On controlling generalized state-space (descriptor) systems

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Abstract
Linear time-invariant control systems of the form Edx/dt = Ax + Bu are considered where E and A are not necessarily square matrices. Any vector x(t0) that can be obtained as a solution of the system for some u(t) and some t = t0, will be termed an ‘admissible initial state’ and it will be assumed that the system has a unique solution for any input and for any admissible initial state. (An efficient numerical method for characterizing admissible inputs and establishing uniqueness of solutions has been developed by Wilkinson (1978).) The problem to be solved is this: find an input u(t) that drives the system from an admissible initial state to a prescribed final state in a given time tf > 0. If the basic assumption regarding uniqueness of solutions holds, the problem of controlling the given generalized state-space (descriptor) system can be reduced to a standard control problem involving an ‘ordinary’ state-space system of particularly simple form. Then, if the ‘ordinary’ system is completely controllable, a desired input can be calculated from a well-known formula. In order to avoid the calculation and integration of matrix exponential functions which are required by the standard formula, in § 6 of this paper an attempt is made to reduce the calculation of the desired input to a two-point (matrix) boundary-value problem whose solution requires only solution of an algebraic matrix equation

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