Build a response from tiny impulses
The impulse response is an LTI system's reusable response pattern; convolution adds shifted copies of it.
Build a step response by accumulating a known impulse response.
Causality restricts us to present and past inputs. Convolution adds the effect of those past inputs using one impulse response.
Review the previous lesson: A controller cannot read tomorrow's measurementFirst, 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.
Build the idea step by step
An ideal unit impulse is a mathematical input concentrated into an instant, with total area one. Its output is the impulse response. For a linear system, each small piece of input starts a scaled response. Time invariance lets us shift that response to the time when the input arrived. Add these responses to get the total output. The integral below is this addition over continuous time, called convolution. Start with no stored energy so that an initial-state response does not need to be added separately.
Read the equations
For a causal continuous-time LTI system at zero initial state, add the impulse response shifted to each past input time.
What the symbols mean
- output caused by a unit impulse; its units are output units divided by (input units times seconds)
- input signal
- output signal
- past time of one input contribution, in seconds
- current time, seconds; t minus τ is elapsed time since an earlier input
The integration variable marks when each earlier input arrived. The difference between the present time and that earlier time is how long its response has been evolving. This example has no instantaneous input-to-output term; such a term would be added separately.
Each idealized impulse starts a shifted response; an LTI output adds them. The Lab's finite pulse is not an ideal impulse.
A unit step from an impulse response
Take an impulse response that decays exponentially and a unit step input. Time is in seconds and input/output are normalized.
Add every shifted impulse contribution from zero to the current time.
After one second the output is about 0.632. It approaches 1 but does not jump there instantly.
More detail and model assumptions
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.
Connect the example to the lab
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.Why the next lesson follows
The Laplace transform turns this time-domain accumulation and differentiation into algebra.