Nonlinear Control Sy.. - Free

4.10. STABILITY OF DISCRETE-TIME SYSTEMS 131
4.10.2 Discrete-Time Positive Definite Functions
Time-independent positive functions are uninfluenced by whether the system is continuous
or discrete-time. Time-dependent positive function can be defined as follows.
Definition 4.9 A function W : R" X Z --+ P is said to be
Positive semidefinite in D C R' if
(i) W(0,k)=0 Vk>0.
(ii) W(x,k)>0 `dx#0, xED.
Positive definite in D C P" if
(i) W (O, k) = 0 Vk > 0, and
(ii) I a time invariant positive definite function Vi(x) such that
Vl (x) < W (x, k) dx E D V.
It then follows by lemma 3.1 that W (x, k) is positive definite in B,. C D if and
only if there exists a class 1C function such that
al(jjxMI)