What LTI really promises
Linear and time-invariant are two separate tests. Passing both lets us reuse one response to predict many others.
Test linearity and time invariance as two separate claims, with the same starting state.
The tank's local model and the ideal RLC equations become easier to analyze when their coefficients stay fixed. These are examples of linear time-invariant models within their stated range.
Review the previous lesson: RLC dynamics and electrical resonanceFirst, 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.
Build the idea step by step
LTI means linear and time-invariant. Linear means two rules hold: doubling an input doubles its response, and adding inputs adds their responses. Test this from a zero initial state, or for changes from the same equilibrium. Time-invariant means starting the same input later shifts the same response later. The device itself has not changed. A fixed linear model can have both properties; a voltage limit or changing controller gain can break them.
Read the equations
For a zero-state linear system, a weighted sum of inputs produces the same weighted sum of outputs.
What the symbols mean
- input-to-output rule for the system
- two input signals, in the input's units
- unitless weights used in this test
H means the rule that turns an input into an output. It is not a number multiplying the input. The letters a and b are constant weights. We need both linearity and time invariance before using one impulse response to describe every input.
Check scaling in a first-order model
Use the zero-state model: output rate equals minus output plus input. Input, output, and time are normalized and unitless in this example.
A constant unit input gives output 1 minus the decaying exponential.
Doubling the input doubles this output at every time. Saturation would break this scaling test.
More detail and model assumptions
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.
Connect the example to the lab
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.Why the next lesson follows
Even an LTI model must respect time order: an output cannot depend on a future input.