# Optimal Control of an Ultraviolet Water Disinfection System
## 2 A Disinfection UV System
### The Dynamic Model of the UV Disinfection System
#### 2.2.2 Bacterial Abatement Calculation  
Well-known disinfection kinetic models have been proposed or analyzed in many papers (see [8; 9; 22]) to model the germicidal action of UV light. Numerous papers have used simple kinetic models to calculate the bacterial abatement defined below:  
\[A(t)\ =\ \log_{10}\frac{N_{0}}{N(t)}, \tag{5}\]  
\(N_{0}\) and \(N(t)\) are, respectively, the concentrations of viable organisms (expressed in cross flow ultra filtration/100 mL) before and after the germicidal action of UV light. Instead of using a disinfection process model to calculate \(A(t_{\text{exp}})\), we considered and improved a dynamic model for the particular UV system presented in [5]. This paper describes a series of experimental measurements that yielded data concerning the dynamic behavior of the UVDS under consideration. Starting from this data, we propose a nonlinear model whose output is \(A(t)\) and input variables are \(I(t)\) and \(Q(t)\). Using an identification procedure that exploited the stored measurements, the bacterial abatement is expressed according to a linear dependence (see the linear system of Figure 3). If we denote the Laplace transforms of \(I(t)\), \(Q(t)\), and \(A(t)\) by \(I(s)\), \(Q(s)\), and \(A(s)\), respectively, it holds that:  
\[A(s)\ =\ T_{I}(s)\cdot I(s)\ +\ T_{Q}(s)\cdot Q(s). \tag{6}\]  
\(T_{I}(s)\) and \(T_{Q}(s)\) are the transfer functions of the two control channels. In the sequel, we adopt the two transfer functions obtained by system identification:  
\[T_{I}(s)\ =\ \frac{k_{I}\cdot\omega_{n}^{2}}{s^{2}\ +\ 2\xi\omega_{n}s\ +\ \omega_{n}^{2}};\ T_{Q}(s)\ =\ \frac{k_{Q}}{1\ +\ \tau s}. \tag{7}\]  
The values of \(I(t)\) and \(Q(t)\) meet the following constraints:  
\[I_{\text{min}}\ \leq\ I(t)\ \leq\ I_{\text{max}};\ Q_{\text{min}}\ \leq\ Q(t)\ \leq\ Q_{\text{max}} \tag{8}\]  
The parameters \(k_{I},\ k_{Q},\ \xi,\ \tau,\omega_{n}\ I_{\text{min}},\ I_{\text{max}},\ Q_{ \text{min}},\ Q_{\text{max}}\) have the usual definitions, with values given in Appendix A. Considering Remark 3, the output \(A(t)\) has a saturated evolution at the moment \(t_{\text{exp}}\). Therefore, our goal is to construct a nonlinear model that calculates the value \(A(t)\). At the same time, according to Remark 2, this model will be devoted only to control inputs with constant values \(I(t)=I_{0}\) and \(Q(t)=Q_{0}\) over the simulation interval. This model's construction is based on four considerations:  
* The value \(A(t)\) has two terms, like in Equation (6), corresponding to the two inputs. * The dynamic behavior of the two terms is retained, i.e., we retain the values of the poles for the two transfer functions from Equation (7). * We consider different static gains for each pair of values (\(I_{0}\), \(Q_{0}\)) as a direct consequence of Remark 3. These static gains are \(k_{1}(I_{0},Q_{0})\) and \(k_{2}(I_{0},Q_{0})\), which are different from \(k_{I}\) and \(k_{Q}\). * The static gains are determined using experimental measurements. This is based on data extracted from the characteristic curves presented in paper [5] and reproduced in Appendix A.  
Figure 3: The computation of the bacterial abatement. Applying these considerations, Equation (6) takes the nonlinear form (9), which expresses the Laplace transform of the response to the step variation of the two inputs:  
\[A(s,I_{0},Q_{0})\ =\ k_{1}(I_{0},Q_{0})\cdot T_{1}(s)\cdot\frac{1}{s}\ +\ k_{2}(I_{0},Q_{0})\cdot T_{2}(s) \cdot\frac{1}{s}, \tag{9}\]  
where  
\[T_{1}(s)\ =\ \frac{\omega_{n}^{2}}{s^{2}\ +\ 2\xi\omega_{n}s\ +\ \omega_{n}^{2}};\ T _{2}(s)\ =\ \frac{1}{1\ +\ \tau s} \tag{10}\]  
and  
\[I_{\min}\ \leq\ I_{0}\ \leq\ I_{\max};\ Q_{\min}\ \leq\ Q_{0}\ \leq\ Q_{\max}\]  
If the gain values \(k_{1}(I_{0},Q_{0})\), \(k_{2}(I_{0},Q_{0})\) are known, the bacterial abatement \(A(I_{0},\,Q_{0})\) can be calculated by integrating Equation (9).