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Abstract

The asymptotic stability of positive descriptor continuous-time and discrete-time linear systems is considered. New sufficient conditions for stability of positive descriptor linear systems are established. The efficiency of the new stability conditions are demonstrated on numerical examples of continuous-time and discrete-time linear systems.
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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 Bialystok, Poland
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Abstract

The concepts of full-order and reduced-order observers are extended to the fractional linear continuous-time systems. Necessary and sufficient conditions for the existence of the observers for fractional linear systems are established. Procedures for designing of the observers are given and illustrated by examples.
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Authors and Affiliations

Tadeusz Kaczorek
1
ORCID: ORCID

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

The eigenvalues assignment problems for descriptor linear systems with state and its derivative feedbacks are considered herein. Necessary and sufficient conditions for the existence of solutions to the problems are established. The Euler and Tustin approximations of the continuous-time systems are analyzed. Procedures for computation of the feedbacks are given and illustrated by numerical examples.
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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 Bialystok, Poland
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Abstract

Poles and zeros assignment problem by state feedbacks in positive continuous-time and discrete-time systems is analyzed. It is shown that in multi-input multi-output positive linear systems by state feedbacks the poles and zeros of the transfer matrices can be assigned in the desired positions. In the positive continuous-time linear systems the feedback gain matrix can be chosen as a monomial matrix so that the poles and zeros of the transfer matrices have the desired values if the input matrix B is monomial. In the positive discrete-time linear systems to solve the problem the matrix B can be chosen monomial if and only if in every row and every column of the n x n system matrix A the sum of n-1 its entries is less than one. Key words: assignment, pole, zero, transfer matrix, linear, positive, system, state feedback
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Bibliography

[1] E. Antsaklis and A. Michel: Linear Systems. Birkhauser, Boston, 2006.
[2] L. Farina and S. Rinaldi: Positive Linear Systems: Theory and Applications. J. Wiley & Sons, New York, 2000.
[3] T. Kaczorek: Linear Control Systems, vol. 2. Research Studies Press LTD., J. Wiley, New York, 1992.
[4] T. Kaczorek: Positive 1D and 2D Systems. London, UK, Springer-Verlag, 2002.
[5] T. Kaczorek: Selected Problems of Fractional Systems Theory. Berlin, Germany, Springer-Verlag, 2011.
[6] T. Kaczorek and K. Rogowski: Fractional Linear Systems and Electrical Circuits, Studies in Systems, Decision and Control, Vol. 13. Springer, 2015.
[7] T. Kailath: Linear systems. Prentice Hall, Englewood Cliffs, New York, 1980.
[8] R.E. Kalman: Mathematical description of linear systems. J. SIAM Control, 1(2), (1963), 152–192, DOI: 10.1137/0301010.
[9] R.E. Kalman: On the general theory of control systems. Proc. First International Congress on Automatic Control, London, UK, Butterworth, (1960), 481–493,
[10] J. Klamka: Controllability of Dynamical Systems. Kluwer, Acadenic Publ., Dordrecht 1991.
[11] H. Rosenbrock: State-Space and Multivariable Theory. New York, USA, J. Wiley, 1970.
[12] S.M. Zak: Systems and Control. New York, Oxford University Press, 2003.
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Authors and Affiliations

Tadeusz Kaczorek
1
ORCID: ORCID

  1. Białystok University of Technology, Faculty of Electrical Engineering, Wiejska 45D, 15-351 Białystok, 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 asymptotic stability of the convex linear combination of continuous-time and discretetime linear systems is considered. Using the Gershgorin theorem it is shown that the convex linear combination of the linear asymptotically stable continuous-time and discretetime linear systems is also asymptotically stable. It is shown that the above thesis is also valid (even simpler) for positive linear systems.
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Authors and Affiliations

Tadeusz Kaczorek
1
ORCID: ORCID

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

The exponential decay of transient values in discrete-time nonlinear standard and fractional orders systems with linear positive linear part and positive feedbacks is investigated. Sufficient conditions for the exponential decay of transient values in this class of positive nonlinear systems are established. A procedure for computation of gains characterizing the class of nonlinear elements are given and illustrated on simple example.
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Authors and Affiliations

Tadeusz Kaczorek
1
ORCID: ORCID
Andrzej Ruszewski
1
ORCID: ORCID

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

The asymptotic stability of fractional positive descriptor continuous-time and discretetime linear systems is considered. New sufficient conditions for stability of fractional positive descriptor linear systems are established. The efficiency of the new stability conditions are demonstrated on numerical examples of fractional continuous-time and discrete-time linear systems.
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Authors and Affiliations

Tadeusz Kaczorek
1
ORCID: ORCID
Andrzej Ruszewski
1
ORCID: ORCID

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

The Floquet-Lyapunov transformation is extended to fractional discrete-time linear systems with periodic parameters. A procedure for computation of the transformation is proposed and illustrated by a numerical example.
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Authors and Affiliations

Tadeusz Kaczorek
1
ORCID: ORCID
Andrzej Ruszewski
1
ORCID: ORCID

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

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