A sampling zero of a sampled-data model for continuous-time systems with relative degree two

When a continuous-time linear system with relative degree greater than or equal to two is discretized using a zero-order hold, the resulting model includes extra zeros corresponding to the relative degree of the continuous-time system. Such zeros are called sampling zeros. When the relative degree is equal to two, the discrete-time system has one sampling zero, and it is located just on the unit circle in the limiting case as the sampling period tends to zero. Further, in such a case, the sampling zero stays inside (respectively, outside) the unit circle for small sampling periods when the sum of the zeros of the continuous-time system is greater (respectively, smaller) than the sum of the poles. However, the stability condition of the sampling zero for small sampling periods has not been shown when the sum of the zeros is equal to the sum of the poles. This paper analyzes the asymptotic behavior of the sampling zero and derives a condition for the zero to be stable for small sampling periods.