Tracking, disturbance and sensitivity
A target, a load and sensor noise do not enter the feedback loop at the same place.
Read sensitivity and complementary sensitivity at one frequency, with the signal entry point stated.
We have designed controllers and estimates. Now ask which external signal causes the output change: a target, a disturbance, or sensor noise.
Review the previous lesson: LQG: control what you estimateFirst, picture it
A motor can follow a new speed quickly but still dip when a load arrives. Looking only at the target-step plot hides the second task. Write down where each signal enters before deciding which curve matters.
Build the idea step by step
Label where each signal enters the loop before calculating its effect. A target enters at the comparison point. Sensor noise enters the measurement. An external force can enter the plant input. The functions S and T below describe common signal paths in a linear unity-feedback loop. Their sum is one, so they cannot both be tiny at the same frequency. Improving slow disturbance rejection can come with greater sensitivity to measurement noise elsewhere.
Read the equations
For unity negative feedback, S maps an additive output disturbance to output, while T maps reference to output. Measurement noise maps through minus T. A disturbance entering the plant input has a different path.
What the symbols mean
- loop transfer function at the frequency of interest
- sensitivity function
- complementary sensitivity function
The worked example adds a disturbance at the output. The motor lab's load is a torque entering the plant, so its full transfer path is different. During saturation the simple linear S and T relations do not describe the complete run.
High loop gain has a tradeoff
At one frequency, suppose the loop value is the positive real number 9. Use the stated unity-feedback arrangement.
Sensitivity is 0.1 and complementary sensitivity is 0.9.
An additive output disturbance of amplitude 2 produces output amplitude 0.2 at this frequency. A measurement-noise amplitude of 2 produces 1.8 instead. Rejecting one signal does not mean rejecting every signal.
More detail and model assumptions
One loop, several questions
Reference tracking, load rejection and sensor-noise response follow different paths. The S and T functions are a compact way to reason about a linear unity-feedback loop, but first identify where a disturbance enters. The Lab's motor load is a torque entering the plant, not a signal added to the measured output.
S+T=1 means you cannot make both small at the same frequency. This is a design tradeoff, not a command to maximize loop gain everywhere. The next chapter asks whether the design still works when the plant differs from its nominal model.
Draw three arrows on the same loop
Put the target before the error junction, the load torque at the motor's mechanical equation, and sensor noise at the measurement. A controller may reject a slow load while amplifying a noisy high-frequency measurement. The entry point decides which transfer path to calculate; S alone is not the transfer from every possible disturbance to speed.
First compare runs with the same controller and different injected signals. Only after you identify a failing requirement should you retune the controller. Otherwise you cannot tell whether the improvement came from the design or from a changed test.
Connect the example to the lab
Pin the unloaded target response, then add a step load at 4 s without changing the controller. Compare speed error and applied voltage. Repeat with I enabled, checking for saturation.
Compare tracking and load rejectionMake a prediction before moving a control.
- PREDICTDoes a fast target response prove load rejection?
- CHANGE ONE THINGPin an unloaded motor run, then add a load at 4 s without changing gains.
- OBSERVECompare target tracking before 4 s with the speed dip and applied voltage afterward.
- EXPLAINIdentify the load's injection point and which requirement needs its own test.
A good target response is not proof of disturbance rejection or noise tolerance; test all three with the same stated model and constraints.
Check your understanding+
Why can a target-step test miss a load-rejection problem?
The reference and input-load disturbance enter at different points and follow different transfer paths.Why the next lesson follows
These paths were calculated from a model. Robustness asks whether the design still works when that model is wrong.