Lyapunov Matrix Equations for the Stability Analysis of Linear Time-Invariant Descriptor Systems

For the stability analysis of linear time-invariant descriptor systems two different generalizations of the classical Lyapunov matrix equation are considered. The first generalization includes the singular matrix related to the time-derivatives of the descriptor variables in an obviously symmetric form; the second one shows at a first sight no symmetry which additionally has to be asked for explicitly. This second approach is well-known for ‘admissible’ descriptor systems which includes a restriction to systems of index k = 1. In this contribution the second approach will be generalized to systems with arbitrary index k ≥ 1. Both approaches will be compared with each other showing different solvability conditions and different solutions in general. But for the problem of analyzing asymptotic stability the solution behaviors of the two generalized Lyapunov matrix equations coincide. In spite of the different procedures both approaches lead to the same Lyapunov function for the analysis of asymptotic stability of linear time-invariant descriptor systems. The two approaches will be illustrated by the stability analysis of mechanical descriptor systems, i.e. by mechanical systems with holonomic constrains. Although the application of the approaches usually is very costly, they represent suitable tools for the stability analysis of linear time-invariant descriptor systems.