A Note on the Induced $${\varvec{l}}_\infty $$ l ∞ Stability of Fixed-Point Digital Filters Without Overflow Oscillations and Instability Due to Finite Wordlength Effects

Abstract

A recently reported criterion due to Ahn and Lee (Adv Differ Equ 51:1-7, 2012) for the induced l ∞ stability of fixed-point digital filters without overflow oscillations and instability due to finite wordlength effects is reviewed. This article points out that there is a technical error in the main result and at the same time presents an approach to correcting the main result. Moreover, a relaxed version of the criterion is made available. Illustrative examples are given to demonstrate the effectiveness of the proposed result.

Hankel Norm Based Strict Passivity Performance of Digital Filter Using Saturation Nonlinearities

“…Due to this state in the system, there occurs external interference which results in instability and degrades the performance [8,9]. Thus, several results have been reported based on stability of digital filters with overflow nonlinearities when external disturbance is present [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24].…”

Stability of digital filters subject to external interference and state-delay

“…Hence, analysis of externally disturbed digital filters is one of the important research topics and has attracted continuous interest of researchers (Ahn, 2013a, 2013b; Ahn, 2014; Ahn and Shi 2016a, 2016b; Arockiaraj et al 2017; Kokil and Arockiaraj 2017; Kokil and Shinde 2017; Kokil et al, 2012, 2018; Kumar et al, 2019; Rani et al 2017). To address stability problems related to digital filters with disturbances, popular methods such as H ∞ filtering (Kokil et al, 2012, 2018), l 2 − l ∞ (Ahn, 2013a; Rani et al, 2017), induced l ∞ (Kokil and Arockiaraj, 2017; Kokil and Shinde, 2017), input-to-state stability (Ahn, 2014; Kumar et al, 2019) and passivity (Ahn, 2013b; Ahn and Shi, 2016a, 2016b; Arockiaraj et al, 2017) based approaches have been exploited. However, most of the existing results (Ahn, 2013a, 2013b, 2014; Ahn and Shi, 2016a, 2016b; Arockiaraj et al, 2017; Kokil and Arockiaraj, 2017; Kokil and Shinde, 2017; Kokil et al, 2012, 2018; Kumar et al, 2019; Rani et al, 2017<...>…”

“…To address stability problems related to digital filters with disturbances, popular methods such as H ∞ filtering (Kokil et al, 2012, 2018), l 2 − l ∞ (Ahn, 2013a; Rani et al, 2017), induced l ∞ (Kokil and Arockiaraj, 2017; Kokil and Shinde, 2017), input-to-state stability (Ahn, 2014; Kumar et al, 2019) and passivity (Ahn, 2013b; Ahn and Shi, 2016a, 2016b; Arockiaraj et al, 2017) based approaches have been exploited. However, most of the existing results (Ahn, 2013a, 2013b, 2014; Ahn and Shi, 2016a, 2016b; Arockiaraj et al, 2017; Kokil and Arockiaraj, 2017; Kokil and Shinde, 2017; Kokil et al, 2012, 2018; Kumar et al, 2019; Rani et al, 2017) are not applicable to check stability of digital filters in the simultaneous presence of finite wordlength nonlinearities, external interference and state-delay.…”

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

“…Therefore, the study of stability properties of digital filters with finite wordlength non-linearities is important due to its theoretical interest as well as for application to practical system design. Several researchers have extensively investigated the stability of non-linear systems (Ahn, 2011, 2013a, 2013b, 2013c, 2014a, 2014b, 2015; Ahn and Kim, 2013; Ahn and Lee, 2012; Bose and Chen, 1991; Dey and Kar, 2011a, 2011b; Ebert et al, 1969; Kandanvli and Kar, 2008, 2012; Kar, 2007, 2010; Kar and Singh, 1998, 2000, 2001; Kokil, 2014; Kokil and Kar, 2012; Kokil and Shinde, 2015, 2016; Kokil et al, 2012; Li and Zheng, 2012; Liu and Michel, 1992; Sandberg, 1979; Singh, 1985, 2006). In general, high-order recursive digital filters can be very sensitive to finite wordlength non-linearities.…”

“…Criteria for the realization of limit cycle-free state-space direct form digital filters employing saturation arithmetic have been presented in Ahn and Kim (2013) and Kokil and Shinde (2015). By making use of the induced l ∞ approach, stability conditions for the saturation overflow stability of fixed-point state-space interfered digital filters have been derived in Ahn and Lee (2012) and Kokil and Shinde (2016). Recently, a few criteria dealing with H ∞ elimination of overflow oscillations in fixed-point state-space digital filters employing saturation arithmetic and external disturbance have been appeared (Ahn, 2011, 2015; Kokil et al, 2012).…”