Linear Optimal Estimation for Discrete-time and Continuous-time Systems with Multiple Measurement Delays
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ISSN:1598-6446 eISSN:2005-4092 http://www.springer.com/12555
Linear Optimal Estimation for Discrete-time and Continuous-time Systems with Multiple Measurement Delays Na-Na Jin, Shuai Liu*, and Huan-Shui Zhang Abstract: 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. Keywords: Linear optimal estimation, multiple delay systems, Riccati difference equation, Riccati differential equation.
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INTRODUCTION
With the development of society, there are many problems to be estimated in reality. The existence of noise and interference makes the research of estimation method more important, and also makes the estimation problem more complex. For example, in the communication problem, due to the existence of noise such as atmosphere and line, in order to realize the effective transmission of signal, an important problem of the communication system is to separate the useful signal, that is to get the state estimation through the measurement information [1, 2]. In addition, the navigation problem is also one of the fields where the optimal estimation is widely used [3,4]. At the same time, for the problem of statistical filtering of random signals, in the presence of system noise and observation noise, other filters need to be constructed to obtain the effective estimation of the system state [5, 6]. The phenomenon of time delay exists widely in practice, and the element of time delay is an important factor affecting the dynamic characteristics of a system, that is, when the development trend of things is not only related to the current state, but also depends on the state of the past time, this is the phenomenon of time delay. Time-delay systems exist in many engineering fields such as transportation, communication, process engineering and more recently networked control systems. In recent years, time-
delay systems have attracted recurring interests from research community. Linear optimal estimation of systems with delays has been investigated since 1960s [7, 8]. Due to inevitability of time delays, the effects of time delay on systems should be taken into account when designing estimators. There has been many researches dedicated to linear optimal estimation problems with time-delay. For example, an infinite horizon
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