Jens Jordan

Estimation of linear positive systems with unknown time-varying delays

Abstract This paper considers the estimation problem for linear positive systems with time-varying unknown delays. Similar to set-valued estimation approaches, we provide a confident region within which the trajectory of the observed positive system always evolves. Guaranteed upper and lower estimates for the instantaneous states are characterized by means of a special kind of extended Luenberger-type interval observer. We provide constructive conditions for its existence and establish the asymptotic convergence of its associated interval error. In addition, we give an LP-based method which allows one to construct the proposed interval observer solely from the data of the system.

Numerics Versus Control

Numerical matrix eigenvalue methods such as the inverse power iteration or the QR-algorithm can be reformulated as inverse power iterations on homogeneous spaces. In this paper we survey some recent results on controllability properties of the shifted inverse power iteration on flag manifolds. It is shown that the reachable sets are orbits for a semigroup action on the flag manifold. Except for the special case of projective spaces, the algorithm is never controllable. This implies in particular the non-controllability of the shifted QR-algorithm on isospectral matrices. Controllability results for the inverse power iteration on projective space for real or complex shifts are presented, following [20, 22], and a connection with output feedback pole assignability is mentioned. Controllability of the algorithm on Hessenberg flags is shown. This implies controllability of the shifted QR-algorithm on Hessenberg matrices.

Control and Observation of the Matrix Riccati Differential Equation

We explore controllability and observability properties of the matrix Riccati differential equation as a flow on the Grassmann manifold. Using the known classification of transitive Lie group actions on Grassmann manifolds, we derive necessary and sufficient conditions for accessibility of Riccati equations. This also leads to new sufficient Lie-algebraic conditions for controllability of generalized double bracket flows. Observability of Riccati equations with linear fractional output functions is characterized via a generalized Hautus-Popov test, thus making contact with earlier work by Dayawansa, Ghosh, Martin and Rosenthal on perspective observability of linear systems.