# 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:$$\begin{array}{c}\hfill {\left(\right)}_{T}dB=D\phantom{\rule{4pt}{0ex}}\mathrm{dB},\end{array}$$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:$$\begin{array}{c}\hfill {\left(\right)}_{S}dB=B\phantom{\rule{4pt}{0ex}}\mathrm{dB},\end{array}$$
- 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

## References

- Wegert, E. Visual Complex Functions: An Introduction with Phase Portraits; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2012. [Google Scholar]
- Cavicehi, T.J. Phase-Root Locus and relative stability. IEEE Control Syst.
**1996**, 16, 69–77. [Google Scholar] - Wegert, E.; Semmler, G. Phase plots of complex functions: A journey in illustration. Not. AMS
**2010**, 58, 768–780. [Google Scholar] - Mathews, J.H.; Howell, R.W. Complex Analysis for Mathematics and Engineering; Jones & Bartlett Publishers: Burlington, MA, USA, 2012. [Google Scholar]
- Needham, T. Visual Complex Analysis; Oxford University Press: Oxford, UK, 1998. [Google Scholar]
- Poelke, K.; Polthier, K. Lifted domain coloring. In Computer Graphics Forum; Wiley Online Library: Hoboken, NJ, USA, 2009; Volume 28, pp. 735–742. [Google Scholar]
- Garrido, S.; Moreno, L. PM Diagram of the Transfer Function and Its Use in the Design of Controllers. J. Math. Syst. Sci.
**2015**, 5, 138–149. [Google Scholar] - Podlubny, I. Fractional-order systems and PI
^{λ}D^{μ}controllers. IEEE Trans. Autom. Control**1999**, 44, 208–214. [Google Scholar] [CrossRef] - Martín, F.; Monje, C.A.; Moreno, L.; Balaguer, C. DE-based tuning of PI
^{λ}D^{μ}controllers. ISA Trans.**2015**, 59, 398–407. [Google Scholar] [CrossRef] - Storn, R.; Price, K. Differential evolution—A simple and efficient heuristic for global optimization over continuous spaces. J. Glob. Optim.
**1997**, 11, 341–359. [Google Scholar] [CrossRef] - Kuo, B.C. Automatic Control Systems; Prentice Hall: Upper Saddle River, NJ, USA, 1991. [Google Scholar]
- Ogata, K. Modern Control Engineering; Prentice Hall: Upper Saddle River, NJ, USA, 2009. [Google Scholar]
- Monje, C.A.; Chen, Y.; Vinagre, B.M.; Xue, D.; Feliu-Batlle, V. Fractional-Order Systems and Controls: Fundamentals and Applications; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2010. [Google Scholar]
- Monje, C.A.; Ramos, F.; Feliu, V.; Vinagre, B.M. Tip position control of a lightweight flexible manipulator using a fractional order controller. IET Control Theory Appl.
**2007**, 1, 1451–1460. [Google Scholar] [CrossRef] - Podlubny, I.; Petráš, I.; Vinagre, B.M.; O’Leary, P.; Dorčák, L. Analogue realizations of fractional-order controllers. Nonlinear Dyn.
**2002**, 29, 281–296. [Google Scholar] [CrossRef] - Valério, D.; Sá da Costa, J. Introduction to single-input, single-output fractional control. IET Control Theory Appl.
**2011**, 5, 1033–1057. [Google Scholar] [CrossRef] - Afshari, H.; Kalantari, S.; Karapinar, E. Solution of fractional differential equations via coupled fixed point. Electron. J. Differ. Equ.
**2015**, 286, 1–12. [Google Scholar] - Sevinik, R.; Aksoy, U.; Karapinar, E.; Erhan, Y. On the solution of a boundary value problem associated with a fractional differential equation. Math. Methods Appl. Sci.
**2020**. [Google Scholar] [CrossRef] - Alqahtani, B.; Aydi, H.; Karapinar, E.; Rakocevic, V. A Solution for Volterra fractional integral equations by hybrid contractions. Mathematics
**2019**, 7, 694. [Google Scholar] [CrossRef][Green Version] - Valério, D.; Sá da Costa, J. A review of tuning methods for fractional PIDs. In Proceedings of the 4th IFAC Workshop on Fractional Differentiation and Its Applications, Badajoz, Spain, 18–20 October 2010. [Google Scholar]
- Vinagre, B.M. Modelling and control of dynamic systems characterized by integro-differential equations of fractional order (in Spanish). PHD Thesis, UNED, Madrid, Spain, 2001. [Google Scholar]
- Chen, Y.Q.; Dou, H.; Vinagre, B.M.; Monje, C.A. A robust tuning method for fractional order PI controllers. In Proceedings of the 2nd IFAC Workshop on Fractional Differentiation and Its Applications, Porto, Portugal, 19–21 July 2006; pp. 22–27. [Google Scholar]
- Malek, H.; Luo, Y.; Chen, Y.Q. Identification and tuning fractional order proportional integral controllers for time delayed systems with a fractional pole. Mechatronics
**2013**, 23, 746–754. [Google Scholar] [CrossRef] - Li, H.; Luo, Y.; Chen, Y.C. A fractional order proportional and derivative (FOPD) motion controller: Tuning rule and experiments. IEEE Trans. Control Syst. Technol.
**2010**, 18, 516–520. [Google Scholar] [CrossRef] - Luo, Y.; Chen, Y.Q.; Wang, C.; Pi, Y. Tuning fractional order proportional integral controllers for fractional order systems. J. Process Control
**2010**, 20, 823–831. [Google Scholar] [CrossRef] - Tavazoei, M.S.; Tavakoli-Kakhki, M. Compensation by fractional-order phase-lead/lag compensators. IET Control Theory Appl.
**2014**, 8, 319–329. [Google Scholar] [CrossRef] - Chen, Y.Q.; Dou, H.; Vinagre, B.M.; Monje, C.A. Optimal fractional order proportional integral controller for varying time-delay systems. In Proceedings of the 17th IFAC World Congress (2008), Seoul, Korea, 6–11 July 2008; pp. 4910–4915. [Google Scholar]
- Valério, D.; da Costa, J.S. Tuning of fractional PID controllers with Ziegler–Nichols type rules. Signal Proc.
**2006**, 86, 2771–2784. [Google Scholar] [CrossRef] - Valério, D.; da Costa, J.S. Tuning Rules for Fractional PIDs. Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering; Springer: Berlin, Germany, 2007. [Google Scholar]
- Chen, Y.Q.; Bhaskaran, T.; Xue, D. Practical tuning rule development for fractional order proportional and integral controllers. J. Comput. Nonlinear Dyn.
**2008**, 3, 021403(1)–021403(8). [Google Scholar] [CrossRef][Green Version] - Monje, C.A.; Vinagre, B.M.; Feliu, V.; Chen, Y. Tuning and auto-tuning of fractional order controllers for industry applications. Control Eng. Pract.
**2008**, 16, 798–812. [Google Scholar] [CrossRef][Green Version] - Jin, Y.; Chen, Y.Q.; Xue, D. Time-constant robust analysis of a fractional order [proportional derivative] controller. IET Control Theory Appl.
**2011**, 5, 164–172. [Google Scholar] [CrossRef] - Chen, Y.Q.; Moore, K.L. Relay feedback tuning of robust PID controllers with iso-damping property. IEEE Trans. Syst. Man Cybern. Part B
**2005**, 35, 23–31. [Google Scholar] [CrossRef] - Vinagre, B.M.; Monje, C.A.; Calderón, A.J.; Chen, Y.Q.; Feliu, V. The fractional integrator as a reference function. In Proceedings of the 1st IFAC Congress on Fractional Differentiation and its Applications, FDA’04, Bordeaux, France, 19–21 July 2004; pp. 150–155. [Google Scholar]
- Aström, K.J.; Murray, R.M. Feedback Systems: An Introduction for Scientists and Engineers; Princeton University Press: Princeton, NJ, USA, 2010. [Google Scholar]
- Oustaloup, A. CRONE control: Robust control of non-integer order. Paris Hermes
**1991**. [Google Scholar] - Oustaloup, A.; Levron, F.; Nanot, F.; Mathieu, B. Frequency-band complex noninteger differentiator: Characterization and synthesis. IEEE Trans. Circuits Syst. I Fundam. Theory Appl.
**2000**, 47, 25–40. [Google Scholar] [CrossRef] - Tenreiro Machado, J.A. Root locus of fractional linear systems. Commun. Nonlinear Sci. Numer Simulat.
**2011**, 16, 3855–3862. [Google Scholar] [CrossRef][Green Version] - Podlubny, I. Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications; Academic Press: Cambridge, MA, USA, 1998; Volume 198. [Google Scholar]
- Torvik, P.J.; Bagley, R.L. On the appearance of the fractional derivative in the behavior of real materials. J. Appl. Mech.
**1984**, 51, 294–298. [Google Scholar] [CrossRef] - Merrikh-Bayat, F.; Karimi-Ghartemani, M. An efficient numerical algorithm for stability testing of fractional- delay systems. ISA Trans.
**2009**, 48, 32–37. [Google Scholar] [CrossRef] - Kwok, Y.K. Applied Complex Variables for Scientists and Engineers; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Ozturk, N.; Uraz, A. An analytic stability test for a certain class of distributed parameter systems with a distributed lag. IEEE Trans. Autom. Control
**1984**, 29, 368–370. [Google Scholar] [CrossRef]

**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 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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