Abstract
A class of two-parameter discrete systems defined on the ring \(\mathbb{Z}_m = {\mathbb{Z} \mathord{\left/ {\vphantom {\mathbb{Z} {(m)}}} \right. \kern-\nulldelimiterspace} {(m)}}{\text{ }}\) of class of residues of integers \(\mathbb{Z}\) modulo m is studied. All solutions are shown to be periodic, stability conditions (equality of solutions to zero, beginning from a certain instant) and a controllability condition are formulated. Controllability is shown to guarantee stabilizability.
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REFERENCES
Wunsch, G., Zellulare Systeme, Berlin: Akademie-Verlag, 1977.
Yugel, A. and Zlateeva, S., Discrete Cellular Machines, Teor.Prikl.Mekh., 1977, no. 3, pp. 79-88.
Blyumin, S.L. and Faradzhev, R.G., Analysis and Synthesis of Finite Linear Sequential Machines, Avtom.Telemekh., 1981, no. 6, pp. 57-66.
Prangishvili, I.V., Abramov, N.A., Babichev, E.V., and Ignatushenko, V.V., Mikroelektronika i odnorodnye struktury dlya postroeniya logicheskikh vychislitel'nykh ustroistv (Microelectronics and Homogeneous Structures for Designing Logical Computing Devices), Moscow: Nauka, 1967.
Evreinov, E.V. and Khoroshevskii, V.G., Odnorodnye vychislitel'nye sistemy (Homogeneous Computing Systems), Moscow: Nauka, 1978.
Whilsky, A.S., Interrelation between Digital Signal Processing Theory and Control and Estimation Theory, TIIER, 1978, vol. 66, no. 9, pp. 5-33.
Preston, K., Daff, M., Levyaldi, S., Norgren, F., and Torivaki, D., Elements of Cellular Logic with Application of Image Processing in Medicine, TIIER, 1979, vol. 67, no. 5, pp. 149-185.
Mercero, R., Processing of Two-Dimensional Signals under Discretization by Hexogonal Raster, TIIER, 1979, vol. 67, no. 6, pp. 62-84.
Bernshtein, A.S., Popkov, Yu.S., and Faradzhev, R.G., Multidimensional Laplace-Galois Transformation in the Theory of Nonlinear Sequential Machines, Dokl.Akad.Nauk SSSR, 1972, vol. 206, no. 5, pp. 1061-1064.
Kaczorek, T., Two-Dimensional Linear Systems, Berlin: Springer-Verlag, 1985.
Gaishun, I.V., Mnogoparametricheskie sistemy upravleniya (Multiparameter Control Systems), Minsk: Nauka i Tekhnika, 1996.
Faradzhev, R.G. and Feiziev, F.G., Metody i algoritmy resheniya zadachi kvadratichnoi optimizatsii dlya dvoichnykh posledovatel'nostnykh mashin (Methods and Algorithms for solving the Quadratic Optimization Problem for Binary Sequential Machines), Baku: Elm, 1996.
Lidl, R. and Niederreiter, H., Finite Fields, Reading: Addison-Wesley, 1983. Translated under the title Konechnye polya, Moscow: Mir, 1988.
Kwakernaak, H. and Sivan, R., Linear Optimal Control Systems, New York: Wiley, 1972. Translated under the title Lineinye optimal'nye sistemy upravleniya, Moscow: Mir, 1977.
Bourbaki, N., Algebra, Paris: Hermann, 1951. Translated under the title Algebra.Mnogochleny i polya.Uporyadochennye gruppy, Moscow: Nauka, 1965.
Lang, S., Algebra, Reading: Addison-Wesley, 1965. Translated under the title Algebra, Moscow: Mir, 1968.
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Gaishun, I.V. Two-Parameter Discrete Systems over a Residue Ring and Their Properties. Automation and Remote Control 63, 1717–1723 (2002). https://doi.org/10.1023/A:1020943028933
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DOI: https://doi.org/10.1023/A:1020943028933