Coprime factorization and optimal control on the doubly infinite discrete time axis

We study the problem of strongly coprime factorization over H-infinity of the unit disc. We give a necessary and sufficient condition for the existence of such a coprime factorization in terms of an optimal control problem over the doubly infinite discrete-time axis. In particular, we show that an equivalent condition for the existence of such a coprime factorization is that both the control and filter algebraic Riccati equation (of an arbitrary realization) have a solution (in general unbounded and even non densely defined) and that a coupling condition involving these solutions is satisfied

Bounded real and positive real balanced truncation for infinite-dimensional systems

Bounded real balanced truncation for infinite-dimensional systems is considered. This provides reduced order finite-dimensional systems that retain bounded realness. We obtain an error bound analogous to the finite-dimensional case in terms of the bounded real singular values. By using the Cayley transform a gap metric error bound for positive real balanced truncation is subsequently obtained. For a class of systems with an analytic semigroup, we show rapid decay of the bounded real and positive real singular values. Together with the established error bounds, this proves rapid convergence of the bounded real and positive real balanced truncations

State space formulas for a solution of the suboptimal Nehari problem on the unit disc

We give state space formulas for a ("central") solution of the suboptimal Nehari problem for functions defined on the unit disc and taking values in the space of bounded operators in separable Hilbert spaces. Instead of assuming exponential stability, we assume a weaker stability concept (the combination of input-, output- and input-output stability), which allows us to solve the problem for general H-infinity functions

A counter-example to “Positive realness preserving model reduction with H-\infty norm error bounds”

We provide a counter example to the H∞ error bound for the difference of a positive real transfer function and its positive real balanced truncation stated in “Positive realness preserving model reduction with H-\infty norm error bounds” IEEE Trans. Circuits Systems I Fund. Theory Appl. 42 (1995), no. 1, 23–29. The proof of the error bound is based on a lemma from an earlier paper “A tighter relative-error bound for balanced stochastic truncation.” Systems Control Lett. 14 (1990), no. 4, 307–317, which we also demonstrate is false by our counter example. The main result of this paper was already known in the literature to be false. We state a correct H-\infty error bound for the difference of a proper positive real transfer function and its positive real balanced truncation and also an error bound in the gap metric

Non-dissipative boundary feedback for Rayleigh and Timoshenko beams

We show that a non-dissipative feedback that has been shown in the literature to exponentially stabilize an EulerBernoulli beam makes a Rayleigh beam and a Timoshenko beam unstable