## 3 Particle Filter with Weight Adjustment Design for the Compensation of Sensor Noise and Time Delay
### Operator Based Control System Design  
Figure 1 shows the overall control system design scheme, which is based on operator theory, and its BIBO stability is ensured using Bezout identity, where \(u\in U,y\in Y\), \(w\in W\) are the input and output of the plant \(P(U\to Y)\) and the output of a space change operator, respectively, and \(r_{2}\) is the reference input. \(P=ND^{-1}\) and \(P+\Delta P=(N+\Delta N)D^{-1}\) are the nominal plant and uncertain plant with right factorization, respectively, where \(N\) is stable, \(D\) is stable and invertible, and \(\Delta N\) is unknown. Based on Bezout identity in Equation (8), if Equation (9) holds, where \(M\) and \(\tilde{M}\) are unimodular operators, then the  
nonlinear feedback control system with uncertainty is robust stable and the stabilizing operators \(S\) and \(R\) are designed as in Equation (10). \[SN+RD=M \tag{8}\]  
\[\left\{ \begin{array}{l}S(N+\Delta N)+RD=\widetilde{M}\\ \left\|(S(N+\Delta N)-AN)\widetilde{M}^{-1}\right\|<1\end{array} \right. \tag{9}\]  
\[ \begin{array}{l}N:Y\to U\\ b(t)=\frac{1-K}{\frac{A_{1}+A_{5}}{\ln[\frac{T_{\mathrm{b}}(t)-T_{\mathrm{d}}(t )}{T_{\mathrm{h}}i^{-}L_{\mathrm{d}}(t)}]-\frac{A_{4}}{4}\ln[\frac{T_{\mathrm{ h}}i^{-}T_{\mathrm{d}}(t)}{T_{\mathrm{h}}i^{-}L_{\mathrm{d}}(0)}]}-A_{3}}\\ R:U\to U\\ e(t)=\frac{K}{A_{2}}u(t)\end{array}  \tag{10}\]  
where \(K\) is the design parameter, and  
\[ \begin{array}{ll}A_{1}=\frac{2a(\xi\pi)^{3}}{3\lambda}&A_{2}=-\frac{R_{e}P_ {e}A}{N_{e}\varphi_{e}\rho}\quad A_{3}=\frac{\xi}{\lambda}+\frac{1}{h_{c}}\\ A_{4}=\frac{a^{2}(\xi\pi)^{4}\varphi_{e}\rho}{4\lambda}&A_{5}=\frac{a^{2}(\xi \pi)^{4}}{2d_{A}\lambda}\\ N_{\mu}=0.023R_{e}^{0.8}p_{e}^{0.3}&h_{c}=\frac{c_{p}\rho L_{c}N_{\mu}}{R_{e}P_ {e}\lambda}.\end{array}  \tag{11}\]  
where the parameters can be found in Table 1.  
The tracking controller \(C\) is  
\[C(\epsilon_{2})(t)=K_{a}\epsilon_{2}(t)+K_{\beta}\int_{0}^{t}e_{2}(\tau)d\tau. \tag{12}\]  
The above is the robust nonlinear uncertain heat exchange control system design that is not subject to sensor noise and wireless communication delay. Considering sensor noise and time delay, in order to not redesign the controller, the particle filter algorithm with a weight adjustment is incorporated into the system to estimate the state and predict the output from the measurement that is corrupted by sensor noise and time delay. Particle Filter with Weight Adjustment Design to Compensate the Influence of Sensor Noise and Time Delay  
Considering the sensor noise and time delay existing in the system output, the particle filter is put into an uncertain heat exchange control system for state estimation and further output prediction to avoid the reconfiguration of controller. The estimated states and the predicted output values calculated using the particle filter that have mitigated the influence of noise and time delay will be substituted for the corrupted output picked up from sensors and passed through wireless communication to be fed into the feedback loop. Figure 2  
Figure 1: Overall control system design based on operator theory. shows the particle filter with the weight adjustment design scheme of an uncertain heat exchange control system, where GPF is a generic particle filter, WAPF is the particle filter with the weight adjustment, \(\Phi\) is a time-varying time delay component due to the use of wireless communication, \(y\) is the system output and regarded as the measurement value that has not been affected by sensor noise and time delay, \(\overline{y}\) is the output with a random noise and time delay, \(\hat{x}\) and \(\hat{y}\) are the states and output reconstructed by the particle filter from the impaired signal, which will then be conveyed to operator \(S\) as the feedback signal of the system. _Noise_ in this paper is additive Gaussian noise.