# Demonstration of Controlability and Observability

After I studied my Control Engineering courses some years ago, although I was able to interpret the results obtained after calculating the observability and controllability, I haven't been able to find a demonstration of the equations that are related to both terms.

## The equations [1]

Controllability matrix $$\mathcal {C} = \begin{bmatrix} B & AB & A^{{2}}B & \cdots & A^{{n-1}}B \end{bmatrix}$$

Observability matrix $$\mathcal {O} = \begin{bmatrix} C\\CA\\CA^{2}\\\vdots \\CA^{n-1} \end{bmatrix}$$

• Please add the details of the equations you have in mind. What do you mean by demonstration? Do you mean examples? Do you have specific examples in mind?
– AJN
Jun 7 '21 at 11:56
• @AJN the equations of the observability and controlability are $\left[C, CA, CA^2,...,CA^n \right]^T$, and $\left[B, BA, BA^2,...,BA^n \right]$ respectively. I don't get to see how does matrixes and their ranks are able to tell us if it is possible to control/observe certain variables and all the other conclusions and uses that can be found. Jun 8 '21 at 7:49
• IIRC, the equations when derived for discrete time systems are easier to understand intuitively. I will see if I can find a good reference or type one up myself. You will need to know the concept of linearly independent set of vectors and the concept of basis and spanning of vector spaces. Are you familiar with those concepts?
– AJN
Jun 8 '21 at 8:40
• I have learned the equations with the multiplication in the reverse order. $A^k\ B$ etc.
– AJN
Jun 8 '21 at 8:43
• See if this Wikipedia section explains it. The derivation and example subsections both provide good insight IMO.
– AJN
Jun 8 '21 at 9:00