Limits of generalized state space systems under proportional and derivative feedback
Generalized state space systems are described by time-invariant differential and algebraic equations of the form \[ E\dot x(t)= Ax(t)+ Bu(t),\quad t\geq 0,\quad E,A\in F^{q\times n},\quad B\in F^{q\times m},\tag{1} \] where \(F\) is either the field of real numbers or the field of complex numbers. Let \({\mathcal G}\) be the group of transformations of the generalized state space systems that is generated by the left multiplications (in which (1) is transformed to \(LE\dot x(t)= LAx(t)+ LBu(t)\), where \(L\in F^{q\times q}\) is invertible), the state and input changes of coordinates ((1) is transformed to \(ER\dot x(t)= ARx(t)+ BWu(t)\), where \(R\in F^{n\times n}\) and \(W\in F^{m\times m}\) are invertible), and the state feedbacks ((1) is transformed to \(E\dot x(t)= (A+ BK)x(t)+ Bu(t)\), \(K\in F^{m\times n}\)). Let \({\mathcal D}\) be the group of transformations of the generalized state space systems that is generated by \({\mathcal G}\) and the derivative feedbacks ((1) is transformed to \((E+ BK)\dot x(t)= Ax(t)+ Bu(t)\), where \(K\in F^{m\times n}\)). The canonical forms of the generalized state space systems under the actions of \({\mathcal G}\) and of \({\mathcal D}\) have been obtained, and the corresponding invariants fully described, in \textit{J. J. Loiseau}, \textit{K. Özçaldiran}, \textit{M. Malabre} and \textit{N. Karcanias} [Kybernetika 27, No. 4, 289-305 (1991; Zbl 0767.93007)]. The reviewed paper is built on these canonical forms. The high-gain feedback classification problem is studied. The problem consists of determining, in terms of the invariants, the topological closures of the orbits of the \({\mathcal G}\) (or \({\mathcal D}\)) action, where a generalized state space system (1) is identified with the ordered triple of matrices \((E,A,B)\). For regularizable systems, i.e., such that \(\text{det}(sE- (A- BK))\not\equiv 0\) (as a polynomial of \(s\)) for a suitable \(K\in F^{m\times n}\), a complete description of orbit closures under the \({\mathcal G}\) action is obtained. For the regularizable systems, and under the \({\mathcal D}\) action, necessary conditions in terms of the invariants are given for a system to belonging to an orbit closure. It is conjectured that these conditions are also sufficient. High gain classification of the generalized state space systems in full generality remains an open problem.
scientific article; zbMATH DE number 1109546
scientific article