On the Solution of the General Singular Model of 2-D Systems Nikos Karampetakis Department of Mathematics, Aristotle University of Thessaloniki, Thessaloniki 54006, Greece Paper ID : 4736 The main objectives of this work are : to introduce the singular 2-D model and analyze the question of existence and uniqueness of backward and symmetric solutions, to demonstrate the fundamental importance of a) the 2-D backward fundamental matrix sequence and b) the dual singular model. Singular 2-D models (Kaczorek) do not require any notion of causality or recursibility, but are limited by the milder notion of regularity which is required for existence and uniqueness of solutions. Thus, its applications in 2-D signal processing are wider than those of the restricted Roesser/Fornasini- Marchesini state-space models, i.e. discretized 2-D hyperbolic and heat equation with boundary conditions on all sides of a planar region. In Lewis & Mertzios 1991, the forward solution was investigated for implicit Roesser 2-D models. In this case, it is desired to find the semistate sequence given the inputs and the boundary conditions (BCs) along the i- and j- axis. Complementary to this solution is the backward solution, where the inputs and the BCs along the upper and right-hand sides of the planar region are prescribed. Finally, if the inputs are known and the BCs are of the split or two-point form then the solution could be called symmetric solution. 3. Singular 2-D model 3. Methods If is a solution of the dual singular model for the non-zero input, then the sequence is a solution of the singular model for For the backward solution For the symmetric solution Rewrite the equations where and premultiply by where 4. Results Backward solution Symmetric solution If, and wheredenotes the lower bound of the degree of G in terms of s, then where If, and wheredenotes the upper bound of the degree of G in terms of s, then 5. Conclusions The backward and symmetric solution of the singular 2-D model has been proposed in terms of the forward/backward 2-D fundamental matrix sequence, which generalizes the state-space context of the transition matrix. The backward/symmetric closed formula can be used for the establishment of necessary and sufficient conditions for the local reachability and local controllability of singular 2-D models. The relation between the invariants of the matrix pencil G(z 1,z 2 ) and its dual matrix pencil z 1 z 2 G(1/z 1,1/z 2 ) has to be established and connected with the system properties. 6. Further research 2. Objectives Singular 2-D models (Kaczorek) do not require any notion of causality or recursibility, but are limited by the milder notion of regularity which is required for existence and uniqueness of solutions. Thus, its applications in 2-D signal processing are wider than those of the restricted Roesser/Fornasini- Marchesini state-space models i.e. discretized 2-D hyperbolic and heat equation with boundary conditions on all sides of a planar region. In Lewis & Mertzios 1991, the forward solution was investigated for implicit Roesser 2-D models. In this case, it is desired to find the semistate sequence given the inputs and the boundary conditions (BCs) along the i- and j- axis. 1. Introduction