Resilient Submodular Maximization for Control and Sensing

Fundamental applications in control, sensing, and robotics, motivate the design of systems by selecting system elements, such as actuators or sensors, subject to constraints that require the elements not only to be a few in number, but also, to satisfy heterogeneity or interdependency constraints (called matroid constraints). For example, consider the scenarios: (1) (Control ) Actuator placement: In a power grid, how should we place a few generators both to guarantee its stabilization with minimal control effort, and to satisfy interdependency constraints where the power grid must be controllable from the generators? (2) (Sensing) Sensor placement: In medical brain-wearable devices, how should we place a few sensors to ensure smoothing estimation capabilities? (3) ( Robotics) Sensor scheduling: At a team of mobile robots, which few on-board sensors should we activate at each robot —subject to heterogeneity constraints on the number of sensors that each robot can activate at each time— so both to maximize the robots\u27 battery life, and to ensure the robots\u27 capability to complete a formation control task? In the first part of this thesis we motivate the above design problems, and propose the first algorithms to address them. In particular, although traditional approaches to matroid-constrained maximization have met great success in machine learning and facility location, they are unable to meet the aforementioned problem of actuator placement. In addition, although traditional approaches to sensor selection enable Kalman filtering capabilities, they do not enable smoothing or formation control capabilities, as required in the above problems of sensor placement and scheduling. Therefore, in the first part of the thesis we provide the first algorithms, and prove they achieve the following characteristics: provable approximation performance : the algorithms guarantee a solution close to the optimal; minimal running time: the algorithms terminate with the same running time as state-of-the-art algorithms for matroid-constrained maximization; adaptiveness: where applicable, at each time step the algorithms select system elements based on both the history of selections. We achieve the above ends by taking advantage of a submodular structure of in all aforementioned problems —submodularity is a diminishing property for set functions, parallel to convexity for continuous functions. But in failure-prone and adversarial environments, sensors and actuators can fail; sensors and actuators can get attacked. Thence, the traditional design paradigms over matroid-constraints become insufficient, and in contrast, resilient designs against attacks or failures become important. However, no approximation algorithms are known for their solution; relevantly, the problem of resilient maximization over matroid constraints is NP-hard. In the second part of this thesis we motivate the general problem of resilient maximization over matroid constraints, and propose the first algorithms to address it, to protect that way any design over matroid constraints, not only within the boundaries of control, sensing, and robotics, but also within machine learning, facility location, and matroid-constrained optimization in general. In particular, in the second part of this thesis we provide the first algorithms, and prove they achieve the following characteristics: resiliency: the algorithms are valid for any number of attacks or failures; adaptiveness: where applicable, at each time step the algorithms select system elements based on both the history of selections, and on the history of attacks or failures; provable approximation guarantees: the algorithms guarantee for any submodular or merely monotone function a solution close to the optimal; minimal running time: the algorithms terminate with the same running time as state-of-the-art algorithms for matroid-constrained maximization. We bound the performance of our algorithms by using notions of curvature for monotone (not necessarily submodular) set functions, which are established in the literature of submodular maximization. In the third and final part of this thesis we apply our tools for resilient maximization in robotics, and in particular, to the problem of active information gathering with mobile robots. This problem calls for the motion-design of a team of mobile robots so to enable the effective information gathering about a process of interest, to support, e.g., critical missions such as hazardous environmental monitoring, and search and rescue. Therefore, in the third part of this thesis we aim to protect such multi-robot information gathering tasks against attacks or failures that can result to the withdrawal of robots from the task. We conduct both numerical and hardware experiments in multi-robot multi-target tracking scenarios, and exemplify the benefits, as well as, the performance of our approach