LATEST TRENDS <strong>on</strong> SYSTEMS (Volume I)Plenary Lecture 2Multiple Laplace-Z Transformati<strong>on</strong> and Applicati<strong>on</strong>s in the Study of C<strong>on</strong>tinuo<strong>us</strong> - DiscreteSystemsProfessor Valeriu PrepelitaUniversity Politehnica of BucharestDepartment of Mathematics-Informatics ISplaiul Independentei 313, 060042 BucharestROMANIAE-mail: vprepelita@mathem.pub.roAbstract: The Operati<strong>on</strong>al Calcul<strong>us</strong> as a distinct discipline has a history which has exceeded a century. But its rootscan be found in the works of Leibniz, Bernoulli, Lagrange, Laplace, Euler, Fourier, Cauchy and others. Its importanceis determined by its utility in solving complex problems in many domains such as Calcul<strong>us</strong>, Number Theory, SpecialFuncti<strong>on</strong>s, Ordinary Differential Equati<strong>on</strong>s, Mathematical Physics, Heat Transfer, Electr<strong>on</strong>ics, Automatics, etc.In Systems and C<strong>on</strong>trol Theory the frequency domain methods, based <strong>on</strong> Laplace transformati<strong>on</strong> in the c<strong>on</strong>tinuo<strong>us</strong>timecase or <strong>on</strong> Z transformati<strong>on</strong> in the discrete-time case, play a very important role in the study of the "classical" 1D<strong>systems</strong>.In the last two decades the study of two-dimensi<strong>on</strong>al (2D) <strong>systems</strong> (and more generally, of n-dimensi<strong>on</strong>al <strong>systems</strong>)developed as a distinct branch of Systems Theory, due to its applicati<strong>on</strong>s in vario<strong>us</strong> domains as image processing,seismology and geophysics, c<strong>on</strong>trol of multipass processes etc.The two-dimensi<strong>on</strong>al (2D) <strong>systems</strong> were obtained from classical 1D linear dynamical <strong>systems</strong> by generalizing from asingle time variable to two (space) variables. Different state space models for 2D <strong>systems</strong> have been proposed byRoesser, Fornasini and Marchesini, Attasi, Eising and others.A subclass of 2D <strong>systems</strong> is represented by <strong>systems</strong> which are c<strong>on</strong>tinuo<strong>us</strong> with respect to <strong>on</strong>e variable and discretewith respect to another <strong>on</strong>e. The c<strong>on</strong>tinuo<strong>us</strong>-discrete models have applicati<strong>on</strong>s in many problems like the iterativelearning c<strong>on</strong>trol synthesis, repetitive processes or in engineering problems such as metal rolling.In order to extend the frequency domain methods to these multiple hybrid <strong>systems</strong> <strong>on</strong>e needs a generalizati<strong>on</strong> of theLaplace and Z transformati<strong>on</strong>.The aim of this paper is to give a complete analysis of a suitable hybrid Laplace-Z type transformati<strong>on</strong> and toemphasize its applicati<strong>on</strong>s in the study of multidimensi<strong>on</strong>al c<strong>on</strong>tinuo<strong>us</strong>-discrete <strong>systems</strong> or for solving multiple hybridequati<strong>on</strong>s.In secti<strong>on</strong> 2 the c<strong>on</strong>tinuo<strong>us</strong>-discrete original functi<strong>on</strong>s are defined and it is shown that their set is a complexcommutative linear algebra with unity. A multiple hybrid Laplace-Z transformati<strong>on</strong> is defined as a linear operatordefined <strong>on</strong> this algebra and taking values in the set of multivariable functi<strong>on</strong>s which are analytic over a suitabledomain.In secti<strong>on</strong> 3 the main properties of the multiple hybrid Laplace-Z transformati<strong>on</strong> are stated and proved, includinglinearity, homothety, two time-delay theorems, translati<strong>on</strong>, differentiati<strong>on</strong> and difference of the original, differentiati<strong>on</strong>of the image, integrati<strong>on</strong> and sum of the original, integrati<strong>on</strong> of the image, c<strong>on</strong>voluti<strong>on</strong>, product of originals, initial andfinal values.Secti<strong>on</strong> 4 is devoted to the inversi<strong>on</strong> problem. Some formulas and methods for determining the original are given.This hybrid transformati<strong>on</strong> is employed in Secti<strong>on</strong> 5 to obtain transfer matrices for different classes of 2D (and moregenerally (q,r)-D) c<strong>on</strong>tinuo<strong>us</strong>-discrete linear c<strong>on</strong>trol <strong>systems</strong> of Roesser-type, Fornasini-Marchesini-type and Attasitype models, including descriptor and delayed <strong>systems</strong>.The realizati<strong>on</strong> problem is studied in Secti<strong>on</strong> 6. Two can<strong>on</strong>ical c<strong>on</strong>trollable and observable realizati<strong>on</strong>s are provided.An algorithm is proposed which determines a minimal realizati<strong>on</strong> for separable (q,r)-D multi-input-multi-output (MIMO)<strong>systems</strong>. This method generalizes to (q,r)-D <strong>systems</strong> the celebrated Ho-Kalman algorithm. The proposed algorithmcan also be <strong>us</strong>ed for MIMO separable nD discrete-time linear <strong>systems</strong> or for MIMO nD <strong>systems</strong> described by a classof hyperbolic partial differential equati<strong>on</strong>s.ISSN: 1792-4235 19 ISBN: 978-960-474-199-1
LATEST TRENDS <strong>on</strong> SYSTEMS (Volume I)Brief Biography of the Speaker:Valeriu Prepelita graduated from the Faculty of Mathematics-Mechanics of the University of Bucharest in 1964. Heobtained Ph.D. in Mathematics at the University of Bucharest in 1974. He is currently Professor at the Faculty ofApplied Sciences, the University Politehnica of Bucharest, Head of the Department Mathematics-Informatics. Hisresearch and teaching activities have covered a large area of domains such as Systems Theory and C<strong>on</strong>trol,Multidimensi<strong>on</strong>al Systems, Functi<strong>on</strong>s of a Complex Variables, Linear and Multilinear Algebra, Special Functi<strong>on</strong>s,Ordinary Differential Equati<strong>on</strong>s, Partial Differential Equati<strong>on</strong>s, Operati<strong>on</strong>al Calcul<strong>us</strong>, Probability Theory andStochastic Processes, Operati<strong>on</strong>al Research, Mathematical Programming, Mathematics of Finance.Professor Valeriu Prepelita is author of more than 100 published papers in refereed journals or c<strong>on</strong>ferenceproceedings and author or co-author of 12 books. He has participated in many nati<strong>on</strong>al and internati<strong>on</strong>al grants. He ismember of the Editorial Board of some journals, member in the Organizing Committee and the Scientific Committeeof several internati<strong>on</strong>al c<strong>on</strong>ferences, keynote lecturer or chairman of some secti<strong>on</strong>s of these c<strong>on</strong>ferences. He is areviewer for five internati<strong>on</strong>al journals. He received the Award for Distinguished Didactic and Scientific Activity of theMinistry of Educati<strong>on</strong> and Instructi<strong>on</strong> of Romania.ISSN: 1792-4235 20 ISBN: 978-960-474-199-1
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