## 4 Simulation
Figures 7 and 8 display the results when sensor noise is increased, where variance grows from \(\sigma_{s}^{2}=0.3^{2}\) to \(\sigma_{s}^{2}=0.7^{2}\), and further increased relative to the measurement noise modeled in the particle filter. The system can guarantee the stability and tracking performance with the incorporation of a generic particle filter, as shown in Figure 8, while divergence occurs in the system after about 1000 s without a particle filter, as shown in Figure 7. Compared with the results with and without a particle filter, it is clear that a particle filter does reduce the effect of sensor noise and time delay on the control performance. However, when the sensor noise becomes larger to a certain extent (\(\sigma_{s}^{2}=1.5^{2}\)) relative to the measurement noise modeled in the particle filter (\(\sigma_{n}^{2}=0.1^{2}\)), the generic particle filter cannot work normally. According to the analysis in Sections 3.2 and 3.3, this is because the measurement noise pushes nearly all the sampled particles to low likelihoods, which will be removed in resampling and cause particle degeneracy. The particle weights can become extremely small, even exceeding the limit of storage under the reality that the amount of calculation storage is limited; thus, the state cannot be estimated and the particle filter algorithm collapses, which drags the system down to discontinuity, as shown in Figure 3.0 Note that because the noise is generated independently of the control system, it is not consistent with the states, the input and output of the control system in terms of time, as in Figure 9b. Figure 3.1 Control performance is improved with generic particle filter under sensor noise (\(\sigma_{s}^{2}=0.3^{2}\)) and time delay: (**a**) output and input; (**b**) sensor noise and errors; (**c**) states and output. Figure 3.2 Control performance is increasingly unsatisfactory without particle filter as the sensor noise increases (\(\sigma_{s}^{2}=0.7^{2}\)): (**a**) output and input; (**b**) sensor noise, states, and output. Particle weight is adjusted according to Algorithm 1. The proportion of particles that violate the \(3\sigma\) rule to judge if weight adjustment needs to be employed is set as \(prop=70\%\). Figure 10 shows the effectiveness of the weight adjusted particle filter in the reduction in the large noise impact. It is clear that after adopting the particle filter with weight adjustment algorithm, the continuous operation, stability and tracking performance of the system are all guaranteed. Figure 3.3 Control performance can still be guaranteed due to the use of a particle filter to reduce the impact of noise even as the sensor noise is increased (\(\sigma_{s}^{2}=0.7^{2}\)): (**a**) output and input; (**b**) sensor noise and errors; (**c**) states and output. Figure 3.4 Particle filter with weight adjustment algorithm can still work normally even when the sensor noise increases to a certain extent relative to the modeled one. Further, the stability and tracking performance of the overall control system incorporated with particle filter with weight adjustment is guaranteed (\(\sigma_{s}^{2}=1.5^{2}\)): (**a**) output and input; (**b**) sensor noise and errors; (**c**) states and output. Figure 3.5 When the sensor noise is increasing to a certain extent, the particle filter algorithm cannot work normally. Further, the overall control system incorporated with PF is interrupted (\(\sigma_{s}^{2}=1.5^{2}\)): (**a**) output and input; (**b**) sensor noise and errors; (**c**) states and output. Figure 11 shows that when the variance of sensor noise grows (\(\sigma_{\rm s}^{2}=1.7^{2}\)), due to the influence of large sensor noise, the generic particle filter algorithm loses its function and the operation is interrupted, then the whole system is implicated.