# A Systematic Control Design Method with Active Damping Control in Voltage Source Converters
## 3 Design of the Control Gain Matrix
### Weighting Matrixes' Selection
Based on the classification of the states, the states' cost matrix is selected as \(Q=diag(q_{1},q_{1},q_{2},q_{2},q_{1},q_{1},q_{1},q_{2},q_{2},q_{2},q_{2},q_{2 },q_{3})\). First, stability and performance definitions are provided. Then the impact of different values of tuning parameters (\(q_{1}\), \(q_{2}\), \(q_{3}\), and \(r\)) on the system stability and performance and the control gain matrix are studied. The results are summarized in Table 2.  
\begin{table}
\begin{tabular}{|c|c|c|c|c|c|c|} \hline \(q_{1}\) & **norm (\(K_{f}\))** & \(\sigma_{max}\) & \(\xi_{min}\) & \(q_{2}\) & **norm (\(K_{f}\))** & \(\sigma_{max}\) & \(\xi_{min}\) \\ \hline \(1\times 10^{1}\) & \(9.7\times 10^{3}\) & –1.5 & 0.28 & \(1\times 10^{-2}\) & \(1.6\times 10^{3}\) & –42 & 0.03 \\ \hline \(1\times 10^Definitions, stability, and performance indicators:  
The following equations can represent the \(i\)th eigenvalue of the system (\(\lambda_{i}\)) and its damping factor (\(\sigma_{i}\)), frequency of oscillations (\(\omega_{di}\)), and damping ratio (\(\xi_{i}\)):  
\[\lambda_{i}=\sigma_{i}\pm j\omega_{di},\zeta_{i}=\frac{-\sigma_{i}}{\sqrt{ \sigma_{i}{{}^{2}}+\omega_{di}{{}^{2}}}} \tag{12}\]  
It is worth remarking that the system is stable for \(\sigma<0\) and \(\xi>0\), oscillating for \(\sigma=\xi=0\), and unstable for \(\sigma>0\) and \(\xi<0\). The unstable or weakest damped eigenvalue among all eigenvalues, called a critical mode, causes significant oscillations or even system instability. To guarantee stability margin and limit the system overshoot, the damping factor and ratio of the critical mode should fulfill some requirements (\(\sigma\leq\sigma\) & \(\alpha_{minmax}\)). The desired damping ratio depends on the modal frequency and accepted oscillation settling time by different utilities and standards (30,31). It is usually selected to be something between 5 and 20 percent. In this work, a desired 10% damping ratio is accepted, guaranteeing an acceptable overshoot and decay rate of oscillations. Moreover, placing the maximum damping factor far from the imaginary axis can ensure system stability and improve response speed in terms of the rise time. The system's rise time is assigned by the outer voltage control loops that have lower bandwidths among different control loops. For example, a rise time (\(t_{r}\)) lower than 100 ms is usually required for AVC, resulting in a maximum damping factor less than 22 rad/s: (\(\sigma_{max}\approx-2.2/t_{r}=-2.2/100\) ms = \(-22\) rad/s) (23). Selection of \(q_{1}\):  
As it can be seen in Table 2, this parameter has a significant impact on the maximum damping factor. For example, for \(q_{1}=10\), the system is close to the imaginary axis and hazardous area (\(\sigma_{max}=-1.5\) rad/s). By increasing this parameter to \(q_{1}=1\)\(\times\)\(10^{4}\), the critical mode moves far from the imaginary axis to obtain a good stability margin (\(\sigma_{max}=-40\) rad/s). However, by increasing the \(q_{1}\) more, the critical mode is not changed anymore, while the control gain norm increases. Thus, at this point, if the obtained damping factor is acceptable, \(q_{1}\) can be used for the control gain calculation in (10). Otherwise, the new initial values for all tuning parameters should be selected, and the process should be repeated. A good decision is to set a lower value for \(r\) to allow more control input usage. This action may provide a better stability margin in terms of the damping factor. Finally, as it can be seen, the damping ratio is not affected, and it remained constant for different values of \(q_{1}\). Selection of \(q_{2}\):  
Unlike the \(q_{1}\), \(q_{2}\) does not affect the damping factor, while it affects the damping ratio considerably. For a lower value, the \(\xi_{min}\) is close to zero, which is undesirable and causes higher overshoots. Increasing this parameter increases the system damping ratio remarkably in the cost of higher control gains. Therefore, the selection of this parameter is a compromise between damping ratio and control gains norm.