Collaborative Research: Designs and Theory for Interval Contractors and Reference Governors with Aerospace Applications

This project will produce mathematical methods for the control of important classes of dynamical systems that are used in aerospace engineering. The controls will be modeled as forcing functions, which represent the admissible forces that can be applied to the dynamics. The focus will be on producing formulas for controls that ensure that desirable prescribed control objectives are met for dynamical systems that contain significant input or state constraints. Input constraints are restrictions on forces that can be applied to the systems, such as maximum allowable thrusts for an aerial vehicle. State constraints are restrictions on the allowable states of the systems, which can arise from obstacle avoidance requirements or the need to keep the pitch or roll of an aerial vehicle in safe ranges. The research is amenable to significant engineering applications with either incomplete information about the systems' states or surroundings, or where there are time deadlines or latencies when achieving prescribed control goals. Latencies in mathematical models are important for modeling the delayed effects of applying forces to dynamical systems. The potential applications include using aerospace engineering methods for national defense, for search and rescue, or for drones that can deliver medical supplies in developing countries that may not yet have well equipped runways. The methods will be tested in numerical simulations, using mathematical models of quadrotors and fixed wing aircraft, and then in real time in an aerospace engineering lab and outdoors using flying platforms. In addition to generating fundamental knowledge that can enhance the performance of aerospace systems in national defense or other significant domains, the project will train PhD students at the interface of aerospace engineering and mathematics. This will help increase the population of US applied mathematicians who can collaborate with engineers and who can use state-of-the-art mathematical techniques to help solve important societal problems. The project will pursue three research strategies. One strategy will develop interval contractors, which are iterative techniques for estimating solutions of uncertain dynamical systems under incomplete information about the current state of the dynamics, focusing on how uncertainty in state and input measurements affect the contraction of the intervals and deriving conditions under which the intervals contract to small enough tubes around unknown time-varying functions in finite time. This strategy will build on the investigators' preliminary results that illustrate significant improvement in tracking in aerospace models that is achievable when interval contractors are used in conjunction with state augmentation methods and convexity arguments to transform polynomial state constraints into more easily handled linear constraints. A second strategy will develop finite time extremum seeking methods, which identify extrema of uncertain functions using sampled or delayed measurements and where the identification process uses interval contractors, and which are therefore a significant departure from standard optimization methods where the objective function is assumed to be known. A third strategy will be robust reference governors, which are add-on schemes for control systems that prevent input or state violations while maintaining the performance of controls that were designed in the absence of input or state constraints. This strategy will focus on using interval contractors to ensure finite time convergence, under unknown delays in continuous and discrete time and perturbed sampled measurements that can model the effects of image processing. The project would prove theorems that provide sufficient conditions for the methods to satisfy prescribed convergence properties with a view towards findings rates of convergence, which is a significant departure from research in the engineering community that was mainly experimental. The applications will include propellant slosh mitigation, automatic safe landing, and source seeking. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.