Multiple Laplace-Z Transformation and Applications in
the Study of Continuous - Discrete Systems
Professor Valeriu Prepelita
University Politehnica of Bucharest
Department of Mathematics-Informatics I
Splaiul Independentei 313, 060042 Bucharest
Abstract: 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
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
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
Section 4 is devoted to the inversion problem. Some
formulas and methods for determining the original are
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.
Brief Biography of the Speaker:
Valeriu Prepelita graduated from the Faculty of
Mathematics-Mechanics of the University of Bucharest in
1964. He obtained Ph.D. in Mathematics at the University
of Bucharest in 1974. He is currently Professor at the
Faculty of Applied Sciences, the University Politehnica
of Bucharest, Head of the Department
Mathematics-Informatics. His research and teaching
activities have covered a large area of domains such as
Systems Theory and Control, Multidimensional Systems,
Functions of a Complex Variables, Linear and Multilinear
Algebra, Special Functions, Ordinary Differential
Equations, Partial Differential Equations, Operational
Calculus, Probability Theory and Stochastic Processes,
Operational Research, Mathematical Programming,
Mathematics of Finance.
Professor Valeriu Prepelita is author of more than 100
published papers in refereed journals or conference
proceedings and author or co-author of 12 books. He has
participated in many national and international grants.
He is member of the Editorial Board of some journals,
member in the Organizing Committee and the Scientific
Committee of several international conferences, keynote
lecturer or chairman of some sections of these
conferences. He is a reviewer for five international
journals. He received the Award for Distinguished
Didactic and Scientific Activity of the Ministry of
Education and Instruction of Romania.