# Optimal Control

## Optimal Control Framework

Given: A controlled dynamical system：$ x^{n+1} = f(x^n, u^n)$

A cost function：$V = \phi(x^N, \alpha) + \sum^{N-1}_{i=0}L(x^i, u^i, \alpha)$

Goal: Find the sequence of commands that minimizes(maximizes) the cost function

## Bellman’s Principle of Optimality

Optimize it using dynamic programming:

## Linear quadratic regulator

Special Assumption: Linear System Dynamics

Quadratic cost function

Goal: - Bring the system to a setpoint and keep it there - Note: this an also be did with a nonlinear system by a local linearization

- As A linear control law expressed as:

Rewrite the optimal cost at stage i as a quadratic form:

Thus,

## Finite horizon approximation

To be continued…

## Motion Predictive Control

To be continued…

## Fast MPC

To be continued…