**1. Introduction**

The least-squares estimation problem in linear discrete-time stochastic systems in which the signal to be estimated is always present in the observations has been widely treated; as is well known, the Kalman filter [12] provides the least-squares estimator when the additive noises and the initial state are Gaussian and mutually independent.

Nevertheless, in many real situations, usually the measurement device or the transmission mechanism can be subject to random failures, generating observations in which the state appears randomly or which may consist of noise only due, for example, to component or interconnection failures, intermittent failures in the observation mechanism, fading phenomena in propagation channels, accidental loss of some measurements or data inaccessibility at certain times. In these situations where it is possible that information concerning the system state vector may or may not be contained in the observations, at each sampling time, there is a positive probability (called *false alarm probability*) that only noise is observed and, hence, that the observation does not contain the transmitted signal, but it is not generally known whether the observation used for estimation contains the signal or it is only noise. To describe this interrupted observation mechanism (*uncertain observations*), the observation equation, with the usual additive measurement noise, is formulated by multiplying the signal function at each sampling time by a binary random variable taking the values one and zero (Bernoulli random variable); the value one indicates that the measurement at that time contains the signal, whereas the value zero reflects the fact that the signal is missing and, hence, the corresponding observation is only noise. So, the observation equation involves both an additive and a multiplicative noise, the latter modeling the uncertainty about the signal being present or missing at each observation.

Linear discrete-time systems with uncertain observations have been widely used in estimation problems related to the above practical situations (which commonly appear, for example, in

©2012 Linares-Pérez et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2012 Linares-Pérez et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 2 Will-be-set-by-IN-TECH 2 Stochastic Modeling and Control Design of Estimation Algorithms from an Innovation Approach in Linear Discrete-Time Stochastic Systems with Uncertain Observations <sup>3</sup>

Communication Theory). Due to the multiplicative noise component, even if the additive noises are Gaussian, systems with uncertain observations are always non-Gaussian and hence, as occurs in other kinds of non-Gaussian linear systems, the least-squares estimator is not a linear function of the observations and, generally, it is not easily obtainable by a recursive algorithm; for this reason, research in this kind of systems has focused special attention on the search of suboptimal estimators for the signal (mainly linear ones).

the observations are correlated at instants that differ two units of time. This study covers more general practical situations, for example, in sensor networks where sensor failures may happen and a failed sensor is replaced not immediately, but two sampling times after having failed. However, even if it is assumed that any failure in the transmission results from sensor failures, usually the failed sensor may not be replaced immediately but after *m* instants of time; in such situations, correlation among the random variables modeling the uncertainty in the observations at times *k* and *k* + *m* must be considered and new algorithms must be

Design of Estimation Algorithms from an Innovation Approach in Linear Discrete-Time Stochastic Systems with Uncertain Observations

3

The current chapter is concerned with the state estimation problem for linear discrete-time systems with uncertain observations when the uncertainty at any sampling time *k* depends only on the uncertainty at the previous time *k* − *m*; this form of correlation allows us to consider certain models in which the signal cannot be missing in *m* + 1 consecutive

The random interruptions in the observation process are modeled by a sequence of Bernoulli variables (at each time, the value one of the variable indicates that the measurement is the current system output, whereas the value zero reflects that only noise is available), which are correlated only at the sampling times *k* − *m* and *k*. Recursive algorithms for the filtering and fixed-point smoothing problems are proposed by using an innovation approach; this approach, based on the fact that the innovation process can be obtained by a causal and invertible operation on the observation process, consists of obtaining the estimators as a linear combination of the innovations and simplifies considerably the derivation of the estimators

The chapter is organized as follows: in Section 2 the system model is described; more specifically, we introduce the linear state transition model perturbed by a white noise, and the observation model affected by an additive white noise and a multiplicative noise describing the uncertainty. Also, the pertinent hypotheses to address the least-squares linear estimation problem are established. In Section 3 this estimation problem is formulated using an innovation approach. Next, in Section 4, recursive algorithms for the filter and fixed-point smoother are derived, including recursive formulas for the estimation error covariance matrices. Finally, the performance of the proposed estimators is illustrated in Section 5 by a numerical simulation example, where a two-dimensional signal is estimated and the estimation accuracy is analyzed for different values of the uncertainty probability and

Consider linear discrete-time stochastic systems with uncertain observations coming from multiple sensors, whose mathematical modeling is accomplished by the following equations.

where {*xk*; *k* ≥ 0} is an *n*-dimensional stochastic process representing the system state, {*wk*; *k* ≥ 0} is a white noise process and *Fk*, for *k* ≥ 0, are known deterministic matrices.

*xk* = *Fk*−<sup>1</sup>*xk*−<sup>1</sup> + *wk*−1, *<sup>k</sup>* ≥ 1, (1)

due to the fact that the innovations constitute a white process.

several values of the time period *m*.

**2. Model description**

The state equation is given by

deduced.

observations.

In some cases, the variables modeling the uncertainty in the observations can be assumed to be independent and, then, the distribution of the multiplicative noise is fully determined by the probability that each particular observation contains the signal. As it was shown by Nahi [17] (who was the first who analyzed the least-squares linear filtering problem in this kind of systems assuming that the state and observation additive noises are uncorrelated) the knowledge of the aforementioned probabilities allows to derive estimation algorithms with a recursive structure similar to the Kalman filter. Later on, Monzingo [16] completed these results by analyzing the least-squares smoothing problem and, subsequently, [3] and [4] generalized the least-squares linear filtering and smoothing algorithms considering that the additive noises of the state and the observation are correlated.

However, there exist many real situations where this independence assumption of the Bernoulli variables modeling the uncertainty is not satisfied; for example, in signal transmission models with stand-by sensors in which any failure in the transmission is detected immediately and the old sensor is then replaced, thus avoiding the possibility of the signal being missing in two successive observations. This different situation was considered by [9] by assuming that the variables modeling the uncertainty are correlated at consecutive time instants, and the proposed least-squares linear filtering algorithm provides the signal estimator at any time from those in the two previous instants. Later on, the state estimation problem in discrete-time systems with uncertain observations, has been widely studied under different hypotheses on the additive noises involved in the state and observation equations and, also, under several hypotheses on the multiplicative noise modeling the uncertainty in the observations (see e.g. [22] - [13], among others).

On the other hand, there are many engineering application fields (for example, in communication systems) where sensor networks are used to obtain all the available information on the system state and its estimation must be carried out from the observations provided by all the sensors (see [6] and references therein). Most papers concerning systems with uncertain observations transmitted by multiple sensors assume that all the sensors have the same uncertainty characteristics. In the last years, this situation has been generalized by several authors considering uncertain observations whose statistical properties are assumed not to be the same for all the sensors. This is a realistic assumption in several application fields, for instance, in networked communication systems involving heterogeneous measurement devices (see e.g. [14] and [8], among others). In [7] it is assumed that the uncertainty in each sensor is modeled by a sequence of independent Bernoulli random variables, whose statistical properties are not necessarily the same for all the sensors. Later on, in [10] and [1] the independence restriction is weakened; specifically, different sequences of Bernoulli random variables correlated at consecutive sampling times are considered to model the uncertainty at each sensor. This form of correlation covers practical situations where the signal cannot be missing in two successive observations. In [2] the least-squares linear and quadratic problems are addressed when the Bernoulli variables describing the uncertainty in the observations are correlated at instants that differ two units of time. This study covers more general practical situations, for example, in sensor networks where sensor failures may happen and a failed sensor is replaced not immediately, but two sampling times after having failed. However, even if it is assumed that any failure in the transmission results from sensor failures, usually the failed sensor may not be replaced immediately but after *m* instants of time; in such situations, correlation among the random variables modeling the uncertainty in the observations at times *k* and *k* + *m* must be considered and new algorithms must be deduced.

The current chapter is concerned with the state estimation problem for linear discrete-time systems with uncertain observations when the uncertainty at any sampling time *k* depends only on the uncertainty at the previous time *k* − *m*; this form of correlation allows us to consider certain models in which the signal cannot be missing in *m* + 1 consecutive observations.

The random interruptions in the observation process are modeled by a sequence of Bernoulli variables (at each time, the value one of the variable indicates that the measurement is the current system output, whereas the value zero reflects that only noise is available), which are correlated only at the sampling times *k* − *m* and *k*. Recursive algorithms for the filtering and fixed-point smoothing problems are proposed by using an innovation approach; this approach, based on the fact that the innovation process can be obtained by a causal and invertible operation on the observation process, consists of obtaining the estimators as a linear combination of the innovations and simplifies considerably the derivation of the estimators due to the fact that the innovations constitute a white process.

The chapter is organized as follows: in Section 2 the system model is described; more specifically, we introduce the linear state transition model perturbed by a white noise, and the observation model affected by an additive white noise and a multiplicative noise describing the uncertainty. Also, the pertinent hypotheses to address the least-squares linear estimation problem are established. In Section 3 this estimation problem is formulated using an innovation approach. Next, in Section 4, recursive algorithms for the filter and fixed-point smoother are derived, including recursive formulas for the estimation error covariance matrices. Finally, the performance of the proposed estimators is illustrated in Section 5 by a numerical simulation example, where a two-dimensional signal is estimated and the estimation accuracy is analyzed for different values of the uncertainty probability and several values of the time period *m*.
