Answers of e and f are needed only. In form of Matlab code please.

The figure shows the pendulum on a cart system. The position of the cart measured from the origin is z and the linear speed of the cart is . The angle of the pendulum from straight up is given by 0, and the angular velocityis 0. The rod is of length I, and has mass m, and is approximated as being infinitely thin. The cart has mass m2. Gravity acts in the down direction. The only applied force is F, which acts in the direction of z. mi 0 gravity m2 F (External Force) You can use the following simplified equations of motion for the system. (m + m2) -b+ F m19 hi Defining the states as = (1, 7, , 7)"the input as = F, and the measured output as = (2, 2)", find the linear state space equations in the * = AX + Bu y = CX+ D a- Now implement a state feedback controller using the full state. b- Using the values for wnz, & , wng, and selected where the rise time is tr, = 0.5 seconds, and the damping ratio is 4 = 0.707, find the desired closed loop poles. - Use state space matrices A, B, C, D derived in previous question. Verify that the state space system is controllable by checking that: rank(C) = n. d- Find the feedback gain K so that the eigenvalues of (A-BK) are equal to the desired closed loop poles. Modify the state feedback solution to add an integrator with anti-windup to the position feedback. e- Add a constant input disturbance of 0.5 Newtons to the input of the plant and allow the system parameters to change up to 20%. f- Tune the integrator pole (and other gains if necessary) to get good tracking performance. The figure shows the pendulum on a cart system. The position of the cart measured from the origin is z and the linear speed of the cart is . The angle of the pendulum from straight up is given by 0, and the angular velocityis 0. The rod is of length I, and has mass m, and is approximated as being infinitely thin. The cart has mass m2. Gravity acts in the down direction. The only applied force is F, which acts in the direction of z. mi 0 gravity m2 F (External Force) You can use the following simplified equations of motion for the system. (m + m2) -b+ F m19 hi Defining the states as = (1, 7, , 7)"the input as = F, and the measured output as = (2, 2)", find the linear state space equations in the * = AX + Bu y = CX+ D a- Now implement a state feedback controller using the full state. b- Using the values for wnz, & , wng, and selected where the rise time is tr, = 0.5 seconds, and the damping ratio is 4 = 0.707, find the desired closed loop poles. - Use state space matrices A, B, C, D derived in previous question. Verify that the state space system is controllable by checking that: rank(C) = n. d- Find the feedback gain K so that the eigenvalues of (A-BK) are equal to the desired closed loop poles. Modify the state feedback solution to add an integrator with anti-windup to the position feedback. e- Add a constant input disturbance of 0.5 Newtons to the input of the plant and allow the system parameters to change up to 20%. f- Tune the integrator pole (and other gains if necessary) to get good tracking performance