Dynamics/Linearization/Numerical Solutions/Single Variable with MATLAB

Introduction
MATLAB Scripts can be simple and straightforward in linearizing and calculating a numerical solution to a non-linear expression of a single variable.

Using a MATLAB Script
The first two terms of the Taylor Series expansion, or linear approximation, result in the following:


 * $$ f(x) \approx f(a) + f'(a)(x - a)$$

Using MATLAB, there are a few options for performing linear approximations. Using this resource, we can run the following script to linear the following expression:


 * $$ f(x) = x^2+\sin(x)+1 $$

The figure shows a resulting depiction of the linearized function.

The linearized function near $$ x=1 $$ is the following:


 * $$ f(a) \approx \sin(1)+\epsilon (\cos(1)+2)+2 $$

Using MATLAB Simulink
We can substitute $$ t $$ for $$ x $$ in the expression:


 * $$ f(x) = x^2+\sin(x)+1 $$

Then, we create an appropriate block diagram as shown in the Figure.



We can then select the blocks between the ramp and the scope, right-click to create a subsystem block, select the new block, right-click an select "Linear Analysis", specify a point for linearization ($$ t=1 $$ in this case), linearize, and look at the resulting linear analysis in terms of state-space representation.

For our example, we find the "Model Linearizer" gives a static gain of 2.54 for our subsystem block.

Individual or Group Activity

 * Write down your own individual non-linear expression.
 * Use Taylor or binomial series expansion to create an analytical expression for the linearized form of your expression at a specified location.
 * Calculate the slope at that specified location.
 * Write a MATLAB script to create a symbolic linearized expression at the specified location.
 * Use Simulink to calculate the slope at the specified location.
 * Create a presentation in Google Slides titled "Linearized Expression with a Single Variable" to share in 2 minutes with the class.