Use the same system and measurement models from the problem above, and obtain a solution using an Unscented Kalman Filter (UKF)

13.16 A planar model for a satellite orbiting around the earth can be modeled as r=r2r2GM+w=r2r where r is the distance of the satellite from the center of the earth, is the angular position of the satellite in its orbit, G=6.67421011m3/kg/s2 is the universal gravitational constant, M=5.981024kg is the mass of the earth, and w (0,106) is random noise due to space debris, atmospheric drag, outgassing, and so on. a) Write a state-space model for this system with x1=r,x2=r,x3=, and x4=. b) What must be equal to in order for the orbit to have a constant radius when w=0 ? c) Linearize the model around the point r=r0,r=0,=0T,=0. What are the eigenvalues of the system matrix for the linearized system when r0=6.57106m ? What would you estimate to be the largest integration step size that could be used to simulate the system? (Hint: recall that for a second-order transfer function with imaginary poles ja, the time constant is equal to 1/a. d) Suppose that measurements of the satellite radius and angular position are obtained every minute, with error standard deviations of 100 meters and 0.1 radians, respectively. Simulate the linearized Kalman filter for three hours. Initialize the system with x(0)=[r0001.10],x^(0)= x(0), and P(0)=diag(0,0,0,0). Plot the radius estimation error as a function of time. Why is the performance so poor? How could you modify the linearized Kalman filter to get better performance? e) Implement an extended Kalman filter and plot the radius estimation error as a function of time. How does the performance compare with the linearized Kalman filter