# Resilient Minimum Entropy Filter Design for Non-Gaussian Stochastic Systems

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## Abstract

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## 1. Introduction

**Notation.**The notations in this paper are quite standard. $\parallel \xb7\parallel $ means the Euclidean norm in ${R}^{n}$. ${A}^{T}$ is the transpose of the matrix A. $\parallel A\parallel $ is the operator of matrix A, i.e., $\parallel A\parallel =sup\{\parallel Ax\parallel :\parallel x\parallel =1\}=\sqrt{{\lambda}_{max}\left({A}^{T}A\right)}$, where ${\lambda}_{max}(\xb7)$ means the largest eigenvalue of A. Moreover, Ω is the sample space, F is a set of events. Let $(\Omega ,F,P)$ be a complete probability space, and $E\{\xb7\}$ stands for the mathematical expectation operator with respect to the given probability measure P. The expected value of a random variable x is denoted by $E\left\{x\right\}$. $Var\{\xb7\}$ represent the variance of random variables. The star * represents a transpose quantity.

## 2. Problem Formulation

**Definition 1**([19,20]) The dynamics of the estimation error $e\left(k\right)$ is exponentially ultimately bounded in the mean square if there exist constants $a>0$, $b>0$, $c>0$ such that for any initial condition $e\left(0\right)$,

**Remark 1**When the error dynamic is exponentially ultimately bounded in the mean square, the estimation error will initially decrease exponentially in the mean square, and remain within a certain region in the steady state, again in the mean square sense. The stability bound is defined in terms of the norm ${\left(E\right\{\parallel e\left(k\right)\parallel}^{2}{\left\}\right)}^{\frac{1}{2}}$ of the Hilbert space of random vectors, and is specified by the coefficient c.

## 3. The Minimum Entropy Filter Gain Updating Algorithm

**Theorem 1**The recursive filtering gain design algorithm to minimize the performance function $J\left(k\right)$ subject to the estimation model Equation (4) is given by

**Proof.**From Equation (11), it can be seen that

## 4. The Bound of the Filter Gain

**Remark 2.**It should be noted that the function ${K}_{\sigma}$ is a Gaussian kernel function, and the error is bounded in the interval ${[a,\phantom{\rule{0.222222em}{0ex}}b]}^{n}$. This guarantees that the upper bound of $\u2225\frac{\partial {K}_{\sigma}}{\partial {e}_{k}}\u2225$ is easy to obtain.

## 5. Resilient Filter Gain Design

**Lemma 1**([24]) Given matrices Y, M and N. Then

**Theorem 2**Consider the error system in Equation (4), the filter gain variation is given by Equation (26), for given constants ${\epsilon}_{1}>0$, ${\epsilon}_{2}>0$ and $\gamma >0$, there exist a positive scalar β, and matrices $P>0$ and Y such that the following inequalities hold

**Proof.**Define a Lyapunov functional candidate for Equation (28) as

**Remark 3**In Theorem 2, the filter gain updating law $\Delta {L}_{k}$ is a suboptimal recursive strategy that guarantees the strictly decreasing nature of the entropy of the estimation errors. Meanwhile, ${L}_{k}$ is a control law to ensure the mean square exponential stability. Hence, based on the resilient control theory, Theorem 2 establishes the relationship between the entropy performance and the stochastic stability for stochastic distribution control systems.

## 6. Numerical Example

## 7. Conclusions

## Acknowledgements

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**MDPI and ACS Style**

Wang, Y.; Wang, H.; Guo, L.
Resilient Minimum Entropy Filter Design for Non-Gaussian Stochastic Systems. *Entropy* **2013**, *15*, 1311-1323.
https://doi.org/10.3390/e15041311

**AMA Style**

Wang Y, Wang H, Guo L.
Resilient Minimum Entropy Filter Design for Non-Gaussian Stochastic Systems. *Entropy*. 2013; 15(4):1311-1323.
https://doi.org/10.3390/e15041311

**Chicago/Turabian Style**

Wang, Yan, Hong Wang, and Lei Guo.
2013. "Resilient Minimum Entropy Filter Design for Non-Gaussian Stochastic Systems" *Entropy* 15, no. 4: 1311-1323.
https://doi.org/10.3390/e15041311