Investigation of practical applications of H infinity control theory to the design of control systems for large space structures

The applicability of H infinity control theory to the problems of large space structures (LSS) control was investigated. A complete evaluation to any technique as a candidate for large space structure control involves analytical evaluation, algorithmic evaluation, evaluation via simulation studies, and experimental evaluation. The results of analytical and algorithmic evaluations are documented. The analytical evaluation involves the determination of the appropriateness of the underlying assumptions inherent in the H infinity theory, the determination of the capability of the H infinity theory to achieve the design goals likely to be imposed on an LSS control design, and the identification of any LSS specific simplifications or complications of the theory. The resuls of the analytical evaluation are presented in the form of a tutorial on the subject of H infinity control theory with the LSS control designer in mind. The algorthmic evaluation of H infinity for LSS control pertains to the identification of general, high level algorithms for effecting the application of H infinity to LSS control problems, the identification of specific, numerically reliable algorithms necessary for a computer implementation of the general algorithms, the recommendation of a flexible software system for implementing the H infinity design steps, and ultimately the actual development of the necessary computer codes. Finally, the state of the art in H infinity applications is summarized with a brief outline of the most promising areas of current research

Optimally Convergent Quantum Jump Expansion

A method for deriving accurate analytic approximations for Markovian open quantum systems was recently introduced in [F. Lucas and K. Hornberger, Phys. Rev. Lett. 110, 240401 (2013)]. Here, we present a detailed derivation of the underlying non-perturbative jump expansion, which involves an adaptive resummation to ensure optimal convergence. Applying this to a set of exemplary master equations, we find that the resummation typically leads to convergence within the lowest two to five orders. Besides facilitating analytic approximations, the optimal jump expansion thus provides a numerical scheme for the efficient simulation of open quantum systems.Comment: 13 pages, 3 figure

A Multilevel Approach for Stochastic Nonlinear Optimal Control

We consider a class of finite time horizon nonlinear stochastic optimal control problem, where the control acts additively on the dynamics and the control cost is quadratic. This framework is flexible and has found applications in many domains. Although the optimal control admits a path integral representation for this class of control problems, efficient computation of the associated path integrals remains a challenging Monte Carlo task. The focus of this article is to propose a new Monte Carlo approach that significantly improves upon existing methodology. Our proposed methodology first tackles the issue of exponential growth in variance with the time horizon by casting optimal control estimation as a smoothing problem for a state space model associated with the control problem, and applying smoothing algorithms based on particle Markov chain Monte Carlo. To further reduce computational cost, we then develop a multilevel Monte Carlo method which allows us to obtain an estimator of the optimal control with $\mathcal{O}(\epsilon^2)$ mean squared error with a computational cost of $\mathcal{O}(\epsilon^{-2}\log(\epsilon)^2)$. In contrast, a computational cost of $\mathcal{O}(\epsilon^{-3})$ is required for existing methodology to achieve the same mean squared error. Our approach is illustrated on two numerical examples, which validate our theory

A Convergent Approximation of the Pareto Optimal Set for Finite Horizon Multiobjective Optimal Control Problems (MOC) Using Viability Theory

The objective of this paper is to provide a convergent numerical approximation of the Pareto optimal set for finite-horizon multiobjective optimal control problems for which the objective space is not necessarily convex. Our approach is based on Viability Theory. We first introduce the set-valued return function V and show that the epigraph of V is equal to the viability kernel of a properly chosen closed set for a properly chosen dynamics. We then introduce an approximate set-valued return function with finite set-values as the solution of a multiobjective dynamic programming equation. The epigraph of this approximate set-valued return function is shown to be equal to the finite discrete viability kernel resulting from the convergent numerical approximation of the viability kernel proposed in [4, 5]. As a result, the epigraph of the approximate set-valued return function converges towards the epigraph of V. The approximate set-valued return function finally provides the proposed numerical approximation of the Pareto optimal set for every initial time and state. Several numerical examples are provided