Closed-Form PI and PID Tuning of All-Pole Plants up to Third Order for Monotonic Minimum-Settling Step Responses

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 a fixed PI/PID cannot assign the closed-loop poles and the controller zeros independently, realizing this target exactly requires the controller zeros to be cancelled, which forces the controller numerator to divide the plant denominator. It follows 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; beyond that the residual binomial factor acquires a complex pair of damping sqrt(3)/2, 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 or one real pole plus a complex pair (PID). The freedom of the coincident designs is shown to be bounded: a quadratic nonnegativity condition gives the exact window of the design pole for strict monotonicity, which collapses at the pole-ratio-2 changeover for real poles and is nonempty for damping ratios above approximately 0.443 for complex poles. Monotonicity guarantees Mt = 1, hence Ms <= 2, phase margin >= 60 degrees, and gain margin >= 6 dB, tightening to universal constants for the binomial family. Load-disturbance attenuation obeys IAEd = 1/Ki, making the cost of cancellation explicit, and comparisons with SIMC, the CHR zero-overshoot rule, and deadbeat-fitted explicit formulas quantify the trade: at matched maximum sensitivity the proposed design settles faster than SIMC on the third-order example, with markedly lower controller gains and peak control effort.