Local stability analysis and H∞ performance for Lipschitz digital filters with saturation nonlinearity and external interferences
Introduction
Over the past few decades, digital filters have received considerable importance because of their practical applications in several areas, such as signal processing, image processing, wireless sensor networks, telecommunication, computer vision, thermal processes, geophysics, radar, and biomedicine [1], [2]. In particular, much attention has been focused on the stability analysis of discrete-time filters under finite word length consequences like quantization and overflow. Such nonlinear effects are unavoidable when a discrete-time filter is implemented via a digital signal processor using the fixed-point arithmetic and it may cause performance degradation in the form of limit cycles and overflow oscillations. Rounding, magnitude truncation, and value truncation are frequently used schemes for quantization, while common overflow correcting techniques are saturation, zeroing, and 2′s complement [3]. Although, the nonlinear consequences of quantization and overflow are not conceptually decoupled, they can be treated independently provided the quantization steps are sufficiently large. In addition to this, it is worth mentioning that a digital filter with relatively higher order is usually implemented by dividing it into several low order filters. Consequently, any mutual interferences among various low order stages can result in performance degradation of the overall digital filter [4], [5], [6].
Stability of digital filters, overflow oscillation elimination, and attenuation of external interference for smooth implementation of digital algorithms are important research subjects in recent days. A lot of research work deals with the stability analyses of linear digital filters owing to finite word length nonlinearities [7], [8], [9], [10], [11], [12]. The work in [7] ensured input-to-state stability of fixed-point digital filtering methods subject to overflow constraint and disturbance. Later, another approach in [8] has developed a new condition to determine input/output-to-state stability of digital filters with limited word length nonlinearity and external interference. As real digital filters function under both nonlinearities, namely quantization and overflow, the approach in [9] incorporated the combined effect of both nonlinearities and proposed a new criterion to determine global asymptotic stability of two-dimensional Roesser digital filters. To develop a unified stability analysis scheme for fixed-point digital filters under saturation overflow nonlinearity and external perturbations that covers the H∞ performance approach, passivity method, and mixed H∞/ passivity scheme, researchers established a new dissipativity criterion in work [10].
Linear matrix inequalities (LMIs) oriented approach has been developed in [11] to analyze the effect of external interference in fixed-point state-space digital filters having quantization and overflow constraints. The stated approaches for stability analysis of digital filters in [7], [8] and [11] are founded on asymptotical convergence formulations without considering any disturbance. Various global stability schemes have been addressed for 1-D and 2-D digital algorithms (see [7], [8], [9], [10], [11]) that either ignore or incorporate the effects of external perturbations. However, the recent study of Arif et al. [12] presents a local stability criterion of digital filters having saturation overflow nonlinearity using local saturation arithmetic, which can be employed to attain less conservative stability results and efficient estimation of digital hardware parameters for smooth implementation of digital schemes.
The design, testing, analysis, and implementation of nonlinear digital filtering algorithms have received a great research attention in recent years due to the fact that most real physical systems are nonlinear in nature. The work on nonlinear systems is an appealing task due to their wide range applications in online entities counting, thin wall milling chatter, error detection, and nonlinear noising techniques [13], [14], [15], [16]. The work of Coulon et al. [13] examines implementation of a nonlinear filter for an efficient event counting in radioactivity for improving performance of monitoring instrumentation. The feature extraction approach in [14] employs a nonlinear algorithm for response analysis of thin wall milling process. Time-varying digital systems, which can also be modeled as Lipschitz nonlinear digital filters, have been employed for noise removal in image processing [15]. Another example of a nonlinear filter is the median filter [16], which can be straightforwardly employed to the impulsive noise removal. It can be observed from the available literature that the problem of stability analysis under overflow nonlinearity for nonlinear digital filtering systems has been rarely addressed. An exceptional study of Amjad et al. [17] employed global saturation arithmetic to develop stability verification approach for Lipschitz nonlinear digital filtering prototypes by incorporating saturation overflow nonlinearity. The stability criteria are developed both in the presence and absence of external interference by using the properties of saturation nonlinearity and Lipschitz continuity. Several nonlinear systems undergo finite word-length limitations in the form of overflow and quantization nonlinearities when implemented via a digital technology. The Lipschitz nonlinear digital system is an important class of systems, which can be employed for nonlinear filtering, state filtering, control systems, neural networks, and decision making systems. When these systems are implemented via a digital hardware, the complexity arises not only due to hardware limitations but also owing to the inherent complex dynamics of the system and external perturbations. Nevertheless, further research is required to analyze stability of the nonlinear forms of discrete-time filters by incorporating overflow arithmetic owing to the countless applications of nonlinear digital algorithms in nonlinear filtering, state estimation, feedback control, parametric estimation, neural networks, evolutionary computing, telecommunication, and decision making systems.
In this paper, a local approach is exploited to analyze the stability and external interference rejection performance of Lipschitz nonlinear discrete-time filtering methods by including the saturation-type overflow. A new stability criterion is developed by using the Lyapunov theory, local saturation arithmetic, and Lipschitz condition along with the matrix inequality and convex constraints tools. The proposed stability criterion ensures the local asymptotic stability for a class of inherently nonlinear digital filters with zero external interference under saturation overflow. Moreover, the developed approach is extended to guarantee the local H∞ stability subject to energy-bounded external interferences. In comparison with existing (global) techniques, the developed method, incorporating the local overflow arithmetic property, provides a less conservative measure of the interference rejection capability for Lipschitz nonlinear digital filtering schemes. Furthermore, the developed LMI-based criterion can also be used to choose appropriate hardware resources (or the word length) for the specified values of the mutual interferences energy and the word length resolution. It should also be noted that the stability analysis approaches, considered in traditional works like [7], [8], [9], [10], [11], do not allow nonlinear nature of digital filters. To the best of the authors’ knowledge, local stability analysis for the nonlinear discrete-time filters with any form of the hardware overflow constraint by incorporating local overflow arithmetic has been performed for the first time. Comprehensive simulation results for a nonlinear recurrent neural network are presented to depict benefits of the proposed criteria as compared with existing techniques.
The rest of this paper is structured as follows. Section 2 introduces the digital filters under consideration along with local sector condition and assumptions. Two new stability criteria are presented in Section 3. In Section 4, simulations results are presented in order to compare the proposed criteria with exiting approaches. Finally, Section 5 gives the conclusive remarks.
Notations: In order to facilitate understanding of the paper, commonly used notations are employed in this paper. The notations Rp and Rp × q are used to express the space of p × 1 real vectors and p × q real matrices, respectively. The expression Q > 0 (Q ≥ 0) means that Q is symmetric positive definite (semi-positive definite) matrix. For a vector (or a matrix) x, x(i) symbolizes the ith element (or row) of x. The absolute value and Euclidean norm are denoted by |.| and ‖.‖, respectively. Note that the notation ‖.‖ has also been used to denote matrix norm. The symbol ∗ indicates a term induced due to symmetry in a symmetric matrix. Moreover, bold and plain symbols are used to make the distinction between matrices (including vectors) and scalars, respectively.
Section snippets
Problem statement
The nonlinear digital filter under consideration is expressed as where, x(k) ∈ Rn, w(k) ∈ Rp, y(k) ∈ Rm, and g(x(k)) ∈ Rn are state vector, external interference vector, output vector, and (system) nonlinearity, respectively. The function symbolizes the overflow constraint caused by accounting for a digital technology. The matrices A ∈ Rn × n, B ∈ Rn × p, and C ∈ Rmxn denote coefficient matrix, external interferences matrix
Proposed stability criteria
If a nonlinear system (1)-(2) is asymptotically stable, it can be concluded that it will not be affected by the overflow nonlinearity for producing overflow oscillations. Therefore, the following theorem provides a novel regional criterion for asymptotic stability of the digital filter (1)-(2) under Assumption 1 by ignoring the external perturbations.
Theorem 1 Suppose there exists a positive definite symmetric matrix ∈ Rnxn, a positive definite diagonal matrix W ∈ Rnxn, a matrix , and a scalar
Numerical simulation results
In this section, we present a detailed stability and performance analysis for a recurrent neural network [28], which can be employed for filtering applications, in the presence of saturation overflow and external disturbances. It is worth mentioning that recurrent neural networks are inherently nonlinear and, consequently, most of the existing approaches such as [5], [6], [7], [8], [9], [10], [11], [12], assuming linear digital systems, are not applicable to this case. For this, we consider a
Conclusions
This work presented novel local stability criteria for Lipschitz nonlinear digital filtering systems under overflow nonlinearity and disturbances. Two stability conditions have been developed by ignoring and considering the external interferences. The first condition studies the local asymptotical convergence of states of the nonlinear digital filters while the second condition ensures local stability along with H∞ performance under external interferences. It is worth mentioning that the
Novelty of the work
In this paper, a local approach is exploited to analyze the stability and external interference rejection performance of Lipschitz nonlinear discrete-time filtering methods by including the saturation-type overflow. The following are the main contributions of the present study:
- (1)
A new stability criterion is developed by using the Lyapunov theory, local saturation arithmetic, and Lipschitz condition along with the matrix inequality and convex constraints tools.
- (2)
The proposed stability criterion
Acknowledgment
The work of C.K. Ahn was supported in part by the National Research Foundation of Korea through the Ministry of Science, ICT and Future Planning under Grant NRF-2017R1A1A1A05001325.
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