Modify the MATLAB program you developed in 10.20 to do the

Chapter 10, Problem 49

(choose chapter or problem)

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

G(s) = \(\frac{\theta(s)}{U(s)}\)

        = \(\frac{334019}{s^4+5126.16 s^3+2470.7 s^2+428419 s-34040}\)

where \(\theta(s)\) 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.