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Lecture# Optimal Control Theory: OCPs

Description

This lecture covers Optimal Control Theory, focusing on Optimal Control Problems (OCPs). It explains the concept of OCPs, admissible controls, dynamical systems, existence of solutions, performance criteria, physical constraints, equivalence of performance criteria, and path constraints. The lecture also delves into the calculus of variations, gateaux derivative, geometric conditions of optimality, first-order necessary conditions, and admissible directions. It concludes with an overview of geometric optimality considerations and the necessary conditions for optimality.

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Instructors (2)

EE-715: Optimal control

This doctoral course provides an introduction to optimal control covering fundamental theory, numerical implementation and problem formulation for applications.

Related concepts (119)

Related lectures (21)

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Optimal control

Optimal control theory is a branch of mathematical optimization that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized. It has numerous applications in science, engineering and operations research. For example, the dynamical system might be a spacecraft with controls corresponding to rocket thrusters, and the objective might be to reach the moon with minimum fuel expenditure.

Model predictive control

Model predictive control (MPC) is an advanced method of process control that is used to control a process while satisfying a set of constraints. It has been in use in the process industries in chemical plants and oil refineries since the 1980s. In recent years it has also been used in power system balancing models and in power electronics. Model predictive controllers rely on dynamic models of the process, most often linear empirical models obtained by system identification.

Bellman equation

A Bellman equation, named after Richard E. Bellman, is a necessary condition for optimality associated with the mathematical optimization method known as dynamic programming. It writes the "value" of a decision problem at a certain point in time in terms of the payoff from some initial choices and the "value" of the remaining decision problem that results from those initial choices. This breaks a dynamic optimization problem into a sequence of simpler subproblems, as Bellman's “principle of optimality" prescribes.

Calculus of variations

The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in functions and functionals, to find maxima and minima of functionals: mappings from a set of functions to the real numbers. Functionals are often expressed as definite integrals involving functions and their derivatives. Functions that maximize or minimize functionals may be found using the Euler–Lagrange equation of the calculus of variations.

Reinforcement learning

Reinforcement learning (RL) is an area of machine learning concerned with how intelligent agents ought to take actions in an environment in order to maximize the notion of cumulative reward. Reinforcement learning is one of three basic machine learning paradigms, alongside supervised learning and unsupervised learning. Reinforcement learning differs from supervised learning in not needing labelled input/output pairs to be presented, and in not needing sub-optimal actions to be explicitly corrected.

Covers the fundamentals of optimal control theory, focusing on defining OCPs, existence of solutions, performance criteria, physical constraints, and the principle of optimality.

Covers Optimal Control Problems focusing on necessary conditions, existence of optimal controls, and numerical solutions.

Explores Nonlinear Model Predictive Control, covering stability, optimality, pitfalls, and examples.

Covers Nonlinear Model Predictive Control (NMPC) principles, including setpoint stabilization and Pontryagin's Maximum Principle.

Covers normed spaces, dual spaces, Banach spaces, Hilbert spaces, weak and strong convergence, reflexive spaces, and the Hahn-Banach theorem.