Fast numerical methods for robust nonlinear optimal control under uncertainty

This thesis treats different aspects of nonlinear optimal control problems under uncertainty in which the uncertain parameters are modeled probabilistically. We apply the polynomial chaos expansion, a well known method for uncertainty quantification, to obtain deterministic surrogate optimal control problems. Their size and complexity pose a computational challenge for traditional optimal control methods. For nonlinear optimal control, this difficulty is increased because a high polynomial expansion order is necessary to derive meaningful statements about the nonlinear and asymmetric uncertainty propagation. To this end, we develop an adaptive optimization strategy which refines the approximation quality separately for each state variable using suitable error estimates. The benefits are twofold: we obtain additional means for solution verification and reduce the computational effort for finding an approximate solution with increased precision. The algorithmic contribution is complemented by a convergence proof showing that the solutions of the optimal control problem after application of the polynomial chaos method approach the correct solution for increasing expansion orders. To obtain a further speed-up in solution time, we develop a structure-exploiting algorithm for the fast derivative generation. The algorithm makes use of the special structure induced by the spectral projection to reuse model derivatives and exploit sparsity information leading to a fast automatic sensitivity generation. This greatly reduces the computational effort of Newton-type methods for the solution of the resulting high-dimensional surrogate problem. Another challenging topic of this thesis are optimal control problems with chance constraints, which form a probabilistic robustification of the solution that is neither too conservative nor underestimates the risk. We develop an efficient method based on the polynomial chaos expansion to compute nonlinear propagations of the reachable sets of all uncertain states and show how it can be used to approximate individual and joint chance constraints. The strength of the obtained estimator in guaranteeing a satisfaction level is supported by providing an a-priori error estimate with exponential convergence in case of sufficiently smooth solutions. All methods developed in this thesis are readily implemented in state-of-the-art direct methods to optimal control. Their performance and suitability for optimal control problems is evaluated in a numerical case study on two nonlinear real-world problems using Monte Carlo simulations to illustrate the effects of the propagated uncertainty on the optimal control solution. As an industrial application, we solve a challenging optimal control problem modeling an adsorption refrigeration system under uncertainty

Quantum Control Landscapes

Numerous lines of experimental, numerical and analytical evidence indicate that it is surprisingly easy to locate optimal controls steering quantum dynamical systems to desired objectives. This has enabled the control of complex quantum systems despite the expense of solving the Schrodinger equation in simulations and the complicating effects of environmental decoherence in the laboratory. Recent work indicates that this simplicity originates in universal properties of the solution sets to quantum control problems that are fundamentally different from their classical counterparts. Here, we review studies that aim to systematically characterize these properties, enabling the classification of quantum control mechanisms and the design of globally efficient quantum control algorithms.Comment: 45 pages, 15 figures; International Reviews in Physical Chemistry, Vol. 26, Iss. 4, pp. 671-735 (2007

Fast Reachable Set Approximations via State Decoupling Disturbances

With the recent surge of interest in using robotics and automation for civil purposes, providing safety and performance guarantees has become extremely important. In the past, differential games have been successfully used for the analysis of safety-critical systems. In particular, the Hamilton-Jacobi (HJ) formulation of differential games provides a flexible way to compute the reachable set, which can characterize the set of states which lead to either desirable or undesirable configurations, depending on the application. While HJ reachability is applicable to many small practical systems, the curse of dimensionality prevents the direct application of HJ reachability to many larger systems. To address computation complexity issues, various efficient computation methods in the literature have been developed for approximating or exactly computing the solution to HJ partial differential equations, but only when the system dynamics are of specific forms. In this paper, we propose a flexible method to trade off optimality with computation complexity in HJ reachability analysis. To achieve this, we propose to simplify system dynamics by treating state variables as disturbances. We prove that the resulting approximation is conservative in the desired direction, and demonstrate our method using a four-dimensional plane model.Comment: in Proceedings of the IEE Conference on Decision and Control, 201