Contributions to the study of a class of optimal control problems on the orthogonal groups SO(3) and SO(4)
- Authors: Adams, Ross Montague
- Date: 2015
- Language: English
- Type: text , Thesis , Doctoral , PhD
- Identifier: http://hdl.handle.net/10962/64826 , vital:28608
- Description: In this thesis we investigate a class of invariant optimal control problems, and their associated quadratic Hamilton-Poisson systems, on the orthogonal groups SO(3) and SO(4). Specifically, we are concerned with the class of left-invariant control affine systems. We begin by classifying all cost-extended systems on SO(3) under cost equivalence. (Cost-extended systems are closely related to optimal control problems.) A classification of all quadratic Hamilton-Poisson systems on the (minus) Lie-Poisson space so(3)*, under affine equivalence, is also obtained. For the normal forms obtained in our classification (of Hamilton-Poisson systems) we investigate the (Lyapunov) stability nature of the equilibria using spectral and energy-Casimir methods. For a subclass of these systems, we obtain analytic expressions for the integral curves of the associated Hamiltonian vector fields in terms of (basic) Jacobi elliptic functions. The explicit relationship between the classification of cost-extended systems on SO(3) and the classification of quadratic Hamilton- Poisson systems on so(3)* is provided. On SO(4), a classification of all left-invariant control affine systems under L-equivalence is obtained. We then determine which of these representatives are controllable, thus obtaining a classification under detached feedback equivalence. We also obtain a partial classification of quadratic Hamilton-Poisson systems on the Lie-Poisson space so(4)*. An investigation of the stability nature of the equilibria for a subclass of these systems is also done. Several illustrative examples of optimal control problems on the orthogonal group SO(3) are provided. More specifically, we consider an optimal control problem corresponding to a representative of our classification (of cost-extended system) for each possible number of control inputs. For each of these problems, we obtain explicit expressions for the extremal trajectories on the homogeneous space S2 by projecting the extremal trajectories on the group SO(3). The examples provided show how our classifications of cost-extended systems and Hamilton-Poisson systems can be used to obtain the optimal controls and the extremal trajectories corresponding to a large class of optimal control problems on SO(3). An example of a four-input optimal control problem on SO(4) is also provided. This example is provided to show how the solutions of certain problems on SO(4) can be related to the solutions of certain optimal control problems on SO(3).
- Full Text:
- Authors: Adams, Ross Montague
- Date: 2015
- Language: English
- Type: text , Thesis , Doctoral , PhD
- Identifier: http://hdl.handle.net/10962/64826 , vital:28608
- Description: In this thesis we investigate a class of invariant optimal control problems, and their associated quadratic Hamilton-Poisson systems, on the orthogonal groups SO(3) and SO(4). Specifically, we are concerned with the class of left-invariant control affine systems. We begin by classifying all cost-extended systems on SO(3) under cost equivalence. (Cost-extended systems are closely related to optimal control problems.) A classification of all quadratic Hamilton-Poisson systems on the (minus) Lie-Poisson space so(3)*, under affine equivalence, is also obtained. For the normal forms obtained in our classification (of Hamilton-Poisson systems) we investigate the (Lyapunov) stability nature of the equilibria using spectral and energy-Casimir methods. For a subclass of these systems, we obtain analytic expressions for the integral curves of the associated Hamiltonian vector fields in terms of (basic) Jacobi elliptic functions. The explicit relationship between the classification of cost-extended systems on SO(3) and the classification of quadratic Hamilton- Poisson systems on so(3)* is provided. On SO(4), a classification of all left-invariant control affine systems under L-equivalence is obtained. We then determine which of these representatives are controllable, thus obtaining a classification under detached feedback equivalence. We also obtain a partial classification of quadratic Hamilton-Poisson systems on the Lie-Poisson space so(4)*. An investigation of the stability nature of the equilibria for a subclass of these systems is also done. Several illustrative examples of optimal control problems on the orthogonal group SO(3) are provided. More specifically, we consider an optimal control problem corresponding to a representative of our classification (of cost-extended system) for each possible number of control inputs. For each of these problems, we obtain explicit expressions for the extremal trajectories on the homogeneous space S2 by projecting the extremal trajectories on the group SO(3). The examples provided show how our classifications of cost-extended systems and Hamilton-Poisson systems can be used to obtain the optimal controls and the extremal trajectories corresponding to a large class of optimal control problems on SO(3). An example of a four-input optimal control problem on SO(4) is also provided. This example is provided to show how the solutions of certain problems on SO(4) can be related to the solutions of certain optimal control problems on SO(3).
- Full Text:
Reconstructing ionospheric TEC over South Africa using signals from a regional GPS network
- Authors: Opperman, B D L
- Date: 2008
- Subjects: Global Positioning System Global Positioning System -- Data processing Electrons -- South Africa Ionosphere -- South Africa Ionospheric radio wave propagation -- South Africa
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:5487 , http://hdl.handle.net/10962/d1005273
- Description: Radio signals transmitted by GPS satellites orbiting the Earth are modulated as they propagate through the electrically charged plasmasphere and ionosphere in the near-Earth space environment. Through a linear combination of GPS range and phase measurements observed on two carrier frequencies by terrestrial-based GPS receivers, the ionospheric total electron content (TEC) along oblique GPS signal paths may be quantified. Simultaneous observations of signals transmitted by multiple GPS satellites and observed from a network of South African dual frequency GPS receivers, constitute a spatially dense ionospheric measurement source over the region. A new methodology, based on an adjusted spherical harmonic (ASHA) expansion, was developed to estimate diurnal vertical TEC over the region using GPS observations over the region. The performance of the ASHA methodology to estimate diurnal TEC and satellite and receiver differential clock biases (DCBs) for a single GPS receiver was first tested with simulation data and subsequently applied to observed GPS data. The resulting diurnal TEC profiles estimated from GPS observations compared favourably to measurements from three South African ionosondes and two other GPS-based methodologies for 2006 solstice and equinox dates. The ASHA methodology was applied to calculating diurnal two-dimensional TEC maps from multiple receivers in the South African GPS network. The space physics application of the newly developed methodology was demonstrated by investigating the ionosphere’s behaviour during a severe geomagnetic storm and investigating the long-term ionospheric stability in support of the proposed Square Kilometre Array (SKA) radio astronomy project. The feasibility of employing the newly developed technique in an operational near real-time system for estimating and dissimenating TEC values over Southern Africa using observations from a regional GPS receiver network, was investigated.
- Full Text:
- Authors: Opperman, B D L
- Date: 2008
- Subjects: Global Positioning System Global Positioning System -- Data processing Electrons -- South Africa Ionosphere -- South Africa Ionospheric radio wave propagation -- South Africa
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:5487 , http://hdl.handle.net/10962/d1005273
- Description: Radio signals transmitted by GPS satellites orbiting the Earth are modulated as they propagate through the electrically charged plasmasphere and ionosphere in the near-Earth space environment. Through a linear combination of GPS range and phase measurements observed on two carrier frequencies by terrestrial-based GPS receivers, the ionospheric total electron content (TEC) along oblique GPS signal paths may be quantified. Simultaneous observations of signals transmitted by multiple GPS satellites and observed from a network of South African dual frequency GPS receivers, constitute a spatially dense ionospheric measurement source over the region. A new methodology, based on an adjusted spherical harmonic (ASHA) expansion, was developed to estimate diurnal vertical TEC over the region using GPS observations over the region. The performance of the ASHA methodology to estimate diurnal TEC and satellite and receiver differential clock biases (DCBs) for a single GPS receiver was first tested with simulation data and subsequently applied to observed GPS data. The resulting diurnal TEC profiles estimated from GPS observations compared favourably to measurements from three South African ionosondes and two other GPS-based methodologies for 2006 solstice and equinox dates. The ASHA methodology was applied to calculating diurnal two-dimensional TEC maps from multiple receivers in the South African GPS network. The space physics application of the newly developed methodology was demonstrated by investigating the ionosphere’s behaviour during a severe geomagnetic storm and investigating the long-term ionospheric stability in support of the proposed Square Kilometre Array (SKA) radio astronomy project. The feasibility of employing the newly developed technique in an operational near real-time system for estimating and dissimenating TEC values over Southern Africa using observations from a regional GPS receiver network, was investigated.
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Chemical and spectroscopic studies of the capsular polysaccharides of some klebsiella and escherichia coli serotypes
- Authors: Stanley, Shawn Mark Ross
- Date: 1990
- Subjects: Polysaccharides , Klebsiella , Escherichia
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:3736 , http://hdl.handle.net/10962/d1001525
- Description: The work described in this thesis forms part of an international programme concerned with the structure elucidation of the capsular antigens of some Enterobacteriaceae. Many of the Klebsiella and some of the Escherichia coli are pathogenic to man and, hence, they are of interest. The virulence of bacteria is a multifactorial phenomenon, in which characteristic traits of bacteria and their hosts play comparable and complementary roles. It is accepted that pathogens are more virulent when encapsulated, because, nearly all disease causing bacteria have a capsule when freshly isolated from the host. This increase in pathogenicity is related, in part, to the capsular polysaccharides' ability to avoid or attenuate the host defence mechanisms. In the majority of cases the protective aspects of the capsule are overcome in the latter stages of infection when the formation of specific antibodies by the host has occurred. However there are situations in which an immune state of the infected host is virtually never reached, and susceptiblity to the infecting bacteria is maintained even in the advanced stage of an infection. Explanation of this phenomenon becomes possible by analysing the structure of the polysaccharides. The inability of the host to raise an immune response to the capsule may be because the structure of the polysaccharide is similar or identical to the host's carbohydrates. The serological and pathogenic relatedness of encapsulated E. coli and Klebsiella, to the encapsulated strains of other genera, is based on structural identity or similarity of the respective capsules. Capsular polysaccharides are analysed by both chemical and instrumental methods, and, at present, nuclear magnetic resonance spectroscopy is the most important analytical technique
- Full Text:
- Authors: Stanley, Shawn Mark Ross
- Date: 1990
- Subjects: Polysaccharides , Klebsiella , Escherichia
- Language: English
- Type: Thesis , Doctoral , PhD
- Identifier: vital:3736 , http://hdl.handle.net/10962/d1001525
- Description: The work described in this thesis forms part of an international programme concerned with the structure elucidation of the capsular antigens of some Enterobacteriaceae. Many of the Klebsiella and some of the Escherichia coli are pathogenic to man and, hence, they are of interest. The virulence of bacteria is a multifactorial phenomenon, in which characteristic traits of bacteria and their hosts play comparable and complementary roles. It is accepted that pathogens are more virulent when encapsulated, because, nearly all disease causing bacteria have a capsule when freshly isolated from the host. This increase in pathogenicity is related, in part, to the capsular polysaccharides' ability to avoid or attenuate the host defence mechanisms. In the majority of cases the protective aspects of the capsule are overcome in the latter stages of infection when the formation of specific antibodies by the host has occurred. However there are situations in which an immune state of the infected host is virtually never reached, and susceptiblity to the infecting bacteria is maintained even in the advanced stage of an infection. Explanation of this phenomenon becomes possible by analysing the structure of the polysaccharides. The inability of the host to raise an immune response to the capsule may be because the structure of the polysaccharide is similar or identical to the host's carbohydrates. The serological and pathogenic relatedness of encapsulated E. coli and Klebsiella, to the encapsulated strains of other genera, is based on structural identity or similarity of the respective capsules. Capsular polysaccharides are analysed by both chemical and instrumental methods, and, at present, nuclear magnetic resonance spectroscopy is the most important analytical technique
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