3) The equation of motion can be derived for the model of an inverted pendulum on a cart shown in the figure below. In the model, t) is the angle of the pendulum clockwise with respect to the vertical, x(t) is the horizontal position of the cart relative to some arbitrary fixed reference location, 24 is the length of the pendulum, M and m are the masses of the cart and the pendulum, respectively, and I is the moment of inertia of the pendulum about its center of gravity. F is the force applied to the body of the cart. i) ii) e(t) M mg 28 It can be shown that the two coupled differential equations that describe the motion of this system are: (m + M)x+mlcos 0 - meosin 0 = F (1 + m) +mlxcos 0 -mglsin 0 = 0 Linearize these two equations using the first two terms of the Taylor series. Choose the "zero" position of 8o(t) = 0,xo(t) = 0, F = 0. Construct a state-space model from the linearized equations