Collaborative-prediction-based recursive filtering for nonlinear systems with sensor saturation under duty cycle scheduling

Figures & data

Table 1. The IBCP Algorithm.

Algorithm IBCP
S1.	Calculate the similarity between item a and other items based on known item ratings by Equation (8(8) $\mathrm{S}\mathrm{i}\mathrm{m}(a,b)=\frac{\sum _{u\in U_{a,b}}(r_{u,a}-{\overline{r}}_a)(r_{u,b}-{\overline{r}}_b)}{\sqrt{\sum _{u\in U_{a,b}}(r_{u,a}-{\overline{r}}_a{)}^2}\sqrt{\sum _{u\in U_{a,b}}(r_{u,b}-{\overline{r}}_b{)}^2}},$(8) ).
S2.	Obtain the top-N recommended items that are most similar to item a using the top-N approach.
S3.	Compute the predicted values of item a by the target user based on the recommended items obtained in step S2 by Equation (9(9) ${\tilde{r}}_{m,a}={\overline{r}}_a+\frac{\sum _{b\in I_m}\mathrm{S}\mathrm{i}\mathrm{m}(a,b)(r_{m,b}-{\overline{r}}_b)}{\sum _{b\in I_m}|\mathrm{S}\mathrm{i}\mathrm{m}(a,b)|},$(9) ).

Table 2. The measured outputs under the DCS.

k	1	…	11	12	13	14	15	16	17	18	19	20
Sensor1($d_c^1=0.8$)	0.73	…	0.79	–	0.43	0.81	0.45	0.55	0.40	0	0.12	0.39
Sensor2($d_c^2=0.85$)	−0.31	…	0.43	0.81	–	–	–	0.38	0.49	0.94	0.58	0.57
k	21	22	23	24	25	26	27	28	29	30	…	40
Sensor1($d_c^1=0.8$)	−0.11	0.01	0.48	–	–	0.29	0.27	−0.07	0.19	−0.03	…	0.31
Sensor2($d_c^2=0.85$)	1.00	0.21	0.03	0.35	0.28	−0.09	0.20	–	–	–	…	−0.26

Figure 1. The actual output of y and its outputs based on ZOH and CP. ($d_c^1=0.8,d_c^2=0.85$). (a) The actual output of $y_1$ and its outputs based on ZOH and CP. and (b) The actual output of $y_2$ and its outputs based on ZOH and CP.

Figure 2. Actual states x and their estimates $\hat{x}$ based on CP ($d_c^1=0.8,d_c^2=0.85$). (a) Actual state $x_1$ and its estimate ${\hat{x}}_1$. and (b) Actual state $x_2$ and its estimate ${\hat{x}}_2$.

Figure 3. MSE on states based on ZOH and CP($d_c^1=0.8,d_c^2=0.85$). (a) MSE on state $x_1$ based on ZOH and CP. and (b) MSE on state $x_2$ based on ZOH and CP.

Figure 4. Actual states and their estimates based on CP ($d_c^1=0.78,d_c^2=0.87$). (a) Actual state $x_1$ and its estimate ${\hat{x}}_1$. and (b) Actual state $x_2$ and its estimate ${\hat{x}}_2$.