Neural Prescribed Performance Control for Uncertain Marine Surface Vessels without Accurate Initial Errors

Si, Wenjie; Dong, Xunde

doi:https://doi.org/10.1155/2017/2618323

Mathematical Problems in Engineering

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Research Article | Open Access

Volume 2017 | Article ID 2618323 | https://doi.org/10.1155/2017/2618323

Neural Prescribed Performance Control for Uncertain Marine Surface Vessels without Accurate Initial Errors

Wenjie Si¹and Xunde Dong¹

Academic Editor: Asier Ibeas

Received28 Sept 2016

Revised14 Nov 2016

Accepted22 Nov 2016

Published15 Jan 2017

Abstract

This paper deals with the problems concerned with the trajectory tracking control with prescribed performance for marine surface vessels without velocity measurements in uncertain dynamical environments, in the presence of parametric uncertainties, unknown disturbances, and unknown dead-zone. First, only the ship position and heading measurements are available and a high-gain observer is used to estimate the unmeasurable velocities. Second, by utilizing the prescribed performance control, the prescribed tracking control performance can be ensured, while the requirement for the initial error is removed via the preprocessing. At last, based on neural network approximation in combination with backstepping and Lyapunov synthesis, a robust adaptive neural control scheme is developed to handle the uncertainties and input dead-zone characteristics. Under the designed adaptive controller for marine surface vessels, all the signals in the closed-loop system are semiglobally uniformly ultimately bounded (SGUUB), and the prescribed transient and steady tracking control performance is guaranteed. Simulation studies are performed to demonstrate the effectiveness of the proposed method.

1. Introduction

As the demand for offshore exploration and operation, ocean surface vessels have been widely used in the marine industry. With the development of marine industrial technology, control design of marine surface vessels has become a hot topic of research [1, 2]. Control of ocean surface ships is a difficult question; the working environment is often complex (ocean currents and sea breeze); external unpredictable disturbances may degrade control system performance and even damage the stability. Therefore, the tracking control of the marine vessels has attracted much attention.

Various tracking control approaches have been presented for marine surface vessels; the sliding mode tracking control schemes were developed in [3, 4] based on explicit models. For the systems with unknown system dynamics, both fuzzy logic systems (FLSs) and neural networks (NNs) have been proved to be useful in the control design, where their universal approximation properties are employed to model unknown nonlinear functions [5–12]. Recently, many approximation-based adaptive control schemes have also been proposed to handle the control problem for uncertain ocean surface ships [13–15]. The output feedback control and the full-state feedback control were designed based on approximation-based adaptive backstepping of the ship dynamics in [16]. A neural learning control method was presented in [17] to solve the problem of tracking control of unknown ship systems. In [18], the unknown ship dynamics was learned by using deterministic learning theory and then learning problem from neural output feedback control of uncertain ship dynamics was studied. The prescribed performance control was introduced in [19] to deal with the neural learning control of ocean surface vessels via deterministic learning. However, in the working environments of marine surface vessels, the initial errors cannot be obtained in advance, and the control method given by [19] is invalid in most working environments. To solve the aforementioned problem, we propose a new performance function for uncertain marine surface vessels with unknown initial errors.

Dead-zone is one of important input nonlinearities which appears in a wide range of practical engineering. The existence of dead-zone nonlinearities degrades the performance of control system and even may lead to system instability. In [20, 21], the robust adaptive control methods were used for nonlinear systems with parametric uncertainties subject to the input deal-zone, and the systems must satisfy linear parameterized condition. Recently, in order to deal with unknown nonlinear systems with input dead-zone when the knowledge of system functions is unavailable, many adaptive controllers have been proposed by utilizing universal approximation capability of neural networks or some fuzzy logic systems [21, 22]. A robust adaptive NN control design method was proposed in [23] for a kind of strict-feedback nonlinear systems with uncertainties and input dead-zone. An adaptive fuzzy output feedback control was studied in [24] for switched nonlinear systems with uncertainties. In [25], the problem of the adaptive fuzzy backstepping output feedback tracking control was investigated for multi-input and multioutput (MIMO) stochastic nonlinear systems. The problem of adaptive decentralized NN control was investigated in [26] for large-scale stochastic nonlinear time-delay systems with input dead-zone.

In the control design, the tracking error is only required to converge to a small residual set, while the transient and steady-state tracking performance is not considered. The practical engineering often requires certain prespecified performance. More recently, the prescribed performance control (PPC) has been proposed in [27]. Furthermore, in [28, 29], the PPC was used for the position tracking control of robot. Combining PPC with dynamic surface control (DSC), a fuzzy control scheme was studied in [30] to ensure the performance of tracking control. When states were unmeasurable, an output feedback control was presented in [31] for large-scale nonlinear time-delay systems. The prescribed performance control technology was extended to MIMO systems [32, 33]. However, to our best knowledge, by using prescribed performance control, no tracking control methods exit for uncertain ocean surface vessels without the need for the initial error conditions.

Motivated by the aforementioned discussion, to guarantee the predefined performance for ocean surface vessels without velocity measurements in the presence of unknown input dead-zone, we will design an adaptive neural output feedback control scheme. RBF NNs are used to approximate the unknown nonlinearities. The prescribed performance function is designed to ensure the performance of the prescribed tracking control without any consideration for accurate initial errors. Then, based on the backstepping and Lyapunov theory, we propose an adaptive neural tracking control method to ensure the boundedness of the closed-loop system.

Compared with previous works, our paper has the following advantages. The unmeasurable velocities of the ocean surface ship are estimated by employing a high-gain observer. To prevent peaking of the high-gain observer at the initial phase, a new method is used different from the saturation functions employed in [16, 34]. A performance function is given to ensure the tracking control performance, and the requirement for the exact initial error is removed. With the proposed controller, the tracking control for uncertain surface vessels is achieved with only position sensors and the control performance of the system is guaranteed by prescribed performance control.

The rest of the paper is organized as follows. Section 2 presents the preliminaries and problem formulation. An adaptive neural tracking control scheme for uncertain ocean surface vessels with prescribed performance control is given in Section 3. The simulation studies are presented in Section 4 to demonstrate the effectiveness of the proposed method. Section 5 concludes this paper.

2. Problem Formulation and Preliminaries

2.1. Ship Dynamics

Consider multiple-input-multiple-output (MIMO) systems for a three degrees of freedom surface vessel subject to unknown model uncertainties and input dead-zone. The dynamics of the surface vessel is described by [1, 35]where , is the surface vessel position, and is the surface vessel heading; is the rotation matrix; , are velocities of the surge, the sway, and the yaw, respectively; is the vector of control input, denotes the inertia matrix of the ship, is the total Coriolis and centripetal acceleration matrix, is the damping matrix, is the vector of buoyancy/gravitational forces, and is used to model the uncertainties with being unknown.

Precisely, is given by [35]; , , and are given as follows:where , , , , , , and are unknown.

We have the description of the ship dynamics (1) subject to the unknown dead-zone nonlinearity as follows:where is the system output, is the system input and the dead-zone output, and the actuator dead-zone characteristic is described as with being the input of the unknown dead-zone.

The main goal of this control scheme is to present an adaptive NN tracking controller for the system described by (3) to ensure that the system output can track the reference signal and all signals in the closed-loop system remain bounded.

Assumption 1. The system output and its first-order derivative are continuous and bounded, such that and , where and are positive constants.

Assumption 2. The reference signal and its th order are continuous and bounded.

2.2. Dead-Zone Characteristic

The actuator dead-zone nonlinearity can be described as follows [36]:where and are unknown smooth functions.

Assumption 3. The parameters of the dead-zone and are unknown constants and satisfy and .

Assumption 4. For unknown functions and , there exist unknown constants and , such that

Let and . The dead-zone nonlinearity can be rewritten aswherewhere and .

It can be obtained that , .

2.3. Prescribed Performance

This section introduces the concept of the prescribed performance. Then, a performance function is given and it will be chosen to be used in the control design.

Definition 5 (see [27]). A smooth function is called a performance function, if(i) is a strictly positive decreasing function;(ii), where is a positive constant.

According to Definition 5, we choose the performance function as follows:where , and are design parameters. It is obvious that satisfies that . According to the description of the performance function , the tracking error has the following performance bound (as shown in Figure 1):where , .

To represent (9) by an unconstrained form, the following state transformation is employed [27]:whereand is called the transformed error. According to (11), the derivation of is as follows:

Based on , one haswith

2.4. RBF Neural Networks

It has been shown that neural networks are good at modeling unknown nonlinear functions in control design [37]. In the study, RBF NN will be employed to model unknown continuous function over a compact set for a given arbitrary accuracy as follows:where is the input vector with being the input dimension of the neural networks. is the neural weight vector with being the node number of the neural network, and denotes the ideal constant weight vector. is the approximation error, . is the vector of basis function; commonly is Gaussian function which has the following form:where is the center of the receptive field and is the bandwidth of Gaussian function [37].

The ideal constant weight vector is defined aswhere is the estimation of .

Lemma 6 (see [38]). Consider the Gaussian RBF neural networks (15). Let . is the dimension of the neural network input and is the width of the Gaussian function (16); then the following inequality holds:where is an upper bound of and is limited and independent of neural input and the neural weights dimension .

2.5. High-Gain Observer

The system output is available for measurement, and is unavailable. To solve the unmeasured states problem, we need a high-gain observer used in [39].

Lemma 7 (see [16]). Consider the following linear system:where is a small constant, and are system states, and is chosen to ensure that the linear system is stable. According to Assumption (1), the following properties hold:(i)where .(ii)There exist two constants and , which rely on the parameters , , , and , such that , .

Considering that in system (1) and the property , we can use to estimate the state variable .

Lemma 8 (see [39]). For the RBF neural networks, if , , is a bounded vector; thenwhere is the bounded function, ; is a positive constant.

3. Adaptive Neural Control Design

The backstepping design is based on the following coordinate changes:where is the virtual control signal, which is defined later.

Step 1. Considering and , one hasNoting the following the error variable as , we haveFrom the property , is defined aswhere is a design parameter.
We choose a Lyapunov function candidate , and its derivative along (24) yieldswhere the last term will be canceled in the next step.

Step 2. From , we haveNoting is the design constant obtained by [35], we consider the following Lyapunov function candidate:The derivative of along (27) is obtained byDenotewhere , . A RBF NN is employed to approximate the unknown function as follows:where and is the approximation error.
We choose the following control aswhere . ; is a design parameter.
From Lemma 7 and the property , one haswhere . is constant.
According to Lemmas 7 and 8, we haveConsider the following Lyapunov function candidateThe time derivative of is given byDefine the adaptive law as follows:where and are design parameters.
Substituting (32) and (37) into (36) yieldsConsidering , we haveBy completing the squares, we haveSubstituting (39)-(40) into (38) yieldswhere and denote the minimum and maximum eigenvalues of matrix .
Considering , we havewhere denotes a block-diagonal matrix.
Then, we have the following inequality:whereTo ensure , the parameters and are chosen to satisfy that and .

Theorem 9. Consider the ship system composed of system (1), observer (19), the neural controller (32), and the adaptive law (37). Under Assumptions 1 and 2, if the initial conditions are bounded, then all the signals in the closed-loop system are semiglobally uniformly ultimately bounded. Moreover, for all the tracking error is confined within the prescribed performance bounds.

Proof. Multiplying (43) by yieldsIntegrating (45), one hasGiven , the closed-loop error signals , , and will converge asymptotically to the compact sets , , and , defined bywhere denotes the neural network node number.
Therefore, , , and are uniformly ultimately bounded. Then is also bounded. is the transformed error, so is bounded. From Assumption 2, is bounded and is bounded, and thus and are bounded. The control input is bounded since is bounded. Thus, all signals in the closed-loop system remain bounded, and the tracking error not only converges to a small neighborhood of zero, but also achieves the prescribed performances.

Remark 10. To achieve the control design with no need of the initial error , preprocessing is needed before the operation of the controller and identifying which controller is applicable. In the paper, we have assumed that system state variable is unmeasurable. In practice, for most merchant vessels, only the heading and the position are measured. To estimate unmeasured , a high-gain observer (19) is introduced. We use the Exponential function to overcome the peaking phenomenon of the high-gain observer.

4. Simulation Studies

Simulation studies are presented to illustrate the effectiveness of the proposed control method in this section. The following choices of the system (1) are given: , , , , , , , and . We assume that these parameters are unknown for the control input . The ship inertia matrix is a constant matrix and, equally with [35], is assumed to be the model uncertainties with .

The dead-zone nonlinearity is given as

For the state variable , the performance function is chosen aswhere .

For the state variables , the performance function is chosen aswhere .

For the state variable , the performance function is chosen aswhere .

The objective of the control is to put forward an adaptive neural control scheme to ensure that all signals in closed-loop system remain bounded, and the system output tracks the ideal reference trajectory .

During the simulation, the initial conditions are and . , . = . The design parameters are given as follows: , , , , and . , , and . The input of the RBF neural networks is and are constructed using nodes with the centers evenly spaced on , the width being , .

Figures 2–10 show the simulation results for the prescribed performance control of the unknown ship dynamics. Figures 2, 4, and 6 show that the good tracking performances of the closed-loop system are achieved, even though no initial tracking errors are available. In Figures 3, 5, and 7 provide the curves of the tracking errors with PPC. It is obvious that the initial error does not need to be known in advance. Figure 8 illustrates the time trajectory of the state variables , , and and their estimations by using the high-gain observer. It can be seen that the peaking effect of the high-gain observer is eliminated. Figure 9 gives the norms of the adaptive laws , , and , which demonstrates the boundedness of the adaptive weights. The control input is shown in Figure 10.

5. Conclusion

In the paper, we investigate the problem of the tracking control with predefined performance for marine surface vessels without velocity measurements in the presence of unknown dead-zone input. RBF neural networks are used to deal with uncertain ship dynamics, a new performance function is designed with no requirement for the exact initial error, and the unmeasurable ship velocity is estimated by using a high-gain observer. It has been shown that the proposed common controller can ensure that all signals in the closed-loop systems are semiglobally uniformly bounded, and tracking error converges to a predefined small neighborhood of zero.

Competing Interests

The authors declare that they have no competing interests.

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Copyright © 2017 Wenjie Si and Xunde Dong. 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.

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