On the Solution of the General Singular Model of 2-d Systems

The main objective of this work is to provide a closed formula for the forward, backward and symmetric solution of the 2-D general singular model (GSM) given in (Kurek, 1985). Of fundamental importance in our approach is the relative forward and backward fundamental matrix. Copyright c 2005 IFAC. Keywords: 2-D general singular models, implicit 2-D systems, fundamental matrix, forward solution, backward solution, symmetric solution 1. INTRODUCTION Consider the 2-D linear discrete time systems proposed in (Kaczorek, 1988) as a generalization of the 2-D state-space model given in (Kurek, 1985) Ex (i+ 1; j + 1) = A0x (i; j)+ (1) A1x (i+ 1; j) +A2x (i; j + 1) +B0u (i; j)+ +B1u (i+ 1; j) +B2u (i; j + 1) where i; j are integer-value vertical and horizontal coordinates, respectively, x (i; j) 2 R is the local state vector at (i; j), u (i; j) 2 R is the input vector, Ak 2 R ; Bk 2 R ; k = 0; 1; 2 and matrices E;A0 2 R n exists and are not necessarily nonsingular. This model includes similar generalization of other 2-D state space models such as the Fornasini and Marchesini (Fornasini and Marchesini, 1970) and the Roesser 2-D model (Roesser, 1975). If E 6= I we call these models implicit 2-D systems. We shall call (1) the general singular model (GSM) or otherwise the implicit Fornasini-Marchesini (FM) model. If E is non-square or det (E) = 0 we call these models singular 2-D systems. One particular case of (1) is the implicit Roesser model proposed in (Kaczorek, 1987) and (Lewis, 1987) as a generalization of the Roesser 2-D model given in (Roesser, 1975). It is shown in (Kaczorek, 1989) that the implicit Roesser and the implicit FM model are equivalent. Due to the equivalence of the above models we consider in the rest of the paper only the GSM model. An example of the GSM model is providing by discretizing the heat equation using the method of central di¤erences (Karamancioglu, 1991), while an example of the implicit Roesser model is given by the 2-D realization of a nonrecursible mask in digital image processing (Lewis and Mertzios, 1991). Implicit FM models are also arising from the discretization of continuous-time systems that are described by partial di¤erential equations i.e. the standard discretization of the elliptic equation that results in a …ve-point discrete mask or the discretization of the di¤usion equation that results in a four-point discrete mask (Karamancioglu, 1991). According to (Lewis, 1992) there are various ways to specify the boundary conditions (BCs) and the region of interest for the FM and Roesser models. First suppose that the 2-D implicit system has BCs speci…ed along the i and j axes. For the GSM model this means we know : x (i; 0) = xi0; i = 0; 1; :::; N x (0; j) = x0j ; j = 0; 1; :::;M (2) where xi0 and x0j are known vectors. Then, if the region of interest is the rectangle [0; N ] [0;M ] in the (i; j) plane, we are concerned with …nding what could be called a "forward solution ". If the BCs are speci…ed along the upper and right-hand sides of the rectangle : x (i;M) = xiM ; i = 0; 1; :::; N x (N; j) = xNj ; j = 0; 1; :::;M (3) then the solution on [0; N ] [0;M ] could be called "backward solution ". A general case which includes both of these situations is where the BCs are of the split or two-point form : C i;0x (i; 0) + C u i;Mx (i;M) = c u i ; 0 i N C 0;jx (0; j) + C h N;jx (N; j) = c h i ; 0 j M (4) with C i;0 C u i;M and C 0;j C h N;j prescribed matrices of full row rank and ci ; c h i given vectors. If the BCs are of the split form given above or otherwise involve the semistate along all boundaries of the rectangular region [0; N ] [0;M ] then the solution on [0; N ] [0;M ] could be called "symmetric solution ". A complete analysis of solutions and properties in the forward, backward and symmetric case for the 1-D singular systems Ex (i+ 1) = Ax (i) + Bu (i) was given in (Lewis and Mertzios, 1990) in terms of the matrices E;A;B and the forward and backward fundamental matrix of (zE A) . (Lewis and Mertzios, 1991) and (Kaczorek, 1990) have proposed, a forward solution to the 2-D implicit Roesser model and GSM respectively, in terms of the forward fundamental matrix of the system. Following similar methods to those of (Lewis and Mertzios, 1990) , we produce a closed formula for the backward and symmetric solution of the GSM (1) in terms of the forward fundamental matrix Tp;q and backward fundamental matrix ~ Tp;q of (z1z2E A0 A1z1 A2z2) . A generalized Leverrier technique for computing the forward fundamental matrix sequence is available (Mertzios and Lewis, 1988), (Karampetakis et al., 1994), so that we may assume that this matrix sequence is given. We shall show in Section 2 that the backward fundamental matrix sequence of z1z2E A0 A1z1 A2z2 is the forward fundamental matrix sequence of the dual polynomial matrix E A0z1z2 A1z2 A2z1, and thus we may assume that the backward fundamental matrix sequence is also given. 2. PRELIMINARY RESULTS Assuming that the polynomial matrix G (z1; z2) = z1z2E A0 A1z1 A2z2 (5) with E 6= 0; is invertible, the Laurent expansion at in…nity of G (z1; z2) 1 exists, is unique (Karampetakis et al., 1994), and is given by : G (z1; z2) 1 = 1 X