Question

Problem 4 (10 points) Astrom et a1 have presented a simplified second-order transfer function model for bicycle dynamics as C1 Parameters defining the bicycle geometry. Schematic (a) top and (b) rear views of a bicycle The steer angle is δ and the roll angle is φ. The input is δ(t), the steering angle, and the output is Plt), the tilt angle (between the the bicycle longitudinal plane and the ground). In the model, parameter a is the horizontal distance from the bicycle center of mass to the center of the back wheel; b is the wheel base (the horizontal distance between the centers of both wheels); h is the vertical distance from the center of mass to the ground; V is the back wheel velocity (assumed constant); and g is the gravity. It is also assumed that the rider remains at a fixed position with respect to the bicycle so that the steer axis is vertical and that all angle deviations are small. a) Using the transfer function, obtain a state-space representation for the bicycle model. Assume a = 0.5, b = 1, h = 1, V = 10, 1-9 b) Using the A matrix obtained in part (a), find system eigenvalues and eigenvectors and check the systems stability. (Due to free variables in eigenvectors, assume 1 and 2 1) c) Find an appropriate similarity transformation matrix (P) to diagonalize the system and obtain the new similar state-space systems diagonal representation Astrom, Karl J., Richard E. Klein, and Anders Lennartsson. "Bicycle dynamics and control: adapted bicycles for education and research. IEEE Control Systems 25.4 (2005)

Answer 1

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