Optimal Control within the Context of Multidisciplinary Design, Analysis, and Optimization

Multidisciplinary design, analysis and optimization involves modeling the interactions of complex systems across a variety of disciplines. The optimization of such systems can be a computationally expensive exercise with multiple levels of nested nonlinear solvers running under an optimizer.The application of optimal control in project development often involves performing trajectory optimization for fixed vehicle designs or parametric sweeps across some key vehicle properties.This information is then relayed to the subsystem design teams who update their designs and relay some bulk characteristics back to the trajectory optimization procedure.This iteration is then repeated until the design closes.However, with increasing interest in more tightly coupled systems, such as electric and hybrid-electric aircraft propulsion and boundary layer ingestion, this process is prone to ignore subtle coupling between vehicle subsystem designs and vehicle operation on a given mission.Integrating trajectory optimization into a tightly coupled multidisciplinary design procedure can be computationally prohibitive, depending on the complexity of the subsystem analyses and the optimal control technique applied.To address these issues a new optimal control software tool, Dymos, has been developed.Dymos is built upon NASA's OpenMDAO software and can leverage its capabilities to efficiently compute gradients for the optimization and optimize complex models in parallel on distributed memory systems.This report provides some explanation into the numerical methods employed in Dymos and provides several use cases that demonstrate its performance on traditional optimal control problems and improvements ino techniques have been used extensively in recent decades to solve a variety of optimal control problems, typically in the form of aerospace vehicle trajectory optimization

A radial basis function method for solving optimal control problems.

This work presents two direct methods based on the radial basis function (RBF) interpolation and arbitrary discretization for solving continuous-time optimal control problems: RBF Collocation Method and RBF-Galerkin Method. Both methods take advantage of choosing any global RBF as the interpolant function and any arbitrary points (meshless or on a mesh) as the discretization points. The first approach is called the RBF collocation method, in which states and controls are parameterized using a global RBF, and constraints are satisfied at arbitrary discrete nodes (collocation points) to convert the continuous-time optimal control problem to a nonlinear programming (NLP) problem. The resulted NLP is quite sparse and can be efficiently solved by well-developed sparse solvers. The second proposed method is a hybrid approach combining RBF interpolation with Galerkin error projection for solving optimal control problems. The proposed solution, called the RBF-Galerkin method, applies a Galerkin projection to the residuals of the optimal control problem that make them orthogonal to every member of the RBF basis functions. Also, RBF-Galerkin costate mapping theorem will be developed describing an exact equivalency between the Karush–Kuhn–Tucker (KKT) conditions of the NLP problem resulted from the RBF-Galerkin method and discretized form of the first-order necessary conditions of the optimal control problem, if a set of conditions holds. Several examples are provided to verify the feasibility and viability of the RBF method and the RBF-Galerkin approach as means of finding accurate solutions to general optimal control problems. Then, the RBF-Galerkin method is applied to a very important drug dosing application: anemia management in chronic kidney disease. A multiple receding horizon control (MRHC) approach based on the RBF-Galerkin method is developed for individualized dosing of an anemia drug for hemodialysis patients. Simulation results are compared with a population-oriented clinical protocol as well as an individual-based control method for anemia management to investigate the efficacy of the proposed method

Multi-Objective Trajectory Optimization of a Hypersonic Reconnaissance Vehicle with Temperature Constraints

Temperature-constrained optimal trajectories for a scramjet-based hypersonic reconnaissance vehicle were generated by developing an optimal control formulation and solving it using a variable order Gauss-Radau quadrature collocation method. The vehicle was assumed to be an air-breathing reconnaissance aircraft that has specified takeoff/landing locations, airborne refueling constraints, specified no-fly zones, and specified targets for sensor data collections. The aircraft model included fight dynamics, aerodynamics, and thermal constraints. This model was incorporated into an optimal control formulation that includes constraints on both the vehicle as well as mission parameters, such as avoidance of no-fly zones and coverage of high-value targets. Optimal trajectories were be developed using several different performance costs in the optimal control formulation--minimum time, minimum time with control penalties, and maximum range. The resulting analysis demonstrated that optimal trajectories that meet specified mission parameters and constraints can be determined and used for larger-scale operational and campaign planning

A Direct Integral Pseudospectral Method for Solving a Class of Infinite-Horizon Optimal Control Problems Using Gegenbauer Polynomials and Certain Parametric Maps

We present a novel direct integral pseudospectral (PS) method (a direct IPS method) for solving a class of continuous-time infinite-horizon optimal control problems (IHOCs). The method transforms the IHOCs into finite-horizon optimal control problems (FHOCs) in their integral forms by means of certain parametric mappings, which are then approximated by finite-dimensional nonlinear programming problems (NLPs) through rational collocations based on Gegenbauer polynomials and Gegenbauer-Gauss-Radau (GGR) points. The paper also analyzes the interplay between the parametric maps, barycentric rational collocations based on Gegenbauer polynomials and GGR points, and the convergence properties of the collocated solutions for IHOCs. Some novel formulas for the construction of the rational interpolation weights and the GGR-based integration and differentiation matrices in barycentric-trigonometric forms are derived. A rigorous study on the error and convergence of the proposed method is presented. A stability analysis based on the Lebesgue constant for GGR-based rational interpolation is investigated. Two easy-to-implement pseudocodes of computational algorithms for computing the barycentric-trigonometric rational weights are described. Two illustrative test examples are presented to support the theoretical results. We show that the proposed collocation method leveraged with a fast and accurate NLP solver converges exponentially to near-optimal approximations for a coarse collocation mesh grid size. The paper also shows that typical direct spectral/PS- and IPS-methods based on classical Jacobi polynomials and certain parametric maps usually diverge as the number of collocation points grow large, if the computations are carried out using floating-point arithmetic and the discretizations use a single mesh grid whether they are of Gauss/Gauss-Radau (GR) type or equally-spaced.Comment: 33 pages, 19 figure

Optimal Finite Thrust Guidance Methods for Constrained Satellite Proximity Operations Inspection Maneuvers

Algorithms are developed to find optimal guidance for an inspector satellite operating nearby a resident space object (RSO). For a non-maneuvering RSO, methods are first developed for a satellite subject to maximum slew rates to conduct an initial inspection of an RSO, where the control variables include the throttle level and direction of the thrust. Second, methods are developed to optimally maneuver a satellite with on/off thrusters into a natural motion circumnavigation or teardrop trajectory, subject to lighting and collision constraints. It is shown that for on/off thrusters, a control sequence can be parameterized to a relatively small amount of control variables and the relative states can be analytically propagated as a function of those control variables. For a maneuvering RSO, differential games are formulated and solved for an inspector satellite to achieve multiple inspection goals, such as aligning with the Sun vector or matching the RSO\u27s energy. The developed algorithms lead to fuel and time savings which can increase the mission life and capabilities of inspector satellites and thus improve space situational awareness for the U.S. Air Force

Optimal Control of Weakly Forced Nonlinear Oscillators

Optimal control of nonlinear oscillatory systems poses numerous theoretical and computational challenges. Motivated by applications in neuroscience, we develop tools and methods to synthesize optimal controls for nonlinear oscillators described by reduced order dynamical systems. Control of neural oscillations by external stimuli has a broad range of applications, ranging from oscillatory neurocomputers to deep brain stimulation for Parkinson\u27s disease. In this dissertation, we investigate fundamental limits on how neuron spiking behavior can be altered by the use of an external stimulus: control). Pontryagin\u27s maximum principle is employed to derive optimal controls that lead to desired spiking times of a neuron oscillator, which include minimum-power and time-optimal controls. In particular, we consider practical constraints in such optimal control designs including a bound on the control amplitude and the charge-balance constraint. The latter is important in neural stimulations used to avoid from the undesirable effects caused by accumulation of electric charge due to external stimuli. Furthermore, we extend the results in controlling a single neuron and consider a neuron ensemble. We, specifically, derive and synthesize time-optimal controls that elicit simultaneous spikes for two neuron oscillators. Robust computational methods based on homotopy perturbation techniques and pseudospectral approximations are developed and implemented to construct optimal controls for spiking and synchronizing a neuron ensemble, for which analytical solutions are intractable. We finally validate the optimal control strategies derived using the models of phase reduction by applying them to the corresponding original full state-space models. This validation is largely missing in the literature. Moreover, the derived optimal controls have been experimentally applied to control the synchronization of electrochemical oscillators. The methodology developed in this dissertation work is not limited to the control of neural oscillators and can be applied to a broad class of nonlinear oscillatory systems that have smooth dynamics