非线性系统(第三版)(英文版)chapter4[2页][001].pdf

1 Additional Exercises for Chapter 4 1. For each of the following systems, use a quadratic Lyapunov function candidate to show that the origin is asymptotically stable. Then, investigate whether the origin is globally asymptotically stable. (1) ˙ 1 = 1 + 2 ˙2 = 2 2 (2) ˙ = ( )(2 + 2 1) ˙ = ( + )(2 + 2 1) 1 1 2 1 2 2 1 2 1 2 2 (3) ˙ 1 = 1 + 2 ˙2 = 2 + 1 1 2. Using () = 2 + 2 , study stability of the origin of the system 1 2 2 2 2 2 2 2 2 2 2 2 2 2 ˙ 1 = 1 ( ) + 2 ( + + ) ˙2 = 1 ( + + ) + 2 ( ) 1 2 1 2 1 2 1 2 when (a) = 0 and (b) = 0. 3. Using the variable gradient method, find a Lyapunov function () that shows asymptotic stability of the origin of the system ˙ = ˙ = ( + ) sin( + ) 1 2 2 1 2 1 2 4. Consider the system ˙ = ˙ = sat(2 + ) 1 2 2 1 1 2 Show that the origin is asymptotically stable, but not globally asymptotically stable. 5. Show that the origin of the following system is unstable. ˙ 1 = 1 + 6 ˙2 = 3 + 6