DISCRETE APPROXIMATION OF CONTINUOUS OBJECTS WITH MATLAB

This work is dedicated to the study of various discrete approximation methods for continuous links, which is the obligatory step in the digital control systems synthesis for continuous dynamic objects and the guidelines development for performing these opera tions using the MATLAB programming system. The paper investigates such sampling methods as pulse-, step-, and linearly invariant Z-transformations, substitution methods based on the usage of numerical integration various methods and the zero-pole correspond ence method. The paper presents examples of using numerical and symbolic instruments of the MATLAB to perform these opera tions, offers an m-function improved version for continuous systems discretization by the zero-pole correspondence method, which allows this method to approach as step-invariant as linearly invariant Z-transformations; programs for continuous objects discrete approximation in symbolic form have been developed, which allows to perform comparative analysis of sampling methods and sys tems synthesized with their help and to study quantization period influence on sampling accuracy by analytical methods. A compari son of discrete transfer functions obtained by different methods and the corresponding reactions in time to different signals is per formed. Using of the developed programs it is determined that the pulse-invariant Z-transformation can be used only when the input of a continuous object receives pulse signals, and the linear-invariant transformation should be used for intermittent signals at the input. The paper also presents an algorithm for applying the Tustin method, which corresponds to the replacement of analogue inte gration by numerical integration using trapezoidal method. It is shown that the Tustin method is the most suitable for sampling of first-order regulators with output signal limitation. The article also considers the zero-pole correspondence method and shows that it has the highest accuracy among the rough methods of discrete approximation. Based on the performed research, recommendations for the use of these methods in the synthesis of control systems for continuous dynamic objects are given.

EXISTENCE OF SOLUTIONS FOR FRACTIONAL EVOLUTION INCLUSION WITH APPLICATION TO MECHANICAL CONTACT PROBLEMS

The goal of this paper is to study an evolution inclusion problem with fractional derivative in the sense of Caputo, and Clarke’s subgradient. Using the temporally semi-discrete method based on the backward Euler difference scheme, we introduce a discrete approximation system of elliptic type corresponding to the fractional evolution inclusion problem. Then, we employ the surjectivity of multivalued pseudomonotone operators and discrete Gronwall’s inequality to prove the existence of solutions and its priori estimates for the discrete approximation system. Furthermore, through a limiting procedure for solutions of the discrete approximation system, an existence theorem for the fractional evolution inclusion problem is established. Finally, as an illustrative application, a complicated quasistatic viscoelastic contact problem with a generalized Kelvin–Voigt constitutive law with fractional relaxation term and friction effect is considered.

Discrete Approximation by a Dirichlet Series Connected to the Riemann Zeta-Function

In the paper, a Dirichlet series ζuN(s) whose shifts ζuN(s+ikh), k=0,1,⋯, h>0, approximate analytic non-vanishing functions defined on the right-hand side of the critical strip is considered. This series is closely connected to the Riemann zeta-function. The sequence uN→∞ and uN≪N2 as N→∞.