# Design of Fractional Order Controllers Using the PM Diagram

^{*}

## Abstract

**:**

## 1. Introduction

- The PM approach can be applied for the modeling and control of any kind of rational and non-rational system represented by a TF.
- In comparison to other approaches in the literature, the PM tool provides a more complete representation of the TF, which makes it an interesting technique to be used in control systems’ education.
- Its graphical nature makes it simple and intuitive to use.

## 2. Phase Magnitude Diagram

## 3. DE-Based Tuning of Fractional Order PID Controllers

#### Fitness Function

- Limits for gain crossover frequency (${\omega}_{cg}$) and phase margin (${\phi}_{m}$): An acceptance interval for ${\omega}_{cg}$ and a minimum ${\phi}_{m}$ has been set. These two constraints are described in Figure 3, which represents the Bode plot of the open loop system $F\left(s\right)$ (a representation of the magnitude and phase of the system as a function of the frequency). The frequency ${\omega}_{cg}$ is directly related with the speed of the temporal response of the system (the higher the frequency, the higher the speed) and the phase margin ${\phi}_{m}$ directly related with the overshoot (oscillation) of the temporal response (the higher the phase margin, the lower the overshoot). These parameters define then the stability and robustness of the system. The next equations show the relation between ${\omega}_{cg}$ and ${\phi}_{m}$ as described graphically in Figure 3:$$\begin{array}{ccc}\hfill |C\left(j{\omega}_{cg}\right)G\left(j{\omega}_{cg}\right){|}_{dB}& =& 0\phantom{\rule{4pt}{0ex}}\mathrm{dB},\hfill \end{array}$$$$\begin{array}{ccc}\hfill arg\left(C\left(j{\omega}_{cg}\right)G\left(j{\omega}_{cg}\right)\right)& =& -180+{\phi}_{m}.\hfill \end{array}$$
- High frequency noise rejection: A desired feature for a system is to attenuate the noise at high frequencies. The complementary sensitivity function is used to meet this requirement:$$\begin{array}{c}\hfill {\left|T\left(j\omega \right)=\frac{C\left(j\omega \right)G\left(j\omega \right)}{1+C\left(j\omega \right)G\left(j\omega \right)}\right|}_{dB}=D\phantom{\rule{4pt}{0ex}}\mathrm{dB},\\ \hfill \forall \phantom{\rule{4pt}{0ex}}\omega \ge {\omega}_{t}\phantom{\rule{4pt}{0ex}}rad/s\Rightarrow {\left|T\left(j{\omega}_{t}\right)\right|}_{dB}\le D\phantom{\rule{4pt}{0ex}}\mathrm{dB}.\end{array}$$The noise attenuation will be equal to D for frequencies $\omega \ge {\omega}_{t}$ rad/s.
- To ensure a good output disturbance rejection: This requirement is related to the sensitivity function:$$\begin{array}{c}\hfill {\left|S\left(j\omega \right)=\frac{1}{1+C\left(j\omega \right)G\left(j\omega \right)}\right|}_{dB}=B\phantom{\rule{4pt}{0ex}}\mathrm{dB},\\ \hfill \forall \phantom{\rule{4pt}{0ex}}\omega \le {\omega}_{s}\phantom{\rule{4pt}{0ex}}rad/s\Rightarrow {\left|S\left(j{\omega}_{s}\right)\right|}_{dB}\le B\phantom{\rule{4pt}{0ex}}\mathrm{dB},\end{array}$$
- Steady-state error cancellation (no error in the permanent state of the temporal response): the fractional integrator $1/{s}^{\lambda}$ is as efficient as an integer order integrator [31], which means that this constraint is always fulfilled.

## 4. Implementation Using the PM Diagram

- Draw PM diagram for the open-loop TF of controller and system.
- Compute the crossover frequency.
- Define an interval of frequencies where the phase is intended to be flat.
- Compute colors at frequencies of interest.
- Calculate cost using colors.

- Frequency interval ${A}_{dec}$ where the phase is intended to be flat (${\omega}_{f}-{\omega}_{i}$) equal to 1 decade.
- High frequency noise rejection: ${\omega}_{t}=100$ rad/s $\Rightarrow |T\left(j{\omega}_{t}\right){|}_{dB}\le -15\phantom{\rule{4pt}{0ex}}$ dB (7).
- Sensitivity: ${\omega}_{s}=0.01$ rad/s $\Rightarrow |S\left(j{\omega}_{s}\right){|}_{dB}\le -15\phantom{\rule{4pt}{0ex}}$ dB (8).

## 5. Test Systems

**Example**

**3.**

**Example**

**4.**

## 6. Experimental Results

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**PM diagram for $G\left(s\right)=1/(s+1)(s+3)$: Left: Three-dimensional view; Right: Planar projection.

**Figure 2.**PM diagram, features of interest: Left: PM diagram for $G\left(s\right)=1/(s+1)(s+3)$. Grid with the damping ratio $\zeta $ (black dashed lines from the origin) and the natural frequency ${\omega}_{n}$ (circles with different radius) level curves; Right: PM diagram for $G\left(s\right)=40/(s+1)(s+3)(s+5)$. Illustrative example to measure the phase margin, the gain margin, and phases in the Bode plot (colors in red dots).

**Figure 4.**Examples 1 and 2: Top: Example 1 (re-heating furnace, Podlubny); Bottom: Example 2 (immersed plate, Monje); Left: PM diagram without controller; Right: PM diagram with controller.

**Figure 5.**Examples 3 and 4: Top: Example 3 (heat diffusion, Aström); Bottom: Example 4 (Merrikh-Bayat); Left: PM diagram without controller; Right: PM diagram with controller.

**Figure 6.**Example 1 (re-heating furnace, Podlubny). Representation of different variables of interest.

Acronym | Name | Meaning |
---|---|---|

PM | PM diagram | Phase Magnitude diagram |

PID | PID controller | Proportional Integral Derivative controller |

PI${}^{\lambda}$D${}^{\mu}$ | Fractional order PID controller | Fractional order Proportional Integral Derivative controller |

${M}_{dif-PM}$ | ${M}_{dif-PM}$ | Magnitude added at the frequency of interest |

${P}_{dif-PM}$ | ${P}_{dif-PM}$ | Phase added at the frequency of interest |

${\phi}_{m}$ | Phase margin | Stability measurement |

${A}_{dec}$ | Interval where the phase is intended to be flat | Robustness measurement |

${\omega}_{cg}$ | Gain crossover frequency | Gain crossover frequency |

$T\left(j{\omega}_{t}\right)$ | Complementary sensitivity function | High frequency noise rejection |

$S\left(j{\omega}_{t}\right)$ | Sensitivity function | Disturbance rejection |

ISE | Integral of the Squared Error | Performance criterion in stability analysis |

IAE | Integral of the Absolute Error | Performance criterion in stability analysis |

HSV | Hue, Saturation, Value (or Brightness) of an image | Color model |

RGB | Red, Green, Blue levels of an image | Color model |

**Table 2.**TFs of the systems and controllers for the examples in Section 5.

TF | Controller | |
---|---|---|

Example 1, Podlubny (re-heating furnace) | $\frac{1}{0.794{s}^{2.571}+5.238{s}^{0.837}+1.556}$ | $0.33+0.44{s}^{0.27}+3.3{s}^{-0.83}$ |

Example 2, Monje (immersed plate) | $\frac{1}{{s}^{2}+0.5{s}^{1.5}+0.5}$ | $11.269+24.87{s}^{0.582}+4.08{s}^{-1.99}$ |

Example 3, Aström (heat diffusion) | $e}^{-\sqrt{s}$ | $1.31+0.73{s}^{0.884}+1.20{s}^{-0.296}$ |

Example 4, Merrikh-Bayat | $\frac{(\sqrt{s}+1){e}^{-\sqrt{s}}}{s}$ | $2.67+0.60{s}^{1.04}+1.08{s}^{-0.222}$ |

**Table 3.**Parameters of interest for the examples when controlled. ISE and IAE in squared units and units, respectively, according to the variable that is measured in each example when a unit step is the input. ${w}_{cg}$ in radians/second. ${G}_{m}$ is an absolute magnitude. ${\phi}_{m}$ in degrees.

ISE | IAE | ${\mathit{w}}_{\mathit{cg}}$ | ${\mathit{G}}_{\mathit{m}}$ | ${\mathit{\phi}}_{\mathit{m}}$ | |
---|---|---|---|---|---|

Example 1, Podlubny (re-heating furnace) | 2.73 | 4.56 | 0.8 | 2.51 | 57.3 |

Example 2, Monje (immersed plate) | 1.03 | 2.18 | 9.3 | ∞ | 53.15 |

Example 3, Aström (heat diffusion) | 5.08 | 14.89 | 1.8 | 4.47 | 142.91 |

Example 4, Merrikh-Bayat | 1.09 | 1.60 | 2.8 | 9.94 | 74.1 |

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**MDPI and ACS Style**

Garrido, S.; Monje, C.A.; Martín, F.; Moreno, L.
Design of Fractional Order Controllers Using the PM Diagram. *Mathematics* **2020**, *8*, 2022.
https://doi.org/10.3390/math8112022

**AMA Style**

Garrido S, Monje CA, Martín F, Moreno L.
Design of Fractional Order Controllers Using the PM Diagram. *Mathematics*. 2020; 8(11):2022.
https://doi.org/10.3390/math8112022

**Chicago/Turabian Style**

Garrido, Santiago, Concepción A. Monje, Fernando Martín, and Luis Moreno.
2020. "Design of Fractional Order Controllers Using the PM Diagram" *Mathematics* 8, no. 11: 2022.
https://doi.org/10.3390/math8112022