Improvement of the Asymptotic Properties of Zero Dynamics for Sampled-Data Systems in the Case of a Time Delay

It is well known that the existence of unstable zero dynamics is recognized as a major barrier in many control systems, and deeply limits the achievable control performance. When a continuous-time system with relative degree greater than or equal to three is discretized using a zero-order hold (ZOH), at least one of the zero dynamics of the resulting sampled-data model is obviously unstable for sufficiently small sampling periods, irrespective of whether they involve time delay or not. Thus, attention is here focused on continuous-time systems with time delay and relative degree two. This paper analyzes the asymptotic behavior of zero dynamics for the sampled-data models corresponding to the continuous-time systems mentioned above, and further gives an approximate expression of the zero dynamics in the form of a power series expansion up to the third order term of sampling period. Meanwhile, the stability of the zero dynamics is discussed for sufficiently small sampling periods and a new stability condition is also derived. The ideas presented here generalize well-known results from the delay-free control system to time-delay case.