What LTI really promises
Linear and time-invariant are two separate tests. Passing both lets us reuse one response to predict many others.
First, picture it
If two small voltage changes produce two speed changes, linearity says their combined input produces the sum of those changes. Time invariance says the same experiment tomorrow gives the same shape, merely shifted in time, if the plant's parameters have not changed.
What the model says
For an input-output operator H, linearity means H[a u₁+b u₂]=aH[u₁]+bH[u₂] for the chosen zero-state or deviation-variable description. Time invariance means delaying an input by τ delays its output by τ. A constant-coefficient linearized model at a fixed equilibrium can satisfy both. A changing operating point, scheduled gain, saturation or rail contact can break the LTI description of the complete run.
H[a u₁+b u₂]=aH[u₁]+bH[u₂]; H[u(t−τ)]=y(t−τ)The first relation assumes comparable initial conditions; the second assumes the system itself does not change with time.
Which part of the lab is LTI?
The motor's electrical-mechanical equations use fixed coefficients, but the displayed closed-loop experiment also includes a sampled controller, limited voltage and possibly anti-windup. The nominal frequency panel studies a specified linear sampled loop. Its prediction should be compared with a run where the voltage stays within limits and the same operating conditions hold.
LTI is useful precisely because it is a clearly stated model class. A model that fails an assumption may still help locally, but do not use its Bode plot as a global guarantee for a nonlinear trajectory.
Make it concrete
Compare a moderate motor target with a much larger one at fixed gains. If requested and applied voltage separate in the large run, that run has crossed a clear nonlinearity.
Find the voltage limitMake a prediction before moving a control.
- PREDICTWill a larger target simply scale the response?
- CHANGE ONE THINGPin a moderate motor target, then raise only the target.
- OBSERVEInspect requested versus applied voltage before comparing speed.
- EXPLAINState where linear scaling stopped being a justified expectation.
A straight-looking output trace does not prove linearity. Linearity is a relation across multiple input experiments under consistent initial conditions.
Check your understanding+
Can a plant model be linear while a voltage-limited closed-loop run is nonlinear?
Yes. Clipping changes the mapping from the controller command to the applied plant input.