A Study of Linear vs. Nonlinear Control Techniques for the Reconfiguration of Satellite Formations

This thesis investigates several linear and nonlinear feedback control methods for satellite formation reconfigurations and compares them to a near optimal open loop, discrete-time, impulsive maneuver. The reconfigurations are done in terms of a set of relative parameters that define an orbit about the leader satellite (or center reference position if a leader satellite does not exist at the center of the formation). The purpose of the study is two-fold, to compare the control usage of continuous feedback control methods versus a discrete bum method and to determine if nonlinear control techniques offer significant improvement over more conventional linear control laws. Linear Quadratic Regulators (LQR), LQR with linearizing feedback, State Dependent Riccati Equation (SDRE) and sliding mode controllers are considered. Simulations showed that reconfigurations for small relative orbits were adequately controlled using linear techniques

The constrained discrete-time state-dependent Riccati equation technique for uncertain nonlinear systems

The objective of the thesis is to introduce a relatively general nonlinear controller/estimator synthesis framework using a special type of the state-dependent Riccati equation technique. The continuous time state-dependent Riccati equation (SDRE) technique is extended to discrete-time under input and state constraints, yielding constrained (C) discrete-time (D) SDRE, referred to as CD-SDRE. For the latter, stability analysis and calculation of a region of attraction are carried out. The derivation of the D-SDRE under state-dependent weights is provided. Stability of the D-SDRE feedback system is established using Lyapunov stability approach. Receding horizon strategy is used to take into account the constraints on D-SDRE controller. Stability condition of the CD-SDRE controller is analyzed by using a switched system. The use of CD-SDRE scheme in the presence of constraints is then systematically demonstrated by applying this scheme to problems of spacecraft formation orbit reconfiguration under limited performance on thrusters. Simulation results demonstrate the efficacy and reliability of the proposed CD-SDRE. The CD-SDRE technique is further investigated in a case where there are uncertainties in nonlinear systems to be controlled. First, the system stability under each of the controllers in the robust CD-SDRE technique is separately established. The stability of the closed-loop system under the robust CD-SDRE controller is then proven based on the stability of each control system comprising switching configuration. A high fidelity dynamical model of spacecraft attitude motion in 3-dimensional space is derived with a partially filled fuel tank, assumed to have the first fuel slosh mode. The proposed robust CD-SDRE controller is then applied to the spacecraft attitude control system to stabilize its motion in the presence of uncertainties characterized by the first fuel slosh mode. The performance of the robust CD-SDRE technique is discussed. Subsequently, filtering techniques are investigated by using the D-SDRE technique. Detailed derivation of the D-SDRE-based filter (D-SDREF) is provided under the assumption of Gaussian noises and the stability condition of the error signal between the measured signal and the estimated signals is proven to be input-to-state stable. For the non-Gaussian distributed noises, we propose a filter by combining the D-SDREF and the particle filter (PF), named the combined D-SDRE/PF. Two algorithms for the filtering techniques are provided. Several filtering techniques are compared with challenging numerical examples to show the reliability and efficacy of the proposed D-SDREF and the combined D-SDRE/PF

Multi-Tap Extended Kalman Filter for a Periodic Waveform with Uncertain Frequency and Waveform Shape, and Data Dropouts

Gait analysis presents the challenge of detecting a periodic waveform in the presence of time varying frequency, amplitude, DC offset, and waveform shape, with acquisition gaps from partial occlusions. The combination of all of these components presents a formidable challenge. The Extended Kalman Filter for this system model has six states, which makes it weakly identifiable within the standard Extended Kalman Filter network. In this work, a novel robust Extended Kalman Filter-based approach is presented and evaluated for clinical use in gait analysis. The novel aspect of the proposed method is that at each sample, the present and several past observations are used to update the system state, strengthening the state identification. These past observations are referred to as delay-line taps

Linear Control Theory to Introduce PeriodicParameters for a Marine Ecosystem Model

Global ocean biogeochemical models are of great importance for the assessment of the role of the ocean in the global carbon cycle and for the estimation of the impact of the climate change on marine ecosystems. The ocean acts as a major sink of carbon dioxide and takes up about one third of anthropogenic CO2. This characteristic plays a central role with respect to the climate discussion. The CO2-uptake of the ocean refers to biogeochemical processes which are simulated by corresponding parameterized models. The evaluation of these models leads to a sensitivity analysis of the model parameters and a model state estimation using associated measurement data. Usually, the biogeochemical models are coupled to ocean circulation models. This work comprises an investigation and application of control theory and optimization methodologies using discrete linear quadratic optimal control with both closed loop and open loop and the Kalman filter method. The fundamental aim of this work is to explore the potentialities of those proposed methods regarding an enhancement of a one-dimensional non-linear marine ecosystem model of NPZD (N for dissolved inorganic nitrogen, P for phytoplankton, Z for zooplankton and D for detritus) type. This ecosystem model, developed by Oschlies and Garçon, simulates the distribution of nitrogen, phytoplankton, zooplankton and detritus in a water column and is driven by ocean circulation data. The proposed methods are used to introduce annually periodic model parameters in a linearized version of the model. Firstly, I use the closed loop discrete linear quadratic optimal control (LQOC). It will be shown that the obtained version of the model gives a significant reduction of the modeldata misfit, compared to the misfit obtained for the original model with optimized constant parameters. The found inner-annual variability of the optimized parameters provides hints for an improvement of the original model. The obtained optimal periodic parameters are also used in validation and prediction experiments with the original non-linear version of the model. The experiments indicate that the considered method is very suitable for the considered marine ecosystem models. Secondly, I use the obtained periodic parameters in a state estimation applying the Kalman filter method. A comparison with results obtained by using optimized constant parameters in the Kalman filter state estimation approach is performed. We show that the Kalman filter method provides a very reasonable solution with periodic parameters compared to that with constant parameters. Thirdly, I use the open loop discrete linear quadratic optimal control (DOLOC) to investigate the impact of the linearization scheme about the state variables on the model-data-fit by using these periodic parameters in the original non-linear NPZD model. The proposed methodologies, particularly the closed loop discrete linear quadratic optimal control approach, serve as initial steps towards a tool for an efficient enhancement of marine ecosystem models. The investigation of further improvements of the presented algorithms as well as other promising approaches in the framework of linear quadratic optimal control optimization are considered to be highly valuable