Overflow Arithmetic Research Articles

Improved delay-dependent stability analysis of digital filters with generalized overflow arithmetic

This paper introduces a new set of conditions that establish Input-to-State Stability (ISS) for digital filters with external disturbance input and time-varying delays, all within the context of generalized overflow arithmetic. A unique augmented Lyapunov function and a novel summation inequality are applied to formulate a new criterion for ISS. This criterion assesses the ISS of digital filters in the presence of external disturbance, enabling the identification of asymptotic stability even in the absence of such disturbance. The criteria are developed using linear matrix inequalities (LMIs). The efficacy of the results is demonstrated through three numerical examples, highlighting that the proposed criteria are less conservative than some recent results.

Novel Global Asymptotic Stability Conditions for Discrete-Time Systems With Time-Varying Delay and Generalized Overflow Arithmetic

Novel global asymptotic stability (GAS) criterion for discrete-time (DT) systems with generalized overflow arithmetic and time-varying delay is presented in this brief. The criterion is based on two unique lemmas that allow for a more precise characterization of nonlinearities by utilizing system information in greater detail. The approach leads to significantly relaxed stability conditions as compared to a recently reported criterion.

Hankel norm performance of 2-D digital filters described by Roesser model with overflow arithmetic

ABSTRACT This paper is concerned with the problem of asymptotic stability with Hankel norm performance (HNP) of two-dimensional (2-D) digital filter characterised by Roesser model, where the 2-D system includes overflow nonlinearities and external interference or excitation of finite duration. The underlying nonlinearities cover the usual types of overflow arithmetic utilised in practice such as saturation, two’s complement, zeroing and triangular. To solve this problem, new sufficient criterion is proposed by employing quadratic 2-D Lyapunov function. The proposed criterion guarantees that the addressed system has 2-D HNP bound. This criterion can examine the unwanted memory effect (ME) reduction of 2-D interfered digital filters for past excitations. In the absence of external inputs, the asymptotic convergence of 2-D digital filters is established under the proposed criterion. The developed HNP criterion is in linear matrix inequality framework and, therefore, computationally tractable. In addition, the optimisation problem is formulated to obtain the minimum HNP of the 2-D interfered digital filter. To illustrate the effectiveness of the proposed criterion, a numerical example is provided. The criteria addressed in current paper can give a whole analysis framework for the unwanted MEs of filters.

Stability of 2D Lipschitz Nonlinear Digital Filters in Fornasini–Marchesini Second Model with Overflow Arithmetic

Stability of 2D Lipschitz Nonlinear Digital Filters in Fornasini–Marchesini Second Model with Overflow Arithmetic

Criterion for the Global Asymptotic Stability of Fixed-Point Lipschitz Nonlinear Digital Filter with 2’s Complement Overflow Arithmetic

This paper is concerned with the global asymptotic stability (GAS) problem of fixed-point Lipschitz nonlinear digital filters employing 2’s complement overflow arithmetic. Nonlinear digital filtering finds immense applications in various fields such as adaptive systems and controllers, digital controllers and observers for nonlinear systems, realization of neural networks using digital hardware, controllers for feedback linearization, etc. Lipschitz nonlinear digital filter is considered in this paper as it is frequently employed in nonlinear digital filtering, state filtering, neural networks, feedback control, digital controllers, decision-taking systems and so on. Based on Lyapunov theory, the property of 2’s complement overflow arithmetic and Lipschitz condition associated with system nonlinearities, a new criterion for the suppression of overflow oscillations in 2’s complement state variable realizations of digital filters is established. Several examples along with simulation results are provided to highlight the utility of the criterion.

Improved Delay-Dependent Stability Analysis of Fixed-Point State-Space Digital Filters With Time-Varying Delay and Generalized Overflow Arithmetic

This paper is concerned with the stability analysis of fixed-point state-space digital filters with generalized overflow arithmetic and a time-varying delay. This paper aims to derive a delay and nonlinear function bound dependent asymptotical stability criterion with less conservatism. Firstly, a new Lyapunov functional with several augmented terms, including extra free matrices and overflow nonlinear function, is constructed such that it has a relaxed positive condition. Then, for bounding the summation term arising in the forward difference of Lyapunov functional, a new lemma is developed to introduce the terms for linking the delayed states and the overflow nonlinear function, the Wirtinger-based summation inequality and several zero-value terms are applied to add more cross terms. As a result, a stability criterion with less conservatism is established and its conservatism. Finally, several numerical examples are given to illustrate the advantages of the proposed method.

Input-to-State Stability Analysis of Digital Filters With Generalized Overflow Arithmetic and Time-Varying Delay

Input-to-State Stability Analysis of Digital Filters With Generalized Overflow Arithmetic and Time-Varying Delay

Novel Criterion for Preventing Overflow Oscillations in Fixed-Point Digital Filters With State Saturation

This letter presents a novel global asymptotic stability (GAS) criterion for fixed-point digital filters (DFs) employing saturation overflow arithmetic. The criterion is developed using an improved characterization of saturation nonlinearities (as prevailing in the DF under study) and Lyapunov theory. The criterion outperforms a series of existing criteria.

Realisation of overflow oscillation-free fixed-point digital filters with 2’s complement arithmetic

ABSTRACT In this paper, we focus on the problem of overflow oscillation elimination in 2’s complement state variable realisation of fixed-point digital filters. A new global asymptotic stability (GAS) criterion for the zero-input fixed-point state-space digital filters employing 2’s complement overflow arithmetic is proposed. The proposed criterion captures the structural properties of 2’s complement overflow nonlinearities in a greater detail as compared to several existing criteria. A comparative evaluation of the proposed criterion with various previously reported criteria is made. The obtained criterion turns out to be less stringent than several existing results. An example along with simulation results to substantiate the utility of the criterion is also provided.

An Improved Stability Criterion for Digital Filters With Generalized Overflow Arithmetic and Time-Varying Delay

This brief concerns the stability of fixed-point state-space digital filters with generalized overflow arithmetic and a time-varying delay. A relaxed lemma that includes a commonly used lemma as a special case is developed to show the relationship between the state and the overflow arithmetic, and a Lyapunov functional with positive-definite conditions that are more relaxed than usual is constructed. As a result, a less conservative stability criterion is established, and the efficiency and merit of the criterion are illustrated with a numerical example.

Realizability Condition for Digital Filters With Time Delay Using Generalized Overflow Arithmetic

This brief investigates the input-to-state stability problem of digital filters in the presence of both external disturbance and time delay using generalized overflow arithmetic. A sufficient condition, under which the considered digital filter is realizable and input-to-state stable, is first derived with the help of a tighter matrix inequality. Next, based on this realizability condition, an asymptotic stability criterion is obtained. It is shown that the proposed criterion is less conservative than some existing results. Then two numerical examples are presented to demonstrate the efficiency of the obtained realizability condition and stability criterion.

Local stability analysis and H∞ performance for Lipschitz digital filters with saturation nonlinearity and external interferences

Abstract This paper proposes a novel method to analyze the local stability of Lipschitz nonlinear digital filtering schemes under saturation overflow nonlinearity. Conditions for the stability analysis and robust performance estimation are provided in the form of matrix inequalities by utilizing Lyapunov theory, local saturation overflow arithmetic, and Lipschitz condition. The proposed criterion ascertains (local) asymptotic stability in the absence of perturbations. Under the effects of external interferences, a condition for the local stability, ensuring the H∞ performance objective, is developed. The proposed approach offers a less conservative and more accurate estimate of H∞ performance index than the global method by utilizing a bound on the interferences energy. Moreover, the proposed criterion, in contrast to the existing global methods, can be employed to choose an adequate word length of a digital hardware for the specified values of tolerable perturbations energy, H∞ performance index, and fixed-point resolution. It is worth mentioning that analysis approaches have not been completely reported in the literature, in which local stability criteria for nonlinear discrete-time filtering prototypes under both overflow and disturbances have been developed. A detailed stability analysis for a nonlinear recurrent neural network is performed for demonstrating the effectiveness of the proposed scheme.

ISS Criterion for the Realization of Fixed-Point State-Space Digital Filters with Saturation Arithmetic and External Interference

This paper investigates the problem of the input-to-state stability (ISS) of fixed-point state-space digital filters in the presence of saturation overflow arithmetic and external interference. By utilizing an augmented Lyapunov function and the passivity property associated with multiple saturation nonlinearities, a new criterion for the ISS of interfered digital filters with saturation is proposed. When the external interference is present, the criterion can be used to determine ISS of digital filters. Without the external interference, the criterion is capable of detecting the asymptotic stability of digital filters. The proposed ISS criterion is in linear matrix inequality framework and, hence, is computationally tractable. The obtained criterion turns out to be less conservative than a recently reported ISS criterion. The superiority of the presented approach over the existing techniques is confirmed by numerical examples.

Analysis of Memory Effects in Digital Filters with Overflow Arithmetic

<p>This paper deals with the problem of undesired memory effects in nonlinear digital<br />filters owing to the influence of past excitations on future outputs. The nonlinearities under consideration cover the usual types of overflow arithmetic employed in practice. Based on the Hankel norm performance, a new criterion is proposed to ensure the reduction of undesired memory effects in digital filters with overflow arithmetic. In absence of external input, the nonexistence of overflow oscillations is also confirmed by the proposed criterion. A numerical example together with simulation result showing the effectiveness of the criterion is given.</p>

Criterion for limit cycle-free state-space digital filters with external disturbances and generalized overflow non-linearities

This paper investigates the problem of [Formula: see text] elimination of overflow oscillations in fixed-point state-space digital filters using generalized overflow non-linearities and external disturbance. The generalized overflow non-linearities under consideration cover the common types of overflow arithmetic used in practice, for instance zeroing, two’s complement, triangular and saturation. New criteria are established to ensure not only exponential stability, but also reduction in the effect of external disturbance to an [Formula: see text] norm constraint. The obtained criteria are in linear matrix inequality (LMI) framework and, hence, are computationally tractable. The presented approach constitutes a generalization over several previously reported approaches for the [Formula: see text] elimination of overflow oscillations. For saturation non-linearities, the presented result turns out to be less conservative than several existing criteria. Numerical examples are provided to demonstrate the effectiveness of the presented approach.

Stochastic stability analysis for 2-D Roesser systems with multiplicative noise

This paper is concerned with the robust stochastic stability analysis for two-dimensional (2-D) discrete state-multiplicative noisy systems (SMNSs) in the Roesser form. We first derive a new sufficient condition under which linear discrete 2-D SMNSs are 2-D robustly stochastically stable. The underlying problem can then be recast as a convex problem expressed by linear matrix inequalities, which can be facilitated using existing numerical algorithms. We then apply the obtained result to examine the 2-D robust stochastic stability for 2-D digital filters in the Roesser form with random coefficient variation and saturation overflow arithmetic based on free-weighting matrices and diagonally dominant matrices. An illustrative example is presented to verify the usefulness and potential of the proposed results.

Dissipativity analysis for fixed-point interfered digital filters

This paper is concerned with establishing a new criterion for the (Q, S, R)-α-dissipativity of fixed-point interfered state-space digital filters with saturation overflow arithmetic. The objective of this paper is to present the H∞ performance, passivity, and mixed H∞/passivity criteria in a unified framework. By tuning the weight matrices, the proposed criterion reduces to the H∞ performance, passivity, and mixed H∞/passivity criteria. Improved criteria are also proposed for reducing the conservatism of the proposed criterion. These criteria are expressed with linear matrix inequalities (LMIs). A numerical example shows the effectiveness of the proposed results.

A note on the improved LMI-based criterion for global asymptotic stability of 2-D state-space digital filters described by Roesser model using twoʼs complement overflow arithmetic

This paper focuses on some critical issues in a recently reported approach [V. Singh, Improved LMI-based criterion for global asymptotic stability of 2-D state-space digital filters described by Roesser model using twoʼs complement overflow arithmetic, Digital Signal Process. 22 (2012) 471–475] for the global asymptotic stability of two-dimensional (2-D) fixed-point state-space digital filters described by the Roesser model employing twoʼs complement overflow arithmetic. In particular, it is highlighted that the situation where Singhʼs approach can be applied to ensure the global asymptotic stability of digital filters in the presence of twoʼs complement overflow nonlinearities is not conceivable. • We focus on critical issues in a recent approach for stability of 2-D digital filter. • It is hard to conceive any situation where the reported approach can be applied. • Implication of the approach in the context of 1-D digital filter is discussed.

IOSS Criterion for the Absence of Limit Cycles in Interfered Digital Filters Employing Saturation Overflow Arithmetic

In this paper, a new criterion for the absence of limit cycles in fixed-point state-space digital filters with saturation overflow arithmetic and external interference is proposed via an input/output-to-state stability (IOSS) approach. The criterion ensures not only the asymptotic stability, but also the reduction of the effect of the external interference. This criterion is represented by linear matrix inequality (LMI) and, hence, is computationally tractable. Via a numerical example, we demonstrate the effectiveness of the proposed IOSS criterion.

2-D digital filter realization without overflow oscillations

A novel criterion for the elimination of overflow oscillations in 2-D state-space digital filters described by the Roesser model employing two’s complement overflow arithmetic is presented. The criterion takes the form of linear matrix inequality (LMI) and, hence, is computationally tractable. The criterion is a generalization and improvement over an earlier criterion. An example shows the effectiveness of the new criterion.