Advanced Theoretical and Applied Studies of Fractional Differential Equations 2013
View this Special IssueResearch Article  Open Access
António M. Lopes, J. A. Tenreiro Machado, "Root Locus Practical Sketching Rules for FractionalOrder Systems", Abstract and Applied Analysis, vol. 2013, Article ID 102068, 14 pages, 2013. https://doi.org/10.1155/2013/102068
Root Locus Practical Sketching Rules for FractionalOrder Systems
Abstract
For integerorder systems, there are wellknown practical rules for RL sketching. Nevertheless, these rules cannot be directly applied to fractionalorder (FO) systems. Besides, the existing literature on this topic is scarce and exclusively focused on commensurate systems, usually expressed as the ratio of two noninteger polynomials. The practical rules derived for those do not apply to other symbolic expressions, namely, to transfer functions expressed as the ratio of FO zeros and poles. However, this is an important case as it is an extension of the classical integerorder problem usually addressed by control engineers. Extending the RL practical sketching rules to such FO systems will contribute to decrease the lack of intuition about the corresponding system dynamics. This paper generalises several RL practical sketching rules to transfer functions specified as the ratio of FO zeros and poles. The subject is presented in a didactic perspective, being the rules applied to several examples.
1. Introduction
Root locus (RL) analysis is a graphical method that shows how the poles of a closedloop transfer function change with relation to a given system parameter [1, 2]. Usually, the chosen parameter is a proportional gain, , included in a unity feedback closedloop controlled system (Figure 1).
The open and closedloop transfer functions are given by and , respectively. The denominator of is the characteristic equation, and its roots are the system closedloop poles. Every point of the RL simultaneously satisfies the wellknown argument (angle) and magnitude conditions given by
The RL is a classical and powerful tool for the dynamical analysis and design of integerorder linear timeinvariant (LTI) systems [1–6]. Nowadays, there are efficient numerical algorithms, implemented in several software packages (e.g., MATLAB, Octave, Scilab, and FreeMat) [7–10] that take advantage of the powerful digital processors of modern computers to perform RL analysis. For fractionalorder (FO) systems, while several studies addressing RL are available [11–17], the problem is more difficult and researchers have mainly preferred to adopt frequencybased methods.
On the other hand, the ability to quickly sketch RL by hand is invaluable in making fundamental decisions early in the design process. For integerorder systems, there are wellknown practical rules for RL sketching, but those cannot be directly applied to FO systems. Moreover, the existing literature on this topic exclusively focuses on the particular case of commensurate FO systems that occur when truncating real valued integrodifferential orders up to a finite precision [15, 16]. This allows the generalisation of some rules to FO systems, but limiting the precision and the type of symbolic expressions [17, 18]. The rules for commensurate FO systems do not apply to transfer functions expressed as the ratio of FO zeros and poles. However, this is an important case as it is an extension of the classical integerorder problem usually addressed by control engineers when dealing with RL analysis.
In this paper, we extend several practical rules, available to sketch the RL of integerorder systems, to the FO domain. The main contribution is that the practical sketching rules apply to openloop transfer functions expressed as the ratio of FO zeros and poles, contributing to fill the gap in the existing literature about this topic. The subject is presented in a didactic perspective, being the rules applied to several examples that contribute to reduce the lack of intuition about the corresponding system dynamics.
Bearing these ideas in mind, the paper is organized as follows. Section 2 introduces fundamental concepts related to fractional calculus. Section 3 analyses several FO systems and generalises the RL rules to a class of FO systems. Finally, Section 4 draws the main conclusions.
2. Fractional Calculus
Fractional calculus (FC) denotes the branch of calculus that extends the concepts of integrals and derivatives to noninteger and complex orders [19–23]. During the last years, FC was found to play a fundamental role in the modelling of a considerable number of phenomena [24–29] and emerged as an important tool for the study of dynamical systems where classical methods reveal strong limitations. Nowadays, the application of FC concepts includes a wide spectrum of studies [30–33], going from the dynamics of financial markets [34, 35], biological systems [36, 37], earth sciences [38], and DNA sequencing [39] up to mechanical [40–43], electrical [44–46], and control systems [21, 24].
The generalisation of the concept of derivative and integral to noninteger orders, , has been addressed by several mathematicians. The RiemannLiouville, GrünwaldLetnikov, and Caputo definitions of fractional derivative are the most used and are given, respectively, by [47] where () represents Euler’s gamma function, [] is the integer part of , and is a time step.
The Laplace transform applied to (2) yields where and denote the Laplace operator and variable, respectively, and t represents time.
The general LTI, singleinputsingleoutput (SISO), and FO incommensurate system can be represented by [48] where () and () represent the system input and output, respectively, is the derivative operator, ,, , and . Besides, it is considered that , and .
In the Laplace domain, (6) results in a transfer function given by the ratio of two noninteger polynomials:
If , , with and , , then (7) is a commensurate FO system and can be written as
The FO system is said to be rational if .
In general, a polynomial is a multivalued function, the domain of which is a Riemann surface with an infinite number of sheets [48]. Only in the particular case of being rational, the number of sheets will be finite. Such type of function becomes singlevalued when an appropriate cut of the complex plane is assumed. This branch cut is not unique, but the negative real axis is usually chosen. In this case, the origin of the complex plane is a branch point and the first Riemann sheet, , is defined as
For example, Figure 2 depicts two Riemann surfaces corresponding to the function , the roots of which are
(a)
(b)
For and , the Riemann surface has two sheets (Figure 2(a)), and for and , the Riemann surface presents three different sheets (Figure 2(b)). In the former case, there are no roots, and in the latter case, two roots appear on the first sheet. Riemann surfaces are important when dealing with RL of FO systems, as will be seen in Section 3.
3. Root Locus
In this section, we assume that the system openloop transfer function is given by the following: where , and , .
Equation (11) represents a direct extension to the FO domain of the classical integerorder problem usually addressed by control engineers when dealing with RL analysis. Rules for RL sketching applicable to this case are summarised in Table 1. Only the first Riemann sheet will be considered.

In the sequel, several examples are presented, namely, (i) one FO real pole; (ii) two FO real poles; (iii) one FO pole and one FO zero; (iv) a pair of FO complex conjugate poles. The RL plots are generated using the numeric algorithm presented in [17]. The application of the practical sketching rules is detailed for a few examples, and for all cases, the RL plots serve the purpose of elucidating system dynamics. This will help readers to gain intuition about system behaviour as a function of poles and zeros fractional orders.
3.1. One FractionalOrder Real Pole
In this case, the openloop transfer function is given by where the RL corresponds to the roots of the characteristic equation
In general, the RL spreads along several Riemann sheets, meaning that RL branches can begin in one sheet, cross the branch cut, and enter in another sheet. For example, considering with and , we verify that the characteristic (13) has roots in two Riemann sheets (Figure 3(a)). However, choosing results in roots in five different sheets (Figure 3(b)).
(a)
(b)
It is wellknown that just the first Riemann sheet has physical significance [49]. As such, in the sequel, we consider only the RL branches corresponding to the first sheet.
Observing the RL of , we verify that for , there are no closedloop poles. However, for , several graphs are obtained, as shown in Figure 4. Starting from the integer case () represented in Figure 4(a), as the FO pole increases, two branches emerge from the openloop pole and flow towards infinity (Figure 4(b)). For , we get the classical plot with two vertical branches (Figure 4(c)). Increasing _{1} (), two RL branches are still observed (Figure 4(d)). When , the wellknown three branches RL occurs (Figure 4(e)), and finally, when the FO pole is in the interval , four branches emerge. Larger values of the FO pole (i.e., ) were also investigated. We concluded that the RL sketching rules also apply. The results are of the same type, and therefore, we decided not to include them.
(a)
(b)
(c)
(d)
(e)
(f)
The practical rules apply to all FO cases. For example, for the RL shown in Figure 4(f), as , the RL has four branches. The asymptotes centroid and angles are and , , , and , respectively. Solving the characteristic equation for , the RL branches intersect the imaginary axis at , for .
3.2. Two FractionalOrder Real Poles
In this subsection, we consider the openloop transfer function given by
The RL was computed for various values of (, ) and the graphs analysed. It was observed that no RL branches exist when . Several RL examples are depicted in Figures 5 to 7 for . The results are presented in three groups: (i) ; (ii) ; (iii) . Similar results were observed for and . For both cases, the practical sketching rules still apply.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 5 shows the plots from group (i). When , the RL has a single branch in the real axis (Figures 5(a) and 5(b)). As increases (), two branches emerge from the poles or , depending on the values of and , and tend to infinity (Figures 5(c) to 5(f)).
As said in Section 3.1, all practical rules are valid for , (15). Using the case shown in Figure 5(f), for example, we have , meaning that the RL has two branches. As we have two openloop poles, rule 4 must be used to determine the pole from which the branches are departing. Thus, applying the angle condition to the test points and , we obtain and , respectively, indicating that no branches can depart from . Rule 4 can be used in all cases; nevertheless, an easier to use specific rule about RL starting and ending points still requires more research before a definitive statement.
The angle condition is also used to determine the departing angles from pole , resulting in . The asymptotes centroid and angles are and , respectively.
Figure 6 depicts results from group (ii). When , we get the plots represented in Figures 6(a) and 6(b). We observe two RL branches that, as before, depending on the values of and , can depart from one or the other openloop poles. In both cases, the branches tend to infinity with angles . Increasing the value of (), two RL branches are still observed (Figures 6(c) to 6(h)).
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(a)
(b)
(c)
(d)
(e)
(f)
(g)
The results from group (iii) are illustrated in Figure 7. For , the RL of Figures 7(a) to 7(c) shows three branches that depart from the same or different openloop poles and flow to infinity with angles and . Increasing (), four RL branches arise (Figures 7(d) to 7(g)).
The results obtained for two FO real poles are similar to those of a single real pole. This means a similar behavior, both in terms of the number of branches and the type of RL charts, whenever and are close. It should be noted that the RL depends not only on the equivalent order (by means of rules 2, 5, or 6) but also on the FO of each pole. By other words, the same value of may lead to different RL.
3.3. One FractionalOrder Pole and One FractionalOrder Zero
In this case, the openloop transfer function is given by
The RL was obtained for various values of (, ) and the graphs analysed as previously. It was observed that no RL branches exist when . Figures 8 to 10 depict several RL for . As before, for easing the comparison, the results are presented in three groups: (i) ; (ii) ; (iii) . Additional experiments were carried out, both for different values of the FO pole and FO zero and for . We concluded that the sketching rules are valid for all cases and the results are similar to those presented.
(a)
(b)
(c)
Figure 8 shows plots from group (i). We see that each RL has two branches that depend on the difference between the orders of the denominator and numerator, : when , both branches converge to the openloop zero (Figure 8(a)); if , one branch converges to the openloop zero and the other tends to infinity (on the real axis) (Figure 8(b)); for , the two branches flow to infinity (Figure 8(c)).
Applying rule 6 to the case depicted in Figure 8(b), for all real axis points in the line , we have , meaning that this line belongs to the RL. The breakin point is computed using rule 8, resulting in .
Figure 9 depicts several plots from group (ii), that is, . All RL still have two branches, the paths of which depend on the difference between the FO of the openloop pole and zero (Figures 9(a) to 9(e)).
(a)
(b)
(c)
(d)
(e)
(a)
(b)
(c)
(d)
(e)
(f)
(g)
Several RL for group (iii), , are shown in Figure 10. It can be observed that all RL have four branches, and as before, the paths depend on the difference between the orders of the openloop pole and zero.
3.4. One Pair of FractionalOrder Complex Conjugate Poles
The openloop transfer function is given by where and denotes the conjugate of .
Plotting the RL, it can be seen that there are no branches unless . In Figure 11, several RL graphs are shown for . Figure 11(a) depicts the RL for , where we can see that there are gaps between the openloop poles and the points were the branches initiate. Recalling that the RL can spread along several Riemann sheets, meaning that RL branches can begin in one sheet, cross the branch cut, and enter in another sheet, the gaps correspond to points not belonging to the first Riemann sheet. As in the previous examples, when , the RL has two branches (Figures 11(a) to 11(c)). When , the number of branches is four. Even though, for , there are gaps in two branches (Figure 11(d)), and for , two extra small branches depart from the openloop poles and end close to those points, entering in another Riemann sheet (Figure 11(e)). The same qualitative behaviour is observed for (Figures 11(f) to 11(g)). Figure 11(h) depicts the RL for , revealing eight branches departing from the openloop poles.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
To conclude the analysis, we use the case shown in Figure 11(g) to underline that all RL practical rules are applicable, namely, the asymptotes centroid and angles, which are and , , 52.9, and , respectively. The angle condition is used to determine the departing angles from pole , resulting in the values , , 68.8, and .
4. Conclusion
The RootLocus (RL) is a classical method for the analysis and synthesis of linear timeinvariant (LTI) integerorder systems, consisting of the plot of the paths of all possible closedloop poles as a design parameter varies in a given range. Nowadays, there are efficient numerical algorithms devoted to RL analysis, implemented by several packages. For integerorder systems, there are wellknown practical rules for RL sketching, but those cannot be directly applied to FO systems, and the existing literature on this topic almost exclusively focuses on particular cases, namely, the commensurate FO systems.
This paper generalises RL practical rules to a class of FO systems, which are defined by an openloop transfer function expressed as a ratio of FO zeros and poles. As usual, using practical rules, even though the RL sketch might result somewhat incomplete, the ability to quickly sketch RL by hand is invaluable, from the control designer viewpoint, in making fundamental decisions early in the design process.
References
 W. R. Evans, “Graphical analysis of control systems,” Transactions of the American Institute of Electrical Engineers, vol. 67, no. 1, pp. 547–551, 1948. View at: Google Scholar
 W. R. Evans, “Control systems synthesis by root locus method,” Transactions of the American Institute of Electrical Engineers, vol. 69, no. 1, pp. 66–69, 1950. View at: Google Scholar
 A. M. Krall, “The root locus method: a survey,” SIAM Review, vol. 12, pp. 64–72, 1970. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 A. M. Eydgahi and M. Ghavamzadeh, “Complementary root locus revisited,” IEEE Transactions on Education, vol. 44, no. 2, pp. 137–143, 2001. View at: Publisher Site  Google Scholar
 E. Bahar and M. Fitzwater, “Numerical technique to trace the loci of the complex roots of characteristic equations,” SIAM Journal on Scientific and Statistical Computing, vol. 2, no. 4, pp. 389–403, 1981. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 C. I. Byrnes, D. S. Gilliam, and J. He, “Rootlocus and boundary feedback design for a class of distributed parameter systems,” SIAM Journal on Control and Optimization, vol. 32, no. 5, pp. 1364–1427, 1994. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 http://www.mathworks.com/.
 http://www.gnu.org/software/octave/.
 http://www.scilab.org/.
 http://freemat.sourceforge.net/.
 D. Matignon, “Stability properties for generalized fractional differential systems,” in Systèmes Différentiels Fractionnaires, vol. 5, pp. 145–158, 1998. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 D. Matignon, “Stability results on fractional differential equations with applications to control processing,” in Proceedings of the Computational Engineering in Systems Applications (CESA '96) IMACSIEEE/SMC Multiconference, pp. 963–968, Lille, France, 1996. View at: Google Scholar
 C. F. Lorenzo and T. T. Hartley, “Initialization, conceptualization, and application in the generalized (Fractional) calculus,” NASA TP1998208415, 1998. View at: Google Scholar
 I. Petráš, “Stability of fractionalorder systems with rational orders: a survey,” Fractional Calculus & Applied Analysis, vol. 12, no. 3, pp. 269–298, 2009. View at: Google Scholar  MathSciNet
 F. MerrikhBayat, M. Afshar, and M. KarimiGhartemani, “Extension of the rootlocus method to a certain class of fractionalorder systems,” ISA Transactions, vol. 48, no. 1, pp. 48–53, 2009. View at: Publisher Site  Google Scholar
 F. MerrikhBayat and M. Afshar, “Extending the rootlocus method to fractionalorder systems,” Journal of Applied Mathematics, vol. 2008, Article ID 528934, 13 pages, 2008. View at: Publisher Site  Google Scholar
 J. A. T. Machado, “Root locus of fractional linear systems,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 10, pp. 3855–3862, 2011. View at: Publisher Site  Google Scholar
 J. A. T. Machado, “A gallery of root locus of fractional systems,” in Proceedings of the ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference (DETC '13), Portland, Ore, USA, 2013. View at: Google Scholar
 S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, London, UK, 1993. View at: MathSciNet
 K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993. View at: MathSciNet
 I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, San Diego, CA, USA, 1999. View at: MathSciNet
 A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, NorthHolland Mathematics Studies, Elsevier, Amsterdam, The Netherlands, 2006. View at: MathSciNet
 K. Diethelm, The Analysis of Fractional Differential Equations, Springer, Berlin, Germany, 2010. View at: Publisher Site  MathSciNet
 A. Oustaloup, La Commande CRONE: Commande Robuste D’Ordre Non Entier, Hermès, Paris, France, 1991.
 G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, Oxford University Press, New York, NY, USA, 2008. View at: MathSciNet
 R. Magin, Fractional Calculus in Bioengineering, Begell House Publishers, Redding, CA, USA, 2006.
 F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London, UK, 2010. View at: Publisher Site  MathSciNet
 C. A. Monje, Y. Chen, B. M. Vinagre, D. Xue, and V. Feliu, Fractional Order Systems and Controls: Fundamentals and Applications, Springer, London, UK, 2010. View at: Publisher Site  MathSciNet
 J. T. Machado, V. Kiryakova, and F. Mainardi, “Recent history of fractional calculus,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1140–1153, 2011. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 T. J. Anastasio, “The fractionalorder dynamics of brainstem vestibulooculomotor neurons,” Biological Cybernetics, vol. 72, no. 1, pp. 69–79, 1994. View at: Publisher Site  Google Scholar
 J.G. Lu and Y.Q. Chen, “Robust stability and stabilization of fractionalorder interval systems with the fractional order alpha: the $0<\alpha <1$ Case,” IEEE Transactions on Automatic Control, vol. 55, no. 1, pp. 152–158, 2010. View at: Publisher Site  Google Scholar  MathSciNet
 D. Baleanu, A. K. Golmankhaneh, A. K. Golmankhaneh, and R. R. Nigmatullin, “Newtonian law with memory,” Nonlinear Dynamics, vol. 60, no. 12, pp. 81–86, 2010. View at: Publisher Site  Google Scholar
 C. M. Ionescu, J. A. T. Machado, and R. de Keyser, “Modeling of the lung impedance using a fractionalorder ladder network with constant phase elements,” IEEE Transactions on Biomedical Circuits and Systems, vol. 5, no. 1, pp. 83–89, 2011. View at: Publisher Site  Google Scholar
 E. Scalas, R. Gorenflo, and F. Mainardi, “Fractional calculus and continuoustime finance,” Physica A, vol. 284, no. 1–4, pp. 376–384, 2000. View at: Publisher Site  Google Scholar  MathSciNet
 F. B. Duarte, J. A. T. Machado, and G. Monteiro Duarte, “Dynamics of the Dow Jones and the NASDAQ stock indexes,” Nonlinear Dynamics, vol. 61, no. 4, pp. 691–705, 2010. View at: Publisher Site  Google Scholar
 C. M. Ionescu, P. Segers, and R. de Keyser, “Mechanical properties of the respiratory system derived from morphologic insight,” IEEE Transactions on Biomedical Engineering, vol. 56, no. 4, pp. 949–959, 2009. View at: Publisher Site  Google Scholar
 C. Ionescu and J. T. Machado, “Mechanical properties and impedance model for the branching network of the sapping system in the leaf of Hydrangea Macrophylla,” Nonlinear Dynamics, vol. 60, no. 12, pp. 207–216, 2010. View at: Publisher Site  Google Scholar
 J. A. T. Machado and A. M. Lopes, “Dynamical analysis of the global warming,” Mathematical Problems in Engineering, vol. 2012, Article ID 971641, 12 pages, 2012. View at: Publisher Site  Google Scholar
 J. A. T. Machado, A. C. Costa, and M. D. Quelhas, “Fractional dynamics in DNA,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 8, pp. 2963–2969, 2011. View at: Publisher Site  Google Scholar
 W. H. Deng and C. P. Li, “Chaos synchronization of the fractional Lü system,” Physica A, vol. 353, no. 1–4, pp. 61–72, 2005. View at: Publisher Site  Google Scholar
 R. R. Nigmatullin, “'Fractional' kinetic equations and 'universal' decoupling of a memory function in mesoscale region,” Physica A, vol. 363, no. 2, pp. 282–298, 2006. View at: Publisher Site  Google Scholar
 O. P. Agrawal, “Fractional variational calculus in terms of Riesz fractional derivatives,” Journal of Physics A, vol. 40, no. 24, pp. 6287–6303, 2007. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 A. Oustaloup, X. Moreau, and M. Nouillant, “The crone suspension,” Control Engineering Practice, vol. 4, no. 8, pp. 1101–1108, 1996. View at: Publisher Site  Google Scholar
 A. G. Radwan, A. M. Soliman, and A. S. Elwakil, “Design equations for fractionalorder sinusoidal oscillators: four practical circuit examples,” International Journal of Circuit Theory and Applications, vol. 36, no. 4, pp. 473–492, 2008. View at: Publisher Site  Google Scholar
 I. Petráš, “A note on the fractionalorder Chua's system,” Chaos, Solitons and Fractals, vol. 38, no. 1, pp. 140–147, 2008. View at: Publisher Site  Google Scholar
 H. L. Cao, Z. H. Deng, X. Li, J. Yang, and Y. Qin, “Dynamic modeling of electrical characteristics of solid oxide fuel cells using fractional derivatives,” International Journal of Hydrogen Energy, vol. 35, no. 4, pp. 1749–1758, 2010. View at: Publisher Site  Google Scholar
 J. A. T. Machado, “Fractional order modelling of fractionalorder holds,” Nonlinear Dynamics, vol. 70, no. 1, pp. 789–796, 2012. View at: Publisher Site  Google Scholar  MathSciNet
 I. Petras, FractionalOrder Nonlinear Systems: Modeling, Analysis and Simulation, Springer, New York, NY, USA, 2011.
 B. Gross and E. P. Braga, Singularities of Linear System Functions, Elsevier, New York, NY, USA, 1961. View at: MathSciNet
Copyright
Copyright © 2013 António M. Lopes and J. A. Tenreiro Machado. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.