Question

Solve parts a,b,c,d. Show matlab code and all steps. Show detailed explanation for part a

The differential equations describing the motion of the inverted stick on a cart as given in the figure below are: (M + m)X(t) + ml cos θ(t)-θ(t)-m1θ2 (t) sin θ(t)-F(t) m12 θ(t) + ml cos θ(t)-X(t)-mgl sin θ(t)-r(t) For the questions below, assume the control input torque τ(1) is not available (r(1-0) a. Explain how a constant equilibrium angle 6()-30 can be accomplished. b. Express these equations in the linear state-space form around the equilibrium at x-θ = 0 (stationary cart with stick in the upright position) with θ(t) as the output. For the linearization, apply small-deviation approximations directly to the equaions: sin00, cos, 0 c. Determine the transfer function between F(t) and ) around the upright equilibrium at x--0. Conclude on the stability of the inverted stick at its upright position without feedback. d. An artificial-intelligence (AI) expert, who is allergic to mathematics, claims that a rule-based system can be designed to maintain the stick at its upright equilibrium by monitoring the stick angle 0(t) and applying a force F(t) proportional to θ(t), so that when the stick is falling to the right, the cart is pushed to the right, and when the stick is falling to the left, the cart is pushed to the left. Is he correct? Analyze using root locus on whether or not this AI rule-based control system works. If it does not work, suggest a modification to the control scheme to make it work. Provide reasons to justify that the modification will work

Answer 1