Research Article  Open Access
Optimal Disturbances Rejection Control for Autonomous Underwater Vehicles in Shallow Water Environment
Abstract
To deal with the disturbances of wave and current in the heading control of Autonomous Underwater Vehicles (AUVs), an optimal disturbances rejection control (ODRC) approach for AUVs in shallow water environment is designed to realize this application. Based on the quadratic optimal control theory, the AUVs heading control problem can be expressed as a coupled twopoint boundary value (TPBV) problem. Using a recently developed successive approximation approach, the coupled TPBV problem is transformed into solving a decoupled linear state equation sequence and a linear adjoint equation sequence. By iteratively solving the two equation sequences, the approximate ODRC law is obtained. A Luenberger observer is constructed to estimate wave disturbances. Simulation is provided to demonstrate the effectiveness of the presented approach.
1. Introduction
Nowadays, there has been increasing interest in the use of AUVs to expand the ability to survey ocean areas, for example, exploration and exploitation of seafloor minerals, oceanographic mapping, and underwater pipelines tracking [1]. However, due to highly nonlinear and strongly coupled dynamics of AUVs and the environmental disturbances (such as currents and waves), it is always a challenge to design controller for AUVs. Recently, numerous nonlinear control methods have been utilized to achieve improved performances for motion control of AUVs. Typical results include sliding control, adaptive control, and optimal control. In [2–4], the sliding mode control techniques were applied to motion control of AUVs. In [5–7], the backstepping control approach was employed to design controller for path following of underactuated AUVs. In [8, 9], and controllers were presented for motion control of AUVs. In [10–12], the adaptive control methods were utilized for motion control of AUVs to improve robustness of the control systems. In [13], a suboptimal control approach was proposed for motion control of AUVs. In [14], a feedback linearization technique was applied for AUVs tracking problem. In [15], the combined problem of trajectory planning and tracking control for underactuated AUVs in the horizontal plane was addressed, and a backstepping approach was presented. In [16], Lyapunov approach and backstepping control approach were applied to design pathfollowing controller of an AUV in the horizontal plane with constant ocean currents. In [17], an robust faulttolerant controller was designed to improve the security and reliability of navigation and enhance the accuracy and robustness of navigation control system for AUVs.
Note that nonlinearity and disturbances such as currents and wave are unavoidable in AUVs control systems, which affect the performance of the AUVs system. At present, there are many methods to deal with these problems. In [18], an output feedback control approach was proposed for the trajectory tracking control of AUVs, which moved in shallow water areas. In [19], for the problem of dynamic positioning and waypoint tracking of underactuated AUVs in the presence of constant unknown ocean currents and parametric modeling uncertainty, a nonlinear adaptive controller was proposed. In [20], a stable sliding mode controller was designed, which can track AUVs along the desired trajectory in complex sea conditions. In [21], an adaptive output feedback controller was presented for AUVs named ODIN to track a desired trajectory with bounded errors in wave disturbances condition. In [22], a secondorder sliding mode controller was proposed for an AUV, which can compensate for disturbances such as waves, currents, and buoyancy force.
In the field of modern control, one of the key ideas is the use of optimization and optimal control theory to give a systematic procedure for the design of feedback control systems [23–25], LQR approach provides one of the most useful techniques for designing state feedback controllers. In order to overcome disturbances and nonlinearities for the AUV system, in this paper, an ODRC approach is proposed based on the quadratic optimal control theory. Firstly, the AUVs model and wave disturbances model are obtained, and according to disturbances type, an integral unit is introduced to eliminate its ocean current disturbances effect, and a feedforward control is used to reject wave disturbances. Secondly, for the AUVs heading control system, the coupled TPBV problem is derived from the maximum principle of optimal control theory. Then by using a successive approximation approach [26, 27], the coupled TPBV problem is transformed into solving two decoupled linear differential sequences in state vectors and adjoint vectors. By iterative solution, the ODRC law is obtained, and a Luenberger disturbances observer is constructed to make it realizable. The contribution in this paper is the ODRC approach which is applied to design rejection controller for AUVs in shallow water environment, which only requires solving the Riccati equation and the Sylvester matrix equation one time, while mainly solving a recursion formula of adjoint vectors.
This paper is organized as follows. In Section 2, the model of AUVs motion in horizontal plane is introduced, and the system of shallow wave disturbances is constructed. Section 3 presented an ODRC design for AUVs. Simulation validates the effectiveness of the designed controller under wave disturbances in Section 4. Section 5 provides the concluding remarks.
2. System Models
2.1. Mathematical Model of AUVs
A schematic of the sixDOF AUVs model with related coordinate system is shown in Figure 1.
The two reference frames are applied to the model: Earthfixed frame and bodyfixed frame. Descriptions of the parameters are expressed in Table 1 [28–30].

The Earthfixed frame is treated as an inertial frame. In order to facilitate the analysis and synthesis for AUVs, the coupling effect between the roll surface movement and two cases of plane motion is usually ignored, and then the vehicle motion is divided into horizontal and vertical movement. The heading motion equations in horizontal plane are given in dimensional form aswhereIn system (1), is the control rudder angle; , , and , respectively, are the vehicle’s length, quality, weight, and buoyancy; is the density of seawater; , and , respectively, are the vehicle’s moment of inertia about , , and axis; is the external disturbances; and are hydrodynamic coefficients.
Suppose that the axial velocity is a given constant, and the influences of the vertical plane motion and the parameters of the rolling motion are neglected; then we have
Define the heading instruction as a constant , and , heading error is , and thenConsidering the ocean current disturbances, an integral unit is introduced to eliminate the accumulated error
Define the state vector as , the control vector , and the dynamic model of the AUVs heading control system can be rewritten as follows:where, , and are defined in (6), is the initial state, and is a limitary nonlinear vector, which includes errors and uncertainties.
Remark 1. The limitary nonlinear vector satisfies the following Lipschitz conditions:where and are positive constants.
2.2. Disturbances Model of Wave Force
The external disturbances for AUVs are complex. In the near water surface, the wave force disturbances are the most important factor to the AUVs system. But ocean waves are always irregular. In present study, the irregular long storm waves are often simplified [31–33] as follows:where is the th component wave number, is the th component wave frequency, is a random phase angle uniformly distributed within , is the th component wave amplitude, , is the th component ocean power spectrum density (PSD) function, and is the frequency discretization intervals of wave spectrum.
The wave force disturbances have been represented according to the PiersonMoskowitz (PM) spectrum, written as where , , where is the acceleration due to gravity, and is the significant wave height in meters.
Considering the stationary waves on the sea surface, let and , and choose multiple regular wave superposition to get random waves, so the point longcrested waves are written aswhere is the number of superimposed waves, is encounter angle frequency, and is the encounter wave angle.
We construct a system model to describe the irregular wave forces for the AUV in twodimensional horizontal plane.
Define as the horizontal velocity of water particle orbital motion.
Let ; taking the derivative of , we havewhere .
Define ; thenwhere is dimensional unit matrix and is dimensional zero matrix.
According to the linear wave theory, the wave force disturbances for the AUV system are as follows: So the total wave force disturbances for AUVs can be described by the following system:where and are real constant matrices of appropriate dimensions.
3. Controller Design
According to the dynamic model of the AUVs heading control system (6), we select the following average quadratic performance index:where is the weighting matrix for states, symmetric positive semidefinite, and is the weighting matrix for the control inputs, symmetric positive definite. and are properly selected to shape the response characteristics in the closedloop AUVs heading control system (6).
The optimal control problem is to search for a control law for system (6), which makes the value of the average quadratic performance index (16) minimum.
Applying the maximum principle of the optimal control problem in (6) and (16), the optimal control law can be written aswhere is the solution to the following TPBV problem:which is the optimality necessary condition, where and .
For the AUVs heading control system (6) and wave force system (15) with the average quadratic performance index (16), we now state the following theorem.
Theorem 2. Consider the optimal control problem described by systems (6) and (15) with the average quadratic performance index (16). Then the ODRC law is existent and unique, and its form is as follows: where is the unique positivedefinite solution of the following Riccati matrix equation: is the unique solution of the following Sylvester matrix equation:
Proof. Letwhere , , and are the state vector, the wave force disturbances, and the adjoint vector, respectively.
It is well known that is the unique positivedefinite solution of Riccati matrix equation (20). Substituting into (21), then can be solved uniquely; here . When and are got uniquely, therefore, we can obtain the ODRC law uniquely as follows:where and are the solutions of the following equations:and is the solution of the wave forces system (15).
By using the successive approximation approach [26, 27, 31, 34], we construct the adjoint vector sequenceand the state equation sequenceIt can be proved that the adjoint vector solution sequence in (25) uniformly converges to , and the state vector solution sequence in (26) uniformly converges to [26, 27, 31, 34].
When , the limits of and become the optimal state vector and the optimal adjoint vector .
Define ; then the ODRC law is rewritten as (19). This completes the proof.
Remark 3. It is usually impossible to obtain the exact adjoint vector when designing the ODRC law in practice. In many cases, it may be better to choose an as the approximation of where depends on a concrete error coefficient . The thorder ODRC law is as follows:
Remark 4. The thorder ODRC law consists of a feedback term , a feedforward disturbances rejection term , and a nonlinear compensatory term .
Remark 5. The wave disturbances are difficult to be measured and obtained, and is the state vectors of system (15); the ODRC law is physically unrealizable in practice. So, we construct a Luenberger disturbances observer to make it realizable as follows:where is the observation value of and is the observer matrix of appropriate dimensions. We can choose the appropriate dimensions matrix , and the eigenvalues of the matrix have negative real parts. Then the thorder ODRC law is as follows:
4. Example and Simulation
4.1. Simulation Model and Parameters for AUVs and Wave Force Disturbances
In this section, a simulation study is carried out using a typical AUV model [35] to demonstrate the performance of the proposed approach. The hydrodynamic coefficients of an AUV are shown in Table 2.

Assuming that the axial velocity is , the significant wave height is , and encounter angle frequency of wave is 0.6320 and 1.0470, then the PSD of the wave height is shown in Figure 2.
And the wave force can be calculated from (11), which is shown in Figure 3.
The matrices and vector of the AUVs heading control system (6) are as follows:
The parameters for the average quadratic performance index (16) are chosen asBy solving (20) and (21), we obtain
4.2. Simulation Results and Analysis
Based on the AUV model and parameters, the state vector is considered as ; the simulation curves and at different iteration times are shown in Figures 4–8.
The average quadratic performance index values and errors , at different iteration times, are listed in Table 3, and is the control precision.

From Table 3, it can be seen clearly that the average quadratic performance index values decrease as iteration times increase and tend to a deterministic optimal value ultimately. If we choose the control precision , then the relative error of the average quadratic performance index values satisfies . It indicates that the 4th ODRC law is very close to the optimal control law . It is obvious that the proposed approach requires a few iterations to get the approximate ODRC law. And it is more robust about current and wave disturbances. Simulation results show that the proposed approach applied to the AUVs is effective.
5. Conclusion
In this paper, the disturbances rejection control problem for the AUVs heading control system has been considered, and an ODRC approach has been designed. For the design, nonlinearities in the AUVs system are retained, and disturbances for the AUVs system are considered. By using a successive approximation approach, an ODRC law is proposed based on the quadratic optimal control theory, which consists of the optimal feedback item, the feedforward disturbances rejection item, and the nonlinear compensatory item. Finally, the effectiveness of the proposed approach has been illustrated by an AUV model.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This study was supported in part by the National Natural Science Foundation of China (61673357, 41276085, and 61572448) and by the Natural Science Excellence Foundation of Shandong Province (ZR2015FM004 and ZR2014JL043).
References
 R. B. Wynn, V. A. I. Huvenne, T. P. Le Bas et al., “Autonomous Underwater Vehicles (AUVs): their past, present and future contributions to the advancement of marine geoscience,” Marine Geology, vol. 352, pp. 451–468, 2014. View at: Publisher Site  Google Scholar
 G. Bartolini and A. Pisano, “Blackbox position and attitude tracking for underwater vehicles by secondorder slidingmode technique,” International Journal of Robust and Nonlinear Control, vol. 20, no. 14, pp. 1594–1609, 2010. View at: Publisher Site  Google Scholar  Zentralblatt MATH  MathSciNet
 L. Wang, H.M. Jia, L.J. Zhang, and H.B. Wang, “Horizontal tracking control for AUV based on nonlinear sliding mode,” in Proceedings of the IEEE International Conference on Information and Automation (ICIA '12), vol. 1, pp. 460–463, 2012. View at: Google Scholar
 Z.P. Yan, D. Wu, J.J. Zhou, and X. Zhang, “Application of extension integral variable structure control method on simulation of AUV heading control system,” in Proceedings of the 30th Chinese Control Conference, vol. 7, pp. 2617–2621, July 2011. View at: Google Scholar
 J. Ghommam and M. Saad, “Backsteppingbased cooperative and adaptive tracking control design for a group of underactuated AUVs in horizontal plan,” International Journal of Control, vol. 87, no. 5, pp. 1076–1093, 2014. View at: Publisher Site  Google Scholar  MathSciNet
 L. Lapierre and B. Jouvencel, “Robust nonlinear pathfollowing control of an AUV,” IEEE Journal of Oceanic Engineering, vol. 33, no. 2, pp. 89–102, 2008. View at: Publisher Site  Google Scholar
 L. Lapierre and D. Soetanto, “Nonlinear pathfollowing control of an AUV,” Ocean Engineering, vol. 34, no. 1112, pp. 1734–1744, 2007. View at: Publisher Site  Google Scholar
 J. Petrich and D. J. Stilwell, “Robust control for an autonomous underwater vehicle that suppresses pitch and yaw coupling,” Ocean Engineering, vol. 38, no. 1, pp. 197–204, 2011. View at: Publisher Site  Google Scholar
 L. Moreira and C. G. Soares, “${H}_{2}$ and ${H}_{\infty}$ designs for diving and course control of an Autonomous underwater vehicle in presence of waves,” Journal of Oceanic Engineering, vol. 33, no. 2, pp. 69–88, 2008. View at: Publisher Site  Google Scholar
 B. K. Sahu and B. Subudhi, “Adaptive tracking control of an autonomous underwater vehicle,” International Journal of Automation and Computing, vol. 11, no. 3, pp. 299–307, 2014. View at: Publisher Site  Google Scholar
 B. B. Miao, T. S. Li, and W. L. Luo, “A DSC and MLP based robust adaptive NN tracking control for underwater vehicle,” Neurocomputing, vol. 111, pp. 184–189, 2013. View at: Publisher Site  Google Scholar
 L.J. Zhang, X. Qi, and Y.J. Pang, “Adaptive output feedback control based on DRFNN for AUV,” Ocean Engineering, vol. 36, no. 910, pp. 716–722, 2009. View at: Publisher Site  Google Scholar
 B. Geranmehr and S. R. Nekoo, “Nonlinear suboptimal control of fully coupled nonaffine sixDOF autonomous underwater vehicle using the statedependent Riccati equation,” Ocean Engineering, vol. 96, pp. 248–257, 2015. View at: Publisher Site  Google Scholar
 D. W. Kim, “Tracking of REMUS autonomous underwater vehicles with actuator saturations,” Automatica, vol. 58, pp. 15–21, 2015. View at: Publisher Site  Google Scholar  MathSciNet
 F. Repoulias and E. Papadopoulos, “Planar trajectory planning and tracking control design for underactuated AUVs,” Ocean Engineering, vol. 34, no. 1112, pp. 1650–1667, 2007. View at: Publisher Site  Google Scholar
 S.W. Shi, W.S. Yan, J. Gao, and W.B. Li, “Pathfollowing control of an AUV in the horizontal plane with constant ocean currents,” Acta Armamentarii, vol. 31, no. 3, pp. 375–379, 2010. View at: Google Scholar
 X.Q. Cheng, J.Y. Qu, Z.P. Yan, and X.Q. Bian, “H_{∞} robust faulttolerant controller design for an autonomous underwater vehicle's navigation control system,” Journal of Marine Science and Application, vol. 9, no. 1, pp. 87–92, 2010. View at: Publisher Site  Google Scholar
 S. Liu, D. Wang, and E. Poh, “Nonlinear output feedback tracking control for AUVs in shallow wave disturbance condition,” International Journal of Control, vol. 81, no. 11, pp. 1806–1823, 2008. View at: Publisher Site  Google Scholar  MathSciNet
 A. P. Aguiar and A. M. Pascoal, “Dynamic positioning and waypoint tracking of underactuated AUVs in the presence of ocean currents,” International Journal of Control, vol. 80, no. 7, pp. 1092–1108, 2007. View at: Publisher Site  Google Scholar  MathSciNet
 F.D. Gao, C.Y. Pan, Y.Y. Han, and X. Zhang, “Nonlinear trajectory tracking control of a new autonomous underwater vehicle in complex sea conditions,” Journal of Central South University, vol. 19, no. 7, pp. 1859–1868, 2012. View at: Publisher Site  Google Scholar
 L.J. Zhang, X. Qi, Y.J. Pang, and D.P. Jiang, “Adaptive output feedback control for trajectory tracking of AUV in wave disturbance condition,” International Journal of Wavelets, Multiresolution and Information Processing, vol. 11, no. 3, pp. 1–15, 2013. View at: Publisher Site  Google Scholar  MathSciNet
 H. Joe, M. Kim, and S.C. Yu, “Secondorder slidingmode controller for autonomous underwater vehicle in the presence of unknown disturbances,” Nonlinear Dynamics, vol. 78, no. 1, pp. 183–196, 2014. View at: Publisher Site  Google Scholar
 H.G. Zhang, Q.L. Wei, and Y.H. Luo, “A novel infinitetime optimal tracking control scheme for a class of discretetime nonlinear systems via the greedy HDP iteration algorithm,” IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics, vol. 38, no. 4, pp. 937–942, 2008. View at: Publisher Site  Google Scholar
 Q.L. Wei, H.G. Zhang, D.R. Liu, and Y. Zhao, “An optimal control scheme for a class of discretetime nonlinear systems with time delays using adaptive dynamic programming,” Acta Automatica Sinica, vol. 36, no. 1, pp. 121–129, 2010. View at: Publisher Site  Google Scholar  MathSciNet
 Q.X. Qu, Y.H. Luo, and H.G. Zhang, “Robust approximate optimal tracking control of timevarying trajectory for nonlinear affine systems,” Control Theory & Applications, vol. 33, no. 1, pp. 77–84, 2016. View at: Publisher Site  Google Scholar
 G.Y. Tang, “Suboptimal control for nonlinear systems: a successive approximation approach,” Systems and Control Letters, vol. 54, no. 5, pp. 429–434, 2005. View at: Publisher Site  Google Scholar  MathSciNet
 G.Y. Tang and D.X. Gao, “Approximation design of optimal controllers for nonlinear systems with sinusoidal disturbances,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 2, pp. 403–414, 2007. View at: Publisher Site  Google Scholar  MathSciNet
 T. I. Fossen, Guidance and Control of Ocean Vehicles, John Wiley & Sons, New York, NY, USA, 1994.
 T. Prestero, Verification of a SixDegree of Freedom Simulation Model for the REMUS Autonomous Underwater Vehicles, MIT, 2001.
 O.E. Fjellstad and T. I. Fossen, “Position and attitude tracking of AUV’s: a quaternion feedback approach,” IEEE Journal of Oceanic Engineering, vol. 19, no. 4, pp. 512–518, 1994. View at: Publisher Site  Google Scholar
 Q. Yang, H. Su, and G.Y. Tang, “Approximate optimal tracking control for nearsurface AUVs with wave disturbances,” Journal of Ocean University of China, vol. 15, no. 5, pp. 789–798, 2016. View at: Publisher Site  Google Scholar
 H. Ma, G.Y. Tang, and W. Hu, “Feedforward and feedback optimal control with memory for offshore platforms under irregular wave forces,” Journal of Sound and Vibration, vol. 328, no. 45, pp. 369–381, 2009. View at: Publisher Site  Google Scholar
 B.L. Zhang and Q.L. Han, “Networkbased modelling and active control for offshore steel jacket platform with TMD mechanisms,” Journal of Sound and Vibration, vol. 333, pp. 6796–6814, 2014. View at: Google Scholar
 S.Y. Han, D. Wang, Y.H. Chen, G.Y. Tang, and X.X. Yang, “Optimal tracking control for discretetime systems with multiple input delays under sinusoidal disturbances,” International Journal of Control, Automation and Systems, vol. 13, no. 2, pp. 292–301, 2015. View at: Publisher Site  Google Scholar
 X.Q. Cheng, Research on the Robust Heading Control for AUV Based on Feedback Linearization in Near Surface, Harbin Engineering University, Harbin, China, 2008.
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Copyright © 2017 Qing Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.