Testing Backward-Flatness of Nonlinear Discrete-Time Systems

TL;DR

Backward-flatness is identified as a gap in efficiently testing difference flatness for discrete-time nonlinear systems. The authors construct an associated system whose trajectories correspond one-to-one with the original, proving that backward-flatness of the original holds iff the associated system is forward-flat, thus enabling indirect testing via the established geometric forward-flatness framework. Applying Algorithm 1 and related rank results to the associated system yields a forward-flat output, from which a backward-flat output for the original system is obtained through composition with the system map, ensuring an $(x,u)$-flat output for backward-flat systems. The work also derives Jacobian-based rank conditions that relate the parameterizing maps of the original and associated systems, substantiated by an academic example, and provides a feasible path toward computationally efficient analysis of difference flatness in discrete time.

Abstract

Despite ongoing research, testing the flatness of discrete-time systems remains a challenging problem. To date, only the property of forward-flatness - a special case of difference-flatness - can be checked in a computationally efficient manner. In this paper, we propose a systematic approach for testing backward-flatness, which is another special case of difference-flatness, and for deriving a corresponding backward-flat output. Additionally, we discuss the relationship between the Jacobian matrices associated with the flat parameterization of backward- and forward-flat systems and illustrate our results by an academic example.

Figures (2)

• Figure 1: One-to-one correspondence of the trajectories
• Figure 2: Relation between the flatness properties of the systems \ref{['eq:sysEq_ext']} and \ref{['eq:flat_sys_mirrored']}