Article Transactions of the Institute of Measurement and Control 1–13 Ó The Author(s) 2017 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/0142331217709067 journals.sagepub.com/home/tim Criteria for stability of uncertain discrete-time systems with time- varying delays and finite wordlength nonlinearities Siva Kumar Tadepalli 1 , V Krishna Rao Kandanvli 2 and Abhilav Vishwakarma 1 Abstract This paper considers the problem of global asymptotic stability of a class of uncertain discrete-time systems under the influence of finite wordlength nonlinearities (quantization and/or overflow) and time-varying delays. The parameter uncertainties are assumed to be norm-bounded. Utilizing the con- cept of a Wirtinger-based inequality and a reciprocally convex method, two delay-dependent stability criteria are presented. The selection of the cri- teria depends on the type of the nonlinearities, that is, a combination of quantization and overflow or saturation overflow nonlinearities involved in the present systems. The approach presented in this paper yields less conservative results and reduces the computational burden as compared to previ- ously reported criteria. Numerical examples are given to illustrate the effectiveness of the presented approach. Keywords Delay-dependent stability, finite wordlength effect, linear matrix inequality, Lyapunov method, time-varying delay, uncertain system Introduction In the implementation of stable and linear discrete-time dyna- mical systems using finite wordlength processors with fixed- point arithmetic, one commonly encounters several kinds of nonlinearities (e.g. quantization and overflow nonlinearities). These finite wordlength nonlinearities may lead to instability in the designed system. The instability may arise in the form of zero-input limit cycles, which characterize nonlinear sys- tems. While designing a discrete-time system under finite wor- dlength implementation it is, therefore, important to determine the range of system parameters such that the sys- tem is free of limit cycles. The quantization and overflow non- linearities may interact with each other, motivating a considerable number of studies on their combined effects (Kandanvli and Kar, 2009a, 2011; Kar and Singh, 2001; Sim and Pang, 1985; Singh, 2013; Tadepalli and Kandanvli, 2016). However, if the wordlength is large, then the effects of quanti- zation and overflow may be regarded as non-interacting (Claasen et al., 1976; Kandanvli and Kar, 2009a; Singh, 2008; Tadepalli and Kandanvli, 2016). Under this condition, quan- tization effects are neglected when studying the effects of overflow (Chen, 2009; Kandanvli and Kar, 2009b, 2013; Singh, 2013; Tadepalli and Kandanvli, 2016; Tadepalli et al., 2014, 2015) and vice versa (Bose, 1994; Dewasurendra and Bauer, 2008). Saturation overflow nonlinearity, owing to its larger stability region, has also been widely studied (Chen, 2009; Kandanvli and Kar, 2009b, 2013; Kar, 2007; Singh, 2006; Song and Wang, 2013; Tadepalli and Kandanvli, 2016; Tadepalli et al., 2014, 2015). Therefore, the stability analysis of discrete-time systems under the influence of finite wor- dlength nonlinearities is considered to be an important subject of system theoretic study (Chen, 2009; Claasen et al., 1976; Kandanvli and Kar, 2009a,b, 2011, 2013; Kar and Singh, 2001; Sim and Pang, 1985; Singh, 2008, 2013; Tadepalli and Kandanvli, 2016; Tadepalli et al., 2014, 2015). Apart from the instabilities due to finite wordlength imple- mentation, the presence of parameter uncertainties and delays may also lead to instability in discrete-time systems. It is well known that, while deriving delay-dependent stability criteria for such systems, the issue of computational burden may play a vital role (see the recent works by Nam et al., 2015, and Seuret et al., 2015). Several studies have been performed on the stability of discrete-time systems under the simultaneous influence of finite wordlength nonlinearities, parameter uncer- tainties and delays (Kandanvli and Kar, 2008, 2009a,b, 2011, 2013; Tadepalli and Kandanvli, 2016; Tadepalli et al., 2014, 2015). Examples of such systems include digital control 1 Department of Electronics and Telecommunication Engineering, Bhilai Institute of Technology, Durg, India 2 Department of Electronics and Communication Engineering, Motilal Nehru National Institute of Technology Allahabad, Allahabad, India Corresponding author: Siva Kumar Tadepalli, Department of Electronics and Telecommunication Engineering, Bhilai Institute of Technology, Durg, 491001, India. Email: siva.kumar.1678@gmail.com