Linearity

First order

• Linear Circuit

Linear Functions vs. Linear Dynamics

• A function maps a single value whereas a dynamic system maps a signal to a signal.
• Linearity means you follow the super position principle. In a function, this applied to a mapping, if its for a dynamic system, it applies to a signal.
• The super position principal has two components…
  • Additivity
  • Homogeneity (scalability)

Linearizing Functions

This uses an approximation (you can only do this for a single point of interest)

• This is a first order approximation
• Linearization approximations are given by $f_{linear}\left(x\right)=f\left(a\right)+\frac{\partial f}{\partial x}|\left(x-a\right)$ for a point of interest a.

Linearizing Dynamic Systems

• Find all nominal (equilibrium) points if not given
• Find $x_0$, $u_0$ such that $0=f\left(x_o,u_o\right)$
• Set new variables as $x=x_0+\delta x,u=u_0+\delta u$
• Apply first order Taylor approximation as: $f(x,u)\approx f(x_0,u_0)+\frac{\partial f}{\partial x}{|}_{x=x_0,u=u_0}(x-x_0)+\frac{\partial f}{\partial u}{|}_{x=x_0,u=u_0}(u-u_0)$
  • What this means is… For each function we are linearizing, the linear form is a function of the variables in the non-linear equation as a sum of their partial derivatives with the initial values of the variables substituted in. See Goodnotes for an example (Ex. 6 Linearization)
• Redefine the variables as $\delta x\to x$ and $\delta u\to u\Rightarrow x^{\prime }=Ax+Bu$
  • Where x and u are deviations from their original points
  • This just means sub out the $\partial$ with the original variables to form a linear equation
• You need to linearize the whole or specific terms
  • Not sure what this actually means so probs not that important