# Pendulum System with Relative Degree Two

## Model description:

Consider the pendulum system with the relative degree two

\begin{align*} \dot{x}_{1} &= x_{2}\\ \dot{x}_{2} &= -cx_{2}-d\sin x_{1}+u \\ y &= x_{1} \end{align*}

with $c > 0$.

2

## Publication details:

 Title Nonlinear sampled-data models and zero dynamics Publication Type Conference Paper Year of Publication 2009 Authors Nishi, M., Ishitobi M., and Kunimatsu S. Conference Name International Conference on Networking, Sensing and Control, 2009. ICNSC '09. Date Published 03/2009 Publisher IEEE Conference Location Okayama ISBN Number 978-1-4244-3491-6 Accession Number 10646009 Keywords closed loop systems, continuous time systems, control system synthesis, nonlinear control systems, poles and zeros, sampled data systems, stability Abstract One of the approaches to sampled-data controller design for nonlinear continuous-time systems consists of obtaining an appropriate model and then proceeding to design a controller for the model. Hence, it is important to derive a good approximate sampled-data model because the exact sampled-data model for nonlinear systems is often unavailable to the controller designers. Recently, Yuz and Goodwin have proposed an accurate approximate model which includes extra zero dynamics corresponding to the relative degree of the continuous-time nonlinear system. Such extra zero dynamics are called sampling zero dynamics. A more accurate sampled-data model is, however, required when the relative degree of a continuous-time nonlinear plant is two. The reason is that the closed-loop system becomes unstable when the more accurate sampled-data model has unstable sampling zero dynamics and a controller design method based on cancellation of the zero dynamics is applied. This paper derives the sampling zero dynamics of the more accurate sampled-data model and shows a condition which assures the stability of the sampling zero dynamics of the model. DOI 10.1109/ICNSC.2009.4919304