The A, B, C, and D matrices for the cart-pendulum system, with the following state definitions, 21 = Ic X2 = Ic 23 = Op IA = 0p are given in the gantry_lab.m script. Note that the script we use in class, inv-pendulum_ss.m is a useful template for working this problem. (a) (5 points) Numerically confirm that the eigenvalues of A are equal to the poles of Geart. Also, confirm using MATLAB that the system is fully controllable (controllability Gramian has full rank). (b) (5 points) Use the acker() command to design a full-state feedback controller that places all closed-loop poles at -1. Verify that your controller is correct by checking the eigenvalues of the closed-loop state matrix Act- (c) (15 points) Using the LQR methodology, design and tune a full-state feedback regulator to reduce the pendulum angle to less than 0.5 within 10 seconds for a 0.1-meter initial condition in cart position. Provide your cost matrices Q and R. and describe the reasoning behind these selections (may include some trial and error). Check if your controller results, in completing this initial condition test can stay within 15V input signal. If not, adjust your R. matrix and iterate the design.