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1
Content available Fractional descriptor continuous-time linear systems described by the Caputo-Fabrizio derivative
100%
Kaczorek T. , Borawski K.
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tom 26
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nr 3
533-541
EN
The Weierstrass-Kronecker theorem on the decomposition of the regular pencil is extended to fractional descriptor continuous-time linear systems described by the Caputo-Fabrizio derivative. A method for computing solutions of continuous-time systems is presented. Necessary and sufficient conditions for the positivity and stability of these systems are established. The discussion is illustrated with a numerical example.
2
Content available Stability analysis of continuous-time linear systems consisting of n subsystems with different fractional orders
100%
Busłowicz M.
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tom Vol. 60, nr 2
279-284
EN
The stability problem of continuous-time linear systems described by the state equation consisting of n subsystems with different fractional orders of derivatives of the state variables has been considered. The methods for asymptotic stability checking have been given. The method proposed in the general case is based on the Argument Principle and it is similar to the modified Mikhailov stability criterion known from the stability theory of natural order systems. The considerations are illustrated by numerical examples.
3
Content available Comparison of approximation methods of positive stable continuous-time linear systems by positive stable discrete-time systems
86%
Kaczorek T.
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tom Vol. 11
1--10
EN
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.
4
Content available Responses of positive standard and fractional linear systems and electrical circuits with derivatives of their inputs
86%
Kaczorek T.
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tom Vol. 66, nr 4
419--425
EN
Responses of positive standard and fractional continuous-time and discrete-time linear systems with derivatives of their inputs are presented herein. It is shown that the formulae for state vectors and outputs are also valid for their derivatives if the inputs and outputs and their derivatives of suitable order are zero for t = 0. Similar results are also shown for positive standard and fractional discrete-time linear systems.
5
Content available remote A modified state variable diagram method for determination of positive realizations of linear continuous-time systems with delays
86%
Kaczorek T.
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nr 4
897-905
EN
A new modified state variable diagram method is proposed for determination of positive realizations of linear continuoustime systems with delays in state and input vectors. Using the method, it is possible to find a positive realization with reduced numbers of delays for a given transfer matrix. Sufficient conditions for the existence of positive realizations of given proper transfer matrices are established. The proposed method is demonstrated on numerical examples.
6
Content available Design of an Ultra-Low Power CT Σ∆ A/D Modulator in 65nm CMOS for Cardiac Pacemakers: From System Synthesis to Circuit Implementation
86%
Wang Y. , Cai H.
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tom Vol. 60, No. 1
109-115
EN
A high performance, ultra-low power, fully differentia 2nd-order continuous-time Σ∆ analogue-to-digital modulator for cardiac pacemakers is presented in this paper. The entire design procedure is described in detail from the high-level system synthesis in both discrete and continuous-time domain, to the low-level circuit implementation of key functional blocks of the modulator. The power consumption of the designed modulator is rated at 182nA from a 1.2V power supply, meeting the ultra-low power requirement of the cardiac pacemaker applications. A 65nm CMOS technology is employed to implement the Σ∆ modulator. The modulator achieves a simulated SNR of 53.8dB over a 400 Hz signal bandwidth, with 32KHz sampling frequency and an oversampling ratio of 40. The active area of the modulator is 0.45×0.50mm².
7
Content available Synteza obserwatorów funkcyjnych liniowych ciągłych układów niecałkowitego rzędu
86%
Kociszewski R.
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tom R. 18, nr 4
94--99
PL
W pracy rozpatrzono problematykę syntezy obserwatorów funkcyjnych ciągłych układów niecałkowitego rzędu. Pokazano, że synteza obserwatora niecałkowitego rzędu polegająca na wyznaczaniu poszczególnych macierzy występujących w jego równaniu może być najpierw zapisana w postaci warunku LMI, a następnie efektywnie rozwiązana za pomocą dedykowanego oprogramowania do optymalizacji wypukłej. Rozważania zilustrowano przykładem symulacyjnym.
EN
The paper presents a problem of synthesis of functional observers for fractional order continuous-time linear systems. It has been shown that this problem can be formulated and solved by the use of linear matrix inequalities (LMI) method. Necessary and sufficient conditions for solvability of the problem for computation of an unknown matrices of asymptotic stable observers have been given. The proposed approach is illustrated by a numerical example.
8
Content available New stability tests for fractional positive descriptor linear systems
86%
Kaczorek T. , Ruszewski A.
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tom Vol. 33, No. 1
71--81
EN
The asymptotic stability of fractional positive descriptor continuous-time and discrete-time 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.
9
Content available Drazin inverse matrix method for fractional descriptor continuous-time linear systems
86%
Kaczorek T.
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tom Vol. 62, nr 3
409--412
EN
The Drazin inverse of matrices is applied to find the solutions of the state equations of the fractional descriptor continuous-time systems with regular pencils. An equality defining the set of admissible initial conditions for given inputs is derived. The proposed method is illustrated by a numerical example.
10
Content available Stability of positive fractional switched continuous-time linear systems
86%
Kaczorek T.
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tom Vol. 61, nr 2
349--352
EN
The asymptotic stability of positive fractional switched continuous-time linear systems for any switching is addressed. Simple sufficient conditions for the asymptotic stability of the positive fractional systems are established. It is shown that the positive fractional switched systems are asymptotically stable for any switchings if the sum of entries of every column of the matrices of all subsystems is negative.
11
Content available Approximation of positive stable continuous-time linear systems by positive stable discrete-time systems
72%
Kaczorek T.
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2013
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tom R. 17, nr 2
359-364
EN
The positive asymptotically stable continuous-time linear systems are approximated by positive asymptotically stable discretetime linear systems by the use of Pade type approximation. It is shown that the approximation preserves the positivity and asymptotic stability of the systems. The stabilization problem of positive unstable continuous-time and corresponding discrete-time linear systems by state-feedbacks is also addressed.
PL
Dodatnie układy stabilne ciągłe są aproksymowane za pomocą liniowej aproksymacji Pade dodatnimi, stabilnymi układami dyskretnymi. Wykazano, że aproksymacja ta zachowuje dodatniość i stabilność asymptotyczną. Rozważania ogólne zostały zilustrowane przykładem numerycznym.
12
Content available A new method for determination of positive realizations of linear continuous-time systems
72%
Kaczorek T.
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nr 5
13
Content available remote Fractional positive linear systems and electrical circuits
72%
Kaczorek T.
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tom R. 84, nr 9
135-141
EN
Fractional positive linear continuous-time and discrete-time systems are addressed. The solutions to the state equations of the fractional systems are derived using the Laplace and the Z transforms, respectively. Necessary and sufficient conditions are established for the internal and external positivity of the fractional systems. Some applications of the of the fractional systems theory to electrical circuits are given.
PL
W artykule są rozpatrywane dodatnie układy ciągle i dyskretne ułamkowego rzędu. Korzystając z przekształcenia Laplace’a i odpowiednio z przekształcenia Z wyprowadzono rozwiązania równań stanu tych ciągłych i dyskretnych układów ułamkowego rzędu. Sformułowano i udowodniono warunki konieczne i wystarczające dodatniości tych układów. Na przykładzie dwójnika R,C pokazano możliwość zastosowania przedstawionych metod do analizy obwodów elektrycznych.
14
Content available Stability conditions for linear continuous-time fractional-order state-delayed systems
72%
Busłowicz M.
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tom Vol. 64, nr 1
3--7
EN
The stability problem of continuous-time linear fractional order systems with state delay is considered. New simple necessary and sufficient conditions for the asymptotic stability are established. The conditions are given in terms of eigenvalues of the state matrix and time delay. It is shown that in the complex plane there exists such a region that location in this region of all eigenvalues of the state matrix multiplied by delay in power equal to the fractional order is necessary and sufficient for the asymptotic stability. Parametric description of boundary of this region is derived and simple new analytic necessary and sufficient conditions for the stability are given. Moreover, it is shown that the stability of the fractional order system without delay is necessary for the stability of this system with delay. The considerations are illustrated by a numerical example.
15
Content available Absolute stability of a class of nonlinear systems with nonpositive linear parts
72%
Kaczorek T.
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nr 2
16
Content available Zastosowanie LMI do syntezy obserwatorów liniowych ciągłych układów niecałkowitego rzędu
58%
Kociszewski R.
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tom R. 56, nr 5
392-395
PL
W pracy rozpatrzono problem syntezy obserwatorów liniowych układów ciągłych niecałkowitego rzędu. Wykorzystując aparat liniowych nierówności macierzowych (LMI) sformułowano warunki oraz podano procedury do wyznaczania macierzy wzmocnień obserwatorów, dla rzędu ? układu: 0 < α < 1 i 1 < α < 2. Rozważania zilustrowano przykładem liczbowym. Obliczenia i symulacje wykonano w środowisku Matlab/Simulink.
EN
Many sophisticated analytical procedures to control system design are based on the assumption that the full state vector is available for measurement. The example of such control procedure is placement of the unstable system eigenvalues. In many systems of practical importance, however, the entire state vector is not available for measurement. In some systems, for example, measurements may require the use of costly measurement devices and it may be unreasonable to measure all state variables. An auxiliary dynamical system, which reconstructs the state vector, is known as a full-order or an identity observer, and is coupled to the original system through the available system inputs and outputs [14]. The paper presents a problem of synthesis of full-order observers for fractional continuous-time linear systems. It has been shown that this problem can be formulated and solved by the use of linear matrix inequalities (LMI) methods [15]. Two cases of fractional order i.e. 0 < α < 1 and 1 < α < 2 of the system (1) have been considered. Necessary and sufficient conditions (Theorem 1 and 2) for solvability of the problem as well as two procedures (Procedure 1 and 2) for computation of a gain matrix L of asymptotic stable observer (2) have been given. The proposed approach is illustrated with a practical example. Numerical calculations have been performed in the Matlab package and accompanied by public domain software: SeDuMi solver and YALMIP parser. The LMI approach to observer synthesis for fractional continuous-time linear systems has not been considered as yet.
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