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

该项目将为航空航天工程中使用的重要动力系统的控制提供数学方法。控制将被建模为力函数，它表示可以应用于动力学的可接受的力。重点将是产生控制公式，以确保满足包含重要输入或状态约束的动态系统所需的规定控制目标。输入约束是对可以应用于系统的力的限制，例如飞行器的最大允许推力。状态约束是对系统允许状态的限制，这可能源于避障要求或将飞行器的俯仰或侧滚保持在安全范围内的需要。对于系统状态或环境信息不完全的重大工程应用，或者在实现规定的控制目标时存在最后期限或延迟的情况下，这项研究都是适用的。数学模型中的延迟对于对动力系统施加力的延迟效应进行建模是重要的。潜在的应用包括将航空航天工程方法用于国防、搜索和救援，或者用于无人机，这些无人机可以在发展中国家运送医疗物资，这些国家可能还没有配备完善的跑道。这些方法将在数值模拟中进行测试，使用四旋翼和固定翼飞机的数学模型，然后在航空航天工程实验室和户外使用飞行平台进行实时测试。除了产生能够提高航空航天系统在国防或其他重要领域的性能的基础知识外，该项目还将在航空航天工程和数学的交汇点上培训博士生。这将有助于增加美国应用数学家的数量，他们可以与工程师合作，使用最先进的数学技术来帮助解决重要的社会问题。该项目将实施三项研究战略。一种策略是开发区间承包商，这是一种迭代技术，用于在关于动态当前状态的不完全信息下估计不确定动力系统的解，重点关注状态和输入测量的不确定性如何影响区间的收缩，以及在有限时间内区间收缩到围绕未知时变函数的足够小的管的条件。这一战略将建立在研究人员的初步结果的基础上，这些结果表明，在航空航天模型中，当间隔承包商与状态扩充方法和凸性参数相结合，将多项式状态约束转换为更容易处理的线性约束时，可以实现显著的跟踪改进。第二种策略将开发有限时间极值搜索方法，这种方法使用采样或延迟的测量来识别不确定函数的极值，并且识别过程使用区间承包商，因此与假设目标函数已知的标准优化方法有很大不同。第三种策略将是稳健的参考调节器，这是控制系统的附加方案，可以防止输入或状态违规，同时保持在没有输入或状态约束的情况下设计的控制的性能。这一策略将侧重于在连续和离散时间中的未知延迟以及可以模拟图像处理效果的扰动采样测量的情况下，使用区间承包商来确保有限时间收敛。该项目将证明为方法提供充分条件以满足规定的收敛性质的充分条件的定理，以期获得收敛的速度，这与工程界主要是实验的研究有很大的不同。这些申请将包括减少推进剂晃动、自动安全着陆和寻找来源。这一裁决反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.