Linearization (Control Systems)

Given a State Space Representation given by: $\{\begin{array}{l}\dot{x}(t)=f(x(t),u(t))\\ y(t)=h(x(t),u(t))\end{array}$

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

$\{\begin{array}{l}\delta \dot{x}(t)=A\delta x(t)+B\delta u(t)\\ dy(t)=C\delta x(t)+D\delta u(t)\end{array}$

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 $\frac{df}{dx}$.
• B is the matrix $\frac{du}{dx}$.
• C is the matrix $\frac{dh}{dx}$.
• D is the matrix $\frac{dh}{du}$.

Related

• State Space Representation