Analytical PI Tuning for Second-Order Plants with Monotonic Response and Minimum Settling Time

This study presents two analytical closed-form PI controller tuning solutions for second-order plants with real poles, each achieving monotonic step response and minimum settling time. The first solution employs pole-zero cancellation, placing the controller zero at the slower plant pole and reducing the closed-loop dynamics to a critically damped second-order system. The second solution, applicable when the plant pole ratio is less than two, places all three closed-loop poles at a common location without cancelling any plant pole, yielding a closed-loop transfer function with a triple real pole and a zero. Despite retaining a closed-loop zero, this solution achieves strictly faster settling time than the pole-zero cancellation method in its region of applicability. The two solutions coincide at the boundary pole ratio of two and together form a continuous piecewise-analytical tuning covering the full range of plant pole ratios. This study further establishes that closed-loop transfer functions of the form a^n/(s + a)^n possess a maximum sensitivity Ms together with phase margin and gain margin that are independent of the pole location a and depend solely on the order n, yielding universal robustness constants for each n. A closed-form expression GM(n) = 1 + sec^n(pi/n) is established for the gain margin of the family. Numerical verification confirms the analytical results across multiple plant configurations.