Abstract
This paper addresses the problem of trajectory tracking of underactuated surface vessels (USVs) in the presence of thruster failure. Multilayer neural networks (MNNs) are employed to estimate the unknown model parameters and external disturbances. To design a fault-tolerant controller without a fault detection scheme, we use the Nussbaum gain technique. We introduce an additional control to resolve the difficulty arising from having fewer inputs than degrees-of-freedom. Further, an approach angle is proposed to track both a straight and curved path. Stability analysis and simulations are performed to demonstrate the effectiveness of the proposed scheme.
1. Introduction
As land resources are depleted, interest in ocean exploration has increased. There are many marine vehicles designed to explore the ocean. Among these, unmanned marine vehicles (UMVs) such as autonomous underwater vehicles (AUVs) and unmanned surface vessels are useful because they can reduce the cost and the risk of personal injury [1]. Therefore, significant research has been undertaken to perform complex tasks using UMVs.
An important problem in the control of UMVs is failure occurrence on the actuators [2, 3]. If there is no immediate solution for actuator failure, the UMV must abort the mission and return to the base for repair. This results in a waste of time and money. Therefore, to enhance the reliability while performing its mission in the presence of thruster failure, there are more thrusters than the degrees-of-freedom of the UMV. For this reason, the thruster-force allocation problem is important to overcome thruster failure [4]. The allocation of thruster forces of an AUV that can accommodate both thruster faults and saturation was presented in [5, 6]. A fault diagnosis and accommodation system was proposed using weighted pseudoinverse in [7]. Weighted pseudoinverse and quantum particle swarm optimization were introduced for hybrid fault-tolerant control in [8]. The combination of backstepping technique and fuzzy control was presented for tracking control of fully actuated surface vessels in [9]. However, all these works focused on fully or overactuated systems. This implies that the cost and weight of the system are increased because there must be more devices. Moreover, if a fully actuated system is damaged, we then have an underactuated controller [10]. Therefore, it is necessary to develop a fault-tolerant controller for underactuated systems.
Motivated by these observations, a neural network-based fault-tolerant control is presented for underactuated surface vessels (USVs). The proposed algorithm can be applied to AUVs because the equations of motion for surface vessels are the same as those of AUVs in the horizontal plane. For a practical perspective, we assume that the bow and stern are not symmetric and the model parameters for the hydrodynamic terms are unknown. To estimate the unknown model parameters and external disturbances, we employ multilayer neural networks (MNNs) [11]. Unlike other fault-tolerant control methods [12–14], we do not require a fault detection scheme; we design the fault-tolerant controller using the Nussbaum gain technique [15]. Further, an additional control is introduced to resolve the difficulty of controller design of underactuated systems, and an approach angle is proposed to track any trajectory including both a straight- and curved-line path. It is proved that all the signals of the closed-loop system are ultimately, uniformly bounded. Simulation results are presented to demonstrate the effectiveness of the proposed scheme.
The main contributions of our paper are as follows. (1) The fault-tolerant controller is designed without the fault detection scheme. (2) For underactuated surface vessels, we develop the additional control input and analyze the stability including the sway dynamics. (3) The approach angle which is composed only of position information of USVs is introduced. With the help of a novel approach angle, we can track any trajectory including both a straight- and curved-line path. (4) For a practical application, the model parameters for the hydrodynamic terms and external disturbances are assumed to be unknown. These uncertainties are estimated by MNNs.
Throughout this paper, the following notations are used: (1) denotes any suitable vector norm; (2) is the trace; (3) is the Frobenius norm defined by ; (4) and denote bounded ideal neural weights and the estimates of neural weights, respectively; (5) denotes the error between and ; (6) and denote the minimum and maximum eigenvalues of matrix , respectively; (7) is a diagonal matrix.
2. Underactuated Surface Vessel Model
The kinematics and dynamics of USVs are described as follows [16]: where denotes position and yaw angle of the USV in the earth-fixed frame; denotes surge, sway, and yaw velocities of the USV in the body-fixed frame, respectively, denotes the external forces such as wind and ocean currents, and is the control vector of the surge force and the yaw moment . In the above equation, matrices , , , and are given as follows:where , , , , , , , , , , , , , and are the linear and quadratic drag coefficients, is the mass of the USV, , , , , and are the added masses, is the -coordinate of the USV center of gravity in the body-fixed frame, is the inertia with respect to the vertical axis, and is invertible.
Remark 1. Unlike other works, we assume that the parameters and are not zero. This means that the sway dynamics is influenced by the yaw moment and we must control sway and yaw velocities simultaneously using only the yaw moment . When the parameters and are not zero, controller design difficulties are increased.
To address the problem stated in Remark 1, we utilize the following state transformations [17]:where . The transformed equation (4) implies that the virtual center of mass is positioned by in the longitudinal direction. Using (4), the USV dynamics (1) can be expressed by defining state variables and : where In these expressions, we assume that , , and are unknown time-varying functions because they cannot be obtained precisely by measurement or calculation.
When faults occur in the control vector , we can express them as follows: where is a nominal control vector, denotes the partial loss of control vector, and is an unknown bias vector. and are defined as
The control objective is to design a fault-tolerant controller such that it tracks the reference trajectory even though faults exist in the control vector of the USV. The reference trajectory is given as where and denote the desired position; denotes the desired yaw angle, and , , and are the desired velocities of the USV. To generate the reference trajectory, we can use path planning techniques presented in [18, 19].
Assumption 1. The desired velocities , , and are bounded. Further, the first derivatives , , and are available and bounded.
3. Fault-Tolerant Control
3.1. Neural Network Estimation
In this paper, MNN is used to estimate the unknown function . From the universal approximation property for neural networks (NNs) [20], the nonlinear function can be written aswhere is the NN reconstruction error and is the sigmoid function defined as .
Assumption 2. and are bounded such that and for positive constants and , respectively, and is bounded such that for a positive constant .
The Taylor series expansion of in (10) for a given can be written as where , denotes higher order terms, and . From (10) and (11), we can obtain the following result: where .
Property 1. The unknown term in (12) is bounded such that , where , , and is a positive bounding constant.
Proof. See the appendix.
3.2. Neural Network-Based Fault-Tolerant Control
In this section, we design the neural network-based fault-tolerant control system. To avoid the difficulty of detecting the faults, the controller is proposed using the Nussbaum gain technique. The following lemma is used to design the neural network-based fault-tolerant control law.
Lemma 2 (see [21]). Let and be smooth functions defined on with , . For , if the following inequality holds, where and are positive constants, is a Nussbaum-type function, and is a time-varying parameter which takes values in the closed intervals , then , , and must be bounded on .
We now design the neural network-based fault-tolerant control law using the dynamic surface design approach.
Step 1. Define the following errors: where and . An approach angle is defined as where and is a positive constant.
Remark 3. In this paper, we use an approach angle as an alternative for the desired yaw angle . Using the approach angle , we can track any trajectory including both straight-line and curved-line paths. There are other works using the approach angle; however, our approach angle is different from the other studies because it requires only position information.
Using (5) and (9), the time derivative of (14) is given as To stabilize the error dynamics of (16), we choose the virtual controls , , and aswhere and are design parameters.
Step 2. In this step, we use the NNs to estimate uncertain terms , , and . From (12), we can derive the following results: where is the estimate of and . By Property 1, is bounded such that , where is the unknown constant vector and .
Assumption 3. and are bounded such that and for positive constants and , respectively, and is bounded such that for a positive constant .
Define the following errors: where is a positive constant. , , and are signals obtained by the following first-order filters: where is a positive constant. Here, , , and are used to solve the problem that there is no actual control on the sway dynamics as seen from (5) and obtained by the following differential equations: where , , are design parameters and is the estimate of . To design the unified control algorithm from the error dynamics of (19), we introduce a function as where is the threshold which is the criterion for judgment.
Remark 4. The threshold can be derived from the fault detection algorithm. However, the proposed algorithm does not require the threshold because we design the neural network-based fault-tolerant controller under the assumption that the function in (22) is unknown.
Using (5), (7), and (22), the time derivative of (19) is represented as where
Property 2. and are bounded such that and .
Proof. See the appendix.
From (23), we choose the actual controls and as follows: where and are Nussbaum-type functions. For the stability analysis, we define the boundary layer error as Let us consider the following Lyapunov functions:where is positive definite matrix.
Theorem 5. Consider the USV model (5) in the presence of the faults. For any initial conditions and , where and are any positive constants, if Assumptions 1–3 are satisfied and one applies the controllers (17) and (25) with the adaptation laws defined as where is a positive constant, then one can ensure that where is a positive constant.
Proof. See the appendix.
4. Simulation Results
In this section, we simulate a neural network-based fault-tolerant controller to verify the performance of the proposed control system. The parameters of the underactuated surface vessel are as indicated in [22]. The control gains are chosen as , , , , , and , where . The reference trajectory is generated by the following desired velocities:The initial postures for the reference USV and the actual USV are and , respectively. In this simulation, we use three MNNs to estimate the uncertainties and the MNNs are composed of four hidden layers and three inputs. In addition, the weight tuning parameters of the MNNs are chosen as , , and , where . To verify the robustness of the proposed neural network-based fault-tolerance controller, we assume that the faults occur after 20 seconds such that and .
Figure 1(a) presents the trajectory tracking result. The tracking errors are depicted in Figures 1(c) and 1(d). As indicated in Figures 1(c) and 1(d), there are tracking errors when faults occur on ; however these errors are reduced quickly. Further, the tracking errors are bounded for the entire time even though faults exist. From Figure 1(a), we can confirm that the actual USV tracks successfully the reference trajectory including both straight and curved paths. Therefore, we can conclude that the proposed control system possesses robustness for faults, model uncertainties, and external disturbances.
(a)
(b)
(c)
(d)
5. Conclusions
This paper presented a neural network-based fault-tolerant controller for an underactuated USV with faults, model uncertainties, and external disturbances. MNNs were employed to estimate the highly nonlinear uncertain terms, and approach angle was implemented to track any reference trajectory including both straight- and curved-line paths. To avoid the difficulty of detecting the faults, the neural network-based fault-tolerant controller was designed using the Nussbaum gain technique. From the simulation results, it was demonstrated that the proposed control system has good tracking performance in the presence of the faults, model uncertainties, and external disturbances. In the future, the actuator saturation problem will be addressed and evaluate the performance through real experiments.
Appendix
Proof of Property 1. Since and are bounded, the unknown term in (12) can be written aswhere and are positive constants. Using Assumption 2 and triangular inequality, (A.1) is bounded bywhere , , , , and .
Proof of Property 2. Since and are bounded from (8) such that and , one can easily prove that and .
Proof of Theorem 5. We prove Theorem 5 in two steps. The first step is to prove boundness of and . In the second, we prove boundness of .
Step 1. The time derivative of using (16)–(20) and (23)–(26) results inwhere On and , and , where and are positive constants. Therefore, using (28) and Young’s inequality, (A.3) is written asChoosing , , and with yields where and . Upon multiplication of (A.6) by , it becomes Integrating it over , we have where . Therefore, we can conclude that , , and are bounded on according to Lemma 2.
Step 2. The time derivative of using (16)–(20) and (23)–(26) results inwhere On and , , where is a positive constant. Then, using (28) and Young’s inequality, (A.3) is written asChoosing and with yields where and . Similarly, we havewhere . Therefore, we can conclude that , , and are bounded on according to Lemma 2.
Since and are bounded, one can easily prove from (25) that and are bounded. Then, and are bounded such that and , where and are positive constants. From (A.6) and (A.12), on when , and on when . This means that and are invariant sets. Moreover, the boundedness of and implies that , , and are bounded. Therefore, we can conclude from (4) and (15) that the inequality in (29) holds.
Remark A.1. The choice of control gains has some suggestions as follows: (i) increasing , , , and and decreasing and help to increase , subsequently reducing the bound of error; (ii) increasing , , and and decreasing help to increase , subsequently reducing the bound of error.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2012R1A1A1041216).