A Closed Form Solution for the Normal Form and Zero Dynamics of a Class of Nonlinear Systems

TL;DR

This paper introduces a simple algebraic method to derive closed form solutions for the normal form and zero dynamics of certain nonlinear systems, especially in mechanical and aerospace applications, facilitating easier analysis and control design.

Contribution

The paper presents the first straightforward algebraic approach to compute zero dynamics in closed form for a specific class of nonlinear systems, improving upon existing complex algorithms.

Findings

Closed form solutions enable easier zero dynamics analysis.

Application demonstrated on flexible spacecraft dynamics.

Method simplifies control design for nonlinear systems.

Abstract

The normal form and zero dynamics are powerful tools useful in analysis and control of both linear and nonlinear systems. There are no simple closed form solutions to the general zero dynamics problem for nonlinear systems. A few algorithms exist for determining the zero dynamics, but none is straightforward and all are difficult to apply to large dimensional problems. A Closed form solution to the zero dynamics problem would motivate more usage of this powerful technique. The author presents here a simple algebraic methodology for the normal form and zero dynamics calculation of a class of nonlinear systems, mostly found in dynamical mechanical systems. The solution is in closed form so that application of the theorem presented is straight forward. As an illustration, the zero dynamics calculations for the complex dynamics of a flexible spacecraft is presented to demonstrate the simplicity and usefulness of the proposed closed form solution. Keywords: Control, Differential Geometry, Normal Form, Zero Dynamics, Nonlinear Systems, Feedback Linearization, Attitude Dynamics, Flexible Spacecraft.