The Influence of Limited Bit on the Implementation of the Transfer in Digital System

The unexploring problem of digital control systems analyzes in this article — the impact on their behavior of the limited bit resolution of the hardware and, accordingly, the discrete transfer functions coefficients. The research conducted by the zeros/poles and transient characteristics methods using the mathematical application MATLAB with the package Control System Toolbox and confirmed the relevance of this problem.

The research purpose is to determine the minimum sampling period in digital systems, provided there stability and adequacy to the behavior of a continuous system (prototype), given the limited binary resolution and the data accuracy. This will make it possible to offer solutions to reduce the negative impact of the limited bit resolution in the developed digital systems and expand the range of rational sampling rate.

Boundary dependences for the minimum sampling period and inertial time of elementary first- and second-order transfer functions formulated. It is shown that in the case of limited precision arithmetic the order’s growing of the transfer function polynomial increases its sensitivity to the its coefficients accuracy.

The study of the limited arithmetic precision impact carried out on the example of transfer functions corresponding to the fifth and seventh orders binomial forms. The influence of the accuracy of the transfer functions polynomials coefficients and the placement of zeros and poles of discrete transfer functions on the complex plane shown on depending on the sampling step. The transient characteristics of discrete systems with limited bit resolution compared to their continuous analogue.

Studies confirm that the method of a continuous transfer function decomposition for decomposition into elementary components of the first and second orders and their following sampling, allows expanding the bounds of allowable sampling steps in digital systems with limited precision arithmetic. The influence of errors shown on the example of polynomials from the second to the seventh order, where in particular with growing order the polynomial increases its sensitivity to the accuracy of its coefficients setting.