1#. • Basis Weight Control System
fixed, which is an unfavorable limitation to determine the optimal solution. In Refs. [32 33], the authors introduce two dynamical coefficients to optimize the target. The cognitive coefficient c 1 decreases with the increase in the number of iterations, and the social coefficient c 2 increases with the increase in the number of iterations. This ensures that individuals can fully roam the entire target space in the early stage of search, rather than spend a lot of energy clustering near the local optimal solution. In the later search stage, small cognitive coefficient and large social coefficient encourage the particles to converge to the global optimal state before the end of the search. The dynamical cognitive coefficient c 1 and social coefficient c 2 are expressed as [26] : c 1 = ( c 1 f ϖ c 1 i ) iter / iter max + c 1 i (3) c 2 = ( c 2 f ϖ c 2 i ) iter / iter max + c 2 i (4) where the cognitive coefficient c 1 decreases from c 1 i to c 1 f , and the social coefficient c 2 increases from c 2 i to c 2 f . c 1 i and c 2 f are set to 2.5, each, while c 1 f and c 2 i are set to 0.5, each. In addition, the position updating Eq. (2) is slightly modified as: x k + 1 jn = x jn + C v k + 1 jn (5) where C = 2/ (2 ϖ ϕ ϖ ϕ 2 ϖ 4 ϕ ) ; ϕ is set to 4.1. 4.3 Ȟ MPSO algorithm The MPSO is introduced in Ref. [19]. The dynamical cognitive coefficient c 1 and social coefficient c 2 are expressed as: c 1 = ( c 1 f ϖ c 1 i ) e ϖ 70 Ç È w k ϖ w min w max ϖ w min 6 + c 1 i (6) c 2 = ( c 2 f ϖ c 2 i ) e ϖ 70 Ç È w k ϖ w min w max ϖ w min 6 + c 2 i (7) where c 1 i , c 1 f , c 2 i , and c 2 f are in the range of [0, 4] and c 1 i > c 2 f > c 2 i > c 1 f . c 1 i and c 2 f are set to 1.7, each, while c 1 f and c 2 i are set to 1.3, each. In addition, a mutation rule is applied for each particle. If r 3 >0.9, then n =[ N·r 4 ] and x jn = x min +( x max ϖ x min ) r 5 . r 3 , r 4 , and r 5 are random values in the range of [0, 1]. ` 3.03/`ALGORITHM A common feature of the above improved PSO algorithm is to improve the convergence accuracy and speed by altering the cognitive and social coefficients, c 1 and c 2 ,
respectively. However, the influence of linear attenuation weight on the algorithm is not considered. For the linear attenuation weight, the change of w completely depends on the current and maximum number of iterations, which leads to slow convergence in the later stage of the iteration. Therefore, it is difficult to jump out of the local optimal solution. In this section, strange nonchaos is introduced into the PSO algorithm and the SNPSO algorithm is developed. Strange nonchaotic behavior is an unpredictable dynamical behavior in nonlinear systems. Mapping this behavior to some parameters in the algorithm will improve the performance of the algorithm. To obtain strange nonchaotic sequences, a strange nonchaotic system is described as follows [23] :
º ¼ ½ ½ ½ ½ v 1 y ( n ) + a + ε cos(2π φ ( n ) ), if y ( n ) Ш 0 v 2 y ( n ) + a + ε cos(2π φ ( n ) ) + 1, if y ( n ) > 0 »
(8)
y ( n + 1) =
φ ( n + 1) = φ ( n ) + w mod (1) (9) There are four unknown parameter values in Eq. (8), which is not conducive for determining strange nonchaotic sequences. Therefore, we first determine the values of parameters v 1 and v 2 via several experiments, reduce the number of unknown variables, and then determine strange nonchaotic sequences using the remaining variable parameters a and č . v 1 and v 2 are constant parameters and fixed at v 1 =0.7 and v 2 =1.2, a and č are bifurcation parameters, n is the number of iterations, w is the frequency and is fixed at w = ( 5 ϖ 1)/2 . Strange nonchaotic behavior is characterized by the phase diagram and largest Lyapunov exponent. The largest Lyapunov exponent ē max is calculated by: λ x = lim N͖ ∞ 1 N ϕ i = 1 n ln| φ f φ x i | (10) The bifurcation parameters a and č are fixed at a = 0.5965 and č =0.1. Fig. 4(a) illustrates the phase diagram of a fractal attractor, and the largest Lyapunov exponent equals ϖ 0.0014 (nonchaotic). Therefore, the fractal attractor is a strange nonchaotic attractor. The strange nonchaotic sequence has features of randomness and non-repetition as illustrated in Fig. 4(b). To obtain
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