Riccati equations and normalized coprime factorizations for strongly stabilizable infinite-dimensional systems

The first part of the paper concerns the existence of strongly stabilizing solutions to the standard algebraic Riccati equation for a class of infinite-dimensional systems of the form Σ(A,B,S−1/2B*,D), where A is dissipative and all the other operators are bounded. These systems are not exponentially stabilizable and so the standard theory is not applicable. The second part uses the Riccati equation results to give formulas for normalized coprime factorizations over H∞ for positive real transfer functions of the form D+S−1/2B*(author−A)−1,B

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

A Comparison of finite-dimensional controller designs for distributed parameter systems

This paper compares five different approaches to the design of finite-dimensional controllers for linear infinite-dimensional systems. The approaches are varied and include state and frequency domain methods, exact controller designs, controller designs by approximation and robust controller designs

Coprime factorization and robust stabilization for discrete-time infinite-dimensional systems

We solve the problem of robust stabilization with respect to right-coprime factor perturbations for irrational discrete-time transfer functions. The key condition is that the associated dynamical system and its dual should satisfy a finite-cost condition so that two optimal cost operators exist. We obtain explicit state space formulas for a robustly stabilizing controller in terms of these optimal cost operators and the generating operators of the realization. Along the way we also obtain state space formulas for Bezout factors

Exponential stabilization of well-posed systems by colocated feedback

Published versio

Spectral systems

The class of distributed systems generated by spectral operators is an important one and includes the many differential operators arising in boundary-value problems involving non-symmetric linear differential operators, whose eigenfunction expansions converge in much the same way as Fourier series. Thus they enjoy many of the properties of systems generated by self-adjoint operators. We examine here some of the implications in linear systems theory

The spectrum determined growth assumption for perturbations of analytic semigroups

This short paper gives useful information concerning properties of perturbations of infinitesimal generators of analytic semigroups. In particular, it is shown that the spectrum determined growth assumption is retained not only under bounded perturbations but also under a class of unbounded perturbations, which arise in control problems for parabolic systems where the control occurs pointwise or on the boundary

Pole assignment for distributed systems by finite-dimensional control

The question of pole assignment for a class of distributed systems by means of finite dimensional control is considered. A finite dimensional compensator is usually designed based on a reduced order model, and it is not a priori known what effect this will have on the original infinite dimensional system. Explicit formulas are derived which enable one to calculate the eigenvalues of the resulting closed loop system. For a certain compensator design, estimates are derived which can be used to guarantee stability a priori. Numerical results are also presented