**1. Introduction**

Despite the abundance of research in advanced control strategies, the PID (proportionalintegrative-derivative) controller remains the preferred control algorithm in industrial applications [1,2]. To produce the desired effects, PIDs need to be adequately tuned. A mathematical model is usually needed in order to properly tune the controller. However, large industrial plants are characterized by numerous sub-systems and obtaining an accurate process model is not cost effective as it can be difficult and/or time consuming. To overcome this issue, two different approaches for autotuning PIDs were developed, as indicated in Figure 1.

Both approaches use step or sinusoidal input data and collect the process output response. For a direct autotuner the PID parameters are determined directly from process input/output data, while for the indirect PID autotuner, simple process models are first determined and then the PID parameters are computed according to some tuning rules based on the model parameters. The majority of indirect methods use either first-order plus dead time (FOPDT) or second-order plus dead time (SOPDT) models.

Two of the most popular autotuning methods have been developed by Ziegler and Nichols [3]. One of these methods is a direct approach, based on the relay experiment, as indicated in Figure 2. Once the relay test is performed on a process, it will lead to a sinusoidal output signal which is used to estimate the process critical frequency and the corresponding critical gain. Tuning rules based on the process critical frequency and gain are employed to compute the PID controller parameters. The Ziegler-Nichols direct autotuning method is highly popular because of its simplicity and good performance results.

**Citation:** Muresan, C.I.; Birs, I.; Ionescu, C.; Dulf, E.H.; De Keyser, R. A Review of Recent Developments in Autotuning Methods for Fractional-Order Controllers. *Fractal Fract.* **2022**, *6*, 37. https://doi.org/ 10.3390/fractalfract6010037

Academic Editors: Thach Ngoc Dinh, Shyam Kamal and Rajesh Kumar Pandey

Received: 13 December 2021 Accepted: 5 January 2022 Published: 11 January 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

**Figure 1.** Autotuning approaches.

**Figure 2.** Relay experiment.

Several extensions of this approach and alternative solutions have been developed over the years. One of these uses the describing function analysis and a simple relay feedback test to estimate the process critical gain and corresponding frequency [4]. A solution for noisy signals was proposed based on a relay with hysteresis [5]. An artificial time delay is added within the relay closed-loop system in order to determine the process gain and phase at a random oscillation frequency. Then, a PI (proportional-integrative) controller is tuned according to this process data. A modified Ziegler-Nichols method [6]—where the ratio between the integral and derivative time constants is *r* = 4—was also developed. Other research papers discuss the impact the ratio value has upon the control performance [7]. Solutions to improve the robustness of the control system have been addressed [2]. Åström and Hägglund [1] use the relay test to design controllers based on robust loop shaping, with a clear tradeoff between robustness and performance.

The second method developed by Ziegler and Nichols [3] consists in applying a step signal on the process input and collecting the output data. The method is suitable for processes that have FOPDT dynamics or exhibit an S-shaped response, as indicated in Figure 3. The approach goes through an indirect step, where the parameters of the FOPDT model are estimated. Finally, the PID controller parameters are computed using a set of tuning rules that depend on the FOPDT model parameters.

**Figure 3.** S-shaped response.

The demand for better control performance and increased robustness has led to several modifications of the standard PID controller, including a generalization to fractional order [8]. Research on fractional order PID (FO-PID) controllers has demonstrated that this generalization allows for more flexibility in the design, due to the two supplementary tuning parameters involved, the fractional orders of integration and differentiation [9–13]. This flexibility comes with important advantages, such as better closed-loop performance, disturbance rejection capabilities, improved control of time-delay systems and increased robustness [9–14]. The fractional order PID transfer function is given as:

$$\mathcal{L}\_{FO-PID}(\mathbf{s}) = k\_p \left( 1 + \frac{1}{T\_i \mathbf{s}^\lambda} + T\_d \mathbf{s}^\mu \right) \tag{1}$$

where 0 < *λ* < 2 and 0 < *μ* < 1 are the fractional orders of integration and differentiation, respectively, and *kp* is the proportional gain, and *Ti* and *Td* are the integral and derivative time constants. The "classical" tuning rules used to determine the five controller parameters are derived from the following performance specifications [9,12,15–17]:

1. A gain crossover frequency *ωc*. This leads to the magnitude condition:

$$|H\_{ol}(j\omega\_{\mathfrak{C}})| = 1 \tag{2}$$

with *Hol*(s) the open-loop transfer function is defined as: *Hol*(s) *= P*(*s*). *CFO-PID*(s), where *P*(*s*) is the process transfer function;

2. A phase margin PM. This leads to the phase condition:

$$
\angle H\_{ol}(j\omega\_{\mathbb{C}}) = -\pi + PM \tag{3}
$$

3. Iso-damping property (or robustness to gain variations). This is specified through:

$$\left. \frac{d(\angle H\_{ol}(j\omega))}{d\omega} \right|\_{\omega = \omega\_k} = 0 \tag{4}$$

where *ω* denotes the frequency. This last condition ensures that the overshoot of the closed-loop system remains approximately constant in the case of gain variations;

4. Good output disturbance rejection. This leads to a constraint on the sensitivity function *<sup>S</sup>*: /

$$\left| S(j\omega) = \frac{1}{1 + P(j\omega)H\_{FO} - PID(j\omega)} \right| \le B \text{ dB} \tag{5}$$

for frequencies *ω* ≤ *ωs*, with *B* a scalar;

5. High frequency noise rejection. This leads to a constraint on the complementary sensitivity function *T* as:

$$T(j\omega) = \frac{P(j\omega)H\_{FO-PID}(j\omega)}{1 + P(j\omega)H\_{FO-PID}(j\omega)}\Big|\_{} \le A \text{ dB} \tag{6}$$

/ / for frequencies *ω* ≥ *ωT*, with *A* a scalar.

/ /

Further information regarding the tuning, implementation and related topics to fractional order PIDs can be found in some excellent review papers [15,16,18–24]. The phase shaper [25] is among the first automatic controller designs that uses fractional calculus tools. The autotuning method is based on the iso-damping property, but the final controller is an integer order PID. Throughout the past two decades, a couple of FO-PID autotuning methods have emerged. Some of these provide direct and indirect tuning rules for FO-PIDs in general or for fractional order PI (FO-PI) controllers. The purpose of this manuscript is to offer a comprehensive review of these autotuning methods, to compare them and to discuss which method is ranked best for controlling a specific type of process.

The paper is structured as follows. Sections 2 and 3 provide for a review of the most widely known indirect and direct autotuning methods for FO-PIDs, while Section 4 provides for some numerical examples. Possible applications of autotuning methods are reviewed in Section 5, along with a survey on self-tuning FO-PIDs. The last section concludes the paper.
