⚡ electrical engineering

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

This paper presents an explicit, closed-form analytical solution for tuning PI controllers on stable second-order plants with real poles to achieve a monotonic step response with minimum settling time, resulting in a critically damped system with universal robustness properties and no free parameters.

Original authors: Senol Gulgonul

Published 2026-04-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Senol Gulgonul

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are driving a heavy delivery truck (the plant) and you want to park it perfectly in a tight spot without hitting the curb (overshoot) and without taking forever to stop (settling time).

Most drivers (engineers) use a standard set of rules to figure out how hard to press the gas and brake (the PI controller). But often, they have to guess, test, and tweak the settings over and over again. Sometimes they stop too short, sometimes they overshoot and hit the curb, and sometimes they just take too long to get the job done.

This paper by Senol Gulgonul is like discovering a magic formula that tells you exactly how to drive that truck to stop perfectly, every single time, without any guessing.

Here is the breakdown of the "magic trick" using simple analogies:

1. The Problem: The Heavy Truck with Two Speed Limits

The "truck" in this story is a machine that naturally has two speeds of reaction:

The Slow Pole ( $T_1$ ): This is like the heavy cargo in the back. It takes a long time to get moving or stop.
The Fast Pole ( $T_2$ ): This is like the engine's immediate reaction. It reacts quickly.

The goal is to make the truck stop smoothly (no jerking forward or backward) and as fast as possible.

2. The Old Way vs. The New Way

The Old Way: Engineers usually try to balance the gas and brake by trial and error, or by using complex computer simulations to find the "best" settings. It's like trying to tune a radio by turning the dial randomly until the static clears.
The New Way (The Paper's Solution): The author found a specific mathematical "recipe" that works for any truck of this type. You don't need a computer; you just need to know the weight of the cargo and the engine speed.

3. The Secret Trick: "Canceling the Slow Part"

The most brilliant part of this paper is a concept called Pole-Zero Cancellation.

Imagine your truck has a heavy, slow-moving trailer (the slow pole) that drags everything down.

The Insight: Instead of trying to fight the trailer, the author says, "Let's attach a counter-weight to the trailer that perfectly balances it out."
How it works: The controller adds a "zero" (a counter-force) that matches the "slow pole" exactly. It's like if you had a heavy backpack slowing you down, and you found a magical spring that pulled you forward with exactly the same force as the backpack's weight. The backpack effectively disappears from the equation.
The Result: The heavy, slow truck suddenly behaves like a lighter, faster car. The "slow" problem is mathematically erased.

4. The "Goldilocks" Setting: Critical Damping

Once the slow part is canceled, you are left with a system that has two identical fast poles. Now, you have to decide how hard to brake.

Too soft: You stop slowly (too much time).
Too hard: You stop instantly but bounce back and forth (overshoot).
Just right (Critical Damping): This is the "Goldilocks" zone. You brake exactly hard enough to stop in the minimum time without ever bouncing back.

The paper proves that if you set your controller to this "Goldilocks" point, you get the fastest possible stop with zero overshoot.

5. The Magic Numbers (The Recipe)

The paper gives you the exact numbers to plug into your controller. You don't need to guess.

The Integral Time ( $T_i$ ): Set this exactly equal to the Slow Time of your machine. (Match the counter-weight to the heavy cargo).
The Gain ( $K$ ): Set this based on the ratio of the slow time to the fast time. (Adjust the gas pressure based on how heavy the cargo is compared to how fast the engine is).

6. The "Superpower": Universal Robustness

Here is the coolest part. Usually, if you change the weight of your cargo or the type of engine, you have to re-tune your brakes.

But with this specific recipe, the author discovered a Universal Superpower:

No matter what kind of truck you have (as long as it's a standard two-speed type), the safety margins are always the same.
The "Phase Margin" (a measure of how stable the system is) is always 76.35 degrees.
The "Peak Sensitivity" (how much the system might wiggle) is always 1.155.

It's like saying, "If you follow this recipe, your car will handle a snowstorm, a desert, or a mountain road with the exact same stability, no matter what car you are driving."

Summary

This paper is a breakthrough because it stops engineers from guessing. It says:

Cancel the slow part of your machine using a specific setting.
Tune for the "Goldilocks" stop (critical damping).
Result: You get the fastest, smoothest stop possible, and your system is perfectly safe and stable, no matter the specific details of the machine you are controlling.

It turns a complex, messy engineering problem into a simple, one-step instruction manual.

1. Problem Statement

The paper addresses a fundamental challenge in industrial control: tuning Proportional-Integral (PI) controllers for second-order plants (systems with two real negative poles) to achieve two competing objectives simultaneously:

Monotonic Step Response: The output must rise to the setpoint without overshoot or dipping below the final value (strictly non-decreasing). This is critical in applications like precision positioning, thermal control, and dosing where overshoot causes damage or safety violations.
Minimum Settling Time: The system must reach and stay within a specific error band (e.g., 2%) as quickly as possible.

The Gap: While existing methods (e.g., Ziegler-Nichols, optimization-based approaches) exist, they either fail to guarantee strict monotonicity, require numerical optimization (lacking analytical proof), or rely on assumptions that do not yield the absolute minimum settling time. Specifically, the paper seeks to resolve an open question from Astrom and Hagglund regarding the optimal pole placement for this specific scenario.

2. Methodology

The author derives a closed-form analytical solution using pole placement within the framework of Astrom and Hagglund. The methodology proceeds through the following logical steps:

System Model: The plant is modeled as $P(s) = \frac{K_p}{(1+sT_1)(1+sT_2)}$ with $T_1 \ge T_2 > 0$ . The controller is a standard PI: $C(s) = K(1 + \frac{1}{sT_i})$ .
Degree of Freedom Analysis: The closed-loop system is third-order, but the PI controller only offers two tunable parameters ( $K$ and $T_i$ ). Therefore, arbitrary pole placement is impossible because the coefficient of the $s^2$ term is fixed by the plant dynamics.
Key Insight (Forced Cancellation): Instead of assuming pole-zero cancellation, the author demonstrates that designing the closed-loop poles to be slower than the fast plant pole ( $-1/T_2$ $- 1/ T_{2}$ ) mathematically forces the cancellation of the slow plant pole ( $-1/T_1$ $- 1/ T_{1}$ ).
- By setting the integral time $T_i = T_1$ , the controller zero cancels the slow plant pole.
- This reduces the effective system order from third-order to second-order.
Optimization for Speed: Once reduced to a second-order system, the condition for monotonic response requires the damping ratio $\zeta \ge 1$ . To achieve the minimum settling time, the system must be critically damped ( $\zeta = 1$ ).
Derivation: Applying $\zeta = 1$ to the reduced second-order characteristic equation yields unique, explicit formulas for the controller parameters.

3. Key Contributions

The paper provides the following specific contributions:

Explicit Tuning Formulas: The optimal PI parameters are derived analytically with no free parameters:
$K = \frac{T_1}{4 K_p T_2}, \quad T_i = T_1$
These formulas depend solely on the plant's time constants ( $T_1, T_2$ ) and gain ( $K_p$ ).
Proof of Necessity: The paper proves that pole-zero cancellation of the slow pole is not an arbitrary design choice but a necessary consequence of the limited degrees of freedom in a PI controller when aiming for the fastest monotonic response.
Universal Robustness Properties: A major theoretical finding is that the resulting closed-loop system possesses universal robustness properties that are independent of the specific plant parameters ( $K_p, T_1, T_2$ $K_{p}, T_{1}, T_{2}$ ). Regardless of the plant, the tuned system always exhibits:
- Maximum Complementary Sensitivity ( $M_t$ ) = 1.0
- Maximum Sensitivity ( $M_s$ ) = $2/\sqrt{3} \approx 1.155$
- Phase Margin ($PM$) = 76.35°

4. Results and Verification

The proposed method was validated through extensive numerical simulations across six different second-order plant configurations with varying time constant ratios ( $T_1/T_2$ ).

Performance Metrics:
- Overshoot: 0% in all cases.
- Monotonicity: Strictly confirmed for all step responses.
- Settling Time: The settling time ( $T_s$ ) scales linearly with the fast time constant $T_2$ ( $T_s \approx 11.66 T_2$ ), confirming that the dominant dynamics are governed by the repeated pole at $-1/(2T_2)$ .
Robustness Verification:
- Simulations confirmed that $M_t$ , $M_s$ , and $PM$ remained constant across all plant variations, matching the theoretical values derived in Section 4.
- Nyquist plots showed all closed-loop transfer functions lying exactly on the unit circle ( $M_t=1$ ), confirming the theoretical prediction.

5. Significance

This work is significant for both theoretical control engineering and industrial practice:

Analytical Rigor: It moves beyond heuristic or numerical optimization methods, providing a mathematically proven, closed-form solution for a classic control problem.
Simplicity and Reliability: Engineers can tune PI controllers for second-order systems using simple algebraic formulas without needing complex software or iterative tuning.
Guaranteed Safety: The guarantee of a strictly monotonic response makes this method ideal for safety-critical applications where overshoot is unacceptable.
Universal Robustness: The discovery that the robustness margins are invariant to plant parameters suggests a "plug-and-play" stability characteristic, simplifying the design process for families of similar systems.
Resolution of Open Questions: It directly answers the specific inquiry raised by Astrom and Hagglund regarding optimal pole selection for monotonic PI control.

In conclusion, Gulgonul presents a definitive, analytically proven tuning rule that achieves the theoretical limit of performance (minimum settling time) while strictly enforcing safety constraints (monotonicity) for a broad class of industrial processes.