Nonlinear Control Sy.. - Free

198 CHAPTER 7. INPUT-TO-STATE STABILITY
7.6 Exercises
(7.1) Consider the following 2-input system ([70]):
Is it input-to-state stable?
(7.2) Sketch the proof of theorem 7.5.
i= -x3 + x2U1 - XU2 + U1162
(7.3) Provide an alternative proof of Theorem 7.9, using only the definition of input-tostate-stability,
Definition 7.1.
(7.4) Sketch a proof of Theorem 7.10.
(7.5) Consider the following system:
(i) Is it locally input-to-state-stable?
(ii) Is it input-to-state-stable?
(7.6) Consider the following system:
(i) Is it locally input-to-state-stable?
(ii) Is it input-to-state-stable?
.t1 = -x1 + x2Ix2
{ x2 = -x2 + 21x2 + u
I i1 = -21 - 22
ll i2 = x1-x2+u
(7.7) Consider the following cascade connection of systems
(i) Is it locally input-to-state-stable?
(ii) Is it input-to-state-stable?
2 = -x3 + x2u
{ z = -z3 + z(1 + Z2)x
(7.8) Consider the following cascade connection of systems:
1 z
f
2 = -x3 + x2u1 - XU2 + u1u2
-z3 +z 2 X