Optimization Sandbox

I am on a journey to learn optimization, during college I took courses in Numerical Optimization and Optimal Control, needless to say I was fascinated by the subjects.

The plan is to learn enough material to use in a career for development of space vehicles.

I will focus on:

• Trajectory Optimization
• Model Predictive Control
• Monte Carlo Simulations

I plan to use Python and the CasADi library to faciliate my learning experience. This repository will contain example code from their examaple repo and any other ideas I come up with.

Optimal Control Review

• Trajectory Optimization
  • Control as a function of time
  • Single initial state
  • Types
    • Direct Methods
      • Solve a non-linear program
      • Widely used in practice
    • Indirect Methods
      • Solve non-linear root finding problem
      • Rarely used in practice
• Policy Optimization
  • Control as a function of state
  • Arbitrary initial state
  • Types
    • Heuristic Optimization
      • Parameter optimization
      • Arbitrary objective function
    • Dynamic Programming
      • Partial differential equation
      • Dynamics and objective are function of state and control

Trajectory Optimization

Transcription Methods

• Indirect Methods
  • "optimize then discretize"
  • more accurate
  • harder to pose and solve
• Direct Methods
  • "discretize and optimize"
  • less accurate
  • easier to pose and solve

Trajectory Methods

• Shooting Methods
  • Based on simulation
  • Better for problems with simple control and no path constraints
• Collocation Methods
  • Based on function approximation
  • Better for problems with complicated control and/or path constraints

Fine Tunning

• h-methods
  • Low-order method
  • Converge by increasing number of segments
  • Increaseing the number of control units increases accuracy
• p-methods
  • High-order methods
  • Converge by increasing method order
  • Increasing method order provide better fitting, thus increasing accuracy
• Adaptive Methods
  • Uses novel methods to switch between h-methods and p-methods to improve results

Trajectory Optimization Methods

• Collocation Methods
  • Trapezoidal direct collocation
  • Hermite–Simpson direct collocation
  • Global orthogonal collocation (Chebyshev Lobatto)
• Shooting Methods
  • Direct Single shooting
  • Direct multiple shooting (4th-order Runge–Kutta)

Resources

• Trajectory Optimization Tutorial: Slides
• An Introduction to Trajectory Optimization: How to do your Direct Collocation
• Transcription Methods for Trajectory Optimization