Elsevier

Signal Processing

Volume 88, Issue 1, January 2008, Pages 86-98
Signal Processing

A new sufficient condition for the global asymptotic stability of 2-D state-space digital filters with saturation arithmetic

https://doi.org/10.1016/j.sigpro.2007.07.005Get rights and content

Abstract

A new sufficient condition for the global asymptotic stability of two-dimensional (2-D) state-space digital filters described by the Roesser model employing saturation arithmetic is presented. The new condition not only unifies a string of previous stability results, but also yields improvement over them, hence enlarging the overflow stability region of 2-D digital filters.

Introduction

Two-dimensional (2-D) systems have found many applications in many areas such as image processing, seismographic data processing, thermal processes, gas absorption, water stream heating, etc. [1]. While implementing discrete systems, the finite register length of digital hardware or computer generates nonlinear effects such as overflow and quantization. The presence of such nonlinearities may result in the instability of the designed filter. The zero-input limit cycles may possibly occur due to such nonlinearities. The global asymptotic stability of the null solution ensures the absence of limit cycles in the realized filter. In recent years, the stability properties of 2-D discrete systems described by the Roesser model [2] have been investigated extensively [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17]. Several publications [3], [4], [8], [14] relating to Lyapunov equation with constant coefficients for the Roesser model have appeared. It has been shown in [3] that the existence of the positive definite solutions to the 2-D Lyapunov equation is, in general, only sufficient but not necessary for 2-D stability. Based on the properties of strictly bounded real matrices, a necessary and sufficient condition has been developed [3] for the existence of positive definite solutions of the 2-D Lyapunov equation. The stability of 2-D digital filters implemented with two's complement quantization has been investigated in [5]. A few criteria for the global asymptotic stability of digital filters under various combinations of overflow and quantization nonlinearities have been proposed in [6], [7]. The stability properties of 2-D discrete systems based on Roesser model subject to overflow nonlinearities have been studied in [8], [9], [10], [11], [12], [13], [14], [15], [16], [17].

This paper deals with the problem of global asymptotic stability of zero-input 2-D digital filters described by the Roesser model using saturation overflow arithmetic. It is assumed that the effects of quantization are negligible. The paper is organized as follows. Section 2 presents a description of the system under consideration and a brief review of previous related works. In Section 3, we present the main result (Theorem 6) and its two corollaries. These corollaries present computationally tractable stability conditions, which are easy to test using MATLAB linear matrix inequality (LMI) toolbox [18], [19]. In order to establish the significance of the present approach, in Section 4 we conduct extensive comparison of the new criterion with previously reported criteria. In particular, it is established that the presented criterion unifies and improves a string of existing results [10], [14], [15], [16], [17]. In Section 5, two examples highlighting the usefulness of the presented approach are discussed. In Section 6, we provide concluding remarks.

Section snippets

System description and previous criteria

The following notations are used throughout the paper:

Rp×qset of p×q real matrices
Rpset of p×1 real vectors
0null matrix or null vector of appropriate dimension
BTtranspose of the matrix (or vector) B
B> 0B is positive definite symmetric matrix
P1P2direct sum of matrices P1 and P2
det(B)determinant of any square matrix B

The 2-D discrete system to be studied presently is described by the Roesser model employing saturation arithmetic. Specifically, the system under consideration is given byx11(k,l)==,

Main result and its corollaries

In this section, a criterion for the global asymptotic stability of system (1a), (1b), (1c), (1d), (1e), (1f), (1g) is presented. The main result may be stated as follows.

Theorem 6

The zero solution of system (1a), (1b), (1c), (1d), (1e), (1f), (1g) is globally asymptotically stable if there exist positive definite symmetric matrices PhRm×m, PvRn×n satisfying[P-ATG-GTAG+GT-P]>0,where G=[gij]∈R(m+n)×(m+n) is a row diagonally dominant matrix with positive diagonal elements and P=PT=PhPv.

Proof

First note that,

Comparison

In this section, we will compare the main result of this paper with a string of previous results reported in [10], [14], [15], [16], [17].

Proposition 1

Corollary 1 implies Theorem 1.

Proof

By choosing matrix N as a positive definite diagonal matrix D, it can easily be seen that Theorem 1 is recovered from Corollary 1 as a special case. 

Proposition 2

Corollary 1 implies Theorem 2.

Proof

Conditions (3a), (3b) imply that HhRm×m, HvRn×n in Theorem 2 are row diagonally dominant symmetric matrices with positive diagonal entries. Now, by

Examples

To demonstrate the applicability of the present results and compare them with previous results, we now consider two specific examples.

Example 1: Consider a 2-D digital filter of order (2, 1) described by (1) with A=

In the following, it is shown that Theorem 1, Theorem 2, Theorem 5 fail to determine the global asymptotic stability of this system while that is affirmed by both the corollaries (Corollaries 1 and 2).

Substituting A as specified by (33) and positive definite matrices D=[d1000d2000d3],

Conclusion

A new criterion (Theorem 6) for the global asymptotic stability of 2-D state-space digital filters described by the Roesser model employing saturation arithmetic has been established. LMI based sufficient conditions (Corollaries 1 and 2) have been derived to ensure the global asymptotic stability of the 2-D systems. As illustrated in Section 4, the new criterion unifies and yields improvement over a string of previous stability results [10], [14], [15], [16], [17]. The 2-D results discussed in

Acknowledgment

The author wishes to thank the reviewers for their constructive comments and suggestions.

References (20)

  • V. Singh

    New LMI condition for the nonexistence of overflow oscillations in 2-D state-space digital filters using saturation arithmetic

    Digital Signal Process.

    (2007)
  • T. Kaczorek

    Two-dimensional Linear Systems

    (1985)
  • R.P. Roesser

    A discrete state-space model for linear image processing

    IEEE Trans. Automat. Control

    (1975)
  • B.D.O. Anderson et al.

    Stability and the matrix Lyapunov equation for discrete 2-dimensional systems

    IEEE Trans. Circuits Syst.

    (1986)
  • C. Xiao et al.

    Stability and the Lyapunov equation for n-dimensional digital systems

    IEEE Trans. Circuits Syst. I

    (1997)
  • T. Bose

    Asymptotic stability of two-dimensional digital filters under quantization

    IEEE Trans. Signal Process.

    (1994)
  • L.-J. Leclerc et al.

    New criteria for asymptotic stability of one- and multidimensional state-space digital filters in fixed-point arithmetic

    IEEE Trans. Signal Process.

    (1994)
  • H. Kar et al.

    Stability analysis of 1-D and 2-D fixed-point state-space digital filters using any combination of overflow and quantization nonlinearities

    IEEE Trans. Signal Process.

    (2001)
  • N.G. El-Agizi et al.

    Two-dimensional digital filters with no overflow oscillations

    IEEE Trans. Acoust. Speech Signal Process.

    (1979)
  • S.G. Tzafestas et al.

    Two-dimensional digital filters without overflow oscillations and instability due to finite word length

    IEEE Trans. Signal Process.

    (1992)
There are more references available in the full text version of this article.

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  • New results on saturation overflow stability of 2-D state-space digital filters

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    Another approach for dealing with the problem of global asymptotic stability of 2-D Roesser model with saturation has been given in [32]. The approach in [32] leads to an improvement over a string of previous stability results [21,24,26,27,30,31]. Reference [33] brings out a modified version of the criterion in [32].

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