Problem 2. (5 points) Consider the inverted pendulum on a cart given in Figure 1. B-X1 me mg sine u Figure 1. Inverted pendulum schematic. The state variables for this system can be defined to be: x, (1)=0(1), x, (1)=ȧ(1). Assume that the equation of motion for the pendulum can be written as follows: g sin(x) X2 cos(x) m+m 4 mcos(x) 4 m cos(x) 3 m,+m 3 m, +m u where g-9.8m/s², m. is the cart's mass, m is the mass of the pole, / is the half length of the pole, and u is the applied force. a) (2 points) Find a linearized state equation if the desired equilibrium value for each state is zero. b) (3 points) Find a linearized state equation if it is desired to position the pendulum at a 45".