In this paper a new method for the calculation of the root locus of fractional systems is presented. The proposed algorithm takes advantage of present day computational resources and processes directly the characteristic equation. The results demonstrate the feasibility of the method for different types of expressions.N/

Galhano, Alexandra

Tenreiro Machado, J. A.

Scientific Repository of the Polytechnic Institute of Porto

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A Computational Approach for Obtaining the Root Locus of Fractional Systems

Repositório Científico do Instituto Politécnico do Porto

   	    A Computational Approach for Obtaining the Root Locus of Fractional SystemsJ. A. Tenreiro Machado∗ and Alexandra M. Galhano∗∗Institute of Engineering of Porto, Dept. of Electrical Engineering, Porto PortugalSummary. In this paper a new method for the calculation of the root locus of fractional systems is presented. The proposed algorithmtakes advantage of present day computational resources and processes directly the characteristic equation. The results demonstrate thefeasibility of the method for different types of expressions.IntroductionDuring the last decades Fractional calculus (FC) veriﬁed a strong development which is demonstrated by the large volumeof research published. In spite of its popularity, FC requires some efforts towards the development and adoption ofstandard tools often adopted in system analysis and control [7, 8, 14, 9]. This paper addresses the development of analgorithm for obtaining the Root Locus (RL) of fractional order expressions of any type, easy to implement using today’scomputational resources. In this line of thought the next section introduces the algorithm for the calculation of the RL offractional order expressions and presents several examples.Root locus of fractional systemsThe RL is a classical tool for the stability analysis of integer order linear systems [1, 2, 5, 3], but its application in thefractional counterpart poses some difﬁculties. Recently the RL was considered for the stability analysis of fractionalsystems by tacking advantage of commensurable expressions that occur when truncating real valued integrodifferentialorders up to a ﬁnite precision [12]. This strategy allows the use of built in routines available in several engineeringpackages, with a good level of integration of commands and editor, but, on the other hand, limits the precision and thetype of symbolic expressions. Bearing these ideas in mind, it was decided to implement a numerical tool for constructionthe RL without the previous limitations. The algorithm consists in searching the complex plane s for possible solutionsof the closed-loop characteristic equation. In general, in the s-plane it is adopted a two stage calculation scheme, witha large spaced grid of points for a preliminary evaluation of candidate solutions and, for each solution, a more precisecalculation by inserting a local small spaced grid of points. Due to the importance of solutions in the real and imaginaryaxis it was decided to search directly those cases. Therefore, the structure of the algorithm is designed as follows:1. Deﬁnition of the system characteristic equation2. Deﬁnition of parameters:2.1 The limits of the search in the complex plane2.2 The number of points with the large spaced grid for testing possible solutions2.3 The accuracy threshold for entering the high precision root evaluation2.4 The number of points in the small spaced grid for calculating more precise solutions.3. Calculation of solutions in the imaginary axis with two phases3.1 Test solutions with large grid3.2 If feasible solution found, then larger precision evaluation by searching within a local small spaced grid4. Calculation of solutions in the real axis, directly with small spaced grid.5. Calculation of solutions in the complex plane with two phases5.1 Test solutions with large spaced grid5.2 If feasible solution found, then larger precision evaluation by searching within a local small spaced gridFor the implementation of the algorithm were tested both compiler and interpreted computational packages. It was veriﬁedthat, although popular in some areas, interpreted codes were much slower and, therefore, in the sequel the experimentsare based on code implemented with the Lazarus compiler [6].For the purpose of testing the algorithm are considered several fractional characteristic equations Q (s) proposed in theliterature, namely:• Example 1, by F. Merrikh-Bayat et al [10] Q (s) = s2 − 3s1.5 − 2s+ 2s0.5 + 12 + k (s0.5 − 1)• Example 2, by I. Podlubny et al [13] Q (s) = 0.7943s2.5708 + 5.2385s0.8372 + 1.5560 + k• Example 3, by I. Jesus et al [4] Q (s) = 1 + ke−3.0√s0.042   	    Figure 1: Root locus for k ≥ 0 of the fractional-order characteristic equations: Q (s) = s2 − 3s1.5 − 2s+ 2s0.5 + 12+ k (s0.5 − 1),Q (s) = 0.7943s2.5708 + 5.2385s0.8372 + 1.5560 + k, Q (s) = 1 + ke−3.0√s0.042 , Q (s) = s+ k(s0.5 + 1)e−√s• Example 4, by F. Merrikh-Bayat et al [11] Q (s) = s+ k (s0.5 + 1) e−√swhere s represents the Laplace transform variable. Figure 1 depicts the corresponding RL obtained through the proposedcomputation scheme. The results are consistent with those reported based on the adaptation of the standard RL, whenavailable, or with stability conditions based on other methods, for those cases where the RL was not obtained. It was alsoconﬁrmed that the quality, precision and speed can be easily tuned through the parameters deﬁning the grid points andthe precision threshold. From the operational point of view we ﬁnd that we loose the automatisms provided by commonengineering packages, since we have to insert the code for each characteristic equation but, on the other hand, we gaincomputational speed and freedom of analyzing a larger set of fractional-order expressions. In conclusion, the proposedalgorithm provides a simple platform for the stability analysis and control design of closed loop linear fractional systems.References[1] Evans W. R. (1948) Graphical Analysis of Control Systems Trans., AIEE 67:547-551.[2] Evans W. R. (1950) Control Systems Synthesis by Root Locus Method Trans., AIEE 69:66-69.[3] Eydgahi A. M., Ghavamzadeh M. (2001) Complementary root locus revisited IEEE Transactions on Education 44:137-143.[4] Jesus I.S., Machado J. T. (2008) Fractional Control of Heat Diffusion Systems Nonlinear Dynamics 54:263-282.[5] Krall A. M. (1970) The root locus method: a survey SIAM Review 12:64-72.[6] Lazarus web site http://www.lazarus.freepascal.org/[7] Machado J. T. (1997) Analysis and design of fractional-order digital control systems Systems Analysis, Modelling, Simulation 27:107-122.[8] Machado J. T. (1999) Fractional-Order Derivative Approximations in Discrete-Time Control Systems Systems Analysis-Modelling-Simulation34:419-434.[9] Machado J. T. (2001) Discrete-time fractional-order controllers Journal of Fractional Calculus & Applied Analysis 4:47-66.[10] Merrikh-Bayat F., Afshar M. (2008) Extending the Root-Locus Method to Fractional-Order Systems Journal of Applied Mathematics Hindawi,Article ID 528934, 13 pages, doi:10.1155/2008/528934.[11] Merrikh-Bayat F., Karimi-Ghartemani M. (2009) An efﬁcient numerical algorithm for stability testing of fractional-delay systems ISA Transactions48:32-37.[12] Merrikh-Bayat F. (2010) Stability of Fractional-Delay Systems: A Practical Approach, D. Baleanu et al. (eds.), New Trends in Nanotechnology andFractional Calculus Applications, Springer.[13] Podlubny I. (1998) Fractional Differential Equations, Volume 198: An Introduction to Fractional Derivatives, Fractional Differential Equations, toMethods of Their Solution (Mathematics in Science and Engineering), Academic Press.[14] Podlubny I. (1999) Fractional-order systems and PIλDμ-controllers IEEE Transactions on Automatic Control 44:208-213.

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A Computational Approach for Obtaining the Root Locus of Fractional Systems

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