The objective of this research is to develop a comprehensive approach to risk adjusted robust control, starting with the use of experimental data to obtain and validate a plant description and ending with a closed loop system that has a prescribed risk of violating the performance specifications. Its conceptual backbone is a combination of operator theoretic and stochastic tools that emphasizes both computational complexity and practicality of the results. Intellectual Merit: The conventional robust control framework developed in the past two decades has proved to be very successful in addressing robustness and performance issues in systems subject to unstructured uncertainty. This is also true, to a lesser extent, in the case of structured uncertainty, provided that the number of uncertainty blocks is small and the plant under consideration has moderate size, with the limiting factor here being the computational and scaling properties of the resulting optimization problems. An additional limitation stems from the fact that these approaches take a worst case approach both to stability and performance. While a very strong case can be made for the former, in many cases an equally strong case can be made against the use of a worst case approach to performance, since it can lead to overly conservative systems. This proposal is motivated by the possibility (substantiated by the co PIs preliminary work) of addressing these issues through the use of a risk adjusted approach, where the designer is willing to trade off a preassigned risk of violating a performance specification in return for a reduction, often substantial, in both the complexity of the resulting controller and its conservatism. Advantages offered by the proposed framework over currently available techniques include the abilities to: a.- Systematically synthesize low complexity, practically implementable robust systems. b.- Lead to tractable problems, and in cases where the underlying problem is intrinsically hard, to provide for computationally tractable relaxations with risk adjusted certificates. Examples of these cases include model (in)validation under arbitrary uncertainty structures and fixer order controller synthesis, both beyond the ability of hitherto available methods. c.- Indicate the intrinsic limits of performance of the plant, making unavoidable design tradeoffs clear and allowing the control engineer to explicitly make these tradeoffs, without trial and error iterations, gracefully degrading performance when some of the requirements cannot be met. Broader Impact: In addition to advancing the current state of the art in control theory, the proposed research will bring closer to being practical several technologies currently at the proof of concept stage. Examples of these include active vision applications to aware environments and communication networks with improved robustness and quality of service characteristics. Arguably one of the critical factors preventing the deployment of these technologies beyond controlled lab environments is the lack of robustness of the resulting systems. Addressing this fragility is beyond the ability of currently available robust control techniques due to their poor computational and scaling properties. Educational Impact: In addition to benefiting graduate education, we plan to incorporate results from this research in an undergraduate introductory robust control course, that will expose students at an earlier stage to the issues of robustness and computational complexity. Typically this is done at the graduate level, and thus undergraduate students are unaware of these ideas, although it could be argued that they are some of most powerful and better developed assets that our community has, with a potential that extend beyond pure control. Risk adjusted ideas provide an ideal vehicle to accomplish this initial exposure, eliminating the need to wait until students build the background in functional analysis required to tackle graduate level robust control courses.
