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For a system in state-space representation: $\dot{x}(t) = Ax(t) + Bu(t)$; $y(t) = Cx(t) + Du(t)$, we say that a system is controllable if for $\gamma = [B \quad AB \quad \cdots \quad A^{n-1}B]$, rank$(\gamma) = n$.

It is easy to solve for the rank, by hand or by computer algebra system. But I would like to know the intuition behind the term 'controllability'. Does it have anything to do with the way you try to influence (control?) a system?

If the rank is NOT $n$, then it means some of the elements $B, AB, \cdots A^{n-1}B$ can be expressed as a linear combination of another element. We say the system is not controllable. What does 'that' mean?

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You can only reach to the states that is a linear combination of the columns of the matrices $B, AB, \dots, A^{n-1} B$. This fact follows from linear algebra, expansion of $e^{A}$ and Cayley-Hamilton Theorem.

This means whatever input you select, you are bounded in the reachable subspace, which is $\text{Im}(\gamma)$. For example if the reachable subspace is a line in a 2 dimensional system, the states cannot have values outside the line if started on the line. This means you can only change 1 state using the input, the other state changes on its own so you have no control over it. Thus, it is called unreachable/uncontrollable state.

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Try to realize that the state-space representation of a system is splitting your (physical) system up into 4 parts. In which the $A$ matrix represents the state of your system depending on a certain time $t$. $B$ represents the input-matrix which determines where on your system your input vector/scalar $u(t)$ will work. All you're checking with the controllability is to see if (independent of the $u(t)$ you choose) you will be able to control all the states of your system. This is to say, if there is a part of the physical system which you can't control, because you're "missing" an input somewhere.

Here are some definitions of controllability and reachability (which are closely related):

Reachability:

A state $x_1$ is called reachable at time $t_1$ if for some finite initial time $t_0$ there exists an input $u(t)$ that transfers the state $x(t)$ from the origin at $t_0$ to $x_1$.

A system is reachable at time $t_1$ if every state $x_1$ in the state-space is reachable at time $t_1$.

Controllability:

A state $x_0$ is controllable at time $t_0$ if for some finite time $t_1$ there exists an input $u(t)$ that transfers the state $x(t)$ from $x_0$ to the origin at time $t_1$.

A system is called controllable at time $t_0$ if every state $x_0$ in the state-space is controllable.

Try to think of it less in mathematical terms and more in a physical system.

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