Design of observers for time-varying discrete-time descriptor systems

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Abstract
In this note, linear discrete-time time-varying ‘descriptor’ systems, i.e. control systems of the form Ek+1 Xk+1 = AkXk + Bkuk, yk = Ckxk are considered where Ek+1 and Ak+1 are constant, not necessarily square, matrices. The problem is to design a reduced-order observer for the system. Using matrix generalized inverses and singular-value decomposition, a procedure is developed which either leads to a routine observer-design problem involving an ordinary discrete-time state equation or proves that no observer exists for the given ‘descriptor’ system. The method of this note is partly based on an earlier paper of the authors on observer design for continuous-time time invariant systems of the form E dx/dt=Ax + Bu, y=Cx where E and A are square matrices (El-Tohami et al. 1983) and makes use of some results of Luenberger on discrete-time time-varying descriptor systems (Luenberger 1977).