Nearest Descriptor System with Impulsive Initial Conditions

Abstract: Distance problems play a very important role in the analysis of linear control systems. Distance problems are used to study changes in system properties with respect to small changes in system parameters. In this talk, I will discuss about the distance to the nearest descriptor systems having impulsive initial conditions. A descriptor system is defined by the equation Ex˙ = Ax, E, A ∈ R ^n×n, where matrix E is singular. In case of descriptor systems, for certain initial values of the state variable x, the system behavior would contain impulsive trajectories. I related the existence of such impulsive initial conditions to the presence of defective eigenvalues at infinity in the matrix pencil sE−A. I provide closed form expression for the distance when both matrices E and A are perturbed via perturbations having rank-1. I also provide a closed form expression for the distance for the case when both matrices E and A are symmetric and matrix E is perturbed via symmetric perturbation. For the general case, i.e when matrices E and A are perturbed using perturbations having no rank or structure restriction, I formulate the distance as an optimization problem. Also, I compute the value of the distance using numerical packages like Structured Low Rank Approximation (SLRA) and MATLAB Globalsearch.