# Linear System Identification and Vibration Control of End-Effector for Industrial Robots
## 4 Design and Simulation of LQG Controller  
For the discrete state space, the link quality indicator regulator, abbreviated as LQI regulator, obtains the optimal control rate by minimizing the linear quadratic cost function \(J(u)=\sum\limits_{k=0}^{Y=}(z^{T}Qz+u^{T}Ru)\) without considering the noise interference. \(Q\) and \(r\) are the weight matrix of cost function determined by human. While the control strategy of the linear-quadratic-Gaussian controller shorted as LQG controller is to minimize the error quadratic functional of linear systems disturbed by external noise. It does not need the system state to be fully observable. Therefore, it is more suitable for this control system to be calculated by adjusting the value of weight \(Q\) and \(r\) during designing. The state estimator gives the state equation as  
\[\overset{\wedge}{x}[k+1|k]=A\overset{\wedge}{x}[k|k-1]+Bu[k]+L(y[k]-C \overset{\wedge}{x}[k|k-1]-Du[k]) \tag{9}\]  
According to the state estimation of Kalman filter and the optimal control matrix K obtained by the LQI method, a 1-DOF position tracking controller is constructed. The structure diagram is shown in Figure 10.  
Figure 11 illustrates the comparison of the Bode diagram between the identification system and the original system. The amplitude of the identification system is consistent with the amplitude of the Bode diagram of the actual input-output relationship between 0 Hz and 100 Hz. But there is a huge  
Figure 10: Linear-quadratic-Gaussian (LQG) controller block diagram.  
deviation beyond this range. As for the phase, there is an understandable delay between 0 Hz and 100 Hz, but there is no regular pattern during overstep.  
Figures 12 and 13 show the Bode diagram of the closed-loop system and the LQG controller. Generally, the Bode diagram amplitude of the closed-loop system is less than 1 and the phase diagram is basically maintained at 360 degrees when the frequency is below 100 Hz after adding the LQG controller, which indicates that the closed-loop system can effectively suppress the vibration below 38.7 Hz.  
Figure 11: Comparison of Bode diagram between identification system and original system.  
Figure 12: Bode diagram of closed-loop system.  
Figure 13: Bode diagram of LQG Controller.  
Figure 14 shows the results of the response for the open-loop system and the closed-loop system, respectively. It can be seen that the amplitude of the closed-loop system has a rapid reduction which means the dynamic performance of the system is significantly improved after adding the controller.  
The system controller designed by appeal is simulated. The frequency of sinusoidal interference signal is 35 Hz which is near the natural frequency, and the vibration amplitude is \(\pm\)20 \(\upmu\)m. The vibration response of the structure is shown in Figure 15. It can be seen from Figure 15 that the vibration amplitude of the structure is successfully reduced from \(\pm\)20 \(\upmu\)m to \(\pm\)1 \(\upmu\)m after the effective suppression of the control system indicating that the vibration suppression percentage of the vibration suppression system is as high as 95% without high frequency noise. When the vibration frequency of the excitation signal is 50 Hz and the vibration amplitude is \(\pm\)20 \(\upmu\)m, the response of the system reaches about \(\pm\)40 \(\upmu\)m. The results are shown in Figure 16, and the control system completely loses the vibration suppression effect. In conclusion, the LQG vibration suppression system has an obvious suppression effect on vibration interference less than 38.7 Hz, but has no suppression effect on other vibration signals higher than 38.7 Hz.  
Figure 14: Response curve of open and closed loop systems.  
Figure 15: Simulation results of 35 Hz interference.