Exponential Convergence Bounds using Integral Quadratic Constraints

The theory of integral quadratic constraints (IQCs) allows verification of stability and gain-bound properties of systems containing nonlinear or uncertain elements. Gain bounds often imply exponential stability, but it can be challenging to compute useful numerical bounds on the exponential decay rate. In this work, we present a modification of the classical IQC results of Megretski and Rantzer that leads to a tractable computational procedure for finding exponential rate certificates

On the exponential convergence of input-output signals of nonlinear feedback systems

We show that the integral-constraint-based robust feedback stability theorem for certain Lurye systems exhibits the property that the endogenous input-output signals enjoy an exponential convergence rate for all initial conditions of the linear time-invariant subsystem. More generally, we provide conditions under which a feedback interconnection of possibly open-loop unbounded subsystems to admit such an exponential convergence property, using perturbation analysis and a combination of tools including integral quadratic constraints, directed gap measure, and exponential weightings. As an application, we apply the result to first-order convex optimisation methods. In particular, by making use of the Zames-Falb multipliers, we state conditions for these methods to converge exponentially when applied to strongly convex functions with Lipschitz gradients.Comment: This paper has been submitted to Automatic

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

This paper presents a novel framework for characterizing dissipativity of uncertain dynamical systems subject to algebraic constraints. The main results provide sufficient conditions for dissipativity when 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 practical impact of this work is illustrated through a case study that examines performance of the IEEE 39-bus power network with uncertainties used to model a set of potential line failures

Scalable Design of Heterogeneous Networks

A systematic approach to the analysis and design of a class of large dynamical systems is presented. The approach allows decentralised control laws to be designed independently using only local subsystem models. Design can be conducted using standard techniques, including loopshaping based on Nyquist and Popov plots, H$_\infty$ methods, and $\mu$-synthesis procedures. The approach is applied to a range of network models, including those for consensus, congestion control, electrical power systems, and distributed optimisation algorithms subject to delays.Engineering and Physical Sciences Research Council grant number EP/G066477/