Boundary recursion for descriptor variable systems

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Abstract
A linear discrete-time descriptor variable system defined on the time interval [0,N] has the form E(k+1)x(k+1) = A(k)x(k) + B(k)u(k) for k = 0,1,2,...,N-1. For each k, x(k) is an n-dimensional vector of descriptor variables, u(k) is an m-dimensional vector of input variables, and the matrices A(k), E(k+1), B(k) are of compatible dimensions. If the system satisfies the property of solvability [1], it will have an n-dimensional family of solutions. It is shown in this paper that if the system also satisfies the property of conditionability, the solutions will each satisfy an equation of the form Z0x(0) + ZNx(N) = v(N) where the matrix Z = [Z0,ZN] is of rank n and v(N) depends on the input sequence. This equation is called a boundary mapping equation. In the case of solvable and conditionable systems the boundary mapping equation is a complete summary of the system, in the sense that there is a one-to-one correspondence between solutions to this equation and solutions to the original descriptor variable system. It therefore provides a compact representation of the system's influence on the boundary points. It is an (n-dimensional) equation which "jumps over" the intermediate values and directly links x(0) and x(N). In fact, the boundary mapping generalizes the state-transition matrix associated with state variable systems.

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