Impulse minimization, index reduction, and column regularization for descriptor systems

Index of a descriptor system reflects the intensity of impulses in its solutions. The necessary and sufficient condition on descriptor systems having index at most a specific value and column regularity are established. Additionally, a necessary condition is proposed for the existence of an output feedback that can reduce the index of the system to a pre-assigned desired value while simultaneously column-regularizing the system. Furthermore, the necessary and sufficient condition is derived for the existence of an output feedback that can reduce the system's index to at most two. All theoretical conditions are proposed directly in terms of system matrices. Orthogonal transformations are used for design procedure and proofs. Numerical and physical examples are presented to support the outcome of this paper.