Linearity

Info

Broadly means a scaled input produces a scaled output.
Linear means if you take a linear combination of inputs, you get an equivalent linear combination of outputs.

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

Using Linearization
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

Linearizability

In Distributed Systems
We want to make a system appear as if there were only one copy of data and all operations on it are atomic. In a linearizable system, once one client completes a write, all clients reading from the database must be able to see the write. Therefore, this is a recency guarantee where once client A reads the value of a write, all other clients must be able to read that value (no older value can be read).

We can try to make things linearizable at the cost of performance using Read Repair. Linearizability is slow all of the time and is proportional to the uncertainty of the delays in the network.