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Keywords:
$H_\infty $ control; linear matrix inequality; external disturbance; robust control; uncertainties
$H_\infty $ control; linear matrix inequality; external disturbance; robust control; uncertainties
Summary:
A nonlinear $H_\infty$ control using the linear matrix inequality (LMI) for an autonomous underwater vehicle (AUV) that is equipped with parametric uncertainties and external disturbances is discussed. The proposed controller is intended to attain the stability of the AUV system and performance in the occurrence of disturbances and uncertainties. Initially, the asymptotic stability of the nominal AUV system is shown by using the Lyapunov stability function and the LMI condition is attained. In the second scenario, the external disturbance is considered as an input, and accordingly, the LMI condition for a perturbed AUV system, i. e., the nominal AUV system, is achieved in the presence of external disturbances by substituting the state matrix in the obtained LMI. The improvement of the proposed controller is further discussed in the third scenario by determining the asymptotic stability condition of the perturbed uncertain AUV system, i. e., uncertainties are added to the perturbed AUV system. The perturbation is considered a signal that varies with time. The proposed control algorithm exhibits the effective tracking of the desired depth in all three scenarios. Among all the scenarios, it is observed that the depth tracking by AUV is more effective for the perturbed uncertain AUV system when compared with the remaining two scenarios. Most of the existing AUV controllers mainly focus on disturbance rejection, where the parametric uncertainties are not considered, and also lack stability analysis when the uncertainty levels are increased. The proposed control approach presents the novelty in addressing the parametric uncertainties and time-varying disturbances in contrast to the existing methods that do not consider uncertainties or lack stability under varying conditions. A Lyapunov-based LMI approach is adapted to ensure asymptotic stability for the nominal, perturbed, and perturbed uncertain AUV systems demonstrating superior depth tracking performance, notably in the formidable scenario with disturbances as well as uncertainties. The YALMIP tool is used to develop the proposed control strategy within the MATLAB/Simulink framework. In the first scenario, i. e., for a nominal AUV system, the settling time in tracking the desired depth is found to be 58 seconds. For the perturbed AUV system and the perturbed uncertain AUV system, the settling time is found to be 51 seconds and 37 seconds, respectively. It is observed that the settling time for the third scenario is much less than that of the other two scenarios. On the other hand, the steady state error (SSE) for the nominal AUV system and the perturbed uncertain AUV system is found to be 0.83\
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