Harald K. Wimmer

Canonical angles of unitary spaces and perturbations of direct complements

Abstract Let ℂ n = ℒ ⊕ %plane1D;4B2; be a given direct sum decomposition. We determine the largest number r i such that all subspaces ℳ for which the gap θ between ℒ and %plane1D;4B2; satisfies θ(ℒℳ) have the property dim (ℳ∩%plane1D;4B2;). The problem involves angles between subspaces, i.e. singular values of products of projections.

The set of positive semidefinite solutions of the algebraic Riccati equation of discrete-time optimal control

In this paper the algebraic Riccati equation (ARE) of the discrete-time linear-quadratic (LQ) optimal control problem and its set of positive semidefinite solutions is studied under the most general assumption which is output stabilizability. With respect to an appropriate basis, the discrete-time algebraic Riccati equation (DARE) decomposes into a Lyapunov equation and an irreducible Riccati equation. The focus is on the Riccati part which amounts to studying a DARE where all unimodular modes are controllable. A bijection between positive semidefinite solutions and certain well-defined sets of F-invariant subspaces is established which, together with its inverse, is order reversing. As an application, issues concerning positive definite or strong solutions are clarified. Analogous results for negative semidefinite solutions are valid only under an additional assumption on the unobservable subspace.

On the history of the Bezoutian and the resultant matrix

Abstract It is shown how the Bezoutian and the resultant matrix evolved from Eulers work in elimination theory.

A comparison theorem for matrix Riccati difference equations

Abstract Difference equations of the form X(t) = F ∗ (t)X(t − 1)F(t) − F ∗ (t)X(t − 1)G(t)[I + G ∗ (t)X(t − 1)G(t) −1 G ∗ (t)X(t − 1)F(t) + Q(t) and their associated Hermitian matrices H(t) = ( F Q F ∗ −GG ∗ )(t) are studied. Solution of different Riccati equations can be compared if the difference of their corresponding Hermitian matrices is semidefinite for all t . An application to the discrete-time LQ optimal control problem is given.

Unmixed solutions of the discrete-time algebraic Riccati equation

The algebraic Riccati equation of the optimal control problem associated with the discrete-time system

Normal Forms of Symplectic Pencils and the Discrete-Time Algebraic Riccati Equation*

x(k + 1) = Fx(k) + Gu(k)

Decomposition and Parametrization of Semidefinite Solutions of the Continuous-Time Algebraic Riccati Equation

is studied. It is shown that in the case of a controllable system, there exist solutions with prescribed unmixed characteristic polynomial of the corresponding closed-loop matrix. Existence of solutions will also be proved under the weaker condition of modulus-controllability. Maximal solutions are discussed.

Roth's theorems for matrix equations with symmetry constraints

Abstract The solution of the discrete-time algebraic Riccati equation leads to symplectic pencils of matrices. Normal forms of such pencils under symplectic equivalence are determined. Special attention is given to characteristic roots of modulus 1 and their corresponding elementary divisors and inertial invariants.

Brief Monotonicity of the optimal cost in the discrete-time regulator problem and Schur complements

Negative-semidefinite solutions of the ARE

Linear matrix equations: The module theoretic approach

{\cal R}(X)=A^{*}X+XA +XBB^{*}X-C^{*}C=0