Improving the Asymptotic Properties of Discrete System Zeros in Fractional-Order Hold Case

Cheng Zeng,Shan Liang,Yingying Su

doi:10.1155/2013/268684

Abstract

Remarkable improvements in the asymptotic properties of discrete system zeros may be achieved by properly adjusted fractional-order hold (FROH) circuit. This paper analyzes asymptotic properties of the limiting zeros, as the sampling periodTtends to zero, of the sampled-data models on the basis of the normal form representation of the continuous-time systems with FROH. Moreover, when the relative degree of the continuous-time system is equal to one or two, an approximate expression of the limiting zeros for the sampled-data system with FROH is also given as power series with respect to a sampling period up to the third-order term. And, further, the corresponding stability conditions of the sampling zeros are discussed for fast sampling rates. The ideas of the paper here provide a more accurate approximation for asymptotic zeros, and certain known achievements on asymptotic behavior of limiting zeros are shown to be particular cases of the results presented.

Highlights

Zeros, along with poles, are fundamental characteristics of linear time-invariant systems, and the stability of zeros is one of the most important issues in the model matching and adaptive control problems
The normal form of (1) with the relative degree one, m = n − 1 is represented with an input u and an output y [19, 20] as ξ = −dξ + Ku − ω, η = Pη + qξ, (3)
The zeros of a discrete-time system corresponding to the continuous-time transfer function (1) with fractionalorder hold (FROH) are given for T ≪ 1 approximately by the roots of

Summary

Introduction

Along with poles, are fundamental characteristics of linear time-invariant systems, and the stability of zeros is one of the most important issues in the model matching and adaptive control problems. At least one of the zeros lies strictly outside the unit circle if the relative degree of a continuous-time transfer function is greater than or equal to three [12, 18] This fact indicates that even though all the zeros of such a continuous-time system are stable, the corresponding discrete-time system has at least one unstable zero in the limiting case as the sampling period tends to zero. The corresponding discrete-time plants have one or two sampling zero(s) in the case of a FROH when the relative degree of a continuous-time transfer function is one or two In these cases, the sampling zeros are located just on the unit circle, that is, in the marginal case of the stability. We further discuss the stability of the sampling zeros for sufficiently small sampling periods, and some interesting examples are given to validate the main results

Sampled-Data Models with FROH

Main Results

Simulation Examples

Conclusions