Control

# Optimal Control

Optimal control is a method of controlling a system in a way that minimizes a given objective. It’s suited for complex systems and applications.

• It’s Control (stability and robustness of the control system) + performance (optimality)

Optimal control is framing control as an optimization problem.

[!ad-example] Examples of Optimal Control Problems

• finding the optimal trajectory for a spacecraft to reach a distant planet with minimal fuel consumption.
• finding the optimal control inputs for a robot arm to perform a task with minimal error
• finding the optimal control inputs to regulate the temperature of a furnace at a desired setpoint while minimizing energy consumption.

Great reference: Linear Quadratic Methods by Anderson and Moore

Strong similarity with Kalman Filter which is able to compute Bayes Filter updates.

Reference

Let associate state-input pairs with a cost “density”. is the control output, see PID Control for same notation

Define: where for all (T might be infinity).

Optimal control problem is to find such that is minimized.

### Robustness

For the problem of robustness (dealing with uncertainty), there are multiple solutions being researched:

The best method to use depends on the specific application and the nature of the uncertainties and disturbances in the system’s dynamics.

### Software

There is this software casadi that seems to work for it.