Itô–Volterra Optimal State Estimation With Continuous, Multirate, Randomly Sampled, and Delayed Measurements

Abstract

The optimal filter for continuous, linear, stochastic, time-varying systems described by the Itô-Volterra equations with discontinuous measure is derived. With an appropriately selected measure, the result is applicable to a wide range of observation processes, including the hybrid case of observations formed by an arbitrary combination of continuous and discrete measurements, which may be sampled with a priori unknown, changing, and, possibly, random rates and delays. The simultaneous presence of continuous and sampled measurements causes impulsive discontinuity in the inputs of the optimal filter equations, which leads to a discontinuous change in state estimates every time a sampled measurement becomes available. Using the theory of vibrosolutions, the explicit and unique expressions for the jumps in state estimates and estimation error covariance are derived. Several examples illustrate the procedure of modeling hybrid measurement systems by selecting an appropriate discontinuous measure. We further show that the Itô-Volterra model and the main result of the paper can be specialized to several important cases, including state space systems, for which we recover several known state estimation results, and derive a novel optimal filter for continuous LTV systems with an arbitrary combination of continuous and delayed sampled measurements. This optimal filter updates the state estimates for incoming measurements as soon as they become available and does not require prior knowledge of sampling instants and delays, which makes it applicable when deterministic and random changes in sampling and delays are present. Several computational examples demonstrate the implementation of the developed filter and compare its performance to the traditional alternatives using MonteCarlo simulations.Index Terms-Continuous and sampled measurements, delayed measurements, Itô-Volterra systems, multirate and random sampling, optimal state estimation.

Non‐fragile H ∞ control with randomly occurring gain variations, distributed delays and channel fadings

“…In this section, by resorting to the Lyapunov functional method and the stochastic analysis technique, sufficient conditions are provided to guarantee the stability and the H ∞ performance for the closed-loop systems (6).…”

“…Therefore, the system (6) is stochastically stable. Let us now move to the analysis of the H ∞ performance for the system (6). For this purpose, we establish a cost function…”

“…Therefore, some network-induced phenomena have inevitably emerged in the areas of signal processing and control engineering, of which the most popular one is the communication delays that have attracted a great deal of research effort for the control/filtering problems of NCSs, see e.g. [1][2][3][4][5][6][7][8][9][10][11]. Among various types of time-delays, the distributed delays have recently drawn a growing research interest because of its engineering significance [4,12,13], where most corresponding results have been concerned with continuous-time systems with continuously distributed delays described by either a finite or infinite integral.…”

Target Tracking of a Linear Time Invariant System under Irregular Sampling

“…Left multiply For the process model (8) and measurement model (4) with the system parameter (9), we use the Kalman filter to get the estimate of system state at every sampling time as follows:…”

ean-Square Filtering for Polynomial System States Confused with Poisson Noises over Polynomial Observations

“…The complete classification of the "general situation" cases (this means that there are no special assumptions on the structure of state and observation equations and the initial conditions), where the nonlinear finite-dimensional filter exists, is given in Yau (1994). There also exists an extensive bibliography on robust, in particular, H ∞ filtering for linear (Xu and Chen (2003), Mahmoud and Shi (2003) and Xu et al (2005)) and nonlinear (Xie et al (1996), Nguang and Fu (1996), Fridman and Shaked (1997), Shi (1998), Fleming and McEneaney (2001), Yaz and Yaz (2001), Xu and van Dooren (2002), Wang et al (2003), Gao and Wang (2004), Zhang et al (2005), Gao et al (2005), Zhang et al (2007), Gao and Chen (2007), Wang et al (2008), Wang et al (2009), Wei et al (2009) and Shen et al (2009)) stochas-tic systems. Apart form the "general situation," the mean-square finite-dimensional filters have been designed for certain classes of polynomial system states with Gaussian noises over linear observations (Basin (2008), Basin et al (2008) and Basin et al (2009)) and a few results related to nonlinear Poisson systems can be found in Lu et al (2001), Kolmanovsky and Maizenberg (2002a), Hannequin and Mas (2002), Kolmanovsky and Maizenberg (2002b), Zhang et al (2008a), Dupé et al (2008), Zhang et al (2008b), and Basin and Maldonado (2011).…”