Identification of a nonlinear rational model based on bias compensated multi-innovation stochastic gradient algorithm

Figures & data

Figure 1. Block diagram of an ARX-NRM.

Table 1. Computational costs of the SG, MI-SG, BC-MI-SG, and RLS algorithms.

	SG	MI-SG	BC-MI-SG	RLS
Computation cost	$6n+2$	$5n+np+2$	$5n+np+n_a+5$	$4n^2+6n$
$n=10,p=3,$
$n_a=5$	62	82	90	460
$n=100,p=3,$
$n_a=50$	602	802	855	40,600

Figure 2. Curves of the observed data.

Figure 3. Estimation errors using the SG, MI-SG and BC-MI-SG algorithms.

Table 2. Estimates using the SG, MI-SG, RLS and BC-MI-SG algorithms.

Algorithm		50	100	200	400	600	True
SG	$a_1$	0.3565	0.5624	0.7392	0.8683	0.9116	1.0000
	$a_2$	0.0027	0.0085	0.0268	0.0984	0.1741	1.0000
	$b_1$	0.3044	0.3526	0.4077	0.4083	0.4203	0.5000
	$b_2$	0.6456	0.7483	0.8694	0.8782	0.9116	1.0000
	$\delta (\mathrm{\%})$	69.5595	62.2554	56.5903	51.2437	46.5461
MI-SG	$a_1$	0.8569	0.9600	1.1180	0.9686	0.9962	1.0000
	$a_2$	0.3042	0.3439	0.7279	0.8385	0.8885	1.0000
	$b_1$	0.4545	0.4401	0.4617	0.4793	0.4897	0.5000
	$b_2$	0.9159	0.9551	1.1138	0.9904	0.9842	1.0000
	$\delta (\mathrm{\%})$	39.7595	36.6965	17.7479	9.2127	6.2787
RLS	$a_1$	0.9816	0.9803	0.9994	0.9760	0.9802	1.0000
	$a_2$	0.8819	0.9111	0.9975	0.9494	0.9291	1.0000
	$b_1$	0.4808	0.4951	0.5008	0.4907	0.4898	0.5000
	$b_2$	0.9758	0.9799	0.9970	0.9812	0.9842	1.0000
	$\delta (\mathrm{\%})$	6.8482	5.1813	0.2234	3.3156	4.2167
BC-MI-SG	$a_1$	0.8227	0.9661	1.0815	1.0023	1.0045	1.0000
	$a_2$	0.2633	0.3063	0.9366	0.9834	1.0012	1.0000
	$b_1$	0.4503	0.4358	0.4804	0.4912	0.4994	0.5000
	$b_2$	0.8948	0.9599	1.1108	1.0094	1.0007	1.0000
	$\delta (\mathrm{\%})$	42.5233	38.7559	8.4748	1.1752	0.2646

Figure 4. Estimation errors using BC-MI-SG with different noise variances.

Figure 5. Curve of the propylene catalytic oxidation data.

Table 3. Results using the SG, MI-SG and BC-MI-SG algorithms for the propylene catalytic oxidation data.

Algorithm	$\hat{a}(N)(\times {10}^{-3})$	$\hat{b}(N)(\times {10}^{-6})$	$\delta (N)(\mathrm{\%})$
SG	1.7332	−234.4738	231.4576
MI-SG	193.5557	640.9437	28.1679
BC-MI-SG	229.9675	727.8740	0.4642
TrueValue	231.0000	728.0000

Data availability statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.