Mass-Spring Damper State Space Model

Newton’s 2nd Law gives:

$$M\ddot{y}=u-ky-b\dot{y}$$

Choosing states:

$$\begin{array}{r}{x}_1=y\\ x_2=\dot{y}\end{array}$$

Then, we have:

$$\begin{array}{rl}\dot{x_1}& =\dot{y}=x_2\\ x_2& =\ddot{y}=\frac{1}{M}(u-k{x}_1-b{x}_2)\end{array}$$

Then:

$$\dot{x}=\left[\begin{array}{cc}0& 1\\ -\frac{k}{M}& -\frac{b}{M}\end{array}\right]x+\left[\begin{array}{c}0\\ \frac{1}{M}\end{array}\right]u$$

and $y=x_1$ gives us

$$y=\left[\begin{array}{cc}1& 0\end{array}\right]x+0u$$

Extra state

What if we add an extra (unnecessary) state $x_3$? Let us have:

$${\dot{x}}_3=2x_3$$

Then, we would have:

$$\dot{x}=\left[\begin{array}{ccc}0& 1& 0\\ -\frac{k}{M}& -\frac{b}{M}& 0\\ 0& 0& 2\end{array}\right]x+\left[\begin{array}{c}0\\ \frac{1}{M}\\ 0\end{array}\right]u$$

and

$$y=\left[\begin{array}{ccc}1& 0& 0\end{array}\right]x+0u$$

Note that the dimension (number of states) of a state space model is not unique either!