable if there exists a constant $K\ge 0$ such that $\Vert y{\Vert }_{\infty }\le K\Vert x{\Vert }_{\infty }$ for all bounded inputs $x$ and outputs $y$.

i.e., bounded inputs produce bounded outputs

We determine if a system is BIBO stable by giving a bounded input and seeing if the result is unbounded. In terms of a transfer function, BIBO stability requires the poles to be in the open left-hand plane (i.e., negative or zero real part). Equivalently, this requires the eigenvalues of matrix $A$ to be in the OHLP.

A system is marginally stable if only certain bounded input signals result in bounded output signals, i.e., there exists bounded signals such that they result in an unbounded output.

The Routh-Hurwitz criterion gives us a necessary criterion that are satisfied by stable linear systems. This allows us to determine the stability, and compute values that make the system stable.

Footnotes

1. More or less from one of Prof Najm’s many lectures. ↩