# 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:

$$J_i(Xi) = \mathop{arg min}{u_i\in u(xi)}{{L(x^i, u^i, \alpha) + V^*{i+1}x_{(i+1)}}}$$

Special Assumption: Linear System Dynamics $$x^{n+1} = Ax^n + Bu^n$$

Quadratic cost function $$L(x^i, u^i, \alpha) ＝ x^{i^T}Qx^i + u^{i^T}Ru^{i^T}$$

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

\begin{aligned} V^_i(Xi) & = \mathop{arg min}{u_i\in u(x_i)}{{L(x^i, u^i, \alpha) + V^{i+1}x{(i+1)}}} & = \mathop{arg min}_{u_i\in u(xi)}{{x^{i^T}Qx^i + u^{i^T}Ru^{i^T} + V^*{i+1}x{(i+1)}}} & = \mathop{arg min} {u_i\in u(xi)}{{x^{i^T}Qx^i + u^{i^T}Ru^{i^T} + V^*{i+1}(Ax^n + Bu^n)}} \end{aligned}

• As A linear control law expressed as:

$$u^{i^*} = -K^ix^i$$

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

$${V^i}^* = {x^i}^TP^ix^i$$

Thus,

$$V^_i(Xi) = \mathop{arg min}{u_i\in u(x_i)}{ {x^{i^T}Qx^i + u^{i^T}Ru^{i^T} + V^_{i+1}(Ax^n + Bu^n)} }$$

To be continued…

To be continued…

To be continued…