Introduction - a learning manifesto

Hello World! I have been wanting to blog for a long time. My goal would be to dive into topics I find interesting and to do a series of posts for each one. I like to understand things thoroughly myself, but also in the simplest way possible. Simple doesn't mean easy: certain subjects can be intrinsically difficult to grasp, but I think there's always room for crafting a better path to understanding. I personally look for a path that has pedagogical value, allowing to develop intuition and insight gradually, but never sidestepping essential formal details. One pet peeve of mine is mathematics. Let's consider an example:

Controllability of a linear systems Let $$\dot{x}=Ax+Bu$$ be a linear time-invariant system. The system is said to be controllable if it can be steered from any initial state to a desired state in a finite amount of time. This means that there exists a control function $$u: [t_0, t_1] \mapsto R$$ that applied to the system in the state $x(t_0)$ will lead to the desired $x(t_1)$.

Even if you have never seen a linear system of differential equations before, this definition is simple and the concept behind it sounds reasonable. The state may be e.g. the position and speed of a physical object, and we may want to control it.¹

How can we know if the system is controllable given the matrices $A$ and $B$? Here comes an obscure theorem:

Controllability Gramian Define the controllability Gramian

$$ P(t_0, t_1) = \int_{t_0}^{t_1}{e^{A(t_1-\tau)}BB^{T}e^{A^T(t_1-\tau)} d \tau} $$

If P is invertible then the system is controllable in $[t_0, t_1]$ and $u$ is given by

$$ u(t) = B^{T}e^{A^{T}(t_1-t)}P^{-1}(t_0, t_1)(x_f - e^{A(t_1-t_0)}x_i) $$

The bad proof will be based on dropping the expression for $u$ in the equation that describes the evolution of a linear time-invariant system and prove that we reach $x(t_1)=x_f$ from $x(t_0)=x_i$. Unfortunately no insight whatsoever is given into why the controllability Gramian is defined this way, it seems as it appears from thin air. There is a better way to understand and derive this result, but it needs the introduction of more concepts to be understood (and it is connected to the four fundamental subspaces from linear algebra!).

Is the increased learning load justified if a basic proof exists, although less insightful? Striking a balance between ease and depth is what I consider a difficult problem in learning and teaching, and I'll use this blog to have a take at it.

Expect posts about mathematics, programming, computer science, electronics, engineering and anything else.

Happy reading!