0x202 Optimal Control
Dynamics
The general function to represent a smooth system is
where \(x\) is the state and \(u\) is the state. In a mechanical system \(x\) can be \((q, v)\) where \(q\) is configuration (might not be a vector) and \(v\) is velocity.
For example of an pendulum, \(x = (\theta, \dot\theta) \in S^1 \times R\) where \(S^1\) is a circle. this is a cylinder
Definition (control affine systems) Many systems have the following specific structure (can be put into this form)
where \(f_0(x)\) is a drift and \(B(x)\) is the input Jacobian.
Classical Control
mainly about frequency domain analysis, it deals with the behavior of dynamical systems with inputs, and how their behavior is modified by feedback, using the Laplace transform as a basic tool to model such systems.
PID Control
https://www.youtube.com/watch?v=FXSpHy8LvmY&list=PLn8PRpmsu08podBgFw66-IavqU2SqPg_w&index=2
Modern Control
time-domain state space
full state feedback
pole placement
Linear Quadratic Regulator
planning with a known model can be solved with the Algebraic Riccati Equation