# Set-Point Control of a Spatially Distributed Buck Converter
## 1 Introduction  
The buck converter is a DC-DC converter transforming an input voltage into a lower output voltage [1; 2]. The classical topology consists of an inductor and a capacitor combined with a diode and a transistor. Here, the diode and transistor resemble an ideal switch for the input voltage. Applying a pulse-width modulated signal to this switch results in a reduced apparent voltage, where the duty cycle, i.e., the ratio between closed switch position time and the period of the signal, determines the apparent output voltage level. The low pass character of the passive network of inductance and capacitance then results in a smoothing of the switching voltage. Switched converters are a field of active research in power electronics [3]. DC-DC converters are used for consumer electronics, mobile devices, and photovoltaic applications [4; 5]. For energy technology applications, the research focus is on multi-level three-phase converters [6]. Important goals are the design of converters are costs, energy efficiency, reliability, as well as the suppression of overshoots and higher harmonics. To achieve these goals, control has a special role to play [2; 6; 7]. In contrast to the described classical configuration, a new topology replacing the passive elements, i.e., the inductor and the capacitor, with a transmission line has been proposed in [8] and investigated further in [9; 10; 11]. This has been motivated by the transmission line model, which itself consists of distributed inductance and capacitor. The main advantage of this configuration lies in the reduced number of required elements, i.e., no inductor nor capacitor are required. If the load has a spatial distance from the power supply, a cable is needed anyway. We use this cable to replace the reactive network elements ofthe classical buck converter. In some applications, the use of a transmission line may be more advantageous than the use of an integrated inductor having a low inductance [12]. Disadvantages, however, are the significantly increased switching frequencies, due to the small capacitance and inductance values in conventional wires, and increased system complexity, due to the distributed nature of the transmission line. The connection of a boost converter to the load via a transmission line is investigated in [13]. Similarly, the connection of power converters by transmission lines is discussed in [14]. Note that transmission line models are also used in recent publications to model electric machines [15; 16]. In the scientific literature there are a lot of works on the control of classical converter topologies. There, many control strategies are discussed, such as linear control (P/PI control, state feedback, output feedback [2]), as well as non-linear control (e.g., by exact linearization [17], flatness-based control [18], sliding mode [19] etc.). The circuit models of these converters use lumped elements. Unfortunately, the above mentioned control strategies cannot directly be applied to our converter circuit. While previous work has addressed the modeling, analysis, and design of the distributed buck converter, this paper also addresses controller design. More precisely, the mathematical models for distributed buck converters under resistive and inductive load were derived in [8; 9], respectively. In [11], a practical circuit realization was suggested and successfully tested. In this contribution, we will derive first control schemes taking into account the distributed system nature. More precisely, we employ a discrepancy-based control technique [20; 21; 22]. The paper is structured as follows: In Section 2 we derive models for the classical, as well as the distributed buck converter. The open-loop simulation is discussed in Section 3. To derive control laws for this model, the basic concepts of discrepancy-based control are presented in Section 4. Several different control strategies are derived in Section 5. We will discuss our results and suggest further directions of research in Section 6.