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CYBERNETICS AND PHYSICS
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Volume 2, 2013, Number 3. Mechanics, dynamical systems and control: theory and applications (Topical issue)
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Zeros of delayed sampled systems
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Linear SISO time-invariant continuous systems with time delay are considered. A sufficiently rapid zero-order hold sampling of such a system leads to the discrete-time model with two subsets of zeros, namely so-called intrinsic zeros and sampling (or limiting) ones. Intrinsic zeros depend on original zeros almost exponentially, while sampling zeros are asymptotically close to the zeros of Euler polynomials. More accurately, they converge to the zeros of Euler polynomials

in the case of zero time delay. This well known result is extended here to the case of positive time delay. It is shown that limiting zeros depend on the relative degree and on an additional parameter eps which is equal to the fractional part of the quotient of the time delay and the sampling period. Polynomials having those zeros are called here generalized Euler polynomials. They coincide with ordinary Euler polynomials if eps=0. It is shown that all zeros of generalized Euler polynomials are negative and simple. They monotonically vary between the neighboring zeros of the corresponding ordinary Euler polynomial when eps grows from 0 to 1. Since zeros of the ordinary Euler polynomial are pair-wise mutually inverse, we obtain a criterion for a sam-

pled system to be stably invertible.

CYBERNETICS AND PHYSICS, Vol. 2, No. 3, 183–188.

in the case of zero time delay. This well known result is extended here to the case of positive time delay. It is shown that limiting zeros depend on the relative degree and on an additional parameter eps which is equal to the fractional part of the quotient of the time delay and the sampling period. Polynomials having those zeros are called here generalized Euler polynomials. They coincide with ordinary Euler polynomials if eps=0. It is shown that all zeros of generalized Euler polynomials are negative and simple. They monotonically vary between the neighboring zeros of the corresponding ordinary Euler polynomial when eps grows from 0 to 1. Since zeros of the ordinary Euler polynomial are pair-wise mutually inverse, we obtain a criterion for a sam-

pled system to be stably invertible.

CYBERNETICS AND PHYSICS, Vol. 2, No. 3, 183–188.