线性离散系统非高斯噪声二次估计方法

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"这篇研究论文探讨了线性离散时间非高斯系统的输入噪声二次多项式估计问题。文章中提出了一种将非高斯噪声的二次估计转化为线性估计问题的方法，利用克罗内克代数规则处理增强系统的噪声特性，并结合卡尔曼滤波理论设计了非高斯噪声二次估计器。此外，还通过构建等效状态空间模型证明了估计器的稳定性，并通过实例验证了方法的有效性。" 本文主要涉及以下几个关键知识点： 1. **输入噪声二次多项式估计**：在系统分析和控制中，输入噪声通常是不可忽视的因素。当噪声是非高斯分布时，传统的基于高斯假设的估计方法不再适用。二次多项式估计是一种考虑噪声非线性特性的方法，它涉及到噪声功率的二次项，可以提供更精确的估计结果。 2. **线性离散时间系统与非高斯噪声**：线性离散时间系统是指系统的动态方程可以表示为线性关系的系统，其状态随着时间离散变化。非高斯噪声是指其概率分布不遵循正态（高斯）分布的随机变量，这种噪声在实际系统中很常见，例如传感器噪声、通信信道噪声等。 3. **克罗内克代数**：在解决线性离散时变系统的非高斯噪声问题时，克罗内克代数是一种强大的工具。它允许处理矩阵和向量的乘积，从而简化复杂系统的表示和分析。通过克罗内克乘法，可以将非线性问题转化为线性问题，便于进行后续的估计器设计。 4. **非高斯噪声二次去卷积滤波器**：去卷积滤波器是一种用于消除信号中混叠效应的滤波器，它在非高斯噪声环境下，需要考虑到噪声的二次项。通过将非高斯噪声的二次估计问题转化为线性问题，可以设计出有效的去卷积滤波器。 5. **固定滞后平滑器**：在时间序列分析中，固定滞后平滑器用于改善估计质量，通过考虑过去的几个时间步的信息来平滑当前的估计。对于非高斯噪声，固定滞后平滑器的设计也需要适应噪声的非线性特性。 6. **卡尔曼滤波理论**：卡尔曼滤波是一种在线估计方法，适用于线性高斯系统。在这里，它被扩展到处理非高斯噪声的二次估计问题，通过投影公式来求解非高斯噪声二次估计器。 7. **等效状态空间模型**：构建等效状态空间模型是为了分析和控制非高斯系统的稳定性。通过构建这样的模型，可以证明估计器在有不相关噪声的情况下保持稳定。 8. **稳定性分析**：在系统理论中，稳定性是系统是否能保持在期望状态或行为的重要指标。通过构建等效状态空间模型，可以分析非高斯噪声二次估计器在各种条件下的稳定性。该研究提供了处理非高斯噪声的创新方法，对于线性离散时间系统的噪声估计具有重要意义，不仅理论上具有价值，而且在实际应用中，如信号处理、控制系统、通信等领域都有潜在的应用前景。通过实例验证，这些方法能够有效地提高估计精度，降低非高斯噪声的影响。

Non-Gaussian noise quadratic estimation for linear discrete-time

time-varying systems

Huihong Zhao

, Chenghui Zhang

Clean Energy Research and Technology Promotion Center, Dezhou University, No. 566 University Rd. West, Dezhou 253023, PR China

School of Control Science and Engineering, Shandong University, 17923 Jingshi Road, Jinan 250061, PR China

article info

Article history:

Received 21 July 2015

Received in revised form

26 September 2015

Accepted 2 October 2015

Communicated by Ma Lifeng Ma

Available online 22 October 2015

Keywords:

Input noise quadratic polynomial estima-

tion

Kronecker algebra

Deconvolution ﬁlter

Fixed-lag smoother

abstract

This study deals with the input noise quadratic polynomial estimation problem for linear discrete-time

non-Gaussian systems. The design of the non-Gaussian noise quadratic deconvolution ﬁlter and ﬁxed-lag

smoother is ﬁrstly converted into a linear estimation problem in a suitable second-order polynomial

extended system. By employing the Kronecker algebra rules, the stochastic characteristics of the aug-

mented noise in the augmented system are discussed. Then a solution to the non-Gaussian noise

quadratic estimator is obtained through applying the projection formula in Kalman ﬁltering theory. In

addition, the stability is proved by constructing an equivalent state-space model with uncorrelated

noises. Finally, a numerical example is given to show the effectiveness of the proposed method.

1. Introduction

The input noise estimation (also known as deconvolution) has a

rich history and a wide range of applications in image restoration,

oil exploration, speech signal processing, fault detection and so on

[1–4]. The task of the deconvolution problem is to estimate the

intended unknown input noise of a system by utilizing the

obtainable outputs. For the ﬁrst time, an optimal white noise

smoother with application to seismic data processing in oil

exploration was presented in [2]. Applying the polynomial

approach in frequency domain, the optimal deconvolution esti-

mator was derived based on spectral factorization in [5]. Later,

both input and measurement white noise estimators were

Recently, the deconvolution theory was successfully applied to the

multi-sensor linear discrete time systems [7,8] and the systems

with packet dropouts [9–11]. Note that the above results were

obtained based on the input Gaussian noise assumption, however,

in many important technical areas the input noise is non-Gaussian

(see for instance [12–14]). This is the motivation to develop a new

algorithm which permits us to ﬁnd a satisfactory non-Gaussian

noise estimator for linear discrete-time time-varying systems.

The estimation problem for non-Gaussian systems has received

more and more attention and some fundamental results have been

developed, refer to [12,15] and the references therein. For linear

non-Gaussian systems, the conditional expectation giving the

minimum mean square error estimate is an inﬁnite dimensional

problem, and its solution cannot be easily numerically computed

[15]. Although the Kalman ﬁlter is the best afﬁne estimator for the

non-Gaussian case, its estimated accuracy is inadequate in some

cases. Note that the polynomial ﬁltering algorithm [12,15], which

employ both the observations of the original system and their

Kronecker products, is more accurate than the classical Kalman

ﬁlter, while maintaining the characteristics of easy calculability

and recursivity. Therefore, an increasing number of authors have

focussed on the polynomial estimator design for the non-Gaussian

systems. The pioneer work can be traced back to the recursive

arbitrary-degree ﬁnite-memory polynomial estimator design via

the classical Kalman ﬁltering theory [12]. Later, the result was

successfully extended to polynomial ﬁlter for stochastic bilinear

systems [16] and polynomial extended Kalman ﬁlter [17]. When

the state-space model was unknown, the ﬁxed-point, ﬁxed-

interval and ﬁ

xed-lag smoothers from uncertain observations

were presented based on the covariance information of the pro-

cesses in [18]. Recently, this method was applied to the study of

multi-sensor information fusion quadratic ﬁlter for linear systems

with uncertain observations [19]. However, these works have a

limitation that the Non-Gaussian noise polynomial estimator was

not investigated.

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/neucom

Neurocomputing

http://dx.doi.org/10.1016/j.neucom.2015.10.015

Corresponding author.

E-mail addresses: huihong1980@163.com (H. Zhao),

zchui@sdu.edu.cn (C. Zhang).

Neurocomputing 174 (2016) 921 – 927

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