From Morse Triangular Form of ODE Control Systems to Feedback Canonical Form of DAE Control Systems

TL;DR

This paper establishes a connection between the feedback canonical form of differential-algebraic control systems and Morse canonical form of ODE control systems, introducing an explicitation procedure and an algorithm for transformation.

Contribution

It introduces a novel explicitation method linking DACSs with ODECSs and provides an algorithm to convert DACS into feedback canonical form using this connection.

Findings

The explicitation procedure effectively relates DACSs to ODECSs.

The extended Morse forms facilitate the transformation process.

The proposed algorithm successfully transforms DACS into feedback canonical form.

Abstract

In this paper, we relate the feedback canonical form \textbf{FNCF} of differential-algebraic control systems (DACSs) with the famous Morse canonical form \textbf{MCF} of ordinary differential equation control systems (ODECSs). First, a procedure called an explicitation (with driving variables) is proposed to connect the two above categories of control systems by attaching to a DACS a class of ODECSs with two kinds of inputs (the original control input $u$ and a vector of driving variables $v$). Then, we show that any ODECS with two kinds of inputs can be transformed into its extended \textbf{MCF} via two intermediate forms: the extended Morse triangular form and the extended Morse normal form. Next, we illustrate that the \textbf{FNCF} of a DACS and the extended \textbf{MCF} of the explicitation system have a perfect one-to-one correspondence. At last, an algorithm is proposed to transform a given DACS into its \textbf{FBCF} via the explicitation procedure and a numerical example is given to show the efficiency of the proposed algorithm.