We present a Newton-based extremum seeking algorithm for maximizing higher derivatives of unknown maps in the presence of time-varying delays. Dealing with time-varying delays has impact in the predictor design in terms of the transport PDE with variable convection speed functions, the backstepping transformation as well as the conditions imposed on the delay. First, the delay can grow at a rate strictly smaller than one but not indefinitely, the delay must remain uniformly bounded. Second, the delay may decrease at any uniformly bounded rate but not indefinitely, that is, it must remain positive. We incorporate a filtered predictor feedback with a perturbation-based estimate for the Hessian's inverse using a differential Riccati equation, where the convergence rate of the real-time optimizer can be made user-assignable, rather than being dependent on the unknown Hessian of the higher-derivative map. Furthermore, exponential stability and convergence to a small neighborhood of the unknown extremum point are achieved for locally quadratic derivatives by using backstepping transformation and averaging theory in infinite dimensions.

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具有时变延迟的高导数映射的基于牛顿的极值搜索

我们提出了一种基于牛顿的极值搜索算法，用于在存在时变延迟的情况下最大化未知地图的更高导数。在具有可变对流速度函数的传输偏微分方程、反步变换以及对延迟施加的条件方面，处理时变延迟对预测器设计有影响。首先，延迟可以以严格小于 1 的速率增长，但不能无限增长，延迟必须保持一致有界。其次，延迟可以以任何均匀有界的速率减少，但不能无限期地减少，也就是说，它必须保持正值。我们使用微分 Riccati 方程将过滤的预测器反馈与基于扰动的 Hessian 逆估计相结合，其中实时优化器的收敛速度可以由用户分配，而不是依赖于更高导数映射的未知 Hessian。此外，通过在无限维度上使用反步变换和平均理论，局部二次导数实现了指数稳定性和向未知极值点的小邻域的收敛。