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A short review of the method of finitely-convergent algorithms in the
theory of adaptive systems is presented. The method was proposed in (Yakubovich,
V.A., Recurrent Finite-convergent Algorithms to Solve the Systems of Inequalities,
Dokl. Akad. Nauk SSSR, 1966, vol. 166, no. 6, pp. 1308–1311. English translation
in Soviet Math. Dokl., 1966, vol. 7, pp. 300–304). The method consists of reduction
of an adaptive control problem to a countable system of inequalities. A procedure
of their solution plays a role of adaptation algorithm. It should converge in a finite
time in a closed loop system. The method often allows to obtain suboptimal (in
the minimax sense) adaptive control systems.
Basic ideas and achievments of the method as well as some recent results are
considered. Among them are adaptive control of sampled systems with time delay
which is not a multiple of the sampling period. In this relation a problem of limiting
zeros of sampled systems with delay arise. It is shown that limiting values of the
sampling zeros have the same properties as zeros of Euler polynomials.
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