Bounded input, bounded output: what is being promised?
BIBO stability is a statement about every bounded input and the observed output, not about one attractive plot.
First, picture it
A small bounded push should not make the observed response grow without bound. A motor that looks calm for twelve seconds under one load has passed only that test; another input, a longer time or an unobserved internal state may tell a different story.
What the model says
BIBO means: for each input bounded for all time, the selected output remains bounded for all time under the stated zero-state input-output model. For a causal continuous-time LTI system without a problematic direct term, absolute integrability of the impulse response is the test. In a proper rational minimal realization, poles strictly in the left half-plane give the familiar stable case. This input-output property must not be silently substituted for internal stability of every state.
|u(t)|≤M for all t ⇒ |y(t)|≤N; ∫₀∞|h(t)|dt<∞N may depend on M and the system, but not on time. The impulse-response test is for the stated LTI input-output setting.
Two small counterexamples
An ideal integrator has y(t)=∫₀ᵗu(τ)dτ. Feed it the bounded constant input u=1 and its output becomes y=t, which is unbounded. 'The input is harmless-looking' is not enough.
Now imagine two internal states: ẋ₁=−x₁+u, ẋ₂=+x₂, while the sensor reports only y=x₁. The input-to-output relation can look stable, but an initial x₂≠0 grows invisibly. This is why a canceled or unobservable unstable mode matters even when a transfer-function plot looks safe.
What the Lab can and cannot show
You can pin runs, compare bounded load pulses, inspect finite-duration output and watch for rail exit. These are diagnostic experiments, not a proof over every bounded input and infinite time. Use an analytic model and its assumptions for a mathematical claim, then use experiments to challenge those assumptions.
Make it concrete
Apply one small and one larger load pulse to the motor, keeping controller settings fixed. Report finite-run behavior without labeling either result a BIBO proof.
Compare finite load testsMake a prediction before moving a control.
- PREDICTWill this load pulse make the response diverge in the displayed window?
- CHANGE ONE THINGChange only the pulse magnitude and pin both runs.
- OBSERVERecord the window length, output, input limit and any fault.
- EXPLAINSeparate your observation from the unproved all-input BIBO claim.
A stable transfer function can conceal unstable internal dynamics when the realization is not minimal. Conversely, a single diverging nonlinear saturated run is not automatically a proof about an LTI transfer function.
Check your understanding+
Is a bounded 12-second motor trace sufficient to establish BIBO stability?
No. BIBO quantifies over all bounded inputs and all time in a stated input-output model.