1#. • Basis Weight Control System
` 3IMULATION`EXPERIMENTS This section aims to study the proposed SNPSO algorithm to address the identification problem of the actual process. Therefore, we choose the FOPDT and SOPDT as the estimation models to approximate the following real process models: P 1 ( s ) = 1 s + 1 e ϖ s (11) P 2 ( s ) = 1 ( s + 1) 2 e ϖ s (12) P 3 ( s ) = 1 ( s + 1) 8 (13) P 4 ( s ) = 9 ( s + 1) ( s 2 + 2 s + 9) (14) The inclusion of systems P 1 and P 2 is to confirm that when modeling FOPDT or SOPDT systems with FOPDT or SOPDT models, all the procedures realise an exact match. The identification performance of four algorithms (classical PSO, PSO-TVAC, MPSO, and the proposed SNPSO) is compared. The parameters of four algorithms are set as follows: the sampling time t =20 s; sampling interval ě t =0.01 s; input is step signal; additive noise is Gaussian white noise with a mean value of 0 and variance of 0.16. The parameters of the four algorithms are presented in Table 1. Matlab R2018a is adopted to perform the above design steps under the environment of Intel(R) Core(TM) i5-4210M CPU@2.60 GHz. In total, 10 independent operations are conducted for each system, and detailed result analysis is described as follows. 5BCMF Ȟ 1BSBNFUFSTPGDMBTTJDBM140 140 - 57"$ .1ź 40 BOE4/140 Methods Population Maxgen
follows:
K F T F s + 1
G F ( s ) = (15) where the parameters K F , T F , and L F are the estimation parameters. The classical PSO, PSO-TAVC, MPSO, and the proposed SNPSO are adopted to identify the selected estimation model. To verify the identification ability of the above algorithms, experiments are conducted at step response with zero noise, step response with Gaussian noise, and denoised step response. When the output ŷ ( t ) is a step response with zero noise ( n ( t )=0), Table 2 presents the identification accuracy, minimum (min), maximum (max), mean, and standard (std) deviation of the optimal loss function using the four methods ten times, respectively. The accuracy of model identification using the SNPSO algorithm is as high as 90%, which is much higher than the other three algorithms. Fig. 6 illustrates the parameters K F , T F , and L F of the estimation model obtained by running the four algorithms ten times, respectively. The classical PSO, PSO-TAVC, MPSO, and proposed SNPSO algorithms are represented by blue, red, green, and yellow star-lines. SNPSO has excellent average convergence and stability. Fig. 7 illustrates the variations for objective function J using four algorithms. The SNPSO method converges faster and more precisely than other PSO-based algorithms. e ϖ L F s To verify the robustness of the proposed algorithm, a Gaussian white noise n ( t ) with variance of 0.16 is added to the output ŷ ( t ) . Fig. 8 illustrates the parameters K F , T F , and L F of the estimation model obtained from ten experiments using four algorithms. The SNPSO ensures that the proportion of the distance | K F ϖ 1|<0.004 is 60%, distance | T F ϖ 1|<0.4 is 60%, and distance | L F ϖ 1|< 0.02 is 80%. The values of three parameters of the estimation model obtained by the other three algorithms 5BCMF Ȟ 4ZTUFNJEFOUJGJDBUJPOPG'015%VTJOHGPVS140 - CBTFEBMHPSJUINT Methods Classical PSO PSO-TVAC MPSO SNPSO Best parameter K F =1, T F =1, L F =1 K F =1, T F =1, L F =1 K F =1, T F =1, L F =1 K F =1, T F =1, L F =1 Accuracy/% 20 20 30 90 Min 0 0 0 0 Max 927 3296 1072 506 Mean 201 432 284 50 Std 300 1023 426 160
Other parameters c 1 =2 c 2 =2 c 1 i =2.5, c 1 f =0.5, c 2 i =0.5, c 2 f =2.5 c 1 i =1.7, c 1 f =1.3, c 2 i =1.3, c 2 f =1.7 c 1 i =1.7, c 1 f =1.3, c 2 i =1.3, c 2 f =1.7
Inertia weight w max =0.9 w min =0.4 w max =0.9 w min =0.4 w max =0.9 w min =0.4 w max =0.9 w min =0.4
Classical PSO
30
100
PSO-TVAC
30
100
MPSO
30
100
SNPSO
30
100
6.1 Ȟ Transfer function P 1 The real process is assumed to be the transfer function (Eq. (11)). The estimation model (FOPDT model) is as
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