The simplified transfer function model from steering angle

Chapter 8, Problem 48

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The simplified transfer function model from steering angle \(\delta(s)\) to tilt angle \(\varphi(s)\) in a bicycle is given by

G(s) = \(\frac{\varphi(s)}{\delta(s)}\) = \(\frac{a V}{b h} * \frac{s+\frac{V}{a}}{s^2-\frac{g}{h}}\)

In this model, h represents the vertical distance from the center of mass to the floor, so it can be readily verified that the model is open-loop unstable. (Åström, 2005). Assume that for a specific bicycle, a = 0:6 m, b = 1:5 m, h = 0:8 m, and g = 9:8 m/sec. In order to stabilize the bicycle, it is assumed that the bicycle is placed in the closed-loop configuration shown in Figure P8.3 and that the only available control variable is V, the rear wheel velocity.

a. Find the range of V for closed-loop stability.

b. Explain why the methods presented in this chapter cannot be used to obtain the root locus.

c. Use MATLAB to obtain the system’s root locus.