Unscented Filtering from Delayed Observations with Correlated Noises

Abstract

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.

1. Introduction

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. [1] first modified the conventional algorithms to fit this observation model and since then many results based on this model have been reported (see, among others, [2–5] and references therein). Literature on nonlinear filtering from randomly delayed observations is less extensive. Recently, generalizations of extended and unscented Kalman filters, using one- and two-step randomly delayed observations, have been proposed and compared in [6, 7], respectively, for a class of nonlinear discrete-time systems with independent additive noises.

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 [6] to the case of correlated and nonadditive signal and measurement noises.

2. State and Observation Models

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: 𝑥_𝑘+1=𝑓_𝑘(𝑥_𝑘,𝑤_𝑘𝑦),𝑘≥0,_𝑘=ℎ_𝑘(𝑥_𝑘,𝑣_𝑘),𝑘≥1, (2.1) where 𝑥_𝑘 and 𝑦_𝑘 are random vectors which describe the system state and output at time 𝑘, respectively. The process {𝑤_𝑘;𝑘≥0} is the state noise, {𝑣_𝑘;𝑘≥1} is the measurement noise, and, for all 𝑘, 𝑓_𝑘 and ℎ_𝑘 are known analytic (not necessarily linear) functions.

We assume that at time 𝑘=1 the real observation 𝑦₁ is always available for the estimation but, as indicated previously, we go to consider the possibility that the current observation at any time 𝑘>1, 𝑦_𝑘, is either the current system output, 𝑦_𝑘, with probability of 1−𝑝_𝑘, or the previous one, 𝑦_𝑘−1, with probability of 𝑝_𝑘 (delay probability). Thus, the available observations for 𝑘>1 are 𝑦_𝑘=𝑦_𝑘−1,withprobability𝑝_𝑘𝑦_𝑘,withprobability1−𝑝_𝑘, (2.2) and the delayed observation model can be described as [6] 𝑦_𝑘=(1−𝛾_𝑘)𝑦_𝑘+𝛾_𝑘𝑦_𝑘−1,𝑘>1;𝑦₁=𝑦₁, (2.3) where {𝛾_𝑘;𝑘>1} are Bernoulli random variables (binary switching sequence taking the values 0 or 1) with 𝑃(𝛾_𝑘=1)=𝑝_𝑘, which model the delays in the observations. Indeed, if 𝛾_𝑘=1 (which occurs with probability 𝑝_𝑘), then 𝑦_𝑘=𝑦_𝑘−1 and the measurement is delayed by one sampling period; otherwise, 𝛾_𝑘=0 implies that 𝑦_𝑘=𝑦_𝑘 or, equivalently, that the measurement is updated (which occurs with probability 1−𝑝_𝑘).

In applications of communication networks, the noise {𝛾_𝑘;𝑘>1} usually represents the random delay from sensor to controller and the assumption of one-step sensor delay is based on the reasonable supposition that the induced data latency from the sensor to the controller is restricted so as not to exceed the sampling period.

To deal with the state estimation problem, the following assumptions about the processes involved in (2.1) and (2.3) are considered.

Assumption 1. The initial state in (2.1), 𝑥₀, is a random vector with 𝐸[𝑥₀]=𝑥₀ and 𝐸[(𝑥₀−𝑥₀)(𝑥₀−𝑥₀)^𝑇]=𝑃₀.

Assumption 2. The noises {𝑤_𝑘;𝑘≥0} and {𝑣_𝑘;𝑘≥1} are correlated zero-mean white processes with 𝐸[𝑤_𝑘𝑤^𝑇_𝑘]=𝑄_𝑘,𝐸[𝑣_𝑘𝑣^𝑇_𝑘]=𝑅_𝑘, and 𝐸[𝑤_𝑗𝑣_𝑘]=𝑆_𝑘𝛿_{𝑗,𝑘−1}, with 𝛿_{𝑘−1,𝑘−1}=1 and 𝛿_{𝑗,𝑘−1}=0,𝑗≠𝑘−1.

Assumption 3. {𝛾_𝑘;𝑘>1} is a sequence of independent Bernoulli random variables with known probabilities, 𝑃(𝛾_𝑘=1)=𝑝_𝑘, for all 𝑘>1.

Assumption 4. The initial state, 𝑥₀, and the processes ({𝑤_𝑘;𝑘≥0},{𝑣_𝑘;𝑘≥1}) and {𝛾_𝑘;𝑘>1} are mutually independent.

3. Unscented Filtering Algorithm

The unscented transformation (see [8] for details) approximates the distribution of a 𝑁-dimensional random vector 𝑋 by sample distributions with the same mean and covariance, 𝑋 and 𝑃^𝑋. The distributions correspond to a set of 2𝑁+1 sigma-points defined as 𝜒₀=𝜒𝑋,_𝑖=𝑋+((𝑁+𝜆)𝑃^𝑋)_𝑖𝜒,𝑖=1,…,𝑁,_𝑖=𝑋−((𝑁+𝜆)𝑃^𝑋)_𝑖−𝑁,𝑖=𝑁+1,…,2𝑁, (3.1) (expression (𝑃)_𝑖 denotes the 𝑖-th column of the matrix 𝑃) whose mean and covariance are 𝑋 and 𝑃^𝑋, respectively, when the following weights, 𝑊^𝑚_𝑖 for the mean and 𝑊^𝑐_𝑖 for the covariance, are used: 𝑊^𝑚₀=𝜆(𝑁+𝜆),𝑊^𝑐₀=𝜆𝑁+𝜆+(1−𝛼²𝑊+𝛽),_𝑚𝑖=𝑊^𝑐_𝑖=12(𝑁+𝜆),𝑖=1,…,2𝑁. (3.2) The parameters in (3.1) and (3.2) are 𝜆=𝛼²(𝑁+𝜅)−𝑁, where 𝛼 is a scaling parameter determining the spread of the sigma-points, and 𝜅, 𝛽 are tuning parameters. When the mean and covariance of a nonlinear transformation 𝑔(𝑋) are approximated by the sample mean and covariance of the transformed sigma-points, 𝑔(𝜒_𝑖),𝑖=0,…,2𝑁, weighted with 𝑊^𝑚_𝑖 and 𝑊^𝑐_𝑖, respectively, these approximations are accurate up to the second and first term of their Taylor expansion series, respectively.

On the basis of this procedure, unscented filtering uses the state equation (which provides 𝑥_𝑘 as a nonlinear function of 𝑥_𝑘−1 and 𝑤_𝑘−1) to approximate the conditional mean and covariance of 𝑥_𝑘 given 𝑌^𝑘−1=Col(𝑦₁,…,𝑦_𝑘−1) from those of 𝑥_𝑘−1 and 𝑤_𝑘−1. These statistics are then updated with the observation 𝑦_𝑘 using the Kalman equations to obtain approximations of the conditional mean and covariance of 𝑥_𝑘 given 𝑌^𝑘.

For the update, the mean and covariance of 𝑦_𝑘 given 𝑌^𝑘−1, and hence those of 𝑦_𝑘−1=ℎ_𝑘−1(𝑥_𝑘−1,𝑣_𝑘−1) and 𝑦_𝑘=ℎ_𝑘(𝑥_𝑘,𝑣_𝑘), are approximated using again the unscented procedure and, for this purpose, the conditional statistics of the vectors 𝑥_𝑘−1,𝑣_𝑘−1,𝑥_𝑘, and 𝑣_𝑘 must be known. Therefore, in view of the requirements in the prediction and update steps, the starting point for obtaining the filter of 𝑥_𝑘 is the knowledge of the approximated conditional mean and covariance of the vector 𝑋_𝑘−1=Col(𝑥_𝑘−1,𝑣_𝑘−1,𝑤_𝑘−1,𝑣_𝑘) given 𝑌^𝑘−1; these statistics, which are denoted by 𝑋_{𝑘−1/𝑘−1} and 𝑃^𝑋𝑋_{𝑘−1,𝑘−1/𝑘−1}, respectively, provide the approximations of the conditional statistics of 𝑋_𝑘 given 𝑌^𝑘 which, in turn, provide those of 𝑥_𝑘. The procedure is now detailed in the following two steps.

Prediction step
From the independence of the vectors Col(𝑣_𝑘,𝑤_𝑘,𝑣_𝑘+1) and 𝑌^𝑘−1 and the conditional independence of Col(𝑥_𝑘,𝑣_𝑘) and Col(𝑤_𝑘,𝑣_𝑘+1), the conditional mean and covariance of 𝑋_𝑘=Col(𝑥_𝑘,𝑣_𝑘,𝑤_𝑘,𝑣_𝑘+1) given 𝑌^𝑘−1 are given by 𝑋_{𝑘/𝑘−1}=𝑥_{𝑘/𝑘−1}000,𝑃^𝑋𝑋_{𝑘,𝑘/𝑘−1}=𝑃^𝑥𝑥_{𝑘,𝑘/𝑘−1}𝑃^𝑥𝑣_{𝑘,𝑘/𝑘−1}𝑃00_{𝑣𝑥𝑘,𝑘/𝑘−1}𝑅_𝑘0000𝑄_𝑘𝑆_𝑘+100𝑆^𝑇_𝑘+1𝑅_𝑘+1, (3.3) and then, the problem is to obtain approximations for the conditional mean and covariance of 𝑥_𝑘, 𝑥_{𝑘/𝑘−1} and 𝑃^𝑥𝑥_{𝑘,𝑘/𝑘−1}, as well as for the conditional cross-covariance of 𝑥_𝑘 and 𝑣_𝑘, 𝑃^𝑥𝑣_{𝑘,𝑘/𝑘−1}.
Since 𝑥_𝑘=𝑓_𝑘−1(𝑥_𝑘−1,𝑤_𝑘−1) and 𝑣_𝑘 are both functions of the vector 𝑋_𝑘−1, in order to approximate their conditional statistics we use the sigma-points 𝜒_{𝑖,𝑘−1}=Col(𝜒^𝑥_{𝑖,𝑘−1},𝜒^𝑣_{𝑖,𝑘−1},𝜒^𝑤_{𝑖,𝑘−1},𝜒^𝑣_𝑖,𝑘),𝑖=0,…,2𝑁, defined from 𝑋_{𝑘−1/𝑘−1} and 𝑃^𝑋𝑋_{𝑘−1,𝑘−1/𝑘−1} as in (3.1) (here 𝑁 is the dimension of the augmented vector 𝑋_𝑘−1), and the required statistics are approximated by those corresponding to the transformed sigma-points, 𝑓^𝑎_𝑘−1(𝜒_{𝑖,𝑘−1})=𝑓_𝑘−1(𝜒^𝑥_{𝑖,𝑘−1},𝜒^𝑤_{𝑖,𝑘−1}), by using the weights defined in (3.2): 𝑥_{𝑘/𝑘−1}=^2𝑁_𝑖=0𝑊^𝑚_𝑖𝑓^𝑎_𝑘−1(𝜒_{𝑖,𝑘−1}𝑃),_{𝑥𝑥𝑘,𝑘/𝑘−1}=^2𝑁_𝑖=0𝑊^𝑐_𝑖(𝑓^𝑎_𝑘−1(𝜒_{𝑖,𝑘−1})−𝑥_{𝑘/𝑘−1})(𝑓^𝑎_𝑘−1(𝜒_{𝑖,𝑘−1})−𝑥_{𝑘/𝑘−1})^𝑇,𝑃^𝑥𝑣_{𝑘,𝑘/𝑘−1}=^2𝑁_𝑖=0𝑊^𝑐_𝑖𝑓^𝑎_𝑘−1(𝜒_{𝑖,𝑘−1})𝜒^𝑣𝑇_𝑖,𝑘. (3.4)

Update step
As previously commented, the obtaining of 𝑋_𝑘/𝑘 and 𝑃^𝑋𝑋_{𝑘,𝑘/𝑘} from 𝑋_{𝑘/𝑘−1} and 𝑃^𝑋𝑋_{𝑘,𝑘/𝑘−1} is carried out by using the Kalman filter equations and hence, the mean and covariance of 𝑦_𝑘 given 𝑌^𝑘−1, as well as the conditional cross-covariance of 𝑦_𝑘 and 𝑋_𝑘, need to be approximated; next, we describe the approximation procedure.
Taking into account (2.3) and since, from the independence, 𝑃(𝛾_𝑘=1/𝑌^𝑘−1)=𝑝_𝑘, the conditional statistics of 𝑦_𝑘 given 𝑌^𝑘−1 are expressed in terms of those corresponding to 𝑦_𝑘−1 and 𝑦_𝑘 as follows: 𝑦_{𝑘/𝑘−1}=(1−𝑝_𝑘)𝑦_{𝑘/𝑘−1}+𝑝_𝑘𝑦_{𝑘−1/𝑘−1},𝑃^𝑦𝑦_{𝑘,𝑘/𝑘−1}=(1−𝑝_𝑘𝑦𝑦)𝑃_{𝑘,𝑘/𝑘−1}+𝑝_𝑘𝑃𝑦𝑦_{𝑘−1,𝑘−1/𝑘−1}+𝑝_𝑘(1−𝑝_𝑘)(𝑦_{𝑘/𝑘−1}−𝑦_{𝑘−1/𝑘−1})(𝑦_{𝑘/𝑘−1}−𝑦_{𝑘−1/𝑘−1})^𝑇,𝑃^𝑋𝑦_{𝑘,𝑘/𝑘−1}=(1−𝑝_𝑘)𝑃^𝑋_𝑘,𝑘𝑦_/𝑘−1+𝑝_𝑘𝑃^𝑋_𝑘,𝑘𝑦_{−1/𝑘−1}, (3.5) where again applying the independence hypotheses of the model, 𝑃^𝑋_𝑘,𝑘𝑦_/𝑘−1=𝑃^𝑥_𝑘,𝑦_{𝑘/𝑘−1}𝑃^𝑣_𝑘𝑦_{,𝑘/𝑘−1}00,𝑃^𝑋_𝑘,𝑘𝑦_{−1/𝑘−1}=𝑃^𝑥_𝑘,𝑦_{𝑘−1/𝑘−1}000. (3.6)

As in the prediction step, the conditional statistics of 𝑦_𝑘−1=ℎ_𝑘−1(𝑥_𝑘−1,𝑣_𝑘−1) are approximated from the sigma-points 𝜒_{𝑖,𝑘−1} associated with 𝑋_{𝑘−1/𝑘−1} and 𝑃^𝑋𝑋_{𝑘−1,𝑘−1/𝑘−1}, by defining ℎ^𝑎_𝑘−1(𝜒_{𝑖,𝑘−1})=ℎ_𝑘−1(𝜒^𝑥_{𝑖,𝑘−1},𝜒^𝑣_{𝑖,𝑘−1}):

𝑦_{𝑘−1/𝑘−1}=^2𝑁_𝑖=0𝑊^𝑚_𝑖ℎ^𝑎_𝑘−1(𝜒_{𝑖,𝑘−1}𝑃𝑦𝑦),_{𝑘−1,𝑘−1/𝑘−1}=^2𝑁_𝑖=0𝑊^𝑐_𝑖(ℎ^𝑎_𝑘−1(𝜒_{𝑖,𝑘−1})−𝑦_{𝑘−1/𝑘−1})(ℎ^𝑎_𝑘−1(𝜒_{𝑖,𝑘−1})−𝑦_{𝑘−1/𝑘−1})^𝑇,𝑃^𝑥_𝑘,𝑦_{𝑘−1/𝑘−1}=^2𝑁_𝑖=0𝑊^𝑐_𝑖(𝑓^𝑎_𝑘−1(𝜒_{𝑖,𝑘−1})−𝑥_{𝑘/𝑘−1})(ℎ^𝑎_𝑘−1(𝜒_{𝑖,𝑘−1})−𝑦_{𝑘−1/𝑘−1})^𝑇. (3.7)

However, to approximate the statistics of 𝑦_𝑘=ℎ_𝑘(𝑥_𝑘,𝑣_𝑘), which is a function of the vector Col(𝑥_𝑘,𝑣_𝑘), we use the information given in (3.3) and (3.4) about their conditional statistics. Thus, we consider a set of sigma-points, Col(𝜉^𝑥_𝑖,𝑘,𝜉^𝑣_𝑖,𝑘),𝑖=0,…,2𝑀 (𝑀 being the dimension of the vector Col(𝑥_𝑘,𝑣_𝑘)), defined in a similar way to those in (3.1) for the two first block components of 𝑋_{𝑘/𝑘−1} and 𝑃^𝑋𝑋_{𝑘,𝑘/𝑘−1}, with weights 𝑤^𝑚_𝑖 and 𝑤^𝑐_𝑖 defined as in (3.2), and the following approximations are used: 𝑦_{𝑘/𝑘−1}=^2𝑀_𝑖=0𝑤^𝑚_𝑖ℎ_𝑘(𝜉^𝑥_𝑖,𝑘,𝜉^𝑣_𝑖,𝑘𝑃𝑦𝑦),_{𝑘,𝑘/𝑘−1}=^2𝑀_𝑖=0𝑤^𝑐_𝑖(ℎ_𝑘(𝜉^𝑥_𝑖,𝑘,𝜉^𝑣_𝑖,𝑘)−𝑦_{𝑘/𝑘−1})(ℎ_𝑘(𝜉^𝑥_𝑖,𝑘,𝜉^𝑣_𝑖,𝑘)−𝑦_{𝑘/𝑘−1})^𝑇,𝑃^𝑥_𝑘,𝑦_{𝑘/𝑘−1}=^2𝑀_𝑖=0𝑤^𝑐_𝑖(𝜉^𝑥_𝑖,𝑘−𝑥_{𝑘/𝑘−1})(ℎ_𝑘(𝜉^𝑥_𝑖,𝑘,𝜉^𝑣_𝑖,𝑘)−𝑦_{𝑘/𝑘−1})^𝑇,𝑃^𝑣_𝑘𝑦_{,𝑘/𝑘−1}=^2𝑀_𝑖=0𝑤^𝑐_𝑖𝜉^𝑣_𝑖,𝑘(ℎ_𝑘(𝜉^𝑥_𝑖,𝑘,𝜉^𝑣_𝑖,𝑘))^𝑇. (3.8)

The conditional statistics of 𝑦_𝑘−1 and 𝑦_𝑘 are substituted in (3.5) and (3.6) to obtain those of 𝑦_𝑘, which are used in the following equations providing the filter of 𝑋_𝑘 and the error covariance: 𝑋_𝑘/𝑘=𝑋_{𝑘/𝑘−1}+𝑃^𝑋𝑦_{𝑘,𝑘/𝑘−1}(𝑃^𝑦𝑦_{𝑘,𝑘/𝑘−1})⁻¹(𝑦_𝑘−𝑦_{𝑘/𝑘−1}𝑃),_{𝑋𝑋𝑘,𝑘/𝑘}=𝑃^𝑋𝑋_{𝑘,𝑘/𝑘−1}−𝑃^𝑋𝑦_{𝑘,𝑘/𝑘−1}(𝑃^𝑦𝑦_{𝑘,𝑘/𝑘−1})⁻¹(𝑃^𝑋𝑦_{𝑘,𝑘/𝑘−1})^𝑇. (3.9)

The initial conditions of the proposed algorithm are given by 𝑋_0/0=𝑥₀000,𝑃^𝑋𝑋_0,0/0=𝑃₀000000000𝑄₀𝑆₁00𝑆^𝑇₁𝑅₁, (3.10) which is easily obtained from the independence hypotheses and initial conditions of the model.

Summarizing, given 𝑋_{𝑘−1/𝑘−1} and 𝑃^𝑋𝑋_{𝑘−1,𝑘−1/𝑘−1}, the computation procedure of the proposed unscented filter is as follows:

Step 1. Compute the sigma-points defined from 𝑋_{𝑘−1/𝑘−1} and 𝑃^𝑋𝑋_{𝑘−1,𝑘−1/𝑘−1} as in (3.1), and(i)compute 𝑥_{𝑘/𝑘−1}, 𝑃^𝑥𝑥_{𝑘,𝑘/𝑘−1}, and 𝑃^𝑥𝑣_{𝑘,𝑘/𝑘−1} by (3.4), and compute 𝑋_{𝑘/𝑘−1} and 𝑃^𝑋𝑋_{𝑘,𝑘/𝑘−1} by (3.3).(ii)compute 𝑦_{𝑘−1/𝑘−1}, 𝑃𝑦𝑦_{𝑘−1,𝑘−1/𝑘−1}, and 𝑃^𝑥_𝑘,𝑦_{𝑘−1/𝑘−1} by (3.7).

Step 2. Compute the sigma-points defined from 𝑋_{𝑘/𝑘−1} and 𝑃^𝑋𝑋_{𝑘,𝑘/𝑘−1} as in (3.1), and compute 𝑦_{𝑘/𝑘−1}, 𝑃𝑦𝑦_{𝑘,𝑘/𝑘−1}, 𝑃^𝑥_𝑘,𝑦_{𝑘/𝑘−1}, and 𝑃^𝑣_𝑘𝑦_{,𝑘/𝑘−1} by (3.8).

Step 3. Compute 𝑃^𝑋_𝑘,𝑘𝑦_/𝑘−1 and 𝑃^𝑋_𝑘,𝑘𝑦_{−1/𝑘−1} by (3.6).

Step 4. Compute 𝑦_{𝑘/𝑘−1}, 𝑃^𝑦𝑦_{𝑘,𝑘/𝑘−1} and 𝑃^𝑋𝑦_{𝑘,𝑘/𝑘−1} by (3.5).

Step 5. Compute 𝑋_𝑘/𝑘 and 𝑃^𝑋𝑋_{𝑘,𝑘/𝑘} by (3.9).

Finally, by extracting the first block components of 𝑋_𝑘/𝑘 and 𝑃^𝑋𝑋_{𝑘,𝑘/𝑘}, the filter of the original vector 𝑥_𝑘 and the error covariance are obtained.

4. Simulation Example

To illustrate the performance of the proposed unscented filter, we consider the following logistic type of transition and measurement equations, used previously in [9] to compare the performance of various nonlinear filters in the case of mutually independent noises and nondelayed observations: 𝑥_𝑘+1=exp(𝑥_𝑘)exp(𝑥_𝑘)+exp(𝑤_𝑘),𝑘≥0;𝑦_𝑘=exp(𝑥_𝑘)exp(𝑥_𝑘)+exp(𝑣_𝑘),𝑘≥1, (4.1) where the initial state 𝑥₀ is a random variable with uniform distribution between zero and one; the state and observation noises are assumed to be zero-mean Gaussian joint processes with 𝑄_𝑘=1, 𝑅_𝑘=1 and known 𝑆_𝑘=𝑆, for all 𝑘.

To apply the proposed algorithm, we assume that the observations available for the estimation can be randomly delayed by one sampling period; that is, 𝑦_𝑘=(1−𝛾_𝑘)𝑦_𝑘+𝛾_𝑘𝑦_𝑘−1,𝑘>1;𝑦₁=𝑦₁, (4.2) and that the noise {𝛾_𝑘;𝑘>1} modeling the delays is a sequence of independent Bernoulli variables with known delay probability 𝑃(𝛾_𝑘=1)=𝑝, for all 𝑘.

We have implemented a MATLAB program which simulates the state, 𝑥_𝑘, and the real, 𝑦_𝑘, and delayed measurements, 𝑦_𝑘, for 𝑘=1,…,50, for different values of 𝑆 and 𝑝, and which provides the unscented filtering estimates of 𝑥_𝑘. The root mean square error (RMSE) criterion has been used to quantify the performance of the estimates.

Considering 1000 independent simulations and denoting by {𝑥_𝑘^(𝑠),𝑘=1,…,50} the 𝑠th set of the artificially simulated states and by 𝑥^(𝑠)_𝑘/𝑘 the filtering estimate at time 𝑘 in the 𝑠th simulation run, the RMSE of the filter at time 𝑘 is calculated by RMSE_𝑘1=(1000¹⁰⁰⁰_𝑠=1(𝑥_𝑘^(𝑠)−𝑥^(𝑠)_𝑘/𝑘)²)^1/2. (4.3)

Let us first examine the performance of the algorithm with respect to different values of 𝑆; Figure 1 illustrates the RMSE_𝑘 when the delay probability is 𝑝=0.5 and different values of 𝑆 are considered; specifically, 𝑆=0,0.3,0.5,0.7 and 𝑆=0.9; this figure shows, as expected, that the higher the value of 𝑆 (which means that the correlation between the state and the observations increases) the smaller that of RMSE_𝑘 and, consequently, the performance of the estimators is better. Analogous results are obtained for other values of 𝑝 and 𝑆.

Figure 1 

RMSE𝑘 when 𝑝=0.5 and 𝑆=0,0.3,0.5,0.7,0.9.

Moreover, in order to compare the performance of the estimators as a function of the delay probability 𝑝, the arithmetic average corresponding to the 50 iterations of RMSE_𝑘 was calculated for 𝑝=0.1,…,0.9. The results of this are shown in Figure 2, from which it is apparent that the means increase as 𝑝 increases, the increase being greater when 𝑆 is greater, and consequently, as expected, the performance of the estimators deteriorates as the delay probability 𝑝 rises. From this figure, it is also inferred that, for each fixed value of 𝑝, the means decrease with increasing 𝑆, which extends the result in Figure 1 to different values of 𝑝.

Figure 2 

Mean of RMSE𝑘 when 𝑆=0,0.3,0.5,0.7,0.9, versus 𝑝.

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 𝑝 and 𝑆. The results show that the proposed algorithm outperforms the EKF algorithm and the improvement is greater when the delay probability 𝑝 is greater and, also, when the correlation 𝑆 increases. Table 1, showing the means of RMSE_𝑘 for both algorithms considering 𝑝=0.3,0.5,0.7,0.9 and 𝑆=0.7,0.9, illustrates this fact.

Table 1 

Mean of RMSE_k for the proposed unscented filter (UF) and the extended Kalman filter (EKF) when p =0.3,0.5,0.7,0.9 and 𝑆=0.7,0.9.

Acknowledgment

This work has been partially supported by the Ministerio de Ciencia e Innovación and the Junta de Andalucía through Projects MTM2008-05567 and P07-FQM-02701, respectively.