# A Hybrid Linear Quadratic Regulator Controller for Unmanned Free-Swimming Submersible
## 6 Experimental Results
### Results  
Concerning Section 4, it has been observed that the steady-state error is of a considerable margin on the unity gain. Similarly, the overshoot is not up to the required mark. Consequently, it results in a large settling time which is intolerable for a control system. When disturbance occurs in the system, it behaves disorderly and does not settle to its indigenous state. We have introduced PID, lead-compensator, and HLQR in UFSS to revamp the transient response. They tend to minimize the difference of ratio between the actual value and the desired value of the UFSS. After integrating the aforementioned controllers in the UFSS, the behavior of UFSS in response to the unit step is observed. Both PID and HLQR eliminate steady-state error resulting in a final value equal to 1. However, the lead-compensator only improves the response of the system. It does not eliminate the steady-state error. For the pitch system, the steady-state error is quite significant. Therefore, it is reduced by the lead-compensator. Nevertheless, 12% error remained in it that is visible in Figure 10. The head system does not possess a steady-state error in it. It implies that there is a zero steady-state error for all controller, as shown in Figure 11.  
The analysis of these controllers has shown that all of them have achieved the desired requirements for both overshoot and settling time. However, the response graph is not  
Figure 11: Analysis of heading control system with controllers.  
Figure 10: Analysis of pitch control system with controllers.  
the same for each type. Moreover, the HLQR controller has shown that it has handled disturbances and attained wanted requirements more briskly. It leads to a faster response time and lesser rise time. On the other hand, the PID controller has shown comparatively less response time. Similarly, the LQR and the lead-compensator are slowest among all. It has resulted in the greatest rise time which is shown in both Figures 10 and 11. In order to provide a better comparison over controllers, the detailed parameters for the observation of pitch and head systems are provided in Tables 6 and 7, respectively.  
The numerical values shown in Tables 6 and 7 reveal that the HLQR has an advantage over PID and lead-compensator controllers. The HLQR method involves more mathematical computations as compared to the PID and lead-compensator. On the other hand, these mathematical computations are rather simple to calculate with multiple-output systems. The PID is also a versatile controller that only uses gain values \(K_{p}\), \(K_{i}\), and \(K_{d}\) to achieve the result. The HLQR has an upper edge in performance as compared to the PID due to its matrix method. The lead-compensator is not preferred among all of these due to its lacking ability to eliminate the steady-state error (shown in the last column of Table 7). In HLQR, we can achieve a better response than other controllers as we have fixed the control criteria (depending on the control parameters). Usually, the control criteria in PID are achieved by varying the PID parameters. Here, we come from the opposite side with a fixed control criteria. However, the change in PID parameters may lead to a combined effect on the control parameters. To summarize, the HLQR is robust as compared to PID and lead-compensator controllers.