20-03-2014, 01:00 PM
A novel criterion for the global asymptotic stability of 2-D discrete systems described by Roesser model using saturation arithmetic
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ABSTRACT
A novel criterion for the global asymptotic stability of two-dimensional (2-D) discrete
systems described by the Roesser model employing saturation arithmetic is presented. The
criterion is compared with previously reported criteria. Numerical examples showing the
effectiveness of the present criterion are given.
Introduction
In recent years, due to the rapid increase of the applicability of two-dimensional (2-D) systems theory in many areas such
as signal filtering, image processing, seismographic data processing, thermal processes in chemical reactors, gas absorption,
water stream heating, etc. [1], there has emerged a continuously growing interest in the system theoretic problems of 2-D
discrete systems. While implementing discrete systems using fixed-point arithmetic on a digital computer or on special
purpose digital hardware, nonlinearities due to finite wordlength, namely, overflow and quantization are generated. The
presence of such nonlinearities may result in the instability of the designed system. When dealing with the design and
implementation of 2-D systems, it is, therefore, important to know the conditions under which the system will be globally
asymptotically stable.
Comparison and examples
It is worth comparing Theorem 1 with Theorem 2. In Theorem 1, positive definite symmetric matrices Ph and P v are re-
quired to be diagonally dominant. Observe that, for the situations where p = m and q = n, Theorem 2 is same as Theorem 1.
When p = m and/or q = n, a close examination of (4), (5a) and (5b) reveals that the matrix P of Theorem 2 is more general
than the matrix P of Theorem 1. Thus, Theorem 2 is less stringent than Theorem 1.
Further, it is shown in [13] that the criterion [12] can be obtained directly from Theorem 1 as a special case. Conse-
quently, Theorem 2 is less restrictive than the criterion [12]
Conclusion
A novel criterion (Theorem 2) for the global asymptotic stability of 2-D state-space digital filters described by the Roesser
model employing saturation arithmetic has been established. The criterion obtained in this paper is less restrictive than
[9,12] and may provide improved results over [6,14] and [16, Corollary 1]. The 2-D result discussed in this paper can be
easily extended to m-D (m > 2) systems.
[attachment=60311]
ABSTRACT
A novel criterion for the global asymptotic stability of two-dimensional (2-D) discrete
systems described by the Roesser model employing saturation arithmetic is presented. The
criterion is compared with previously reported criteria. Numerical examples showing the
effectiveness of the present criterion are given.
Introduction
In recent years, due to the rapid increase of the applicability of two-dimensional (2-D) systems theory in many areas such
as signal filtering, image processing, seismographic data processing, thermal processes in chemical reactors, gas absorption,
water stream heating, etc. [1], there has emerged a continuously growing interest in the system theoretic problems of 2-D
discrete systems. While implementing discrete systems using fixed-point arithmetic on a digital computer or on special
purpose digital hardware, nonlinearities due to finite wordlength, namely, overflow and quantization are generated. The
presence of such nonlinearities may result in the instability of the designed system. When dealing with the design and
implementation of 2-D systems, it is, therefore, important to know the conditions under which the system will be globally
asymptotically stable.
Comparison and examples
It is worth comparing Theorem 1 with Theorem 2. In Theorem 1, positive definite symmetric matrices Ph and P v are re-
quired to be diagonally dominant. Observe that, for the situations where p = m and q = n, Theorem 2 is same as Theorem 1.
When p = m and/or q = n, a close examination of (4), (5a) and (5b) reveals that the matrix P of Theorem 2 is more general
than the matrix P of Theorem 1. Thus, Theorem 2 is less stringent than Theorem 1.
Further, it is shown in [13] that the criterion [12] can be obtained directly from Theorem 1 as a special case. Conse-
quently, Theorem 2 is less restrictive than the criterion [12]
Conclusion
A novel criterion (Theorem 2) for the global asymptotic stability of 2-D state-space digital filters described by the Roesser
model employing saturation arithmetic has been established. The criterion obtained in this paper is less restrictive than
[9,12] and may provide improved results over [6,14] and [16, Corollary 1]. The 2-D result discussed in this paper can be
easily extended to m-D (m > 2) systems.