Georgia Institute of Technology College of Sciences

Adaptive control problems arise in many engineering applications in which one needs to design feedback controllers that ensure tracking of desired reference trajectories while at the same time identify unknown parameters such as control gains. This talk will summarize the speaker's work on adaptive tracking and parameter identification, including an application to curve tracking problems in robotics. The talk will be understandable to those familiar with the basic theory of ordinary differential equations. No prerequisite background in systems and control will be needed to understand and appreciate this talk.

This talk was cancelled due to the current status. The following is the original abstract for the talk. The celebrated Brill-Noether theorem says that the space of degree $d$ maps of a general genus $g$ curve to $\mathbb{P}^r$ is irreducible. However, for special curves, this need not be the case. Indeed, for general $k$-gonal curves (degree $k$ covers of $\mathbb{P}^1$), this space of maps can have many components, of different dimensions (Coppens-Martens, Pflueger, Jensen-Ranganathan). In this talk, I will introduce a natural refinement of Brill-Noether loci for curves with a distinguished map $C \rightarrow \mathbb{P}^1$, using the splitting type of push forwards of line bundles to $\mathbb{P}^1$. In particular, studying this refinement determines the dimensions of all irreducible components of Brill-Noether loci of general $k$-gonal curves.

TBD

One of the most challenging aspects of designing human sensitive systems is in designing systems that assist decision makers in applying an effective intervention to a large group of individuals. This design challenge becomes especially difficult when the decision maker must operate under scarce resources and only partial knowledge of how each individual will react to the intervention.

In this talk, I will consider this problem from the perspective of a clinician that is designing a personalized weight loss program. Despite this focus, the precision analytics framework I propose for designing these interventions is quite general and can apply to many settings where a single coordinator must influence agents who make decisions by maximizing utility functions that depend on prior system states, inputs, and other parameters that are initially unknown. This precision analytics framework involves three steps: first, a predictive model that effectively captures the decision-making process of an agent; second, an optimization algorithm to estimate this model’s parameters for each agent and predict their future decisions; and third, an optimization model that uses these predictions to optimize a set of incentives that will be provided to each agent. A key advantage of this framework is that the calculated incentives are adapted as new information is collected. In the case of personalized weight loss interventions, this means that the framework can leverage patient level data from mobile and wearable sensors over the course of the intervention to personalize the recommended treatment for each individual.

  I will present theoretical results that show that the incentives computed by this approach are asymptotically optimal with respect to a loss function that describes the coordinator's objective.  I will also present an effective decomposition scheme to optimize the agent incentives, where each sub-problem solves the coordinator's problem for a single agent, and the master problem is a pure integer program. To validate this method I will present a numerical study that shows this proposed framework is more cost efficient and clinically effective than simple heuristics in a simulated environment. I will conclude by discussing the results of a randomized control trial (RCT) and pilot study where this precision analytics framework was applied for personalizing exercise goals for UC Berkeley staff and students. The results of these trials show that using personalized step goals calculated by the precision analytics algorithm result in a significant improvement over existing state of the art approaches in a real world setting.

TBA

The motion of a forced vibro-impacting inclined energy harvester is investigated in parameter regimes with asymmetry in the number of impacts on the bottom and top of the device. This motion occurs beyond a grazing bifurcation, at which alternating top and bottom impacts are supplemented by a zero velocity impact with the bottom of the device. For periodic forcing, we obtain semi-analytical expressions for the asymmetric periodic motion with a ratio of 2:1 for the impacts on the device bottom and top, respectively. These expressions are derived via a set of nonlinear maps between different pairs of impacts, combined with impact conditions that provide jump dis continuities in the velocity. Bifurcation diagrams for the analytical solutions are complemented by a linear stability analysis around the 2:1 asymmetric periodic solutions, and are validated numerically. For smaller incline angles, a second grazing bifurcation is numerically detected, leading to a 3:1 asymmetry. For larger incline angles, period doubling bifurcations precede this bifurcation. The converted electrical energy per impact is reduced for the asymmetric motions, and therefore less desirable under this metric.

We consider rooted subgraphs in random graphs, i.e., extension counts such as (i) the number of triangles containing a given vertex or (ii) the number of paths of length three connecting two given vertices. 
In 1989, Joel Spencer gave sufficient conditions for the event that, with high probability, these extension counts are asymptotically equal for all choices of the root vertices.  
For the important strictly balanced case, Spencer also raised the fundamental question whether these conditions are necessary. 
We answer this question by a careful second moment argument, and discuss some intriguing problems that remain open.