PI and PID Tuning of Plants up to Third Order for a Monotonic Minimum Settling Time Solution

Abstract

A unified, closed-form analytical PI/PID tuning method is presented for all-pole plants up to third order that yields a strictly monotonic (zero-overshoot) step response with minimum settling time. The design target is the binomial closed loop p^n/(s+p)^n, which is monotonic with robustness depending only on the order n. Because adding a left-half-plane zero to a fixed pole pattern only slows the response, the minimum-settling solution requires the controller zeros to be cancelled, which forces the controller numerator to divide the plant denominator. Carrying this principle through shows that an exact, real-gained solution exists for any stable plant precisely up to second order with a PI controller and third order with a PID controller; the residual binomial factor acquires a complex pair beyond that, which a generic plant does not contain. Explicit gains are derived for first-order plants (PI), second-order plants with real and complex poles (PI and PID), and third-order plants with three real poles and with one real pole plus a complex pair (PID). The second-order PI case is treated in full as the lowest-order instance. Monotonicity guarantees Mt = 1, hence Ms less then 2, phase margin above 60 degree, and gain margin above 6 dB, tightening to universal constants for the binomial family. Numerical verification confirms the results.