On complex stability radii for implicit discrete time systems (Q1762809)

For linear, discrete-time, regular, asymptotically stable descriptor systems \[ A x_{n+1}=Bx_n\tag{1} \] with constant coefficients \(A\) and \(B\), complex stability radii w.r.t. structured perturbations in \(A\) and \(B\) are investigated. To be more precise, let \(A,B,E,F\) be complex matrices, where \(A\), \(B\) and \(EF\) are square and of the same size and (1) is asymptotically stable, i.e., \(\rho(A,B)<1\), where \(\rho(A,B)\) is the spectral radius of the pencil \((A,B)\). Complex stability radii are defined by \[ r_C=\inf\{\|\Delta\|\,|\,\rho(A,B+E\Delta F)\geq 1, \Delta\text{ a complex matrix}\} \] and \[ r_L=\inf\{\|\Delta\|\,|\,\rho(A+E\Delta F,B)\geq 1, \Delta\text{ a complex matrix}\}, \] where \(\|\cdot\|\) is an arbitrary operator norm. Formulas for \(r_C\) and \(r_L\) in terms of the transfer function \(s\mapsto F(sA-B)^{-1}E\) are presented, and further aspects of \(r_C\) and \(r_L\) are discussed.