Assessing the stability of linear time-invariant continuous interval dynamic systems

It is known that stability analysis of linear time-invariant dynamic systems under parameter uncertainties can be equated to estimating the range of the eigenvalues of matrices whose elements are intervals. In this note, first the problem of finding tight outer bounds on the eigenvalue ranges is considered. A method for computing such bounds is suggested which consists, essentially, of setting up and solving a system of n mildly nonlinear algebraic equations, n being the size of the interval matrix investigated. The main result of the note, however, is a method for determining the right end-point of the exact eigenvalue ranges. The latter makes use of the outer bounds. It is applicable if certain computationally verifiable monotonicity conditions are fulfilled. The methods suggested can be applied for robust stability analysis of both continuous- and discrete-time systems. Numerical examples illustrating the applicability of the new methods are also provided.