# Set-Point Control of a Spatially Distributed Buck Converter
## 5 Closed-Loop Control of the Distributed Buck Converter
### Implementation by Approximation  
Both control laws (12) and (13) derived in Sections 5.1 and 5.2 contain the integral  
\[\int_{0}^{l}(i_{d}-i)\,\mathrm{d}\,z \tag{19}\]which is defined on a spatially distributed signal. The integral (19) is essentially the mean current deviation along the transmission line. In practice, it is not feasible to measure the current distribution along the transmission line. To overcome this problem three very simple approximations, which can be applied to approximate this integral, are: the left, the midpoint, and the right rectangular approximations.  
\[\int_{0}^{l}(i_{d}-i(z,t))\,\mathrm{d}z\approx\left\{\begin{array}{cc}(i_{d}- i(0,t))\cdot l&\text{(left rectangular approximation)}\\ (i_{d}-i(l/2,t))\cdot l&\text{(midpoint rectangular approximation)}\\ (i_{d}-i(l,t))\cdot l&\text{(right rectangular approximation)}\end{array}\right. \tag{20}\]  
Using the left rectangular approximation would only require the signal value \(i(0,t)\) on the left boundary of the transmission line, where the semiconductor switch and the controller are located. Other approximations, such as the midpoint and the right rectangular approximation, as well as trapezoidal or Simpson's rule would require current measurements with a spatial distance from the controller. Therefore, the left rectangular approximation is preferable and will be used in the following. Since \(l>0\), the left rectangular approximation applied to (13) yields the control law  
\[d(t)=\begin{cases}1&\text{if }(i_{d}(0)-i(0,t))>0,\\ 0&\text{if }(i_{d}(0)-i(0,t))\leq 0.\end{cases} \tag{21}\]  
To simulated the controlled system we use the parameters as discussed in Section 3. The desired output voltage is set to \(v_{d}=6\,\mathrm{V}\). This corresponds to a desired current  
\[i_{d}=\frac{u_{d}}{Z}=\frac{6\,\mathrm{V}}{10\,\mathrm{\Omega}}=0.6\,\mathrm{ A}\]  
required by the control law (21). The simulation results are shown in Figure 7, where the initial (discretized) voltage and current distribution were set to zero. In the beginning, the current \(i(0,t)\) on the left boundary is less then the desired current \(i_{d}\). In particular, we have  
\[i(0,t)=\frac{E}{Z_{0}}=\frac{12\,\mathrm{V}}{50\,\mathrm{\Omega}}=0.24\, \mathrm{A}\quad\text{for}\quad t<2\,\tau l\approx 60\,\mathrm{ns}\]  
until the reflected wave reaches the left boundary. According to (21), the switch is in the position \(d=1\), see Figure 7 (top left). When the current exceeds the desired current at \(t\approx 0.12\,\mathrm{\SIUnitSymbolMicro s}\), the switch changes between the positions \(d=0\) and \(d=1\) at a very high frequency as shown in Figure 7 (bottom left). After the transients, we have very small voltage ripples in the output voltage \(v(l,t)\), see Figure 7 (top right).  
**Remark 3**.: _The (discontinuous) two-point control laws (13) and (21) can be regarded as sliding mode control [19; 38] or, more specifically, as discrepancy-based sliding mode control [39]. As a matter of fact, sliding mode is a common approach to control power converters [17; 19; 29]. In a classical sliding mode control, we would have a (theoretically) infinite switching frequency when the sliding surface is reached. In practice, the switching is not perfect, resulting in a high-frequency control action known as_ chattering_[38; 40]. When the control signal \(d\) given by (21) changes, the voltage wave would travel forward an be reflected on the right boundary. When the backward travelling wave reaches the left bound, the original control action may result in the next change of the control signal. This consideration should result in the switching frequency \(f=1/2\,\tau l\). Due to numerical deviations from the exact solution we also see a high frequency control action for the time between the reflected waves, see Figure 7 (bottom left) starting at \(t\approx 0.12\,\mu\mathrm{s}\). This behavior is also expected in practice due to noise._