A method for analysing the stability of dynamical systems is proposed, based on the introduction of a weighted phase volume and time rescaling by a positive function. The advantage of the method is the ability to set the contraction properties of the phase volume by choosing the weighting function and the scaling factor, while preserving the topology of the phase portrait. Integral dissipativity conditions are derived, leading to new definitions of integral stability, asymptotic stability, and exponential stability. For quadratic weighting functions, covering and inner ellipsoids are constructed, providing geometric estimates of reachable sets. The connection between the proposed approach and classical Lyapunov stability is established. The efficiency of the method is demonstrated through numerical examples.
Reachability analysis of safety-critical systems requires computing rigorous enclosures of all possible state trajectories. Taylor Model (TM)-based methods have proved effective at mitigating the so-called wrapping effect which leads to overly conservative enclosures of reachable…
Hamilton-Jacobi Reachability (HJR) is an important framework for controlling safety-critical systems despite uncertainty. Its theoretical underpinnings are rooted in Hamilton-Jacobi Partial Differential Equations, which provide the value function used for controller synthesis. Th…
This paper develops a data-driven reachability framework for linear systems whose disturbances are modeled by probabilistic zonotopes (PZs), combining bounded deterministic and Gaussian stochastic components. In contrast to methods that require a precisely known disturbance model…
This paper develops a computationally efficient framework for reachability analysis of transmission-level power system dynamics with synchronous generators, grid-forming and grid-following inverters, and uncertain power injections/withdrawals. Starting from reduced-order device m…
We consider the uniform exponential stability analysis of infinite-dimensional impulsive systems defined on a Banach or Hilbert space, whose flow is governed by a fixed $C_0$-semigroup generator and whose jumps occur at a prescribed time sequence. While the flow and jump maps are…
This work presents an examination of the main interactions between grid-following (GFL) and grid-forming (GFM) voltage source converters (VSCs) and synchronous generators (SGs), capturing the dynamics of a real power grid and pointing out the limitations of considering an ideal o…