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

合作研究：延迟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用于模拟许多重要的工程系统，例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，以及行动障碍患者的康复机制。然而，许多此类工程应用涉及输入延迟、状态约束和不确定性，这可能使它们超出现有控制器设计和理论的范围。当控制目标包括避免与障碍物碰撞等不良情况时，状态约束就会出现，而输入延迟通常是由传感器设计或传输现象引起的，这使得测量系统的当前状态变得困难。此外，在许多工程应用中，控制机制必须是自治的。确保自主性的一项重要技术是使用反馈控制，这意味着控制值必须从系统状态的过去值确定。该项目将开发最前沿的反馈控制设计和理论，以帮助解决上述挑战，并将在实时实验中演示这些技术。一项关键技术将涉及预测，它提供了一种方法，使用过去的观察从控制系统来计算未来的动态状态和未来的控制值，即使当系统涉及长输入延迟或相当大的不确定性。正如在pi的背景中所反映的那样，该项目将富有洞察力的工程与复杂的数学相结合，其目标是产生在延迟或状态约束下具有严格性能保证的实际有用的控制。要解决的问题是控制工程界最具挑战性和最重要的问题之一。该项目将力求变革的方法，并将追求三个理论策略。第一部分将寻求偏微分方程的广义Lyapunov函数构造，它可以包括将当前为常微分方程构建严格Lyapunov函数的方法扩展到更困难的双曲型偏微分方程。第二种策略将涉及将鲁棒预测控制表示为积分延迟方程的解，以及用扰动一阶双曲型偏微分方程表示的对偶表示。结合Lyapunov函数构造，可以为预测控制的非线性常微分方程提供鲁棒跟踪，并为相应的偏微分方程提供鲁棒结果。第三种策略将使用鲁棒前向不变性，其中包括指定状态约束，以方便计算最大允许摄动集，以确保在不确定情况下的安全运行。该项目将以前沿工程应用为指导，以帮助确保所有项目结果的实用性。普通微分方程应用将涉及神经肌肉电刺激，这是一种康复方法，可以帮助运动神经元紊乱的人恢复运动，以及控制一类自主海洋机器人，用于深海测量或监测水质。偏微分方程应用将包括用于制造平板显示器或光刻的激光脉冲整形系统，以及有助于减轻石油生产中段塞效应的多相流系统。

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.