Define the following as states: $x,x^{},,{}^{}$ then re$-$write the equations in terms of state variables.

Determine the four state equations

For the crane model above, if we assume that the pendulum angle is small, we can use a linear

approximation

$(1-\frac{m}{m_G})x^{}+\frac{b}{m_G}{x}^{}-\frac{m}{m_G}g=\frac{u}{m_G}$

$(1-\frac{m}{m_G}){}^{}-\frac{b}{l{m}_G}{x}^{}+\frac{g}{l}=\frac{u}{l{m}_G}$

Write the four state equations, and additionally write the state space

representation. Let the output be pendulum angle $$