Groebner Basis Methods for Multichannel Sampling with Unknown Offsets

In multichannel sampling, several sets of sub-Nyquist sampled signal values are acquired. The offsets between the sets are unknown, and have to be resolved, just like the parameters of the signal itself. This problem is nonlinear in the offsets, but linear in the signal parameters. We show that when the basis functions for the signal space are related to polynomials, we can express the joint offset and signal parameter estimation as a set of polynomial equations. This is the case for example with polynomial signals or Fourier series. The unknown offsets and signal parameters can be computed exactly from such a set of polynomials using Gröbner bases and Buchberger’s algorithm. This solution method is developed in detail after a short and tutorial overview of Gröbner basis methods. We then address the case of noisy samples, and consider the computational complexity, exploring simplifications due to the special structure of the problem

Control Systems: New Approaches to Analysis and Design

This dissertation deals with two open problems in control theory. The first problem concerns the synthesis of fixed structure controllers for Linear Time Invariant (LTI) systems. The problem of synthesizing fixed structure/order controllers has practical importance when simplicity, hardware limitations, or reliability in the implementation of a controller dictates a low order of stabilization. A new method is proposed to simplify the calculation of the set of fixed structure stabilizing controllers for any given plant. The method makes use of computational algebraic geometry techniques and sign-definite decomposition method. Although designing a stabilizing controller of a fixed structure is important, in many practical applications it is also desirable to control the transient response of the closed loop system. This dissertation proposes a novel approach to approximate the set of stabilizing Proportional-Integral-Derivative (PID) controllers guaranteeing transient response specifications. Such desirable set of PID controllers can be constructed upon an application of Widder's theorem and Markov-Lukacs representation of non-negative polynomials. The second problem explored in this dissertation handles the design and control of linear systems without requiring the knowledge of the mathematical model of the system and directly from a small set of measurements, processed appropriately. The traditional approach to deal with the analysis and control of complex systems has been to describe them mathematically with sets of algebraic or differential equations. The objective of the proposed approach is to determine the design variables directly from a small set of measurements. In particular, it will be shown that the functional dependency of any system variable on any set of system design parameters can be determined by a small number of measurements. Once the functional dependency is obtained, it can be used to extract the values of the design parameters

Biomechanical Analysis of Concealed Pack Load Influences on Terrorist Gait Signatures Derived from Gröbner Basis Theory

This project examines kinematic gait parameters as forensic predictors of the influence associated with individuals carrying concealed weighted packs up to 20% of their body weight. An initial inverse dynamics approach combined with computational algebra provided lower limb joint angles during the stance phase of gait as measured from 12 human subjects during normal walking. The following paper describes the additional biomechanical analysis of the joint angle data to produce kinetic and kinematic parameters further characterizing human motion. Results include the rotational velocities and accelerations of the hip, knee, and ankle as well as inertial moments and kinetic energies produced at these joints. The reported findings indicate a non-statistically significant influence of concealed pack load, body mass index, and gender on joint kinetics (p\u3e0.05). Ratios of loaded to unloaded kinematics, however, identified some statistical influence on gait (

Exploring the potential energy landscape over a large parameter-space

Solving large polynomial systems with coefficient parameters are ubiquitous and constitute an important class of problems. We demonstrate the computational power of two methods - a symbolic one called the Comprehensive Grobner basis and a numerical one called the cheater's homotopy - applied to studying both potential energy landscapes and a variety of questions arising from geometry and phenomenology. Particular attention is paid to an example in flux compactification where important physical quantities such as the gravitino and moduli masses and the string coupling can be efficiently extracted

A study of some molecular interactions on alumina surfaces by inelastic electron tunnelling spectroscopy

Further developments have been carried out to improve the resolution and sensitivity of the spectrometer by introducing a dual phase lock-in amplifier and using new software to enhance the flexibility of the computer interfaced with the spectrometer. The spectrometer has been utilised to study a variety of molecular orientations on an alumina substrate. These have included an investigation to distinguish optical and geometrical isomers together with some alkynes in order to explore the validity of the previously proposed Selection Rule. The new observation that the triple bond is detected even when parallel to the substrate surface is reported. An attempt to study the polymerisation of ethylene on an alumina substrate has been carried out and some evidence is presented to support an increase in polymerisation with time. It has been shown that formic acid is produced 'in situ' within an aluminium-aluminium oxide-lead tunnelling junction from atmospheric carbon dioxide and water. A mechanism to account for this reaction is proposed. Junction structure has been studied particularly by utilising a modified crystal oscillator thickness monitor to investigate the influence of electrode and insulating oxide thickness both on junction electrical integrity and the mechanism of doping completed tunnelling junctions

Algebraic synthesis methods for linear multivariable control systems

The mathematical formulation of various control synthesis problems , (such as Decentralized Stabilization Problem , (DSP) , Total Finite Settling Time Stabilization for discrete time linear systems, (TFSTS) , Exact Model Matching Problem, (EMMP), Decoupling and Noninteracting Control Problems) , via the algebraic framework of Matrix Fractional Representation . (MFR) - i.e. the representation of the transfer matrices of the system as matrix fractions over the ring of interest - results to the study of matrix equations over rings , such as : A . X + B . Y = C , (X. A + Y . B =C) (1) A· X = B , (y. A = B) (2) A·X·B = C (3) A·X + Y·B = C, X·A + B·Y = C, A·X·B + C·Y·D = E (4) The main objective of this dissertation is to further investigate conditions for existence and characterization of certain types of solutions of equation (1) ; develop a unifying algebraic approach for solvability and characterization of solutions of equations (1) - (4), based on structural properties of the given matrices, over the ring of interest. The standard matrix Diophantine equation (1) is associated with the TFSTS for discrete time linear systems and issues concerning the characterization of solutions according 'to the Extended McMillan Degree, (EMD) , (minimum EMD , or fixed EMD) , of the stabilizing controllers they define , are studied . A link between the issues in question and topological properties of certain families of solutions of (1) is established . Equation (1) is also studied in association with the DSP and Diagonal DSP (DDSP) , for continuous time linear systems . Conditions for characterizing block diagonal solutions of (1) , (which define decentralized stabilizing controllers) , are derived and a closed form description of the families of diagonal and two blocks diagonal decentralized stabilizing controllers is introduced. The set of matrix equations (1) - (4) is assumed over the field of fractions of the ring of interest , ℛ , (mainly a Euclidean Domain, (ED) , and thus a Principal Ideal Domain , (PID)) , and solvability as well as parametrization of solutions over ℛ is investigated under the unifying algebraic framework of extended non square matrix divisors , projectors and annihilators of the known matrices over ℛ . In practice the ring of interest is either the ring of polynomials ℝ [s] , or the rings of proper