On the stability radius of a generalized state-space system

The concept of “distance to instability” of a system matrix is generalized to system pencils which arise in descriptor (semistate) systems. Difficulties arise in the case of singular systems, because the pencil can be made unstable by an infinitesimal perturbation. It is necessary to measure the distance subject to restricted, or structured, perturbations. In this paper a suitable measure for the stability radius of a generalized state-space system is defined, and a computable expression for the distance to instability is derived for regular pencils of index less than or equal to one. For systems which are strongly controllable it is shown that this measure is related to the sensitivity of the poles of the system over all feedback matrices assigning the poles.

Robust pole assignment in systems subject to structured perturbations

The problem of robust pole assignment by feedback in a linear, multivariable, time-invariant system which is subject to structured perturbations is investigated. A measure of robustness, or sensitivity, of the poles to a given class of perturbations is derived, and a reliable and efficient computational algorithm is presented for constructing a feedback which assigns the prescribed poles and optimizes the robustness measure.

Robust eigenstructure assignment in quadratic matrix polynomials: nonsingular case

Feedback design for a second-order control system leads to an eigenstructure assignment problem for a quadratic matrix polynomial. It is desirable that the feedback controller not only assigns specified eigenvalues to the second-order closed loop system but also that the system is robust, or insensitive to perturbations. We derive here new sensitivity measures, or condition numbers, for the eigenvalues of the quadratic matrix polynomial and define a measure of the robustness of the corresponding system. We then show that the robustness of the quadratic inverse eigenvalue problem can be achieved by solving a generalized linear eigenvalue assignment problem subject to structured perturbations. Numerically reliable methods for solving the structured generalized linear problem are developed that take advantage of the special properties of the system in order to minimize the computational work required. In this part of the work we treat the case where the leading coefficient matrix in the quadratic polynomial is nonsingular, which ensures that the polynomial is regular. In a second part, we will examine the case where the open loop matrix polynomial is not necessarily regular.

Robust control system design for generalized state-space systems

Robustness in multi-variable control system design requires that the solution to the design problem be insensitive to perturbations in the system data. In this paper we discuss measures of robustness for generalized state-space, or descriptor, systems and describe algorithmic techniques for optimizing robustness for various applications.

Algorithms for Robust Pole Assignment in Singular Systems

The solution of the pole assignment problem by feedback in singular systems is parameterized and conditions are given which guarantee the regularity and maximal degree of the closed loop pencil. A robustness measure is defined, and numerical procedures are described for selecting the free parameters in the feedback to give optimal robustness.

Stability results for the time-harmonic Maxwell equations with impedance boundary conditions

We consider the time-harmonic Maxwell equations with constant coefficients in a bounded, uniformly star-shaped polyhedron. We prove wavenumber-explicit norm bounds for weak solutions. This result is pivotal for convergence proofs in numerical analysis and may be a tool in the analysis of electromagnetic boundary integral operators.

A process for analysis of microarray comparative genomics hybridisation studies for bacterial genomes

Background: Microarray based comparative genomic hybridisation (CGH) experiments have been used to study numerous biological problems including understanding genome plasticity in pathogenic bacteria. Typically such experiments produce large data sets that are difficult for biologists to handle. Although there are some programmes available for interpretation of bacterial transcriptomics data and CGH microarray data for looking at genetic stability in oncogenes, there are none specifically to understand the mosaic nature of bacterial genomes. Consequently a bottle neck still persists in accurate processing and mathematical analysis of these data. To address this shortfall we have produced a simple and robust CGH microarray data analysis process that may be automated in the future to understand bacterial genomic diversity. Results: The process involves five steps: cleaning, normalisation, estimating gene presence and absence or divergence, validation, and analysis of data from test against three reference strains simultaneously. Each stage of the process is described and we have compared a number of methods available for characterising bacterial genomic diversity, for calculating the cut-off between gene presence and absence or divergence, and shown that a simple dynamic approach using a kernel density estimator performed better than both established, as well as a more sophisticated mixture modelling technique. We have also shown that current methods commonly used for CGH microarray analysis in tumour and cancer cell lines are not appropriate for analysing our data. Conclusion: After carrying out the analysis and validation for three sequenced Escherichia coli strains, CGH microarray data from 19 E. coli O157 pathogenic test strains were used to demonstrate the benefits of applying this simple and robust process to CGH microarray studies using bacterial genomes.