Problem 3 (25 points) You may use MATLAB for this problem. Please include your code when you do. The linearized and normalized equations of motion for a satellite are: -23-3a=u B+2+3=0 where a is the roll angle, is the pitch angle, and u is the engine thrust in the roll direction. 1. Express the system in state space form where x = [, , 3, 3] and y = a. Y(s) 2. Find the transfer function H(s) = (8) 3. Verify that the system is unstable, controllable and observable. 4. Create a state feedback controller with all closed loop poles at -7, calculate K, to remove steady state error and plot the step response. 5. Determine the equivalent discrete-time state space description of the plant for a sample period T = 0.18 x[n+1] = Ax[n] + Bu[n] [n] = C [n] +Dun Do this by finite difference method (In+1 =2n+T). Compare your calculation with the result of MATLAB's continuous to discrete time converter (c2d) and find the difference between the norms of the resulting matrices. For example, if Ad is your calculation and Ae is MATLAB's, calculate norm(Ad - Ae). 6. Using your calculated Ad, determine the full state feedback gain vector K that will place all the discrete-time closed loop poles at 0.5 (this is approximately the equivalent pole to s = -7, the pole for the continuous system. The transformation is z = eT). 7. Calculate Krd to remove steady state error 8. Simulate the discrete time state space control implementation. Plot the step response y(t). To simulate a discrete time system, you can write code similar to this: In = xoj data = [0; x0]; for t = times end xn= = Adn + Bauni Yn = Cdxn + Daun; data = [data, [t; yn]]; (1) 3