Please make sure to answer 2.1, 2.2, and 2.3 Also when you compute the plot please use Matlab and include the screenshot of both the plot and the code used to get the plot. Please and thank you.

2) Control Using Proportional Feedback In order to stabilize the system using feedback, the cart must be moved back and forth in response to the angle the pendulum makes with the vertical in order to keep the pendulum from tipping over. We first use proportional feedback in the form ke (), i.e., H(s) -ky Inverted Pendulum Motor & Cart (1) Proportional feedback Physically, the output e (1) can be measured by placing one terminal of a rheostat on the hinge and the other terminal on the pendulum. The resistance of the tapped rheostat is directly proportional to 8(t) = 2. Thus, if 8 (1) =-72, the resistance is zero, and maximum if (0) = 72. The resistance can be used to control a motor attached to the cart wheels, and thus, control the cart position x(+). Since x(t) is measured in unit of length, kp has a unit of length'radian 2.1 Obtain the closed-loop transfer function of the inverted pendulum system above. 2.2 Determine the bounds for the ky that produce stability using the Routh-Hurwitz criterion. 2.3 Using the inverse Laplace transform, analytically compute and plot its step response for ko = 122.5 cm. What conclusion can be drawn from the plot in terms of stability? What is the amplitude and period of the pendulum swing? 2) Control Using Proportional Feedback In order to stabilize the system using feedback, the cart must be moved back and forth in response to the angle the pendulum makes with the vertical in order to keep the pendulum from tipping over. We first use proportional feedback in the form ke (), i.e., H(s) -ky Inverted Pendulum Motor & Cart (1) Proportional feedback Physically, the output e (1) can be measured by placing one terminal of a rheostat on the hinge and the other terminal on the pendulum. The resistance of the tapped rheostat is directly proportional to 8(t) = 2. Thus, if 8 (1) =-72, the resistance is zero, and maximum if (0) = 72. The resistance can be used to control a motor attached to the cart wheels, and thus, control the cart position x(+). Since x(t) is measured in unit of length, kp has a unit of length'radian 2.1 Obtain the closed-loop transfer function of the inverted pendulum system above. 2.2 Determine the bounds for the ky that produce stability using the Routh-Hurwitz criterion. 2.3 Using the inverse Laplace transform, analytically compute and plot its step response for ko = 122.5 cm. What conclusion can be drawn from the plot in terms of stability? What is the amplitude and period of the pendulum swing