# Linearization (Control Systems)

Given a State Space Representation given by: ${x˙(t)=f(x(t),u(t))y(t)=h(x(t),u(t)) $

The following is a Linear Time Invariant (LTI) system known as the linearization of the above state space around $(x,u)$:

${δx˙(t)=Aδx(t)+Bδu(t)dy(t)=Cδx(t)+Dδu(t) $

We get a linear system. Here, $A,B,C,D$ represent partial derivatives.

How do we find $A,B,C,D$? We essentially compute the partial derivatives evaluated at a point, that’s how we linearize around the point.

- A is the matrix $dxdf $.
- B is the matrix $dxdu $.
- C is the matrix $dxdh $.
- D is the matrix $dudh $.