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 
.
