Gröbner basis techniques for certain problems in coding and systems theory

There is much common ground between the areas of coding theory and systems theory. Fitzpatrick has shown that a Göbner basis approach leads to efficient algorithms in the decoding of Reed-Solomon codes and in scalar interpolation and partial realization. This thesis simultaneously generalizes and simplifies that approach and presents applications to discrete-time modeling, multivariable interpolation and list decoding. Gröbner basis theory has come into its own in the context of software and algorithm development. By generalizing the concept of polynomial degree, term orders are provided for multivariable polynomial rings and free modules over polynomial rings. The orders are not, in general, unique and this adds, in no small way, to the power and flexibility of the technique. As well as being generating sets for ideals or modules, Gröbner bases always contain a element which is minimal with respect tot the corresponding term order. Central to this thesis is a general algorithm, valid for any term order, that produces a Gröbner basis for the solution module (or ideal) of elements satisfying a sequence of generalized congruences. These congruences, based on shifts and homomorphisms, are applicable to a wide variety of problems, including key equations and interpolations. At the core of the algorithm is an incremental step. Iterating this step lends a recursive/iterative character to the algorithm. As a consequence, not all of the input to the algorithm need be available from the start and different "paths" can be taken to reach the final solution. The existence of a suitable chain of modules satisfying the criteria of the incremental step is a prerequisite for applying the algorithm.

Combining and choosing case base maintenance algorithms

Case-Based Reasoning (CBR) uses past experiences to solve new problems. The quality of the past experiences, which are stored as cases in a case base, is a big factor in the performance of a CBR system. The system's competence may be improved by adding problems to the case base after they have been solved and their solutions verified to be correct. However, from time to time, the case base may have to be refined to reduce redundancy and to get rid of any noisy cases that may have been introduced. Many case base maintenance algorithms have been developed to delete noisy and redundant cases. However, different algorithms work well in different situations and it may be difficult for a knowledge engineer to know which one is the best to use for a particular case base. In this thesis, we investigate ways to combine algorithms to produce better deletion decisions than the decisions made by individual algorithms, and ways to choose which algorithm is best for a given case base at a given time. We analyse five of the most commonly-used maintenance algorithms in detail and show how the different algorithms perform better on different datasets. This motivates us to develop a new approach: maintenance by a committee of experts (MACE). MACE allows us to combine maintenance algorithms to produce a composite algorithm which exploits the merits of each of the algorithms that it contains. By combining different algorithms in different ways we can also define algorithms that have different trade-offs between accuracy and deletion. While MACE allows us to define an infinite number of new composite algorithms, we still face the problem of choosing which algorithm to use. To make this choice, we need to be able to identify properties of a case base that are predictive of which maintenance algorithm is best. We examine a number of measures of dataset complexity for this purpose. These provide a numerical way to describe a case base at a given time. We use the numerical description to develop a meta-case-based classification system. This system uses previous experience about which maintenance algorithm was best to use for other case bases to predict which algorithm to use for a new case base. Finally, we give the knowledge engineer more control over the deletion process by creating incremental versions of the maintenance algorithms. These incremental algorithms suggest one case at a time for deletion rather than a group of cases, which allows the knowledge engineer to decide whether or not each case in turn should be deleted or kept. We also develop incremental versions of the complexity measures, allowing us to create an incremental version of our meta-case-based classification system. Since the case base changes after each deletion, the best algorithm to use may also change. The incremental system allows us to choose which algorithm is the best to use at each point in the deletion process.

Improved kinematic sensing for motion control applications

New compensation methods are presented that can greatly reduce the slit errors (i.e. transition location errors) and interval errors induced due to non-idealities in optical incremental encoders (square-wave). An M/T-type, constant sample-time digital tachometer (CSDT) is selected for measuring the velocity of the sensor drives. Using this data, three encoder compensation techniques (two pseudoinverse based methods and an iterative method) are presented that improve velocity measurement accuracy. The methods do not require precise knowledge of shaft velocity. During the initial learning stage of the compensation algorithm (possibly performed in-situ), slit errors/interval errors are calculated through pseudoinversebased solutions of simple approximate linear equations, which can provide fast solutions, or an iterative method that requires very little memory storage. Subsequent operation of the motion system utilizes adjusted slit positions for more accurate velocity calculation. In the theoretical analysis of the compensation of encoder errors, encoder error sources such as random electrical noise and error in estimated reference velocity are considered. Initially, the proposed learning compensation techniques are validated by implementing the algorithms in MATLAB software, showing a 95% to 99% improvement in velocity measurement. However, it is also observed that the efficiency of the algorithm decreases with the higher presence of non-repetitive random noise and/or with the errors in reference velocity calculations. The performance improvement in velocity measurement is also demonstrated experimentally using motor-drive systems, each of which includes a field-programmable gate array (FPGA) for CSDT counting/timing purposes, and a digital-signal-processor (DSP). Results from open-loop velocity measurement and closed-loop servocontrol applications, on three optical incremental square-wave encoders and two motor drives, are compiled. While implementing these algorithms experimentally on different drives (with and without a flywheel) and on encoders of different resolutions, slit error reductions of 60% to 86% are obtained (typically approximately 80%).