A filtering algorithm based on the unscented transformation is proposed to estimate the state of a nonlinear system from noisy measurements which can be randomly delayed by one sampling time. The state and observation noises are perturbed by correlated nonadditive noises, and the delay is modeled by independent Bernoulli random variables.

The signal estimation problem in time-delay stochastic systems plays an important role in different application fields. For example, in engineering applications involving communication networks with a heavy network traffic, the measurements available may not be uptodate. Although the delay can sometimes be interpreted as a known deterministic function of the time, the numerous sources of uncertainty make it preferable to interpret it as a stochastic process, including its statistical properties in the system model. This fact must be considered in the study of the signal estimation problem since the conventional algorithms are then not applicable.

In the past few years, attention has been focused on investigating estimation problems from measurements subject to a random delay which does not exceed one sampling time, modeling the delay values by a zero-one white noise with known probabilities indicating that the measurements either arrive on time or are delayed. In linear systems, Ray et al. [

In this paper, we address the problem of estimating from nonlinear measurements subject to a random delay which does not exceed one sampling time, and when the last available measurement is used for the estimation at any time. This situation is modelled by considering Bernoulli random variables whose value of one indicates that the corresponding observation is not updated. Concretely, we propose an extension of the unscented filter in [

In this section, we present the nonlinear systems with one-step randomly delayed observations to be considered and we describe the assumption about the underlying processes.

The considered nonlinear discrete-time model is represented by the equations:

We assume that at time

In applications of communication networks, the noise

To deal with the state estimation problem, the following assumptions about the processes involved in (

The initial state in (

The noises

The initial state,

The unscented transformation (see [

On the basis of this procedure, unscented filtering uses the state equation (which provides

For the update, the mean and covariance of

From the independence of the vectors

Since

As previously commented, the obtaining of

Taking into account (

As in the prediction step, the conditional statistics of

However, to approximate the statistics of

The conditional statistics of

The initial conditions of the proposed algorithm are given by

Summarizing, given

Compute the sigma-points defined from

compute

compute

Compute the sigma-points defined from

Compute

Compute

Compute

Finally, by extracting the first block components of

To illustrate the performance of the proposed unscented filter, we consider the following logistic type of transition and measurement equations, used previously in [

To apply the proposed algorithm, we assume that the observations available for the estimation can be randomly delayed by one sampling period; that is,

We have implemented a MATLAB program which simulates the state,

Considering 1000 independent simulations and denoting by

Let us first examine the performance of the algorithm with respect to different values of

Moreover, in order to compare the performance of the estimators as a function of the delay probability

Mean of

Finally, to compare the performance of the proposed and the EKF algorithms, the latter was applied to the observation data of the simulation example for different values of

Mean of RMSE_{k}

This work has been partially supported by the