## 2 Problem Description
### CFDL Model for WWTP  
Taking the DO concentration as an example, the nonlinear discrete system of the \(S_{O,5}\) in a WWTP can be expressed as:  
\[y(k+1)=f\big{(}y(k),y(k-1),...y\big{(}k-n_{y}\big{)},u(k),u(k-1),...,u(k-n_{u} )\big{)} \tag{3}\]  
where \(u(k)\in R\) represents the system input \(K_{La5}\), \(y(k)\in R\) indicates the system output \(S_{O,5}\), \(n_{u}\) and \(n_{y}\) indicate the input and output order of the system, and \(f(\cdot)\) represents a nonlinear function which is unknown. Based on the CFDL theory [25], reasonable assumptions are introduced for the WWTP system shown in Equation (2). **Assumption 1**. : _The partial derivative of \(f(\cdot)\) in (3) is continuous under the action of the control input \(K_{La5}\)._  
**Assumption 2**. : _The generalized Lipschitz condition is met by the WWTP system (3) that is, at any particular moment k, if \(\Delta u(k)\
eq 0\), then \(|y(k+1)-y(k)|\leqslant\mathsf{b}|\Delta u(k)|\), where the constant \(\mathsf{b}\) is positive._  
**Remark 1**. : _Indeed, the input-output change rate of a nonlinear control system for a WWTP is typically constrained by Assumption 1. A bounded input energy change should result in a bounded output energy change inside the system, since the WWTP complies with the rule of energy conservation, satisfying Assumptions 1 and 2._  
For a WWTP system satisfying the above assumptions, we have Lemma 1 as follows. \begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Definition** \\ \hline \(S_{I}\) & Soluble inert organic matter \\ \(S_{S}\) & Readily biodegradable substrate \\ \(X_{I}\) & Particulate inert organic matter \\ \(X_{S}\) & Slowly biodegradable substrate \\ \(X_{B,H}\) & Active heterotrophic biomass \\ \(X_{B,A}\) & Active autotrophic biomass \\ \(X_{P}\) & Particulate products arising from biomass decay \\ \(S_{O}\) & Oxygen \\ \(S_{NO}\) & Nitrate and nitrite nitrogen \\ \(S_{NH}\) & NH\({}_{4}\)\({}^{*}\) + NH\({}_{3}\) nitrogen \\ \(S_{ND}\) & Soluble biodegradable organic nitrogen \\ \(X_{ND}\) & Particulate biodegradable organic nitrogen \\ \(S_{ALK}\) & Alkalinity \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of ASM1 variables. **Lemma 1**. : _For the system (3) of \(S_{O,5}\) in a WWTP satisfying Assumptions 1 and 2, when \(|\Delta u(k)|\
eq 0\) at time \(k\), there is a time-varying parameter \(\phi_{c}(k)\in R\) referred to as the pseudo partial derivative (PPD), that system (2) can be converted into the following model:_  
\[\Delta y(k+1)=\phi_{c}(k)\Delta u(k) \tag{4}\]  
_where \(\phi_{c}(k)\) is bounded for any time \(k\)._  
For the unknown PPD \(\phi_{c}(k)\) in the data model (4) with dynamic linearized, an improved projection algorithm is used for estimation, considering the following PPD estimation criterion function:  
\[J(\phi_{c}(k))=|y(k)-y(k-1)-\phi_{c}(k)\Delta u(k-1)|^{2}+\mu\big{|}\phi_{c}(k )-\hat{\phi}_{c}(k-1)\big{|}^{2} \tag{5}\]  
where \(\mu>0\) is the weighting factor. Let \(\partial J/\partial\phi_{c}(k)=0\), then the estimation formula of \(\phi_{c}(k)\) can be computed in the following way:  
\[\hat{\phi}_{c}(k)=\hat{\phi}_{c}(k-1)+\frac{\eta\Delta u(k-1)\big{(}\Delta y( k)-\hat{\phi}_{c}(k-1)\Delta u(k-1)\big{)}}{\mu+\Delta u(k-1)^{2}} \tag{6}\]  
where \(\eta\in(0,1]\) denotes the weighting factor, which is aimed to make the algorithm more flexible and general. \(\hat{\phi}_{c}(k)\) represents the estimation of \(\phi_{c}(k)\). Due to the fact that the WWTP is a typical time-varying system with significant parameters change, in order to enhance the tracking ability of PPD estimation algorithm (6) for time-varying parameters, when \(\big{|}\hat{\phi}_{c}(k)\big{|}\leqslant\varepsilon\) or \(|\Delta u(k-1)|\leqslant\varepsilon\), make  
\[\hat{\phi}_{c}(k)=\hat{\phi}_{c}(1) \tag{7}\]  
where \(\varepsilon\) is a sufficiently small positive number usually taken as \(10^{-5}\), and \(\hat{\phi}_{c}(1)\) is the initial value of \(\hat{\phi}_{c}(k)\).