Hamiltonian (control theory)

Résumé

The Hamiltonian is a function used to solve a problem of optimal control for a dynamical system. It can be understood as an instantaneous increment of the Lagrangian expression of the problem that is to be optimized over a certain time period. Inspired by—but distinct from—the Hamiltonian of classical mechanics, the Hamiltonian of optimal control theory was developed by Lev Pontryagin as part of his maximum principle. Pontryagin proved that a necessary condition for solving the optimal control problem is that the control should be chosen so as to optimize the Hamiltonian. Problem statement and definition of the Hamiltonian Consider a dynamical system of n first-order differential equations :\dot{\mathbf{x}}(t) = \mathbf{f}(\mathbf{x}(t),\mathbf{u}(t),t) where \mathbf{x}(t) = \left[ x_{1}(t), x_{2}(t), \ldots, x_{n}(t) \right]^{\mathsf{T}} denotes a vector of state variables, and \mathbf{u}(t) = \left[ u_{1}(t), u_{2}(t), \ldots,

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Optimal control of the Cahn-Hilliard equation

Stefano Guarino

The goal of the project is to study an optimal control problem for the Cahn-Hilliard equation. To this end, we proceed in three steps: we first introduce the Cahn-Hilliard equation and how it is derived, and describe some of its applications; then, we approximate it with the finite element method and solve it with FreeFem++. Finally, we formulate the optimal control problem and solve it with FreeFem++.

2015

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Convective weather, and thunderstorm development in particular, represents a major source of disruption, delays and safety hazards in the Air Traffic Management system. Thunderstorms are challenging to forecast and evolve on relatively rapid timescales; therefore, aircraft trajectory planning tools need to consider the uncertainty in the forecasted evolution of these convective phenomena. In this work, we use data from a satellite-based product, Rapidly Developing Thunderstorms, to estimate a model of the uncertain evolution of thunderstorms. We then introduce a methodology based on numerical optimal control to generate avoidance trajectories under uncertain convective weather evolution. We design a randomized procedure to initialize the optimal control problem, explore the different resulting local optima, and identify the best trajectory. Finally, we demonstrate the proposed methodology on a realistic test scenario, employing actual forecast data and an aircraft performance model.

2019

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We investigate the potential for aggregations of residential thermostatically controlled loads (TCLs), such as air conditioners, to arbitrage intraday wholesale electricity market prices via non-disruptive direct load control. Since wholesale electricity prices reflect power system conditions, arbitrage provides a service to the grid, helping to balance real-time supply and demand. While previous work on the energy arbitrage problem has used simple energy storage models, we use high fidelity TCL-specific models which allow us to understand and quantify the full capabilities and constraints of these time-varying systems. We explore two optimization/control frameworks for solving the arbitrage problem, both based on receding horizon linear programming. Since we find that the first approach requires significant computation, we develop a second approach involving decomposition of the optimal control problem into separate optimization and control problems. Simulation results show that TCLs could save on the order of 10% of wholesale energy costs via arbitrage, with savings decreasing with price forecast error. © 2013 EUCA.

2013

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