## 3 Results
#### Identification of the Model Components  
A wealth of methods has been proposed and reported to ease the identification of electrochemical processes from frequency response data, ranging from graphical analysis of Nyquist [35] and Bode diagrams [36] to multidimensional analysis [37] and software solutions [38]. There are indications that, over the last two decades [39], the trend in electrochemical sensing [40], measurement [41; 42] and analysis has been directed towards real-time EIS analysis of dynamic systems. For instance, [PERSON] et al. [43] present an Impedance Real-Time Monitoring (IRTM) procedure for estimating Randles cell equivalent electrical circuits (EEC) of a polymer electrolyte fuel cell. [PERSON] et al. use five impedance measurements to obtain three sets of EEC parameters [43]; the authors reduce the time required for data acquisition and analysis by averaging the calculated EEC parameters, and claim that the IRTM procedure yields accuracies close to those obtained using iterative fitting procedures. The method presented here is directed towards obtaining a mathematically-derived indication that the EEC parameters represent the state-space model. After the state-space model has been identified as a feasible state-space representation, the next step is to identify which component correlates to a particular electrochemical process. As a departure point, consider the transfer function of the resulting Trimethoprim/Sulfamethoxazole model in the frequency range 20 Hz-30 MHz:  
\[G_{TRIM}(s)=\frac{1.765\,\times 10^{-5}\,s^{2}+135.8\,s+704.3}{3.755\,\times 10^{ -6}s^{2}+1.353\,s+1} \tag{12}\]  
which can be written in terms of DC Gain, poles and zeros as:  
\[G_{TRIM}(s)=DC\;Gain\left(\frac{\left(\frac{s}{Z_{1}}+1\right)\left(\frac{s}{ Z_{2}}+1\right)}{\left(\frac{s}{P_{1}}+1\right)\left(\frac{s}{P_{2}}+1\right)}\right) \tag{13}\]  
\[G_{TRIM}(s)=3.1357\,\times 10^{-2}\left(\frac{\left(\frac{s}{0.19286}+1\right) \left(\frac{s}{1.30\,\times 10^{7}}+1\right)}{\left(\frac{s}{1.35281}+1 \right)\left(\frac{s}{2.78\times 10^{6}}+1\right)}\right) \tag{14}\]  
The bode plot is the resulting effect (sum of the frequency response magnitude and frequency) of the five terms: DC Gain, Poles (\(P_{1}\), \(P_{2}\)) and Zeroes (\(Z_{1}\), \(Z_{2}\)). Figure 10 shows the resulting frequency response of each individual terms as well as that of the resulting bode plot of the complete model. In order to ease the explanation of how each component relates to a particular electrochemical process, consider the Nyquist plot of the Trimethoprim/Sulfamethoxazole model (Figure 11). In Figure 11, from left to right, the distance between the origin and the first semicircle corresponds to the bulk resistance, R\({}_{0}\), consistent with the medium resistivity for the test conditions. The first semicircle from suggests that there is interfacial resistance originated by decomposition of the electrolyte, and corresponds to the effect of R\({}_{1}\) and C\({}_{1}\), whereas the second semicircle represents the charge transfer resistance and double-layer capacitance (R\({}_{2}\), C\({}_{2}\)). For all drugs studied, increasing the number of parameters appears to yield an improved approximation. In contrast, the results indicate that a large number of equivalent electrical cells does not lead to a minimal realization of the state-space model. Thus, a five-element equivalent impedance model yields a reduced fit but corresponds to a minimal state-space realization. As for diffusion processes, which may drive the frequency response at low frequencies, a reduced number of equivalent model components seems to represent the process. In contrast, in ambroxol, there appears to be a physicochemical process, which may be described as a CPE that extends at higher frequencies; although a seven-parameter model appears to yield good approximation, the analysis of singular values indicates an uncontrollable state of the fitting model and thus is not feasible linear time-invariant representation.