Sensitivity analysis of stable generalized Lyapunov equations

In this paper we study the sensitivity of Lyapunov equations which are encountered in generalized state-space systems of the form Ex˙=Ax, where E is nonsingular, and the system stable. Generalized systems as opposed to state-variable systems have different domain and codomain. To treat this problem, we first define operators for the domain and codomain and also introduce norms induced from these operators and secondly, we show that the norms of the domain and the codomain are equivalent and as such it suffices to study the sensitivity of the generalized Lyapunov equation only in one space. Further, we relate the norms of the inverse operators to the minimal L₂ damping of the generalized system in both spaces. These results lead to bounds on perturbations that guarantee the stability of a generalized stable pencil. To further motivate our work, we study several examples including one that clearly shows how we can obtain misleading bounds in the case when state-variable methods are applied on the matrix E^-1 A