Lecture4 : TRAJECTORY TRACKING CONTROL & INTRODUCTION TO NON-LINEAR CONTROL
Positive Definite Function (pdf):
A function is said to be
positive definite function (pdf) if
It is continuous.
.
There exist a function of class K such that
hold for all
.
Examples:
For the function to be positive definite function it must satisfy following conditions.
(i)
Continuous.
(ii)
V(t,0) = 0 for all t.
(iii)
a function condition
on V(t,x) for nonzero x
.
Function 1 & 2 satisfy all three conditions but function 3 violates condition no.2 because as it does not contain so function will be zero for . Basically PDF functions are used in Lyapunov theory analysis. This is very important consideration while dealing with Lyapunov theory..A function is said to to be PDF if it contain all the components of vector X.
Other Definitions:
A function V is decrescent if there exist a contstant
r > 0 and a function of class K such that
V is radially unbounded if for all
and for some continuous function with additional property that.
V is a locally negative definite function if –V is an lpdf and a negative definite function if –V is a positive definite function.
.
of class K such that 
hold for all
so function will be zero for 
. Basically PDF functions are used in Lyapunov theory analysis. This is very important consideration while dealing with Lyapunov theory..A function 
of class K such that
for all
and for some continuous function 
.
