A new sufficient condition for the global asymptotic stability of 2-D state-space digital filters with saturation arithmetic
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×q set of p×q real matrices Rp set of p×1 real vectors 0 null matrix or null vector of appropriate dimension BT transpose of the matrix (or vector) B B> 0 B is positive definite symmetric matrix P1⊕P2 direct 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 by
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 Ph∈Rm×m, Pv∈Rn×n satisfyingwhere G=[gij]∈R(m+n)×(m+n) is a row diagonally dominant matrix with positive diagonal elements and P=PT=Ph⊕Pv. 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 Hh∈Rm×m, Hv∈Rn×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
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
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.
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