Optimal Control

Formulation Let model be a function $x(k+1) = f(x(k), u(k), k): \mathbb{R}^n \times \mathbb{R}^m \times \mathbb{Z}_{\geq 0} \to \mathbb{R}^n$ where $x \in \mathbb{R}^n$

Name	Equation
Objective Function	$J(x) = \varphi(x(N)) + \sum_{k=0}^{N-1} L(x(k), u(k), k)$
Lagrange	$\overline{J} = J + \sum_{k=0}^{N-1} \lambda(k+1) (f(x,u,k) - x(k+1)$
Hamiltonian	$H(x,u,\lambda,k)=L(x,u,k) + \lambda^T f(x,u,k)$

Ricatti Equation
LQ
MPC

\[\begin{align} x_{2} &= Ax_1 + Bu_1 + B_dd_1 \nonumber\\ &=A(Ax_0+Bu_0+B_dd_0) + Bu_1 + B_dd_1 \nonumber\\ &=A^2x_0+ABu_0+AB_dd_0+Bu_1+B_dd_1 \nonumber\\ &=A^2x_0+ \nonumber \begin{bmatrix} AB&B \end{bmatrix} \begin{bmatrix} u_0\\ u_1 \end{bmatrix} + \begin{bmatrix} AB_d&B_d \end{bmatrix} \begin{bmatrix} d_0\\ d_1 \end{bmatrix}\nonumber \end{align}\] \[\begin{align} J(U)=& (H(Fx_0+GU+SD)-Y_{ref})^TQ(H(Fx_0+GU+SD)-Y_{ref}) \nonumber \\ &+(U-U_{ref})^TR(U-U_{ref}) \nonumber \\ =&U^T(G^TH^TQHG+R)U+2\{(H(Fx_0+SD)-Y_{ref})^TQHG-U_{ref}^TR\}U \nonumber\\ &+ const \nonumber \end{align}\]