## 3 Particle Filter with Weight Adjustment Design for the Compensation of Sensor Noise and Time Delay
### Operator Based Control System Design
Inspired by the motivation of the auxiliary variable particle filter [28], where the most promising particles (high prediction likelihood) at time \(t-1\) are expanded to time \(t\), thereby merging the information of posterior probability to generate sampled particles, the posterior information at the previous time instant \(y_{T_{co},t-1}\) is integrated into the particle sampling procedure at time \(t\). Furthermore, the influence of the posterior information at time \(t-1\) on the sampled particles is applied with an appropriate proportion. In this work, the proportion is set as \((m+\Delta m)/(m+\Delta m+M)\), which is exactly in line with the physical laws of reality. Taking the time delay element into account yields  
\[z_{t}=\Phi(\bar{y}_{t})=\bar{y}(t-d), \tag{15}\]  
where \(d\) represents the time delay. Please refer to [14] for the method of generating the time delay. If the sensor noise (denoted here as \(m\)) is put into the expression of the output from sensors and flowing through the wireless path, then the signal received by the particle filter is  
Figure 2:0  
which is exactly the expression of the two interference items in the signal input to the particle filter algorithm. The particle filter is adopted for processing the signal subject to noise and time delay to acquire the approximation of system states and further the system output that reduced the impact of the above two factors. The residuals between the time-delayed output due to the wireless communication and undelayed particles contribute to the particle weight update. Therefore, the processing of the measurement values with noise and time delay is consistent with that of the general particle filter. That is, the two interference items are dealt with comprehensively in this work. During the control process incorporated with a generic particle filter, it is found that large noise appearing in the measurement makes the posterior calculation less accurate. It is because measurements that are realistic given the high amount of noise obtain very low likelihoods, the measurement model judges such amounts of noise as highly unlikely, and the associated particles will be removed during the resampling step [21]. In detail, the prediction pdf of each particle is updated based on the residual \(|res^{i}_{t}|\) shown in Equation (17). Higher residuals result in a lower likelihood, meaning that the sampled particle is less likely to represent the true state and vice versa under the situation where the measurement is not polluted by actual large noise. However, when the actual measurement noise is much larger than the noise modeled in the measurement model of the particle filter, the residual will also become larger, resulting in a small particle weight. This will cause the state estimation in a generic particle filter to be no longer applicable, and the traditional resampling method to be meaningless. More extreme, if almost all particle weights are becoming smaller to a certain extent, system control incorporated with a particle filter cannot proceed normally, which will be shown in the simulation in Section 4. Figure 2:1  
For coping with the unsuitable particle weights for state estimation and subsequent resampling owing to the actual given large noise, it is a reasonable consideration to adopt a particle weight adjustment scheme. Precisely, when the realistic given system output \(\overline{y}_{t}\) measured by sensors and transmitted through the wireless communication path is available, the residual \(|\textit{res}_{t}^{i}|\), which represents the difference between the actual measurement and sampled particles, is first calculated and the \(3\sigma\) rule is exploited to determine whether \(|\textit{res}_{t}^{i}|\) is large so that the particle weight is pushed to a low likelihood. A small particle weight may indicate that the current measurement is affected by realistic large noises, but this judgment from only one particle is not absolutely accurate.