¼ —ðAgðkÞþ f ðgðkÞÞÞ PðAgðkÞþ f ðgðkÞÞÞ — gðkÞ PgðkÞþ ðAgðkÞþ f ðgðkÞÞÞ PAgðkÞ

T T    T T

þ ðAgðkÞþ f ðgðkÞÞÞ Pf ðgðkÞÞ þ gðkÞ A PðAgðkÞþ f ðgðkÞÞÞ þ f ðgðkÞÞ PðAgðkÞþ f ðgðkÞÞÞ ð36Þ

Use the inequality xT y þ yT x 6 xT x þ yT y for any x; y 2 Rn  to obtain

T       2 T T T

MV ðgðkÞÞ 6 ðAgðkÞþ f ðgðkÞÞÞ  ðP   — PÞðAgðkÞþ f ðgðkÞÞÞ þ gðkÞ  ð-H  H — PÞgðkÞþ ðAgðkÞþ f ðgðkÞÞÞ  PAgðkÞ

T    T

þ gðkÞ A PðAgðkÞþ f ðgðkÞÞÞ

In order to use Lemma 2, we rewrite (39)  as

;  P2  ¼ - I,  and  P3  ¼ ð 0     HQ Þ。  By  Lemma  2,  the  matrix  inequality  can  be  written  in  the  form  of  the

linear matrix inequality in Theorem 2。 Hence, the system (26) is asymptotically stable, and is also finite-time stable since it is the discrete-time version of the FPSO algorithm, which means that the discrete-time FPSO algorithm (28) converges over a finite-time interval。      Q

Remark 10。 In fact, if the parameters a; b; c; a, and x satisfy the LMI (30), the discrete-time FPSO algorithm (28) converges within a finite-time interval。 However, how to choose the parameters is not an easy task。 In order to deal with this problem, we first select the parameters c and a where c is used to control exploration ability and a does not influence the search ability of the discrete-time FPSO algorithm。 Then, we choose a and x from Xd in (29), which can influence the convergence of the discrete-time FPSO algorithm。 Finally, by using the Matlab LMI toolbox to solve the LMI (30), we can get the parameter b which is used to control convergence   time。

4。Numerical simulation

From the aforementioned theory analysis, one can see that the proposed FPSO algorithm can satisfy the requirements gi- ven in the Introduction。 Therefore, in this section, we will illustrate the optimization characteristics and performance capa- bilities of the discrete-time FPSO algorithm through two ill-posed functions and twenty-five benchmark functions, respectively。

4。1。Two  ill-posed functions

In this subsection, we will illustrate the characteristics of the proposed discrete-time FPSO (DFPSO) algorithm based on two ill-posed functions: Griewank and Rastrigin with two dimensions。 A similar analysis has been given for the PSO algo- rithm and the GPSO algorithm in [13,14]。 Moreover, the maximum number of function evaluations is limited to 4000 and the population size is 40。 The parameters x and a are given such  that  the  corresponding  algorithms converge。 For  the PSO algorithm, 0 < a < 4 and —1 < x < 1。 For the GPSO algorithm, 0 < a < 16; —3 < x < 1, and Dt ¼ 0:5。 For the DFPSO algorithm, 0 < a < 32; —7 < x < 1; Dt ¼ 0:5; c ¼ 0:5, and b ¼ 0:01。 The success rates and average iterations over 100 sim-