A controller cannot read tomorrow's measurement
Causality says the present output can depend on present and past inputs, but not on an input that has not happened.
Explain why a later disturbance cannot change an earlier trace under the same initial conditions.
Linearity tells us how responses combine. Causality tells us which input times may affect the output now.
Review the previous lesson: What LTI really promisesFirst, picture it
A motor cannot react to a load pulse before the pulse reaches it unless the controller already knows the planned event. A forecast is a model-based guess, not a future sensor reading. This distinction matters whenever an offline plot or filter seems to anticipate a change.
Build the idea step by step
A causal system uses only information available now or earlier. Compare two runs from the same initial state. If their inputs have matched up to this instant, their outputs must match up to this instant too. The output may depend on the input immediately; it just cannot use a future measurement. Causality does not tell us whether the response stays small. We will study that separate question under stability.
Read the equations
If two inputs are identical up to now, a causal system must have the same output now when it starts from the same state.
What the symbols mean
- a time at or before the current time, in seconds
- current time, in seconds
- two possible input histories
- outputs produced from the same initial state
The two runs must start with the same stored state. A controller may use a known schedule, but that schedule is already available information, not a measurement from the future.
With identical initial conditions and settings, adding a later load cannot change the earlier output.
A load added later
Use a first-order normalized system that starts at zero. Keep the input at zero until 4 seconds.
One trial remains at zero. In another, input becomes 1 at 4 seconds. Both outputs are zero before that change.
At 5 seconds the second output is about 0.632, while the first remains zero. The later input changes the future, not the past.
More detail and model assumptions
Prediction is allowed; future measurements are not
A controller can use a schedule known in advance, such as a planned reference change, or predict future states from a model. It cannot use the actual future load measurement before that load occurs. A zero-phase filter can use samples on both sides of a time point when processing a completed recording; the same operation cannot run in real time without delay or approximation.
Look for the event boundary
If two experiments have the same seed, initial state and controls, and one has an added load at 4 s, their states before 4 s should agree within numerical tolerance. Divergence before the event would suggest changed setup, a replay bug or a noncausal calculation—not a genuine early physical response.
Connect the example to the lab
Pin a motor run without a load, add a load event at 4 s, and compare both traces before and after the event.
Compare before and after a loadMake a prediction before moving a control.
- PREDICTShould the loaded and unloaded traces differ before the load event?
- CHANGE ONE THINGKeep the motor setup fixed and add only a 4 s load event.
- OBSERVECompare traces on both sides of 4 s with the same replay cursor.
- EXPLAINExplain any early difference by checking changed conditions first.
Causal does not mean fast, stable or physically realizable under every other constraint. It only limits which input times may affect the present output.
Check your understanding+
Does a model-based prediction make a controller noncausal?
No. A prediction based on current and past information is causal; reading an actual future measurement would not be.Why the next lesson follows
For a causal LTI system, its response to one ideal impulse can build the response to any suitable input.