Distributed Extremum-Seeking Control with Applications to High-Altitude Balloons

The real-time optimization of large-scale systems is a difficult problem due to the need for complex models involving uncertain parameters and the high computational cost of solving such problems by a decentralized approach. Extremum-seeking control (ESC) is a model-free real-time optimization technique which can estimate unknown parameters and can optimize nonlinear time-varying systems using only a measurement of the cost function to be minimized. In this thesis, we develop a distributed version of extremum-seeking control which allows large-scale systems to be optimized without models and with minimal computing power. First, we develop a continuous-time distributed extremum-seeking controller. It has three main components: consensus, parameter estimation, and optimization. The consensus provides each local controller with an estimate of the cost to be minimized, allowing them to coordinate their actions. Using this cost estimate, parameters for a local input-output model are estimated, and the cost is minimized by following a gradient descent based on the estimate of the gradient. Next, a similar distributed extremum-seeking controller is developed in discrete-time. Finally, we consider an interesting application of distributed ESC: formation control of high-altitude balloons for high-speed wireless internet. These balloons must be steered into a favourable formation where they are spread out over the Earth and provide coverage to the entire planet. Distributed ESC is applied to this problem, and is shown to be effective for a system of 1200 ballons subjected to realistic wind currents. The approach does not require a wind model and uses a cost function based on a Voronoi partition of the sphere. Distributed ESC is able to steer balloons from a few initial launch sites into a formation which provides coverage to the entire Earth and can maintain a similar formation as the balloons move with the wind around the Earth.

Variation in offspring size towards the high elevational range edge of a montane annual

Understanding the ecological determinants of species’ distribution is a fundamental goal of ecology, and is increasingly important with changing limits to species’ range. Species often reach distributional limits on gradients of resource availability, but the extent to which offspring provisioning varies towards range limits is poorly understood. Selection is generally expected to favour higher provisioning of individual offspring in environments with short growing seasons and limited moisture, nutrients, or hosts for parasitism. However, individual provisioning may decline if parent size is limited by resources. This thesis focuses on three major questions: 1) does seed size vary over an elevational gradient? 2) does this variation respond adaptively towards the range limit? and 3) is potential elevational variation environmentally or genetically controlled? I tested variation in seed investment towards the upper elevational limit of the hemiparasitic annual herb Rhinanthus minor, sampled across an elevational range of 1,000m in the Rocky Mountains of Alberta, Canada. I also used a reciprocal transplant experiment to address the heritability of seed mass. Seed mass increased marginally towards higher elevations, while seed number and plant size declined. There was a strong elevational increase in seed mass scaled by overall plant size. Therefore, investment in individual seeds was higher towards the upper range edge, indicating potential adaptation of the reproductive strategy to allow for establishment in marginal environments. Genetic, environmental, and genotype-by-environment interactions were observed in transplanted populations, but the relative proportions of these effects on seed size were unclear.