## 4 Simulation  
In this section, the simulation results of the particle filter with weight adjustment design for the robust nonlinear uncertain heat exchange process in Figure 2 will be given to demonstrate the effectiveness of the method. The simulation time is set as 2401 s, the sampling time is set as \(\Delta t=1\) s, the parameter values used are listed in Table 1, the number of particles is \(N=200\), and the time delay is assumed to be subject to Gaussian distribution. Because the effect of time delay on the control system has been verified in [14], it will not be repeated, and the effect of sensor noise and time delay will be shown comprehensively here. In the simulation, the method of generating the time delay and its distribution remain unchanged, and only the variance of sensor noise is changed to obtain  
Figure 4: Equivalent block-diagram of Figure 3.  
Figure 3: Equivalent block-diagram of Figure 2.  
sensor noise of different scales relative to that modeled in the measurement model in the particle filter. The following set of equations are considered an illustrative example:  
\[p(x_{t}\mid x_{t-1}) =\mathcal{N}\Big{(}x_{t};f(x_{t-1},u_{t}),\sigma_{v}^{2}\Big{)}\] \[p(y_{t}\mid x_{t}) =\mathcal{N}\Big{(}y_{t};h(x_{t},u_{t}),\sigma_{n}^{2}\Big{)} \tag{26}\] \[p(n_{s}) =\mathcal{N}\Big{(}\mu_{s},\sigma_{s}^{2}\Big{)},\]  
where \(\sigma_{v}^{2}\) and \(\sigma_{n}^{2}\) are variances of process noise and measurement noise modeled in the particle filter, respectively, and \(\sigma_{v}^{2}=0.001^{2}\) and \(\sigma_{n}^{2}=0.1^{2}\). \(f(\cdot)\) and \(h(\cdot)\) are the process model and measurement model without noise and the same as in Equation (1). \(\mu_{s}\) and \(\sigma_{s}^{2}\) are the mean and variance of sensor noise, and \(\mu_{s}=0\) and \(\sigma_{s}^{2}\) is set as different values. The mean of process noise and measurement noise modeled in the particle filter is 0. Figure 5 shows the result under sensor noise (\(\sigma_{s}^{2}=0.3^{2}\)) and time delay without particle filter, where Figure 5a shows the system output \(T_{co}\), the target value \(r_{2}\) and the output affected by sensor noise and time delay, Figure 5b shows the sensor noise imposed on the output. The system states \(T_{ho}\), \(T_{hi}\) and system output \(T_{co}\) are also shown in Figure 5b, where since the value of hot fluid input \(T_{hi}\) is a constant, it is not displayed. It can be observed that the control performance is not satisfactory due to sensor noise and time delay. There is an oscillation in both output and input values, and there is even a tendency to deviate from the target value after about 1550 s.  
Figure 6 shows the result using a generic particle filter to reduce the effect of noise and time delay with the same noise variance as in Figure 5. It is clear that the particle filter does improve the control performance under noise and time delay. Figure 6a shows the output and input results as illustrated in Figure 5a, and it can be seen that the stability and the tracking performance can be guaranteed. Figure 6b shows the sensor noise imposed on the system output, the error between states estimated using GPF and that from the system (\(T_{ho}\), error and \(T_{ci}\) error), and the error between output predicted using GPF and that from the system (\(T_{co}\) error). From Figure 6b, the variance of the error between the states output from GPF and that from the system has been reduced compared with the given sensor noise, which demonstrates that the influence of sensor noise has decreased. Figure 6c shows the states and output constructed from particle filter and that from the system, which is not imposed sensor noise and time delay, where the states and output using GPF is very close to that of the system. Figure 5: Control performance is not satisfactorywithout particle filter under sensor noise (\(\sigma_{s}^{2}=0.3^{2}\)) and time delay: (**a**) output and input; (**b**) sensor noise, states, and output.