Model-free practical PI-Lead control design by ultimate sensitivity principle

Abstract: Practical design and tuning of feedback controllers has to do often without any model of the given dynamic process. Only some general assumptions about the process, in this work type-one stable behavior, can be available for engineers, in particular in motion control systems. This paper proposes a practical and simple in realization procedure for designing a robust PI-Lead control without modeling. The developed method derives from the ultimate sensitivity principles, known in the empirical Ziegler-Nichols tuning of PID control, and makes use of some general characteristics of loop shaping. A three-steps procedure is proposed to determine the integration time constant, control gain, and Lead-element in a way to guarantee a sufficient phase margin, while all steps are served by only experimental observations of the output value. The proposed method is also evaluated with experiments on a noise-perturbed electro-mechanical actuator system with translational motion.

• The paper presents a model-free methodology that empirically determines the PI controller's parameters using the ultimate sensitivity principle.
• It tunes the integrator time constant and gain by analyzing oscillation frequencies and step response overshoot to ensure balanced robustness and responsiveness.
• The approach enhances transient response and disturbance rejection with a Lead-compensator, demonstrating practical effectiveness in noise-perturbed systems.

Model-Free PI-Lead Controller Design via Ultimate Sensitivity Principle

Problem Context and Motivation

The design and tuning of feedback controllers for dynamic processes without explicit mathematical models frequently arise in control engineering, notably in motion control systems. Such scenarios are characterized by unknown plant dynamics—with only general behavioral assumptions, such as type-one dynamics (integrating behavior), process stability, and minimum-phase property—while sensor noise and unmodeled nonlinearities further complicate tuning procedures. The persistent demand for model-free, practically accessible controller tuning approaches is underscored by the limitations of model-based methods and the operational challenges in industrial settings.

This work introduces a systematic, entirely empirical method for designing a robust PI-Lead controller, leveraging ultimate sensitivity principles widely associated with Ziegler-Nichols PID tuning, but extending them to achieve improved robustness and transient performance with direct experimental determination of control parameters.

Control Strategy and Methodology

The proposed methodology employs a three-step model-free experimental procedure, centered around empirical loop shaping:

1. Integrator Time Constant Determination:

The first phase involves gradually decreasing the integrator time constant $T_i$ of a PI controller (initialized with gain $K_p=1$) while observing the closed-loop output during a setpoint step. The method is designed to exploit the integrating nature of the type-one process, where the interaction between the controller and plant integrator leads to a distinctive phase response. When $T_i$ is reduced to a critical value, the closed-loop exhibits persistent oscillations, demarcating zero phase margin—the so-called ultimate condition. The frequency of these oscillations, alongside the corresponding controller corner frequency, are used to assign $T_i$ via:

$$T_i = \frac{10}{\max\{\bar{\omega}_{gc}, \bar{\omega}^c_{pi}\}}$$

where $\bar{\omega}_{gc}$ is the empirically measured frequency of oscillations and $\bar{\omega}^c_{pi} = 1/\bar{T}_i$ at the ultimate condition. This assignment ensures sufficient phase margin without unnecessary reduction of control bandwidth.

2. Control Gain Adjustment:

Upon fixing $T_i$, the controller gain $K_p$ is experimentally adapted by monitoring the transient output overshoot $M$ in response to step inputs. The gain is varied within a prescribed range, and the value attaining a desired overshoot ($M \in [30, 40]$%) is identified. This empirical target directly relates to closed-loop conjugate-complex pole damping ratios and phase margins, analytically tied to the overshoot via:

$$M = \exp\left(-\frac{\pi \zeta}{\sqrt{1 - \zeta^2}}\right)$$

The procedure results in a tuned $K_p$ that achieves a $\zeta$ corresponding to a phase margin in the range $[30^\circ, 40^\circ]$, balancing robustness and responsiveness.

3. Lead Phase Enhancement:

To further improve transient response and robustness, a Lead-compensator $L(s)$ is analytically specified for additional phase advance in the critical frequency range. Its standard form:

$$L(s) = K_L \frac{\tau s + 1}{\alpha \tau s + 1}, \quad 0 < \alpha < 1$$

is parameterized with $\alpha=0.1$, maximizing attainable phase lead at the frequency:

$$\omega_{\max(\varphi)} = \frac{1}{\sqrt{\alpha} \tau}$$

With the PI controller's $T_i$, the Lead-compensator frequency is assigned to center its phase advance within one half-decade above $1/T_i$, and the gain $K_L$ set to unity to preserve low-frequency dynamics.

This three-step procedure requires only input-output experimental measurements, dispensing with explicit modeling or identification.

Experimental Results

The methodology was validated on a noise-perturbed electro-mechanical actuator with a translational voice-coil motor, incorporating relevant nonlinearities (gravity compensation, input-gain nonlinearity, friction). All tuning was performed using output measurements under step and disturbance conditions.

The tuned PI-Lead controller exhibited numerically confirmed:

• Integrator time constant $T_i = 0.031$ sec
• Controller gain $K_p = 450$
• Lead-compensator $L(s) = \frac{0.031 s + 1}{0.0031 s + 1}$

Strong experimental results demonstrated:

• Accurate tracking for multiple step magnitudes, with overshoot tightly constrained by the tuning targets
• Enhanced disturbance rejection with the Lead-compensator present
• Robustness to sensor noise and unmodeled nonlinearities
• Clear differentiation from conventional PID tuned via Ziegler-Nichols: PID yielded higher aggressiveness and excessive overshoot/undershoot, while the PI-Lead achieved more controlled transient performance and practical signal behaviors

Practical and Theoretical Implications

The methodology's reliance on the ultimate sensitivity principle for each control component enables effective design of feedback controllers without modeling, streamlining implementation for a wide range of type-one (integrating) processes. It is inherently robust against measurement noise and reduces the risk of windup and over-aggressive controller action, often seen in heuristic PID tuning. The Lead extension allows for improved loop shaping—and addresses phase margin constraints unique to integrating systems—potentially facilitating adaptation to systems with oscillatory or time-delay elements.

The empirical steps can generalize to type-zero and other classes, creating opportunities for further extensions to systems with oscillatory modes, distributed delays, or zero dynamics. Integration with modern automatic tuning routines and embedded controller platforms is direct, given the method's measurement-centric design.

Conclusion

This work presents a robust, model-free approach for PI-Lead controller design in type-one dynamic systems, leveraging the ultimate sensitivity principle. The three-step experimental procedure enables precise loop shaping and transient response control without explicit plant knowledge, confirmed through rigorous experimental evaluation on a noise-affected actuator. The proposed method advances practical controller tuning, offering potential extensions for broader classes of dynamic systems and fostering improved control performance in realistic, unmodeled environments.