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Abstract

In the paper positive fractional continuous-time linear systems are considered. Positive fractional systems without delays and positive fractional systems with a single delay in control are studied. New criteria for approximate and exact controllability of systems without delays as well as a relative controllability criterion of systems with delay are established and proved. Numerical examples are presented for different controllability criteria. A practical application is proposed.
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Bibliography

[1] A. Abdelhakim and J. Tenreiro Machado: A critical analysis of the conformable derivative, Nonlinear Dynamics, 95 (2019), 3063–3073, DOI: 10.1007/s11071-018-04741-5.
[2] K. Balachandran, Y. Zhou and J. Kokila: Relative controllability of fractional dynamical systems with delays in control, Communications in Nonlinear Science and Numerical Simulation, 17 (2012), 3508–3520, DOI: 10.1016/j.cnsns.2011.12.018.
[3] K. Balachandran, J. Kokila, and J.J. Trujillo: Relative controllability of fractional dynamical systems with multiple delays in control, Computers and Mathematics with Apllications, 64 (2012), 3037–3045, DOI: 10.1016/j.camwa.2012.01.071.
[4] P. Duch: Optimization of numerical algorithms using differential equations of integer and incomplete orders, Doctoral dissertation, Lodz University of Technology, 2014 (in Polish).
[5] C. Guiver, D. Hodgson and S. Townley: Positive state controllability of positive linear systems. Systems and Control Letters, 65 (2014), 23–29, DOI: 10.1016/j.sysconle.2013.12.002.
[6] R.E. Gutierrez, J.M. Rosario and J.T. Machado: Fractional order calculus: Basic concepts and engineering applications, Mathematical Problems in Engineering, 2010 Article ID 375858, DOI: 10.1155/2010/375858.
[7] T. Kaczorek: Positive 1D and 2D Systems, Communications and Control Engineering, Springer, London 2002.
[8] T. Kaczorek: Fractional positive continuous-time linear systems and their reachability, International Journal of Applied Mathematics and Computer Science, 18 (2008), 223–228, DOI: 10.2478/v10006-008-0020-0.
[9] T. Kaczorek: Positive linear systems with different fractional orders, Bulletin of the Polish Academy of Sciences: Technical Sciences, 58 (2010), 453–458, DOI: 10.2478/v10175-010-0043-1.
[10] T. Kaczorek: Selected Problems of Fractional Systems Theory, Lecture Notes in Control and Information Science, 411, 2011.
[11] T. Kaczorek: Constructability and observability of standard and positive electrical circuits, Electrical Review, 89 (2013), 132–136.
[12] T. Kaczorek: An extension of Klamka’s method of minimum energy control to fractional positive discrete-time linear systems with bounded inputs, Bulletin of the Polish Academy of Sciences: Technical Sciences, 62 (2014), 227–231, DOI: 10.2478/bpasts-2014-0022.
[13] T. Kaczorek: Minimum energy control of fractional positive continuoustime linear systems with bounded inputs, International Journal of Applied Mathematics and Computer Science, 24 (2014), 335–340, DOI: 10.2478/amcs-2014-0025.
[14] T. Kaczorek and K. Rogowski: Fractional Linear Systems and Electrical Circuits, Springer, Studies in Systems, Decision and Control, 13 2015.
[15] T. Kaczorek: A class of positive and stable time-varying electrical circuits, Electrical Review, 91 (2015), 121–124. DOI: 10.15199/48.2015.05.29.
[16] T. Kaczorek: Computation of transition matrices of positive linear electrical circuits, BUSES – Technology, Operation, Transport Systems, 24 (2019), 179–184, DOI: 10.24136/atest.2019.147.
[17] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, 2006.
[18] J. Klamka: Controllability of Dynamical Systems, Kluwer Academic Publishers, 1991.
[19] T.J.Machado,V. Kiryakova and F. Mainardi: Recent history of fractional calculus, Communications in Nonlinear Science and Numerical Simulation, 6 (2011), 1140–1153, DOI: 10.1016/j.cnsns.2010.05.027.
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[21] A. Monje, Y. Chen, B.M. Viagre, D. Xue and V. Feliu: Fractional-order Systems and Controls. Fundamentals and Applications, Springer-Verlag, 2010.
[22] K. Nishimoto: Fractional Calculus: Integrations and Differentiations of Arbitrary Order, University of New Haven Press, 1989.
[23] K.B. Oldham and J. Spanier: The Fractional Calculus, Academic Press, 1974.
[24] I. Podlubny: Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, In: Mathematics in Science and Engineering, Academic Press, 1999.
[25] S.G. Samko, A.A. Kilbas and O.I. Marichev: Fractional Integrals and Derivatives: Theory and Applications, Gordan and Breach Science Publishers, 1993.
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[27] B. Sikora: Controllability of time-delay fractional systems with and without constraints, IET Control Theory & Applications, 10 (2016), 1–8, DOI: 10.1049/iet-cta.2015.0935.
[28] B. Sikora: Controllability criteria for time-delay fractional systems with a retarded state, International Journal of Applied Mathematics and Computer Science, 26 (2016), 521–531, DOI: 10.1515/amcs-2016-0036.
[29] B. Sikora and J. Klamka: Constrained controllability of fractional linear systems with delays in control, Systems and Control Letters, 106 (2017), 9–15, DOI: 10.1016/j.sysconle.2017.04.013.
[30] B. Sikora and J. Klamka: Cone-type constrained relative controllability of semilinear fractional systems with delays, Kybernetika, 53 (2017), 370–381, DOI: 10.14736/kyb-2017-2-0370.
[31] B. Sikora: On application of Rothe’s fixed point theorem to study the controllability of fractional semilinear systems with delays, Kybernetika, 55 (2019), 675–689, DOI: 10.14736/kyb-2019-4-0675.
[32] T. Schanbacher: Aspects of positivity in control theory, SIAM J. Control and Optimization, 27 (1989), 457–475.
[33] B. Trzasko: Reachability and controllability of positive fractional discretetime systems with delay, Journal of Automation Mobile Robotics and Intelligent Systems, 2 (2008), 43–47.
[34] J. Wei: The controllability of fractional control systems with control delay, Computers and Mathematics with Applications, 64 (2012), 3153–3159, DOI: 10.1016/j.camwa.2012.02.065.
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Authors and Affiliations

Beata Sikora
1
ORCID: ORCID
Nikola Matlok
1

  1. Department of Applied Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland
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Abstract

The cyclicity of the state matrices of positive linear electrical circuits with the chain structure is considered. Two classes of positive linear electrical circuits with the chain structure and cyclic Metzler state matrices are analyzed. Some new properties of these classes of positive electrical circuits are established. The results are extended to fractional linear electrical circuits.
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Bibliography

  1.  A. Berman and R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences. Philadelphia: SIAM, 1994.
  2.  L. Farina and S. Rinaldi, Positive Linear Systems; Theory and Applications. New York: J. Wiley, 2000.
  3.  T. Kaczorek, Positive 1D and 2D Systems. London: Springer-Verlag, 2002.
  4.  T. Kaczorek, “Positive linear systems with different fractional orders,” Bull. Pol. Acad. Sci. Tech. Sci., vol. 58, no. 3, pp. 453–458, 2010.
  5.  T. Kaczorek, “Normal fractional positive linear systems and electrical circuits,” in Proc. Conf. Automation 2019, Warsaw, 2020, pp. 13–26.
  6.  T. Kaczorek, Selected Problems of Fractional Systems Theory. Berlin: Springer, 2011.
  7.  T. Kaczorek and K. Rogowski, Fractional Linear Systems and Electrical Circuits. Cham: Springer, 2015.
  8.  W. Mitkowski, “Dynamical properties of metzler systems,” Bull. Pol. Acad. Sci. Tech. Sci., vol. 54, no. 4, pp. 309–312, 2008.
  9.  W. Mitkowski, Outline of Control Theory. Kraków: Publishing House AGH, 2019.
  10.  P. Ostalczyk, Discrete Fractional Calculus. River Edge, NJ: World Scientific, 2016.
  11.  I. Podlubny, Fractional Differential Equations. San Diego: Academic Press, 1999.
  12.  T. Kaczorek, “Reachability and observability of positive discrete-time linear systems with integer positive and negative powers of the state frobenius matrices,” Arch. Control Sci., vol. 28, no. 1, pp. 5–20, 2018.
  13.  M.D. Ortigueira and J. A. Tenreiro Machado, “New discrete-time fractional derivatives based on the bilinear transformation: definitions and properties,” J. Adv. Res., vol. 25, pp. 1–10, 2020.
  14.  A. Ruszewski, “Stability of discrete-time fractional linear systems with delays,” Arch. Control Sci., vol. 29, no. 3, pp. 549–567, 2019.
  15.  L. Sajewski, “Stabilization of positive descriptor fractional discrete-time linear systems with two different fractional orders by decentralized controller,” Bull. Pol. Acad. Sci. Tech. Sci., vol. 65, no. 5, pp. 709–714, 2017.
  16.  R. Stanisławski, K. Latawiec, and M. Łukaniszyn, “A comparative analysis of laguerre-based approximatiors to the grunwald-letnikov fractional-order difference,” Math. Probl. Eng., vol. 2015, 2015.
  17.  F.G. Gantmacher, The Theory of Matrices. London: Chelsea Pub. Comp., 1959.
  18.  T. Kaczorek and K. Borawski, “Stability of continuoustime and discrete-time linear systems with inverse state matrices,” Meas. Autom. Monit., vol. 62, no. 4, pp. 132–135, 2016.
  19.  T. Kaczorek, Polynomial and Rational Matrices. London: Springer, 2007.
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Authors and Affiliations

Tadeusz Kaczorek
1
ORCID: ORCID

  1. Bialystok University of Technology, Faculty of Electrical Engineering, Wiejska 45D, 15-351 Białystok, Poland
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Abstract

The positivity and cyclicity of descriptor linear electrical circuits with chain structure is considered. Two classes of descriptor linear electrical circuits are analyzed. Some new properties of these classes of electrical circuits are established. The results are extended to fractional descriptor linear electrical circuits.
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Bibliography

  1.  A. Berman and R.J. Plemmons, Nonnegative Matrices in the Mathematical Sciences. Philadelphia: SIAM, 1994.
  2.  L. Farina and S. Rinaldi, Positive Linear Systems; Theory and Applications. New York: J. Wiley, 2000.
  3.  T. Kaczorek, Positive 1D and 2D Systems. London: Springer-Verlag, 2002.
  4.  T. Kaczorek, “Positive linear systems with different fractional orders,” Bull. Pol. Acad. Sci. Tech. Sci., vol. 58, no. 3, pp. 453– 458, 2010.
  5.  T. Kaczorek, Selected Problems of Fractional Systems Theory. Berlin: Springer, 2011.
  6.  T. Kaczorek, “Normal fractional positive linear systems and electrical circuits,” in Proc. Conf. Automation 2019, Warsaw, 2020, pp. 13–26.
  7.  T. Kaczorek and K. Rogowski, Fractional Linear Systems and Electrical Circuits. Cham: Springer, 2015.
  8.  W. Mitkowski, “Dynamical properties of Metzler systems,” Bull. Pol. Acad. Sci. Tech. Sci., vol. 54, no. 4, pp. 309–312, 2008.
  9.  W. Mitkowski, Outline of Control Theory. Kraków: Publishing House AGH, 2019.
  10.  P. Ostalczyk, Discrete Fractional Calculus. River Edge, NJ: World Scientific, 2016.
  11.  I. Podlubny, Fractional Differential Equations. San Diego: Academic Press, 1999.
  12.  T. Kaczorek, “Reachability and observability of positive discrete-time linear systems with integer positive and negative powers of the state frobenius matrices,” Arch. Control Sci., vol. 28, no. 1, pp. 5–20, 2018.
  13.  M.D. Ortigueira and J.A. Tenreiro Machado, “New discrete-time fractional derivatives based on the bilinear transformation: definitions and properties,” J. Adv.Res., vol. 25, pp. 1–10, 2020.
  14.  A. Ruszewski, “Stability of discrete-time fractional linear systems with delays,” Arch. Control Sci., vol. 29, no. 3, pp. 549– 567, 2019.
  15.  L. Sajewski, “Stabilization of positive descriptor fractional discrete-time linear systems with two different fractional orders by decentralized controller,” Bull. Pol. Acad. Sci. Tech. Sci., vol. 65, no. 5, pp. 709–714, 2017.
  16.  R. Stanisławski, K. Latawiec, and M. Łukaniszyn, “A Comparative Analysis of Laguerre-Based Approximatiors to the Grunwald-Letnikov Fractional-Order Difference,” Math. Probl. Eng., vol. 2015, p. 512104, 2015.
  17.  L. Dai, Singular Control Systems, ser. Lecture Notes in Control and Information Sciences. Berlin: Springer, 1989, vol. 118.
  18.  D. Guang-Ren, Analysis and Design of Descriptor Linear Systems. New York: Springer, 2010.
  19.  T. Kaczorek, Linear Control Systems. vol. 1. New York, USA: J. Wiley, 1992.
  20.  T. Kaczorek and K. Borawski, Descriptor Systems of Integer and Fractional Orders, ser. Studies in Systems, Decision and Control. Cham: Springer, 2021, vol. 367.
  21.  K. Borawski, “Superstabilization of Descriptor ContinuousTime Linear Systems via State-Feedback Using Drazin Inverse Matrix Method,” Symmetry, vol. 12, no. 6, p. 940, 2020.
  22.  M. Rami and D. Napp, “Characterization and Stability of Autonomous Positive Descriptor Systems,” IEEE Trans. Autom. Contr., vol. 57, no. 10, pp. 2668–2673, 2012.
  23.  E. Virnik, “Stability analysis of positive descriptor systems,” Linear Algebra Appl., vol. 429, no. 10, pp. 2640–2659, 2008.
  24.  F.G. Gantmacher, The Theory of Matrices. London: Chelsea Pub. Comp., 1959.
  25.  T. Kaczorek, “Positive electrical circuits with the chain structure and cyclic Metzler state matrices,” Bull. Pol. Acad. Sci. Tech. Sci., vol. 69, no. 4, pp. 1–5, 2021.
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Authors and Affiliations

Tadeusz Kaczorek
1
ORCID: ORCID
Kamil Borawski
1

  1. Bialystok University of Technology, Faculty of Electrical Engineering, ul. Wiejska 45D, 15-351 Białystok, Poland

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