Stability and stabilization of a class of ill-conditioned second order differential linear repetitive processes

This paper considers differential linear repetitive processes which are a distinct class of 2D systems whose dynamics evolve over a subset of the upper right quadrant of the 2D plane. In particular, information propagation in one direction only occurs over a finite duration and is governed by a matrix differential linear equation. A stability theory exists for these processes but a problem arises in its application to second order processes due to the possible ill-conditioning of the leading coefficient matrix in the state-space model. Here we derive new results on stability analysis and control law design for this case by first transforming the second-order state-space model to an equivalent first-order descriptor system, thus avoiding the necessity of inverting a possibly ill-conditioned matrix. The results developed here can be computed using linear matrix inequalities (LMIs).