## 3 Justification and Formulation of Extremum-Seeking Control for Series Resonant Converters  
The suitability of ESC for PFM converters such as resonant converters arises from the fact that their output power exhibits a smooth, unimodal dependence on switching frequency. This is supported by first harmonic approximation (FHA) analysis, which models the resonant tank as a frequency-dependent impedance. This characteristic aligns well with the assumptions required for ESC to converge to the true extremum. Furthermore, the derivation in this paper establishes a specific formulation of ESC where the switching frequency serves as the control input and the PV power as the output to be maximized. The resulting system should satisfy the ESC design conditions: time scale separation, a smooth input-output map, and unimodal behavior. Controlling resonant converters presents unique challenges due to their highly nonlinear and frequency-dependent behavior. Unlike traditional PWM converters, resonant converters exhibit a complex relationship between switching frequency and output voltage  
Figure 9: The proposed design procedure for the series resonant converter with capacitive galvanic isolation. or current, which is strongly influenced by the load and the resonant tank parameters. This nonlinearity complicates the development of accurate small-signal models, particularly under variable operating conditions such as changing load or input voltage. Furthermore, the resonant gain characteristics can shift significantly due to component tolerances and parasitic elements, making precise analytical control design difficult. Traditional linear control techniques often require extensive system identification and parameter tuning, which may not be robust to environmental variations or aging effects. Extremum-seeking control (ESC) offers an effective solution to these challenges by providing a model-free optimization approach. Instead of relying on a detailed mathematical model, ESC continuously perturbs the system input, such as the switching frequency, and observes the output response to iteratively locate the optimal operating point, typically the maximum power point (MPP) in PV applications. This real-time adaptation enables ESC to handle the inherent nonlinearities and uncertainties of resonant converters, ensuring stable and efficient operation without the need for complex modeling or controller redesign, even as system parameters change over time. Moreover, ESC is particularly suitable for PV maximum power point tracking (MPPT) because the PV power-voltage curve is generally smooth and unimodal under uniform irradiance, forming a static map that does not exhibit fast dynamics. The static and smooth nature of the PV characteristic curve ensures that the ESC algorithm can reliably and efficiently converge to the true maximum power point without being misled by transient disturbances, which makes it an ideal control strategy for the combined PV-resonant converter system. A general extremum-seeking control (ESC) system is formulated based on the principle of real-time optimization through periodic perturbation and gradient estimation. The core idea of ESC involves applying a small sinusoidal perturbation to the system input, measuring the system output response, and demodulating this response to estimate the gradient of the output with respect to the input. Specifically, if the control input is denoted by \(u(t)\) and the system output by \(y(t)\), ESC perturbs the input as  
\[u(t)=\hat{u}(t)+a\times\sin(\omega t) \tag{8}\]  
where \(\hat{u}(t)\) is the slowly varying nominal input, \(a\) is the perturbation amplitude, and \(\omega\) is the perturbation frequency. The output \(y(t)\) is then multiplied by \(\sin(\omega t)\) and passed through a low-pass filter to extract an estimate of the gradient \(\frac{d(y)}{d(u)}\). This estimate is used to update \(\hat{u}(t)\) through an integrator and drive the system toward the extremum of the input-output map.