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Nilpotency indices of state matrices and linear dependence on time of state variable of electrical circuits with state feedbacks

Tadeusz Kaczorek
Archives of Electrical Engineering | 2010 | vol. 59 | No 1 - 2 September | 21-34 | DOI: 10.2478/s10171-010-0002-3
Keywords electrical circuits nilpotency indices state-feedback
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Abstract

Electrical circuits with state-feedbacks are addressed. It is shown that by suitable choice of the gain matrices of state-feedbacks it is possible to obtain the closed-loop system matrices with nilpotency indices equal to two and their state variables are linear functions of time. The considerations are illustrated by linear electrical circuits.

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Authors and Affiliations

Tadeusz Kaczorek

Minimum energy control of fractional positive electrical circuits

Tadeusz Kaczorek
Archives of Electrical Engineering | 2016 | vol. 65 | No 2 June | 191-201 | DOI: 10.1515/aee-2016-0013
Keywords fractional positive electrical circuits minimum energy control bounded inputs procedure
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Abstract

Minimum energy control problem for the fractional positive electrical circuits is formulated and solved. Sufficient conditions for the existence of solution to the problem are established. A procedure for solving of the problem is proposed and illustrated by an example of fractional positive electrical circuit.

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Authors and Affiliations

Tadeusz Kaczorek

Electric circuit analysis by means of optimization criteria. Part I - the simple circuits.

M. Siwczyński M. Jaraczewski
Bulletin of the Polish Academy of Sciences Technical Sciences | 2004 | vol. 52 | No 4 | 359-367
Keywords electrical circuits power theory optimization in Hilbert spaces
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Abstract

One of the main problems of electrical power quality is to ensure a constant power ?ux from the supply system to the receiver, keeping in the same time the undisturbed wave form of the current and voltage signals. Distortion of signals are caused by nonlinear or time varying receivers, voltage changes or power losses in a supply system. The wave-form of the voltage of the source may also be deformed. This study seeks the optimal current and voltage wave-form by means of an optimization criteria. The optimization problem is de?ned in Hilbert space and the special functionals are minimized. The source inner impedance operator is linear and time-varying. Some examples of calculations are presented.

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Authors and Affiliations

M. Siwczyński
M. Jaraczewski

Invariance of reachability and observability for fractional positive linear electrical circuit with delays

Tong Yuan Hongli Yang
Archives of Electrical Engineering | 2021 | vol. 70 | No 3 | 513-530 | DOI: 10.24425/aee.2021.137571
Keywords invariance observability positive linear electrical circuits with delays reachability
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Abstract

This paper focuses on the invariance of the reachability and observability for fractional order positive linear electrical circuits with delays and their checking methods. By derivation and comparison, it shows that conditions and checking methods of reachability and observability for integer and fractional order positive linear electrical circuits with delays are invariant. An illustrative example is presented at the end of the paper.
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Bibliography

[1] Dorf C.R., Svoboda A.J., Introduction to electric circuits, John Wiley & Sons (2010).
[2] Kaczorek T., Rogowski K., Fractional linear systems and electrical circuit, Springer (2015).
[3] Federico M., Power system modelling and scripting, Springer (2010).
[4] Kaczorek T., Selected problems of fractional systems theory, Springer (2011).
[5] Xin Z., Wenru L. et al., Application of fractional calculus in iterative sliding mode synchronization control, Archives of Electrical Engineering, vol. 69, no. 3, pp. 499–519 (2020).
[6] Piotrowska E., Analysis of linear continuous-time systems by the use of the conformable fractional calculus and Caputo, Archives of Electrical Engineering, vol. 67, no. 3, pp. 629–639 (2018).
[7] Francisco G.A.J., Juan R.G., Fractional RC and LC electrical circuits, Ingeniera, Investigacin y Tecnologa, vol. 15, no. 2, pp. 311–319 (2014).
[8] Sikora R., Fractional derivatives in electrical circuit theory-critical remarks, Archives of Electrical Engineering, vol. 66, no. 1, pp. 155–163 (2017).
[9] Sikora R., Pawłowski S., Problematic Applications of Fractional Derivatives in Electrotechnics and Electrodynamics, Conference on Selected Issues of Electrical Engineering and Electronics, Szczecin, Poland, pp. 1–5 (2018).
[10] Sikora R., Pawłowski S., Fractional derivatives and the laws of electrical engineering, COMPEL-The international journal for computation and mathematics in electrical and electronic engineering, vol. 37, no. 4, pp. 1384–1391 (2018).
[11] Muthana T.A., Mohamed Z., On the control of time delay power systems, International Journal of Innovative Computing, Information and Control, vol. 9, no. 2, pp. 769–792 (2013).
[12] Zhaoyan L., Jun Q., A simple method to compute delay margin of power system with single delay, Automation of Electric Power System, vol. 32, no. 18, pp. 8–13 (2008).
[13] Jianjun Z., Yonggao Z., Research on optimal configuration of fault current limiter based on reliability in large power network, Archives of Electrical Engineering, vol. 69, no. 3, pp. 661–677 (2020).
[14] Kaczorek T., Stability of positive continuous-time linear systems with delays, Bulletin of The Polish Academy of Sciences-technical Sciences, vol. 57, no. 4, pp. 395–398 (2009).
[15] Kaczorek T., Stability tests of positive fractional continuous-time linear systems with delays, TransNav, the International Journal on Marine Navigation and Safety of Sea Transportation, vol. 7, no. 2, pp. 211–215 (2013).
[16] Xianming Z., Min W., On delay-dependent stability for linear systems with delay, Journal of Circuit and Systems, vol. 8, no. 3, pp. 118–120 (2003).
[17] Hai Z., Daiyong W., Stability analysis for fractional-order linear singular delay differential systems, Discrete Dynamics in Nature and Society, vol. 2014, no. 2014, pp. 1–8 (2014).
[18] Xianggeng Z., Yuxia L. et al., Stability analysis of fractional-order Langford systems, Journal of Shandong University of Science and Technology (Natural Science), vol. 38, no. 3, pp. 65–71(2019).
[19] Wei J., Zhicheng W., Controllability of singular control systems with delay, Journal of Hunan University, vol. 26, no. 4, pp. 6–9 (1999).
[20] Qiong W., Wei J., The complete controllability, after all controllability, ultimate controllability and quasi controllability of delay control system, College Mathematic, vol. 19, no. 3, pp. 63–66 (2003).
[21] Wei J., The controllability of delay degenerate control systems with independent subsystems, Applied Mathematics and Mechanics, vol. 24, no. 6, pp. 706–713 (2003).
[22] Peng L.,Wenlong W. et al., Alternate Charging and Discharging of Capacitor to Enhance the Electron Production of Bioelectrochemical Systems, Environmental Science and Technology, vol. 45, no. 15, pp. 6647–6653 (2011).
[23] Kaczorek T., Reachability and controllability to zero of positive fractional discrete-time systems, European Control Conference, Kos, Greece, pp. 1708–1712 (2007).
[24] Xindong S., Hongli Y., A new method for judgement computation of stability and stabilization of fractional order positive systems with constraints, Journal of Shandong University of Science and Technology (Natural Science), vol. 40, no. 1, pp. 12–20 (2021).
[25] Kaczorek T., Invariant properties of positive linear electrical circuit, Archives of Electrical Engineering, vol. 68, no. 4, pp. 875–890 (2019).
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Authors and Affiliations

Tong Yuan
1
ORCID: ORCID
Hongli Yang
1
  1. Shandong University of Science and Technology, China

Positive electrical circuits with the chain structure and cyclic Metzler state matrices

Tadeusz Kaczorek
Bulletin of the Polish Academy of Sciences Technical Sciences | 2021 | 69 | 4 | e137731 | DOI: 10.24425/bpasts.2021.137731
Keywords cyclic matrix positive electrical circuit fractional Metzler matrix
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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

Positivity and cyclicity of descriptor electrical circuits with chain structure

Tadeusz Kaczorek Kamil Borawski
Bulletin of the Polish Academy of Sciences Technical Sciences | 2022 | 70 | 1 | e139955 | DOI: 10.24425/bpasts.2021.139955
Keywords cyclic matrix descriptor positive electrical circuit fractional Metzler matrix
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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

A new extension of the Cayley-Hamilton theorem to fractional different orders linear systems

Tadeusz Kaczorek
Bulletin of the Polish Academy of Sciences Technical Sciences | 2022 | 70 | 2 | e139960 | DOI: 10.24425/bpasts.2021.139960
Keywords Cayley-Hamilton theorem electrical circuit extension fractional different order
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Abstract

The classical Cayley–Hamilton theorem is extended to fractional different order linear systems. The new theorems are applied to different orders fractional linear electrical circuits. The applications of new theorems are illustrated by numerical examples.
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Authors and Affiliations

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

Modelling of Dielectric Properties of BiNbO4-Based Microwave Ceramics

D. Czekaj A. Lisińska-Czekaj B. Garbarz-Glos W. Bąk
Archives of Metallurgy and Materials | 2020 | vol. 65 | No 3 | 1177-1181 | DOI: 10.24425/amm.2020.133236
Keywords BiNbO4 ceramics impedance spectroscopy simulation and modelling equivalent electric circuit method
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Abstract

In the present paper results of the studies devoted to computer simulations of dielectric response of electroceramics in a frequency domain as well as analysis of the experimental data are given. As an object of investigations BiNbO4-based microwave ceramics was taken. Simulations of the hypothetical impedance response of the ceramic system were performed under assumption of the brick-layer model. A strategy for analysis and modelling of the impedance data for microwave electroceramics was discussed. On the base of the discussed strategy modelling of the dielectric response of BiNbO4 ceramics was performed with the electric equivalent circuit method. The Voigt’s and Maxwell’s circuits were taken as electric models. Parameters of the electric components of the circuits were determined and related to parameters of the ceramic object under study. It was found that fitting quality was good and changed within the range χ2 = 6.78 × 10–4 – 6.77 × 10–5 depending on the model.

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Authors and Affiliations

D. Czekaj
A. Lisińska-Czekaj
B. Garbarz-Glos
W. Bąk

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