## 1 Introduction  
Increasingly tighter emission limits, along with the pressure on the automotive industry to reduce fuel consumption, require new concepts for controlling and monitoring of emission reduction systems [1,2]. To abate exhaust gas emission of gasoline engines, three-way catalysts (TWC) are used. They consist of ceramic honeycomb monoliths, mostly of cordierite, with small channels that are coated with a so-called washcoat. A typical washcoat consists of carrier oxides of stabilized alumina, highly dispersed precious metals (Pt, Rh, and Pd) and an oxygen buffer of ceria zirconia solid solutions [3]. The washcoat provides specific surfaces in the range of 100 m\({}^{2}\)/g. For stoichiometrically operated gasoline engines, the air-fuel ratio (\(\lambda\)) is determined in the exhaust gas by means of a first lambda probe. When a deviation from the nominal stoichiometric value of \(\lambda=1\) occurs, the air-fuel ratio is adjusted. As an average over time, a stoichiometric value of about \(\lambda=1\) must be ensured. An optimal conversion occurs only as long as the catalyst is still in good condition, _i.e._, as long as it is not \"aged\". A decreasing catalyst quality leads to a decrease in the conversion of harmful and limited exhaust components like hydrocarbons (HC), carbon monoxide (CO), and nitrogen oxides (NO\({}_{x}\)), and to an increase in the light-off temperature. These effects are correlated with the ability of the TWC to store oxygen. A second lambda-probe arranged downstream of the TWC is able to detect this. By an indirect method, the state of the oxygen storage catalyst, more exactly the oxidation state of the applied ceria in the washcoat, is deduced from the signals of the two lambda-probes applying complicated models. More details on the present state-of-the-art can be found in the literature [4,5]. A relatively new approach to determine the state of catalysts _in situ_ uses the so-called microwave cavity perturbation method. It measures directly the ceria oxidation state as has been shown previously by careful titrations experiments [6]. It allows also to find more exactly the point of best operation, which depends slightly on the type of reducing agents and hence on the type and quality of the fuel (gas, gasoline, bio fuel, _etc._) and on the combustion processes. Even more, a TWC model from [7] could have been quantitatively validated [8]. First attempts to control a TWC-based exhaust gas aftertreatment system have also been published [9,10]. This microwave-based method has been successfully studied in lab experiments [11] as well as in an engine dynamometer test benches [12]. Even for novel exhaust catalyst systems, like NH\({}_{3}\)-SCR catalysts, where the amount of stored ammonia needs to be detected [13] or for diesel particulate filters, where the amount of stored soot is an important measurand [14], this microwave-based technique can be applied. A recent overview on this technique is given in [15]. For a simplified arrangement of the microwave-based technique, the canning of the catalyst forms a cavity resonator, which is filled with the TWC monolith including its washcoat. As a \"sensing element\", one coaxial antenna is installed into the canning. The resonance behavior of such a device depends on the electrical material parameters (conductivity and complex permittivity) of the catalystdevice. Because the electrical properties of the corderite honeycomb are not dependent on the exhaust composition, but the properties of the catalytically active washcoat depend on the state of the catalyst, the resonance behavior can be used as a measure for the oxidation state of ceria in the washcoat, which correlates fully with the conductivity of ceria in the ceria-zirconia storage material [16]. The resonance parameters can be determined by the \(S\)-parameters [17] in transmission (\(S_{21}\) or \(S_{12}\)) if one uses two antennas or in reflection with only antenna as it is the case in this study.