ISS Criterion for Lipschitz Nonlinear Interfered Fixed-Point Digital Filters with Saturation Overflow Arithmetic

Abstract

In this paper, we study the problem of input-to-state stability (ISS) of Lipschitz nonlinear fixed-point digital filters in a state-space realization with saturation arithmetic and external disturbance. Using the Lyapunov theory along with the ‘passivity property’ of saturation nonlinearities and the Lipschitz condition, a new criterion for the ISS of such digital filters is established. The criterion can be used to confirm the diminishing effect of external interference as well as to guarantee the asymptotic stability of digital filters in the absence of interference. Suitable examples are given to illustrate the efficacy of the proposed approach.

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Data Availability

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

Abbreviations

$${\varvec{R}}^{\mu }$$ :

Set of $$\mu \times 1$$ real vectors

$${\varvec{R}}^{\mu \times \tau }$$ :

Set of $$\mu \times \tau$$ real matrices

$${\mathbb{R}}^{ + }$$ :

Set of nonnegative real numbers

$${\varvec{\varPi}}^{{{T}}}$$ :

Transpose of a vector (or matrix) $${\varvec{\varPi}}$$

$${\mathbf{0}}$$ :

Null vector or null matrix of appropriate dimension

$${\varvec{\varPi}} > {\mathbf{0}}$$ :

Matrix $${\varvec{\varPi}}$$ is symmetric and positive definite

$${\varvec{\varPi}} < {\mathbf{0}}$$ :

Matrix $${\varvec{\varPi}}$$ is symmetric and negative definite

$$\left\| {\; \cdot \;} \right\|\;$$ :

Any vector norm or matrix norm

$$\lambda_{\;\min } ({\varvec{\varPi}})$$ :

Minimum eigenvalue of matrix $${\varvec{\varPi}}$$

$$\lambda_{\;\max } ({\varvec{\varPi}})$$ :

Maximum eigenvalue of matrix $${\varvec{\varPi}}$$

References

1. 1.

N. Agarwal, H. Kar, New results on saturation overflow stability of 2-D state-space digital filters. J. Frankl. Inst. 353(12), 2743–2760 (2016)

2. 2.

S. Ahmad, M. Rehan, On observer-based control of one-sided Lipschitz systems. J. Frankl. Inst. 353(4), 903–916 (2016)

3. 3.

C.K. Ahn, Two new criteria for the realization of interfered digital filters utilizing saturation overflow nonlinearity. Signal Process. 95(2), 171–176 (2014)

4. 4.

C.K. Ahn, P. Shi, Dissipativity analysis for fixed-point interfered digital filters. Signal Process. 109(4), 148–153 (2015)

5. 5.

C.K. Ahn, P. Shi, Generalized dissipativity analysis of digital filters with finite-wordlength arithmetic. IEEE Trans. Circuits Syst. II 63(4), 386–390 (2016)

6. 6.

C.K. Ahn, P. Shi, M.V. Basin, Two-dimensional dissipative control and filtering for Roesser model. IEEE Trans. Autom. Control 60(7), 1745–1759 (2015)

7. 7.

C.K. Ahn, P. Shi, H.R. Karimi, Novel results on generalized dissipativity of two-dimensional digital filters. IEEE Trans. Circuits Syst. II 63(9), 893–897 (2016)

8. 8.

C.K. Ahn, L. Wu, P. Shi, Stochastic stability analysis for 2-D Roesser systems with multiplicative noise. Automatica 69(7), 356–363 (2016)

9. 9.

M.U. Amjad, M. Rehan, M. Tufail, C.K. Ahn, H.U. Rashid, Stability analysis of nonlinear digital systems under hardware overflow constraint for dealing with finite word-length effects of digital technologies. Signal Process. 140(11), 139–148 (2017)

10. 10.

A. Antoniou, Digital Signal Processing (McGraw-Hill, Toronto, 2006)

11. 11.

L.A. Aranda, P. Reviriego, J.A. Maestrro, Error detection technique for a median filter. IEEE Trans. Nucl. Sci. 64(8), 2219–2226 (2017)

12. 12.

S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory (SIAM, Philadelphia, 1994)

13. 13.

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)

14. 14.

T.F. Chan, S. Osher, J. Shen, The digital TV filter and nonlinear denoising. IEEE Trans. Image Process. 10(2), 231–241 (2001)

15. 15.

Y.C. Chu, K. Glover, Bounds of the induced norm and model reduction errors for systems with repeated scalar nonlinearities. IEEE Trans. Autom. Control 44(3), 471–483 (1999)

16. 16.

T.A.C.M. Classen, W.F.G. Mecklenbrauker, J.B.H. Peek, Effects of quantization and overflow in recursive digital filters. IEEE Trans. Acoust. Speech Signal Process. 24(6), 517–529 (1976)

17. 17.

R. Coulon, J. Dumazert, V. Kondrasovs, S. Normand, Implementation of a nonlinear filter for online nuclear counting. Radiat. Meas. 87(4), 13–23 (2016)

18. 18.

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)

19. 19.

P. Gahinet, A. Nemirovski, A.J. Laub, M. Chilali, LMI Control Toolbox for Use with MATLAB (The MathWorks Inc., Natick, 1995)

20. 20.

J.P. Hespanha, A.S. Morse, Certainty equivalence implies detectability. Syst. Control Lett. 36(1), 1–13 (1999)

21. 21.

Z.P. Jiang, Y. Wang, Input-to-state stability for discrete-time nonlinear systems. Automatica 37(6), 857–869 (2001)

22. 22.

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)

23. 23.

H. Kar, An improved version of modified Liu–Michel’s criterion for global asymptotic stability of fixed-point state-space digital filters using saturation arithmetic. Digit. Signal Process. 20(4), 977–981 (2010)

24. 24.

C.M. Kellett, A compendium of comparison function results. Math. Control Signals Syst. 26(3), 339–374 (2014)

25. 25.

P. Kokil, S.S. Shinde, Asymptotic stability of fixed-point state-space digital filters with saturation arithmetic and external disturbance: an IOSS approach. Circuits Syst. Signal Process. 34(12), 3965–3977 (2015)

26. 26.

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)

27. 27.

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)

28. 28.

M.K. Kumar, P. Kokil, H. Kar, Novel ISS criteria for digital filters using generalized overflow nonlinearities and external interference. Trans. Inst. Meas. Control 41(1), 156–164 (2019)

29. 29.

Y. Li, H. Li, G. Zhao, Optimal state estimation for finite-field networks with stochastic disturbances. Neurocomputing 414(11), 238–244 (2020)

30. 30.

H. Li, S. Wang, X. Li, G. Zhao, Perturbation analysis for controllability of logical control networks. SIAM J. Control Optim. 58(6), 3632–3657 (2020)

31. 31.

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

32. 32.

X. Liu, J. Xia, J. Wang, H. Shen, Interval type-2 fuzzy passive filtering for nonlinear singularly perturbed PDT-switched systems and its application. J. Syst. Sci. Complex. (2021). https://doi.org/10.1007/s11424-020-0106-9

33. 33.

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

34. 34.

X. Lu, H. Li, A hybrid control approach to $$H_{\infty }$$ problem of nonlinear descriptor systems with actuator saturation. IEEE Trans. Autom. Control (2020). https://doi.org/10.1109/TAC.2020.3046559

35. 35.

R. Marino, P. Tomei, Nonlinear output feedback tracking with almost disturbance decoupling. IEEE Trans. Autom. Control 44(1), 18–28 (1999)

36. 36.

J. Monteiro, R.V. Leuken, Integrated Circuit and System Design: Power and Timing Modeling, Optimization and Simulation (Springer, Berlin, 2010)

37. 37.

T. Ooba, Stability of discrete-time systems joined with a saturation operator on the state-space. IEEE Trans. Autom. Control 55(9), 2153–2155 (2010)

38. 38.

J.G. Proakis, D.G. Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications (Pearson Prentice Hall, New Jersey, 1996)

39. 39.

L.R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, New Jersey, 1975)

40. 40.

P. Rani, P. Kokil, H. Kar, l2l suppression of limit cycles in interfered digital filters with generalized overflow nonlinearities. Circuits Syst. Signal Process. 36(7), 2727–2741 (2017)

41. 41.

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

42. 42.

D. Schlichtharle, Digital Filters: Basics and Design (Springer, Berlin, 2000)

43. 43.

A. Shams, M. Rehan, M. Tufail, C.K. Ahn, W. Ahmed, Local stability analysis and H performance for Lipschitz digital filters with saturation nonlinearity and external interferences. Signal Process. 153(12), 101–108 (2018)

44. 44.

H. Shen, Z. Huang, J. Cao, J.H. Park, Exponential $$H_{\infty }$$ filtering for continuous-time switched neural networks under persistent dwell-time switching regularity. IEEE Trans. Cybern. 50(6), 2440–2449 (2020)

45. 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)

46. 46.

V. Singh, Elimination of overflow oscillations in fixed-point state-space digital filters using saturation arithmetic. IEEE Trans. Circuits Syst. 37(6), 814–818 (1990)

47. 47.

E.D. Sontag, Smooth stabilization implies coprime factorization. IEEE Trans. Autom. Control 34(4), 435–443 (1989)

48. 48.

E.D. Sontag, Input to State Stability: Basic Concepts and Results (Springer, Berlin, 2008)

49. 49.

A.R. Teel, J.P. Hespanha, A. Subbaraman, Equivalent characterization of input-to-state stability for stochastic discrete time systems. IEEE Trans. Autom. Control 59(2), 516–522 (2013)

50. 50.

J. Tsinias, Input to state stability properties of nonlinear systems and applications to bounded feedback stabilization using saturation. ESAIM Control Optim. Calc. Var. 2, 57–87 (1997)

51. 51.

Y. Tsividis, Mixed Analog-Digital VLSI Devices and Technology (World Scientific Publishing, Singapore, 2002)

52. 52.

S. Wu, R. Li, X. Liu, L. Yang, M. Zhu, Experimental study of thin wall milling chatter stability nonlinear criterion, in International Conference on Digital Enterprise Technology, 2016. Procedea CIRP 56 (DET, 2016), pp. 422–427

53. 53.

M. Zhang, Q. Zhu, New criteria of input-to-state stability for nonlinear switched stochastic delayed systems with asynchronous switching. Syst. Control Lett. 129, 43–50 (2019)

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Correspondence to Janmejaya Rout.

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Rout, J., Kar, H. ISS Criterion for Lipschitz Nonlinear Interfered Fixed-Point Digital Filters with Saturation Overflow Arithmetic. Circuits Syst Signal Process (2021). https://doi.org/10.1007/s00034-021-01823-5

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Keywords

• Digital filter
• External interference
• Inherent system nonlinearity
• Input-to-state stability
• Saturation nonlinearity