as A planar model for a satellite orbiting around the earth can be modeled 3: =...

Transcribed Image Text:

as A planar model for a satellite orbiting around the earth can be modeled 3: = r²_ Ö - GM 72 + w -20r T 3 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 = 6.6742 × 10-¹¹m³/kg/s² is the universal gravitational constant, M = 5.98 x 1024 kg is the mass of the earth, and w~ (0, 10-6) 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 = = 0, = 0. 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 = ro, r = 0,0 woT, 0 <= wo. What are the eigenvalues of the system matrix for the linearized system when ro 6.57 x 106 m? What would you estimate to be the largest as A planar model for a satellite orbiting around the earth can be modeled 3: = r²_ Ö - GM 72 + w -20r T 3 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 = 6.6742 × 10-¹¹m³/kg/s² is the universal gravitational constant, M = 5.98 x 1024 kg is the mass of the earth, and w~ (0, 10-6) 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 = = 0, = 0. 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 = ro, r = 0,0 woT, 0 <= wo. What are the eigenvalues of the system matrix for the linearized system when ro 6.57 x 106 m? What would you estimate to be the largest