"Wisdom begins in wonder" – Socrates
Author: Isaac Gendler
Onto Functions
12/10/18
“What happens when all of the mappings in a linear transformation have at least one output?”
Sets of numbers can be changed around in linear transformations. And if every number maps onto at least one other number, then the function is Onto.
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Vector Spaces
12/09/18
“How can we represent a set of vectors?”
Vectors are one of the most important aspects of Linear Algebra. But in order to have a true set of vectors, they must be closed under vector addition and scalar multiplication. This is known as a Vector Space and is vital for a true understanding of Linear Algebra.
Image credit http://3.bp.blogspot.com
Block Diagrams for Controls Systems
12/08/18
“How can we represent control process graphically?”
Control systems are a bedrock of modern day technology. However, sometimes it would be easier to see their processes if visualized. This can be done by using Block Diagrams, which shows each component of the control system in its natural order.
The LQR Function
12/07/18
“How can we create a control function with easy inputs?”
Modern control systems rely on matrix operations. There are multiple ways to create optimal varieties of this. One such way is to create a cost function which assigns different penalties to each input. This is known as the LQR Function and is implemented in a multitude of modern control systems around the world.
Carbon Neutral Economies
12/06/18
“How can an economy not contribute to climate change?”
Traditionally, economies have grown in tandem with carbon emissions. But with the advent of modern-day extreme climate change this is no longer feasible. To solve this, nations will have to adopt Carbon Neutral Economies in which all sectors have slashed their emissions to zero. The transition to a low carbon economy could yield enormous economic growth as well as inequality easing. Multiple governments around the world are committed to this action, with California’s 2045 and the EU’s 2050 initiatives as examples.
Image credit https://assets.weforum.org
Matrix Transposition
12/05/18
“How can we flip rows and columns of a matrix?”
When doing Linear Algebra, there are times where we may want to flip the rows and columns. This can be accomplished through Matrix Transposition. A matrix that has undergone transposition is known as a transposed matrix. If we multiply the transposed matrix by the original matrix then we arrive at the identity matrix.
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Ackermann’s Formula
12/04/18
“What is an efficient method for pole placement in control systems?”
Pole Placement is an excellent way to stabilize a control system. But sometimes we may want an extra efficient way to do this, This is where we can apply Ackermann’s Formula, which automates it nicely for us.
The Span of a Matrix
12/03/18
“How can we represent all possible combinations of a matrix?”
Matrices are great for representing groups of numbers. But sometimes we would like to see all possible results. We can do this using the Span of a Matrix, using weighted values for each part.
Pole Placement
12/02/18
“How can we modify a state-space to become more stable?”
State-spaces are often used in control system design to represent a plant. However, sometimes the system by itself is unstable. So how can we use our engineering mindset to solve this? Well, what we can do is introduce an extra feedback loop with some constants that modify the poles of the system such that they are all in the right-half plane. This is known as Pole Placement and is one of the most commonly used techniques in classical control.
Image credit http://ctms.engin.umich.edu
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