**6. Conclusions and future research**

18 Will-be-set-by-IN-TECH

0.1 0.2 0.3 0.4 0.5 0.12

0.1 0.2 0.3 0.4 0.5 0.09

**Figure 3.** Filtering error variances and smoothing error variances for *N* = 2 of the first state component

0.1 0.2 0.3 0.4 0.5 0.1

γ 2 =0.1, γ 2 =0.2, γ 2 =0.3, γ 2 =0.4, γ 2 =0.5

Probability γ 1

0.1 0.2 0.3 0.4 0.5 0.08

γ 2 =0.1, γ 2 =0.2, γ 2 =0.3, γ 2 =0.4, γ 2 =0.5

**Figure 4.** Filtering error variances and smoothing error variances for *N* = 2 of the second state

component at *k* = 200 versus *γ*<sup>1</sup> with *γ*<sup>2</sup> varying from 0.1 to 0.5 when *m* = 3.

Probability γ 1

γ 2 =0.1, γ 2 =0.2, γ 2 =0.3, γ 2 =0.4, γ 2 =0.5

Probability γ 1

γ 2 =0.1, γ 2 =0.2, γ 2 =0.3, γ 2 =0.4, γ 2 =0.5

Probability γ 1

0.14 0.16 0.18 0.2 0.22

**Filtering error variances**

**Smoothing error variances**

at *k* = 200 versus *γ*<sup>1</sup> with *γ*<sup>2</sup> varying from 0.1 to 0.5 when *m* = 3.

**Filtering error variances**

**Smoothing error variances**

0.1 0.11 0.12 0.13 0.14 0.15

0.12 0.14 0.16 0.18 0.2 0.22

0.1 0.12 0.14 0.16 In this chapter, the least-squares linear filtering and fixed-point smoothing problems have been addressed for linear discrete-time stochastic systems with uncertain observations coming from multiple sensors. The uncertainty in the observations is modeled by a binary variable taking the values one or zero (Bernoulli variable), depending on whether the signal is present or absent in the corresponding observation, and it has been supposed that the uncertainty at any sampling time *k* depends only on the uncertainty at the previous time *k* − *m*. This situation covers, in particular, those signal transmission models in which any failure in the transmission is detected and the old sensor is replaced after *m* instants of time, thus avoiding the possibility of missing signal in *m* + 1 consecutive observations.

By applying an innovation technique, recursive algorithms for the linear filtering and fixed-point smoothing estimators have been obtained. This technique consists of obtaining the estimators as a linear combination of the innovations, simplifying the derivation of these estimators, due to the fact that the innovations constitute a white process.

Finally, the feasibility of the theoretical results has been illustrated by the estimation of a two-dimensional signal from uncertain observations coming from two sensors, for different uncertainty probabilities and different values of *m*. The results obtained confirm the greater effectiveness of the fixed-point smoothing estimators in contrast to the filtering ones and conclude that more accurate estimations are obtained as the values of *m* are lower.

In recent years, several problems of signal processing, such as signal prediction, detection and control, as well as image restoration problems, have been treated using quadratic estimators and, generally, polynomial estimators of arbitrary degree. Hence, it must be noticed that the current chapter can be extended by considering the least-squares polynomial estimation

#### 20 Will-be-set-by-IN-TECH 20 Stochastic Modeling and Control Design of Estimation Algorithms from an Innovation Approach in Linear Discrete-Time Stochastic Systems with Uncertain Observations <sup>21</sup>

problems of arbitrary degree for such linear systems with uncertain observations correlated in instants that differ *m* units of time. On the other hand, in practical engineering, some recent progress on the filtering and control problems for nonlinear stochastic systems with uncertain observations is being achieved. Nonlinearity and stochasticity are two important sources that are receiving special attention in research and, therefore, filtering and smoothing problems for nonlinear systems with uncertain observations would be relevant topics on which further investigation would be interesting.

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