A variety of optimal control, estimation, system identification and design problems can be formulated as functional optimization problems with differential equality and inequality constraints. Since these problems are infinite-dimensional and often do not have a known analytical solution, one has to resort to numerical methods to compute an approximate solution. This paper uses a unifying notation to outline some of the techniques used in the transcription step of simultaneous direct methods (which discretize-then-optimize) for solving continuous-time dynamic optimization problems. We focus on collocation, integrated residual and Runge-Kutta schemes. These transcription methods are then applied to a simulation case study to answer a question that arose during the COVID-19 pandemic, namely: If there are not enough ventilators, is it possible to ventilate more than one patient on a single ventilator? The results suggest that it is possible, in principle, to estimate individual patient parameters sufficiently accurately, using a relatively small number of flow rate measurements, without needing to disconnect a patient from the system or needing more than one flow rate sensor. We also show that it is possible to ensure that two different patients can indeed receive their desired tidal volume, by modifying the resistance experienced by the air flow to each patient and controlling the ventilator pressure.

中文翻译：

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动态优化的直接转录：以COVID-19大流行期间双患者通气为例的教程

可以将各种最优控制，估计，系统识别和设计问题表述为具有差分等式和不等式约束的功能优化问题。由于这些问题是无穷大的，通常没有已知的解析解，因此必须诉诸于数值方法来计算近似解。本文使用统一的符号来概述用于解决连续时间动态优化问题的同时直接方法（先离散化然后优化）的转录步骤中使用的一些技术。我们专注于搭配，集成残差和Runge-Kutta方案。然后将这些转录方法应用于模拟案例研究，以回答在COVID-19大流行期间出现的一个问题，即：如果没有足够的呼吸机，在一台呼吸机上是否可以让多名患者通气？结果表明，原则上可以使用相对少量的流量测量值足够准确地估计单个患者参数，而无需将患者与系统断开连接或需要多个流量传感器。我们还表明，通过改变流向每个患者的气流所承受的阻力并控制呼吸机的压力，可以确保两个不同的患者确实能够接收到所需的潮气量。无需将患者与系统断开连接或需要多个流量传感器。我们还表明，通过改变流向每个患者的气流所承受的阻力并控制呼吸机的压力，可以确保两个不同的患者确实能够接收到所需的潮气量。无需将患者与系统断开连接或需要多个流量传感器。我们还表明，通过改变流向每个患者的气流所承受的阻力并控制呼吸机压力，可以确保两个不同的患者确实能够接收到所需的潮气量。