Differential Flatness of Single-Input Commensurate Delay Systems With Applications to Trajectory Planning, Tracking, and Transformation to Fully Actuated Systems

This paper studies the differential flatness of time-delay systems (TDSs). Based on factorizations in the pseudo-polynomials ring, a flat output for the single-input linear commensurate TDS (namely, the delays in the system are multiplies of a certain unit delay) is constructed explicitly. Then the TDS can be converted into a high-order fully actuated system (HOFAS) model with the generalized state as the flat output, and the finite spectrum assignment (FSA) problem can be solved immediately by using the HOFAS approach, which provides some new insight into the study of the FSA problem. By parameterizing the considered TDS via the constructed flat output and applying the interpolation theory, the state trajectory planning problem and state tracking problem are solved, resulting a two-degree-of-freedom (2DOF) controller. Numerical examples demonstrates the effectiveness of the presented approach.

I. Introduction

The differential flatness concept was originally established by Fliess et al. [12]. There exist many practical systems that are differential flat, such as satellite systems with robotic arms [27]. One of the main properties of differential flat systems is that both the state and the input can be explicitly expressed as functions of the flat output and their derivatives without integrating any differential equations. Such a property is very useful in various control problems, including the trajectory planning and state tracking problem [1], [7], [38]. The essence of trajectory planning is to determine the trajectory of the systems that satisfies specific requirements. Such a problem is difficult to solve in general because it may require iterative operations by numerical methods to find the desired solution that satisfies the prescribed conditions. Therefore, if one can find a system’s flat output, then such system can be parameterized by the flat output, and, consequently, the problem of trajectory planning can be solved easily. Once the desired state has been obtained by the trajectory planning, the state tracking problem, which involves designing a control so that the system’s state can track the desired one, can be transformed into a stabilization problem.