State-Space Models

A state-space model describes a system by its state, the few numbers that capture everything about it right now, and by matrices that say how the state changes. It handles many inputs and outputs at once, which makes it the language of modern robot control.

Spring stiffness k Damping b Input force u

The model

For a mass $m$ on a spring $k$ with damping $b$, choose the state $\mathbf{x} = (p, v)$, position and velocity. Then

$$\dot{\mathbf{x}} = \underbrace{\begin{bmatrix} 0 & 1 \\ -k/m & -b/m \end{bmatrix}}_{A}\mathbf{x} + \underbrace{\begin{bmatrix} 0 \\ 1/m \end{bmatrix}}_{B} u \qquad y = \underbrace{\begin{bmatrix} 1 & 0 \end{bmatrix}}_{C}\mathbf{x}$$

The eigenvalues of $A$ are the system's poles.

What to notice

The right plot is the state space itself: position across, velocity up. Every moment of the motion is one point, and the system traces a path. With no damping it is a closed loop, and with damping it spirals into the rest point. A steady input force moves the rest point, since the spring then stretches to balance it. The same form scales to a full robot arm, where the state holds every joint angle and speed.