Valeriu Prepelita - Academia.edu (original) (raw)

Papers by Valeriu Prepelita

WSEAS TRANSACTIONS on SYSTEMS archive, 2010

A class of 2D hybrid boundary-value time-variable systems is studied, in the general approach of ... more A class of 2D hybrid boundary-value time-variable systems is studied, in the general approach of the coefficient matrices, states, inputs and controls over spaces of functions of bounded variation or of regulated functions. A generalized variation-of-parameters formula is obtained for differential-difference equations of the considered type and it is used to derive the formulae of the state and of the output of these systems. The state space representation of the adjoints of these 2D hybrid systems is introduced and their input-output map is obtained. The duality between the 2D hybrid boundary-value systems and their adjoints is expressed by the means of two bilinear forms.

International Conference on System Science and Simulation in Engineering, Oct 17, 2009

2D generalized hybrid systems are considered, having the coefficient matrices of bounded variatio... more 2D generalized hybrid systems are considered, having the coefficient matrices of bounded variation or regulated matrix functions, the controls being regulated vector functions. The formulae of the state and of the input-output map of these systems are obtained. Some 2D separable kernels are associated to these systems and a 2D continuous-discrete Wiener-Hopf equation is studied in this framework. The realization problem is discussed for 2D separable kernels and necessary and sufficient conditions for the minimality of a realization are provided.

International Conference on Mathematical methods, Computational techniques and Intelligent systems, Jul 1, 2011

Linear one-dimensional (1D) acausal systems, i.e. systems with boundary conditions have been intr... more Linear one-dimensional (1D) acausal systems, i.e. systems with boundary conditions have been introduced in state space representation by A.J. Krener [12], [13], in connection with the modeling of boundary value regulation. M.B. Adams, A.S. Willsky and B.C. Levy [1], [2] have obtained important results in the linear estimation of stochastic processes governed by time varying systems with boundary conditions. T. Kailath has tackled this topic in an input-output approach in a series of papers on linear estimation theory [11]. H. Bart, I. Gohberg and M.A. Kaashoek have come to linear systems with boundary conditions motivated by the analysis of Wiener-Hopf integral equation and related convolution equations [4], [5]. The theory of systems with boundary conditions has been developed by I. Gohberg and M.A. Kaashoek in a series of papers [7], [8], [9], [10]. They have brought this theory to the level of the classical theory of the causal linear systems. For instance, the characterization of the classes of irreducible and minimal systems has been obtained and it has been emphasized that these classes and the class of controllable and observable systems are different (whereas in the case of causal systems they coincide). In the same time, in the framework of Systems Theory, different state space models of two-dimensional 2D systems has been proposed by Roesser [19], Fornasini and Marchesini [6], Attasi [3] and others. The study of 2D (and nD) systems has known an important development in the last three decades due to their significant applications in various areas as image processing, seismology, geophysics or computer tomography. The above mentioned papers and the subsequent ones have studied the causal systems, i.e. systems whose states and outputs are determined by the inputs and the initial states. In this paper we present the extension of these results (see [14]-[18]) to multidimensional (nD, n?2) boundary-value systems, by introducing a class of systems which represents the continuous-time time-varying counterpart of Attasi's discrete-time 2D model [3]. The state-space representation of the considered nD boundary-value systems is given, including well-posed boundary conditions. The formulas of the state and of the input-output map of the nD boundaryvalue systems are obtained, by means of a suitable variation-of-parameters formula. Generalized nD separable kernels are associated to these systems. The realization problem is discussed for nD separable kernels and necessary and sufficient conditions for minimality are presented. The adjoints of nD boundary-value systems are introduced and the input-output maps of the adjoint systems are derived. Two inner products are defined and they are used to obtain the relationship between the input-output operators of the boundary-value systems and their adjoints.

WSEAS Transactions on Systems and Control archive, Feb 1, 2010

2D hybrid continuous-discrete systems with boundary conditions are studied, in the general approa... more 2D hybrid continuous-discrete systems with boundary conditions are studied, in the general approach of the coefficient matrices and controls over spaces of functions of bounded variation or of regulated functions. The formulae of the state and of the general response of these systems are provided, both in the case of causal and acausal cases. It is shown that the behaviour of the systems with boundary conditions is characterized by some generalized 2D semiseparable kernels. The existence of realizations of generalized 2D semiseparable kernels is proved and necessary and sufficient conditions for the minimality of the realizations are obtained.

WSEAS Transactions on Systems and Control archive, Sep 1, 2008

Generalized linear systems are considered, which contain in their state-space representation matr... more Generalized linear systems are considered, which contain in their state-space representation matrices with elements functions of bounded variation and controls in the space of regulated functions. The Perron-Stieltjes integral is used to obtain a variation-of-parameters formula. On this basis the formula of the state of the system as well as the input-output map are derived. The fundamental concepts of controllability and reachability are analysed in this approach by means of two controllability and reachability Gramians. An optimal control is provided which solves the problem of the minimum energy transfer. The observability of these generalized systems is studied. In the case of completely observable systems a formula is obtained which recovers the initial state from the exterior data. The duality between the concepts of controllability and observability is emphasized as well as Kalman's canonical form.

International Conference on System Science and Simulation in Engineering, Oct 17, 2009

2D generalized continuous-discrete systems are studied, in the general case of the coefficient ma... more 2D generalized continuous-discrete systems are studied, in the general case of the coefficient matrices and controls over spaces of functions of bounded variation or of regulated functions. Systems with boundary conditions are considered and their behavior is described. The adjoints of these systems are defined and their input-output map is obtained. These results are employed to emphasize the relationship between the inputs and the outputs of adjoint generalized systems.

A model of multidimensional time-invariant hybrid systems is studied, which depends on q continuo... more A model of multidimensional time-invariant hybrid systems is studied, which depends on q continuous-time variables and on r discrete-time ones. The general response of the model is provided and some necessary and sufficient conditions of controllability and reachability are obtained by means of a suitable controllability matrix and a reachability Gramian. The relationship between the concepts of controllability and reachability is emphasized. The geometric characterization of the subspace of reachable states is given and a list of controllability and reachability criteria is derived.

WSEAS TRANSACTIONS on SYSTEMS archive, 2009

A class of multidimensional time-invariant hybrid systems is studied, which depends on q continuo... more A class of multidimensional time-invariant hybrid systems is studied, which depends on q continuoustime variables and on r discrete-time ones. The formula of the input-output map of these systems is given. A multiple (q, r)-hybrid Laplace transformation is defined and it is used to determine the transfer matrices of the considered systems. The minimal realization problem is analysed and an algorithm is given which provides minimal realizations.

International Conference on System Science and Simulation in Engineering, Oct 17, 2009

A model of generalized linear control systems is considered, which is represented by matrices wit... more A model of generalized linear control systems is considered, which is represented by matrices with elements functions of bounded variation and controls over the space of regulated functions (i.e. functions which possesses finite one-sided limits on a given interval). The Perron-Stieltjes integral with respect to the set of regulated functions (which include the set of functions of bounded variation) was defined in [12]. This integral is equivalent to the Kurzweil integral (see [2], [8] and [9]). In this paper, using the results of M. Tvrdy ([10], [11]) concerning the properties of the Perron-Stieltjes integral with respect to regulated functions and the differential equation in this space, the formulas of the states and of the general response of the control systems are obtained. This allows us to extend in this framework the concepts of reachability and observability (see for instance [1] and [6]). These fundamental concepts are analysed by means of two suitable controllability and observability Gramians. The duality between the concepts of controllability and observability is emphasized as well as Kalman's canonical form. The spaces of reachable and observable states are described. The minimal energy transfer is studied and the optimal control is provided. In the case of completely observable systems a formula is obtained which recovers the initial state from the exterior data. It is emphasized that these systems are generalizations of the classical linear systems described by differential equations with controls. The considered approach seems to be the most general framework in which the linear control systems can be studied. Linear boundary value (acausal) systems are studied in the same framework [3]. Semiseparable kernels are associated to acausal systems with well-posed boundary conditions. Minimal realizations of semiseparable kernels are characterized as well as the irreducibility of the acausal systems. Adjoint systems are defined and an input-output operator is provided. A Peano-Baker type formula is obtained for the calculus of the fundamental matrix of the generalized linear differential systems [4]. This study can be continued in many directions such as stability, positivity, multidimensional generalized systems [5], 2D generalized differential-difference systems [7], linear quadratic optimal control etc.

International Conference on Mathematical and Computational Methods in Science and Engineering, Nov 7, 2008

In the last two decades the study of two-dimensional (2D) systems (and more generally, of n-dimen... more In the last two decades the study of two-dimensional (2D) systems (and more generally, of n-dimensional systems) developed as a distinct branch of system theory, due to its applications in various domains as image processing, seismology and geophysics, control of multipass processes etc. The two-dimensional (2D) systems were obtained from classical linear dynamical systems by generalizing from a single time variable to two (space) variables. Different state space models for 2D systems have been proposed by Roesser, Fornasini and Marchesini, Attasi, Eising and others. A subclass of 2D systems is represented by systems which are continuous with respect to one variable and discrete with respect to another one. The continuous-discrete models have applications in many problems like the iterative learning control synthesis or repetitive processes. The aim of this paper is to develop a complete theory for a class of time-variable 2D systems, which are the continuous-discrete counterpart of Attasi's 2D discrete time-invariant models. In Section 2 variation of parameters formula is established for 2D continuous-discrete (2Dcd) systems and the formulae of the state and of the output of the systems are derived. The concept of controllability which is fundamental in control theory was introduced by Kalman under the stimulation of the engineering problems of time optimal control. The notion of reachability was derived from controllability by reversing the time. Reachability of time-variable 2Dcd systems is analyzed in Section 3 by introducing a 2D reachability Gramian. Time-invariant 2Dcd systems are studied and several necessary and sufficient conditions of complete reachability and complete controllability are derived. It results that the considered class is the closest one to that of classical 1- dimensional systems, since all the known criteria of reachability for 1D systems can be extended to 2Dcd systems. Other advantages of this framework are that the analysed reachability is global and that time-variable systems can be successfully studied. The notion of observability is defined and analysed in Section 4 for 2D time-varying continuous-discrete separable systems. An observability Gramian is introduced and completely observable systems are characterized by means of the rank of this Gramian. For completely observable systems a formula is derived which provides the initial state by knowing the control and the corresponding output. For 2D time-invariant continuous-discrete systems a list of necessary and sufficient conditions of observability is established. A geometric characterization of the subspace of unobservable states is given in terms of invariant subspaces included in the kernel of the output matrix. The duality between the concepts of reachability and observability is emphasized as well as their connection with the minimality of these systems. Section 5 is devoted to the study of stability of the time-invariant 2D continuous-discrete systems. Necessary and sufficient conditions of asymptotic stability are obtained, which extend the conditions for 1D continuous-time and 1D discrete-time systems, including a suitable Liapunov function. A necessary condition is expressed by using a generalized Liapunov equation. In section 6 a multiple hybrid Laplace transformation is defined and the main properties of this transformation are stated and proved, including linearity, homothety, two time-delay theorems, translation, differentiation and difference of the original, differentiation of the image, integration and sum of the original, integration of the image, convolution, product of originals, initial and final values. Some formulas for determining the original are given. This hybrid transformation is employed to obtain transfer matrices for different classes of 2D continuous-discrete linear control systems of Roesser-type, Fornasini-Marchesini-type and Attasi type models, including descriptor and delayed systems. The realization problem is studied in Section 7. An algorithm is proposed which determines a minimal realization for separable 2D multi-input-multi-output (MIMO) systems. This method generalizes to 2D systems the celebrated Ho-Kalman algorithm. The proposed algorithm can also be used for MIMO separable 2D discrete-time linear systems or for MIMO 2D systems described by a class of hyperbolic partial differential equations.

International Conference on Mathematical and Computational Methods in Science and Engineering, Nov 7, 2008

A class of multidimensional time-invariant MIMO hybrid systems is studied, which depends on q con... more A class of multidimensional time-invariant MIMO hybrid systems is studied, which depends on q continuous-time variables and on r discrete-time ones. The formula of the input-output map of these systems is obtained. A multiple (q, r)-hybrid Laplace transformation is defined and it is used to determine the transfer matrices of the considered systems. The minimal realization problem is analysed and an algorithm is given which provides minimal realizations.

Annals of the Academy of Romanian Scientists Series on Mathematics and Its Application

The controllability of a class of 2D linear time varying continuous time control systems is studi... more The controllability of a class of 2D linear time varying continuous time control systems is studied. The state space representation is provided and the formulas of the states and the input-output map of these systems are derived. The fundamental concepts of controllability and reachability are analysed and suitable controllability and reachability Gramians are constructed to characterize the controllable and the reachable time varying systems. In the case of time invariant 2D systems, some algorithms are developed to calculate different controllability Gramians as solutions of adequate Lyapunov type equations. Corresponding Matlab programs are implemented to solve these Lyapunov equations.

An algorithm is provided to determine the minimal subspace which is invariant with respect to som... more An algorithm is provided to determine the minimal subspace which is invariant with respect to some commutative matrices and which includes a given subspace. Reachability criteria are obtained for 2D continuous-discrete time-variable Attasi type systems by using a suitable 2D reachability Gramian. Necessary and sufficient conditions of reachability are derived for LTI 2D systems. The presented algorithm is used to determine the subspace of the reachable states of a 2D system.

International Conference on Systems, Jul 22, 2010

The Operational Calculus as a distinct discipline has a history which has exceeded a century. But... more The Operational Calculus as a distinct discipline has a history which has exceeded a century. But its roots can be found in the works of Leibniz, Bernoulli, Lagrange, Laplace, Euler, Fourier, Cauchy and others. Its importance is determined by its utility in solving complex problems in many domains such as Calculus, Number Theory, Special Functions, Ordinary Differential Equations, Mathematical Physics, Heat Transfer, Electronics, Automatics, etc. In Systems and Control Theory the frequency domain methods, based on Laplace transformation in the continuous-time case or on Z transformation in the discrete-time case, play a very important role in the study of the "classical" 1D systems. In the last two decades the study of two-dimensional (2D) systems (and more generally, of n-dimensional systems) developed as a distinct branch of Systems Theory, due to its applications in various domains as image processing, seismology and geophysics, control of multipass processes etc. The two-dimensional (2D) systems were obtained from classical 1D linear dynamical systems by generalizing from a single time variable to two (space) variables. Different state space models for 2D systems have been proposed by Roesser, Fornasini and Marchesini, Attasi, Eising and others. A subclass of 2D systems is represented by systems which are continuous with respect to one variable and discrete with respect to another one. The continuous-discrete models have applications in many problems like the iterative learning control synthesis, repetitive processes or in engineering problems such as metal rolling. In order to extend the frequency domain methods to these multiple hybrid systems one needs a generalization of the Laplace and Z transformation. The aim of this paper is to give a complete analysis of a suitable hybrid Laplace-Z type transformation and to emphasize its applications in the study of multidimensional continuous-discrete systems or for solving multiple hybrid equations. In section 2 the continuous-discrete original functions are defined and it is shown that their set is a complex commutative linear algebra with unity. A multiple hybrid Laplace-Z transformation is defined as a linear operator defined on this algebra and taking values in the set of multivariable functions which are analytic over a suitable domain. In section 3 the main properties of the multiple hybrid Laplace-Z transformation are stated and proved, including linearity, homothety, two time-delay theorems, translation, differentiation and difference of the original, differentiation of the image, integration and sum of the original, integration of the image, convolution, product of originals, initial and final values. Section 4 is devoted to the inversion problem. Some formulas and methods for determining the original are given. This hybrid transformation is employed in Section 5 to obtain transfer matrices for different classes of 2D (and more generally (q,r)-D) continuous-discrete linear control systems of Roesser-type, Fornasini-Marchesini-type and Attasi type models, including descriptor and delayed systems. The realization problem is studied in Section 6. Two canonical controllable and observable realizations are provided. An algorithm is proposed which determines a minimal realization for separable (q,r)-D multi-input-multi-output (MIMO) systems. This method generalizes to (q,r)-D systems the celebrated Ho-Kalman algorithm. The proposed algorithm can also be used for MIMO separable nD discrete-time linear systems or for MIMO nD systems described by a class of hyperbolic partial differential equations.

On donne deux methodes pour le calcul de la matrice fondamentale associee aux equations different... more On donne deux methodes pour le calcul de la matrice fondamentale associee aux equations differentielles lineaires generalisees sur l'espace de Banach des fonctions a variation bornee. Le resultat principal de l'article generalise dans ce cadre la formule de Peano-Baker.

În lucrare sunt prezentate aplicaţii ale transformării hibride multiple de tip Laplace şi z studi... more În lucrare sunt prezentate aplicaţii ale transformării hibride multiple de tip Laplace şi z studiate în partea I şi Partea a II-a. Această transformare este utilizată pentru rezolvarea unor ecuaţii multidimensionale diferenţiale cu diferenţe şi integrale. Se deduc soluţiile unor astfel de probleme care apar în teoria aşteptării. Cele mai importante aplicaţii ale acestei transformări Laplace multiple se referă la posibilitatea utilizării metodelor frecvenţiale la sistemele de comandă multidimensional hibride. Se obţin matricele de transfer ale diferitelor clase de astfel de sisteme, incluzând modele de tip Roesser, Fornasini-Marchesini şi Attasi. Some applications of the multiple hybrid Laplace and z-type transform studied in Parts I and II are presented. This transform is used to solve multidimensional differential-difference and integral equations. The solutions of such problems which appear in Queueing theory are derived. The most important applications of this multiple Laplace tr...

The Geometric Approach techniques are extended to a class of multidimensional (rD, ≥ 2) linear sy... more The Geometric Approach techniques are extended to a class of multidimensional (rD, ≥ 2) linear systems. An algorithm is provided for determining the maximal subspace which is invariant with respect to r commuting drift matrices and is included in a given subspace. When this subspace is the kernel of the output matrix, this algorithm determines the subspace of the unobservable states of the system. A Matlab program is presented, which implements the algorithm and computes an orthonormal basis of this maximal invariant subspace. MSC2000: primary 93B07, 93C35; secondary 93C05, 93B25, 93C55.

WSEAS TRANSACTIONS on SYSTEMS archive, 2010

International Conference on System Science and Simulation in Engineering, Oct 17, 2009

International Conference on Mathematical methods, Computational techniques and Intelligent systems, Jul 1, 2011

WSEAS Transactions on Systems and Control archive, Feb 1, 2010

WSEAS Transactions on Systems and Control archive, Sep 1, 2008

International Conference on System Science and Simulation in Engineering, Oct 17, 2009

WSEAS TRANSACTIONS on SYSTEMS archive, 2009

International Conference on System Science and Simulation in Engineering, Oct 17, 2009

International Conference on Mathematical and Computational Methods in Science and Engineering, Nov 7, 2008

Annals of the Academy of Romanian Scientists Series on Mathematics and Its Application

International Conference on Systems, Jul 22, 2010