Robust controllability of non-stationary differential-algebraic equations

P. S. Petrenko

The Bulletin of Irkutsk State University. Series Mathematics, Tome 25 (2018), pp. 79-92

Résumé

We consider linear time-varying system of first order ordinary differential equations with identically degenerate matrix of the derivative of the unknown function. Such systems are called differential-algebraic equations (DAE). The unsolvability measure with respect to the derivatives for some DAE is an integer that is called the index of the DAE. The analysis is carried out under the assumption of the existence of a structural form with separated differential and algebraic subsystems. This structural form is equivalent to the initial system in the sense of solution, and the operator which transformes the DAE into the structural form possesses the left inverse operator. The finding of the structural form is constructive and do not use a change of variables. In addition the problem of consistency of the initial data is solved automatically. The approach uses the concept of $r$-derivative array equations, where $r$ is the unsolvability index of the DAE. The existence of a nonsingular minor of order $n(r + 1)$ in the matrix describing derivative array equations is a necessary and sufficient condition for the existence of this structural form ($n$ is the dimension of DAE system). We investigate robust controllability of non-stationary DAE with perturbations given by matrix norms (unstructured uncertainty), which are present in matrices with the unknown function and control function. The problem of the robust controllability is to find the conditions under which the perturbed system will remain completely or $R$-controllable on some interval in the presence of this property of the initial DAE system. It is constructed a structural form for the perturbed DAE system and based on it's analysis sufficient conditions for robust complete and $R$-controllability of the DAE of the indeces $1$ and $2$ are obtained.

Détail

Keywords: differential-algebraic equations, descriptor systems, perturbed systems, robust controllability.

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