Build a response from tiny impulses
The impulse response is an LTI system's reusable response pattern; convolution adds shifted copies of it.
First, picture it
Imagine a bell tapped at several times. Each tap starts its own ring-down. If the bell behaves linearly, the sound you hear is the sum of those overlapping ring-downs. A continuously changing input can be thought of as many tiny taps.
What the model says
For a zero-initial-state continuous-time LTI system, h(t) is the output to an ideal unit impulse. An input u(t) can be assembled from weighted, time-shifted impulses; the output is y(t)=∫h(t−τ)u(τ)dτ. For a causal system the integral only needs the past and present. A step is the accumulation of impulses, so its response is the integral of h under suitable conditions.
y(t)=(h*u)(t)=∫₀ᵗ h(t−τ)u(τ)dτ [causal, zero-state]This integral form assumes ordinary causal signals and no separate direct-feedthrough impulse term; state the model when using it.
Each idealized impulse starts a shifted response; an LTI output adds them. The Lab's finite pulse is not an ideal impulse.
A one-line example
For a first-order system with h(t)=e^(−t) for t≥0, a unit step u(t)=1 for t≥0 gives y(t)=∫₀ᵗ e^(−(t−τ))dτ=1−e^(−t). The step curve starts at zero and approaches one. This is the same dynamics viewed through two different test inputs.
In a discrete-time controller the integral becomes a weighted sum. The plant can still evolve continuously between controller updates, so do not confuse the controller's sampling period with the physics integration step.
A finite tap is not a Dirac impulse
The Lab's motor tap load is a finite-duration torque pulse. It is useful for seeing how a disturbance starts and ends, but its area and shape depend on magnitude and duration. Shortening the pulse at fixed height also reduces its total impulse; it does not automatically approximate a unit Dirac impulse.
Make it concrete
Pin a motor run with one tap load. Change only its duration and compare when speed first dips, how far it dips and when it recovers.
Change a finite load pulseMake a prediction before moving a control.
- PREDICTWill a longer pulse usually affect the motor for longer?
- CHANGE ONE THINGPin one tap-load run, then change only pulse duration.
- OBSERVECompare disturbance interval, speed dip and recovery.
- EXPLAINDistinguish a finite pulse from the mathematical unit impulse.
Convolution with one fixed h is a global prediction only while the same LTI model and zero-state conditions hold. Saturation or changing controller gains during a run violates that simple setup.
Check your understanding+
Why does a step response contain information about the impulse response?
A step can be represented as accumulated impulses; for a suitable causal LTI system its response is the time integral of h.