Research Article | Open Access

Wei Zhao, Li Tang, Yan-Jun Liu, "Disturbance Observer-Based Adaptive Neural Network Control of Marine Vessel Systems with Time-Varying Output Constraints", *Complexity*, vol. 2020, Article ID 6641758, 12 pages, 2020. https://doi.org/10.1155/2020/6641758

# Disturbance Observer-Based Adaptive Neural Network Control of Marine Vessel Systems with Time-Varying Output Constraints

**Academic Editor:**Qiuye Sun

#### Abstract

This article investigates an adaptive neural network (NN) control algorithm for marine surface vessels with time-varying output constraints and unknown external disturbances. The nonlinear state-dependent transformation (NSDT) is introduced to eliminate the feasibility conditions of virtual controller. Moreover, the barrier Lyapunov function (BLF) is used to achieve time-varying output constraints. As an important approximation tool, the NN is employed to approximate uncertain and continuous functions. Subsequently, the disturbance observer is structured to observe time-varying constraints and unknown external disturbances. The novel strategy can guarantee that all signals in the closed-loop system are semiglobally uniformly ultimately bounded (SGUUB). Finally, the simulation results verify the benefit of the proposed method.

#### 1. Introduction

In recent years, the adaptive control [1, 2] of nonlinear systems [3–7] is attracted more and more attention. As two main approximation tools, neural networks (NNs) and fuzzy logic systems (FLSs) [8, 9] are used to dispose some uncertain variables and unknown functions. For example, in [10], by using NNs, the neural network adaptive fault-tolerant control method under multiobjective constraints is designed. An effective finite-frequency *H*−/ fault detection method is proposed for the descriptor system in [11]. It is noted that the aforementioned studies neglect the influence of constraints on nonlinear systems.

In general, because of physical limitations and safety factors of the system, multifarious constraints exist in most practical systems, such as flexible string system [12], flexible-joint robot manipulators [13], nonuniform gantry crane systems [14], and flexible aerial refueling hose [15]. The barrier Lyapunov function (BLF) is presented in [16] for strict feedback systems. Since then, as a main tool, the BLF plays a crucial role in dealing with constraints. Subsequently, full state constraints adaptive controller is designed based on BLF and backstepping technique in [17]. In order to handle the control problems of state-constrained nonlinear systems better, in [18, 19], the authors present an integral BLF. It is worth noting that none of the above articles involve time-varying constraints. In [20–22], the adaptive controller of time-varying full-state constraint is constructed. Thus, the constant constraints expand the time-varying constraints. However, there is a limitation in the adaptive control design based on BLF, i.e., the feasibility of the virtual controller, which will increase the cost of control design. Hence, how to eliminate the feasibility of the controller is an urgent problem. It is noted that few of them involve the feasibility conditions of intermediate controllers to marine vessel systems, which is also a challenge of this paper.

More recently, artificial intelligence has made impressive progress. More and more adaptive tracking control of marine vessel is widely studied. Subsequently, a number of significant results are proposed around the adaptive control design of the ship. The adaptive NN controller for ships is designed in [23]. Afterward, much effort has been made in addressing the situation where the ship system parameters are unknown. For the sliding mode method based on the model, an adaptive control approach is proposed in [24]. The precise recognition and learning control of marine ships under an unknown dynamic environment is further developed in [25]. Based on the above description, we propose an adaptive NN control method of marine vessel with time-varying output constraints.

In this article, we try our best to propose an adaptive NN control method for the marine vessel system with disturbance observer and time-varying output constraints. Furthermore, NNs are developed to deal with uncertain parameters and unknown functions. As is known to all, in the existing literature, this kind of marine vessel system with time-varying state constraints is rarely dealt with by removing the feasibility of virtual controller. A novel transformation function is proposed to ensure that the states do not violate the bounds. The main work and contributions of this article are as follows:(1)As is known to all, most of the existing constraint control methods involve feasibility conditions. In this article, a new coordinate transformation is designed to resolve the time-varying position constraint of marine surface vessel without involving feasibility conditions. The coordinate transformation is used to get a control scheme that can avoid the feasibility condition, which is conducive to the realization of the control scheme.(2)An exponential convergence disturbance observer is designed to deal with time-varying constraints and uncertain external disturbances, which are created by wind, waves, currents, and so on.

The specific organization of this paper is as follows. First, the dynamic model of the surface vessels is introduced. Then, the detailed procedure of BLF derivation for dealing with time-varying output constraints is given, in which the disturbance observer, the adaptive controller, and the adaptive laws are designed. Finally, simulation results verify the effectiveness of the proposed method.

#### 2. Preliminaries

We consider the dynamics of 3 degree-of-freedom (3DOF) surface ships with model uncertainty and external disturbance. The dynamic model between the vessel’s position and velocity can be expressed as follows:where the vector stands for the output of earth-frame positions and heading. is the position of the vessel, and denotes the heading, respectively. The vector represents the velocity vector of the vessel. The rotation matrix can be described as follows:

In this paper, the output of position and heading of marine vessel system is constrained bywhere

The relationship between the vessel’s velocity and the force acting on the vessel can be described by the dynamic model of vessel [26] as follows:where is unknown and time-varying external disturbance caused by wind and waves, and is unknown restorative force caused by the ship’s gravity and buoyancy. The control input is . Besides, denotes the positive definite inertia matrix, represents the torques of Coriolis and centripetal, and stands for damping matrix. They are, respectively,wherewith , , , , , , , , , , , , , , , , , , , and are design parameters.

Define and , and then (1) and (5) can be transformed into the following equation:

The main task of this article is to design controller and adaptive laws so that the system output of vessel tracks the expected trajectory as effectively as possible. Meanwhile, all signals are bounded in the closed-loop systems and their constraints are not violated forever.

As important approximation methods in nonlinear adaptive control, the radial basis function neural networks (RBFNNs) are usually utilized to deal with unknown and continuous functions. In this article, the RBFNNs are applied to approximate the unknown and continuous function . The following relation holds:where represents approximation error and satisfies , which will be used in the following derivation. is the input vector, the optimal weight matrix is denotes the number of neurons. The known continuous basis function vector is and is the Gaussian function withwhere stands for the center of the receptive field, and the width of the Gaussian basis function is .

*Remark 1. *As usual, the neural network chosen in this article has some special properties. Firstly, with the increase of the number of neurons, approximation error will gradually decrease. Secondly, the approximate area of the NNs is uniformly covered by the receptive field . Finally, based on the equation , we can get the width of the Gaussian function and denotes the maximal Euclid distance of the approximation region.

It is necessary to research the marine vessel position constraints because marine environmental disturbances are time-varying and marine conditions are complex. In [27], for the nonlinear system control algorithm based on BLF, the state constraint is converted into the constraint of tracking error, which increases the restriction on the initial conditions of the system. It should be noted that most of the previous research studies on constraints were carried out in simple cases. To deal with time-varying output constraints gracefully, we consider the following nonlinear state-dependent transformation (NSDT):where the initial states , in which . From (11), it should be pointed that tends to infinity when is close to , for any initial condition . That isTherefore, it is noted that for any , if , then can be guaranteed for any , that is only need , , are bounded, and time-varying output constraints will not be violated. In other words, the issue of output constraints attributed to making sure that for is bounded.

*Assumption 1. *Time-varying function , and , are bounded and continuous, as well as its first derivative is also bounded and continuous.

The time derivative of is given aswherefor

Then, (13) can also be rewritten as follows:whereIn this paper, feasibility condition in the control design is removed. Accordingly, it is much better to implement control methods. Most of the existing vessel constraint methods dispose the constant state constraints. For comprehensive consideration, the adaptive NN control of time-varying output constraints is studied in this article.

The different coordinate transformation is introduced as follows:withwhere is the intermediate controller and it will be defined later. It should be clear that is bounded in the set of .

*Assumption 2. *The expected tracking signal , is bounded, and the derivative of with respect to time is also bounded.

Then, the time derivative of is given as follows:wherefor , and according to the definition of and , we know that and are known and computable and will be used in the design of the controller in the later.

*Remark 2. *For uncertain vessel nonlinear system, to make sure that the state constraints are never violated their bounded, the virtual controller must satisfy the feasibility condition:where is the virtual controller and and are smooth functions. Although the virtual controller is related to state variables, it also depends on other design parameters. The feasibility conditions of (22) must be satisfied, and only in this way can the control scheme be implemented effectively. However, it is unrealistic and even difficult to find such parameters.

*Remark 3. *It should be pointed that according to the actual need, the external disturbances are required to be bounded. Note that all of these boundary conditions are not necessary to implement adaptive control, and they can only be used in analysis.

*Remark 4. *If and are the nonnegative real numbers, , is nonzero real number, and . Then, we can getIn this article, the following disturbance observer is constructed to deal with unknown disturbance:where is an observer gain positive definite matrix, is the introduced intermediate variable, and the definition of is given in the following equation:Then, differentiating , it yieldswhere represents estimation error vector and , vector denotes disturbance estimation, and vector is external disturbance. Define , where is a positive constant.

The detailed design steps are as follows: the following Lyapunov function is chosen: Based on in (17), and the definition of system (8), then taking derivation of , one obtains Based on (28), the time derivative of is given by The virtual controller is designed as follows: Substituting (30) into (29), we have the Lyapunov function is designed as follows: where denotes the NN weight error, is the estimation of weight matrix , and is positive definite matrix.Taking the derivative of in (18), and based on , it yieldsSubstituting (33) into (32), it leads toThe unknown function is defined bywhere is an unknown and continuous function and are the inputs of the NNs. With the help of the NN approximation capability, the unknown function can be rewritten as follows:The actual controller is constructed as follows:Substituting (37) and (36) into (34), one obtainsSubstituting (26) into (38), we haveThe adaptive laws are selected as follows:where is the design parameter.

Using Young’s inequality, we have the following inequalities:where and are the design parameters.

Substituting all the above inequalities into (39), we haveConsider the Lyapunov function as follows:From (31) and (42), we can obtainwherewith and standing for the largest and minimum eigenvalue of a matrix, respectively.

Theorem 1. *Consider the marine vessel nonlinear system with unknown time-varying disturbances (8) under the virtual controller (30), actual controller (37), and adaptive laws (40). Then, we have the following: (1) error signal undulates in a small neighborhood close to zero; (2) all signals in the closed-loop system are SGUUB; and (3) in the meantime, the output signal will not violate the time-varying boundary.*

*Proof. *According to (44), we can obtainFrom (44), it is clear that the signals , , , and are SGUUB. Since is bounded by with being also bounded. As well, is bounded. Because is bounded in the set , from , we can infer that . Based on (12), for any initial condition , one deduces that is bounded. Because , , , and are bounded, gradually, it is obtained that the virtual controller is also bounded. According to , we can infer that is bounded.

#### 3. Simulation Example

In this section, to illustrate the effectiveness of the proposed method, we conduct the simulation on the model of Cybership II, which is a 1 : 70 scale supply vessel replica setup in a marine control laboratory in the Norwegian University of Science and Technology [28]. The following expected trajectory is considered:where the matrices are chosen as follows:

The disturbance vector is given as follows:

The initial state variables are selected as and . The state variables , , and satisfy , , and with , , , , , and .

The initial value of the observer is selected as . The design parameters are given as . The parameter matrices are given as , , , and . The control gains are set as and .

Figures 1–11 show the simulation results. Figures 1–3 show the output signal , reference signal , and constraints interval and with , which indicates that the tracking trajectory is perfect, and the state variable does not violate the constraint interval. Figure 4 indicates the trajectory of the state variable . Figures 5 and 6 are given to illustrate the trajectories of tracking error and , and it can be seen that the error is small enough. The trajectory of actual controller is displayed in Figure 7. From Figure 8, it indicates that the disturbance observer is constructed to deal with the external disturbance better. Meanwhile, Figures 9 and 10 are used to explain the trajectory of adaptive laws. We can clearly see that the adaptive laws are bounded. Figure 11 shows the trajectory of virtual controller. The results show that the adaptive control method is effective.

**(a)**

**(b)**

**(c)**

**(a)**

**(b)**

**(c)**

**(a)**

**(b)**

**(c)**

**(a)**

**(b)**

**(c)**

**(a)**

**(b)**

**(c)**

**(a)**

**(b)**

**(c)**

**(a)**

**(b)**

**(c)**

**(a)**

**(b)**

**(c)**

#### 4. Conclusion

The adaptive NN control method is studied for vessel nonlinear systems with disturbance observer and time-varying output constraints in this paper. Compared with the existing research on vessel, the method studied in this article is more practical. First, the feasibility condition of virtual controller is eliminated by introducing nonlinear state correlation function and new coordinate transformation. Only in this way, we can deal with control design gracefully. Then, with the help of NNs, the unknown and continuous function is approximated to simplify the design of the controller. Next, the disturbance observer is constructed to dispose the disturbances. Finally, the control performance is tested through simulation examples. In the future work, we will apply the constraint form of this paper to the position and speed constraints of flexible manipulator.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This work was supported in part by the National Natural Science Foundation of China under Grants 61973147 and 61903168, in part by the Doctoral Research Initiation of Foundation of Liaoning Province under Grant 2020-BS-243, and in part by the Natural Scientific Research Foundation Guiding Plan of Liaoning Province under Grant 2019-ZD-0701.

#### References

- Y. Xiao, C. L. P. Chen, and B. Wang, “Bandwidth degradation QoS provisioning for adaptive multimedia in wireless/mobile networks,”
*Computer Communications*, vol. 25, no. 13, pp. 1153–1161, 2002. View at: Publisher Site | Google Scholar - L. Cheng, Y. Liu, Z. G. Hou, M. Tan, D. J. Du, and M. R. Fei, “A rapid spiking neural network approach with an application on hand gesture recognition,”
*IEEE Transactions on Cognitive and Developmental Systems*, 2019. View at: Publisher Site | Google Scholar - T. Li, R. Zhao, C. L. P. Chen, L. Fang, and C. Liu, “Finite-time formation control of under-actuated ships using nonlinear sliding mode control,”
*IEEE Transactions on Cybernetics*, vol. 48, no. 11, pp. 3243–3253, 2018. View at: Publisher Site | Google Scholar - T. Li, Z. Li, D. Wang, and C. L. Chen, “Output-feedback adaptive neural control for stochastic nonlinear time-varying delay systems with unknown control directions,”
*IEEE Transactions on Neural Networks and Learning Systems*, vol. 26, no. 6, pp. 1188–1201, 2015. View at: Publisher Site | Google Scholar - J. Yu, B. Chen, H. Yu, C. Lin, and L. Zhao, “Neural networks-based command filtering control of nonlinear systems with uncertain disturbance,”
*Information Sciences*, vol. 426, pp. 50–60, 2018. View at: Publisher Site | Google Scholar - L.-B. Wu and J. H. Park, “Adaptive fault-tolerant control of uncertain switched nonaffine nonlinear systems with actuator faults and time delays,”
*IEEE Transactions on Systems, Man, and Cybernetics: Systems*, vol. 50, no. 9, pp. 3470–3480, 2020. View at: Publisher Site | Google Scholar - L. Tang, D. Ma, and J. Zhao, “Adaptive neural control for switched non-linear systems with multiple tracking error constraints,”
*IET Signal Processing*, vol. 13, no. 3, pp. 330–337, 2019. View at: Publisher Site | Google Scholar - S. Tong, X. Min, and Y. Li, “Observer-based adaptive fuzzy tracking control for strict-feedback nonlinear systems with unknown control gain functions,”
*IEEE Transactions on Cybernetics*, vol. 50, no. 9, pp. 3903–3913, 2020. View at: Publisher Site | Google Scholar - L. Liu, Y.-J. Liu, D. Li, S. Tong, and Z. Wang, “Barrier Lyapunov function-based adaptive fuzzy FTC for switched systems and its applications to resistance-inductance-capacitance circuit system,”
*IEEE Transactions on Cybernetics*, vol. 50, no. 8, pp. 3491–3502, 2020. View at: Publisher Site | Google Scholar - Q. Cui, Y. Wang, and Y. Song, “Neuroadaptive fault-tolerant control under multiple objective constraints with applications to tire production systems,”
*IEEE Transactions on Neural Networks and Learning Systems*, pp. 1–10, 2020. View at: Publisher Site | Google Scholar - Z. Wang, P. Shi, and C.-C. Lim, “H−∕H∞ fault detection observer in finite frequency domain for linear parameter-varying descriptor systems,”
*Automatica*, vol. 86, pp. 38–45, 2017. View at: Publisher Site | Google Scholar - S. Zhang, D. Huang, and W. He, “Active vibration control for a flexible string system with input backlash,”
*IET Control Theory & Applications*, vol. 10, no. 7, pp. 800–805, 2016. View at: Publisher Site | Google Scholar - H. Liu and Y. Huang, “Robust adaptive output feedback tracking control for flexible-joint robot manipulators based on singularly perturbed decoupling,”
*Robotica*, vol. 36, no. 6, pp. 822–838, 2018. View at: Publisher Site | Google Scholar - W. He and S. S. Ge, “Cooperative control of a nonuniform gantry crane with constrained tension,”
*Automatica*, vol. 66, no. 4, pp. 146–154, 2016. View at: Publisher Site | Google Scholar - Z. Liu, J. Liu, and W. He, “Modeling and vibration control of a flexible aerial refueling hose with variable lengths and input constraint,”
*Automatica*, vol. 77, pp. 302–310, 2017. View at: Publisher Site | Google Scholar - K. P. Tee, S. S. Ge, and E. H. Tay, “Barrier Lyapunov functions for the control of output-constrained nonlinear systems,”
*Automatica*, vol. 45, no. 4, pp. 918–927, 2009. View at: Publisher Site | Google Scholar - D.-P. Li, Y.-J. Liu, S. Tong, C. L. P. Chen, and D.-J. Li, “Neural networks-based adaptive control for nonlinear state constrained systems with input delay,”
*IEEE Transactions on Cybernetics*, vol. 49, no. 4, pp. 1249–1258, 2019. View at: Publisher Site | Google Scholar - K. P. Tee and S. S. Ge, “Control of state-constrained nonlinear systems using integral barrier Lyapunov functionals,” in
*Proceedings of the 2012 IEEE 51st IEEE Conference on Decision and Control (CDC)*, pp. 3239–3244, Maui, HI, USA, December 2012. View at: Google Scholar - L. Liu, Y. J. Liu, S. C. Tong, and C. L. P. Chen, “Integral barrier Lyapunov function based adaptive control for switched nonlinear systems,”
*Science China Information Sciences*, vol. 63, no. 3, Article ID 132203, 2020. View at: Publisher Site | Google Scholar - L. Liu and X. Li, “Event-triggered tracking control for active seat suspension systems with time-varying full-state constraints,”
*IEEE Transactions on Systems, Man, and Cybernetics: Systems*, p. 1, 2020. View at: Publisher Site | Google Scholar - D. Li, C. L. P. Chen, Y.-J. Liu, and S. Tong, “Neural network controller design for a class of nonlinear delayed systems with time-varying full-state constraints,”
*IEEE Transactions on Neural Networks and Learning Systems*, vol. 30, no. 9, pp. 2625–2636, 2019. View at: Publisher Site | Google Scholar - Y.-J. Liu, L. Ma, L. Liu, S. Tong, and C. L. P. Chen, “Adaptive neural network learning controller design for a class of nonlinear systems with time-varying state constraints,”
*IEEE Transactions on Neural Networks and Learning Systems*, vol. 31, no. 1, pp. 66–75, 2020. View at: Publisher Site | Google Scholar - Z.-P. Jiang, “Global tracking control of underactuated ships by Lyapunov’s direct method,”
*Automatica*, vol. 38, no. 2, pp. 301–309, 2002. View at: Publisher Site | Google Scholar - N. Khaled and N. Chalhoub, “A dynamic model and a robust controller for a fully-actuated marine surface vessel,”
*Journal of Vibration and Control*, vol. 17, no. 6, pp. 801–812, 2011. View at: Publisher Site | Google Scholar - S.-L. Dai, C. Wang, and F. Luo, “Identification and learning control of ocean surface ship using neural networks,”
*IEEE Transactions on Industrial Informatics*, vol. 8, no. 4, pp. 801–810, 2012. View at: Publisher Site | Google Scholar - S. Yin and B. Xiao, “Tracking control of surface ships with disturbance and uncertainties rejection capability,”
*IEEE/ASME Transactions on Mechatronics*, vol. 22, no. 3, pp. 1154–1162, 2016. View at: Google Scholar - K. Zhao, Y. D. Song, T. D. Ma, and L. He, “Prescribed performance control of uncertain Euler-Lagrange systems subject to full state constraints,”
*IEEE Transactions on Neural Networks and Learning Systems*, vol. 29, no. 8, pp. 3478–3489, 2017. View at: Publisher Site | Google Scholar - K. P. Tee and S. S. Ge, “Control of fully actuated ocean surface vessels using a class of feedforward approximators,”
*IEEE Transactions on Control Systems Technology*, vol. 14, no. 4, pp. 750–756, 2006. View at: Publisher Site | Google Scholar

#### Copyright

Copyright © 2020 Wei Zhao 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.