Control Systems, Stability, and Robustness
Feedback as a Structural Principle
Closed-Loop Behavior
Feedback changes system behavior by comparing desired and actual outputs and correcting the difference. In classical form, a plant and controller produce the loop transfer function The closed-loop transfer is Feedback can stabilize unstable plants, reject disturbances, and improve tracking, but it can also amplify noise if poorly designed.
What is the closed-loop transfer function for unity negative feedback?
For unity negative feedback, the closed-loop map is .
Correct answer: C:
What does internal stability mean in a closed-loop system?
Internal stability is stronger than input-output stability because it includes hidden states and interconnections.
Correct answer: All internal signals remain bounded for bounded initial conditions and bounded inputs
Stability Margins and Frequency Response
Gain, Phase, and Loop Shaping
Frequency-domain analysis uses the loop transfer to study disturbance rejection and robustness. Gain margin and phase margin quantify how much uncertainty the system can tolerate before becoming unstable. Designers shape to achieve adequate bandwidth, low steady-state error, and acceptable robustness.
Control Design Concepts
| Concept | Purpose | Typical metric |
|---|---|---|
| Bandwidth | Speed of response | Closed-loop cutoff frequency |
| Phase margin | Robustness to delay | Degrees at gain crossover |
| Steady-state error | Tracking accuracy | Error constants |
| Sensitivity | Disturbance/uncertainty amplification |
Why is phase margin important?
Phase margin indicates how much additional phase lag is needed to destabilize the loop, so it is a robustness indicator.
Correct answer: B: It quantifies robustness to phase lag and delays
Define the sensitivity function for a unity-feedback system.
Sensitivity captures how uncertainties and disturbances propagate through the loop.
Correct answer: S(s) = 1/(1+L(s))
Advanced Practice
Robust control design often uses loop-shaping ideas together with structured uncertainty models, not just nominal pole placement.