0018-9286 (c) 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See
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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
M
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
2
, 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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