Robust Analysis of Linear Systems with Uncertain Delays using PIEs

This paper establishes a PIE (Partial Integral Equation)-based technique for the robust stability and H_∞ performance analysis of linear systems with interval delays. The delays considered are time-invariant but uncertain, residing within a bounded interval excluding zero. We first propose a structured class of PIE systems with parametric uncertainty, then propose a Linear PI Inequality (LPI) for robust stability and H_∞ performance of PIEs with polytopic uncertainty. Next, we consider the problem of robust stability and H_∞ performance of multi-delay systems with interval uncertainty in the delay parameters and show this problem is equivalent to robust stability and performance of a given PIE with parametric uncertainty. The robust stability and H_∞ performance of the uncertain time-delay system are then solved using the LPI solver in the MATLAB PIETOOLS toolbox. Numerical examples are given to prove the effectiveness and accuracy of the method. This paper adds to the expanding field of PIE approach and can be extended to linear partial differential equations.