Post on 08-Apr-2020
Chapter 4Chapter 4Lyapunov Stabilityy p y
The Invariance PrincipleThe Invariance PrincipleTime Varying Systems
3: ( ) 0 0E V x b x x : ( ) 0 0: ( , ) (0,0)
E V x b x xM x x
• Assume M contains a point with a nonzero
iti th th l ti t th t
( , ) ( , )
position x, then the acceleration at that point is 30 1 0k k
This implies that the trajector ill mo e
30 1 0x x xm m
• This implies that the trajectory will move out of E. Not invariant. Contradiction!
Derivative of VDerivative of V
( ), ( ),lim
V x t t t t V x t tdVd
( ), ( ), ( ), ( ),lim
t
t
dt tV x t t t t V x t t t V x t t t V x t t
t
( ), ( ), ( ), ( ),lim lim
t
t t
tV x t t t t V x t t t V x t t t V x t t
t t
( , )
t tV V V Vx f x tt x t x