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

该项目将产生用于航空航天工程中使用的动力系统的重要类别的控制的数学方法。控制将被建模为强制功能，这代表了可接受的力量，可以施加到动态。重点将是生产公式的控制，确保所需的规定的控制目标得到满足的动态系统，包含显着的输入或状态约束。输入约束是对可以施加到系统的力的限制，例如飞行器的最大允许推力。状态约束是对系统的允许状态的限制，其可以由避障要求或将飞行器的俯仰或滚转保持在安全范围内的需要引起。该研究适合于重要的工程应用，无论是不完整的信息系统的状态或环境，或有时间期限或延迟时，实现规定的控制目标。数学模型中的时滞对于建模施加力到动力系统的延迟效应是重要的。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，或用于无人机，可以在可能尚未拥有装备良好的跑道的发展中国家提供医疗用品。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真实的时间测试。除了生成可以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，该项目还将培养航空航天工程和数学接口的博士生。这将有助于增加美国应用数学家的人口，他们可以与工程师合作，并可以使用最先进的数学技术来帮助解决重要的社会问题。一种策略将开发区间承包商，这是一种迭代技术，用于在动态当前状态的不完整信息下估计不确定动态系统的解，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，并推导出区间在有限时间内围绕未知时变函数收缩到足够小的管的条件。该策略将建立在研究人员的初步结果的基础上，这些结果说明了在航空航天模型中跟踪的显着改善，当区间承包商与状态增强方法和凸性参数结合使用时，可以将多项式状态约束转换为更容易处理的线性约束。第二种策略将开发有限时间极值搜索方法，该方法使用采样或延迟测量来识别不确定函数的极值，并且其中识别过程使用区间承包商，并且因此与目标函数被假定为已知的标准优化方法显著偏离。第三种策略是鲁棒的参考调节器，它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性能。这一战略将侧重于使用区间承包商，以确保有限的时间收敛，在连续和离散时间和扰动采样测量，可以模拟图像处理的效果未知的延迟。该项目将证明为各种方法满足规定的收敛性质提供充分条件的定理，以期发现收敛速度，这与工程界主要是实验性的研究有很大的不同。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值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.