A robust neural network approximation-based prescribed performance output-feedback controller for autonomous underwater vehicles with actuators saturation

Engineering Applications of Artificial Intelligence 88 (2020) 103382

Contents lists available at ScienceDirect

Engineering Applications of Artificial Intelligence journal homepage: www.elsevier.com/locate/engappai

A robust neural network approximation-based prescribed performance output-feedback controller for autonomous underwater vehicles with actuators saturation✩ Omid Elhaki a , Khoshnam Shojaei a,b ,∗ a b

Department of Electrical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran Digital Processing and Machine Vision Research Center, Najafabad Branch, Islamic Azad University, Najafabad, Iran

ARTICLE

INFO

Keywords: Actuator saturation Adaptive robust controller Autonomous underwater vehicles High-gain observer Prescribed performance technique Multi-layer neural networks

ABSTRACT A robust neural network approximation-based output-feedback tracking controller is proposed for autonomous underwater vehicles (AUVs) in six degrees-of-freedom in this paper. The prescribed performance technique is employed to obtain some pre-defined maximum overshoot/undershoot, convergence speed and ultimate tracking accuracy for the tracking errors. A high-gain observer is used to approximate unavailable velocity vector which is crucial to design the output-feedback controller. A robust multi-layer neural network and adaptive robust techniques are combined to simultaneously compensate for the unmodeled dynamics, system nonlinearities, exogenous kinematic and dynamic disturbances, and reduce the risk of the actuator saturation. Then, the uniform ultimate boundedness stability of the closed-loop control system is proved via a Lyapunovbased stability synthesis. It is demonstrated that the posture tracking errors converge to a vicinity of the origin with a guaranteed prescribed performance during the tracking mission without velocity measurements. Finally, simulation results with a comparative study verify the theoretical findings.

1. Introduction Neural networks (NNs) are powerful tools to design intelligent controllers for uncertain robotic systems. Among many robotic systems, the intelligent control problem of autonomous underwater vehicles (AUVs) has been an interesting field of the research in the past years because of their applications such as sea floor mapping, exploration, underwater archaeology, finding of ship and airplane wreckage (Krupínski et al., 2017; Cui et al., 2017; Fischer et al., 2014; Zhong et al., 2018; Shen et al., 2018; Shojaei, 2018b; Xiang et al., 2018; Yu et al., 2018; Shojaei, 2016). AUVs operate in a rough environment that includes ocean currents, waves, obstacles, and high pressure of the water that raise difficulties to maintain tracking accuracy in six degrees-of-freedom. Accordingly, the problem of designing nonlinear controllers for AUVs is very challenging because of their highly nonlinear, coupled, and uncertain dynamics against a turbulent environment. To this end, novel robust and neural network-based controllers are designed for such uncertain systems (Chu et al., 2018; Peng et al., 2019; Qiao and Zhang, 2018; van de Ven et al., 2005; Fujii and Ura, 1991; Du et al., 2014; Zhu et al., 2018; Ahmed and Hasegawa, 2013). In general, the positions and velocities of AUVs are necessary in the controller design procedure.

Nonetheless, AUVs may not be equipped with velocity sensors due to the economic and other limitations. Even if velocity sensors are available, their measurements are not accurate because of the noise. As a result, the alternative methods are of high priority. Hence, the use of high-gain observers is an effective way for the estimation of unavailable states to tackle the aforementioned problem (Ge and Zhang, 2003; Tee et al., 2008). The high-gain observer was used for the design of outputfeedback controllers for linear systems in Doyle and Stein (1979). Then, it was extended to the design of observer-based controllers for nonlinear systems (Tornambè, 1989; Behtash, 1990). In Shojaei (2018a), a formation tracking controller was pointed out for a group of tractor–trailer systems by using the high-gain observer. In Ren et al. (2010), a neural adaptive observer-based backstepping controller has been presented for uncertain output-feedback systems. Shojaei (2017) has developed an output-feedback controller for a group of wheeled mobile robots. Li et al. (2015) introduced finite-time output-feedback controller for 6-degree-of-freedom (DOF) AUVs by employing a finitetime convergent observer. Another issue with regard to the control of AUVs is that the classical nonlinear control design approaches just can guarantee the stability of the closed-loop system, however, they cannot satisfy prespecified transient and steady-state behaviors of the tracking

✩ No author associated with this paper has disclosed any potential or pertinent conflicts which may be perceived to have impending conflict with this work. For full disclosure statements refer to https://doi.org/10.1016/j.engappai.2019.103382. ∗ Corresponding author at: Department of Electrical Engineering, Najafabad Branch, Islamic Azad University, Najafabad, Iran. E-mail address: [email protected] (K. Shojaei).

https://doi.org/10.1016/j.engappai.2019.103382 Received 31 March 2019; Received in revised form 6 November 2019; Accepted 14 November 2019 Available online xxxx 0952-1976/© 2019 Elsevier Ltd. All rights reserved.

O. Elhaki and K. Shojaei

Engineering Applications of Artificial Intelligence 88 (2020) 103382

the external disturbances in both kinematic and dynamic levels and neural network approximation errors. Hence, the robustness of the proposed neural control scheme will be guaranteed at every level. The upcoming parts of this paper are arranged in the following order. In the next section, the problem formulation and some introductory concepts are stated. In Section 3, a new tracking controller is designed and its closed-loop stability is deeply investigated. Simulation results are demonstrated in Section 4, and conclusions are finally drawn in the last section.

errors such as the convergence rate, maximum undershoot∕overshoot, and final tracking accuracy. To this end, the performance bound technique (Bechlioulis and Rovithakis, 2008) can be applied to prescribe the transient and steady-state behaviors of the tracking errors. This method ensures the stability with a guaranteed prespecified performance that is necessary to design nonlinear controllers in the presence of model uncertainties and time-varying external forces (Wang et al., 2017). The aforesaid problem becomes more difficult when the AUV actuators cannot endure every amount of input signals in the practical applications, and actuator saturation occurs. A saturated controller is designed for 5-DOF underactuated AUVs in Shojaei (2016) without guaranteeing a prescribed performance. Shojaei (2018b) has developed an output-feedback controller for 5-DOF underactuated AUVs, but the proposed controller cannot guarantee a prescribed performance. Some novel robust trajectory tracking controllers are designed for 5-DOF underactuated AUVs in Shojaei and Dolatshahi (2017) and Shojaei (2019), though both controllers ignore the advantages of output-feedback and prescribed performance-based methods. In Bechlioulis et al. (2017), a robust controller is designed with a guaranteed prescribed performance for 6-DOF underactuated AUVs. Du et al. (2015) has developed a robust output-feedback controller for fully actuated ships without using the benefits of the prescribed performance method. In Dai et al. (2018), a full state feedback platoon formation controller is developed for fully actuated marine ships. A non-prescribed performance-based controller is studied in Park et al. (2017) for underactuated surface vessels by integrating neural networks and output-feedback approaches. A neural target tracking controller is introduced in Elhaki and Shojaei (2018) for 5-DOF underactuated AUVs with a prescribed performance. However, the proposed controllers in Bechlioulis et al. (2017) and Elhaki and Shojaei (2018) cannot be implemented without velocity measurements, and both controllers ignore the advantages of the output-feedback method which is the main topic of this paper. Compared with all of formerly proposed controllers for AUVs, the problem of designing robust neural network-based controller for 6-DOF fully-actuated AUVs with a prescribed performance without velocity measurements with the actuators saturation compensation is addressed by this paper which is not found in the literature to the best of authors’ knowledge. Hence, in the light of the foregoing discussions, the main contributions of this work are summed up as follows to introduce a more evolved intelligent control solution for AUVs compared with all the previous works (Krupínski et al., 2017; Cui et al., 2017; Fischer et al., 2014; Zhong et al., 2018; Shen et al., 2018; Shojaei, 2018b; Xiang et al., 2018; Yu et al., 2018; Shojaei, 2016; Chu et al., 2018; Peng et al., 2019; Qiao and Zhang, 2018; van de Ven et al., 2005; Fujii and Ura, 1991; Du et al., 2014; Zhu et al., 2018; Ahmed and Hasegawa, 2013; Ge and Zhang, 2003; Tee et al., 2008; Doyle and Stein, 1979; Tornambè, 1989; Behtash, 1990; Shojaei, 2018a; Ren et al., 2010; Shojaei, 2017; Li et al., 2015; Bechlioulis and Rovithakis, 2008; Wang et al., 2017; Shojaei and Dolatshahi, 2017; Shojaei, 2019; Bechlioulis et al., 2017; Du et al., 2015; Dai et al., 2018; Park et al., 2017; Elhaki and Shojaei, 2018): (1) In contrast to Bechlioulis et al. (2017) that proposed a novel state feedback controller for AUVs recently, this work develops a fully output-feedback neural dynamic surface controller for AUVs under model uncertainties and external forces by employing a high-gain observer to estimate the velocity vector of the vehicle. It does not need a prior knowledge of the system dynamics, and only depends on the measurable postures of the vehicle. (2) By employing the prescribed performance bound approach, some restrictions will be imposed on the system posture errors to satisfy prespecified transient and steadystate behaviors of the tracking errors such as the convergence rate, maximum overshoot∕undershoot and final tracking accuracy. (3) A projection-type adaptive multi-layer neural network is applied to cope with the structured uncertainties, nonlinear-in-parameter nonlinearities, and saturation nonlinearity at every level. (4) In spite of most previous works, adaptive robust controllers (ARCs) are utilized in both kinematic and dynamic control laws to simultaneously compensate for

2. Problem formulation and preliminaries 2.1. Notations The following notations are used in this article. For every given matrix 𝐴, the Frobenius norm is illustrated by ‖𝐴‖2𝑓 = 𝑡𝑟(𝐴𝑇 𝐴), where 𝑡𝑟(.) stands for 𝑡𝑟𝑎𝑐𝑒 operator. Euclidean norm of an arbitrary vector 𝑥 ∈ ℜ𝑛 is denoted by ‖𝑥‖2 = 𝑥𝑇 𝑥, ℜ+ defines a set of real positive numbers, 𝜆𝑚𝑎𝑥 (.) and 𝜆𝑚𝑖𝑛 (.) demonstrate the largest and smallest eigenvalues of a matrix, respectively, 𝑑𝑒𝑡(.) stands for 𝑑𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑛𝑡 operator, 𝑃 𝑟𝑜𝑗(.) points out 𝑃 𝑟𝑜𝑗𝑒𝑐𝑡𝑖𝑜𝑛 operator, and 𝑑𝑖𝑎𝑔[.] also stands for a diagonal matrix. 2.2. AUV kinematics and dynamics The complete nonlinear mathematical kinematic and dynamic models of a 6-DOF AUV are as follows (Do and Pan, 2009; Fossen, 2002): 𝑞̇ = 𝐽 (𝑞)𝜈 + 𝛿

(1)

𝑀 𝜈̇ = −𝐶(𝜈)𝜈 − 𝐷𝜈 − 𝑔(𝑞) + 𝑢𝑎 + 𝐻(𝜈) + 𝛥 where 𝑞 = [𝑥, 𝑦, 𝑧, 𝜙, 𝜃, 𝜓]𝑇 represents the position and orientation of the vehicle. The term 𝛿 ∈ ℜ6×1 describes the vector of kinematic disturbances which show the effects of waves and ocean currents. Such disturbances cause to unwanted drifts in the vehicle velocities and, as a result, they cause to posture errors. It is assumed that 𝛿 is upper-bounded as ‖𝛿‖ ≤ 𝛾 where 𝛾 ∈ ℜ+ is an unknown scalar. 𝑀 ∈ ℜ6×6 is a positive-definite symmetric inertia matrix, 𝐷 ∈ ℜ6×6 is a matrix of damping forces and torques which is strictly positive, the skew-symmetric matrix 𝐶(𝜈) ∈ ℜ6×6 contains Coriolis and centripetal terms, 𝜈 = [𝑢, 𝑣, 𝑤, 𝑝, 𝑞, 𝑟]𝑇 is the vector of velocities, 𝑔(𝑞) ∈ ℜ6×1 is the vector of the gravity and buoyancy forces/moments, 𝐻(𝜈) ∈ ℜ6×1 is the vector of unmodeled dynamics, nonlinear-in-parameter (NLIP) uncertain functions, and high order terms in terms of velocities, 𝑢𝑎 ∈ ℜ6×1 is the vector of actuator inputs with saturation, 𝛥 = [𝛥1 , … , 𝛥6 ]𝑇 represents the vector of unwanted external disturbance torques and forces that are caused by ocean currents and waves, and they lead to unwanted drifts in input forces and torques. This term is bounded by |𝛥𝑗 | ≤ 𝛿𝑗∗ , ∀𝑗 = 1, … , 6 where 𝛿𝑗∗ are unknown positive constants, and [ ] 𝐽1 (𝑞2 ) 03×3 𝐽 (𝑞) = , 03×3 𝐽2 (𝑞2 ) where 𝐽1 and 𝐽2 are defined in Do and Pan (2009). Assumption 1. The pitch angle fulfills |𝜃(𝑡)| < 𝜋∕2, ∀𝑡 ≥ 0. Remark 1 (Do and Pan, 2009; Fossen, 2002). AUVs do not presumably get into the vicinity of 𝜃 = ±𝜋∕2 because of the metacentric restoring forces existence. Thus, |𝜃| ≤ 𝜃𝑚𝑎𝑥 < 𝜋∕2, ∀𝑡 ≥ 0. Assumption 2. The desired reference trajectory and its first and second derivatives are bounded. Remark 2 (Do and Pan, 2009). In this paper, it is assumed that the center of gravity and the center of buoyancy are situated vertically on the 𝑂𝑏 𝑍𝑏 axis in the body-fixed frame for model (1). Therefore, there are no interaction and couplings in the matrices 𝑀 and 𝐷. Besides, modeling errors and unmodeled dynamics are collected in the terms 𝐻(𝜈) and 𝛥. 2

O. Elhaki and K. Shojaei

Engineering Applications of Artificial Intelligence 88 (2020) 103382

𝜂𝑖 (𝑡) = (𝜂𝑖0 −𝜂𝑖∞ )𝑒−𝑎𝑖 𝑡 +𝜂𝑖∞ , where 𝜂𝑖0 > 𝜂𝑖∞ > 0, and 𝑎𝑖 > 0 demonstrates some lower bound on the convergence speed of the tracking errors. To provide a guarantee for inequality (2), strictly increasing smooth functions 𝑌𝑖 (𝜖𝑒1𝑖 ) are chosen in the following form:

Assumption 3. The posture measurements are only available for the feedback in real-time and the velocity signals are not measurable. Remark 3. In practice, high-quality velocity signals are not easily measurable due to the noise contamination and the noisy signals degrade the feedback controller performance. Since the time-derivative of position measurements greatly increases the signal-to-noise ratio, it is not possible to use the time-derivative of the position to obtain velocities. Moreover, velocity sensors increase the weight and the cost of the vehicle. In consequence, the development of a controller which only needs position measurements is of great importance in many applications. Therefore, the design of observer-based controllers is widely recommended to obviate the need for velocity measurements. Since the separation principle does not often hold for nonlinear systems, the design of the output-feedback or observer-based controllers is a challenging task. This problem is more troublesome in the presence of unmodeled dynamics and external disturbances. This paper copes with this important problem.

𝛽𝑖 𝑒

(𝜖𝑒 +𝑣𝑖 ) 1𝑖

− 𝛼𝑖 𝑒

−(𝜖𝑒 +𝑣𝑖 ) 1𝑖

, 𝑖 = 1, 2, … , 6 (3) 𝑒 1𝑖 + 𝑒 1𝑖 where 𝑣𝑖 = 0.5 ln(℘𝑖 ∕(𝛽𝑖 + 𝛼𝑖 − ℘𝑖 )), 𝑖 = 1, 2, 3, ℘𝑖 are some positive constants, and 𝑣𝑖 = 0.5 ln(𝛼𝑖 ∕𝛽𝑖 ), 𝑖 = 4, 5, 6. Due to the strict monotonicity feature of 𝑌𝑖 and 𝜂𝑖 (𝑡) ≠ 0, the transformed tracking errors 𝜖𝑒1𝑖 = 𝑌𝑖−1 (𝑒1𝑖 (𝑡)∕𝜂𝑖 (𝑡)) exist, which are obtained in the following forms:

𝑌𝑖 (𝜖𝑒1𝑖 ) =

(𝜖𝑒 +𝑣𝑖 )

−(𝜖𝑒 +𝑣𝑖 )

𝜖𝑒1𝑖 =0.5 ln(𝑒̄1𝑖 + 𝛼𝑖 ) + 0.5 ln(𝛽𝑖 + 𝛼𝑖 − ℘𝑖 )

(4)

− 0.5 ln(𝛽𝑖 ℘𝑖 − 𝑒̄1𝑖 ℘𝑖 ), 𝑖 = 1, 2, 3 (5)

𝜖𝑒1𝑖 =0.5 ln(𝑒̄1𝑖 𝛽𝑖 + 𝛼𝑖 𝛽𝑖 ) − 0.5 ln(𝛼𝑖 𝛽𝑖 − 𝑒̄1𝑖 𝛼𝑖 ), 𝑖 = 4, 5, 6

where 𝑒̄1𝑖 = 𝑒1𝑖 (𝑡)∕𝜂𝑖 (𝑡) is 𝑖th component of the normalized tracking error vector. Now, differentiating (4) and (5) lead to the transformed error dynamics in vectorial form as follows:

The matrix 𝐽1 (𝑞2 ) is orthogonal and globally invertible since 𝐽1−1 (𝑞2 ) = 𝐽1𝑇 (𝑞2 ). Note that 𝑑𝑒𝑡(𝐽2 (𝑞2 )) = 1∕𝑐𝑜𝑠(𝜃). Thus, by recalling Assumption 1, it can be deduced that 𝐽2 (𝑞2 ) is also globally invertible. The model (1) has the following properties (Do and Pan, 2009):

(6)

𝜖̇ 𝑒 = 𝜁(𝑒2 − 𝛩𝑒1 ) where 𝜁 = 𝑑𝑖𝑎𝑔[𝜁1 , 𝜁2 , … , 𝜁6 ], 𝜁𝑖 = 𝛩 = 𝑑𝑖𝑎𝑔[𝜂̇ 1∕𝜂1 , … , 𝜂̇ 6∕𝜂6 ], 𝑒2 = [𝑞̇ 1 − 𝑞𝑑1 , … , 𝑞6 − 𝑞𝑑6 ]𝑇 .

Property 1. The inertia matrix 𝑀 is always symmetric and positive-definite such that: • 𝑀(𝑞) = 𝑀 𝑇 (𝑞) > 0 • 𝜆𝑚 ‖𝑥‖2 ≤ 𝑥𝑇 𝑀𝑥 ≤ 𝜆𝑀 ‖𝑥‖2 , ∀𝑥 ∈ ℜ6 where 𝜆𝑚 < 𝜆𝑀 < ∞, 𝜆𝑚 = min 𝜆𝑚𝑖𝑛 {𝑀(𝑞)}, and 𝜆𝑀 = max 𝜆𝑚𝑎𝑥 {𝑀(𝑞)}

1 ( 1 2𝜂𝑖 𝑒̄1𝑖 +𝛼𝑖

− 𝑒̄ 1−𝛽 ), 𝑖 = 1, 2, … , 6, 1𝑖 𝑖 𝑞̇ 𝑑1 , … , 𝑞̇ 6 − 𝑞̇ 𝑑6 ]𝑇 , and 𝑒1 = [𝑞1 −

Assumption 4. Initial conditions of the constrained tracking errors meet −𝛼𝑖 𝜂𝑖 (0) < 𝑒1𝑖 (0) < 𝛽𝑖 𝜂𝑖 (0), 𝑖 = 1, 2, … , 6. It is simple to validate that (3) has the following conditions: 𝐶1 ∶ −𝛼𝑖 < 𝑌𝑖 (𝜖𝑒1𝑖 ) < 𝛽𝑖

Property 2. The Coriolis matrix 𝐶(𝜈) includes the following properties:

𝐶2 ∶

• 𝑥𝑇 (𝑀̇ − 2𝐶(𝜈))𝑥 = 0, ∀𝑥 ∈ ℜ6 , ∀𝜈 ∈ ℜ6 • 𝐶(𝜈) = −𝐶 𝑇 (𝜈), ∀𝜈 ∈ ℜ6

lim 𝑌 (𝜖 ) 𝜖𝑒 →+∞ 𝑖 𝑒1𝑖

= 𝛽𝑖

lim 𝑌 (𝜖 ) 𝜖𝑒 →−∞ 𝑖 𝑒1𝑖

= −𝛼𝑖

1𝑖

𝐶3 ∶

1𝑖

Property 3. The hydrodynamic damping matrix 𝐷 preserves the following features:

𝐶4 ∶ 𝑌𝑖 (0) = −𝛼𝑖 + ℘𝑖 𝑖 = 1, 2, 3,

Thus, if the designed controller ensures that 𝜖𝑒1𝑖 ∈ 𝐿∞ , then, the tracking errors will stay within these performance bounds as 𝑡 → ∞, and constraints (2) are always fulfilled so that −𝛼𝑖 𝜂𝑖 (∞) < 𝑒1𝑖 (∞) < 𝛽𝑖 𝜂𝑖 (∞) which points out that the tracking errors converge to a small area including the origin with a prescribed performance.

• 𝐷(𝑞) = 𝐷𝑇 (𝑞) > 0 • 𝜆𝑑 ‖𝑥‖2 ≤ 𝑥𝑇 𝐷𝑥 ≤ 𝜆𝐷 ‖𝑥‖2 , ∀𝑥 ∈ ℜ6 where 𝜆𝑑 < 𝜆𝐷 < ∞, 𝜆𝑑 = min 𝜆𝑚𝑖𝑛 {𝐷}, and 𝜆𝐷 = max 𝜆𝑚𝑎𝑥 {𝐷} For the exact information about AUV’s kinematics, dynamics modeling, and model properties, readers are referred to Do and Pan (2009) and Fossen (2002). Bring into mind the following Property of hyperbolic function tanh(.) that is effectively used in this paper (Polycarpou, 1996; Shojaei, 2015):

2.4. High-gain observer Because only the postures (𝑖.𝑒. 𝑞(𝑡)) of the vehicle are provided for the feedback, and the posture derivatives such as velocities and accelerations are unavailable in practice, the high-gain observer is employed as a good candidate to provide unavailable velocity estimations by the following Lemma:

Property 4. Take the following features of tanh(.) into account that are employed to design the proposed control laws in the sequel: 1. 2. 3. 4.

𝑌𝑖 (0) = 0 𝑖 = 4, 5, 6,

ℎ‖𝑥‖ − 𝑥𝑇 ℎ tanh(0.2785ℎ𝑥∕𝛾𝑑 ) ≤ 𝑛𝛾𝑑 , ∀𝛾𝑑 > 0, ∀𝑥 ∈ ℜ𝑛 ‖ tanh(𝑥)‖2 ≤ ‖𝑥‖2 | tanh(𝑘1 𝑥)| ≤ 𝑘1 |𝑥| | tanh( 𝑎𝑏 )| < 1

Lemma 1 (Ge and Zhang, 2003). Presume that the system output variable 𝑞(𝑡) and its first to 𝑛 − 1 derivatives are bounded, that is ‖𝑞 (𝑘) ‖ ≤ 𝐵𝑘 , 𝑘 = 1, … , 𝑛 − 1 with 𝐵𝑘 ∈ ℜ+ . Take the following linear system into account: 𝑜𝜚̇ 𝑖 = 𝜚𝑖+1 ,

𝑖 = 1, 2, 3, … , 𝑛 − 1

(7)

2.3. Prescribed performance and error dynamics transformation

𝑜𝜚̇ 𝑛 = −𝜆1 𝜚𝑛 − 𝜆2 𝜚𝑛−1 − ⋯ − 𝜆𝑛−1 𝜚2 − 𝜚1 + 𝑞(𝑡)

If the tracking errors (𝑖.𝑒. 𝑒1 = 𝑞 − 𝑞𝑑 ) approach to a residual set and evolve strictly within a decreasing limited function of time, a prescribed performance is realized. The mathematical equations of the predefined tracking performance are defined by Bechlioulis and Rovithakis (2008) and Wang et al. (2017):

where 𝑜 ∈ ℜ+ is a small design scalar, the parameters 𝜆1 to 𝜆𝑛−1 are selected to make the polynomial 𝑠𝑛 + 𝜆1 𝑠𝑛−1 + ⋯ + 𝜆𝑛−1 𝑠 + 1 be Hurwitz, and 𝜚𝑖 , 𝑖 = 1, … , 𝑛, are the state variables of the observer. Afterward, some useful properties are valid which are given as follows: 𝜚𝑘+1 (𝑖) − 𝑞 (𝑘) = −𝑜𝜛 (𝑘+1) , 𝑘 = 0, 1, … , 𝑛 − 1 𝑜𝑘

− 𝛼𝑖 𝜂𝑖 (𝑡) < 𝑒1𝑖 (𝑡) < 𝛽𝑖 𝜂𝑖 (𝑡), 𝑖 = 1, 2, … , 6

(2)

where 𝜛 = 𝜚𝑛 + 𝜆1 𝜚𝑛−1 + ⋯ + 𝜆𝑛−1 𝜚1 with 𝜛 (𝑘) shows 𝑘th derivative of 𝜛. (𝑖𝑖) One may find constants 𝑡1 ∈ ℜ+ and 𝐺𝑘 ∈ ℜ+ so that for all 𝑡 > 𝑡1 we have |𝜛 (𝑘) | ≤ 𝐺𝑘 .

where 𝛼𝑖 > 0 and 𝛽𝑖 > 0 are some positive design scalars, and 𝜂𝑖 (𝑡) is a bounded, smooth, and decreasing function of time that can be chosen as 3

O. Elhaki and K. Shojaei

Engineering Applications of Artificial Intelligence 88 (2020) 103382

Notice that 𝜚𝑘+1 ∕𝑜𝑘 tends to 𝑞 (𝑘) with a limited error by utilizing items (𝑖) and (𝑖𝑖) of Lemma 1 which does not need any prior knowledge about the system dynamic model, and only depends on measurable postures of the vehicle. These advantages and simplicities of the highgain observer make it a powerful and strong tool in designing nonlinear output-feedback controllers for the nonlinear systems.

Now, define error variables as: 𝑧1 = 𝜖𝑒 ,

(11)

𝑧2 = 𝜈 − 𝑧𝑑𝑓

(12)

where 𝑧𝑑𝑓 defines the virtual control signal that is passed through the following filter (Swaroop et al., 2000):

2.5. Actuator saturation nonlinearity

(13)

𝜏𝑏 𝑧̇ 𝑑𝑓 + 𝑧𝑑𝑓 = 𝑧𝑑 , 𝑧𝑑𝑓 (0) = 𝑧𝑑 (0)

From a practical viewpoint, if the designed control law produces large amplitude signals more than the actuator limits, the actuator saturation inevitably happens. Then, the control signals are saturated such that the generated input signals cannot be fully implemented by the actuators. Since they are operating at their ultimate range, this situation may ruin the performance and efficiency of the controller during the tracking mission and the actuators may become damageable (Gao and Selmic, 2006). The saturated torque input is defined by:

where 𝑧𝑑 is the virtual controller, and 𝜏𝑏 ∈ Then, the following error is defined:

ℜ+

is a design parameter. (14)

𝑧𝑑𝑒 = 𝑧𝑑𝑓 − 𝑧𝑑 (1∕2)𝑧𝑇1 𝑧1

(1∕2)𝛤𝛾−1 𝛾̃ 2

Choose the Lyapunov function as 𝐿1 (𝑡) = + where 𝛾̃ = 𝛾 − 𝛾̂ . Now, differentiating (11) along (12), (14), and first term of Eq. (1) yields:

⎧ 𝐵 ∶ 𝑙𝜏 ≥ 𝐵 𝑀 ⎪ 𝑀 𝑢𝑎 (𝑡) = ⎨ 𝑙𝜏 ∶ 𝐵𝑚 < 𝑙𝜏 < 𝐵𝑀 ⎪ 𝐵𝑚 ∶ 𝑙𝜏 ≤ 𝐵𝑚 ⎩

𝑧̇ 1 = 𝜁(𝐽 (𝑞)(𝑧2 + 𝑧𝑑𝑒 ) + 𝐽 (𝑞)𝑧𝑑 + 𝛿 − 𝑞̇ 𝑑 − 𝛩𝑒1 )

(15)

and select the virtual controller as: 𝑧𝑑 = (𝜁 𝐽 (𝑞))−1 (−𝐾1 𝑧1 + 𝜁 𝑞̇ 𝑑 + 𝜁 𝛩𝑒1 − ‖𝜁‖̂𝛾 tanh(0.2785‖𝜁‖̂𝛾 𝑧1 ∕𝛾𝑑 )) (16)

where 𝑢𝑎 , 𝜏, 𝐵𝑀 , and 𝐵𝑚 are the vectors of saturated torques, control torques, maximum and minimum ultimate range of the actuators, respectively, and 𝑙 is a ratio between 𝑢𝑎 and 𝜏. The amount of control torques that cannot be put into effect by the AUV actuators is given by the vector 𝑑𝜏 (𝜏) = 𝑢𝑎 − 𝜏 where:

where 𝐾1 ∈ ℜ6×6 is a positive diagonal proportional gain matrix, and 𝛾𝑑 ∈ ℜ+ is a design parameter. Then, 𝛾̂ is estimated by the following rule: 𝛾̂̇ = 𝑃 𝑟𝑜𝑗𝛾̂ (𝜉𝛾 )

⎧ 𝐵 − 𝜏 ∶ 𝑙𝜏 ≥ 𝐵 𝑀 ⎪ 𝑀 𝑑𝜏 (𝜏) = ⎨ (𝑙 − 1)𝜏 ∶ 𝐵𝑚 < 𝑙𝜏 < 𝐵𝑀 ⎪ 𝐵𝑚 − 𝜏 ∶ 𝑙𝜏 ≤ 𝐵𝑚 ⎩

(17)

𝜉𝛾 = 𝛤𝛾 ‖𝑧1 ‖‖𝜁‖ − 𝛿𝛾 𝛤𝛾 𝛾̂ { 𝑃 𝑟𝑜𝑗𝛾̂ (𝜉𝛾 ) =

Since the actuator saturation nonlinearity 𝑑𝜏 (𝜏) is nonlinear-in-parameter, it will be compensated by the neural network controller in the next section. Control Objective: The control goal of this paper is to develop the control law 𝜏 for 6-DOF fully-actuated AUVs described by system (1) such that we achieve the followings:

𝜉𝛾 , 𝑖𝑓 |̂𝛾 | < 𝛾𝑚𝑎𝑥 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

where 𝛿𝛾 ∈ ℜ+ is a design constant, 𝛤𝛾 stands for the adaptation gain, and 𝛾𝑚𝑎𝑥 defines the maximum upper bound of 𝛾̂ . Next, by differentiating 𝐿1 with respect to the time, and substituting (15)–(17), one has: 𝐿̇ 1 (𝑡) ≤𝑧𝑇1 𝜁 𝐽 (𝑞)(𝑧2 + 𝑧𝑑𝑒 ) − 𝜆𝑚𝑖𝑛 {𝐾1 }‖𝑧1 ‖2 + ‖𝑧1 ‖‖𝜁‖̂𝛾 − 𝑧𝑇1 ‖𝜁‖̂𝛾 tanh(0.2785‖𝜁‖̂𝛾 𝑧1 ∕𝛾𝑑 ) + 𝛿𝛾 𝛾̃ 𝛾̂

1. The vehicle asymptotically tracks the desired trajectory in the presence of unmodeled dynamics and nonlinearities without velocity measurements, and the possibility of the actuator saturation can be reduced to preclude a weak tracking performance. 2. All variables in the closed-loop control system are uniformly ultimately bounded, and the prescribed steady-state and transient behaviors of the tracking errors in the sense of (2) are ensured.

+ 𝛾̃ 𝛤𝛾−1 (𝛤𝛾 ‖𝑧1 ‖‖𝜁‖ − 𝑃 𝑟𝑜𝑗𝛾̂ (𝛤𝛾 ‖𝑧1 ‖‖𝜁‖))

(18)

Next, one has the following equation by bringing (13), (14), and 𝜁𝑖 > 0 as 𝑡 → ∞ into mind: 1 𝑧̇ 𝑑𝑒 = 𝑧̇ 𝑑𝑓 − 𝑧̇ 𝑑 = − 𝑧𝑑𝑒 − 𝑧̇ 𝑑 (19) 𝜏𝑏 Therefore, one may find that 𝑧 ‖𝑧̇ 𝑑𝑒 + 𝑑𝑒 ‖ ≤ 𝜒 𝜏𝑏

3. Main results In this section, the adaptive robust neural network output-feedback controller will be developed based on the dynamic surface control method, that diminishes the complexity of the proposed controller, to attain the aforementioned control objectives. Then, the stability of the proposed closed-loop system is investigated by the Lyapunov stability theory.

(20)

where 𝜒 is the continuous function of closed-loop variables and their derivatives that is used to derive the following inequality: 𝑧𝑇𝑑𝑒 𝑧̇ 𝑑𝑒 ≤ −

1 1 ‖𝑧 ‖2 + ‖𝑧𝑑𝑒 ‖2 + 𝜒 2 𝜏𝑏 𝑑𝑒 4

(21)

where (21) will be used in the stability analysis in the sequel. Then, by multiplying both sides of (12) by 𝑀, taking its time derivative along the second item of Eqs. (1) and (12), and taking the actuator nonlinearity in Section 2.5 into account, one arrives at:

3.1. Projection-type adaptive neural output-feedback controller design Due to the reality that the velocities of the AUV are unavailable, a high-gain observer is efficiently employed to obtain an approximation of the velocity vector 𝜈(𝑡). Now, by recalling the nominal kinematic model of AUVs, i.e. 𝑞̇ = 𝐽 (𝑞)𝜈, and applying Lemma 1, one has: 𝜚 𝜈̂ = 𝐽 −1 (𝑞)( 2 ) (8) 𝑜 with the following observer: 𝑜𝜚̇ 1 = 𝜚2 ,

(10)

𝑜𝜚̇ 2 = −𝜆1 𝜚2 − 𝜚1 + 𝑞(𝑡).

(22)

𝑀 𝑧̇ 2 = − 𝐶(𝜈)𝑧2 − 𝐷𝑧2 + 𝜏 + 𝛥 − 𝐶(𝜈)𝑧𝑑𝑓 − 𝐷𝑧𝑑𝑓 − 𝑔(𝑞) + 𝐻(𝜈) + 𝑑𝜏 (𝜏) − 𝑀 𝑧̇ 𝑑𝑓 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ 𝜍𝑝

Since 𝜍𝑝 in (22) inherently includes NLIP uncertainties, a three layer neural network is employed based on Fig. 2 to estimate the unknown entities as 𝜍𝑝 = 𝑆 𝑇 𝜎(𝐹 𝑇 𝑥𝑛 ) + 𝜀(𝑥𝑛 ), ∀𝑥𝑛 ∈ 𝛺𝑥 ⊂ ℜ𝑁𝑖 +1 , where 𝛺𝑥 shows a compact set and 𝑆 and 𝐹 are the ideal weight matrices, 𝜀(𝑥𝑛 ) =

(9) 4

O. Elhaki and K. Shojaei

Engineering Applications of Artificial Intelligence 88 (2020) 103382

𝑆̂̇ =𝑃 𝑟𝑜𝑗𝑆̂ (𝜉𝑠 ) 𝜉𝑠 =𝛤𝑠 (𝜎(𝐹̂ 𝑇 𝑥̂ 𝑛 ) − 𝜎( ́ 𝐹̂ 𝑇 𝑥̂ 𝑛 )𝐹̂ 𝑇 𝑥̂ 𝑛 )𝑧̂ 𝑇2 − 𝛿𝑠 𝛤𝑠 𝑆̂

[𝜀1 , … , 𝜀6 ]𝑇 is the NN functional estimation error vector that is bounded over a compact set such that |𝜀𝑗 (𝑥𝑛 )| ≤ 𝜄̄𝑗 , ∀𝑥𝑛 ∈ 𝛺𝑥 , 𝑗 = 1, … , 6 where 𝜄̄𝑗 ∈ ℜ+ , 𝑆 𝑇 ∈ ℜ𝑁𝑜 ×(𝑁ℎ +1) , 𝐹 𝑇 ∈ ℜ𝑁ℎ ×(𝑁𝑖 +1) , 𝑁𝑖 , 𝑁ℎ , and 𝑁𝑜 are the numbers of input-layer, hidden-layer, and output-layer neurons, respectively, 𝜎(𝐹 𝑇 𝑥𝑛 ) = [1, 𝜎(𝐹 ̄ 𝑟𝑇 𝑥𝑛 ), … , 𝜎(𝐹 ̄ 𝑟𝑇 𝑥𝑛 )]𝑇 where 𝐹𝑟𝑇 , 𝑗 = 1

𝑁ℎ

𝑇 0 ⎧ 𝜉𝑠 , 𝑖𝑓 𝑆̂ ∈ 𝑃𝑠 𝑜𝑟 𝑛𝑠̂ 𝜉𝑠 ≤ 0 ⎪ 𝑎𝑛𝑑 𝑆̂ ∈ 𝛿(𝑃𝑠 ) 𝑃 𝑟𝑜𝑗𝑆̂ (𝜉𝑠 ) = ⎨ 𝑛𝑠̂ 𝑛𝑇𝑠̂ ⎪ (𝐼𝑠 − 𝛤𝑠 𝑇 )𝜉 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 ⎩ 𝑛𝑠̂ 𝛤𝑠 𝑛𝑠̂ 𝑠

𝑗

1, … , 𝑁ℎ is the jth row of 𝐹 𝑇 , 𝜎(𝑥) ̄ = 1∕(1 + 𝑒−𝑥 ) (Lewis et al., 2004; Ge et al., 2013), where 𝑆 and 𝐹 are given by: ⎡ 𝑠1,1 ⎢ 𝑠 𝑆 = ⎢ 2,1 ⎢ ⋮ ⎢𝑠 ⎣ (𝑁ℎ +1),1 ⎡ 𝑓1,1 ⎢ 𝑓 2,1 𝐹 =⎢ ⎢ ⋮ ⎢𝑓 ⎣ (𝑁𝑖 +1),1

𝑠1,2 𝑠2,2 ⋮ 𝑠(𝑁ℎ +1),2 𝑓1,2 𝑓2,2 ⋮ 𝑓(𝑁𝑖 +1),2

⋯ ⋯ ⋱ ⋯

⎤ ⎥ ⎥ ⎥ 𝑠(𝑁ℎ +1),𝑁𝑜 ⎥⎦

⋯ ⋯ ⋱ ⋯

⎤ ⎥ ⎥ ⎥ 𝑓(𝑁𝑖 +1),𝑁ℎ ⎥⎦

(28)

𝑠1,𝑁𝑜 𝑠2,𝑁𝑜 ⋮

𝐹̂̇ =𝑃 𝑟𝑜𝑗𝐹̂ (𝜉𝑓 ) ́ 𝐹̂ 𝑇 𝑥̂ 𝑛 ) − 𝛿𝑓 𝛤𝑓 𝐹̂ 𝜉𝑓 =𝛤𝑓 𝑥̂ 𝑛 𝑧̂ 𝑇 𝑆̂ 𝑇 𝜎(

(29)

2

⎧ 𝜉 , 𝑖𝑓 𝐹̂ ∈ 𝑃 0 𝑜𝑟 𝑛𝑇 𝜉 ≤ 0 𝑓 𝑓̂ 𝑓 ⎪ 𝑓 ⎪ 𝑎𝑛𝑑 𝐹̂ ∈ 𝛿(𝑃𝑓 ) 𝑃 𝑟𝑜𝑗𝐹̂ (𝜉𝑓 ) = ⎨ 𝑛𝑓̂ 𝑛𝑇̂ 𝑓 ⎪ (𝐼 − 𝛤 )𝜉 , 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝑓 𝑓 𝑇 𝑛 ̂ 𝛤𝑓 𝑛𝑓̂ 𝑓 ⎪ 𝑓 ⎩ where 𝑄, 𝛩 ∈ ℜ6×6 are adaptation gains, 𝑃 0 is an arbitrary design vector, 𝑃𝑠 = {𝑆 ∈ ℜ(𝑁ℎ +1)×𝑁𝑜 ∶ 𝑘(𝑆) ≤ 0} and 𝑃𝑓 = {𝐹 ∈ ℜ(𝑁𝑖 +1)×𝑁ℎ ∶ 𝑘(𝐹 ) ≤ 0} are bounded convex sets that the parameter estimates are forced to be kept within 𝑃𝑠 and 𝑃𝑓 , 𝑘(𝑆) and 𝑘(𝐹 ) stand for constraint functions on 𝑆 and 𝐹 , respectively, 𝑃𝑠0 , 𝛿(𝑃𝑠 ), 𝑃𝑓0 , and 𝛿(𝑃𝑓 ) represent the interior and the boundary subsets of 𝑃𝑠 and 𝑃𝑓 , respectively, 𝑛𝑠̂ = ̂ and 𝑛 ̂ = ∇𝑘(𝐹̂ ) indicate the outward unit normal vectors at ∇𝑘(𝑆) 𝑓 ̂ 𝑆̂ ∈ 𝛿(𝑃𝑠 ) and 𝐹̂ ∈ 𝛿(𝑃𝑓 ), respectively, 𝑆(0) ∈ 𝑃𝑠 , 𝐹̂ (0) ∈ 𝑃𝑓 , 𝛤𝑠 = 𝛤𝑠𝑇 > 0 ∈ ℜ(𝑁ℎ +1)×(𝑁ℎ +1) and 𝛤𝑓 = 𝛤𝑓𝑇 > 0 ∈ ℜ(𝑁𝑖 +1)×(𝑁𝑖 +1) represent the estimation gain matrices, 𝛿𝑠 , 𝛿𝑓 ∈ ℜ+ are design scalars, 𝐾2 ∈ ℜ6×6 is a positive diagonal proportional gain matrix. Now, in the light of the second item of Lemma 1, and ‖𝐽 −1 (𝑞)‖ ≤ 𝐵𝑗 where 𝐵𝑗 ∈ ℜ+ is an unknown constant, the following inequalities are valid: 𝜚 ̇ = ‖𝐽 −1 (𝑞)𝑜𝜛‖ ̈ ≤ 𝑜𝐺2 ‖𝐽 −1 (𝑞)‖ ‖𝑧̃ 2 ‖ =‖𝐽 −1 (𝑞)( 2 − 𝑞)‖ 𝑜

𝑓1,𝑁ℎ 𝑓2,𝑁ℎ ⋮

Assumption 5. The matrices of ideal NN weights are bounded over the compact set 𝛺𝑥 so that ‖𝑆‖𝑓 ≤ 𝜄̄𝑠 and ‖𝐹 ‖𝑓 ≤ 𝜄̄𝑓 , where 𝜄̄𝑠 , 𝜄̄𝑓 ∈ ℜ+ (Lewis et al., 2004; Ge et al., 2013). Assumption 6. it is assumed that |𝛥𝑗 + 𝜀𝑗 | ≤ 𝑝∗𝑗 , ∀𝑗 = 1, … , 6 where 𝑝∗𝑗 ∈ ℜ+ define unknown constants and 𝑃 ∗ = [𝑝∗1 , … , 𝑝∗6 ]𝑇 . Remark 4. From a detailed review of the literature Hagan and Demuth (1999), Narendra and Parthasarathy (1990), Uçak and Günel (2019), Jafari and Xu (2019), Juman et al. (2019), Pradeep et al. (2016) and Radac and Precup (2016), there exist different control architecture which employ neural networks and other artificial intelligence methods to design high-performance feedback controllers. Compared with the previous works in the literature, one of the main contributions of this paper is that an effective combination of a robust multi-layer neural network and an adaptive robust technique is utilized to approximate NLIP uncertain functions, unmodeled dynamics and the actuator saturation nonlinearity while simultaneously compensating for environmental kinematic and dynamic disturbances and NN errors without velocity measurements.

≤ 𝑜𝐺2 𝐵𝑗 ∶= 𝐵𝑧 𝜚 ̇ 2 = ‖𝐽 −1 (𝑞)𝑜𝜛‖ ̈ 2 ≤ 𝑜2 𝐺22 ‖𝐽 −1 (𝑞)‖2 ‖𝑧̃ 2 ‖ =‖𝐽 −1 (𝑞)( 2 − 𝑞)‖ 𝑜

(30)

2

≤ 𝑜2 𝐺22 𝐵𝑗2 ∶= 𝐵𝑧2

Since the ideal NN weight matrices are impossible to be meticulously calculated, it is usual that their approximations are used to obtain 𝜍̂ 𝑝 = 𝑆̂ 𝑇 𝜎(𝐹̂ 𝑇 𝑥̂ 𝑛 ), where 𝑥̂ 𝑛 = [𝜈̂ 𝑇 , 𝑧𝑇𝑑𝑓 , 𝑧̇ 𝑇𝑑𝑓 , 𝑞 𝑇 , 𝜏 𝑇 ]𝑇 and 𝑆̂ and 𝐹̂ denote the estimates of 𝑆 and 𝐹 , respectively. Thus, 𝑆̃ = 𝑆 − 𝑆̂ and 𝐹̃ = 𝐹 − 𝐹̂ are used to define weights matrices estimation errors. For the NN weights estimation purpose, an online training process is used in this paper. The estimation error can be stated as (Tee et al., 2008):

(31)

Finally, substituting Eq. (25) into Eq. (22) leads to the following closed-loop error dynamics: 𝑀 𝑧̇ 2 = − 𝐶(𝜈)𝑧2 − 𝐷𝑧2 + 𝑆̃ 𝑇 (𝜎(𝐹̂ 𝑇 𝑥̂ 𝑛 ) − 𝜎( ́ 𝐹̂ 𝑇 𝑥̂ 𝑛 )𝐹̂ 𝑇 𝑥̂ 𝑛 ) + 𝑆̂ 𝑇 𝜎( ́ 𝐹̂ 𝑇 𝑥̂ 𝑛 )𝐹̃ 𝑇 𝑥̂ 𝑛 − 𝐻 𝑃̂

(32)

+ 𝑟𝑡 − 𝐾2 𝑧̂ 2 − 𝐽 𝑇 (𝑞)𝜁 𝑧1 − 𝑘3 ℎ̄ 𝑧̂ 2 + 𝛥 + 𝜀(𝑥𝑛 )

(23)

A summary of the proposed control law is illustrated by Table 1, and the closed-loop diagram of the proposed controller is shown by Fig. 1.

where 𝜎( ́ 𝐹̂ 𝑇 𝑥̂ 𝑛 ) = [0𝑁ℎ ×1 , 𝑑𝑖𝑎𝑔[𝜎́ 1 , … , 𝜎́ 𝑁ℎ ]]𝑇 , with 𝜎́ 𝑗 = 𝑑 𝜎(𝑠)∕𝑑𝑠| ̄ , 𝑗 = 1, 2, … , 𝑁ℎ , and the remaining term 𝑟𝑡 in 𝑠=𝐹̂ 𝑇 𝑥̂

Remark 5. In this paper, the weights update rules are gradient descentbased adaptive laws which are a direct result of the Lyapunov-based stability analysis in the next section. The convergence rate of the proposed NN laws is dependent on adaptive gains 𝛤𝑓 and 𝛤𝑠 . By increasing the value of 𝛤𝑓 and 𝛤𝑠 , the user obtains a faster convergence. According to the literature Lewis et al. (2004) and Ioannou and Sun (1996), the Least-square or integral cost function gradient-based adaptive laws provide better convergence properties which are devoted to the future works.

𝑆 𝑇 𝜎(𝐹 𝑇 𝑥𝑛 ) − 𝑆̂ 𝑇 𝜎(𝐹̂ 𝑇 𝑥̂ 𝑛 ) =𝑆̃ 𝑇 (𝜎(𝐹̂ 𝑇 𝑥̂ 𝑛 ) − 𝜎( ́ 𝐹̂ 𝑇 𝑥̂ 𝑛 )𝐹̂ 𝑇 𝑥̂ 𝑛 ) 𝑇 𝑇 + 𝑆̂ 𝜎( ́ 𝐹̂ 𝑥̂ 𝑛 )𝐹̃ 𝑇 𝑥̂ 𝑛 + 𝑟𝑡 + 𝜀(𝑥𝑛 )

𝑟𝑗 𝑛

(23) is bounded by: ‖𝑟𝑡 ‖ ≤ ‖𝑆‖𝑓 (‖𝜎( ́ 𝐹̂ 𝑇 𝑥̂ 𝑛 )𝐹̂ 𝑇 𝑥̂ 𝑛 ‖ + ‖𝜎(𝐹̂ 𝑇 𝑥̂ 𝑛 )‖) + ‖𝐹 ‖𝑓 ‖𝑥̂ 𝑛 ‖‖𝑆̂ 𝑇 𝜎( ́ 𝐹̂ 𝑇 𝑥̂ 𝑛 )‖𝑓 (24) Now, the following control law is proposed here: 𝜏 = −𝐾2 𝑧̂ 2 − 𝐽 𝑇 (𝑞)𝜁 𝑧1 − 𝑆̂ 𝑇 𝜎(𝐹̂ 𝑇 𝑥̂ 𝑛 ) − 𝑘3 ℎ̄ 𝑧̂ 2 − 𝐻 𝑃̂

(25)

where 𝑘3 ∈ 𝐻 = 𝑑𝑖𝑎𝑔[tanh(𝑧̂ 21 ∕𝑚1 ), … , tanh(𝑧̂ 26 ∕𝑚6 )], 𝑃̂ is the upper bound estimation of unknown parameters, and the auxiliary function ℎ̄ is given by:

3.2. Stability analysis

ℎ̄ = (‖𝜎( ́ 𝐹̂ 𝑇 𝑥̂ 𝑛 )𝐹̂ 𝑇 𝑥̂ 𝑛 ‖ + ‖𝜎(𝐹̂ 𝑇 𝑥̂ 𝑛 )‖)2 + ‖𝑥̂ 𝑛 ‖2 ‖𝑆̂ 𝑇 𝜎( ́ 𝐹̂ 𝑇 𝑥̂ 𝑛 )‖2𝑓

The principal results of this paper are briefly given in this section by using the following inequalities ∀𝑥, 𝑦 ∈ ℜ𝑛 , 𝑁 = 𝑁 𝑇 , 𝑎, 𝑏 ∈ ℜ, 𝜖 ∈ ℜ+ (Ioannou and Sun, 1996; Khalil, 2002):

ℜ+ ,

(26)

Also, the following update formulas are taken into account (Ioannou and Sun, 1996; Cheng et al., 2009): 𝑃̂̇ = 𝑄[𝐻(𝑧̂ 2 )𝑧̂ 2 − 𝛩(𝑃̂ − 𝑃 0 )]

• 𝜆𝑚𝑖𝑛 {𝑁}‖𝑥‖2 ≤ 𝑥𝑇 𝑁𝑥 ≤ 𝜆𝑚𝑎𝑥 {𝑁}‖𝑥‖2 2 2 • 𝑎𝑏 ≤ 𝑎2𝜖 + 𝜖𝑏2

(27) 5

O. Elhaki and K. Shojaei

Engineering Applications of Artificial Intelligence 88 (2020) 103382

Fig. 1. The schematic diagram of the proposed AUV control system. Table 1 A summary of the proposed controller equations. Defined error variables: 𝑧1 = 𝜖𝑒 , 𝑧2 = 𝜈 − 𝑧𝑑𝑓 , 𝑧̂ 2 = 𝐽 −1 (𝑞)(𝜚2 ∕𝑜) − 𝑧𝑑𝑓 Control laws: 𝑧𝑑 = (𝜁 𝐽 (𝑞))−1 (−𝐾1 𝑧1 + 𝜁 𝑞̇ 𝑑 + 𝜁𝛩𝑒1 − ‖𝜁‖̂𝛾 tanh(0.2785‖𝜁‖̂𝛾 𝑧1 ∕𝛾𝑑 )) 𝜏𝑏 𝑧̇ 𝑑𝑓 + 𝑧𝑑𝑓 = 𝑧𝑑 , 𝑧𝑑𝑓 (0) = 𝑧𝑑 (0) 𝜏 = −𝐾2 𝑧̂ 2 − 𝐽 𝑇 (𝑞)𝜁𝑧1 − 𝑆̂ 𝑇 𝜎(𝐹̂ 𝑇 𝑥̂ 𝑛 ) − 𝑘3 ℎ̄ 𝑧̂ 2 − 𝐻 𝑃̂ 𝑥̂ 𝑛 = [𝜈̂ 𝑇 , 𝑧𝑇𝑑𝑓 , 𝑧̇ 𝑇𝑑𝑓 , 𝑞 𝑇 , 𝜏 𝑇 ]𝑇 ℎ̄ = (‖𝜎( ́ 𝐹̂ 𝑇 𝑥̂ 𝑛 )𝐹̂ 𝑇 𝑥̂ 𝑛 ‖ + ‖𝜎(𝐹̂ 𝑇 𝑥̂ 𝑛 )‖)2 + ‖𝑥̂ 𝑛 ‖2 ‖𝑆̂ 𝑇 𝜎( ́ 𝐹̂ 𝑇 𝑥̂ 𝑛 )‖2𝑓 𝐻 = 𝑑𝑖𝑎𝑔[tanh(𝑧̂ 21 ∕𝑚1 ), … , tanh(𝑧̂ 26 ∕𝑚6 )] Observer and parameters update laws: 𝑜𝜚̇ 1 = 𝜚2 𝑜𝜚̇ 2 = −𝜆1 𝜚2 − 𝜚1 + 𝑞(𝑡) ̂ 𝑆̂̇ = 𝑃 𝑟𝑜𝑗𝑆̂ {𝛤𝑠 (𝜎(𝐹̂ 𝑇 𝑥̂ 𝑛 ) − 𝜎( ́ 𝐹̂ 𝑇 𝑥̂ 𝑛 )𝐹̂ 𝑇 𝑥̂ 𝑛 )𝑧̂ 𝑇2 − 𝛿𝑠 𝛤𝑠 𝑆} 𝐹̂̇ = 𝑃 𝑟𝑜𝑗𝐹̂ {𝛤𝑓 𝑥̂ 𝑛 𝑧̂ 𝑇2 𝑆̂ 𝑇 𝜎( ́ 𝐹̂ 𝑇 𝑥̂ 𝑛 ) − 𝛿𝑓 𝛤𝑓 𝐹̂ } 𝑃̂̇ = 𝑄[𝐻(𝑧̂ 2 )𝑧̂ 2 − 𝛩(𝑃̂ − 𝑃 0 )] 𝛾̂̇ = 𝑃 𝑟𝑜𝑗𝛾̂ (𝛤𝛾 ‖𝑧1 ‖‖𝜁‖ − 𝛿𝛾 𝛤𝛾 𝛾̂ )

• 𝑡𝑟{𝑥𝑇 𝑦} ≤

1 ‖𝑥‖2𝑓 2𝑘2

+

𝑘2 ‖𝑦‖2𝑓 2 ≤ 2𝑘12 𝜆𝑚𝑎𝑥 {𝑁}‖𝑥‖2

• 𝑥𝑇 𝑁𝑦 ≤ ‖𝑥‖‖𝑁‖‖𝑦‖ • 𝜃̃ 𝑇 𝛤 −1 (𝜉 − 𝑝𝑟𝑜𝑗 ̂ (𝜉)) ≤ 0

Fig. 2. The schematic diagram of the three layer neural network.

+

𝑘2 2

𝜆𝑚𝑎𝑥 {𝑁}‖𝑦‖2 with:

𝜃

𝑥𝑧 = [𝑧𝑇1 , 𝑧𝑇2 , 𝑠̃11 , … , 𝑠̃(𝑁ℎ +1)𝑁𝑜 , 𝑓̃11 , … , 𝑓̃(𝑁𝑖 +1)𝑁ℎ , 𝑃̃ 𝑇 , 𝛾̃ , 𝑧𝑇𝑑𝑒 ]

The following theorem summarizes our main results:

𝜆𝑢 = 0.5 max{1, 𝜆𝑀 , 𝜆𝑚𝑎𝑥 (𝛤𝑠−1 ), 𝜆𝑚𝑎𝑥 (𝛤𝑓−1 ), 𝜆𝑚𝑎𝑥 (𝑄−1 ), 𝛤𝛾−1 }

Theorem 1. Bring into mind the kinematic and dynamic models of AUVs which are given by Eq. (1), and the transformed system (15), and (22). Under Assumptions 1–6 and the initial conditions −𝛼𝑖 𝜂𝑖 (0) < 𝑒1𝑖 (0) < 𝛽𝑖 𝜂𝑖 (0), the proposed adaptive neural output-feedback controller (25), virtual controller (16), with the estimation laws (17) and (27)–(29), and the high-gain observer (8)–(10), ensure that all variables in the closed-loop observer-controller system are uniformly ultimately bounded (UUB) as 𝑡 → ∞, and tracking errors 𝑒1𝑖 converge to a small region including the origin with a prescribed performance.

𝜆𝑚 = 0.5 min{1, 𝜆𝑚 , 𝜆𝑚𝑖𝑛 (𝛤𝑠−1 ), 𝜆𝑚𝑖𝑛 (𝛤𝑓−1 ), 𝜆𝑚𝑖𝑛 (𝑄−1 ), 𝛤𝛾−1 } On the other side, ∀𝐵, 𝑁 > 0 the sets: 𝛱1 = {(𝑞𝑑 , 𝑞̇ 𝑑 , 𝑞̈𝑑 ) ∶ 𝑞𝑑𝑇 𝑞𝑑 + 𝑞̇ 𝑑𝑇 𝑞̇ 𝑑 + 𝑞̈𝑑𝑇 𝑞̈𝑑 ≤ 𝐵}, 𝛱2 = {𝑥𝑧 (𝑡) ∶ 𝐿(𝑡) ≤ 𝑁} are compact in ℜ18 and ℜ25+(𝑁ℎ +1)𝑁𝑜 +(𝑁𝑖 +1)𝑁ℎ , respectively, and 𝛱3 = 𝛱1 × 𝛱2 is a compact set in ℜ43+(𝑁ℎ +1)𝑁𝑜 +(𝑁𝑖 +1)𝑁ℎ . As a result, 𝜒 in (20) is upper-bounded such that 𝜒 ≤ 𝜒𝑀 . By differentiating 𝐿(𝑡) with respect to the time and taking into account 𝑧𝑇2 (𝑀̇ − 2𝐶(𝜈))𝑧2 = 0, (18), and (32), one has:

Proof. Take the following Lyapunov function candidate into account:

̇ 𝐿(𝑡) ≤𝑧𝑇2 𝑆̃ 𝑇 (𝜎(𝐹̂ 𝑇 𝑥̂ 𝑛 ) − 𝜎( ́ 𝐹̂ 𝑇 𝑥̂ 𝑛 )𝐹̂ 𝑇 𝑥̂ 𝑛 ) 𝑇 ̂𝑇 𝑇 + 𝑧 𝑆 𝜎( ́ 𝐹̂ 𝑥̂ 𝑛 )𝐹̃ 𝑇 𝑥̂ 𝑛 + 𝑧𝑇 𝑟𝑡 + 𝛿𝛾 𝛾̃ 𝛾̂

1 1 ̃ + 1 𝑡𝑟{𝐹̃ 𝑇 𝛤 −1 𝐹̃ } 𝐿(𝑡) =𝐿1 (𝑡) + 𝑧𝑇2 𝑀𝑧2 + 𝑡𝑟{𝑆̃ 𝑇 𝛤𝑠−1 𝑆} 𝑓 2 2 2 1 ̃ 𝑇 −1 ̃ 1 𝑇 + 𝑃 𝑄 𝑃 + 𝑧𝑑𝑒 𝑧𝑑𝑒 (33) 2 2 where 𝑃̃ = 𝑃̂ − 𝑃 ∗ , and (33) is bounded by the following inequality:

2

2

𝜆𝑚 ‖𝑥𝑧 (𝑡)‖2 ≤ 𝐿(𝑡) ≤ 𝜆𝑢 ‖𝑥𝑧 (𝑡)‖2

2

− 𝜆𝑚𝑖𝑛 {𝐾1 }‖𝑧1 ‖2 − 𝑧𝑇2 𝐷𝑧2 − 𝑧𝑇2 𝐾2 𝑧̂ 2 ̂̇ − 𝑡𝑟{𝐹̃ 𝑇 𝛤 −1 𝐹̂̇ } − 𝑧𝑇 𝑘3 ℎ̄ 𝑧̂ 2 − 𝑡𝑟{𝑆̃ 𝑇 𝛤𝑠−1 𝑆} 𝑓 2 + 𝑧𝑇 (𝛥 + 𝜀(𝑥 ) − 𝐻 𝑃̂ ) + 𝑃̃ 𝑇 𝑄−1 𝑃̂̇ 𝑛

+ ‖𝑧1 ‖‖𝜁‖̂𝛾 − 𝑧𝑇1 ‖𝜁‖̂𝛾 tanh(0.2785‖𝜁‖̂𝛾 𝑧1 ∕𝛾𝑑 )

(34) 6

(35)

O. Elhaki and K. Shojaei

Engineering Applications of Artificial Intelligence 88 (2020) 103382

Fig. 3. (a) Trajectory of the AUV in xyz plane, (b) Evolution of norms of online-estimated approximations 𝑆̂ and 𝐹̂ .

Fig. 4.

(a) AUV trajectory in xy plane, (b) AUV trajectory in yz plane.

Fig. 5. (a) AUV trajectory in xz plane, (b) Velocity estimation error 𝜈̃ = 𝜈̂ − 𝜈.

Table 2 The transient and steady-state parameters numerical results for the proposed controller. Tracking errors

Overshoot

peak time

Settling time (0.05 criteria)

RMS

𝑒𝑓

𝑥𝑒 𝑦𝑒 𝑧𝑒 𝜙𝑒 𝜃𝑒 𝜓𝑒

9 9 12 0.13 0.24 0.79

0 0 0 2.98 0.7 0.21

31 31.96 24.75 5.11 5.02 12.45

22.91 24.42 27.22 0.27 0.31 1.34

0.0243 0.0252 0.0045 2.845 × 10−4 0.0024 0.0063

+ 𝛾̃ 𝛤𝛾−1 (𝛤𝛾 ‖𝑧1 ‖‖𝜁‖ − 𝑃 𝑟𝑜𝑗𝛾̂ (𝛤𝛾 ‖𝑧1 ‖‖𝜁‖)) + 𝑧𝑇𝑑𝑒 𝑧̇ 𝑑𝑒

̇ 𝐿(𝑡) ≤ − 𝑧𝑇2 𝐷𝑧2 − 𝑧𝑇2 𝐾2 𝑧2 − 𝑧𝑇2 𝐾2 𝑧̃ 2

+ 𝑧𝑇1 𝜁 𝐽 𝑧𝑑𝑒

+ 𝑡𝑟{𝑆̃ 𝑇 (𝜎(𝐹̂ 𝑇 𝑥̂ 𝑛 ) − 𝜎( ́ 𝐹̂ 𝑇 𝑥̂ 𝑛 )𝐹̂ 𝑇 𝑥̂ 𝑛 )𝑧𝑇2 } + 𝛿𝛾 𝛾̃ 𝛾̂ + 𝑡𝑟{𝐹̃ 𝑇 𝑥̂ 𝑛 𝑧𝑇2 𝑆̂ 𝑇 𝜎( ́ 𝐹̂ 𝑇 𝑥̂ 𝑛 )} + 𝑧𝑇2 𝑟𝑡 − 𝑧𝑇2 𝑘3 ℎ̄ 𝑧̂ 2

Next, by using 𝑧̃ 2 = 𝑧̂ 2 − 𝑧2 and (21), one obtains: 7

(36)

O. Elhaki and K. Shojaei

Engineering Applications of Artificial Intelligence 88 (2020) 103382

Fig. 6. Posture tracking errors with performance bounds (a) 𝑥𝑒 and (b) 𝑦𝑒 with enlarged view of (c) 𝑥𝑒 and (d) 𝑦𝑒 .

Fig. 7. Posture tracking errors with performance bounds: (a) 𝑧𝑒 , (b) 𝜙𝑒 , (c) 𝜃𝑒 , and (d) 𝜓𝑒 . Table 3 The transient and steady-state parameters numerical results for the NPPOFC. Tracking errors

Overshoot

peak time

Settling time (0.05 criteria)

RMS

𝑒𝑓

𝑥𝑒 𝑦𝑒 𝑧𝑒 𝜙𝑒 𝜃𝑒 𝜓𝑒

9 9 12 0.32 0.41 0.81

0 0 0 3.68 3.34 0.49

inf inf inf inf inf inf

17.09 16.79 28.81 1.32 0.90 2.01

0.2268 0.2979 0.0427 0.0434 0.0241 0.0847

8

O. Elhaki and K. Shojaei

Engineering Applications of Artificial Intelligence 88 (2020) 103382

Fig. 8. (a) The control forces, and (b) the control torques.

following inequalities in the light of Property 4, (17), (24), (28)– (31) (Khalil, 2002; Tee et al., 2008; Ioannou and Sun, 1996; Cheng et al., 2009; Du et al., 2015): 1 1 ‖𝑆‖2𝑓 + (‖𝜎( ́ 𝐹̂ 𝑇 𝑥̂ 𝑛 )𝐹̂ 𝑇 𝑥̂ 𝑛 ‖ + ‖𝜎(𝐹̂ 𝑇 𝑥̂ 𝑛 )‖)2 ‖𝑧2 ‖2 2 2 1 1 + ‖𝐹 ‖2𝑓 + ‖𝑥̂ 𝑛 ‖2 ‖𝑆̂ 𝑇 𝜎( ́ 𝐹̂ 𝑇 𝑥̂ 𝑛 )‖2𝑓 ‖𝑧2 ‖2 , 2 2 1 ̃ 2 − 𝑡𝑟{𝑆̃ 𝑇 (𝜎(𝐹̂ 𝑇 𝑥̂ 𝑛 ) − 𝜎( ́ 𝐹̂ 𝑇 𝑥̂ 𝑛 )𝐹̂ 𝑇 𝑥̂ 𝑛 )𝑧̃ 𝑇2 } ≤ ‖𝑆‖ 𝑓 2 𝐵𝑧2 + (‖𝜎(𝐹̂ 𝑇 𝑥̂ 𝑛 )‖ + ‖𝜎( ́ 𝐹̂ 𝑇 𝑥̂ 𝑛 )𝐹̂ 𝑇 𝑥̂ 𝑛 ‖)2 , 2 𝐵2 1 ́ 𝐹̂ 𝑇 𝑥̂ 𝑛 )‖2𝑓 , − 𝑡𝑟{𝐹̃ 𝑇 𝑥̂ 𝑛 𝑧̃ 𝑇2 𝑆̂ 𝑇 𝜎( ́ 𝐹̂ 𝑇 𝑥̂ 𝑛 )} ≤ ‖𝐹̃ ‖2𝑓 + 𝑧 ‖𝑥̂ 𝑛 ‖2 ‖𝑆̂ 𝑇 𝜎( 2 2 − 𝑧𝑇2 𝐾2 𝑧̃ 2 ≤ 0.5𝜆𝑚𝑎𝑥 {𝐾2 }‖𝑧2 ‖2 + 0.5𝜆𝑚𝑎𝑥 {𝐾2 }‖𝑧̃ 2 ‖2 , 𝑡𝑟{𝑆̃ 𝑇 𝛤 −1 (𝛤𝑠 (𝜎(𝐹̂ 𝑇 𝑥̂ 𝑛 ) − 𝜎( ́ 𝐹̂ 𝑇 𝑥̂ 𝑛 )𝐹̂ 𝑇 𝑥̂ 𝑛 )𝑧̂ 𝑇 − 𝑃 𝑟𝑜𝑗𝑠̂ (𝛤𝑠

𝑧𝑇2 𝑟𝑡 ≤

𝑠

2

(𝜎(𝐹̂ 𝑇 𝑥̂ 𝑛 ) − 𝜎( ́ 𝐹̂ 𝑇 𝑥̂ 𝑛 )𝐹̂ 𝑇 𝑥̂ 𝑛 )𝑧̂ 𝑇2 ))} ≤ 0, 𝑡𝑟{𝐹̃ 𝑇 𝛤𝑓−1 (𝛤𝑓 𝑥̂ 𝑛 𝑧̂ 𝑇2 𝑆̂ 𝑇 𝜎( ́ 𝐹̂ 𝑇 𝑥̂ 𝑛 ) − 𝑃 𝑟𝑜𝑗𝑓̂ (𝛤𝑓 𝑥̂ 𝑛 𝑧̂ 𝑇2 𝑆̂ 𝑇 𝜎( ́ 𝐹̂ 𝑇 𝑥̂ 𝑛 )))} ≤ 0, Fig. 9.

1 1 )‖𝐹̃ ‖2𝑓 + 𝛿𝑓 𝑘2 ‖𝐹 ‖2𝑓 , 𝛿𝑓 𝑡𝑟{𝐹̃ 𝑇 𝐹̂ } ≤ −𝛿𝑓 (1 − 2 2𝑘2 ̂ ≤ −𝛿𝑠 (1 − 1 )‖𝑆‖ ̃ 2 + 1 𝛿𝑠 𝑘2 ‖𝑆‖2 , 𝛿𝑠 𝑡𝑟{𝑆̃ 𝑇 𝑆} 𝑓 𝑓 2 2𝑘2

Upper bounding estimation of unknown variables 𝑃 ∗ .

̂̇ − 𝑡𝑟{𝐹̃ 𝑇 𝛤 −1 𝐹̂̇ } − 𝑡𝑟{𝑆̃ 𝑇 𝛤𝑠−1 𝑆} 𝑓

𝑃̃ 𝑇 [𝐻(𝑧̂ 2 )𝑧̂ 2 − 𝛩(𝑃̂ − 𝑃 0 )] + 𝑧𝑇2 (𝑃 ∗ − 𝐻 𝑃̂ ) ≤ 3 1 0.2785[𝑚1 , … , 𝑚6 ]𝑃 ∗ + 𝐵𝑧2 + ‖𝑃 ∗ ‖2 + ‖𝑃̃ ‖2 2 2 1 1 ∗ 2 0 𝑇 ∗ ̃ − 𝜆𝑚𝑖𝑛 {𝛩}‖𝑃 ‖ + (𝑃 − 𝑃 ) 𝛩(𝑃 − 𝑃 0 ), 2 2 ‖𝑧1 ‖‖𝜁‖̂𝛾 − 𝑧𝑇1 ‖𝜁‖̂𝛾 tanh(0.2785‖𝜁‖̂𝛾 𝑧1 ∕𝛾𝑑 ) ≤ 𝑛𝛾𝑑 , 1 1 𝛿𝛾 𝛾̃ 𝛾̂ ≤ 𝛿𝛾 𝛾 2 − 𝛿𝛾 𝛾̃ 2 , 2 2 𝛾̃ 𝛤𝛾−1 (𝛤𝛾 ‖𝑧1 ‖‖𝜁‖ − 𝑃 𝑟𝑜𝑗𝛾̂ (𝛤𝛾 ‖𝑧1 ‖‖𝜁‖)) ≤ 0,

+ 𝑧𝑇2 (𝛥 + 𝜀(𝑥𝑛 ) − 𝐻 𝑃̂ ) + 𝑃̃ 𝑇 𝑄−1 𝑃̂̇ + ‖𝑧1 ‖‖𝜁‖̂𝛾 − 𝑧𝑇1 ‖𝜁‖̂𝛾 tanh(0.2785‖𝜁‖̂𝛾 𝑧1 ∕𝛾𝑑 ) 1 2 + 𝛾̃ 𝛤𝛾−1 (𝛤𝛾 ‖𝑧1 ‖‖𝜁‖ − 𝑃 𝑟𝑜𝑗𝛾̂ (𝛤𝛾 ‖𝑧1 ‖‖𝜁‖)) + 𝜒𝑀 4 1 − ( − 1 − 0.5𝜆𝑚𝑎𝑥 {𝜁 𝐽 })‖𝑧𝑑𝑒 ‖2 𝜏𝑏 − (𝜆𝑚𝑖𝑛 {𝐾1 } − 0.5𝜆𝑚𝑎𝑥 {𝜁 𝐽 })‖𝑧1 ‖2 ̂̇ Now, 𝑆̂̇ and 𝐹̂̇ in 𝑡𝑟{𝑆̃ 𝑇 ((𝜎(𝐹̂ 𝑇 𝑥̂ 𝑛 ) − 𝜎( ́ 𝐹̂ 𝑇 𝑥̂ 𝑛 )𝐹̂ 𝑇 𝑥̂ 𝑛 )𝑧𝑇2 − 𝛤𝑠−1 𝑆)} ̇ 𝑇 −1 𝑇 𝑇 𝑇 + 𝑡𝑟{𝐹̃ (𝑥̂ 𝑛 𝑧2 𝑆̂ 𝜎́ (𝐹̂ 𝑥̂ 𝑛 ) − 𝛤𝑓 𝐹̂ )} should be selected such that these

the inequality (36) is rewritten as follows: ̃ 2 − 𝑑4 ‖𝐹̃ ‖2 − 𝑑5 ‖𝑃̃ ‖2 𝐿̇ ≤ − 𝑑1 ‖𝑧1 ‖2 − 𝑑2 ‖𝑧2 ‖2 − 𝑑3 ‖𝑆‖ 𝑓 𝑓

terms are removed in the above inequality. Then, by applying the

Fig. 10. (a) Time evolution of 𝑠̂1,1 , … , 𝑠̂(𝑁ℎ +1),𝑁𝑜 , (b) Time evolution of 𝑓̂1,1 , … , 𝑓̂(𝑁𝑖 +1),𝑁ℎ . 9

(37)

O. Elhaki and K. Shojaei

Engineering Applications of Artificial Intelligence 88 (2020) 103382

Fig. 11. (a) Upper bounding estimation of kinematic disturbances, (b) Time evolution of transformed tracking errors 𝑧1,1 and 𝑧1,2 , (c) 𝑧1,3 and 𝑧1,4 , (d) 𝑧1,5 and 𝑧1,6 .

Fig. 12. (a) AUV trajectory tracking in the presence of noise, (b) noise added to the system, (c) the control forces, and (d) the control torques in the presence of noise.

− 𝑑6 𝛾̃ 2 − 𝑑7 ‖𝑧𝑑𝑒 ‖2 + 𝛯 + 𝜉𝑐 ℎ̄

0.5𝜆𝑚𝑎𝑥 {𝜁 𝐽 }, 𝜆𝑚𝑖𝑛 {𝐷+𝐾2 } > 0.5𝜆𝑚𝑎𝑥 {𝐾2 }, 𝑘 ∈ ℜ+ , 𝛿𝑠 (1−0.5∕𝑘2 )−0.5 > 0, 𝛿𝑓 (1 − 0.5∕𝑘2 ) − 0.5 > 0, 1∕𝜏𝑏 > 1 + 0.5𝜆𝑚𝑎𝑥 {𝜁 𝐽 }, 𝜆𝑚𝑖𝑛 {𝛩} > 1, 𝛯 is a term

By recalling (31), one infers that ‖𝑧̃ 2 ‖2 is bounded. Moreover, 𝑑1 = 𝜆𝑚𝑖𝑛 {𝐾1 } − 0.5𝜆𝑚𝑎𝑥 {𝜁 𝐽 }, 𝑑2 = 𝜆𝑚𝑖𝑛 {𝐷 + 𝐾2 } − 0.5𝜆𝑚𝑎𝑥 {𝐾2 }, 𝑑3 = 𝛿𝑠 (1 − 0.5∕𝑘2 ) − 0.5, 𝑑4 = 𝛿𝑓 (1 − 0.5∕𝑘2 ) − 0.5, 𝑑5 = 0.5𝜆𝑚𝑖𝑛 {𝛩} − 0.5, 𝑑6 = 0.5𝛿𝛾 , and 𝑑7 = 1∕𝜏𝑏 − 1 − 0.5𝜆𝑚𝑎𝑥 {𝜁 𝐽 } are positive constants when 𝜆𝑚𝑖𝑛 {𝐾1 } >

that is bounded, and 𝜉𝑐 is given as follows: 𝜉𝑐 = −𝑧𝑇2 𝑘3 𝑧̂ 2 + 10

𝐵𝑧2 2

+

‖𝑧2 ‖2 2

(38)

O. Elhaki and K. Shojaei

Engineering Applications of Artificial Intelligence 88 (2020) 103382

Fig. 13. Trajectories of the AUVs in (a) xyz plane, and (b) xy plane.

Fig. 14.

Trajectories of the AUVs in (a) yz plane, (b) xz plane, (c) yz plane (magnified view), and (d) xz plane (magnified view).

where (38) could be extended as follows by considering 𝑧̂ 2𝑖 = 𝑧̃ 2𝑖 + 𝑧2𝑖 : 𝜉𝑐 ℎ̄ =(−𝑧𝑇2 𝑘3 𝑧̃ 2 − 𝑧𝑇2 𝑘3 𝑧2 +

𝐵𝑧2 2

+

where 𝑑 = min{𝑑1 , 𝑑2 , 𝑑3 , 𝑑4 , 𝑑5 , 𝑑6 , 𝑑7 }. From (34) and (41), it is clear ̇ that 𝐿(𝑡) is strictly negative outside the compact set: √ 𝛺𝑧 = {𝑥𝑧 (𝑡)|0 ≤ ‖𝑥𝑧 (𝑡)‖ ≤ max {𝐿(𝑡0 ), 𝜆𝑢 𝛯∕𝑑}∕𝜆𝑚 }

‖𝑧2 ‖2 )ℎ̄ 2

𝑘 1 ≤ (−( 𝑘3 − 0.5)‖𝑧2 ‖2 + ( 3 + 0.5)𝐵𝑧2 )ℎ̄ 2 2

This points out that ‖𝑥𝑧 (𝑡)‖ is decreasing each time that 𝑥𝑧 (𝑡) is outside the compact set 𝛺𝑧 , and this indicates that ‖𝑥𝑧 (𝑡)‖ is UUB and approaches to a small neighborhood of the origin. Furthermore,

(39)

From the above inequality, if 𝑘3 > 1, the condition 𝜉𝑐 ℎ̄ ≤ 0 is ensured

𝑧𝑇1 , 𝑧𝑇2 , 𝑠̃11 , … , 𝑠̃(𝑁ℎ +1)𝑁𝑜 , 𝑓̃11 , … , 𝑓̃(𝑁𝑖 +1)𝑁ℎ , 𝑃̃ 𝑇 , 𝛾̃ , 𝑧𝑇𝑑𝑒 ∈ 𝐿∞

by considering the following region: √ ‖𝑧2 ‖ > 𝐵𝑧

𝑘3 + 1 𝑘3 − 1

which imply that the transformed tracking errors and NN weight matrices errors are UUB. In addition, by recalling Assumptions 5 and 6, (30), (31), and control law (25), one concludes that

(40)

𝜖𝑒 , 𝑧̂ 2 , 𝑠̂11 , … , 𝑠̂(𝑁ℎ +1)𝑁𝑜 , 𝑓̂11 , … , 𝑓̂(𝑁𝑖 +1)𝑁ℎ , 𝜏, 𝑃̂ , 𝛾̂ ∈ 𝐿∞ .

Then, one has: ̇ 𝐿(𝑡) ≤ −𝑑‖𝑥𝑧 ‖2 + 𝛯 ≤ −𝑑𝐿(𝑡)∕𝜆𝑢 + 𝛯

Since 𝜖𝑒 ∈ 𝐿∞ and −𝛼𝑖 𝜂𝑖 (0) < 𝑒1𝑖 (0) < 𝛽𝑖 𝜂𝑖 (0), the condition (2) in Section 2.3 is fulfilled. By recalling Condition 𝐶4 in Section 2.3, since

(41) 11

O. Elhaki and K. Shojaei

Engineering Applications of Artificial Intelligence 88 (2020) 103382

Fig. 15. Posture errors with performance bounds: (a) 𝑥𝑒 , (b) 𝑦𝑒 , (c) 𝑧𝑒 , and (d) 𝜙𝑒 .

Fig. 16. Posture errors with performance bounds: (a) 𝜃𝑒 and (b) 𝜓𝑒 .

𝜖𝑒 (𝑡) converges to a neighborhood of the origin, the tracking errors 𝑒1𝑖 also tend to a ball around the zero with a prescribed performance and the proof is finished. ■ 4. Computer numerical simulations 4.1. Simulation results

0 ⎡ 𝐶1 (𝜈) = ⎢ −30𝑤 ⎢ ⎣ 17.5𝑣

30𝑤 0 −25𝑢

⎡ 0 𝐶2 (𝜈) = ⎢ −15𝑟 ⎢ ⎣ 22.5𝑞

15𝑟 0 −22.5𝑝

−17.5𝑣 25𝑢 0 −22.5𝑞 22.5𝑝 0

⎤ ⎥, ⎥ ⎦ ⎤ ⎥ ⎥ ⎦

𝑀 = 𝑏𝑙𝑜𝑐𝑘𝑑𝑖𝑎𝑔[𝑀1 , 𝑀2 ], 𝐷 = 𝑏𝑙𝑜𝑐𝑘𝑑𝑖𝑎𝑔[𝐷1 , 𝐷2 ] In this section, simulations are performed to validate the performance of the adaptive NN output-feedback controller. The AUV’s matrices are chosen as:

𝑔(𝑞) = [0, 0, 0, − sin(𝜙) cos(𝜃), −5 sin(𝜃), 0]𝑇 [ ] 03×3 𝐶1 (𝜈) 𝐶(𝜈) = 𝐶1 (𝜈) 𝐶2 (𝜈)

⎡ 25 𝑀1 = ⎢ 0 ⎢ ⎣ 0

0 17.5 0

⎡ 30 𝐷1 = ⎢ 0 ⎢ ⎣ 0

0 30 0

The control parameters are chosen as 𝐾1 = 3 × 𝑑𝑖𝑎𝑔[1, 1, 1, 1, 1, 1], 𝐾2 = 2 × 𝑑𝑖𝑎𝑔[1, 1, 0.1, 0.1, 10, 0.01], 𝐵𝑀 = 80, 𝐵𝑚 = −80, 𝑙 = 1, 𝛿𝛾 = 0.002, 𝛾𝑚𝑎𝑥 = 0.2, 𝛤𝛾 = 0.01, 𝛾𝑑 = 10, the observer parameters are selected as 𝑜 = 0.02, 𝜆1 = 4, the ARC parameters are set by 𝑄 = 2 × 𝑑𝑖𝑎𝑔[1, 1, 1, 1, 1, 1], 𝛩 = 5 × 𝑑𝑖𝑎𝑔[1.5, 1.5, 1.5, 1.5, 1.5, 1.5], 𝑚𝑘 = 0.5 (𝑘 = 1, … , 6), 𝑃 0 = [1.5, 1.5, 1.5, 1.5, 1.5 , 1.5]𝑇 , the performance function parameters are selected as 𝜂10 = 𝜂20 = 𝜂30 = 𝜂40 = 𝜂50 =

0 0 30 0 0 30

⎤ ⎡ 22.5 ⎥ , 𝑀2 = ⎢ 0 ⎥ ⎢ ⎦ ⎣ 0

⎤ ⎡ 30 ⎥ , 𝐷2 = ⎢ 0 ⎥ ⎢ ⎦ ⎣ 0

0 20 0

0 22.5 0 0 0 20

0 0 15

⎤ ⎥ ⎥ ⎦

⎤ ⎥ ⎥ ⎦ 12

O. Elhaki and K. Shojaei

Engineering Applications of Artificial Intelligence 88 (2020) 103382

Fig. 17. (a) Velocity estimation errors 𝜈̃ = 𝜈̂ − 𝜈, (b) the evolution of norms of 𝑆̂ and 𝐹̂ , (c) the control forces, and (d) the control torques.

Fig. 18. AUV trajectories in (a) xyz plane, and (b) xz plane..

where

1, 𝜂60 = 2, 𝜂1∞ = 𝜂2∞ = 𝜂3∞ = 0.002, 𝜂4∞ = 𝜂5∞ = 𝜂6∞ = 0.02, 𝑎1 = 𝑎2 = 𝑎3 = 𝑎4 = 𝑎5 = 𝑎6 = 0.2, 𝛼1 = 𝛼2 = 𝛼3 = 10, 𝛼4 = 𝛼5 = 0.3, 𝛼6 = 0.5, 𝛽1 = 𝛽2 = 𝛽3 = 15, 𝛽4 = 𝛽5 = 0.3, 𝛽6 = 0.5, ℘1 = ℘2 = ℘3 = 20, the NN parameters are given by 𝑁ℎ = 5, 𝛤𝑠 = 1.5𝐼𝑁ℎ +1 , 𝑁𝑖 = 30, 𝛤𝑓 = 𝐼𝑁𝑖 +1 , 𝛿𝑠 = 0.6, 𝛿𝑓 = 0.7, 𝑁𝑜 = 6, the initial values of 𝑆̂ 𝑇 ∈ ℜ6×6 , 𝐹̂ 𝑇 ∈ ℜ5×31 are set to zero in the simulations, the initial conditions are set as 𝑞(0) = [10, 10, 10, 0, 0, 45𝜋∕180]𝑇 , 𝑞𝑑 (0) = [1, 1, −2, 0, 0, 0]𝑇 , 𝜏𝑑 = [8, 0, 0, 0, 2, 2]𝑇 , 𝜚1 (0) = [1, 1, 1, 1, 1, 1]𝑇 , 𝜚2 (0) = [1, 1, 1, 1, 1, 1]𝑇 , 𝛾̂ (0) = 0, 𝑃̂ (0) = [0, 0, 0, 0, 0, 0]𝑇 . The vector of uncertain dynamics and exogenous forces in (1), i.e.

𝛿𝑗 = 𝑇𝑘 𝑠𝑖𝑛(0.2𝑡 + 𝜋∕𝑘), 𝑗 = 𝑥, 𝑦, 𝑧, 𝜙, 𝜃, 𝜓, 𝑘 = 1, … , 6 𝛿𝑘 = 𝐴𝑓 𝑠𝑖𝑔𝑛(𝑖) + 𝐴𝑤 𝑠𝑖𝑛(0.1𝑡) + 𝑇𝑑 𝑠𝑖𝑛(0.5𝑡), 𝑖 = 𝑢, 𝑣, 𝑤, 𝑝, 𝑞, 𝑟, 𝑘 = 1, … , 6 with 𝐴𝑓 = 𝐴𝑤 = 0.3, 𝑇𝑘 = 0.1 and 𝑇𝑑 = 1. Figs. 3(a), 4, and 5(a) display that the proposed controller has forced the AUV to accomplish the tracking of the desired trajectory well. It is shown in Figs. 6 and 7 that the tracking errors have evolved within the performance bounds which are converged to the zero, and their prescribed transient response specifications such as the maximum overshoot, convergence rate and final tracking accuracy are guaranteed as proved in Theorem 1. The velocity estimation errors tend to the origin as shown by Fig. 5(b) which imply that the approximated velocities converge to the real velocities with a bounded error. As shown by Figs. 3–11, all variables of the control system are bounded, and control objectives are met against the ocean disturbances and uncertain dynamics. In Figs. 9 and 11(a), the upper bounds parameters are estimated and their effects are compensated during the tracking mission. The nonsaturated input command signals are depicted in Fig. 8 in the presence of unknown nonlinearities and external forces. Fig. 3(b) also displays

𝐻(𝜈) = [𝐻𝑢 (𝑢), 𝐻𝑣 (𝑣), 𝐻𝑤 (𝑤), 𝐻𝑝 (𝑝), 𝐻𝑞 (𝑞), 𝐻𝑟 (𝑟)]𝑇 , is added to the system such that 𝐻𝑖 (𝑖) = 0.5𝑖 + 0.25𝑖|𝑖| + 0.15𝑖3 , 𝑖 = 𝑢, 𝑣, 𝑤, 𝑝, 𝑞, 𝑟. Moreover, external disturbances in kinematic and dynamic levels are appended to the system as follows: 𝛿 = [𝛿𝑥 , 𝛿𝑦 , 𝛿𝑧 , 𝛿𝜙 , 𝛿𝜃 , 𝛿𝜓 ]𝑇 𝛥 = [𝛿1 , 𝛿2 , 𝛿3 , 𝛿4 , 𝛿5 , 𝛿6 ]𝑇 13

O. Elhaki and K. Shojaei

Engineering Applications of Artificial Intelligence 88 (2020) 103382

Fig. 19. Actuator saturation compensation for (a) 𝜏𝑢 , (b) 𝜏𝑣 , (c) 𝜏𝑤 , and (d) AUV trajectories in xyz plane in magnified view.. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

closed-loop signals. Figs. 13 and 14 show that the AUV that is equipped with the proposed controller has tracked the desired trajectory better than the AUV equipped with NPPOFC. Figs. 14(c) and 14(d) display that the unmodeled dynamics and external disturbances have affected the tracking performance of NPPOFC, and the AUV equipped with NPPOFC has tracked the desired trajectory with oscillations. But the proposed controller has repulsed the effects of external forces and nonlinearities. Thus, the AUV equipped with the proposed controller has tracked the reference trajectory with a high accuracy. The performance bounds, posture, and orientation tracking errors are depicted in Figs. 15 and 16. They imply that the proposed controller preserves a better transient and steady-state behavior of the tracking errors. In Fig. 17(c)– (d), the comparison between generated control signals are depicted that shows the actuator saturation for NPPOFC. To evaluate both controllers performance further, another simulation with a higher level of disturbance (𝑇𝑑 = 3) has been performed that is depicted in Fig. 18. It implies that this amount of the disturbance has degraded the NPPOFC performance greatly, while our proposed controller has still preserved its robustness against the higher level of disturbances. Consequently, the proposed controller is more successful than NPPOFC. To test the actuator saturation compensation ability of the proposed controller (with and without saturation compensation), both AUVs initial coordinates are set to 𝑞(0) = [−10, 10, 10, 0, 0, 45𝜋∕180]𝑇 , and another simulation has been carried out that is shown in Fig. 19. Simulation results show that the control forces under the proposed controller (with saturation compensation) in Fig. 19 have not breached the saturation constrains and the AUV with actuator saturation compensation has tracked the desired reference trajectory well (blue solid line in Fig. 19(d)) which has met the control objective in this research work. But the performance of the controller without actuator saturation compensation (red solid line in Fig. 19(d)) is not satisfactory in the tracking mission. Hence, the simulation results point out that the tracking performance has been ameliorated by the proposed control scheme. For a precise evaluation, some quantitative comparisons between two controllers are provided by Tables 2 and 3. Besides, the following performance indexes are used:

the Frobenius norms of the matrices of the estimated NN weights. In Fig. 12, a Gaussian white noise with zero mean and standard deviation 0.01 is added to posture measurements, i.e. 𝑞, in the kinematic and dynamic models of AUV to simulate the behavior of real sensors. Since the sensor noise is a non-parametric uncertainty, its effect may be included in the terms 𝛿 and 𝛥 in Eq. (1). As shown by Fig. 12(a), the tracking performance is acceptable, but further simulations confirm that the performance of controller is reduced by increasing the level of the noise. All the simulations verify that the proposed NN outputfeedback controller has effectively forced the AUV to track the desired trajectory with the predetermined performance specifications. 4.2. A comparative experiment To study the advantages of mixing the prescribed performance method, neural network, and adaptive robust controllers, a comparative simulation is carried out with the following non-prescribed performance-based output-feedback controller (NPPOFC) without actuator saturation compensation that is inspired by Tee and Ge (2006) and Peng et al. (2013): 𝑧1 = 𝑒1 𝑧̂ 2 = 𝜈̂ − 𝑧𝑑 𝑧𝑑 = 𝐽 −1 (−𝑘1 𝑧1 + 𝑞̇𝑑 ) 𝜏 = −𝐾2 𝑧̂ 2 − 𝐽 𝑇 (𝑞)𝑧1 − 𝑆̂ 𝑇 𝜎(𝐹̂ 𝑇 𝑥) ̂ − 𝑘3 ℎ̄ 𝑧̂ 2

(42)

where 𝐾1 = 3 × 𝑑𝑖𝑎𝑔[1, 1, 1, 1, 1, 1], 𝐾2 = 2 × 𝑑𝑖𝑎𝑔[1, 1, 0.1, 0.1, 10, 0.01], 𝑜 = 0.02, 𝜆1 = 4 and 𝜈̂ is updated by (8)–(10), 𝑁ℎ = 5, 𝛤𝑠 = 1.5𝐼𝑁ℎ +1 , 𝑁𝑖 = 30, 𝛤𝑓 = 𝐼𝑁𝑖 +1 , 𝛿𝑠 = 0.6, 𝛿𝑓 = 0.7, 𝑁𝑜 = 6, 𝑆 𝑇 ∈ ℜ6×6 , 𝐹 𝑇 ∈ ℜ5×31 . Note that the above controller is a version of the proposed controller without using prescribed performance bound and ARC techniques based on Tee and Ge (2006) and Peng et al. (2013), and the stability analysis is similar to the proof in Section 3.2. This study aims to show the effectiveness of utilizing neural network-based adaptive robust controller and performance bound methods during the tracking mission. To this end, similar initial conditions are set for both controllers to analyze the 14

O. Elhaki and K. Shojaei

Engineering Applications of Artificial Intelligence 88 (2020) 103382

√

𝑇

(1∕𝑇𝑓 ) ∫0 𝑓 ∣ ∙(𝑡) ∣2 𝑑𝑡 is the root mean square of • 𝑅𝑀𝑆(∙(𝑡)) = the tracking errors. • 𝑒𝑓 = max(𝑇𝑓 − 5 ≤ 𝑡 ≤ 𝑇𝑓 {| ∙ (𝑡)|}) demonstrates the final tracking accuracy in the last five seconds.

Do, K.D., Pan, J., 2009. Control of Ships and Underwater Vehicles: Design for Underactuated and Nonlinear Marine Systems. Springer Science & Business Media. Doyle, J., Stein, G., 1979. Robustness with observers. IEEE Trans. Autom. Control 24 (4), 607–611. Du, J., Abraham, A., Yu, S., Zhao, J., 2014. Adaptive dynamic surface control with nussbaum gain for course-keeping of ships. Eng. Appl. Artif. Intell. 27, 236–240. Du, J., Hu, X., Liu, H., Chen, C., 2015. Adaptive robust output feedback control for a marine dynamic positioning system based on a high-gain observer. IEEE Trans. Neural Netw. Learn. Syst. 26 (11), 2775–2786. Elhaki, O., Shojaei, K., 2018. Neural network-based target tracking control of underactuated autonomous underwater vehicles with a prescribed performance. Ocean Eng. 167, 239–256. Fischer, N.R., Hughes, D., Walters, P., Schwartz, E.M., Dixon, W.E., 2014. Nonlinear RISE-based control of an autonomous underwater vehicle. IEEE Trans. Robot. 30 (4), 845–852. Fossen, T.I., 2002. Marine control system-guidance, navigation and control of ships, rigs and underwater vehicles. Mar. Cybemetics. Fujii, T., Ura, T., 1991. Neural-network-based adaptive control systems for AUVs. Eng. Appl. Artif. Intell. 4 (4), 309–318. Gao, W., Selmic, R.R., 2006. Neural network control of a class of nonlinear systems with actuator saturation. IEEE Trans. Neural Netw. 17 (1), 147–156. Ge, S.S., Hang, C.C., Lee, T.H., Zhang, T., 2013. Stable Adaptive Neural Network Control, Vol. 13. Springer Science & Business Media. Ge, S.S., Zhang, J., 2003. Neural-network control of nonaffine nonlinear system with zero dynamics by state and output feedback. IEEE Trans. Neural Netw. 14 (4), 900–918. Hagan, M.T., Demuth, H.B., 1999. Neural networks for control. In: Proceedings of the 1999 American Control Conference (Cat. No. 99CH36251), Vol. 3. IEEE, pp. 1642–1656. Ioannou, P.A., Sun, J., 1996. Robust Adaptive Control, Vol. 1. PTR Prentice-Hall Upper Saddle River, NJ. Jafari, M., Xu, H., 2019. A biologically-inspired distributed fault tolerant flocking control for multi-agent system in presence of uncertain dynamics and unknown disturbance. Eng. Appl. Artif. Intell. 79, 1–12. Juman, M.A., Wong, Y.W., Rajkumar, R.K., Kow, K.W., Yap, Z.W., 2019. An incremental unsupervised learning based trajectory controller for a 4 wheeled skid steer mobile robot. Eng. Appl. Artif. Intell. 85, 385–392. Khalil, H.K., 2002. Nonlinear Systems, third ed. Prentice Hall, New Jewsey, 9 (42). Krupínski, S., Allibert, G., Hua, M.-D., Hamel, T., 2017. An inertial-aided homographybased visual servo control approach for (almost) fully actuated autonomous underwater vehicles. IEEE Trans. Robot. 33 (5), 1041–1060. Lewis, F.L., Dawson, D.M., Abdallah, C.T., 2004. Robot Manipulator Control Theory and Practice, second ed. Revised and Expanded Marcel Dekker, New York. Li, S., Wang, X., Zhang, L., 2015. Finite-time output feedback tracking control for autonomous underwater vehicles. IEEE J. Ocean. Eng. 40 (3), 727–751. Narendra, K.S., Parthasarathy, K., 1990. Identification and control of dynamical systems using neural networks. IEEE Trans. Neural Netw. 1 (1), 4–27. Park, B.S., Kwon, J.-W., Kim, H., 2017. Neural network-based output feedback control for reference tracking of underactuated surface vessels. Automatica 77, 353–359. Peng, Z., Wang, D., Chen, Z., Hu, X., Lan, W., 2013. Adaptive dynamic surface control for formations of autonomous surface vehicles with uncertain dynamics. IEEE Trans. Control Syst. Technol. 21 (2), 513–520. Peng, Z., Wang, J., Wang, J., 2019. Constrained control of autonomous underwater vehicles based on command optimization and disturbance estimation. IEEE Trans. Ind. Electron. 66 (5), 3627–3635. Polycarpou, M.M., 1996. Stable adaptive neural control scheme for nonlinear systems. IEEE Trans. Automat. Control 41 (3), 447–451. Pradeep, D., Noel, M.M., Arun, N., 2016. Nonlinear control of a boost converter using a robust regression based reinforcement learning algorithm. Eng. Appl. Artif. Intell. 52, 1–9. Qiao, L., Zhang, W., 2018. Adaptive second-order fast nonsingular terminal sliding mode tracking control for fully actuated autonomous underwater vehicles. IEEE J. Ocean. Eng. (99), 1–23. Radac, M.-B., Precup, R.-E., 2016. Three-level hierarchical model-free learning approach to trajectory tracking control. Eng. Appl. Artif. Intell. 55, 103–118. Ren, B., Ge, S.S., Tee, K.P., Lee, T.H., 2010. Adaptive neural control for output feedback nonlinear systems using a barrier Lyapunov function. IEEE Trans. Neural Netw. 21 (8), 1339–1345. Shen, C., Shi, Y., Buckham, B., 2018. Trajectory tracking control of an autonomous underwater vehicle using Lyapunov-based model predictive control. IEEE Trans. Ind. Electron. 65 (7), 5796–5805. Shojaei, K., 2015. Neural adaptive robust output feedback control of wheeled mobile robots with saturating actuators. Internat. J. Adapt. Control Signal Process. 29 (7), 855–876. Shojaei, K., 2016. Neural network formation control of underactuated autonomous underwater vehicles with saturating actuators. Neurocomputing 194, 372–384. Shojaei, K., 2017. Output-feedback formation control of wheeled mobile robots with actuators saturation compensation. Nonlinear Dynam. 89 (4), 2867–2878. Shojaei, K., 2018a. Neural network formation control of a team of tractor–trailer systems. Robotica 36 (1), 39–56.

5. Conclusion and discussion In this paper, an adaptive robust neural output-feedback dynamic surface controller with a guaranteed prescribed performance has been addressed for fully-actuated AUVs by incorporating a high-gain observer, three-layer NN, ARC, and performance bound technique. The uniform ultimate boundedness of tracking errors and all the closedloop variables were achieved. By the prescribed performance bound technique, some constraints were imposed on the system tracking errors that have guaranteed the prespecified transient and final tracking accuracy. Based on the advantages of the multi-layer NNs and ARCs, unknown nonlinearities, unmodeled dynamics, external forces, disturbances in kinematic and dynamic levels, and actuator nonlinearities were compensated. A high-gain observer was utilized to estimate velocities of the AUV to remove the velocity sensors. A comparative experiment has been performed to illustrate a better tracking performance of the proposed controller. Simulation results have verified the efficiency and better performance of the proposed control scheme. It should be noted that although most AUVs are underactuated in practice due to their weight and cost reduction, fully-actuated AUVs have their own advantages in practical applications such as their better maneuverability and every path feasibility. Albeit, the design of tracking controllers is more challenging for the underactuated vehicles from the control viewpoint, there exist some open problems that have not been solved for fully-actuated ones yet. As reported in Section 1, some of these important problems have been addressed recently for fully-actuated vehicles in the literature Krupínski et al. (2017), Qiao and Zhang (2018), Du et al. (2015) and Dai et al. (2018) and the researches on both classes are still active. This paper, in turn, has proposed a tracking controller with novel features which have not been addressed in all previous works. However, the proposed controller is not applicable to 6-DOF underactuated AUVs. The extension of this work to underactuated AUVs and their platoon control will be devoted to our future works. Acknowledgments The authors would like to thank the anonymous reviewers and associate editor for their valuable comments and suggestions. References Ahmed, Y.A., Hasegawa, K., 2013. Automatic ship berthing using artificial neural network trained by consistent teaching data using nonlinear programming method. Eng. Appl. Artif. Intell. 26 (10), 2287–2304. Bechlioulis, C.P., Karras, G.C., Heshmati-Alamdari, S., Kyriakopoulos, K.J., 2017. Trajectory tracking with prescribed performance for underactuated underwater vehicles under model uncertainties and external disturbances. IEEE Trans. Control Syst. Technol. 25 (2), 429–440. Bechlioulis, C.P., Rovithakis, G.A., 2008. Robust adaptive control of feedback linearizable MIMO nonlinear systems with prescribed performance. IEEE Trans. Automat. Control 53 (9), 2090–2099. Behtash, S., 1990. Robust output tracking for non-linear systems. Internat. J. Control 51 (6), 1381–1407. Cheng, L., Hou, Z.-G., Tan, M., 2009. Adaptive neural network tracking control for manipulators with uncertain kinematics, dynamics and actuator model. Automatica 45 (10), 2312–2318. Chu, Z., Xiang, X., Zhu, D., Luo, C., Xie, D., 2018. Adaptive fuzzy sliding mode diving control for autonomous underwater vehicle with input constraint. Int. J. Fuzzy Syst. 20 (5), 1460–1469. Cui, R., Yang, C., Li, Y., Sharma, S., 2017. Adaptive neural network control of AUVs with control input nonlinearities using reinforcement learning. IEEE Trans. Syst. Man Cybern. 47 (6), 1019–1029. Dai, S.-L., He, S., Lin, H., Wang, C., 2018. Platoon formation control with prescribed performance guarantees for usvs. IEEE Trans. Ind. Electron. 65 (5), 4237–4246. 15

O. Elhaki and K. Shojaei

Engineering Applications of Artificial Intelligence 88 (2020) 103382 van de Ven, P.W., Flanagan, C., Toal, D., 2005. Neural network control of underwater vehicles. Eng. Appl. Artif. Intell. 18 (5), 533–547. Wang, W., Huang, J., Wen, C., 2017. Prescribed performance bound-based adaptive path-following control of uncertain nonholonomic mobile robots. Internat. J. Adapt. Control Signal Process. 31 (5), 805–822. Xiang, X., Yu, C., Lapierre, L., Zhang, J., Zhang, Q., 2018. Survey on fuzzy-logic-based guidance and control of marine surface vehicles and underwater vehicles. Int. J. Fuzzy Syst. 20 (2), 572–586. Yu, C., Xiang, X., Zhang, Q., Xu, G., 2018. Adaptive fuzzy trajectory tracking control of an under-actuated autonomous underwater vehicle subject to actuator saturation. Int. J. Fuzzy Syst. 20 (1), 269–279. Zhong, Z., Zhu, Y., Ahn, C.K., 2018. Reachable set estimation for Takagi-Sugeno fuzzy systems against unknown output delays with application to tracking control of AUVs. ISA Trans. 78, 31–38. Zhu, M., Hahn, A., Wen, Y.-Q., 2018. Identification-based controller design using cloud model for course-keeping of ships in waves. Eng. Appl. Artif. Intell. 75, 22–35.

Shojaei, K., 2018b. Three-dimensional tracking control of autonomous underwater vehicles with limited torque and without velocity sensors. Robotica 36 (3), 374–394. Shojaei, K., 2019. Three-dimensional neural network tracking control of a moving target by underactuated autonomous underwater vehicles. Neural Comput. Appl. 31 (2), 509–521. Shojaei, K., Dolatshahi, M., 2017. Line-of-sight target tracking control of underactuated autonomous underwater vehicles. Ocean Eng. 133, 244–252. Swaroop, D., Hedrick, J.K., Yip, P.P., Gerdes, J.C., 2000. Dynamic surface control for a class of nonlinear systems. IEEE Trans. Autom. Control 45 (10), 1893–1899. Tee, K.P., Ge, S.S., 2006. Control of fully actuated ocean surface vessels using a class of feedforward approximators. IEEE Trans. Control Syst. Technol. 14 (4), 750–756. Tee, K.P., Ge, S.S., Tay, F.E., 2008. Adaptive neural network control for helicopters in vertical flight. IEEE Trans. Control Syst. Technol. 16 (4), 753–762. Tornambè, A., 1989. Use of asymptotic observers having-high-gains in the state and parameter estimation. In: Decision and Control, 1989., Proceedings of the 28th IEEE Conference on. IEEE, pp. 1791–1794. Uçak, K., Günel, G.Ö., 2019. Model free adaptive support vector regressor controller for nonlinear systems. Eng. Appl. Artif. Intell. 81, 47–67.

16