Global analysis of piecewise linear systems using impact maps and surface Lyapunov functions

This paper presents an entirely new constructive global analysis methodology for a class of hybrid systems known as piecewise linear systems (PLS). This methodology infers global properties of PLS solely by studying the behavior at switching surfaces associated with PLS. The main idea is to analyze impact maps, i.e., maps from one switching surface to the next switching surface. Such maps are known to be "unfriendly" maps in the sense that they are highly nonlinear, multivalued, and not continuous. We found, however, that an impact map induced by an linear time-invariant flow between two switching surfaces can be represented as a linear transformation analytically parametrized by a scalar function of the state. This representation of impact maps allows the search for surface Lyapunov functions (SuLF) to be done by simply solving a semidefinite program, allowing global asymptotic stability, robustness, and performance of limit cycles and equilibrium points of PLS to be efficiently checked. This new analysis methodology has been applied to relay feedback, on/off and saturation systems, where it has shown to be very successful in globally analyzing a large number of examples. In fact, it is still an open problem whether there exists an example with a globally stable limit cycle or equilibrium point that cannot be successfully analyzed with this new methodology. Examples analyzed include systems of relative degree larger than one and of high dimension, for which no other analysis methodology could be applied. This success in globally analyzing certain classes of PLS has shown the power of this new methodology, and suggests its potential toward the analysis of larger and more complex PLS

Towards an Algebra for Cascade Effects

We introduce a new class of (dynamical) systems that inherently capture cascading effects (viewed as consequential effects) and are naturally amenable to combinations. We develop an axiomatic general theory around those systems, and guide the endeavor towards an understanding of cascading failure. The theory evolves as an interplay of lattices and fixed points, and its results may be instantiated to commonly studied models of cascade effects. We characterize the systems through their fixed points, and equip them with two operators. We uncover properties of the operators, and express global systems through combinations of local systems. We enhance the theory with a notion of failure, and understand the class of shocks inducing a system to failure. We develop a notion of mu-rank to capture the energy of a system, and understand the minimal amount of effort required to fail a system, termed resilience. We deduce a dual notion of fragility and show that the combination of systems sets a limit on the amount of fragility inherited.Comment: 31 page

Overview of robust stability and performance methods of systems with structured mixed perturbations

Robust stability and performance analysis results for systems in the presence of structured mixed perturbations are outlined. Attention is limited to scalar perturbations. The goal is to develop succinctly an overall description of state of the art techniques in analyzing systems with mixed perturbations, and to point the reader to sources in the literature where more details and proofs can be found

On the behavior of threshold models over finite networks

We study a model for cascade effects over finite networks based on a deterministic binary linear threshold model. Our starting point is a networked coordination game where each agent's payoff is the sum of the payoffs coming from pairwise interaction with each of the neighbors. We first establish that the best response dynamics in this networked game is equivalent to the linear threshold dynamics with heterogeneous thresholds over the agents. While the previous literature has studied such linear threshold models under the assumption that each agent may change actions at most once, a study of best response dynamics in such networked games necessitates an analysis that allows for multiple switches in actions. In this paper, we develop such an analysis. We establish that agent behavior cycles among different actions in the limit, we characterize the length of such limit cycles, and reveal bounds on the time steps required to reach them. We finally propose a measure of network resilience that captures the nature of the involved dynamics. We prove bounds and investigate the resilience of different network structures under this measure.Irwin Mark Jacobs and Joan Klein Jacobs Presidential FellowshipSiebel ScholarshipUnited States. Air Force Office of Scientific Research (Grant FA9550-09-1-0420)United States. Army Research Office (Grant W911NF-09-1-0556

2016 Convocation

Welcome: Robert Hernandez, Ph.D., Treasurer, Executive Director of Student Affairs Pledge of Allegiance: Madison Dong, Student Council President Opening Remarks: José M. Torres, Ph.D., President Opening Remarks: Marie Dillon Dahleh, Ph.D., Principal Featured Musical Selection Cello: Emily Camras, Class of 2013 Keynote Address: Yuanxia Ding, Class of 2000, Senior Policy Advisor to the Under Secretary of Education at the U.S. Department of Education Closing Remarks: Marie Dillon Dahleh, Ph.D., Principa

Commencement of the Class of 2016

It’s amazing how fast three years passes. And in the end, IMSA is so much more than just a building. Others may never truly understand what it means to be a part of this community, but we will know. Years from now, we’ll remember the thrill of Clash or the nostalgic warmth of Carnival. We’ll remember late nights spent laughing with best friends, and we’ll remember what it felt like to belong here. It’s true, IMSA gave us the building blocks for academic and professional success. But it also gave us each other. And as we stand here as a class for the last time, I thank IMSA for bringing us together. Class of 2016, never stop learning and growing. Never lose that drive or that passion, and never forget what you have gained from IMSA. As we move on to another chapter of our lives, we will continue to expand our brick-and-mortar walls. We will gain countless more chances, and we will learn countless more things. It is my hope that we, as a class, will continue to seek out new opportunities, and I hope that we continue to find better ways of cementing it all together. Yet no matter how far we go, part of us will always belong in a residence hall at 1500 Sullivan Road. And no matter how far apart we may end up, we can find comfort in knowing that we have built something beautiful together Heidi Dong, Student Council Presiden

Beable trajectories for revealing quantum control mechanisms

The dynamics induced while controlling quantum systems by optimally shaped laser pulses have often been difficult to understand in detail. A method is presented for quantifying the importance of specific sequences of quantum transitions involved in the control process. The method is based on a ``beable'' formulation of quantum mechanics due to John Bell that rigorously maps the quantum evolution onto an ensemble of stochastic trajectories over a classical state space. Detailed mechanism identification is illustrated with a model 7-level system. A general procedure is presented to extract mechanism information directly from closed-loop control experiments. Application to simulated experimental data for the model system proves robust with up to 25% noise.Comment: Latex, 20 pages, 13 figure