Output regulator problem of time-invariant discrete-time descriptor systems

Abstract

Linear time-invariant discrete-time systems of the form Exk +1 = Axk + Buk , yk = Cxt are considered where E and A are square matrices and E may be singular. The investigations are restricted to systems for which there exist non-singular matrices Π and Θ such that ΠEΘ = diag {lr , 0} and ΠAΘ= equation pending where F 2 has full row rank. Such systems are termed 'regular’ by Luenberger (1977), and can be decomposed into two equations of the respective forms ζ k +1= Ψ1ζ k + Ψ2η k and +Δ2uk= 0 where ζ k , η k and xk are related by the equation xk = equation pending thus the descriptor system might be reduced to a state-space system whose state vector is ζ k A reasonable choice for C is C= Γ[I, 0] Θ -1where Γ is some m x r matrix of full row rank, which yields yk = Γζ k . The problem is this: given an initial descriptor vector xo, find, without reducing the descriptor system to a state-space system, a sequence of control vectors u o, u 1,..,uN -1 Hy, that will yield yN = 0 for some N. A necessary and sufficie...