Collaborative Research: Designs and Theory of State-Constrained Nonlinear Feedback Controls for Delay and Partial Differential Equation Systems

合作研究：时滞和偏微分方程系统的状态约束非线性反馈控制的设计和理论控制系统被用来模拟许多重要的工程系统，如涉及激光的电子制造过程，可以监测水污染的海洋机器人，石油钻探和炼油，以及运动障碍患者的康复机制。然而，许多这些工程应用涉及输入延迟，状态约束和不确定性，可以把它们的范围之外的现有控制器的设计和理论。当控制目标包括避免与障碍物碰撞等不期望的情况时，会出现状态约束，而输入延迟通常来自传感器设计或传输现象，这些现象使得难以测量系统的当前状态。而且，在许多工程应用中，控制机制必须是自主的，这是固有的。 确保自主性的一个重要技术是使用反馈控制，这意味着控制值必须根据系统状态的过去值来确定。这个项目将开发尖端的反馈控制设计和理论，可以帮助解决前面的挑战，并将在真实的时间实验中演示的技术。一个关键技术将涉及预测，它提供了一种方法来使用控制系统的过去观测来计算未来的动态状态和未来的控制值，即使系统涉及长输入延迟或相当大的不确定性。正如PI的背景所反映的那样，该项目将富有洞察力的工程与复杂的数学相结合，目标是产生实际有用的控制，这些控制在延迟或状态约束下具有严格的性能保证。要解决的问题是在控制工程界最具挑战性和重要的。该项目将致力于变革性的方法，并将追求三个理论战略。第一个将寻求广义李雅普诺夫函数构造偏微分方程，其中可以包括扩展目前的方法建立严格的李雅普诺夫函数的常微分方程更困难的双曲型偏微分方程的情况下。第二种策略将涉及代表鲁棒预测控制的积分延迟方程的解决方案，并在扰动的一阶双曲型偏微分方程的双重表示。结合李雅普诺夫函数的构造，这可以提供预测控制的非线性常微分方程的鲁棒跟踪和相应的偏微分方程的鲁棒性结果。第三种策略将使用鲁棒的前向不变性，这涉及指定状态约束，以便于计算最大允许扰动集，以确保在不确定性下的安全操作。该项目将以尖端工程应用为指导，以帮助确保所有项目成果的实用性。常微分方程的应用将涉及神经肌肉电刺激，这是一种康复方法，可以帮助恢复运动神经元障碍的人的运动，以及用于水深测量或监测水质的一类自主海洋机器人的控制。偏微分方程的应用将涉及激光脉冲整形系统，用于制造平板显示器或光刻，以及多相流系统，可以帮助减轻在石油生产段塞流的不利影响。

Collaborative Research: Designs and Theory of State-Constrained Nonlinear Feedback Controls for Delay and Partial Differential Equation SystemsControl systems are used to model many important engineering systems, such as electronics manufacturing processes involving lasers, marine robots that can monitor water pollution, oil drilling and refining, and rehabilitation mechanisms for patients with mobility disorders. However, many of these engineering applications involve input delays, state constraints, and uncertainties that can put them outside the scope of existing controller designs and theory. State constraints occur when the control objectives include avoiding undesirable situations such as collisions with obstacles, while input delays often arise from sensor designs or transport phenomena that make it difficult to measure the current state of the system. Also, it is inherent in many engineering applications that the control mechanisms must be autonomous. One important technique for ensuring autonomy is by using feedback control, which means that the control values must be determined from past values of the state of the system. This project will develop cutting edge feedback control designs and theory that can help address the preceding challenges, and will demonstrate the techniques in real time experiments. One key technique will involve prediction, which provides a way to use past observations from the control system to compute future states of the dynamics and future control values, even when the system involves long input delays or considerable uncertainty. As reflected in the backgrounds of the PIs, the project combines insightful engineering with sophisticated mathematics, with the goal of producing practically useful controls that have rigorous performance guarantees under delays or state constraints. The problems to be addressed are among the most challenging and significant ones in the control engineering community. The project will strive for transformative methods, and will pursue three theoretical strategies. The first will seek generalized Lyapunov function constructions for partial differential equations, which can include extensions of current approach for building strict Lyapunov functions for ordinary differential equations to much more difficult hyperbolic partial differential equation cases. The second strategy will involve representing robust predictive controls as solutions of integral delay equations, and a dual representation in terms of perturbed first-order hyperbolic partial differential equations. Combined with the Lyapunov function constructions, this can provide robust tracking for predictively controlled nonlinear ordinary differential equations and robustness results for the corresponding partial differential equations. The third strategy will use robust forward invariance, which involves specifying the state constraints to facilitate computing maximal allowable perturbation sets to ensure safe operation under uncertainty. The project will be guided by cutting edge engineering applications, to help ensure the practical usefulness of all of the project results. The ordinary differential equation applications will involve neuromuscular electrical stimulation, which is a rehabilitation method that can help restore movement in humans with motor neuron disorders, and the control of a class of autonomous marine robots that are used for bathymetric surveys or to monitor water quality. The partial differential equation applications will involve laser pulse shaping systems that are used in the manufacture of flat panel displays or in photolithography, and a multi-phase flow system that can help mitigate the adverse effects of slugging in oil production.