ECE PhD Proposal Review: Jared Miller

PhD Proposal Review: Nonlinear and Time-Delay Systems Analysis using Occupation Measures

Jared Miller

Location: Zoom Link

Abstract: Techniques to analyze nonlinear systems include peak and reachable set estimation. The reachable set of a system is the set of states accessible by trajectories of a dynamical system at specified times given initial conditions. The peak estimation problem finds extreme values of a state function along trajectories. Examples of peak estimation include finding the maximum height of a wave, voltage on a power line, speed of a vehicle, and infection rate of an epidemic. These problems may be posed as infinite dimensional linear programs (LP) in occupation measures, where occupation measures are Borel measures that contain all information about trajectories. Under mild assumptions, a sequence of Linear Matrix Inequalities (LMI) in increasing degree will converge from outside to the LP optimum, which is in turn equal to the true optimum of the program in trajectories.
The first part of this thesis expands upon the occupation measure formulation for peak estimation. The safety of trajectories with respect to an unsafe set may be quantified by measuring the constraint violation (safety margins), which is a maximum peak estimation problem. The distance of closest approach between trajectories and an unsafe set may be bounded through a modification of the peak estimation problem. Peak estimation may be applied to dynamics possessing a broad class of uncertainties, which includes the data-driven setting of black-box polynomial dynamics. A modular MATLAB toolbox is developed to solve and interpret these variations on peak estimation problems.
The second part of this thesis introduces an occupation measure framework for analysis and control of time-delay systems. The evolution of time delay systems depends on present and past values of the state. Some instances of time delay systems with their associated delays include epidemic models (incubation period), population dynamics (gestation time), and fluid modeling (transport time of fluid moving in a pipe). An occupation measure framework is developed to define weak solutions over a finite time interval of nonlinear time-delay systems with a finite number of bounded discrete delays. Applications of this time-delay weak solution include optimal control (including dead-time), peak estimation, and reachable set estimation of time delay systems.