Editing State-space representation (section)

=== Moving object example ===
A classical linear system is that of one-dimensional movement of an object (e.g., a cart).
[[Newton's laws of motion]] for an object moving horizontally on a plane and attached to a wall with a spring:

<math display="block">m \ddot{y}(t) = u(t) - b\dot{y}(t) - k y(t)</math>

where
*<math>y(t)</math> is position; <math>\dot y(t)</math> is velocity; <math>\ddot{y}(t)</math> is acceleration
*<math>u(t)</math> is an applied force
*<math>b</math> is the viscous friction coefficient
*<math>k</math> is the spring constant
*<math>m</math> is the mass of the object

The state equation would then become

<math display="block">\begin{bmatrix} \dot{\mathbf x}_1(t) \\ \dot{\mathbf x}_2(t) \end{bmatrix}
= \begin{bmatrix} 0 & 1 \\ -\frac{k}{m} & -\frac{b}{m} \end{bmatrix} \begin{bmatrix} \mathbf{x}_1(t) \\ \mathbf{x}_2(t) \end{bmatrix}
  + \begin{bmatrix} 0 \\ \frac{1}{m} \end{bmatrix} \mathbf{u}(t)</math>
<math display="block">\mathbf{y}(t) = \left[ \begin{matrix} 1 & 0 \end{matrix} \right] \left[ \begin{matrix} \mathbf{x_1}(t) \\ \mathbf{x_2}(t) \end{matrix} \right]</math>

where
*<math>x_1(t)</math> represents the position of the object
*<math>x_2(t) = \dot{x}_1(t)</math> is the velocity of the object
*<math>\dot{x}_2(t) = \ddot{x}_1(t)</math> is the acceleration of the object
*the output <math>\mathbf{y}(t)</math> is the position of the object

The [[controllability]] test is then

<math display="block">\begin{bmatrix} B & AB \end{bmatrix}
= \begin{bmatrix} \begin{bmatrix} 0 \\ \frac{1}{m} \end{bmatrix} & \begin{bmatrix} 0 & 1 \\ -\frac{k}{m} & -\frac{b}{m} \end{bmatrix}  \begin{bmatrix} 0 \\ \frac{1}{m} \end{bmatrix} \end{bmatrix}
= \begin{bmatrix} 0 & \frac{1}{m} \\ \frac{1}{m} & -\frac{b}{m^2} \end{bmatrix}</math>

which has full rank for all <math>b</math> and <math>m</math>. This means, that if initial state of the system is known (<math>y(t)</math>, <math>\dot y(t)</math>, <math>\ddot{y}(t)</math>), and if the <math>b</math> and <math>m</math> are constants, then there is a force <math>u</math> that could move the cart into any other position in the system.

The [[observability]] test is then

<math display="block">\begin{bmatrix} C \\ CA \end{bmatrix}
= \begin{bmatrix} \begin{bmatrix} 1 & 0 \end{bmatrix} \\ \begin{bmatrix} 1 & 0 \end{bmatrix} \begin{bmatrix} 0 & 1 \\ -\frac{k}{m} & -\frac{b}{m} \end{bmatrix} \end{bmatrix}
= \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}</math>

which also has full rank. Therefore, this system is both controllable and observable.