Control System Design

Design workflow

Translate requirements into measurable tracking error, overshoot, settling time, disturbance rejection, noise sensitivity and actuator limits. Choose a model valid over the operating range, design the controller, then verify nominal and perturbed behavior. A controller meeting one nominal step-response target may fail under delay, saturation or model error.

Useful relationships

For unity negative feedback, sensitivity S = 1/(1 + L) and complementary sensitivity T = L/(1 + L), with S + T = 1. Reducing sensitivity over one band does not remove all design tradeoffs. High loop gain can improve low-frequency tracking but increase control effort and sensitivity to high-frequency noise. For a canonical second-order closed loop ωn²/(s² + 2ζωn s + ωn²), with 0 < ζ < 1 and no extra zeros, fractional overshoot is exp(−πζ/√(1 − ζ²)). The common 2% settling-time estimate is 4/(ζωn); it is an approximation.

Worked example

A target of roughly 5% overshoot suggests ζ ≈ 0.69. Choosing ωn = 3 rad/s gives estimated settling time 4/(0.69 × 3) = 1.93 s and poles −2.07 ± j2.17. These are target poles, not a completed controller design. The actual plant, zeros, realizability and actuator limits determine whether a controller can achieve them. After designing a controller, calculate the actual closed-loop transfer and test the requirements again.

Verification checklist

Test reference steps and ramps relevant to the application, load disturbances, measurement noise, initial-condition responses, saturation and parameter uncertainty. Confirm internal stability and adequate robustness margins; monitor control effort as well as output. For implementation, include sampling, computation and sensor delays.

Quick check

1. Is low overshoot alone sufficient? No; speed, accuracy, robustness and effort also matter.
2. Does the second-order estimate apply to arbitrary high-order systems? Only as a justified approximation when dominant-mode assumptions hold.
3. Why test disturbances separately? Their transfer path can differ from the reference path.