Abstract
The problem of global asymptotic stability (GAS) of fixed-point state-delayed uncertain discrete-time systems combined with a saturation operator in state-space is investigated in this paper. The uncertainties in the system are presumed to be norm-bounded, which has been frequently employed in robust control for uncertain systems. This paper proposes a new GAS criterion for the considered system. The saturation nonlinearities associated with the system, which operate exclusively in the linear region, are identified using a unique methodology. Such identification leads to an improved characterization of saturation nonlinearities involved in the system. The proposed criterion utilizes an upper bound of parameter uncertainties, an asymptotic bound on the system’s states, and a precise characterization of saturation nonlinearities involving system parameters. The merit of the criterion is also exemplified through examples along with simulation results.
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References
N. Agarwal, H. Kar, New results on saturation overflow stability of 2-D state-space digital filters. J. Franklin Inst. 353(12), 2743–2760 (2016)
N. Agarwal, H. Kar, Overflow oscillation-free realization of digital filters with saturation. COMPEL Int. J. Comput. Math. Electr. Electron. Eng. 37(6), 2050–2066 (2018)
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)
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)
S. Boyd, L. El Ghaoui, E. Feron, V. Balakrishnan, Linear matrix inequalities in system and control theory (SIAM, Philadelphia, 1994)
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, 1988)
M.O. Camponez, A.C.S. Simmer, M. Sarcinelli-Filho, Low-noise zero-input, overflow, and constant-input limit cycle-free implementation of state space digital filters. Digit. Signal Process. 17(1), 335–344 (2007)
T.A.C.M. Claasen, W.F.G. Mecklenbräuker, J.B.H. Peek, Effects of quantization and overflow in recursive digital filters. IEEE Trans. Acoust. Speech Signal Process. ASSP-24(6), 517–529 (1976)
I.I. Delice, R. Sipahi, Delay-independent stability test for systems with multiple time-delays. IEEE Trans. Autom. Control 57(4), 963–972 (2012)
P. Gahinet, A. Nemirovski, A.J. Laub, M. Chilali, LMI control toolbox for use with matlab (The MATH Works Inc., Natick, 1995)
H. Gao, J. Lam, C. Wang, Y. Wang, Delay-dependent output-feedback stabilisation of discrete-time systems with time-varying state delay. IEE Proc. Control Theory Appl. 151(6), 691–698 (2004)
K. Gu, S.I. Niculescu, Survey on recent results in the stability and control of time-delay systems. J. Dyn. Syst. Meas. Control 125(2), 158–165 (2003)
X. Ji, T. Liu, Y. Sun, H. Su, Stability analysis and controller synthesis for discrete linear time-delay systems with state saturation nonlinearities. Int. J. Syst. Sci. 42(3), 397–406 (2011)
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)
V.K.R. Kandanvli, H. Kar, Delay-dependent stability criterion for discrete-time uncertain state-delayed systems employing saturation nonlinearities. Arab. J. Sci. Eng. 38, 2911–2920 (2013)
V.K.R. Kandanvli, H. Kar, Global Asymptotic stability of 2-D digital filters with a saturation operator on the state-space. IEEE Trans. Circuits Syst. II 67(11), 2742–2746 (2020)
H. Kar, V. Singh, A new criterion for the overflow stability of second-order state-space digital filters using saturation arithmetic. IEEE Trans. Circuits Syst. I 45(3), 311–313 (1998)
H. Kar, V. Singh, Elimination of overflow oscillations in fixed-point state-space digital filters with saturation arithmetic: an LMI approach. IEEE Trans. Circuits Syst. II 51(1), 40–42 (2004)
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)
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)
P. Kokil, S. Jogi, C.K. Ahn, H. Kar, An improved local stability criterion for digital filters with interference and overflow nonlinearity. IEEE Trans. Circuits Syst. II 67(3), 595–599 (2020)
S. Koshita, M. Abe, M. Kawamata, Variable state-space digital filters using series approximations. Digit. Signal Process. 60, 338–349 (2017)
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)
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)
G. Li, L. Meng, Z. Xu, J. Hua, A novel digital filter structure with minimum roundoff noise. Digit. Signal Process. 20(4), 1000–1009 (2010)
W. Ling, Nonlinear digital filters: analysis and applications (Elsevier, California, 2007)
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)
J. Löfberg, YALMIP: a toolbox for modeling and optimization in MATLAB, in International Symposium on CACSD, 2004. Proceedings of the 2004 (IEEE, 2004), pp. 284–289
M.S. Mahmoud, Robust control and filtering for time-delay systems (Marcel Dekker, New York, 2000)
M.S. Mahmoud, Stabilization of interconnected discrete systems with quantization and overflow nonlinearities. Circuits Syst. Signal Process. 32(2), 905–917 (2013)
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)
X. Meng, J. Lam, B. Du, H. Gao, A delay-partitioning approach to the stability analysis of discrete-time systems. Automatica 46(3), 610–614 (2010)
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)
T. Ooba, Asymptotic stability of two-dimensional discrete systems with saturation nonlinearities. IEEE Trans. Circuits Syst. I 60(1), 178–188 (2013)
R.M. Palhares, C.E. de Souza, P.L.D. Peres, Robust filtering for uncertain discrete-time state-delayed systems. IEEE Trans. Signal Process. 49(8), 1696–1703 (2001)
H. Pan, W. Sun, H. Gao, X. Jing, Disturbance observer-based adaptive tracking control with actuator saturation and its application. IEEE Trans. Autom. Sci. Eng. 13(2), 868–875 (2016)
J.P. Richard, Time-delay systems: an overview of some recent advances and open problems. Automatica 39(10), 1667–1694 (2003)
J.H.F. Ritzerfeld, Noise gain expressions for low noise second-order digital filter structures. IEEE Trans. Circuits Syst. II 52(4), 223–227 (2005)
J. Rout, H. Kar, New ISS result for Lipschitz nonlinear interfered digital filters under various concatenations of quantization and overflow. Circuits Syst. Signal Process. 40, 1852–1867 (2021)
T. Shen, Z. Yuan, X. Wang, Stability analysis for digital filters with multiple saturation nonlinearities. Automatica 48(10), 2717–2720 (2012)
V. Singh, A new realizability condition for limit cycle-free state-space digital filters employing saturation arithmetic. IEEE Trans. Circuits Syst. 32(10), 1070–1071 (1985)
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)
V. Singh, Modified form of Liu–Michel’s criterion for global asymptotic stability of fixed-point state-space digital filters using saturation arithmetic. IEEE Trans. Circuits Syst. II 53(12), 1423–1425 (2006)
V. Singh, New criterion for stability of discrete-time systems joined with a saturation operator on the state-space. AEU Int. J. Electron. Commun. 66(6), 509–511 (2012)
X. Su, P. Shi, L. Wu, S.K. Nguang, Induced filtering of fuzzy stochastic systems with time-varying delays. IEEE Trans. Cybernetics 43(4), 1251–1264 (2013)
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)
M. Wu, Y. He, J.-H. She, Stability analysis and Robust control of time-delay systems (Springer, Berlin, 2010)
W. Xia, W.X. Zheng, S. Xu, Extended dissipativity analysis of digital filters with time delay and Markovian jumping parameters. Signal Process. 152, 247–254 (2018)
W. Xia, W.X. Zheng, S. Xu, Realizability condition for digital filters with time delay using generalized overflow arithmetic. IEEE Trans. Circuits Syst. II 66(1), 141–145 (2019)
S. Xu, J. Lam, A survey of linear matrix inequality techniques in stability analysis of delay systems. Int. J. Syst. Sci. 39(12), 1095–1113 (2008)
L. Zhang, P. Shi, E.-K. Boukas, output-feedback control for switched linear discrete-time systems with time-varying delays. Int. J. Control 80(8), 1354–1365 (2007)
C.K. Zhang, K.Y. Xie, Y. He, Q.G. Wang, M. Wu, An improved stability criterion for digital filters with generalized overflow arithmetic and time-varying delay. IEEE Trans. Circuits Syst. II 67(10), 2099–2103 (2020)
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Agarwal, N., Kar, H. Robust Stability Criterion for State-Delayed Discrete-Time Systems Combined with a Saturation Operator on the State-Space. Circuits Syst Signal Process 41, 5392–5413 (2022). https://doi.org/10.1007/s00034-022-02037-z
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DOI: https://doi.org/10.1007/s00034-022-02037-z