$H_{\infty }$ Fixed-Interval Smoothing Estimation for Time-Delay Systems

This paper is concerned with the H_∞ fixed-interval smoothing estimation for time-delay systems which include continuous-time case and discrete-time case. In the case of discrete-time systems, the problem can be solved by using the conventional state augmentation approach. However, this approach is not suitable for the continuous-time case. In this paper, we will propose a unified approach to study the H_∞ fixed-interval smoothing problem for both continuous-time and discrete-time systems with l output delays. By introducing a suitable stochastic linear time-delay models in an indefinite space, it is shown that the H_∞ fixed-interval smoother can be obtained by calculating H₂ fixed-interval smoother for time-delay systems in an indefinite space. Therefore, based on the orthogonal projection theory in an indefinite space, the H_∞ fixed-interval smoothers for time-delay systems are designed by performing l+1 Riccati equations with the same dimension as the original systems. Moreover, a necessary and sufficient conditions for the existence of the H_∞ fixed-interval smoother will also be given.