We present an extremum seeking (ES)-based robust observer design for thermal-fluid systems, pursuing an application to efficient energy management in buildings. The model is originally described by Boussinesq equations which is given by a system of two coupled partial differential equations (PDEs) for the velocity field and temperature profile constrained to incompressible flow. Using proper orthogonal decomposition, the PDEs are reduced to a set of nonlinear ordinary differential equations. Given a set of temperature and velocity point measurements, a nonlinear state observer is designed to reconstruct the entire state under the error of initial states, and model parametric uncertainties. We prove that the closed loop system for the observer error state satisfies an estimate of L₂ norm in a sense of locally input-to-state stability with respect to parameter uncertainties. Moreover, the uncertain parameters estimate used in the designed observer are optimized through iterations of a data-driven ES algorithm. Numerical simulation of a two-dimensional Boussinesq PDE illustrates the performance of the proposed adaptive estimation method.

中文翻译：

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基于极值搜索的热流体耦合系统降阶模型观测器设计

我们提出了一种用于热流体系统的基于极值寻找 (ES) 的稳健观测器设计，以寻求在建筑物中进行高效能源管理的应用。该模型最初由 Boussinesq 方程描述，该方程由两个耦合偏微分方程 (PDE) 的系统给出，用于速度场和温度分布，约束为不可压缩流。使用适当的正交分解，将偏微分方程简化为一组非线性常微分方程。给定一组温度和速度点测量值，设计非线性状态观测器以在初始状态误差和模型参数不确定性下重建整个状态。我们证明观察者误差状态的闭环系统满足L ₂的估计相对于参数不确定性的局部输入到状态稳定性意义上的范数。此外，设计观测器中使用的不确定参数估计通过数据驱动的 ES 算法的迭代进行了优化。二维 Boussinesq PDE 的数值模拟说明了所提出的自适应估计方法的性能。