The recent addition of autonomous formation flying spacecraft to the world's satellite fleet provides new motivation to study feedback control techniques. In this thesis, we develop nonlinear orbit control laws for use in spacecraft orbital maneuvers, and spacecraft formation flying. We apply these new control laws to a number of sample maneuvers, including formation stablishment and formation keeping maneuvers for NASA-Goddard's Leonardo-BRDF formation, and coupled orbit, and attitude maneuvers for HokieSat, a spacecraft designed, and built by students at Virginia Tech to fly in the Ionospheric Observation Nanosatellite Formation (ION-F). To provide target orbit states for feedback control, we develop and apply an algorithm to calculate a formation master orbit representing the geometric center of the formation. We also define a new technique for choosing orbital element feedback gains which appropriately scales the gains for orbit maintenance, and provides an excellent starting point for gain optimization. The orbital element feedback control law, augmented by mean motion control, and applied with appropriate gains, forces asymptotic convergence to a spacecraft target orbit, for a large variety of spacecraft maneuvers. / Master of Science

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The Finite Element Method (FEM) based computational mechanics is applied to simulate the optimal attenuation of vibrations in actively controlled structures. The simulation results provide the forces to be generated by actuators, as well as the structures response. Vibrations can be attenuated by applying either open loop or closed loop control strategies. In open loop control, the control forces for a given initial (or disturbed) configuration of the structure are determined in terms of time, and can be preprogrammed in advance. On the other hand, the control forces in closed loop control depend only on the current state of the system, which should be continuously monitored. Optimal attenuation is obtained by solving the optimality equations for the problem derived from the Pontryagins principle. These equations together with the initial and final boundary conditions constitute the two-point-boundary-value (TPBV) problem. <p>Here the optimal solutions are obtained by applying an analogy (referred to as the beam analogy) between the optimality equation and the equation for a certain problem of static beams in bending. The problem of analogous beams is solved by the standard FEM in the spatial domain, and then the results are converted into the solution of the optimal vibration control problem in the time domain. The concept of the independent-modal-space-control (IMSC) is adopted, in which the number of independent actuators control the same number of vibrations modes. <p>The steps of the analogy are programmed into an algorithm referred to as the Beam Analogy Algorithm (BAA). As an illustration of the approach, the BAA is used to simulate the open loop vibration control of a structure with several sets of actuators. Some details, such as an efficient meshing of the analogous beams and effective solving of the target condition are discussed. <p> Next, the BAA is modified to handle closed loop vibration control problems. The algorithm determines the optimal feedback gain matrix, which is then used to calculate the actuator forces required at any current state of the system. The methods accuracy is also analyzed.

The Finite Element Method (FEM) based computational mechanics is applied to simulate the optimal attenuation of vibrations in actively controlled structures. The simulation results provide the forces to be generated by actuators, as well as the structures response. Vibrations can be attenuated by applying either open loop or closed loop control strategies. In open loop control, the control forces for a given initial (or disturbed) configuration of the structure are determined in terms of time, and can be preprogrammed in advance. On the other hand, the control forces in closed loop control depend only on the current state of the system, which should be continuously monitored. Optimal attenuation is obtained by solving the optimality equations for the problem derived from the Pontryagins principle. These equations together with the initial and final boundary conditions constitute the two-point-boundary-value (TPBV) problem. <p>Here the optimal solutions are obtained by applying an analogy (referred to as the beam analogy) between the optimality equation and the equation for a certain problem of static beams in bending. The problem of analogous beams is solved by the standard FEM in the spatial domain, and then the results are converted into the solution of the optimal vibration control problem in the time domain. The concept of the independent-modal-space-control (IMSC) is adopted, in which the number of independent actuators control the same number of vibrations modes. <p>The steps of the analogy are programmed into an algorithm referred to as the Beam Analogy Algorithm (BAA). As an illustration of the approach, the BAA is used to simulate the open loop vibration control of a structure with several sets of actuators. Some details, such as an efficient meshing of the analogous beams and effective solving of the target condition are discussed. <p> Next, the BAA is modified to handle closed loop vibration control problems. The algorithm determines the optimal feedback gain matrix, which is then used to calculate the actuator forces required at any current state of the system. The methods accuracy is also analyzed.

This dissertation is concerned with the potential multistability of protein concentrations in the cell that can arise in biochemical networks. That is, situations where one, or a family of, proteins may sit at one of two or more different steady state concentrations in otherwise identical cells, and in spite of them being in the same environment. Models of multisite protein phosphorylation have shown that this mechanism is able to exhibit unlimited multistability. Nevertheless, these models have not considered enzyme docking, the binding of the enzymes to one or more substrate docking sites, which are separate from the motif that is chemically modified. Enzyme docking is, however, increasingly being recognised as a method to achieve specificity in protein phosphorylation and dephosphorylation cycles. Most models in the literature for these systems are deterministic i.e. based on Ordinary Differential Equations, despite the fact that these are accurate only in the limit of large molecule numbers. For small molecule numbers, a discrete probabilistic, stochastic, approach is more suitable. However, when compared to the tools available in the deterministic framework, the tools available for stochastic analysis offer inadequate visualisation and intuition. We firstly try to bridge that gap, by developing three tools: a) a discrete `nullclines' construct applicable to stochastic systems - an analogue to the ODE nullcines, b) a stochastic tool based on a Weakly Chained Diagonally Dominant M-matrix formulation of the Chemical Master Equation and c) an algorithm that is able to construct non-reversible Markov chains with desired stationary probability distributions. We subsequently prove that, for multisite protein phosphorylation and similar models, in the deterministic domain, enzyme docking and the consequent substrate enzyme-sequestration must inevitably limit the extent of multistability, ultimately to one steady state. In contrast, bimodality can be obtained in the stochastic domain even in situations where bistability is not possible for large molecule numbers. We finally extend our results to cases where we have an autophosphorylating kinase, as for example is the case with $Ca^{2+}$/calmodulin-dependent protein kinase II (CaMKII), a key enzyme in synaptic plasticity.