Interval type-2 fuzzy neural network based constrained GPC for NH$_{3}$ flow in SCR de-NO$_{x}$ process

Abstract

To overcome the difficulty of controlling ammonia (NH$_{3}$) flow in the selective catalytic reduction (SCR) nitrogen oxides (NO$_{x}$) decomposition (de-NO$_{x}$) process with time delay, modeling uncertainties and time-varying parameters, a constrained generalized predictive control (GPC) based on interval type-2 fuzzy neural network (IT2FNN) is proposed in this paper. First, the proportional-integral (PI) controller cannot solve the time delay in the SCR de-NO$_x$ process due to long de-NO$_x$ reaction time, therefore this paper proposes a constrained GPC controller to predict the multistep outlet NO$_x$ concentration, where the predictive time domain is greater than the time delay. Second, an accurate process model used in GPC plays an important role in NO$_{x}$ control. Thus, this paper designs a novel IT2FNN model as the SCR de-NO$_{x}$ process model. IT2FNN which adopts interval type-2 fuzzy set (IT2FS) could deal with the modeling uncertainties owing to catalyst activity, uniformity of flue gas and other factors. Meanwhile, to cope with the time-varying parameters because of the fluctuation of load, the parameters of the proposed IT2FNN are updated by the derived algorithms in real time. For reducing computational complexity, this paper adopts the Nie–Tan (NT)-type reduction (TR) operation instead of the Karnik–Mendel (KM) method. Third, under the proposed control scheme, it is theoretically proved that the SCR de-NO$_{x}$ system is stable. Finally, the comparative simulations are given to demonstrate the effectiveness and superiorities of the proposed method.

Appendix A

The chain rule is adopted to compute the derivative terms of ${\partial {\hat{r}}_{out}(k)}/{\partial m_{j}^{i}(k)}$, ${\partial {\hat{r}}_{out}(k)}/{\partial \sigma_{j}^{i}(k)}$, ${\partial {\hat{r}}_{out}(k)}/{\partial \delta_{j}^{i}(k)}$:

$$\begin{aligned} \begin{aligned} \frac{\partial {\hat{r}}_{out}(k)}{\partial m_{j}^{i}(k)}=&\frac{\partial {\hat{r}}_{out}(k)}{\partial {\overline{f}}^{i}}\frac{\partial {\overline{f}}^{i}}{\partial \overline{\mu }_{{\tilde{F}}_{j}^{i}}(x_{j})}\frac{\partial \overline{\mu }_{{\tilde{F}}_{j}^{i}}(x_{j})}{\partial m_{j}^{i}(k)}+\frac{\partial {\hat{r}}_{out}(k)}{\partial {\underline{f}}^{i}}\frac{\partial {\underline{f}}^{i}}{\partial {\underline{\mu }}_{{\tilde{F}}_{j}^{i}}(x_{j})}\frac{\partial {\underline{\mu }}_{{\tilde{F}}_{j}^{i}}(x_{j})}{\partial m_{j}^{i}(k)}\\ =&\frac{{\hat{r}}_{out}^{i}(k)\left[ \sum_{i=1}^{M}({\overline{f}}^{i}+{\underline{f}}^{i})\right] -\sum_{i=1}^{M}\left[ ({\overline{f}}^{i}+{\underline{f}}^{i}){\hat{r}}_{out}^{i}(k)\right] }{\left( \sum_{i=1}^{M}({\overline{f}}^{i}+{\underline{f}}^{i})\right) ^2}{\overline{f}}^{i}\frac{(x_{j}-m_{j}^{i}(k))}{(\sigma_{j}^{i}(k)+\delta_{j}^{i}(k))^2}\\&+\frac{{\hat{r}}_{out}^{i}(k)\left[ \sum_{i=1}^{M}({\overline{f}}^{i}+{\underline{f}}^{i})\right] -\sum_{i=1}^{M}\left[ ({\overline{f}}^{i}+{\underline{f}}^{i}){\hat{r}}_{out}^{i}(k)\right] }{\left( \sum_{i=1}^{M}({\overline{f}}^{i}+{\underline{f}}^{i})\right) ^2}{\underline{f}}^{i}\frac{(x_{j}-m_{j}^{i}(k))}{(\sigma_{j}^{i}(k)-\delta_{j}^{i}(k))^2}\\ =&\frac{{\hat{r}}_{out}^{i}(k)-{\hat{r}}_{out}(k)}{\sum_{i=1}^{M}({\overline{f}}^{i}+{\underline{f}}^{i})}\left( \frac{{\overline{f}}^{i}(x_{j}-m_{j}^{i}(k))}{(\sigma_{j}^{i}(k)+\delta_{j}^{i}(k))^2}+\frac{{\underline{f}}^{i}(x_{j}-m_{j}^{i}(k))}{(\sigma_{j}^{i}(k)-\delta_{j}^{i}(k))^2}\right) \\ \end{aligned} \end{aligned}$$

(58)

$$\begin{aligned} \begin{aligned} \frac{\partial {\hat{r}}_{out}(k)}{\partial \sigma_{j}^{i}(k)}&=\frac{\partial {\hat{r}}_{out}(k)}{\partial {\overline{f}}^{i}}\frac{\partial {\overline{f}}^{i}}{\partial \overline{\mu }_{{\tilde{F}}_{j}^{i}}(x_{j})}\frac{\partial \overline{\mu }_{{\tilde{F}}_{j}^{i}}(x_{j})}{\partial \sigma_{j}^{i}(k)}+\frac{\partial {\hat{r}}_{out}(k)}{\partial {\underline{f}}^{i}}\frac{\partial {\underline{f}}^{i}}{\partial {\underline{\mu }}_{{\tilde{F}}_{j}^{i}}(x_{j})}\frac{\partial {\underline{\mu }}_{{\tilde{F}}_{j}^{i}}(x_{j})}{\partial \sigma_{j}^{i}(k)}\\ &=\frac{{\hat{r}}_{out}^{i}(k)\left[ \sum_{i=1}^{M}({\overline{f}}^{i}+{\underline{f}}^{i})\right] -\sum_{i=1}^{M}\left[ ({\overline{f}}^{i}+{\underline{f}}^{i}){\hat{r}}_{out}^{i}(k)\right] }{\left( \sum_{i=1}^{M}({\overline{f}}^{i}+{\underline{f}}^{i})\right) ^2}{\overline{f}}^{i}\frac{(x_{j}-m_{j}^{i}(k))^2}{(\sigma_{j}^{i}(k)+\delta_{j}^{i}(k))^3}\\ \quad \quad \quad \quad &\quad+\frac{{\hat{r}}_{out}^{i}(k)\left[ \sum_{i=1}^{M}({\overline{f}}^{i}+{\underline{f}}^{i})\right] -\sum_{i=1}^{M}\left[ ({\overline{f}}^{i}+{\underline{f}}^{i}){\hat{r}}_{out}^{i}(k)\right] }{\left( \sum_{i=1}^{M}({\overline{f}}^{i}+{\underline{f}}^{i})\right) ^2}{\underline{f}}^{i}\frac{(x_{j}-m_{j}^{i}(k))^2}{(\sigma_{j}^{i}(k)-\delta_{j}^{i}(k))^3}\\ \quad \quad \quad \quad \ &=\frac{{\hat{r}}_{out}^{i}(k)-{\hat{r}}_{out}(k)}{\sum_{i=1}^{M}({\overline{f}}^{i}+{\underline{f}}^{i})}\left( \frac{{\overline{f}}^{i}(x_{j}-m_{j}^{i}(k))^2}{(\sigma_{j}^{i}(k)+\delta_{j}^{i}(k))^3}+\frac{{\underline{f}}^{i}(x_{j}-m_{j}^{i}(k))^2}{(\sigma_{j}^{i}(k)-\delta_{j}^{i}(k))^3}\right) \\ \end{aligned} \end{aligned}$$

(59)

$$\begin{aligned} \begin{aligned} \frac{\partial {\hat{r}}_{out}(k)}{\partial \delta_{j}^{i}(k)}=&\frac{\partial {\hat{r}}_{out}(k)}{\partial {\overline{f}}^{i}}\frac{\partial {\overline{f}}^{i}}{\partial \overline{\mu }_{{\tilde{F}}_{j}^{i}}(x_{j})}\frac{\partial \overline{\mu }_{{\tilde{F}}_{j}^{i}}(x_{j})}{\partial \delta_{j}^{i}(k)}+\frac{\partial {\hat{r}}_{out}(k)}{\partial {\underline{f}}^{i}}\frac{\partial {\underline{f}}^{i}}{\partial {\underline{\mu }}_{{\tilde{F}}_{j}^{i}}(x_{j})}\frac{\partial {\underline{\mu }}_{{\tilde{F}}_{j}^{i}}(x_{j})}{\partial \delta_{j}^{i}(k)}\\ =&\frac{{\hat{r}}_{out}^{i}(k)\left[ \sum_{i=1}^{M}({\overline{f}}^{i}+{\underline{f}}^{i})\right] -\sum_{i=1}^{M}\left[ ({\overline{f}}^{i}+{\underline{f}}^{i}){\hat{r}}_{out}^{i}(k)\right] }{\left( \sum_{i=1}^{M}({\overline{f}}^{i}+{\underline{f}}^{i})\right) ^2}{\overline{f}}^{i}\frac{(x_{j}-m_{j}^{i}(k))^2}{(\sigma_{j}^{i}(k)+\delta_{j}^{i}(k))^3}\\&+\frac{{\hat{r}}_{out}^{i}(k)\left[ \sum_{i=1}^{M}({\overline{f}}^{i}+{\underline{f}}^{i})\right] -\sum_{i=1}^{M}\left[ ({\overline{f}}^{i}+{\underline{f}}^{i}){\hat{r}}_{out}^{i}(k)\right] }{\left( \sum_{i=1}^{M}({\overline{f}}^{i}+{\underline{f}}^{i})\right) ^2}{\underline{f}}^{i}\frac{-(x_{j}-m_{j}^{i}(k))^2}{(\sigma_{j}^{i}(k)-\delta_{j}^{i}(k))^3}\\ =&\frac{{\hat{r}}_{out}^{i}(k)-{\hat{r}}_{out}(k)}{\sum_{i=1}^{M}({\overline{f}}^{i}+{\underline{f}}^{i})}\left( \frac{{\overline{f}}^{i}(x_{j}-m_{j}^{i}(k))^2}{(\sigma_{j}^{i}(k)+\delta_{j}^{i}(k))^3}-\frac{{\underline{f}}^{i}(x_{j}-m_{j}^{i}(k))^2}{(\sigma_{j}^{i}(k)-\delta_{j}^{i}(k))^3}\right) \\ \end{aligned} \end{aligned}$$

(60)