Numerical methods for optimal control of hybrid systems and applications to the national airspace system

Jinhua Li, Purdue University

Abstract

The research presented in this dissertation is motivated by the need to model, control and optimize complex systems and large-scale systems such as hybrid systems and the national airspace system. Hybrid systems couple the continuous dynamics with discrete logic. Novel numerical methods are developed to find the optimal control laws for both general dynamic systems and a class of hybrid systems, denoted as switched linear systems. The proposed numerical methods are based on Differential Transformation (DT), which is a linear operator that transforms differential equations into algebraic equations. The DT-based method solves the finite-time or infinite-time horizon optimal control problems of general dynamic systems formulated as either the Two-Point Boundary Value Problem (TPBVP) or the Hamilton-Jacobi-Bellman (HJB) equation in a unified framework. The DT-based numerical method can also solve more difficult Switched Linear Quadratic (SLQ) optimal control problems. The SLQ optimal control problem can be formulated as a Multi-Point Boundary Value Problem (MPBVP) using the calculus of variations, in which there are transverse conditions at the unknown switching times. The MPBVP is then transformed into a system of algebraic equations using DT, in which the switching times are variables. By solving the system of algebraic equations, we not only find the optimal control laws represented by the finite-term Taylor series but also the optimal switching times. The proposed DT-based method is shown to be simple to implement and computationally more efficient than the other recent numerical methods. The National Airspace System (NAS) is a complicated large-scale network, consisting of thousands of airports, aircraft, and people, which enables safe and expeditious air travel in the U.S. A new numerical algorithm is proposed to compute the tightest bounding boxes of the reachable sets of an uncertain linear system that describes the macro air traffic flow in the en-route airspace. The result is important because it can predict air traffic congestion in the presence of system uncertainties. The proposed algorithm is based on interval analysis, which is proved to be computationally feasible for the NAS by utilizing the special structure of a system matrix in the model. Finally, the current airspace is divided into sectors for air traffic control. However, such static airspace configuration can not meet the continued growth of air traffic. In response to the initiative of the Next-Generation Air Transportation System (NextGen) by the Joint Planning and Development Office (JPDO), a new model and algorithm are developed to design airspace sectorization dynamically based on graph theory. The Dynamic Airspace Configuration (DAC) algorithm can generate the optimal number of sectors in response to time-varying air traffic. The DAC algorithm is tested with real air traffic data called Enhanced Traffic Management System (ETMS) data. Simulation results show that new sectors computed by the proposed DAC algorithm outperform the current operational sectors for both the average and worst-case performances.

Degree

Ph.D.

Advisors

Hwang, Purdue University.

Subject Area

Aerospace engineering|Electrical engineering|Systems science

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