Certifying Stability and Performance of Uncertain Differential–Algebraic Systems: A Dissipativity Framework

This article presents a novel framework for characterizing the dissipativity of uncertain systems whose dynamics evolve according to differential–algebraic equations. Sufficient conditions for dissipativity (specializing to, e.g., stability or $L_{2}$ gain bounds) are provided in the case that uncertainties are characterized by integral quadratic constraints. For polynomial or linear dynamics, these conditions can be efficiently verified through sum-of-squares or semidefinite programming. The performance analysis of the IEEE 39-bus power network with a set of potential line failures modeled as an uncertainty set provides an illustrative example that highlights the computational tractability of this approach; conservatism introduced in this example is shown to be quite minimal.