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0x202 Optimal Control


The general function to represent a smooth system is

\[\dot{x} = f(x,u)\]

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)

\[\dot{x} = f_0(x) + B(x) u\]

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

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