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1
Content available Application of the Drazin inverse to the analysis of descriptor fractional discrete-time linear systems with regular pencils
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
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2013
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Vol. 23, no. 1
29--33
EN
The Drazin inverse of matrices is applied to find the solutions of the state equations of descriptor fractional discrete-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.
2
Content available Equivalent descriptions of a discrete-time fractional-order linear system and its stability domains
Ostalczyk P.
International Journal of Applied Mathematics and Computer Science
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2012
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Vol. 22, no. 3
533-538
EN
Two description forms of a linear fractional-order discrete system are considered. The first one is by a fractional-order difference equation, whereas the second by a fractional-order state-space equation. In relation to the two above-mentioned description forms, stability domains are evaluated. Several simulations of stable, marginally stable and unstable unit step responses of fractional-order systems due to different values of system parameters are presented.
3
Content available An SQP trust region method for solving the discrete-time linear quadratic control problem
Mostafa E. M. E.
International Journal of Applied Mathematics and Computer Science
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2012
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Vol. 22, no. 2
353-363
EN
In this paper, a sequential quadratic programming method combined with a trust region globalization strategy is analyzed and studied for solving a certain nonlinear constrained optimization problem with matrix variables. The optimization problem is derived from the infinite-horizon linear quadratic control problem for discrete-time systems when a complete set of state variables is not available. Moreover, a parametrization approach is introduced that does not require starting a feasible solution to initiate the proposed SQP trust region method. To demonstrate the effectiveness of the method, some numerical results are presented in detail.
4
Content available remote Novel optimal recursive filter for state and fault estimation of linear stochastic systems with unknown disturbances
Khémiri K., Ben Hmida F., Ragot J., Gossa M.
International Journal of Applied Mathematics and Computer Science
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2011
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Vol. 21, no 4
629-637
EN
This paper studies recursive optimal filtering as well as robust fault and state estimation for linear stochastic systems with unknown disturbances. It proposes a new recursive optimal filter structure with transformation of the original system. This transformation is based on the singular value decomposition of the direct feedthrough matrix distribution of the fault which is assumed to be of arbitrary rank. The resulting filter is optimal in the sense of the unbiased minimum-variance criteria. Two numerical examples are given in order to illustrate the proposed method, in particular to solve the estimation of the simultaneous actuator and sensor fault problem and to make a comparison with the existing literature results.
5
Content available Reduction and Decomposition of Singular Fractional Discrete-Time Linear Systems
Kaczorek T.
Acta Mechanica et Automatica
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2011
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Vol. 5, no. 4
62-66
EN
Reduction of singular fractional systems to standard fractional systems and decomposition of singular fractional discrete-time linear systems into dynamic and static parts are addressed. It is shown that if the pencil of singular fractional linear discrete-time system is regular then the singular system can be reduced to standard one and it can be decomposed into dynamic and static parts The proposed procedures are based on modified version of the shuffle algorithm and illustrated by numerical examples.
6
Content available Positive Realization of SISO 2D Different Orders Fractional Discrete-Time Linear Systems
Sajewski Ł.
Acta Mechanica et Automatica
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2011
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Vol. 5, no. 2
122-127
EN
The realization problem for single-input single-output 2D positive fractional systems with different orders is formulated and a method based on the state variable diagram for finding a positive realization of a given proper transfer function is proposed. Sufficient conditions for the existence of a positive realization of this class of 2D linear systems are established. A procedure for computation of a positive realization is proposed and illustrated by a numerical example.
7
Content available Fuzzy Predictive Control of Fractional-Order Nonlinear Discrete-Time Systems
Domek S.
Acta Mechanica et Automatica
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2011
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Vol. 5, no. 2
23-26
EN
At the end of the 19th century Liouville and Riemann introduced the notion of a fractional-order derivative, and in the latter half of the 20th century the concept of the so-called Grünewald-Letnikov fractional-order discrete difference has been put forward. In the paper a predictive controller for MIMO fractional-order discrete-time systems is proposed, and then the concept is extended to nonlinear processes that can be modelled by Takagi-Sugeno fuzzy models. At first nonlinear and linear fractional-order discrete-time dynamical models are described. Then a generalized nonlinear fractional-order TS fuzzy model is defined, for which equations of a predictive controller are derived.
8
A discrete-time sine wave delay estimator
Hermanowicz E., Rojewski A.
Elektronika : konstrukcje, technologie, zastosowania
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2010
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Vol. 51, nr 3
38-41
EN
In the paper a novel open-loop solution of a sine wave sub sample time delay estimator operating in the discrete-time domain is presented. The proposed solution is based on a concept of Hilbert transforming the reference signal in order to estimate the phase difference relative to the received noised signal whose initial phase and delay are unknown. The engineered estimator works well around the normalized centre frequency equal to 0.25 of sampling rate typical of some communication applications, with SNR (signal to noise ratio) higher than 15 dB. It allows for some slight detuning of the received signal from the above mentioned centre frequency. The performance of the time delay estimator is analysed and illustrated by the results of its operation in presence of noise.
PL
W artykule przedstawiono nowe rozwiązanie dyskretno-czasowego estymatora opóźnienia międzypróbkowego sinusoidy, działającego z otwartą pętlą sprzężenia zwrotnego. Proponowane rozwiązanie wykorzystuje koncepcję przekształcenia Hilberta sygnału odniesienia w celu estymacji różnicy faz względem zaszumionego sygnału odebranego o nieznanej fazie początkowej i nieznanym opóźnieniu. Estymator ten jest przeznaczony do pracy z sygnałem o znormalizowanej częstotliwości środkowej równej jednej czwartej częstotliwości próbkowania - typowej dla zastosowań telekomunikacyjnych, ze stosunkiem mocy sygnału do szumu (SNR) większym od 15dB. Pozwala on na niewielkie odstrojenie sygnału odebranego od tej częstotliwości. Zamieszczono analizę charakterystyk i właściwości proponowanego estymatora opóźnienia i zilustrowano wynikami eksperymentów w obecności szumu.
9
Content available remote Simple conditions for practical stability of positive fractional discrete-time linear systems
Busłowicz M., Kaczorek T.
International Journal of Applied Mathematics and Computer Science
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2009
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Vol. 19, no 2
263-269
EN
In the paper the problem of practical stability of linear positive discrete-time systems of fractional order is addressed. New simple necessary and sufficient conditions for practical stability and for practical stability independent of the length of practical implementation are established. It is shown that practical stability of the system is equivalent to asymptotic stability of the corresponding standard positive discrete-time systems of the same order. The discussion is illustrated with numerical examples.
10
Content available remote Positive partial realization problem for linear discrete-time systems
Kaczorek T.
International Journal of Applied Mathematics and Computer Science
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2007
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Vol. 17, no 2
165-171
EN
A partial realization problem for positive linear discrete-time systems is addressed. Sufficient conditions for the existence of its solution are established. A procedure for the computation of a positive partial realization for a given finite sequence of the values of the impulse response is proposed. The procedure is illustrated by four numerical examples.
11
Content available remote Estimation of the output deviation norm for uncertain, discrete-time nonlinear systems in a state dependent form
Orłowski P.
International Journal of Applied Mathematics and Computer Science
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2007
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Vol. 17, no 4
505-513
EN
Numerical evaluation of the optimal nonlinear robust control requires estimating the impact of parameter uncertainties on the system output. The main goal of the paper is to propose a method for estimating the norm of an output trajectory deviation from the nominal trajectory for nonlinear uncertain, discrete-time systems. The measure of the deviation allows us to evaluate the robustness of any designed controller. The first part of the paper concerns uncertainty modelling for nonlinear systems given in the state space dependent form. The method for numerical estimation of the maximal norm of the output trajectory deviation with applications to robust control synthesis is proposed based on the introduced three-term additive uncertainty model. Theoretical deliberations are complemented with a numerical, water-tank system example.
12
Content available remote Numerically robust synthesis of discrete-time H[infinity] estimators based on dual J-lossless factorisations
Suchomski P.
Control and Cybernetics
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2003
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Vol. 32, no 4
761-802
EN
An approach to the numerically reliable synthesis of the H[infinity] suboptimal state estimators for discretised continuous-time processes is presented. The approach is based on suitable dual J-lossless factorisations of chain-scattering representations of estimated processes. It is demonstrated that for a sufficiently small sampling period the standard forward shift operator techniques may become ill-conditioned and numerical robustness of the design procedures can be significantly improved by employing the so-called delta operator models of the process. State-space models of all H[infinity] sub-optimal estimators are obtained by considering the suitable delta-domain algebraic Riccati equation and the corresponding generalised eigenproblem formulation. A relative condition number of this equation is used as a measure of its numerical conditioning. Both regular problems concerning models having no zeros on the boundary of the delta-domain stability region and irregular (non-standard) problems of models with such zeros are examined. For the first case, an approach based on a dual J-lossless factorisation is proposed while in the second case an extended dual J-lossless factorisation based on a zero compensator technique s required. Two numerical examples are given to illustrate some properties of the considered delta-domain approach.
13
Content available Simple stable discrete-time generalised predictive control with anticipated filtration of control error
Kowalczuk Z., Suchomski P.
Control and Cybernetics
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2002
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Vol. 31, no 1
17-41
EN
It is shown that under some specific conditions, the solution of the generalised predictive control (GPC) design using the concept of anticipated filtering (AF) of the control error always exists, and that such a design leads to stable control systems with definite closed-loop characteristics. The plant cancellation issue is taken into account, and it is demonstrated that certain bounds on GPC design parameters have to be considered. An iterative procedure for simultaneous determination of the three basic design-tuning parameters: the control horizon, the controller gain, and the order of plant cancellation, is also supplied. An important feature of this approach is that the anticipated filtering makes it possible to reduce a disagreeable control effort associated with GPC and to make the [lambda]-tuning mechanism practicable. The bounds on the GPC design parameters are discussed, and certain optimal tuning rules are proposed and validated via simulated experiments.
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