We study two kinds of linearization (internal and external) of nonlinear differential-algebraic equations DAEs of semi-explicit SE form. The difference of external and internal linearization is illustrated by an example of a mechanical system. Moreover, we define different levels of external equivalence for two SE DAEs. The proposed explicitation procedure allows us to treat a given SE DAE as a control system defined up to feedback transformation (a class of control systems). Then sufficient and necessary conditions, expressed via explicitation procedure, are given to describe when a given SE DAE is level-3 externally equivalent to a linear SE DAE of some specific forms. At last, we show by an example that level-2 external linearization of a DAE can be achieved if its explicitation is level-2 input-output linearizable. —

We relate the feedback canonical form FNCF of differential-algebraic controlsystems (DACSs) with the famous Morse canonical form 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 itsextended MCF via two intermediate forms: the extended Morse triangular form and the extended Morse normal form. Next, we illustrate that the FNCF of a DACS and the extended 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 FBCF via the explicitation procedure and a numerical example is given to show the efficiency of the proposed algorithm.

We discuss two notions of index, i.e., the geometric index and the differentiation index for nonlinear differential-algebraic equations (DAEs). First, we analyze solutions of nonlinear DAEs by revising a geometric reduction method (see e.g. Rabier and Rheinboldt (2002),Riaza (2008)). Then we show that although both of the geometric index and the differentiation index serve as a measure of difficulties for solving DAEs, they are actually related to the existence and uniqueness of solutions in a different manner. It is claimed in (Campbell and Gear, 1995) that the two indices coincide when sufficient smoothness and assumptions are satisfied, we elaborate this claim and show that the two indices indeed coincide if and only if a condition of uniqueness of solutions is satisfied (under certain constant rank assumptions). Finally, an example of a pendulum system is used to illustrate our results on the two indices.

In this talk, we connect the notion of differentiation index of DAEs with that of relative degree of control systems, via a concept called the explicitation of DAEs. The explicitation attaches a class of control systems to a given DAE, we show that the relative degree of the systems in the explicitation class is invariant in some sense and that the differentiation index of the original DAE coincides with the maximum of the relative degree of the explicitation systems.

Undergraduate course, University of Groningen, Faculty of Science and Engineering, 2020

I was a lecturer, together with Prof. Stephan Trenn, of the course “Advanced Systems Theory” for bachelor students of Applied Mathematic at University of Groningen.