有限频率下二维数字滤波器模型简化

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"这篇技术论文关注的是在有限频率范围内对二维(2-D)数字滤波器的模型简化问题。文章采用Fornasini-Marchesini局部状态空间(FM LSS)模型来描述2-D数字滤波器，并利用2-D系统的广义Kalman-Yakubovich-Popov (GKYP)引理，推导出有限频率模型简化问题的充分条件。与全频域方法相比，提出的有限频率方法能获得更好的近似效果。" 本文是关于在特定有限频率范围内对二维数字滤波器进行模型简化的研究。二维滤波器在图像处理、信号处理等领域有广泛应用，其复杂性可能导致计算效率降低。因此，模型简化成为优化计算性能的关键。首先，2-D数字滤波器通过Fornasini-Marchesini局部状态空间模型表示。这是一种数学工具，用于描述和分析多变量动态系统，特别是当系统具有局部交互性质时，该模型尤其适用。FM LSS模型能够捕捉2-D滤波器的时空特性，提供了一种结构化的分析框架。然后，论文利用2-D系统的广义Kalman-Yakubovich-Popov (GKYP)引理来建立模型简化的条件。GKYP引理是控制理论中的一个关键工具，它为稳定性分析和控制器设计提供了基础。在2-D系统中，GKYP引理扩展了经典的一维系统理论，使得在有限频率范围内分析和简化模型成为可能。作者推导出的充分条件，为在保持系统性能的同时，减少2-D滤波器的状态空间维度提供了理论依据。这使得在特定频段内，滤波器的计算复杂度得以降低，而不会显著牺牲其滤波性能。与传统的全频域模型简化方法相比，有限频率方法更注重在选定频段内的表现，因此在特定应用中可能更为高效。这种方法特别适用于那些只关心特定频带性能的系统，例如针对特定噪声频率的降噪滤波器。这篇论文为2-D数字滤波器的优化设计提供了新的视角和工具，对于提升实际系统性能，特别是在资源受限的嵌入式系统中，具有重要的实践价值。同时，该研究也为后续的滤波器设计和模型简化算法的发展奠定了理论基础。

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI

10.1109/TAC.2014.2359305, IEEE Transactions on Automatic Control

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. XX, NO. X, XXXX 2014 1

Finite-Frequency Model Reduction of

Two-Dimensional Digital Filters

Da-Wei Ding, Xin Du, and Xiaoli Li

Abstract—This paper is concerned with the model reduction

problem of two-dimensional (2-D) digital ﬁlters over ﬁnite-

frequency ranges. The 2-D digital ﬁlter is described by the

Fornasini-Marchesini local state-space (FM LSS) model. With

the aid of the generalized Kalman-Yakubovich-Popov (GKYP)

lemma for 2-D systems, sufﬁcient conditions for the ﬁnite-

frequency model reduction problem are derived. Compared with

full-frequency methods, the proposed ﬁnite-frequency method

can get a better approximation performance over ﬁnite-frequency

ranges. An example is given to demonstrate the effectiveness of

the proposed method.

Index Terms—Model reduction, 2-D systems, FM LSS model,

ﬁnite frequency.

I. INTRODUCTION

ODEL reduction is one of the fundamental problems

in the ﬁeld of systems and control theory, which has

been extensively investigated in the past several decades [1],

[2], [6]. The model reduction problem can be formulated as

follows: For a given full-order model of a dynamic system, ﬁnd

a reduced-order model such that these two models are close in

some sense, such as L

∞

, H

∞

, H

, etc. As for one-dimensional

(1-D) linear systems, various effective approaches, such as

the balanced truncation method [3], the optimal Hankel-norm

approximation method [4], the aggregation method [5], the

Krylov subspace techniques [6], [7], to mention a few, have

been reported in the literature. It is worth noticing that many

model reduction problems are inherently frequency dependent,

i.e., the requirement on the approximation accuracy over some

frequency ranges is more important than others. To cope with

these problems, different approaches have been developed

for 1-D systems, such as the frequency weighted balanced

truncation method [8]- [11], and the ﬁnite-frequency H

∞

method [12], [13], etc.

On the other hand, two-dimensional (2-D) systems [14]-

[18] have drawn great attention due to their extensive appli-

cations in process control, multi-dimensional digital ﬁltering,

image data processing and transmission, signal processing,

etc. In many engineering applications, such as 2-D signal

and image processing, high order 2-D models are frequently

used to describe physical systems. This leads to high storage

Manuscript received April 14, 2013; revised February 16, 2014. This work

was supported in part by the National Natural Science Foundation of China

under Grants 61104013, 61473032, 61304143, and 61473034, and by Program

for New Century Excellent Talents in University under Grant NCET-13-0662.

D.-W. Ding and X. Li are with School of Automation and Electrical

Engineering, University of Science and Technology Beijing, Beijing 100083,

P. R. China (e-mail: dingdawei@ustb.edu.cn; lixiaoli@hotmail.com).

X. Du is with School of Mechatronic Engineering and Automation, Shang-

hai University, Shanghai 200072, P. R. China (email: duxin@shu.edu.cn).

requirements and expensive computations, and makes it difﬁ-

cult to analyze, simulate or design such large scale systems.

Therefore it is necessary to reduce the size of the 2-D system

models to manageable orders. Model reduction thus plays

an important role in analysis and design of 2-D systems.

There are several model reduction methods developed for 2-

D systems, such as the balanced truncation method [19], [20],

the weighted balanced truncation method [21], and LMI-based

methods [22], [23], etc. It is noted that, as in 1-D systems,

model reduction of 2-D systems over ﬁnite-frequency ranges

is of practical signiﬁcance. For instance, when the frequency

ranges of input signals are known, the reduced-order 2-D ﬁlter

is needed to capture the input-output behavior of the original

ﬁlter over the pre-known frequency ranges. This problem,

however, has not fully been investigated up to now. In [21], a

weighted balanced truncation approach is employed to address

this problem. Although it is useful in practice, the weighted

method suffers from some drawbacks. First, the additional

weights tend to increase the system complexity. Second, the

process of selecting appropriate weights can be tedious and

time-consuming [24]. This motivates us to develop a new

model reduction method for 2-D systems over ﬁnite-frequency

ranges.

In this paper, a ﬁnite-frequency model reduction method

is proposed for 2-D digital ﬁlters described by the FM LSS

model. In detail, a ﬁnite-frequency index is adopted to mea-

sure the approximation performance over pre-speciﬁed ﬁnite-

frequency ranges. By the newly-developed GKYP lemma for

2-D FM LSS systems [25], sufﬁcient conditions are derived for

the model reduction problem over low-frequency and middle-

frequency ranges, respectively. Based on these conditions,

two algorithms are developed to describe how to obtain the

reduced-order models. The proposed ﬁnite-frequency method

can get a better approximation accuracy than the existing full-

frequency ones over ﬁnite-frequency ranges. An example is

given to illustrate the effectiveness of the proposed method.

The rest of the paper is organized as follows. Section II

gives the problem statement. Section III gives the main results,

where a new ﬁnite-frequency model reduction method for 2-

D FM LSS model is proposed in detail. Section IV gives an

example to illustrate the effectiveness of the proposed method.

Finally, conclusions are given in Section V.

Notations. For a matrix M, M

∗

, M

⊥

denote its conjugate

transpose and orthogonal complement, respectively. M > 0

(M < 0) means that M is positive deﬁnite (negative deﬁnite).

The Hermitian part of a square matrix M is denoted by

He(M) := M+M

∗

. The symbol ? will be used in some matrix

expressions to induce a symmetric structure. σ(G), σ

max

(G)

Limited circulation. For review only

Preprint submitted to IEEE Transactions on Automatic Control. Received: September 16, 2014 01:53:13 PST

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