1. Khalil, Exercise 12.5.
Hint: Start with and apply the procedure (12.13), (12.14).
2. Khalil, Exercise 12.8.
Hint: Apply Theorem 12.1. For inverse dynamics, use and .
3. Khalil, Exercise 12.31, Part (1).
Hint: For involutivity, employ the result of Exercise 12.27.
4. Consider the nonlinear system
(a) Show that the system has relative degree two.
(b) Transform the system into the normal form. Hint: For the zero dynamics use
(c) Show that the origin of the zero dynamics is globally asymtotically stable.
(d) Design a feedback law that achieves global asymptotic stability.
and apply the procedure (12.13), (12.14).
and 
.
