Uwe Helmke

Numerical Gradient Algorithms for Eigenvalue and Singular Value Calculations

Recent work has shown that the algebraic question of determining the eigenvalues, or singular values, of a matrix can be answered by solving certain continuous-time gradient flows on matrix manifolds. To obtain computational methods based on this theory, it is reasonable to develop algorithms that iteratively approximate the continuous-time flows. In this paper the authors propose two algorithms, based on a double Lie-bracket equation recently studied by Brockett, that appear to be suitable for implementation in parallel processing environments. The algorithms presented achieve, respectively, the eigenvalue decomposition of a symmetric matrix and the singular value decomposition of an arbitrary matrix. The algorithms have the same equilibria as the continuous-time flows on which they are based and inherit the exponential convergence of the continuous-time solutions.

Balanced realizations via gradient flow techniques

Abstract The task of finding a class of balanced minimal realizations is shown to be equivalent to finding limiting solutions of certain gradient flow differential equations. By viewing such algebraic tasks in the context of calculus, they are amenable to analog computational solutions, or parallel processing machines, perhaps even neural networks. The convergence rates of the differential equations is exponential, and consequentially convergence is rapid and numerical stability properties are attractive.

Singular-Value Decomposition via Gradient and Self-Equivalent Flows

The task of finding the singular-value decomposition (SVD) of a finite-dimensional complex linear operator is here addressed via gradient flows evolving on groups of complex unitary matrices and associated self-equivalent flows. The work constitutes a generalization of that of Brockett on the diagonalization of real symmetric matrices via gradient flows on orthogonal matrices and associated isospectral flows. It complements results of Symes, Chu, and other authors on continuous analogs of the classical QR algorithm as well as earlier work by the authors on SVD via gradient flows on positive definite matrices.

CRITICAL POINTS OF MATRIX LEAST SQUARES DISTANCE FUNCTIONS

Abstract A classical problem in matrix analysis and total least squares estimation is that of finding a best approximant of a given matrix by lower rank ones. In this paper the critical points and the local minimum of the distance function f A ( X ) = ∥ A − X ∥ 2 on varieties of fixed rank symmetric, skew-symmetric, and rectangular matrices X are determined. Our results extend earlier ones of Eckart and Young and of Higham.

Topology of the moduli space for reachable linear dynamical systems: the complex case

This paper deals with the algebraic topology of the space Σn,m of complex reachable linear dynamical systems. Topological invariants such as the singular homology groups of Σn,m are explicitly computed and they are shown to coincide with those of a certain Grassmann manifold. From this some new results on the topology of rational transfer matrices with fixed McMillan degree are obtained.

Isospectral flows on symmetric matrices and the Riccati equation

Abstract Brockett has studied the ordinary differential equation H dot = [H, [H, N]] , with [A, B] = AB − BA , evolving on the space of symmetric matrices. The flow asymptotically diagonalizes symmetric matrices and generalizes the Toda flow. We show that Brocketts flow can be interpreted as a flow on a flag manifold. In a special case the flow is shown to be equivalent to a Riccati equation.

Stability and Robustness Properties of Universal Adaptive Controllers for First Order Linear Systems

Quite recently, examples of non-linear adaptive controllers have been found that adaptively stabilize any scalar first-order linear system. In this paper we describe two general classes of such ‘universal’ adaptive stabilizers that include the previously proposed controllers as special cases, Different types of perturbations of these controllers are studied and we report on extensive simulation experiments. Our results demonstrate considerable qualitative differences in the dynamical behaviour of various adaptive universal stabilizers. While the previously proposed controllers all exhibit the bursting phenomena, we give an explicit example of a stabilizing controller that does not show bursting behaviour.

Hermitian pencils and output feedback stabilization of scalar systems

Abstract Necessary and sufficient semi-algebraic conditions for (a) the stabilization of scalar transfer functions, and (b) the assignability of real poles by static output feedback, are given in terms of the Weierstrass invariants of an associated hermitian matrix pencil. An explicit graphical test for output feedback stabilizability is derived which is equivalent to the Nyquist criterion.

A canonical form for controllable singular systems

Abstract A new canonical form for the action of restricted system equivalence on controllable singular systems is given. The construction of this form is based on the Weierstrass decomposition of the singular system into a slow and a fast subsystem. Both subsystems are transformed into Hermite canonical form. The resulting Hermite canonical form for singular systems has a particularly simple structure and is expected to be useful for e.g. identification purposes. Continuity properties of the Hermite form are investigated and the nonexistence of a globally defined continuous canonical form for controllable singular systems is shown.

Gradient algorithms for principal component analysis

The problem of princip~l component analysis of a symmetric matrix (finding a p-dimensional eigenspace associated with the largest p eigenvalues) can be viewed as a smooth optimization problem on a homogeneous space. A solution in terms of the limiting value of a continuous-time dynamical system is presented, A discretization of the dynamical system is proposed that exploits the geometry of the homogeneous space. The relationship between the proposed algorithm and classical methods are investigated.