Observers for PDEs are themselves PDEs. Therefore, producing real time estimates with such observers is computationally burdensome. For both finite-dimensional and ODE systems, moving-horizon estimators (MHE) are operators whose output is the state estimate, while their inputs are the initial state estimate at the beginning of the horizon as well as the measured output and input signals over the moving time horizon. In this paper we introduce MHEs for PDEs which remove the need for a numerical solution of an observer PDE in real time. We accomplish this using the PDE backstepping method which, for certain classes of both hyperbolic and parabolic PDEs, produces moving-horizon state estimates explicitly. Precisely, to explicitly produce the state estimates, we employ a backstepping transformation of a hard-to-solve observer PDE into a target observer PDE, which is explicitly solvable. The MHEs we propose are not new observer designs but simply the explicit MHE realizations, over a moving horizon of arbitrary length, of the existing backstepping observers. Our PDE MHEs lack the optimality of the MHEs that arose as duals of MPC, but they are given explicitly, even for PDEs. In the paper we provide explicit formulae for MHEs for both hyperbolic and parabolic PDEs, as well as simulation results that illustrate theoretically guaranteed convergence of the MHEs.

翻译：偏微分方程的状态观测器本身也是偏微分方程，因此使用这类观测器进行实时估计的计算负担较大。对于有限维系统和常微分方程系统，滚动时域估计器（MHE）是一种算子，其输出为状态估计值，输入为时域初始时刻的状态估计值以及滚动时域上的测量输出和输入信号。本文针对偏微分方程提出了滚动时域估计器，从而无需在实时计算中数值求解观测器偏微分方程。我们通过偏微分方程反步法实现了这一目标，该方法对于某些双曲型和抛物型偏微分方程，能够显式地给出滚动时域状态估计。具体而言，为了显式生成状态估计值，我们采用反步变换将难以求解的观测器偏微分方程转化为一个可显式求解的目标观测器偏微分方程。我们提出的滚动时域估计器并非全新的观测器设计，而是对现有反步观测器在任意长度滚动时域上的显式滚动时域实现。虽然本文的偏微分方程滚动时域估计器不具备与模型预测控制对偶的滚动时域估计器的最优性，但它们是显式给出的，即使对于偏微分方程也是如此。本文提供了双曲型和抛物型偏微分方程滚动时域估计器的显式公式，并给出了仿真结果，验证了滚动时域估计器理论上保证的收敛性。