Browsing by Subject "Optimal Control"
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Item A Weighted Residual Framework for Formulation and Analysis of Direct Transcription Methods for Optimal Control(2012-02-14) Singh, BaljeetIn the past three decades, numerous methods have been proposed to transcribe optimal control problems (OCP) into nonlinear programming problems (NLP). In this dissertation work, a unifying weighted residual framework is developed under which most of the existing transcription methods can be derived by judiciously choosing test and trial functions. This greatly simplifies the derivation of optimality conditions and costate estimation results for direct transcription methods. Under the same framework, three new transcription methods are devised which are particularly suitable for implementation in an adaptive refinement setting. The method of Hilbert space projection, the least square method for optimal control and generalized moment method for optimal control are developed and their optimality conditions are derived. It is shown that under a set of equivalence conditions, costates can be estimated from the Lagrange multipliers of the associated NLP for all three methods. Numerical implementation of these methods is described using B-Splines and global interpolating polynomials as approximating functions. It is shown that the existing pseudospectral methods for optimal control can be formulated and analyzed under the proposed weighted residual framework. Performance of Legendre, Gauss and Radau pseudospectral methods is compared with the methods proposed in this research. Based on the variational analysis of first-order optimality conditions for the optimal control problem, an posteriori error estimation procedure is developed. Using these error estimates, an h-adaptive scheme is outlined for the implementation of least square method in an adaptive manner. A time-scaling technique is described to handle problems with discontinuous control or multiple phases. Several real-life examples were solved to show the efficacy of the h-adaptive and time-scaling algorithm.Item Dynamics and real-time optimal control of satellite attitude and satellite formation systems(Texas A&M University, 2006-10-30) Yan, HuiIn this dissertation the solutions of the dynamics and real-time optimal control of magnetic attitude control and formation flying systems are presented. In magnetic attitude control, magnetic actuators for the time-optimal rest-to-rest maneuver with a pseudospectral algorithm are examined. The time-optimal magnetic control is bang-bang and the optimal slew time is about 232.7 seconds. The start time occurs when the maneuver is symmetric about the maximum field strength. For real-time computations, all the tested samples converge to optimal solutions or feasible solutions. We find the average computation time is about 0.45 seconds with the warm start and 19 seconds with the cold start, which is a great potential for real-time computations. Three-axis magnetic attitude stabilization is achieved by using a pseudospectral control law via the receding horizon control for satellites in eccentric low Earth orbits. The solutions from the pseudospectral control law are in excellent agreement with those obtained from the Riccati equation, but the computation speed improves by one order of magnitude. Numerical solutions show state responses quickly tend to the region where the attitude motion is in the steady state. Approximate models are often used for the study of relative motion of formation flying satellites. A modeling error index is introduced for evaluating and comparing the accuracy of various theories of the relative motion of satellites in order to determine the effect of modeling errors on the various theories. The numerical results show the sequence of the index from high to low should be Hill's equation, non- J2, small eccentricity, Gim-Alfriend state transition matrix index, with the unit sphere approach and the Yan-Alfriend nonlinear method having the lowest index and equivalent performance. A higher order state transition matrix is developed using unit sphere approach in the mean elements space. Based on the state transition matrix analytical control laws for formation flying maintenance and reconfiguration are proposed using low-thrust and impulsive scheme. The control laws are easily derived with high accuracy. Numerical solutions show the control law works well in real-time computations.Item Intervention in gene regulatory networks(Texas A&M University, 2006-10-30) Choudhary, AshishIn recent years Boolean Networks (BN) and Probabilistic Boolean Networks (PBN) have become popular paradigms for modeling gene regulation. A PBN is a collection of BNs in which the gene state vector transitions according to the rules of one of the constituent BNs, and the network choice is governed by a selection distribution. Intervention in the context of PBNs was first proposed with an objective of avoid- ing undesirable states, such as those associated with a disease. The early methods of intervention were ad hoc, using concepts like mean first passage time and alteration of rule based structure. Since then, the problem has been recognized and posed as one of optimal control of a Markov Network, where the objective is to find optimal strategies for manipulating external control variables to guide the network away from the set of undesirable states towards the set of desirable states. This development made it possible to use the elegant theory of Markov decision processes (MDP) to solve an array of problems in the area of control in gene regulatory networks, the main theme of this work. We first introduce the optimal control problem in the context of PBN models and review our solution using the dynamic programming approach. We next discuss a case in which the network state is not observable but for which measurements that are probabilistically related to the underlying state are available. We then address the issue of terminal penalty assignment, considering long term prospective behavior and the special attractor structure of these networks. We finally discuss our recent work on optimal intervention for the case of a family of BNs. Here we consider simultaneously controlling a set of Boolean Models that satisfy the constraints imposed by the underlying biology and the data. This situation arises in a case where the data is assumed to arise by sampling the steady state of the real biological network.Item Minimum Time/Minimum Fuel Control of an Axisymmetric Rigid Body(2014-05-19) Torres, Jonathan FarinaMany times it is necessary to reorient an aerial vehicle during flight in a minimum time or minimum fuel fashion. This thesis will present a minimum time/fuel control solution to reorienting an axisymmetric rigid body using eigenaxis maneuvers. Any fixed desired attitude can be achieved by rotating the rigid body about its eigenaxis. While an eigenaxis is not a time-optimal maneuver, it will produce the shortest angular trajectory between the rigid bodys current attitude and the desired attitude. Using the eigenaxis, a reference frame will be defined with the third unit vector direction parallel to the eigenaxis. In this reference frame, the controls and the equations of motion will be developed. A control weight will dictate whether the controller will drive the vehicle to the desired orientation in minimum time, minimum fuel, or a hybrid between minimum time and fuel. The controls can then be translated to the body frame through an attitude matrix that relates the body frame to the eigenaxis frame. After a minimum time/fuel controller has been developed, a minimum time/energy controller will then be designed. This minimum time/energy controller will then be compared against the minimum time/fuel controller by examining two fuel performance indices. Comparing these two controllers results in the most efficient controller being dependent on the cost function that describes the actuator type. Therefore, it is not feasible to select one controller over another independent of the fuel cost function. A minimum time/fuel controller has been designed using an eigenaxis maneuver in order to reorient itself. A comparison between the minimum time/fuel and minimum time/energy controller has been investigated using the two cost functions resulting in neither controller being the most efficient independent of the fuel cost function.