# Alternative to BIBO stability of a system

In DSP textbooks a system is stable in the BIBO (Bounded-Input, Bounded Output) sense if and only if every bounded input sequence produces a bounded output sequence. After stating this definition stable always means BIBO-stable.

I'd like to know if there exists other form of stability for a system?

• There's numerical stability when dealing with quantization, but that's a pretty different concept. – endolith Feb 10 '15 at 18:00

For linear systems, BIBO stability is the most useful and practical criterion. For systems described by rational transfer functions it coincides with the condition that all transfer function poles must be located in the left half of the $s$-plane (for continuous-time systems), or inside the unit circle of the $z$-plane (for discrete-time systems). For linear systems, I do not know of any other stability criterion that makes any sense and does not coincide with the BIBO-stability criterion. Since most basic DSP texts focus on linear (and especially time-invariant) systems, you will only find the notion of BIBO-stability there.

Things look a bit different for non-linear systems. There you have several reasonable stability criteria that do not all lead to the same basic criterion as is the case for linear systems. One important notion for non-linear systems is input-to-state stability, which basically means that for zero input, the system is stable about its zero state, and that well-behaved and bounded input signals produce a bounded state trajectory. This article reviews some of these concepts.

But if you're mainly interested in linear systems, BIBO-stability is all you need.

It has been some time since I've looked at Chaos theory. Since Chaos theory is mainly examining non-linear equations, the idea of BIBO isn't sufficient. One of the ways to characterize a non-linaer system is via Lyapunov exponents. It is a way of characterizing the predictability of the system.

This is definitely not the usual thing that's covered in DSP, since DSP is mostly concerned with Linear Systems or systems that have been linearized.

Another version of stability comes from Compressive Sensing (CS) or Sparse Reconstruction. In CS you are looking at solving $\mathbf{Ax}=\mathbf{b}$ where there are more equations than unknowns but we want an vector $\mathbf{x}$ to be sparse i.e. only have a few non-zero elements (the other elements should be zero or small in comparison to the few large elements). Note that because the system is under-determined there are multiple solutions for $\mathbf{Ax}=\mathbf{b}$.

A concern in CS is the stability of the answer i.e. if we recover a sparse vector we want other sparse solutions to be nearby, so if we don't have an exact answer we are at least within a known error bound of the correct answer. If we can find the largest real number $\alpha$ and the smallest real number $\beta$ such that $$\alpha||\mathbf{x}_1-\mathbf{x}_2|| \leq A(\mathbf{x}_1-\mathbf{x}_2) \leq \beta||\mathbf{x}_1-\mathbf{x}_2||$$

for all vectors $\mathbf{x}_1$ and $\mathbf{x}_2$. Then we can bound the error on the recovered vector in cases of: noise-free, noisy, and modelling error i.e. error in the $\mathbf{A}$ matrix. The above equation is referred to as the bi-Lipchitz condition and leads directly to idea of the Restricted Isometry Constraint (RIC) - which is calculating the values of $\alpha$ and $\beta$ when the vectors $\mathbf{x}$ are constrained to be s-sparse i.e. have s elements or fewer that are non-zero. Note that RIC is a sufficient condition but not a necessary one. Other characterizations include: the Null Space Property, the Spark, and Mutual Coherence etc. Good overviews of the theory are: Blumensath, and Davenport

Again this is a highly specialized area and beyond the usual scope of DSP, but you asked the question :)