Optimal Control

MaRDI QIDQ6534338

branch of control theory that deals with finding a control for a dynamical system over a period of time such that an objective function is optimized

Description

Finding controls, e.g. external forces|fields, such that they drive a given control system from a given initial state to a specific final state (target), typically with constraints such as using not too high fields (cost).

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List of mathematical expressions with quantities

Optimal Control Cost	$J[u]\equiv \sum _k{\alpha }_k\underset{0}{\overset{T}{\int }}dt{u}_k^2(t)$
$T$ symbol represents Control System Duration
$t$ symbol represents Time
$u$ symbol represents Control System Input
$J$ symbol represents Optimal Control Cost
$\alpha$ symbol represents Optimal Control Penalty Factor
Optimal Control Target	$J[u,x]\equiv x^{†}(T)Dx(T)$
$D$ symbol represents Control System Matrix D
$T$ symbol represents Control System Duration
$u$ symbol represents Control System Input
$x$ symbol represents Control System State
$J$ symbol represents Optimal Control Target
Optimal Control Backward	$\dot{z}(t)=-(A^{†}+\sum _k{u}_k(t)N_k^{†})z(t)-B^{†}u(t)$
$A$ symbol represents Control System Matrix A
$B$ symbol represents Control System Matrix B
$N$ symbol represents Control System Matrix N
$t$ symbol represents Time
$u$ symbol represents Control System Input
$z$ symbol represents Control System Lagrange Multiplier
Optimal Control Forward	$\dot{x}(t)=(A+\sum _k{u}_k(t)N_k)x(t)+Bu(t)$
$A$ symbol represents Control System Matrix A
$B$ symbol represents Control System Matrix B
$N$ symbol represents Control System Matrix N
$t$ symbol represents Time
$u$ symbol represents Control System Input
$x$ symbol represents Control System State
Optimal Control Update	$u_k(t)=-\frac{1}{{\alpha }_k}\mathrm{\Im }(z^{†}(t)N_{k}x(t))$
$N$ symbol represents Control System Matrix N
$\alpha$ symbol represents Optimal Control Penalty Factor
$t$ symbol represents Time
$u$ symbol represents Control System Input
$x$ symbol represents Control System State
$z$ symbol represents Control System Lagrange Multiplier
Optimal Control Final	$z(t=T)=Dx(t=T)$
$D$ symbol represents Control System Matrix D
$T$ symbol represents Control System Duration
$t$ symbol represents Time
$x$ symbol represents Control System State
$z$ symbol represents Control System Lagrange Multiplier
Optimal Control Initial	$x(t=0)=x_0$
$t$ symbol represents Time
$x$ symbol represents Control System State
$x_0$ symbol represents Initial Control State
Optimal Control Constraint	$J[u,x,z]=2\mathrm{\Re }\underset{0}{\overset{T}{\int }}dt{z}^{†}(t)({\partial }_t-A-\sum _k{u}_k(t)N_k)x(t)$
$A$ symbol represents Control System Matrix A
$N$ symbol represents Control System Matrix N
$T$ symbol represents Control System Duration
$t$ symbol represents Time
$u$ symbol represents Control System Input
$x$ symbol represents Control System State
$z$ symbol represents Control System Lagrange Multiplier

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