Consider the cart–inverted-pendulum system shown in Figure 5.79. Assume that the mass of the cart is 0.8 kg, the mass of the pendulum is 0.2 kg, and the length of the pendulum is 0.6 m.
a. Determine the poles of the linearized system. Is it stable or unstable?
b. Design a full-state feedback controller for the linearized system using the pole placement method. Assume that two of the closed-loop poles are complex conjugate, with a natural frequency of 3.6 rad/s and a damping ratio of 0.6. They dominate the effect of the other two poles, which are assumed at −10 and −20.
c. Assume that the initial angle of the inverted pendulum is 5° away from the vertical reference line. Using the state feedback gain matrix K obtained in Part (b), examine the responses of the nonlinear and linearized closed-loop systems by using Simulink.