In mathematical systems theory, a multidimensional system or m-D system is a system in which not only one independent variable exists (like time), but there are several independent variables. Important problems such as factorization and stability of m-D systems (m > 1) have recently attracted the interest of many researchers and practitioners. The reason is that the factorization and stability is not a straightforward extension of the factorization and stability of 1-D systems because, for example, the fundamental theorem of algebra does not exist in the ring of m-D (m > 1) polynomials.
Applications Multidimensional systems or m-D systems are the necessary mathematical background for modern digital image processing with many applications in biomedicine, X-ray technology and satellite communications. There are also some studies combining m-D systems with partial differential equations (PDEs).
Linear multidimensional state-space model A state-space model is a representation of a system in which the effect of all "prior" input values is contained by a state vector. In the case of an m-d system, each dimension has a state vector that contains the effect of prior inputs relative to that dimension. The collection of all such dimensional state vectors at a point constitutes the total state vector at the point. Consider a uniform discrete space linear two-dimensional (2d) system that is space invariant and causal. It can be represented in matrix-vector form as follows: Represent the input vector at each point ( i , j ) {\displaystyle (i,j)} by u ( i , j ) {\displaystyle u(i,j)} , the output vector by y ( i , j ) {\displaystyle y(i,j)} the horizontal state vector by R ( i , j ) {\displaystyle R(i,j)} and the vertical state vector by S ( i , j ) {\displaystyle S(i,j)} . Then the operation at each point is defined by:
R ( i + 1 , j ) = A 1 R ( i , j ) + A 2 S ( i , j ) + B 1 u ( i , j ) S ( i , j + 1 ) = A 3 R ( i , j ) + A 4 S ( i , j ) + B 2 u ( i , j ) y ( i , j ) = C 1 R ( i , j ) + C 2 S ( i , j ) + D u ( i , j ) {\displaystyle {\begin{aligned}R(i+1,j)&=A_{1}R(i,j)+A_{2}S(i,j)+B_{1}u(i,j)\\S(i,j+1)&=A_{3}R(i,j)+A_{4}S(i,j)+B_{2}u(i,j)\\y(i,j)&=C_{1}R(i,j)+C_{2}S(i,j)+Du(i,j)\end{aligned}}}
where A 1 , A 2 , A 3 , A 4 , B 1 , B 2 , C 1 , C 2 {\displaystyle A_{1},A_{2},A_{3},A_{4},B_{1},B_{2},C_{1},C_{2}} and D {\displaystyle D} are matrices of appropriate dimensions. These equations can be written more compactly by combining the matrices:
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