# A Systematic Control Design Method with Active Damping Control in Voltage Source Converters
## 3 Design of the Control Gain Matrix
### Control Gain Matrix Simplification  
The state-feedback-based active damping can considerably improve system performance and robustness, as presented in previous subsections. However, grid-side current sensors are required, and computations are increased due to employing all states.  
Therefore, it would be worth investigating how the number of sensors and calculations can be reduced. A rational solution is employing a full- or reduced-order Luenberger observer and Kalman filter to eliminate the grid-side sensors. However, it increases the computations and adds additional observer gains that should be tuned, leading to the higher complexity of the control system design and implementation. In this subsection, a more straightforward solution is proposed. First, it is assumed that all states are available, and the optimal control gain matrix is calculated.  
Then it is assumed that the \(m\)th state is not available, so the related feedback gains are set to zero \(\left(x_{m}:\mathrm{off}\to K_{f}(;m)=0\right)\). Finally, the impact of eliminating this state on the system stability and performance is investigated by eigenvalues' analysis. If the critical mode and stability requirements have remained unchanged, this state can be simply excluded. Otherwise, this state and the related gains must be used. For example, if an eliminating grid-side current \(\left(i^{\prime}_{gal},i^{\prime}_{gal}\right)\) is required, the thirteenth and fourteenth columns of \(K_{f}\) must be set to zeros \(\left(x_{13}\delta x_{14}:\mathrm{off}\to K_{f}(;13:14)=0\right)\). Moreover, based on the new control gain matrix, the analysis of the eigenvalues can be carried out again.  
Figure 8 shows the critical mode location when one of the states and its related gains are kept at zeros. The number next to each mode expresses the state number that is omitted. Moreover, 00 is equal to the case that all states are used. For example, as shown, eliminating states 8, 9, or 15 causes system instability, whereas when states 11, 13, or 14 are eliminated, the system remains stable. To examine the analytical results, time-domain simulations are presented in Figure 9.  
This figure illustrates the system outputs under three different control gain matrixes, where all states are available, \(x_{13}\) is not available \(\left(x_{13}:\mathrm{off}\to K_{f}(;13)=0\right)\), and \(x_{9}\) is not available \(\left(x_{9}:\mathrm{off}\to K_{f}(;9)=0\right)\). As shown, eliminating \(x_{13}\) does not lead to system instability. On the other hand, eliminating \(x_{9}\) does.  
As shown in Figure 8, there are seven states that eliminating them does not lead to system instability. It would be worth investigating the changes of critical mode locations where all of them are off simultaneously. This action reduces the number of employed states from 15 to 8.  
Moreover, 46% of the control gain matrix elements become zeros, reducing computations: \(x_{3},x_{4},x_{7},x_{11},x_{12},x_{13}\delta x_{14}:\mathrm{off}\to K_{f}(;3 :4)=K_{f}(;7)=K_{f}(;11:14)=0\).  
Figure 8: Critical mode locations of a grid-connected VSC in the complex s-plane (a) SCR = 1.5, and (b) SCR = 5.  
Figure 9: Time-domain simulations showing the system response under different control gain matrixes.  
The impact of grid SCR variation on the critical mode under the conventional control method and the proposed full and reduced state-feedback control methods are examined in Figure 10. The system robustness is a little worse than full state feedback by reduced state feedback; as expected, however, it is still considerably better than the conventional one. In summary, the system remains stable under the reduced state feedback control, with a good stability margin over a wide range of grid SCR variations. At the same time, the possibility of implementing a high-performance PLL is achieved.