## 4 Case Study: MIMO Magnetic Levitation System
### System Analysis and Results
#### 4.2.2 Design Validation and Analysis  
Figure 7 shows the tracking errors obtained with the proposed design for the on-diagonal element of the closed loop uncertain system. It shows that the magnitudes of the closed loop system (for 6561 randomly selected plants) remain within the limits specified by the tolerance (\(B_{e}\)) in equation (18). The closed loop responses of the uncertain system (for 6561 random plants) for a unit step are shown in Figure 8. The responses with the proposed feedback controller design (Equation (19)) along with the feedforward controllers (refer to Table 1) for channels-11 and -22 track the desired model (\(M\)) responses within the tolerances (\(B_{e}\)).  
Figure 9 compares the control effort with the proposed design (\(G+G_{R}\)) and the pure feedback controller (\(G_{fb}\)) for both the channels. It is seen that the control effort with the proposed feedback controller alone (\(G\)) was much less than the pure feedback controller (\(G_{fb}\)) and the combined feedback + inversion feedforward (\(G+G_{R}\)) control. The biggest share of the control effort produced by the proposed feedback + inversion feedforward design was due to the inversion feedforward control as shown in Figure 9. In practice, this implies a reduction in actuator fatigue, savings in energy, and avoidance of noise-induced saturation as mentioned in Reference [21]. It was noted that the demand for the feedback control was very less because of the feedforward action. This reduces the feedback cost measured in terms of high frequency gain. Hence, the main purpose of the QFT design can be achieved.  
Figure 7: Frequency domain validation of the proposed design for the model matching specification. The plots show the error tolerances \(B_{e}\) (dashed red line) and the closed loop magnitude (solid line) of diagonal element over all the 6561 plants for (**a**) channel-11 (lower magnet), and (**b**) channel-22 (upper magnet).  
To analyze the interaction effect, a unit step input signal was applied at time instant t = 0 s and t = 4 s, for channels-11 and -22, respectively. The closed loop responses of the uncertain system (for 6561 random plants) with the proposed feedback controller design (equation 19) along with the feedforward controllers (equation 19 and Table 1) for channels-11 and -22 track the desired model (\(M\)) responses within the tolerances (\(B_{e}\)) as shown in Figure 10, while the interaction effects are within the limits of specification (<0.1).  
Figure 8: The closed loop (6561 plants) response of the designed control system for (**a**) channel-11 (lower magnet), (**b**) channel-22 (upper magnet).  
Figure 9: Comparison of control actions (6561 plants) in time domain under unit step input for pure feedback and proposed designs; (**a**) channel-11, (**b**) channel-22.  
Figure 11 compares the frequency response magnitude of the feedback controllers between the proposed method and the pure feedback method for both the channels. The proposed feedback controller (\(G\)) gives a high frequency (HF) gain reduction of 20 dB over the pure feedback design (\(G_{fb}\)) for both the channels. This implies that the proposed design is less sensitive to the sensor noise amplification problem (termed as \"cost of feedback\" in QFT). It is interesting to explore the experimental validation of the proposed design in the magnetic levitation system.  
Figure 10: Interaction analysis: the closed loop (6561 plants) response of the designed control system; (**a**) channel-11 and (**b**) channel-22.