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

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

Problem Statement
The paper addresses the control of stable, all-pole plants up to third order using standard Proportional-Integral (PI) and Proportional-Integral-Derivative (PID) controllers. The primary objective is to achieve a step response that is strictly monotonic (zero overshoot) while simultaneously minimizing the settling time. These objectives often compete; achieving zero overshoot typically requires slower dynamics, while minimizing settling time often invites overshoot. The paper seeks a unified, closed-form analytical tuning method that satisfies both constraints without relying on numerical fitting or state-space structures, adhering instead to standard fixed-structure PID controllers.

Methodology
The proposed methodology targets a specific closed-loop transfer function shape: the binomial form $T^*(s) = p^n / (s + p)^n$. This form is selected because its impulse response is non-negative for all $n$, guaranteeing a monotonic step response, and its robustness metrics depend solely on the order $n$, not the speed parameter $p$.

To achieve this target with minimum settling time, the paper establishes a "cancellation principle." It argues that for a fixed pole pattern, adding a left-half-plane zero (from the controller) only slows the response. Therefore, the minimum-settling solution requires the controller's zeros to be exactly cancelled by the plant's poles. This forces the controller numerator $N(s)$ to divide the plant denominator $M(s)$.

The derivation proceeds by equating the closed-loop characteristic polynomial to the desired binomial form. This leads to two distinct construction branches:

1. Cancellation Construction: Used for plants with well-separated real poles. The controller zeros are placed exactly on the plant poles to cancel them, leaving a pure binomial closed-loop system.
2. Coincident Placement Construction: Used for plants with clustered real poles or complex conjugate poles. Here, all $m+1$ closed-loop poles (where $m$ is the plant order) are placed at a single location $-\sigma$. The controller zeros are retained but positioned to the left of $-\sigma$ to maintain monotonicity.

The paper derives explicit gain formulas for:

• First-order plants: PI control with pole cancellation.
• Second-order plants (Real poles): A boundary exists at a pole ratio $T_1/T_2 = 2$. For ratios $\ge 2$, pole-zero cancellation is used; for ratios $< 2$ (clustered), coincident placement is used.
• Second-order plants (Complex poles): PI control is feasible only if damping $\zeta > \sqrt{3}/2$. For lower damping, PID control is required to maintain feasibility.
• Third-order plants: PID control is used. For three real poles, cancellation is used for well-separated poles, and coincident placement for clustered poles. For one real pole and a complex pair, coincident placement is used, provided the complex pair is not too lightly damped.

Key Contributions

• Unified Analytical Tuning: The paper provides explicit, closed-form gain formulas for PI and PID controllers for all-pole plants up to third order, covering both real and complex poles.
• Identification of Order Limits: The work rigorously demonstrates that an exact, real-gained solution for a strictly monotonic, minimum-settling response exists only up to second order with PI and third order with PID. Beyond third order, the required residual factor in the polynomial division acquires complex roots (damping $\sqrt{3}/2$) that generic plants do not possess, making exact cancellation impossible with standard controllers.
• Boundary Conditions: The paper explicitly defines the transition boundaries between cancellation and coincident strategies (e.g., the $T_1/T_2 = 2$ ratio for second-order real plants and the $\zeta = \sqrt{3}/2$ limit for second-order complex plants).
• Robustness Guarantees: By enforcing monotonicity, the method guarantees universal robustness bounds: Maximum Sensitivity $M_s \le 2$, Phase Margin $\ge 60^\circ$, and Gain Margin $\ge 6$ dB. For the specific binomial family, these values are universal constants dependent only on the system order.

Results
Numerical verification confirms that the derived gains produce strictly monotonic step responses with $M_t = 1$ (maximum complementary sensitivity).

• Settling Time: The coincident placement strategy for clustered poles yields faster settling times than simple pole-zero cancellation, with reductions of up to 17.1% for second-order plants with equal time constants.
• Robustness: The derived designs achieve the theoretical robustness limits of the binomial family. For example, a second-order binomial system achieves a phase margin of $76.35^\circ$ and a gain margin of $\infty$, while third and fourth-order systems maintain margins above $68^\circ$ and $14$ dB respectively.
• Feasibility: The paper confirms that for third-order plants with a complex pair, if the damping is too low, the PID gains become negative, indicating the limit of the method for that specific plant configuration without model reduction.

Significance and Claims
The paper claims to provide a complete, closed-form solution for the "minimum settling time with zero overshoot" problem for plants up to third order. Its significance lies in:

1. Simplicity and Rigor: It replaces heuristic or numerical tuning with exact algebraic solutions derived from first principles.
2. Clear Boundaries: It precisely identifies where the standard PID structure fails to achieve the ideal binomial target (i.e., beyond third order), offering a clear theoretical limit.
3. Practical Application: For higher-order plants, the paper suggests a practical route of model-order reduction (e.g., using the half-rule) to approximate the plant as a third-order surrogate, thereby allowing the application of these guaranteed monotonic tuning rules.

The work positions itself as a definitive analytical framework for low-order all-pole systems, ensuring that the trade-off between speed and overshoot is resolved optimally within the constraints of standard industrial PID controllers.