<p style='text-indent:20px;'>The problem of estimating (reconstructing) an unknown input for a system of nonlinear differential equations with the Caputo fractional derivative is considered. Information on the position of the system is available for observations and only a part of system's parameters can be measured. The case of measuring all phase coordinates is also presented. The measurements are continuous and the data obtained in them are noisy. The considered problem is ill-posed and, to solve it, we use the method of dynamic inversion. It is based on regularization methods and constructions of positional control theory. In particular, we use the Tikhonov regularization method also known as the smoothing functional method and the Krasovskii extremal aiming method. The approach to estimating an unknown input implies introducing an auxiliary system (a model) with an appropriate rule of forming a control. The proposed estimation algorithm gives approximations of an unknown input and is stable under informational noises and computational errors. As an example illustrating the elaborated technique, a biological model of human immunodeficiency virus disease is used for simulation. The simulation results demonstrate the importance of the approach to on-line estimating unobservable parameters in real processes.</p>

中文翻译：

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具有 Caputo 分数导数的系统中噪声输入的动态估计。连续测量部分相位坐标的情况

<p style='text-indent:20px;'>考虑使用 Caputo 分数导数估计（重构）非线性微分方程系统的未知输入的问题。系统位置信息可用于观测，只能测量系统参数的一部分。还介绍了测量所有相位坐标的情况。测量是连续的，并且从中获得的数据是嘈杂的。考虑的问题是不适定的，为了解决它，我们使用动态反演的方法。它基于正则化方法和位置控制理论的构造。特别是，我们使用了 Tikhonov 正则化方法，也称为平滑函数方法和 Krasovskii 极值瞄准方法。估计未知输入的方法意味着引入具有适当规则的辅助系统（模型）来形成控制。所提出的估计算法给出了未知输入的近似值，并且在信息噪声和计算错误下是稳定的。作为说明详细技术的一个例子，人类免疫缺陷病毒疾病的生物模型被用于模拟。仿真结果证明了在线估计实际过程中不可观测参数的重要性。</p> 使用人类免疫缺陷病毒病的生物学模型进行模拟。仿真结果证明了在线估计实际过程中不可观测参数的重要性。</p> 使用人类免疫缺陷病毒病的生物学模型进行模拟。仿真结果证明了在线估计实际过程中不可观测参数的重要性。</p>