; 0 < c 6 1; 0 < a < 1

cDk2

ð29Þ

where a is a constant in the deterministic case。 The discrete-time FPSO algorithm converges over a finite-time interval if

T

there exist a positive-definite matrix Q  ¼ Q

> 0 and a parameter b such   that

Fig。  8。  The  convergence   curves  of   the  states   in   the  discrete-time   FPSO   algorithm   (28)   (x ¼ 0:8;  a ¼ 2:2;  b ¼ 0:5;  c ¼ 0:5;  Dk ¼ 0:5;  xi ð0Þ ¼ 5,  and

vi ð0Þ ¼ —9)。

A   ¼ 1 — cDk2 a

A     ¼ Dk — cDk2 ð1 — xÞ

A21 ¼ —cDka

A22  ¼ 1 — cDkð1 — xÞ

Proof。 Since the discrete-time model of the FPSO algorithm (28) can be derived in terms of (26), we write the system (26) in the  Matrix–Vector form:

Set gðkÞ ¼  y1 ðkÞ2 ð Þ

and f ðgðkÞÞ ¼    —bDk  sigð/l ðkÞÞa

—bDksigð/l ðkÞÞ

。 The system (31) can be rewritten  as

gðk þ DkÞ¼ AgðkÞþ f ðgðkÞÞ ð32Þ

If the nonlinear item f ðgðkÞÞ ¼ 0, the stability region of the system (32) is the part of the space ðx; aÞ, where the roots of the characteristic     equation

k2  þ ðacDk2  þ cDkð1 — xÞ— 2Þk þ ð1 — cDkð1 — xÞÞ ¼ 0

are in the unit circle。 Applying the Routh-Hurwitz criterion, this region turns to be (29)。 If the nonlinear item f ðgðkÞÞ – 0, we have

f T ðgðkÞÞf ðgðkÞÞ ¼ ðb2 Dk4  þ b2Dk2Þj/ j2a

Consider the case of  j/a j > 1。 We  obtain

2 T     T

f T ðgðkÞÞf ðgðkÞÞ 6 ðb2Dk4  þ b2 Dk2 Þj/ j

¼ dgðkÞ H HgðkÞ ð33Þ

where

2 4 2 2

d ¼ b Dk   þ b Dk

H 。 a  1 — x 。

¼   0 0

Consider another case  of  0 < j/l j 6 1。  There exits  a  positive constant e > 0  such that

f T ðgðkÞÞf ðgðkÞÞ 6 dej/l j

ð34Þ

Therefore, - ¼ maxfd; deg。 Choose a Lyapunov function candidate as

VðgðkÞÞ ¼ gðkÞT PgðkÞ ð35Þ

where P ¼ PT  > 0 and P 2 R2×2 。

T T

MV ðgðkÞÞ ¼ V ðgðk þ DkÞÞ — VðgðkÞÞ ¼ ðAgðkÞþ f ðgðkÞÞÞ PðAgðkÞþ f ðgðkÞÞÞ — gðkÞ PgðkÞ

T T