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Abstract

The positive asymptotically stable continuous-time linear systems are approximated by corresponding asymptotically stable discrete-time linear systems. Two methods of the approximation are presented and the comparison of the methods is addressed. The considerations are illustrated by three numerical examples and an example of positive electrical circuit.

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Authors and Affiliations

Tadeusz Kaczorek
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Abstract

The practical and asymptotic stabilities of delayed fractional discrete-time linear systems described by the model without a time shift in the difference are addressed. The D-decomposition approach is used for stability analysis. New necessary and sufficient stability conditions are established. The conditions in terms of the location of eigenvalues of the system matrix in the complex plane are given.

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Authors and Affiliations

A. Ruszewski
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Abstract

In this paper, we show that signal sampling operation can be considered as a kind of all-pass filtering in the time domain, when the Nyquist frequency is larger or equal to the maximal frequency in the spectrum of a signal sampled. We demonstrate that this seemingly obvious observation has wideranging implications. They are discussed here in detail. Furthermore, we discuss also signal shaping effects that occur in the case of signal under-sampling. That is, when the Nyquist frequency is smaller than the maximal frequency in the spectrum of a signal sampled. Further, we explain the mechanism of a specific signal distortion that arises under these circumstances. We call it the signal shaping, not the signal aliasing, because of many reasons discussed throughout this paper. Mainly however because of the fact that the operation behind it, called also the signal shaping here, is not a filtering in a usual sense. And, it is shown that this kind of shaping depends upon the sampling phase. Furthermore, formulated in other words, this operation can be viewed as a one which shapes the signal and performs the low-pass filtering of it at the same time. Also, an interesting relation connecting the Fourier transform of a signal filtered with the use of an ideal low-pass filter having the cut frequency lying in the region of under-sampling with the Fourier transforms of its two under-sampled versions is derived. This relation is presented in the time domain, too.

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Authors and Affiliations

Andrzej Borys
ORCID: ORCID
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Abstract

The global stability of discrete-time nonlinear systems with descriptor positive linear parts and positive scalar feedbacks is addressed. Sufficient conditions for the global stability of standard and fractional nonlinear systems are established. The effectiveness of these conditions is illustrated on numerical examples.

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Authors and Affiliations

Tadeusz Kaczorek
ORCID: ORCID
Andrzej Ruszewski
ORCID: ORCID
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Abstract

The stability analysis for discrete-time fractional linear systems with delays is presented. The state-space model with a time shift in the difference is considered. Necessary and sufficient conditions for practical stability and for asymptotic stability have been established. The systems with only one matrix occurring in the state equation at a delayed moment have been also considered. In this case analytical conditions for asymptotic stability have been given. Moreover parametric descriptions of the boundary of practical stability and asymptotic stability regions have been presented.

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Authors and Affiliations

Andrzej Ruszewski
ORCID: ORCID
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Abstract

The Bulletin of the Polish Academy of Sciences: Technical Sciences (Bull.Pol. Ac.: Tech.) is published bimonthly by the Division IV Engineering Sciences of the Polish Academy of Sciences, since the beginning of the existence of the PAS in 1952. The journal is peer‐reviewed and is published both in printed and electronic form. It is established for the publication of original high quality papers from multidisciplinary Engineering sciences with the following topics preferred: Artificial and Computational Intelligence, Biomedical Engineering and Biotechnology, Civil Engineering, Control, Informatics and Robotics, Electronics, Telecommunication and Optoelectronics, Mechanical and Aeronautical Engineering, Thermodynamics, Material Science and Nanotechnology, Power Systems and Power Electronics.

Journal Metrics: JCR Impact Factor 2018: 1.361, 5 Year Impact Factor: 1.323, SCImago Journal Rank (SJR) 2017: 0.319, Source Normalized Impact per Paper (SNIP) 2017: 1.005, CiteScore 2017: 1.27, The Polish Ministry of Science and Higher Education 2017: 25 points.

Abbreviations/Acronym: Journal citation: Bull. Pol. Ac.: Tech., ISO: Bull. Pol. Acad. Sci.-Tech. Sci., JCR Abbrev: B POL ACAD SCI-TECH Acronym in the Editorial System: BPASTS.

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Authors and Affiliations

Tadeusz Kaczorek
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Abstract

The positivity of descriptor continuous-time and discrete-time linear systems with regular pencils are addressed. Such systems can be reduced to standard linear systems and can be decomposed into dynamical and static parts. Two definitions of the positive systems are proposed. It is shown that the definitions are not equivalent. Conditions for the positivity of the systems and the relationship between two classes of positive systems are established. The considerations are illustrated by examples of electrical circuits and numerical examples.

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Authors and Affiliations

Tadeusz Kaczorek
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Abstract

A new notion of a realization of transfer matrix of (P;Q; V)-cone-system for discrete-time linear systems is proposed. Necessary and sufficient conditions for the existence of the realizations are established. A procedure is proposed for computation of a realization of a given proper transfer matrix T(z) of (P;Q; V)-cone-system. It is shown that there exists a realization of T(z) of (P;Q; V)-cone-system if and only if there exists a positive realization of T(z) = V T(z)Q!1, where V;Q and P are non-singular matrices generating the cones V;Q and P respectively.

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Authors and Affiliations

T. Kaczorek
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Abstract

New tests (criterions) for checking the reachability and the observability of positive linear-discrete-time systems are proposed. The tests do not need checking of rank conditions of the reachability and observability matrices of the systems. Simple sufficient conditions for the unreachability and unobservability of the systems are also established.

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Authors and Affiliations

T. Kaczorek
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Abstract

A new class of positive fractional 2D hybrid linear systems is introduced. The solution of the hybrid system is derived. The classical Cayley-Hamilton theorem is extended for fractional 2D hybrid systems. Necessary and sufficient conditions for the positivity are established.

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Authors and Affiliations

T. Kaczorek
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Abstract

A new concept (notion) of the practical stability of positive fractional discrete-time linear systems is introduced. Necessary and sufficient conditions for the practical stability of the positive fractional systems are established. It is shown that the positive fractional systems are practically unstable if corresponding standard positive fractional systems are asymptotically unstable.

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Authors and Affiliations

T. Kaczorek
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Abstract

In the paper there has been made an advantage of the non-classical operational calculus to determination of the response of the certain discrete time-systems. The Z-transform is often used to analysis of the stationary discrete time-systems. However, the use of the Z-transform to determination of the response especially of the non-stationary discrete time-systems is doubtful or may cause complications. This method leads to differential equations of n-th order of variable coefficients, whose solutions are very difficult or impossible. The non-classical operational calculus can be used to analysis both of the stationary and non-stationary discrete time-systems. The presented method with the use of the Heaviside operator soon leads to the target without unnecessary differential equations.

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Authors and Affiliations

E. Mieloszyk
A. Milewska
M. Magulska
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Abstract

Tuning rules for PID and PI-PI servo controllers are developed using a pole placement approach with a multiple pole, i.e. a triple one in the case of PID and a quadruple for PI-PI. The controllers involve complex roots in the numerators of the transfer functions. This is not possible in the classical P-PI structure which admits real roots only. The settling time of the servos determined by the multiple time constant is the only design parameter. Nomograms to read out discrete controller settings in terms of the time constant and control cycle are given. As compared to the classical structures, the upper limit on the control cycle is now twice longer in the case of PID, and four times in the case of PI-PI. This implies that the settling times can be shortened by the same ratios. Responses of a PLC-controlled servo confirm the validity of the design.
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Bibliography

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  7.  T. Tarczewski, M. Skiwski, L.J. Niewiara, and L.M. Grzesiak, “High-performance PMSM servo-drive with constrained state feedback position controller,” Bull. Pol. Acad. Sci. Tech. Sci., vol. 66, pp. 49–58, 2018.
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  10.  T. Żabiński and L. Trybus, “Tuning P-PI and PI-PI controllers for electrical servos,” Bull. Pol. Acad. Sci. Tech. Sci., vol. 58, pp. 51–58, 2010.
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  12.  C. Grimholt and S. Skogestad, “Optimal PI and PID control of first-order plus delay processes and evaluation of the original and improved SIMC rules,” J. Process Control, vol. 70, pp. 36–46, 2018.
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Authors and Affiliations

Andrzej Bożek
1
ORCID: ORCID
Leszek Trybus
1
ORCID: ORCID

  1. Department of Computer and Control Engineering, Rzeszów University of Technology, W. Pola 2, 35-959 Rzeszów, Poland
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Abstract

Necessary and sufficient conditions for the pointwise completeness and the pointwise degeneracy of linear discrete-time different fractional order systems are established. It is shown that if the fractional system is pointwise complete in one step (q = 1), then it is also pointwise complete for q = 2, 3…

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Authors and Affiliations

T. Kaczorek
Ł. Sajewski
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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 positivity and absolute stability of a class of nonlinear continuous-time and discretetime systems are addressed. Necessary and sufficient conditions for the positivity of this class of nonlinear systems are established. Sufficient conditions for the absolute stability of this class of nonlinear systems are also given.

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Authors and Affiliations

Tadeusz Kaczorek
ORCID: ORCID
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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 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 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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Abstract

The minimum energy control problem for the positive descriptor discrete-time linear systems with bounded inputs by the use of Weierstrass-Kronecker decomposition is formulated and solved. Necessary and sufficient conditions for the positivity and reachability of descriptor discrete-time linear systems are given. Conditions for the existence of solution and procedure for computation of optimal input and the minimal value of the performance index is proposed and illustrated by a numerical example.
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Authors and Affiliations

Tadeusz Kaczorek
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Abstract

The positive (minimal) realization problem for a class of singular discrete-time linear single-input, single-output systems with delays in state and delays in control is addressed. Solvability conditions for the positive (minimal) realization problem are established. It is shown that there exists a positive (minimal) realization of an improper transfer function T(z) = n(z) / d(z) if the coefficients of polynomial n(z) are non-negative and of the polynomial d(z) are non-positive except the leading one, which should be positive. A procedure for computation of the positive (minimal) realization of the transfer function is proposed and illustrated by an example.

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Authors and Affiliations

T. Kaczorek

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