State-Space Model of a Force-Driven Mass

Figure 1.6: Ideal mass $m$ on frictionless surface driven by force $f(t)$ .

For the simple example of a mass $m$ driven by external force $f$ along the $x$ axis, we have the system of Fig.1.6. We should choose the state variable to be velocity $v={\dot x}$ so that Newton's $f=ma$ yields

$$\dot{v} \eqsp \frac{1}{m} f.$$

This is a first-order system (no vector needed). We'll look at a simple vector example below in §1.3.7.