In order to self-balance a bicycle, its open-loop transfer function is found to be (Lam, 2011): where (s) is the angle of the bicycle with

In order to self-balance a bicycle, its open-loop transfer function is found to be (Lam, 2011):

where θ(s) is the angle of the bicycle with respect to the vertical, and U(s) is the voltage applied to the motor that drives a flywheel used to stabilize the bicycle. Note that the bicycle is open-loop unstable with one open-loop pole in the right half-plane.

a. Draw the Nyquist diagram of the system.

b. Find the system’s gain and phase margins.

c. Assuming a unit feedback system, find the range of K for closed-loop stability if the forward path transfer function is KG(s).

d. Assuming a second-order approximation, what is the expected %OS if K = 0.141?

e. Use a computer program to simulate your system for a unit-step response using the value of K in Part d.

O(s) G(s) = U(s) 334019 sA +5126.16s + 2470.7s2 + 428419s 34040