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1
Content available Approximate controllability of second order infinite dimensional systems
Klamka Jerzy, Khurshudyan Asatur Zh.
Archives of Control Sciences
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2021
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Vol. 31, No. 1
165--184
EN
In the paper approximate controllability of second order infinite dimensional system with damping is considered. Applying linear operators in Hilbert spaces general mathematical model of second order dynamical systems with damping is presented. Next, using functional analysis methods and concepts, specially spectral methods and theory of unbounded linear operators, necessary and sufficient conditions for approximate controllability are formulated and proved. General result may be used in approximate controllability verification of second order dynamical system using known conditions for approximate controllability of first order system. As illustrative example using Green function approach approximate controllability of distributed dynamical system is also discussed.
2
Content available Non integer order, state space model of heat transfer process using Atangana-Baleanu operator
Oprzędkiewicz K.
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2020
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Vol. 68, nr 1
43--50
EN
In the paper a new, state space, non integer order model of an one-dimensional heat transfer process is proposed. The model uses a new operator with Mittag-Leffler kernel, proposed by Atangana and Beleanu. The non integer order spatial derivative is expressed by Riesz operator. Analytical formula of the step response is given, the convergence of the model is discussed too. Theoretical results are verified by experiments.
3
Content available Non integer order, state space model of heat transfer process using Caputo-Fabrizio operator
Oprzędkiewicz K.
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2018
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Vol. 66, nr 3
249--255
EN
The paper is intended to show a new state space, non integer order model of an one-dimensional heat transfer process. The proposed model derives directly from time continuous, state space semigroup model. The fractional order derivative with respect to time is by a new operator proposed by Caputo and Fabrizio, the non integer order spatial derivative is expressed by Riesz operator. The Caputo-Fabrizio operator can be directly implementated using MATLAB, because it does not require us to apply any approximation. Analytical formulae of step response are given, the system decomposition was discussed also. Main results from the paper show that the use of Caputo Fabrizio operator allows us to obtain the simple in implementation and analysis model of the considered heat transfer process. The accuracy of the proposed model in the sense of a MSE cost function is satisfying.
4
Content available On controllability of second order dynamical systems – a survey
Klamka J., Wyrwał J., Zawiski R.
Bulletin of the Polish Academy of Sciences. Technical Sciences
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2017
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Vol. 65, nr 3
279--295
EN
The paper presents a survey of recent results in the area of controllability of second order dynamical systems. Controllability problem for finite and infinite dimensional, linear, semilinear, deterministic and stochastic dynamical systems (with delays and undelayed) is taken into consideration. Different types of controllability are discussed.
5
Content available Discrete-time feedback stabilization
Mitkowski W., Bauer W., Zagórowska M.
Archives of Control Sciences
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2017
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Vol. 27, no. 2
309--322
EN
This paper presents an algorithm for designing dynamic compensator for infinitedimensional systems with bounded input and bounded output operators using finite dimensional approximation. The proposed method was then implemented in order to find the control function for thin rod heating process. The optimal sampling time was found depending on discrete output measurements.
6
Content available A non integer order, state space model for one dimensional heat transfer proces
Oprzedkiewicz K., Gawin E.
Archives of Control Sciences
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2016
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Vol. 26, no. 2
261--275
EN
In the paper a new, state space, non integer order model for one dimensional heat transfer process is presented. The model is based on known semigroup model. The derivative with respect to time is described by the non integer order Caputo operator, the spatial derivative is described by integer order operator. The elementary properties of the state operator are proven. The solution of state equation is calculated with the use of Laplace transform. Results of experiments show, that the proposed model is more accurate than analogical integer order model in the sense of square cost function.
7
Content available remote High-gain feedback and sliding modes in infinite dimensional systems
Levaggi L.
Control and Cybernetics
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2004
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Vol. 33, no 1
33-50
EN
This paper focuses on the connection between sliding motions and low frequency modes of high-gain feedback systems in an infinite dimensional framework. We study a particular class of abstract control systems in a Hilbert space setting and analyse their high-gain behaviour through singular perturbations. We show that the "slow" motion derived from the reduced model approximates the evolution of the closed loop after a fast transient. Moreover we prove a relation between this slow component of the high-gain feedback system and sliding motions, in the spirit, of the analogous result in the finite dimensional setting by Young, Kokotovic and Utkin (Young et al., 1977).
8
Content available remote Different Models of Chemotherapy Taking Into Account Drug Resistance Stemming from Gene Amplification
Śmieja J., Świerniak A.
International Journal of Applied Mathematics and Computer Science
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2003
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Vol. 13, no 3
297-305
EN
This paper presents an analysis of some class of bilinear systems that can be applied to biomedical modelling. It combines models that have been studied separately so far, taking into account both the phenomenon of gene amplification and multidrug chemotherapy in their different aspects. The mathematical description is given by an infinite dimensional state equation with a system matrix whose form allows decomposing the model into two interacting subsystems. While the first one, of a finite dimension, can have any form, the other is infinite dimensional and tridiagonal. A methodology of the analysis of such models, based on system decomposition, is presented. An optimal control problem is defined in the l1 space. In order to derive necessary conditions for optimal control, the model description is transformed into an integro-differential form. Finally, biomedical implications of the obtained results are discussed.
9
Content available remote Adaptive compensators for pertrubed positive real infinite-dimensional systems
Curtain R. F., Demetriou M. A., Ito K.
International Journal of Applied Mathematics and Computer Science
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2003
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Vol. 13, no 4
441-452
EN
The aim of this investigation is to construct an adaptive observer and an adaptive compensator for a class of infinite-dimensional plants having a known exogenous input and a structured perturbation with an unknown constant parameter, such as the case of static output feedback with an unknown gain. The adaptive observer uses the nominal dynamics of the unperturbed plant and an adaptation law based on the Lyapunov redesign method. We obtain conditions on the system to ensure uniform boundedness of the estimator dynamics and the parameter estimates, and the convergence of the estimator error. For the case of a known periodic exogenous input we design an adaptive compensator which forces the system to converge to a unique periodic solution. We illustrate our approach with a delay example and a diffusion example for which we obtain convincing numerical results.
10
Content available Asymptotic analysis of three random branching walk models arising in molecular biology
Świerniak A., Polański A., Śmieja J., Kimmel M., Rzeszowska-Wolny J.
Control and Cybernetics
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2003
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Vol. 32, no 1
147-162
EN
Using asymptotic techniques based on Laplace transforms, spectral analysis and theory of feedback systems, we characterise the asymptotic behaviour of the repeat loci in microsatellite DNA and cancer cells with increasing number of copies of genes responsible for coding proteins causing drug removal or metabolisation as well as telomeres shortening, which is supposed to be the mechanism of ageing and death. These three problems are described by models in the form of infinitely many differential linear or bilinear first order equations, resulting from branching random walk processes used to represent the evolution of particles in these problems.
PL
Wykorzystując techniki asymptotyczne oparte na transformatach Laplace'a, analizę spektralną oraz teorię układów ze sprzężeniem zwrotnym w artykule scharakteryzowano zachowanie asymptotyczne powtórek w DNA mikrosatelitarnym oraz w komórkach rakowych z rosnącą liczbą kopii genów odpowiedzialnych za kodowanie białek powodujących usuwanie lub przemianę metaboliczną leków, a także skracanie telomerów, o którym się sądzi, że jest mechanizmem starzenia się i śmierci. Te trzy zagadnienia są opisywane przy pomocy modeli w postaci nieskończonej liczby liniowych lub biliniowych równań pierwszego rzędu, wynikających z procesów błądzenia, stosowanych do opisu ewolucji cząstek w tych zagadnieniach.
11
Content available Optimal multidrug treatment in the presence of drug resistance stemming from gene amplification
Śmieja J., Świerniak A.
Journal of Medical Informatics & Technologies
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2002
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Vol. 3
MI13--19
EN
The paper is concerned with development of optimal treatment protocols that take into account both action of several drugs and the evolution of drug resistance. It is a result of analysis of evolution of drug resistance in cancer population but presented methodology can be applied in any case involving drug resistance stemming from gene amplification. First, a biological background is given. In subsequent sections of the paper, the developed technique is presented and some early analytical results, which form a basis for more precise modeling, are shown. Afterwards, the model description is transformed into a vector integro-differential equation, which makes it possible to define necessary conditions of optimal solution to the minimization problem arising from the search for the optimal treatment. Finally, some remarks on the model applicability are presented.
12
Content available remote Metody optymalizacji parametrycznej nieskończenie wymiarowych systemów dynamicznych
Górecki H.
Elektrotechnika i Elektronika
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2001
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T. 20, z. 1
31--38
PL
W artykule przedstawiono metodę obliczenia całki z kwadratu błędu dla układów dynamicznych nieskończenie wymiarowych, to jest układów opisywanych równaniami różniczkowymi z opóźnionym argumentem oraz układów opisywanych równaniami różniczkowymi cząstkowymi. Podano również przykład ilustrujący zastosowanie przedstawionej metody oraz przeprowadzono optymalizację parametryczną.
EN
The theory of calculation integral square error for linear systems with transcendental transfer functions is presented in the paper. This enables to obtain analytical results in the parametric optimization of linear infinite dimensional systems such as described by difference - differential equations or partial differential equations. The concrete example is also given.
13
Content available Qualitative analysis of controlled drug resistance model - inverse Laplace and and semigroup approach
Świerniak A., Polański A., Kimmel M., Bobrowski A., Śmieja J.
Control and Cybernetics
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1999
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Vol. 28, no 1
61-73
EN
In this paper we study some properties of infinite models of the controlled evolution of drug resistance. We combine asymptotic techniques used in previous studies of similar models with methods of control theory and of semigroup theory. It enables us to find conditions for stability of the model both when the sensitive population is annihilated and when there exists a permanent influx from the sensivite compartment into drug resistant one. The conditions are expressed in terms of relationships between amplification and deamplification ratios as well as average life times of cells and intensity of anticancer drug action.
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