Blog de Gabriel: optimal control

Optimal control by: Sergio Gabriel Aldana 22136204

Optimal control theory, an extension of the calculus of variations, is a mathematical optimization method for deriving control policies.

Optimal control deals with the problem of finding a control law for a given system such that a certain optimality criterion is achieved. A control problem includes a cost functional that is a function of state and control variables. An optimal control is a set of differential equations describing the paths of the control variables that minimize the cost functional. The optimal control can be derived using Pontryagin's maximum principle (a necessary condition), or by solving the Hamilton-Jacobi-Bellman equation (a sufficient condition).

Control problems usually include ancillary constraints. A proper cost functional is a mathematical expression giving the traveling time as a function of the speed, geometrical considerations, and initial conditions of the system. It is often the case that the constraints are interchangeable with the cost functional.

$J=\Phi(\textbf{x}(t_0),t_0,\textbf{x}(t_f),t_f) + \int_{t_0}^{t_f} \mathcal{L}(\textbf{x}(t),\textbf{u}(t),t) \,\operatorname{d}t$ A more abstract framework goes as follows. Minimize the continuous-time cost functional

subject to the first-order dynamic constraints


	$\dot{\textbf{x}}(t) = \textbf{a}(\textbf{x}(t),\textbf{u}(t),t),$

the algebraic path constraints


	$\textbf{b}(\textbf{x}(t),\textbf{u}(t),t) \leq \textbf{0},$

and the boundary conditions


	$\boldsymbol{\phi}(\textbf{x}(t_0),t_0,\textbf{x}(t_f),t_f)$

$(\textbf{x}^*(t^*),\textbf{u}^*(t^*),t^*)$ where $\textbf{x}(t)$ is the state, $\textbf{u}(t)$ is the control, t is the independent variable (generally speaking, time), t₀ is the initial time, and t_f is the terminal time. The terms Φ and $\mathcal{L}$ are called the endpoint cost and Lagrangian, respectively. Furthermore, it is noted that the path constraints are in general inequality constraints and thus may not be active (i.e., equal to zero) at the optimal solution. It is also noted that the optimal control problem as stated above may have multiple solutions (i.e., the solution may not be unique). Thus, it is most often the case that any solution to the optimal control problem is locally minimizing.

Blog de Gabriel

viernes, 15 de octubre de 2010

optimal control

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