Why is it necessary to have two state variables

Question

I am learning about control theory.
Let's consider this system.

$$ m a(t) + b v(t) + k x(t) = f(t) $$

$a$ is acceleration
$v$ is velocity
$x$ is displacement
$f$ is external force

In my textbook, in chapter about "state space model" , two state variables ($x$ and $v$) are necessary
to completely describe this system.
However, I think $x$ can completely describe this mass's motion by itself.
If we know $x$, $v$ can derived from $x$.
So I have two questions.

1. Why two variables are needed.
2. Why "$x$ and $v$" ? Can "$v$ and $a$" or "$x$ and $a$" also describe this system ?

Answer 1

Let's rewrite your system as

$$ m \ddot{x}(t) + b \dot{x}(t) + k x(t) = f(t) $$

then you can see what you're saying: Everything I need to know is in $x(t)$!

But is it really?

State space systems are predicated on having a single, first order differential equation to solve. As you can see in the rewritten equation above, there is a double-derivative term.

In order to write that equation as a first order equation, we need to have two state variables: $$ X(t) = \left[ \begin{array}{c} x\\ \dot{x} \end{array} \right] $$ and so $$ A\dot{X}(t) + B X(t) = f(t) $$ where $A=[b\ m]$ and $B=[k\ 0]$.

Let's try to do the same thing using $x$ and $\ddot{x}$.

$$ X'(t) = \left[ \begin{array}{c} x\\ \ddot{x} \end{array} \right] $$ and so $$ A'\dot{X'}(t) + B' X'(t) = f(t) $$ where $A'=[b\ 0]$ and $B'=[k\ m]$.

So I suppose you could chose the state variables to be $x$ and $\ddot{x}$... but it would mean an unused $\dddot{x}$ variable.

Some examples of first order systems can be found in this document.

Answer 2

$v$ cannot be derived from a single $x$ at $t_0$. Once you use require more information than one $x$ at $t_0$ to compute $v$, and thus the momentum, you've added to what you need to describe the state of the system.