Plenary lecture X: topics in multidimensional continuous - discrete systems theory
Authors: Valeriu Prepelita
Venue: Proceedings of the 10th WSEAS international conference on Mathematical and computational methods in science and engineering (2008)

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 formulæ 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.