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. Unstable systems are also dangerous — and can lead to component damage or failure.¹

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

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. A system is asymptotically stable if the time response converges to 0 as time approaches infinity for every $x(0)$.

Determining stability

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). Some other criteria:

• The system is stable if and only if all components of $\exp (At)={\mathcal{L}}^{-1}(sI-A{)}^{-1}$ are bounded functions.
• The system is asymptotically stable if and only if all the eigenvalues of $A$ have a negative real part.
• It is BIBO stable if $A$ has all eigenvalues in OLHP (i.e., the left-hand plane excluding $\text{Re}(s)\ge 0$). Then, for any $B,C,D$, $G(s)$ is BIBO stable.
• The system is unstable if $A$ has at least one eigenvalue with a positive real part.
  • Note that it can still be unstable without this condition.
• A system is BIBO stable if and only if all the poles of $G(s)$ have a negative real part.
• It is BIBO unstable if and only if $G(s)$ has at least one pole with real part $\ge$ 0.
• If a system is asymptotically stable, it is BIBO stable. The inverse is not necessarily true.

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. The Nyquist stability criterion is an important visual criterion satisfied by feedback systems.

A closed-loop system is BIBO stable if and only if all the transfer functions in the matrix below are BIBO stable (with all poles in OLHP) and there are no unstable zero-pole cancellations in $C(s)G(s)$, i.e., a zero in one function cancels out a pole in the other.

$$\left[\begin{array}{c}E(s)\\ U(s)\end{array}\right]=\left[\begin{array}{cc}{F}_{11}(s)& F_{12}(s)\\ F_{21}(s)& F_{22}(s)\end{array}\right]\left[\begin{array}{c}R(s)\\ D(s)\end{array}\right]$$

Footnotes

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