kð1—aÞ

Remark 1。 In Lemma 1, the function f ðxðtÞÞ denotes the change law of the state and  is  given  in  advance  in  the  light  of  the system structure。 V ðxðtÞÞ is a Lyapunov function which is a continuously differentiable and positive definite function。 The relationship  between  f ðxðtÞÞ  (x 2 D)  and  V ðxðtÞÞ  is  that  we  need  to  find  a  Lyapunov  function  such  that  V ðxðtÞÞ > 0  in

D— f0g  and  Vð0Þ ¼ 0,  and  along  the  system  trajectory  x_ ¼ f ðxðtÞÞ,  we  have  V_  þ kVa   6 0。  Hence,  it  is  worth  noting  that

f ðxðtÞÞ is  different from VðxðtÞÞ。

Fig。 1。  The decision-control block diagram of the PSO    algorithm。

Lemma  2  (Schur  Complement)。  Given  constant  matrices  P1 ; P2 ,   and  P3     with  appropriate   dimensions,  where  PT  ¼ P1          and

PT T    —1

2 ¼ P2 > 0, then P1 þ P3 P2   P3 < 0 if and only if

  P1 PT    !

P3    —P2

< 0    or —P2    P3 < 0

PT P1

Remark 2。 In Lemma 2, it should be pointed out that, for any symmetric matrix P, we write P > 0 if P is a positive definite matrix and P < 0 if P is a negative definite matrix。

3。Finite-time particle swarm optimization

In this section, first, we will briefly describe and analyze the continuous-time model of the PSO algorithm。 Then, we will derive a continuous-time FPSO algorithm and prove its finite-time convergence。 Finally, as part of the FPSO algorithm, we will propose a discrete-time version of the FPSO algorithm and give the corresponding convergence condition。

3。1。The continuous-time model of particle swarm     optimization

The stochastic differential model of the PSO algorithm, i。e。 the continuous-time model of the PSO algorithm, is derived by Fernández Martínez et al。 [15] in terms of physical analogy with a damped mass-spring oscillator and is given by

€xiðtÞþ ð1 — xÞx_ i ðtÞþ axiðtÞ¼ a1 xl ðtÞþ a2 xg ðtÞ ð6Þ

with a ¼ a1 þ a2; xi ð0Þ ¼ x0, and x_ i ð0Þ ¼ v0 , where xl ðtÞ and xg ðtÞ are the trajectories of the local and global best positions associated with the ith ði 2 lN Þ particle, respectively; aj  (j ¼ 1; 2) are random variables; xi ð0Þ  and x_ i ð0Þ  are the initial  states at  time  t  ¼ 0。

Introduce the following discretization  scheme

( xi ðtÞ—xi ðt—DtÞ

x_ i ðtÞ’ Dt

xi ðtþDtÞ—2xi ðtÞþxi ðt—DtÞ

ð7Þ

€xi ðtÞ’ 

Dt2

Consider the case of Dt ¼ 1 and t ¼ k。 Then, we  have