Editing State-space representation (section)

== Nonlinear systems ==
The more general form of a state-space model can be written as two functions.

<math display="block">\dot\mathbf{x}(t) = \mathbf{f}(t, x(t), u(t))</math>
<math display="block">\mathbf{y}(t) = \mathbf{h}(t, x(t), u(t))</math>

The first is the state equation and the latter is the output equation.
If the function <math>f(\cdot,\cdot,\cdot)</math> is a linear combination of states and inputs then the equations can be written in matrix notation like above.
The <math>u(t)</math> argument to the functions can be dropped if the system is unforced (i.e., it has no inputs).

=== Pendulum example ===
A classic nonlinear system is a simple unforced [[pendulum]]

<math display="block">m\ell^2 \ddot\theta(t) = - m\ell g\sin\theta(t) - k\ell\dot\theta(t)</math>

where
*<math>\theta(t)</math> is the angle of the pendulum with respect to the direction of gravity
*<math>m</math> is the mass of the pendulum (pendulum rod's mass is assumed to be zero)
*<math>g</math> is the gravitational acceleration
*<math>k</math> is coefficient of friction at the pivot point
*<math>\ell</math> is the radius of the pendulum (to the center of gravity of the mass <math>m</math>)
The state equations are then

<math display="block">\dot{x}_1(t) = x_2(t)</math>
<math display="block">\dot{x}_2(t) = - \frac{g}{\ell}\sin{x_1}(t) - \frac{k}{m\ell}{x_2}(t)</math>

where
*<math>x_1(t) = \theta(t)</math> is the angle of the pendulum
*<math>x_2(t) = \dot{x}_1(t)</math> is the rotational velocity of the pendulum
*<math>\dot{x}_2 = \ddot{x}_1</math> is the rotational acceleration of the pendulum

Instead, the state equation can be written in the general form

<math display="block">\dot{\mathbf{x}}(t) = \begin{bmatrix} \dot{x}_1(t) \\ \dot{x}_2(t) \end{bmatrix} = \mathbf{f}(t, x(t)) = \begin{bmatrix} x_2(t) \\  - \frac{g}{\ell}\sin{x_1}(t) - \frac{k}{m\ell}{x_2}(t) \end{bmatrix}.</math>

The [[Mechanical equilibrium|equilibrium]]/[[stationary point]]s of a system are when <math>\dot{x} = 0</math> and so the equilibrium points of a pendulum are those that satisfy

<math display="block">\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} n\pi \\ 0 \end{bmatrix}</math>

for integers ''n''.