In this paper, we investigate the linear optimal estimation problems of discrete-time and continuous-time systems with multiple state delays in measurements. For discrete-time systems, we obtain the linear optimal estimation of state by direct calculation of optimal gain in terms of the solution to a retarded Riccati-like difference equation instead of a group of Riccati difference equations. For continuous-time systems, we also obtain the analytical expression of linear optimal estimation without resorting to Riccati partial differential equations. All the Riccati equations are of the same dimension as the system to be estimated and the computational cost is much saved. Infinite horizon case is also studied by stability analysis. Kalman filter can be recovered from our result when delays disappear. A numerical example is provided to demonstrate the results.

中文翻译：

---

具有多个测量延迟的离散时间和连续时间系统的线性最优估计

在本文中，我们研究了在测量中具有多个状态延迟的离散时间和连续时间系统的线性最优估计问题。对于离散时间系统，我们通过根据延迟的类 Riccati 差分方程而不是一组 Riccati 差分方程的解直接计算最佳增益来获得状态的线性最优估计。对于连续时间系统，我们也无需求助于Riccati偏微分方程，就可以得到线性最优估计的解析表达式。所有Riccati方程与待估计系统的维数相同，大大节省了计算成本。还通过稳定性分析研究了无限视界情况。当延迟消失时，可以从我们的结果中恢复卡尔曼滤波器。