= Bounded-input bounded-output stability
= BIBO stability
{c}
{synonym}
= BIBO stable
{c}
{synonym}
<Bounded-input bounded-output stability> means that every bounded <plant input> produces a bounded <plant output> from zero initial state. For a causal continuous-time <linear time-invariant system> with a <proper rational transfer function>, <BIBO stability> is equivalent to every <pole> of the reduced <transfer function> having strictly negative <real part>. Cancel common factors first: an apparent denominator root may be absent from the reduced <transfer function>. For $K/(\tau s+1)$ with $\tau>0$, the only <pole> is $-1/\tau$, so the <first-order low-pass filter> is <BIBO stable>. https://ocw.mit.edu/courses/6-241j-dynamic-systems-and-control-spring-2011/996025f6db0d90b00f11c44fc49b85f9_MIT6_241JS11_textbook.pdf[Dahleh, Dahleh and Verghese, section 15.3.1 and example 15.5: input/output versus internal stability]{ref}
Internal <asymptotic stability> also requires every unforced internal state to decay. A canceled factor can hide a growing internal state. For example, consider the <linear system of ordinary differential equations>
$$
\dot x_1=-x_1+u,\qquad \dot x_2=x_2+u,\qquad y=x_1.
$$
Its zero-state <transfer function> can be written $(s-1)/((s+1)(s-1))=1/(s+1)$ and has <BIBO stability>. Yet from zero <initial conditions> and a unit-step <plant input>, $x_2(t)=e^t-1$ grows without bound while the <plant output> remains bounded. The reduced <transfer function> conceals that second state. Even a <transfer function> with no visible cancellations can be implemented with additional hidden states; its <poles> alone do not establish internal <asymptotic stability> of an arbitrary implementation.
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