Skip to main content
Log in

Limit Cycle-Free Realization of Discrete-Time Delayed Systems with External Interference and Finite Wordlength Nonlinearities

  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

Abstract

The paper studies the problem of exponential stability and \(H_{\infty }\) performance of externally interfered discrete-time systems (DTS) with time-varying delays and finite wordlength nonlinearities. The presented approach uses Wirtinger-based inequality (WBI) and reciprocally convex inequality (RCI) that are suitable to approximate the sum terms in the forward difference of Lyapunov–Krasovskii functional (LKF). The results obtained can be used to determine the exponential stability and to diminish the effects of external interference. Examples are presented to illustrate the usefulness of the obtained results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data Availability

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. A.A. Abd El-Latif, B. Abd-El-Atty, M. Amin, A.M. Iliyasu, Quantum-inspired cascaded discrete-time quantum walks with induced chaotic dynamics and cryptographic applications. Scientific Reports 1930, 1–16 (2020)

  2. N. Agarwal, H. Kar, Improved Criterion for Robust Stability of Discrete-Time State-Delayed Systems with Quantization/Overflow Nonlinearities. Circuits Syst. Signal Process. 38(11), 4959–4980 (2019)

    Article  Google Scholar 

  3. C.K. Ahn, Criterion for the elimination of overflow oscillations in fixed-point digital filters with saturation arithmetic and external disturbance. AEU-Int. J. Electron. Commun. 65(9), 750–752 (2011)

    Article  Google Scholar 

  4. C.K. Ahn, A new realization criterion for 2-D digital filters in the Fornasini–Marchesini second model with interference. Signal Process. 104, 225–231 (2014)

    Article  Google Scholar 

  5. M. Amin, AA Abd El-Latif, Efficient modified RC5 based on chaos adapted to image encryption. J. Electron. Imaging 19(1), 013012 (2010)

    Article  Google Scholar 

  6. T. Bose, Asymptotic stability of two-dimensional digital filters under quantization. IEEE Trans. Signal Process. 42(5), 1172–1177 (1994)

    Article  Google Scholar 

  7. S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear matrix inequalities in systems and control theory (SIAM, Philadelphia, 1994)

    Book  MATH  Google Scholar 

  8. H. J. Butterweck, J.H.F. Ritzerfeld, M.J. Werter, Finite wordlength effects in digital filters: A review. EUT report 88-E-205 (Eindhoven University of Technology, Eindhoven, The Netherlands, 1988)

  9. K. Chakrabarty, S.S. Iyengar, H. Qi, E. Cho, Grid coverage for surveillance and target location in distributed sensor networks. IEEE Trans. Comput. 51(12), 1448–1453 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. T.L. Chang, Suppression of limit cycles in digital filters designed with one magnitude-truncation quantizer. IEEE Trans. Circuits Syst. 28(2), 107–111 (1981)

    Article  Google Scholar 

  11. S.F. Chen, Asymptotic stability of discrete-time systems with time-varying delay subject to saturation nonlinearities. Chaos Solitons Fractals 42(2), 1251–1257 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Diksha, P. Kokil, H. Kar, Criterion for the limit cycle free state-space digital filters with external disturbances and quantization/overflow nonlinearities. Eng. Comput. 33(1), 64–73 (2016)

    Article  Google Scholar 

  13. P. Gahinet, A. Nemirovskii, A.J. Laub, M. Chilali, LMI control toolbox—For use with Matlab (The MATH Works Inc., Natick, 1995)

    Google Scholar 

  14. H. Gao, T. Chen, New results on stability of discrete-time systems with time-varying state delay. IEEE Trans. Automatic Control 52(2), 328–334 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  15. V.K.R. Kandanvli, H. Kar, Robust stability of discrete-time state-delayed systems with saturation nonlinearities: linear matrix inequality approach. Signal Process. 89(2), 161–173 (2009)

    Article  MATH  Google Scholar 

  16. V.K.R. Kandanvli, H. Kar, An LMI condition for robust stability of discrete-time state-delayed systems using quantization/overflow nonlinearities. Signal Process. 89(11), 2092–2102 (2009)

    Article  MATH  Google Scholar 

  17. V.K.R. Kandanvli, H. Kar, Delay-dependent LMI condition for global asymptotic stability of discrete-time uncertain state-delayed systems using quantization/overflow nonlinearities. Int. J. Robust Nonlinear Control 21(14), 1611–1622 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. H. Kar, An LMI based criterion for the nonexistence of overflow oscillations in fixed-point state-space digital filters using saturation arithmetic. Digit. Signal Process. 17(3), 685–689 (2007)

    Article  Google Scholar 

  19. P. Kokil, S.X. Arockiaraj, Novel results for induced \(l_{\infty}\) stability for digital filters with external noise. Fluct. Noise Lett. 16(4), 1750032 (2017)

    Article  Google Scholar 

  20. P. Kokil, V.K.R. Kandanvli, H. Kar, A note on the criterion for the elimination of overflow oscillations in fixed-point digital filters with saturation arithmetic and external disturbance. AEU-Int. J. Electron. Commun. 66(9), 780–783 (2012)

    Article  Google Scholar 

  21. P. Kokil, C.G. Parthipan, S. Jogi, H. Kar, Criterion for realizing state-delayed digital filters subjected to external interference employing saturation arithmetic. Clust. Comput. 22(6), 15187–15194 (2019)

    Article  Google Scholar 

  22. M.K. Kumar, H. Kar, ISS Criterion for the Realization of Fixed-Point State-Space Digital Filters with Saturation Arithmetic and External Interference. Circuits Syst. Signal Process. 37(12), 5664–5679 (2018)

    Article  MathSciNet  Google Scholar 

  23. M.K. Kumar, P. Kokil, H. Kar, A new realizability condition for fixed-point state-space interfered digital filters using any combination of overflow and quantization nonlinearities. Circuits Syst. Signal Process. 36(8), 3289–3302 (2017)

    Article  MATH  Google Scholar 

  24. O.M. Kwon, M.J. Park, J.H. Park, S.M. Lee, E.J. Cha, Improved delay-dependent stability criteria for discrete-time systems with time-varying delays. Circuits Syst. Signal Process. 32(4), 1949–1962 (2013)

    Article  MathSciNet  Google Scholar 

  25. J. Lee, Constructive and discrete versions of the Lyapunov’s stability theorem and the LaSalle’s invariance theorem. Commun. Korean Math. Soc. 17(1), 155–164 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  26. A. Lepschy, G.A. Mian, U. Viaro, Effects of quantization in second-order fixed-point digital filters with two’s complement truncation quantizers. IEEE Trans. Circuits Syst. 35(4), 461–466 (1988)

    Article  Google Scholar 

  27. L. Li, B. Abd-El-Atty, A.A. Abd El-Latif, A. Ghoneim, Quantum color image encryption based on multiple discrete chaotic systems, in 2017 Federated Conference on Computer Science and Information Systems (FedCSIS), Prague, Czech Republic (IEEE, 2017), pp. 555–559.

  28. T. Li, N. Sun, Q. Lin, J. Li, Improved criterion for the elimination of overflow oscillations in digital filters with external disturbance. Adv. Difference Equ. 2012(1), 197 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  29. D. Liu, A.N. Michel, Asymptotic stability of discrete-time systems with saturation nonlinearities with applications to digital filters. IEEE Trans. Circuits Syst. I Fundam. Theory Appl. 39(10), 798–807 (1992)

    Article  MATH  Google Scholar 

  30. R. Liu, H. Xu, E. Zheng, Y. Jiang, Adaptive filtering for intelligent sensing speech based on multi-rate LMS algorithm. Clust. Comput. 20(2), 1493–1503 (2017)

    Article  Google Scholar 

  31. J. Liu, J. Zhang, Note on stability of discrete-time time-varying delay systems. IET Control Theory & Applic. 6(2), 335–339 (2012)

    Article  MathSciNet  Google Scholar 

  32. J. Lofberg, YALMIP: a toolbox for modelling and optimization in MATLAB, in International Symposium on CACSD, 2004. Proceedings of the 2004, Taipei, Taiwan (IEEE, 2004), pp. 284–289

  33. M.S. Mahmoud, Stabilization of interconnected discrete systems with quantization and overflow nonlinearities. Circuits Syst. Signal Process. 32(2), 905–917 (2013)

    Article  MathSciNet  Google Scholar 

  34. T.J. Mary, P. Rangarajan, Delay-dependent stability analysis of microgrid with constant and time-varying communication delays. Electric Power Comp. Syst. 44(13), 1441–1452 (2016)

    Article  Google Scholar 

  35. P.T. Nam, P.N. Pathirana, H. Trinh, Discrete Wirtinger-based inequality and its application. J. Frankl. Inst. 352(5), 1893–1905 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  36. Z.T. Njitacke, J. Kengne, H.B. Fotsin, Coexistence of multiple stable states and bursting oscillations in a 4D Hopfield neural network. Circuits Syst. Signal Process. 39(7), 3424–3444 (2020)

    Article  MATH  Google Scholar 

  37. N.S. Nise, Control Systems Engineering, 6th edn. (John Wiley & Sons Inc, USA, 2010)

    MATH  Google Scholar 

  38. V.C. Pal, R. Negi, Q. Zhu, Stabilization of discrete-time delayed systems in presence of actuator saturation based on wirtinger inequality. Math. Probl. Eng. (2019). https://doi.org/10.1155/2019/5954642

    Article  MathSciNet  MATH  Google Scholar 

  39. P. Park, J.W. Ko, C. Jeong, Reciprocally convex approach to stability of systems with time-varying delays. Automatica 47(1), 235–238 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  40. C.G. Parthipan, P. Kokil, Overflow oscillations free implementation of state-delayed digital filter with saturation arithmetic and external disturbance. Trans. Inst. Meas. Control 42(2), 188–197 (2020)

    Article  Google Scholar 

  41. J. Peng, A.A. Abd El-Latif, A. Belazi, Z. Kotulski, Efficient chaotic nonlinear component for secure cryptosystems, in Ninth International Conference on Ubiquitous and Future Networks (ICUFN), Milan, Italy (IEEE, 2017), pp. 989–993.

  42. P. Rani, P. Kokil, H. Kar, \(l_{2} - l_{\infty}\) Suppression of limit cycles in interfered digital filters with generalized overflow nonlinearities. Circuits Syst. Signal Process. 36(7), 2727–2741 (2017)

    Article  MATH  Google Scholar 

  43. M. Rehan, M. Tufail, M.T. Akhtar, On elimination of overflow oscillations in linear time-varying 2-D digital filters represented by a Roesser model. Signal Process. 127, 247–252 (2016)

    Article  Google Scholar 

  44. I. Sandberg, The zero-input response of digital filters using saturation arithmetic. IEEE Trans. Circuits Syst. 26(11), 911–915 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  45. H. Shen, J. Wang, J.H. Park, Z.G. Wu, Condition of the elimination of overflow oscillations in two-dimensional digital filters with external interference. IET Signal Process. 8(8), 885–890 (2014)

    Article  Google Scholar 

  46. P.K. Sim, K.K. Pang, Design criterion for zero-input asymptotic overflow-stability of recursive digital filters in the presence of quantization. Circuits Syst. Signal Process. 4(4), 485–502 (1985)

    Article  MATH  Google Scholar 

  47. G. Strang, Introduction to applied mathematics (Wellesley-Cambridge Press, Wellesley, 1986)

    MATH  Google Scholar 

  48. S.K. Tadepalli, V.K.R. Kandanvli, Improved stability results for uncertain discrete-time state-delayed systems in the presence of nonlinearities. Trans. Inst. Meas. Control 38(1), 33–43 (2016)

    Article  Google Scholar 

  49. S.K. Tadepalli, V.K.R. Kandanvli, H. Kar, A new delay-dependent stability criterion for uncertain 2-D discrete systems described by Roesser model under the influence of quantization/overflow nonlinearities. Circuits Syst. Signal Process. 34(8), 2537–2559 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  50. S.K. Tadepalli, V.K.R. Kandanvli, A. Vishwakarma, Criteria for stability of uncertain discrete-time systems with time-varying delays and finite wordlength nonlinearities. Trans. Inst. Meas. Control 40(9), 2868–2880 (2018)

    Article  Google Scholar 

  51. J. Yu, Z. Deng, M. Yu et al., Design of multiple controllers for networked control systems with delays and packet losses. Trans. Inst. Meas. Control 35(6), 720–729 (2013)

    Article  Google Scholar 

  52. D. Zhang, S.K. Nguang, D. Srinivasan, L. Yu, Distributed filtering for discrete-time T-S fuzzy systems with incomplete measurements. IEEE Trans. Fuzzy Syst. 26(3), 1459–1471 (2017)

    Article  Google Scholar 

  53. D. Zhang, Z. Xu, H.R. Karimi, Q.G. Wang, Distributed filtering for switched linear systems with sensor networks in presence of packet dropouts and quantization. IEEE Trans. Circuits Syst. I 64(10), 2783–2796 (2017)

    Article  Google Scholar 

  54. D. Zhang, Q.G. Wang, D. Srinivasan, H. Li, L. Yu, Asynchronous state estimation for discrete-time switched complex networks with communication constraints. IEEE Trans. Neural Netw. Learn. Syst. 29(5), 1732–1746 (2018)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The authors would like to express their gratitude to the Editor-in-Chief, the Editors and the anonymous reviewers for their constructive comments and suggestions to improve the manuscript.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Kalpana Singh.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

Proof of Theorem 1

Choose the following LKF [50] as

$$ \begin{aligned} V({\varvec{x}}(v)) = & {{\varvec{\Pi}}}^{\text{T}} (v){\varvec{P}}{{\varvec{\Pi}}}(v) + \sum\limits_{{i = v - d_{l} }}^{v - 1} {{\varvec{x}}^{{\text{T}}} (i){\varvec{Z}}_{1} {\varvec{x}}(i)} + \sum\limits_{{i = v - d_{u} }}^{v - 1} {{\varvec{x}}^{{\text{T}}} (i){\varvec{Z}}_{2} {\varvec{x}}(i)} + \sum\limits_{{j = - d_{u} }}^{{ - d_{l} }} {\sum\limits_{i = v + j}^{v - 1} {{\varvec{x}}^{{\text{T}}} (i){\varvec{Z}}_{3} {\varvec{x}}(i)} } \\ & + \sum\limits_{{\theta = - d_{l} + 1}}^{0} {\sum\limits_{j = v - 1 + \theta }^{v - 1} {d_{l} {{\varvec{\upsigma}}}^{{\text{T}}} (j){\varvec{R}}_{1} {{\varvec{\upsigma}}}(j)} } + \sum\limits_{{\theta = - d_{u} + 1}}^{{ - d_{l} }} {\sum\limits_{j = v - 1 + \theta }^{v - 1} {(d_{u} - d_{l} ){{\varvec{\upsigma}}}^{{\text{T}}} (j){\varvec{R}}_{2} {{\varvec{\upsigma}}}(j)} } , \\ \end{aligned} $$
(25)

where \({{\varvec{\Pi}}}^{\text{T}} (v) = [\begin{array}{*{20}c} {{\varvec{x}}^{\text{T}} (v)} & {\sum\limits_{{s = v - d_{l} }}^{v - 1} {{\varvec{x}}^{\text{T}} (s)} } & {\sum\limits_{{s = v - d_{u} }}^{{v - d_{l} - 1}} {{\varvec{x}}^{\text{T}} (s)} } \\ \end{array} ]\) and \({{\varvec{\upsigma}}}(v) = {\varvec{x}}(v + 1) - {\varvec{x}}(v) = {{\varvec{\Omega}}}({\varvec{y}}(v)){ + }{\varvec{e}}(v) - {\varvec{x}}(v).\)

Utilizing Lemmas 1, 2 and following [50], it can be shown that

$$ \begin{aligned} \Delta V({\varvec{x}}(v)) & = V({\varvec{x}}(v + 1)) - V({\varvec{x}}(v)) \\ & \le {{\varvec{\upkappa}}}^{{\text{T}}} (v){{\varvec{\Theta}}}_{1} (d(v)){{\varvec{\upkappa}}}(v) - {\varvec{x}}^{\text{T}} (v){\varvec{Sx}}(v) + \gamma^{2} {\varvec{e}}^{\text{T}} (v){\varvec{e}}(v) - 2\delta \,, \\ \end{aligned} $$
(26)

where

$$\begin{array}{l} {{\varvec{\kappa }}^{\text{T}}}(v) = [\begin{array}{*{20}{c}} {{\varvec{x}^{\text{T}}}(v)}&{{\varvec{x}^{\text{T}}}(v - d{{(}}v{{)}})}&{{\varvec{x}^{\text{T}}}(v - {d_l})}&{{\varvec{x}^{\text{T}}}(v - {d_u})}&{{\varvec{x}^{\text{T}}}(v,0,{d_l})} \end{array}\\ \begin{array}{*{20}{c}} {\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\varvec{x}^{\text{T}}}(v,{d_l},d{{(}}v{{)}})}&{{\varvec{x}^{\text{T}}}(v,d{{(}}v{{)}},{d_u})}&{{{\varvec{\Omega }}^{\text{T}}}({\varvec {y}}(v))}&{{{\varvec{e}}^{\text{T}}}(v)} \end{array}],\end{array}$$
(27a)
$$ \delta = [k_{q} {\varvec{y}}(v) - {{\varvec{\Omega}}}({\varvec{y}}(v))]^{{\text{T}}} {\varvec{G}}\,[{{\varvec{\Omega}}}({\varvec{y}}(v)) - k_{o} {\varvec{y}}(v)]\,. $$
(27b)

In view of (3), the quantity ‘\(\delta\)’ given by (27b) is nonnegative [16, 17]. If the condition \({{\varvec{\Theta}}}_{1} (d(v)) < {\varvec{0}}\) along with (8) is fulfilled, then

$$ \Delta V({\varvec{x}}(v)) < - {\varvec{x}}^{\text{T}} (v){\varvec{Sx}}(v) + \gamma^{2} {\varvec{e}}^{\text{T}} (v){\varvec{e}}(v). $$
(28)

By taking summation on both sides of (28) from 0 to \(\infty\), we obtain

$$ V({\varvec{x}}(\infty )) - V({\varvec{x}}(0)) < - \sum\limits_{v = 0}^{\infty } {{\varvec{x}}^{\text{T}} (v){\varvec{Sx}}(v)} + \gamma^{2} \sum\limits_{v = 0}^{\infty } {{\varvec{e}}^{\text{T}} (v){\varvec{e}}(v)} . $$
(29)

As \(V({\varvec{x}}(\infty )) \ge {0}\) and\(V({\varvec{x}}(0)) = {0,}\) one can see that relation (4) always holds true in view of (29).

In the following, the system (1)–(3) with \({\varvec{e}}(v) = {\varvec{0}}\) is shown to be exponentially stable under the condition \({{\varvec{\Theta}}}_{1} (d(v)) < {\varvec{0}}\). Note that \(V({\varvec{x}}(v))\) satisfies Rayleigh inequality [47]

$$ \begin{aligned} &\left[ {\left\{ {1 + d_{l}^{2} \tau^{2} + (d_{u} - d_{l} )^{2} \tau^{2} } \right\}\lambda_{\min } ({\varvec{P}}) + d_{l} \,\tau^{2} \lambda_{\min } ({\varvec{Z}}_{1} ) + d_{u} \,\tau^{2} \lambda_{\min } ({\varvec{Z}}_{2} )}\right.\hfill \\ &\quad + {(d_{u} - d_{l} + 1)\frac{{(d_{u} + d_{l} )}}{2}\,\tau^{2} \,\lambda_{\min } ({\varvec{Z}}_{3} )} \hfill \\ &\quad + \left. {d_{l}^{2} \frac{{(d_{l} + 1)}}{2}(\tau - 1)^{2} \,\lambda_{\min } ({\varvec{R}}_{1} ) + (d_{u} - d_{l} )^{2} \frac{{(d_{u} + d_{l} + 1)}}{2}(\tau - 1)^{2} \,\lambda_{\min } ({\varvec{R}}_{2} )} \right]\left\| {{\varvec{x}}(v)} \right\|^{2}\hfill \\ & \le V({\varvec{x}}(v)) \le \left[ {\left\{ {1 + d_{l}^{2} \tau^{2} + (d_{u} - d_{l} )^{2} \tau^{2} } \right\}\lambda_{\max } ({\varvec{P}}) + d_{l} \,\tau^{2} \lambda_{\max } ({\varvec{Z}}_{1} ) + d_{u} \,\tau^{2} \lambda_{\max } ({\varvec{Z}}_{2} )} \right. \hfill \\ &\quad+ (d_{u} - d_{l} + 1)\frac{{(d_{u} + d_{l} )}}{2}\tau^{2} \,\,\lambda_{\max } ({\varvec{Z}}_{3} ) + d_{l}^{2} \frac{{(d_{l} + 1)}}{2}(\tau - 1)^{2} \,\lambda_{\max } ({\varvec{R}}_{1} ) \hfill \\& \quad \left. +\, (d_{u} - d_{l} )^{2} \frac{{(d_{u} + d_{l} + 1)}}{2}(\tau - 1)^{2} \,\lambda_{\max } ({\varvec{R}}_{2} ) \right]\left\| {{\varvec{x}}(v)} \right\|^{2} . \hfill \\ \end{aligned} $$
(30)

With \({\varvec{e}}(v) = {\varvec{0}}{,}\) (28) becomes \(\Delta V({\varvec{x}}(v)) < - {\varvec{x}}^{\text{T}} (v){\varvec{Sx}}(v) \le - \lambda_{\min } ({\varvec{S}})\left\| {{\varvec{x}}(v)} \right\|^{2} .\) According to Theorem 3.1 in [25], this condition together with (30) guarantees the exponential stability of the system (1)–(3). With respect to the time-varying delay \(d(v)\), it is obvious that \({{\varvec{\Theta}}}_{1} (d(v))\) is an affine matrix function. Therefore, \({{\varvec{\Theta}}}_{1} (d(v)) < {\varvec{0}}\) if (9) is met. This completes the proof of Theorem 1.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Singh, K., Kandanvli, V.K.R. & Kar, H. Limit Cycle-Free Realization of Discrete-Time Delayed Systems with External Interference and Finite Wordlength Nonlinearities. Circuits Syst Signal Process 41, 4438–4454 (2022). https://doi.org/10.1007/s00034-022-02007-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-022-02007-5

Keywords

Navigation