ABSTRACT
In order to compare Natural Tracking Control (NTC) with Lyapunov Tracking Control (LTC) we accept the tracking operator T(:) to determine the tracking algorithm (12.5) (Section 12.1),
T
t; "; "(1); :::; "(k);
"dt
= T
t; "k;
"dt
= 0N ;8t 2 T0;
k 2 f1; 2; :::g : (13.1)
In view of Theorem 364 (Section 12.1) we accepted the modi…ed tracking goal 359 (Section 12.1):
T
tN ; "k;
EdtN
! = 0N ; 8tN 2 TN0 if T
0N ; "
= 0N ;
"k t(k+1)N ; "k0
= fk
t(k+1)N ; fk0
;
8t1(k+1)N 2 TN0 ; if T 0N ; "
6= 0N ;
k 2 f0; 1; 2; :::g ; (13.2)
where we refer to (12.19) through (12.25) (Section 12.1) so that:
k(t(k+1)N ) = "k(t(k+1)N )+
+
8>><>>: 0(k+1)N ;
8t(k+1)N 2 T(k+1)N0
if T 0(k+1)N ; "
= 0N ;
fk(t(k+1)N ; fk0 );
8t(k+1)N 2 T(k+1)N0
if T 0(k+1)N ; "
6= 0N ; 9>>=>>; : (13.3)
This inspires us to introduce both T t; k;
dt and the time-varying set
T k (t) in terms of k such that (13.2) holds,
T
tN ;
tN ; (1)
tN ; :::; (k)
tN ;
tN dtN
! =
= T
tN ; k
t(k+1)N
;
tN dtN
! = 0N ; 8t(k+1)N 2 T(k+1)N0 ;
(13.4)
T k (t) = k : T
t; k;
dt
= 0N
R(k+1)N ;8t 2 T0; (13.5)
and the following form of a tentative Lyapunov function v(:) on RN :
v(T) = 1
2 kTk2 : (13.6)
It is globally positive de…nite and radially unbounded on RN as the function of the vector T 2RN . However, we can treat it as the function of k;
v(k) = 1
Tt; k;Z t t0=0
dt
2 ; k 2 f1; 2; :::g : (13.7) It is globally positive (not necessarily positive de…nite [232]) with respect to the time-varying set T k (t) (13.5) that is the time-varying hypersurface in R(k+1)N . For the sake of simplicity, let the operator T (:) be time-invariant, hence independent of the integral of ,
T t; k
= T
k ;8t 2 T0: (13.8)
Now, the set T k (t) is also time-invariant,
T k = k : T k = 0N R(k+1)N ; (13.9) and the function v(:) (13.7), (13.8) is globally positive de…nite on R(k+2)N relative to the time-invariant set T k (13.9), which is the time-invariant hypersurface in R(k+2)N ;
T k = T
; (1); :::; (k)
= 0N : (13.10)
Examples 366 through 372 (Section 12.1) illustrate (13.10).
