Spacecraft relative motion modeling and control promises to enable or augment a wide range of missions for scientific research, military applications, and space situational awareness. This dissertation focuses on the development of novel, optimization-based, control design for some representative relative-motion-enabled missions. Spacecraft relative motion refers to two (or more) satellites in nearly identical orbits. We examine control design for relative configurations on the scale of meters (for the purposes of proximity operations) as well as on the scale of tens of kilometers (representative of science gathering missions). Realistic control design for satellites is limited by accurate modeling of the relative orbital perturbations as well as the highly constrained nature of most space systems. We present solutions to several types of optimal orbital maneuvers using a variety of different, realistic assumptions based on the maneuver objectives. Initially, we assume a perfectly circular orbit with a perfectly spherical Earth and analytically solve the under-actuated, minimum-energy, optimal transfer using techniques from optimal control and linear operator theory. The resulting open-loop control law is guaranteed to be a global optimum. Then, recognizing that very few, if any, orbits are truly circular, the optimal transfer problem is generalized to the elliptical linear and nonlinear systems which describe the relative motion. Solution of the minimum energy transfer for both the linear and nonlinear systems reveals that the resulting trajectories are nearly identical, implying that the nonlinearity has little effect on the relative motion. A continuous-time, nonlinear, sliding mode controller which tracks the linear trajectory in the presence of a higher fidelity orbit model shows that the closed-loop system is both asymptotically stable and robust to disturbances and un-modeled dynamics. Next, a novel method of computing discrete-time, multi-revolution, finite-thrust, fuel-optimal, relative orbit transfers near an elliptical, perturbed orbit is presented. The optimal control problem is based on the classical, continuous-time, fuel-optimization problem from calculus of variations, and we present the discrete-time analogue of this problem using a transcription-based method. The resulting linear program guarantees a global optimum in terms of fuel consumption, and we validate the results using classical impulsive orbit transfer theory. The new method is shown to converge to classical impulsive orbit transfer theory in the limit that the duration of the zero-order hold discretization approaches zero and the time horizon extends to infinity. Then the fuel/time optimal control problem is solved using a hybrid approach which uses a linear program to solve the fuel optimization, and a genetic algorithm to find the minimizing time-of-flight. The method developed in this work allows mission planners to determine the feasibility for realistic spacecraft and motion models. Proximity operations for robotic inspection have the potential to aid manned and unmanned systems in space situational awareness and contingency planning in the event of emergency. A potential limiting factor is the large number of constraints imposed on the inspector vehicle due to collision avoidance constraints and limited power and computational resources. We examine this problem and present a solution to the coupled orbit and attitude control problem using model predictive control. This control technique allows state and control constraints to be encoded as a mathematical program which is solved on-line. We present a new thruster constraint which models the minimum-impulse bit as a semi-continuous variable, resulting in a mixed-integer program. The new model, while computationally more expensive, is shown to be more fuel-efficient than a sub-optimal approximation. The result is a fuel efficient, trajectory tracking, model predictive controller with a linear-quadratic attitude regulator which tracks along a pre-computed ``safe'' trajectory in the presence of un-modeled dynamics on a higher fidelity orbital and attitude model. / Ph. D.

This thesis aims to address the real-time planning and nonlinear control of quadrupedal locomotion such that the resulting gaits are robust to various kinds of disturbances. Specifically, this work addresses two scenarios. Namely, a quasi-static formulation in which an inertial appendage (i.e., a tail) is used to assist the quadruped in negating external push disturbances, and an agile formulation which is derived in a manner such that an appendage could easily be added in future work to examine the affect of tails on agile and high-speed motions. Initially, this work presents a unified method in which bio-inspired articulated serpentine robotic tails may be integrated with walking robots, specifically quadrupeds, in order to produce stable and highly robust locomotion. The design and analysis of a holonomically constrained 2 degree of freedom (DOF) tail is shown and its accompanying nonlinear dynamic model is presented. The model created is used to develop a hierarchical control scheme which consists of a high-level path planner and a full-order nonlinear low-level controller. The high-level controller is based on model predictive control (MPC) and acts on a linear inverted pendulum (LIP) model which has been extended to include the forces produced by the tail by augmenting the LIP model with linearized tail dynamics. The MPC is used to generate center of mass (COM) and tail trajectories and is subject to the net ground reaction forces of the system, tail shape, and torque saturation of the tail in order to ensure overall feasibility of locomotion. At the lower level, a full-order nonlinear controller is implemented to track the generated trajectories using quadratic program (QP) based input-output (I-O) feedback linearization which acts on virtual constraints. The analytical results of the proposed approach are verified numerically through simulations using a full-order nonlinear model for the quadrupedal robot, Vision60, augmented with a tail, totaling at 20 DOF. The simulations include a variety of disturbances to show the robustness of the presented hierarchical control scheme. The aforementioned control scheme is then extended in the latter portion of this thesis to achieve more dynamic, agile, and robust locomotion. In particular, we examine the use of a single rigid body model as the template model for the real-time high-level MPC, which is linearized using variational based linearization (VBL) and is solved at 200 Hz as opposed to an event-based manner. The previously defined virtual constraints controller is also extended so as to include a control Lyapunov function (CLF) which contributes to both numerical stability of the QP and aids in stability of the output dynamics. This new hierarchical scheme is validated on the A1 robot, with a total of 18 DOF, through extensive simulations to display agility and robustness to ground height variations and external disturbances. The low-level controller is then further validated through a series of experiments displaying the ability for this algorithm to be readily transferred to hardware platforms. / Master of Science / This thesis aims to address the real-time planning and nonlinear control of four legged walking robots such that the resulting gaits are robust to various kinds of disturbances. Initially, this work presents a method in which a robotic tail can be integrated with legged robots to produce very stable walking patterns. A model is subsequently created to develop a multi-layer control scheme which consists of a high-level path planner, based on a reduced-order model and model predictive control techniques, that determines the trajectory for the quadruped and tail, followed by a low-level controller that considers the full-order dynamics of the robot and tail for robust tracking of the planned trajectory. The reduced-order model considered here enforces quasi-static motions which are slow but generally stable. This formulation is validated numerically through extensive full-order simulations of the Vision60 robot. This work then proceeds to develop an agile formulation using a similar multi-layer structure, but uses a reduced-order model which is more amenable to dynamic walking patterns. The low-level controller is also augmented slightly to provide additional robustness and theoretical guarantees. The latter control algorithm is extensively numerically validated in simulation using the A1 robot to show the large increase in robustness compared to the quasi-static formulation. Finally, this work presents experimental validation of the low-level controller formulated in the latter half of this work.

This thesis presents collision avoidance integrated with lane keeping and adaptive cruise control for a car. Collision avoidance is the ability to avoid obstacles that are in the vehicle's path, without causing damage to the obstacle or car. There are three types of collision avoidance controllers, passive, active, and semi-active. This thesis is designed using active collision avoidance controllers. There are two controllers developed for collision avoidance in this paper. They are an optimal controller and a fuzzy controller. The optimal vehicle trajectory, which maximizes the distance to an obstacle and changes lanes, is derived. The optimal collision avoidance controller is a closed loop controller; with the decisions based on the current state. The fuzzy controller makes decisions based on the system rules. A simulation environment was created to compare these two controllers as viable solutions for collision avoidance. The environment uses MATLAB/Simulink for simulation of the vehicle as well as the optimal and fuzzy controllers. The simulation incorporates system blocks of the kinematics of a car, navigation, states, control law, and velocity controller. Once the controllers are fully developed and tested in the simulation environment, they are implemented and tested on the platform vehicle. This verifies the real world performance of the controllers. The platform vehicle is a modified radio controlled car. This car is completely autonomous. The car has onboard sensors that allow it to follow a white piece of tape as well as detect obstacles. / Master of Science

A compact model representing the dynamics between piezoelectric voltage inputs and cantilever probe positioning, including nonlinear surface interaction forces, for atomic force microscopes (AFM) is considered. By considering a relatively large cantilever stiffness, singular perturbation methods reduce complexity in the model and allows for faster responses to Van der Waals interaction forces experienced by the cantilever's tip and measurement sample. In this study, we outline a nonlinear near-optimal feedback control approach for non-contact mode imaging designed to move the cantilever tip laterally about a desired trajectory and maintain the tip vertically about the equilibrium point of the attraction and repulsion forces. We also consider the universal instance when the tip-sample interaction force is unknown, and we construct cascaded high-gain observers to estimate these forces and multiple AFM dynamics for the purpose of output feedback control. Our proposed output feedback controller is used to accomplish the outlined control objective with only the piezotube position available for state feedback. / Master of Science / In this thesis, the idea of an atomic force microscope (AFM), specifically the applications of the non-contact mode, will be discussed. An atomic force microscope (AFM) is a tool that measures the surface height of nanometer sized samples. To improve the speed and precision of the machine under a non-contact mode objective, a controller is designed based on optimality and is applied to the system. The system contains a series of equations designed to steer the system towards a desired trajectory and minimal vibrations. Given the complexity of the system, resulting from nonlinearities, we will apply singular perturbation principles on the system's stiffness property to separate the larger problem into two smaller ones. These two problems are inserted into a near-optimal controller and a series of simulations are conducted to demonstrate performance. Alongside this, we will outline an observer to estimate the unknown dynamics of the system. These estimates are then applied to our controller to demonstrate that only the AFM's piezotube position is to be known in order to estimate and control the remaining dynamics of the system.

<p> The abort mission refers to the mission where the landing vehicle needs to terminate the landing mission when an anomaly happens and be safely guided to the desired orbit. Missions involving crew on board demands for a robust and efficient abort strategy. This thesis focuses on solving the time-optimal abort guidance (TOAG) problem in real-time via the feature-based learning method. First, according to the optimal control theory, the features are identified to represent the optimal solutions of TOAG using a few parameters. After that, a sufficiently large dataset of time-optimal abort trajectories is generated offline by solving the TOAG problems with different initial conditions. Then the features are extracted for all generated cases. To find the implicit relationships between the initial conditions and identified features, neural networks are constructed to map the relationships based on the generated dataset. A successfully trained neural network can generate solution in real time for a reasonable initial condition. Finally, experimental flight tests are conducted to demonstrate the onboard computation capability and effectiveness of the proposed method. </p>

This thesis is concerned with mathematical modelling and optimal control of constrained systems. Each of the systems under consideration is a system that can be controlled by one of the variables, and this control is subject to constraints. First, we consider middle-distance running where a runner's horizontal propulsive force is the control which is constrained to be within a given range. Middle-distance running is typically a strategy-intensive race as slipstreaming effects come into play since speeds are still relatively fast and runners can leave their starting lane. We formulate a two-runner coupled model and determine optimal strategies using optimal control theory. Second, we consider two applications of control systems with delay related to R&D expenditure. The first of these applications relates to the defence industry. The second relates to the pharmaceutical industry. Both applications are characterised by a long delay between initial investment in R&D and seeing the benefits of R&D realised. We formulate models tailored to each application and use optimal control theory to determine the optimal proportion of available funds to invest in R&D over a given time horizon. Third, we consider a mathematical model of urban burglary based on the Short model. We make some modifications to this model including the addition of deterrence due to police officer presence. Police officer density is the control variable, which is constrained due to a finite number of police officers. We look at different control strategies for the police and their effect on burglary hot-spot formation.

The problem of designing control systems for entire plants is studied. A review of previous works, available techniques and current research challenges is presented, followed by the description of some theoretical tools to improve plantwide control, including the proposal of an augmented lexicographic multi-objective optimization procedure. With these, an augmented hierarchical plantwide control design technique and an optimal multi-objective technique for integrated control structure selection and controller tuning are proposed. The main contributions of these proposed techniques are the inclusion of system identification and optimal control tuning as part of the plantwide design procedure for improved results, support to multi-objective control specifications and support to any type of plant and controllers. Finally, the proposed techniques are applied to industrial benchmarks to demonstrate and validate its applicability. / O problema de projetar sistemas de controle para plantas inteiras é estudado. Uma revisão de trabalhos anteriores, técnicas disponíveis e atuais desafios de pesquisa é apresentada, seguida da descrição de algumas ferramentas teóricas para melhorar o controle plantwide, incluindo a proposta de um procedimento de otimização multi-objetivo lexicográfico aumentado. Com tais elementos, são propostas uma nova técnica hierárquica aumentada de projeto de sistemas de controle plantwide e uma técnica multi-objetivo para seleção de estrutura de controlador integrada à sintonia ótima do controlador. As principais contribuições das técnicas propostas são a inclusão de identificação de sistemas e sintonia ótima de controladores como parte do procedimento de projeto de controle plantwide para melhores resultados, suporte a especificações multi-objetivo e suporte a quaisquer tipos de plantas e controladores. Finalmente, as técnicas propostas são aplicadas a benchmarks industriais para demonstrar e validar sua aplicabilidade.

The primary goal of this thesis is to study finite element based a priori and a posteriori error estimates of optimal control problems of various kinds governed by linear elliptic PDEs (partial differential equations) of second and fourth orders. This thesis studies interior and boundary control (Neumann and Dirichlet) problems. The initial chapter is introductory in nature. Some preliminary and fundamental results of finite element methods and optimal control problems which play key roles for the subsequent analysis are reviewed in this chapter. This is followed by a brief literature survey of the finite element based numerical analysis of PDE constrained optimal control problems. We conclude the chapter with a discussion on the outline of the thesis. An abstract framework for the error analysis of discontinuous Galerkin methods for control constrained optimal control problems is developed in the second chapter. The analysis establishes the best approximation result from a priori analysis point of view and delivers a reliable and efficient a posteriori error estimator. The results are applicable to a variety of problems just under the minimal regularity possessed by the well-posedness of the problem. Subsequently, the applications of p p - interior penalty methods for a boundary control problem as well as a distributed control problem governed by the bi-harmonic equation subject to simply supported boundary conditions are discussed through the abstract analysis. In the third chapter, an alternative energy space based approach is proposed for the Dirichlet boundary control problem and then a finite element based numerical method is designed and analyzed for its numerical approximation. A priori error estimates of optimal order in the energy norm and the m norm are derived. Moreover, a reliable and efficient a posteriori error estimator is derived with the help an auxiliary problem. An energy space based Dirichlet boundary control problem governed by bi-harmonic equation is investigated and subsequently a l y - interior penalty method is proposed and analyzed for it in the fourth chapter. An optimal order a priori error estimate is derived under the minimal regularity conditions. The abstract error estimate guarantees optimal order of convergence whenever the solution has minimum regularity. Further an optimal order l l norm error estimate is derived. The fifth chapter studies a super convergence result for the optimal control of an interior control problem with Dirichlet cost functional and governed by second order linear elliptic PDE. An optimal order a priori error estimate is derived and subsequently a super convergence result for the optimal control is derived. A residual based reliable and efficient error estimators are derived in a posteriori error control for the optimal control. Numerical experiments illustrate the theoretical results at the end of every chapter. We conclude the thesis stating the possible extensions which can be made of the results presented in the thesis with some more problems of future interest in this direction.

The problem of designing control systems for entire plants is studied. A review of previous works, available techniques and current research challenges is presented, followed by the description of some theoretical tools to improve plantwide control, including the proposal of an augmented lexicographic multi-objective optimization procedure. With these, an augmented hierarchical plantwide control design technique and an optimal multi-objective technique for integrated control structure selection and controller tuning are proposed. The main contributions of these proposed techniques are the inclusion of system identification and optimal control tuning as part of the plantwide design procedure for improved results, support to multi-objective control specifications and support to any type of plant and controllers. Finally, the proposed techniques are applied to industrial benchmarks to demonstrate and validate its applicability. / O problema de projetar sistemas de controle para plantas inteiras é estudado. Uma revisão de trabalhos anteriores, técnicas disponíveis e atuais desafios de pesquisa é apresentada, seguida da descrição de algumas ferramentas teóricas para melhorar o controle plantwide, incluindo a proposta de um procedimento de otimização multi-objetivo lexicográfico aumentado. Com tais elementos, são propostas uma nova técnica hierárquica aumentada de projeto de sistemas de controle plantwide e uma técnica multi-objetivo para seleção de estrutura de controlador integrada à sintonia ótima do controlador. As principais contribuições das técnicas propostas são a inclusão de identificação de sistemas e sintonia ótima de controladores como parte do procedimento de projeto de controle plantwide para melhores resultados, suporte a especificações multi-objetivo e suporte a quaisquer tipos de plantas e controladores. Finalmente, as técnicas propostas são aplicadas a benchmarks industriais para demonstrar e validar sua aplicabilidade.

<p>Numerical trajectory optimization is helpful not only for mission planning but also design</p> <p>space exploration and quantifying vehicle performance. Direct methods for solving the opti-</p> <p>mal control problems, which first discretize the problem before applying necessary conditions</p> <p>of optimality, dominate the field of trajectory optimization because they are easier for the</p> <p>user to set up and are less reliant on a forming a good initial guess. On the other hand,</p> <p>many consider indirect methods, which apply the necessary conditions of optimality prior to</p> <p>discretization, too difficult to use for practical applications. Indirect methods though provide</p> <p>very high quality solutions, easily accessible sensitivity information, and faster convergence</p> <p>given a sufficiently good guess. Those strengths make indirect methods especially well-suited</p> <p>for generating large data sets for system analysis and worth revisiting.</p> <p>Recent advancements in the application of indirect methods have already mitigated many</p> <p>of the often cited issues. Automatic derivation of the necessary conditions with computer</p> <p>algebra systems have eliminated the manual step which was time-intensive and error-prone.</p> <p>Furthermore, regularization techniques have reduced problems which traditionally needed</p> <p>many phases and complex staging, like those with inequality path constraints, to a signifi-</p> <p>cantly easier to handle single arc. Finally, continuation methods can circumvent the small</p> <p>radius of convergence of indirect methods by gradually changing the problem and use previ-</p> <p>ously found solutions for guesses.</p> <p>The new optimal control problem solver Giuseppe incorporates and builds upon these</p> <p>advancements to make indirect methods more accessible and easily used. It seeks to enable</p> <p>greater research and creative approaches to problem solving by being more flexible and</p> <p>extensible than previous solvers. The solver accomplishes this by implementing a modular</p> <p>design with well-defined internal interfaces. Moreover, it allows the user easy access to and</p> <p>manipulation of component objects and functions to be use in the way best suited to solve</p> <p>a problem.</p> <p>A new technique simplifies and automates what was the predominate roadblock to using</p> <p>continuation, the generation of an initial guess for the seed solution. Reliable generation of</p> <p>a guess sufficient for convergence still usually required advanced knowledge optimal contrtheory or sometimes incorporation of an entirely separate optimization method. With the</p> <p>new method, a user only needs to supply initial states, a control profile, and a time-span</p> <p>over which to integrate. The guess generator then produces a guess for the “primal” problem</p> <p>through propagation of the initial value problem. It then estimates the “dual” (adjoint)</p> <p>variables by the Gauss-Newton method for solving the nonlinear least-squares problem. The</p> <p>decoupled approach prevents poorly guessed dual variables from altering the relatively easily</p> <p>guess primal variables. As a result, this method is simpler to use, faster to iterate, and much</p> <p>more reliable than previous guess generation techniques.</p> <p>Leveraging the continuation process also allows for greater insight into the solution space</p> <p>as there is only a small marginal cost to producing an additional nearby solutions. As a</p> <p>result, a user can quickly generate large families of solutions by sweeping parameters and</p> <p>modifying constraints. These families provide much greater insight in the general problem</p> <p>and underlying system than is obtainable with singular point solutions. One can extend</p> <p>these analyses to high-dimensional spaces through construction of compound continuation</p> <p>strategies expressible by directed trees.</p> <p>Lastly, a study into common convergence explicates their causes and recommends mitiga-</p> <p>tion strategies. In this area, the continuation process also serves an important role. Adaptive</p> <p>step-size routines usually suffice to handle common sensitivity issues and scaling constraints</p> <p>is simpler and out-performs scaling parameters directly. Issues arise when a cost functional</p> <p>becomes insensitive to the control, which a small control cost mitigates. The best perfor-</p> <p>mance of the solver requires proper sizing of the smoothing parameters used in regularization</p> <p>methods. An asymptotic increase in the magnitude of adjoint variables indicate approaching</p> <p>a feasibility boundary of the solution space.</p> <p>These techniques for indirect methods greatly facilitate their use and enable the gen-</p> <p>eration of large libraries of high-quality optimal trajectories for complex problems. In the</p> <p>future, these libraries can give a detailed account of vehicle performance throughout its flight</p> <p>envelope, feed higher-level system analyses, or inform real-time control applications.</p>