Active eigen-orbit control (Q2725701)

The author's ``eigenorbit method'' (cf. the book [Theory of eigenvalue orbits (1998; Zbl 0941.93002)]) consists of considering or influencing the time-varying eigenvalues of the matrix \(H(t)= \dot Y(t) Y(t)^{-1}\) where the components of \(Y(t)\) are \(y_k(t)= y(t- (k-1)\tau)\); \(\tau\) is a fixed delay and \(y\) is the measured output of a (possibly controlled) dynamical system. Discrete versions are presented and controllers are sought for in this case.NEWLINENEWLINENEWLINEThe author ignores some basic fact, such as the lack of regularity of the eigenstructure depending on a parameter (On page 56, several quantities are differentiated although it is not allowed to do so. In particular, this happens for the matrix \(\Lambda\) of eigenvalues, and the dot is less than five cm away from the nearest kink in the graphical representation of the evolution of an eigenvalue just above.), and he overlooks the gap between stability criteria in the autonomous and time-varying contexts (page 57).NEWLINENEWLINENEWLINEThe pole placement-type equation depends nonlinearly on the control and a linearization perturbs only the eigenvalues. Another controller called ``\(\lambda\)-feedback'' is not really a feedback due to its open-loop nature, and it is moreover computationally demanding (diagonalization and inversion of matrices at each step). Furthermore, no criterion for the choice of the control law is presented. One picks an eigenvalue from the previous step (Which one? How?).NEWLINENEWLINENEWLINETo start with the method is advocated as a modelling tool, but no word can be found in the text about the choice of \(\tau\) or of the number of components of \(Y\) for adequate order determination.NEWLINENEWLINENEWLINEActually, one has the impression that the author does not know from where he comes from and where he is going.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00020].