Abstract
In this paper, we study the problem of stability robustness of two-dimensional discrete systems in a local state-space setting. Two methods are proposed for efficient numerical evaluation of the exact complex perturbation bound ν. The first method combines Byers' bisection method with a three-point-pattern optimization technique to compute ν. The second method utilizes a direct optimization technique to find the bound. In addition, a 2-D Lyapunov approach is proposed to obtain two lower bounds of ν, and numerical techniques for solving the constant 2-D Lyapunov equation involved are presented. The paper is concluded with an example illustrating the main results obtained.
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References
W.-S. Lu, A. Antoniou, and P. Agathoklis, “Stability of 2-D Digital Filters under Parameter Variations,”IEEE Trans. Circuits Syst., vol. 33, no. 5, pp. 476–482, 1986.
K.D. Kim and N.K. Bose, “Invariance of the Strict Hurwitz Property for Bivariate Polynomials under Coefficient Perturbations,”IEEE Trans. Automat. Contr., vol. AC-32, pp. 1172–1174, 1988.
S. Basu, “On the Multidimensional Generalization of Robustness of Scattering Hurwitz Property of Complex Polynomials,”IEEE Trans. Circuits Syst., vol. 36, pp. 1159–1167, 1989.
J. Kogan, “Computation of Stability Radius for Families of Bivariate Polynomials,”Multidimensional Syst. Signal Process., vol. 4, pp. 151–165, 1993.
W.-S. Lu, “Stability Robustness of Two-Dimensional Discrete Systems and its Computation,”IEEE Trans. Circuits Syst., vol. 36, pp. 285–288, 1989.
R.V. Patel and M. Toda, “Quantitative Measures of Robustness of Multivariable Systems,” inProc. ACC, San Francisco, CA, Paper TD3-A, 1980.
R.K. Yedavalli, “Perturbation Bounds for Robust Stability in Linear State Space Models,”Int. J. Control, vol. 42, pp. 1507–1517, 1985.
M.E. Sezer and D.D. Siljak, “Robust Stability of Discrete Systems,”Int. J. Control, vol. 48, no. 5, pp. 2055–2063, 1988.
P. Agathoklis, E.I. Jury, and M. Mansour, “The Discrete-Time Strictly Bounded-real Lemma and its Computation of Positive Definite Solutions to the 2-D Lyapunov Equations,”IEEE Trans. Circuits Syst., vol. 36, pp. 820–837, 1989.
B.D.O. Anderson, P. Agathoklis, E.I. Jury, and M. Mansour, “Stability and the Matrix Lyapunov Equation for Discrete 2-D Systems,”IEEE Trans. Circuits Syst., vol. 33, pp. 261–267, 1986.
R. Byers, “A Bisection Method for Measuring the Distance of a Stable Matrix to the Unstable Matrices,”SIAM J. Sci. Statist. Comput., vol. 9, no. 5, pp. 875–881, 1988.
J.H. Lodge and M.M. Fahmy, “Stability and Overflow Oscillations in 2-D State-Space Digital Filters,”IEEE Trans. Acoust., Speech, Signal Processing, vol. 29, pp. 1161–1172, 1981.
R. Fletcher,Practical methods of Optimization, Chichester, UK: Wiley, 1987.
W.H. Lee, “Robustness Analysis for State Space Models,” Report TP-151, Alphatech Inc. 1982.
J.M. Martin, “State-Space Measure for Stability Robustness,”IEEE Trans. Automat. Control, vol. 32, pp. 509–512, 1987.
G.H. Golub and C.F. Van Loan,Matrix Computations, 2nd ed., Baltimore: Johns Hopkins University Press, 1989.
D.G. Luenberger,Linear and Nonlinear Programming, Reading, MA: Addison-Wesley, 1984.
G.W. Stewart,Introduction to Matrix Computations, New York: Academic Press, 1973.
N.G. El-Agizi and M.M. Fahmy, “Two-Dimensional Digital Filters with No Overflow oscillations,”IEEE Trans. Acoust. Signal Process., vol. 27, pp. 465–469, 1979.
W.-S. Lu and E.B. Lee, “Stability Analysis for 2-D Systems,”IEEE Trans. Circuits Syst., vol. 30, pp. 455–461, 1983.
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Lu, W.S. Some new results on stability robustness of two-dimensional discrete systems. Multidim Syst Sign Process 5, 345–361 (1994). https://doi.org/10.1007/BF00989278
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DOI: https://doi.org/10.1007/BF00989278