Root locii for systems defined on Hilbert spaces

The root locus is an important tool for analysing the stability and time constants of linear finite-dimensional systems as a parameter, often the gain, is varied. However, many systems are modelled by partial differential equations or delay equations. These systems evolve on an infinite-dimensional space and their transfer functions are not rational. In this paper a rigorous definition of the root locus for infinite-dimensional systems is given and it is shown that the root locus is well-defined for a large class of infinite-dimensional systems. As for finite-dimensional systems, any limit point of a branch of the root locus is a zero. However, the asymptotic behaviour can be quite different from that for finite-dimensional systems. This point is illustrated with a number of examples. It is shown that the familiar pole-zero interlacing property for collocated systems with a Hermitian state matrix extends to infinite-dimensional systems with self-adjoint generator. This interlacing property is also shown to hold for collocated systems with a skew-adjoint generator

STRONG STABILIZATION OF (ALMOST) IMPEDANCE PASSIVE SYSTEMS BY STATIC OUTPUT FEEDBACK

The plant to be stabilized is a system node E with generating triple (A, B, C) and transfer function G, where A generates a contraction semigroup on the Hilbert space X. The control and observation operators B and C may be unbounded and they are not assumed to be admissible. The crucial assumption is that there exists a bounded operator E such that, if we replace G(s) by G(s) + E, the new system Sigma(E) becomes impedance passive. An easier case is when G is already impedance passive and a special case is when Sigma has colocated sensors and actuators. Such systems include many wave, beam and heat equations with sensors and actuators on the boundary. It has been shown for many particular cases that the feedback u = -kappa y + v, where u is the input of the plant and kappa > 0, stabilizes Sigma, strongly or even exponentially. Here, y is the output of Sigma and v is the new input. Our main result is that if for some E is an element of L(U), Sigma(E) is impedance passive, and Sigma is approximately observable or approximately controllable in infinite time, then for sufficiently small kappa the closed-loop system is weakly stable. If, moreover, sigma(A)boolean AND iR is countable, then the closed-loop semigroup and its dual are both strongly stable

Stability and robust regulation of passive linear systems

We study the stability of coupled impedance passive regular linear systems under power-preserving interconnections. We present new conditions for strong, exponential, and nonuniform stability of the closed-loop system. We apply the stability results to the construction of passive error feedback controllers for robust output tracking and disturbance rejection for strongly stabilizable passive systems. In the case of nonsmooth reference and disturbance signals we present conditions for nonuniform rational and logarithmic rates of convergence of the output. The results are illustrated with examples on designing controllers for linear wave and heat equations, and on studying the stability of a system of coupled partial differential equations.acceptedVersionPeer reviewe

Structured Perturbations of Semigroup Generators: Theory and Applications

Wir verwenden einen Störungsansatz, um lineare Operatoren und deren Spektral- und Generatoreigenschaften zu untersuchen. Dazu betrachten wir strukturierte Störungen, welche auch im Kontext linearer Systeme mit geschlossenem Feedback betrachtet wurden