Adaptive Neural Network Control with Backstepping for Surface Ships with Input Dead-Zone

Abstract

This paper addresses the problem of adaptive neural network controller with backstepping technique for fully actuated surface vessels with input dead-zone. The combination of approximation-based adaptive technique and neural network system is used for approximating the nonlinear function of the ship plant. Through backstepping and Lyapunov theory synthesis, an indirect adaptive network controller is derived for dynamic positioning ships without dead-zone property. In order to improve the control effect, a dead-zone compensator is derived using fuzzy logic technique to handle the dead-zone nonlinearity. The main advantage of the proposed controller is that it can be designed without explicit knowledge about the ship motion model, and dead-zone nonlinearity is well compensated. A set of simulations is carried out to verify the performance of the proposed controller.

1. Introduction

Ship moving in the sea performs strong nonlinear and coupled property. It is difficult for researchers to obtain the ship motion model. Most modern nonlinear control theories, such as sliding mode control, backstepping technique, and feedback linearization, are mostly based on the knowledge about system models. For these control strategies, the controller cannot be derived without explicit knowledge about the system model. Meanwhile, dead-zone characteristic is quite common in actuators, that is, rudders or propellers. But few works consider the influences on control performance brought by the dead-zone characteristic.

Fuzzy logic system and neural network are commonly used for approximating the unknown terms or uncertainties in the systems [1–6]. An adaptive fuzzy decentralized output-feedback control problem was discussed for a class of nonlinear large-scale systems, and fuzzy logic systems were employed to approximate the unknown nonlinear functions in the control design in [1]. Zhu and Li proposed a stable decentralized adaptive fuzzy sliding mode control scheme for reconfigurable modular manipulators to satisfy the concept of modular software in [2], and a first-order Takagi-Sugeno fuzzy logic system was introduced to approximate the unknown dynamics of subsystem by using adaptive algorithm. The fuzzy basis function was used to approximate an unknown nonlinear function according to some adaptive laws in [3], and then the state observer was designed for estimating the states of the plant, upon which an adaptive fuzzy sliding mode controller was investigated. In [4], a radial basis function (RBF) network were employed to estimate and compensate the uncertainties of ship dynamics and disturbances in controller design. In [5], an adaptive neural network control scheme for robot manipulators with actuator nonlinearities was presented, and the RBF network was introduced to emulate the unknown parameters. An adaptive RBF neural network controller was adopted to learn the unknown upper bound of model uncertainties and external disturbances in [6]. The approach taken in this paper tries to overcome the necessity for ship’s mathematical models by using an adaptive RBF network control algorithm to estimate the unknown nonlinear functions. Meanwhile, adaptive RBF network can improve the robustness of the control systems.

In real applications, the dead-zone characteristic is quite common in actuators. Sometimes the dead-zone nonlinearity affects the control system performance. Generally, the dead-zone parameters are unknown and time-variant, which cause challenging problem for the control system. Dead-zone commonly affects all practical systems, such as mechanics and electronics, so the study on dead-zone has drawn much interest in the control community for a long time [7–12]. To solve the problem brought by the dead-zone, adaptive dead-zone inverses were proposed in [7–9]. Reference [7] built a continuous dead-zone inverse for linear system with unmeasurable dead-zone outputs, while asymptotical adaptive cancellation of an unknown dead-zone was achieved analytically under the condition that the output of a dead-zone was measurable in [8]. For a general nonlinear actuator dead-zone of unknown width, [9] presented a compensation scheme using neural network. In [11], adaptive control with adaptive dead-zone inverse has been introduced by giving a matching condition to the reference model. By utilizing the integral-type Lyapunov function and introducing an adaptive compensation for the upper bound of the optimal approximation error and the dead-zone disturbance, a robust adaptive neural controller for a single input single output nonlinear system was derived in [12]. By utilizing a description of a dead-zone feature, an adaptive law was used to estimate the properties of the dead-zone model intuitively and mathematically, without constructing a dead-zone inverse in [13]. Departing from existing approximate adaptive dead-zone compensations, [14] used indirect parameter estimation algorithms along with online condition monitoring to obtain an accurate estimation of the unknown dead-zone when certain relaxed persistent-excitation conditions are satisfied—a theoretical result that cannot be achieved with the existing methods.

This paper proposes the work on adaptive neural network control with backstepping technique for fully actuated surface vessels with constraint in the actuator. The main contributions of this paper are as follows.(i)To the best of our knowledge, it is the first time in the literature that both input saturation and dead-zone are considered during controller design for fully actuated surface ships motion control problems. (ii)For handling the limit of propellers, the auxiliary design system is introduced to analyze the input saturation, and the effect of dead-zone nonlinearity is compensated by using fuzzy logic system (FLS). (iii)The adaptive method and radial basis function neural network (RBFNN) system are combined to estimate the unknown nonlinear functions of the ship plant.(iv)Under the assumptions of the inexistence of the input saturation and the dead-zone characteristic in actuator, an adaptive RBFNN controller using backstepping technique for fully actuated surface ships is derived without knowing the ship motion model.

The main advantage of the proposed method is that the controller can be designed without explicit knowledge about the ship motion model, and control input saturation and dead-zone nonlinearity are well compensated.

The rest of this paper is organized as follows. In Section 2, a model of fully actuated surface vessels with dead-zone is established. Section 3 contains the design of adaptive RBFNN controller with backstepping, the auxiliary design system is introduced to handle the input saturation, and an FLS is utilized to approximate the nonlinearity of the dead-zone in actuators. Then a set of simulations is taken in Section 4 to verify the control effect of the proposed method. Finally, conclusions are made in Section 5.

2. Model of Surface Vessels

2.1. Ship Motion Model for Surface Vessels

For the horizontal motion of a fully actuated surface vessel, the kinematics and dynamics models can be described by (1), for more details see [15]. ̇(𝜼)𝐌̇(𝝂)(𝝂)𝜼=𝐉𝝂,𝝂+𝐂𝝂+𝐃𝝂=𝝉+𝐑^Τ(𝜼)𝐛, (1) where 𝜼=[𝑥,𝑦,𝜓]^Τ, 𝝂=[𝑢,𝑣,𝑟]^Τ, and 𝑥, 𝑦 denote the coordinates of the vessel in the earth-fixed frame 𝑋_𝐸𝑌_𝐸𝑍_𝐸 (see Figure 1), and ψ  is the heading angle of the ship. 𝑢, 𝑣, and 𝑟 express the velocities in surge, sway, and yaw, respectively in the body-fixed reference frame 𝑋𝑌𝑍, 𝝉=[𝜏_𝑥,𝜏_𝑦,𝜏_𝜓]^Τ represents the control inputs, that is, forces and moments produced by the propellers, 𝐌 stands for the mass and inertia matrix, 𝐂(𝝂) denotes the Coriolis-centripetal matrix, 𝐃(𝝂) is the damping matrix, and 𝐛 is a bias term representing slowly varying environmental forces and moments caused by the wind, second-order waves, and currents. 𝐉(𝜼) is a state dependent transformation matrix which can be written as 𝐉(𝜼)=𝐉𝜓=cos𝜓−sin𝜓0sin𝜓cos𝜓0001. (2)

Figure 1 

Definition of the earth-fixed 𝑋𝐸𝑌𝐸𝑍𝐸 and the body-fixed 𝑋𝑌𝑍 reference frames.

Note that the matrix 𝐉(𝜼) is nonsingular for all 𝜼∈ℝ³ and  𝐉⁻¹(𝜼)=𝐉^Τ(𝜼); for example, 𝐉(𝜼)𝐉^Τ(𝜼)=𝐈.

The system model can be rewritten as [16] 𝐌_𝜂(𝜼)𝜼̈+𝐂_𝜂(𝝂,𝜼)𝜼̇+𝐃_𝜂(𝝂,𝜼)𝜼̇=𝐉^−Τ(𝜼)𝝉+𝐛, (3) where 𝐌_𝜂(𝜼)=𝐉^−Τ(𝜼)𝐌𝐉⁻¹(𝜼),𝐂_𝜂(𝝂,𝜼)=𝐉^−Τ(𝜼)𝐂(𝝂)−𝐌𝐉⁻¹(𝜼)𝐉̇(𝜼)𝐉⁻¹(𝜼),𝐃_𝜂(𝝂,𝜼)=𝐉^−Τ(𝜼)𝐃(𝝂)𝐉⁻¹(𝜼). (4)

Considering the presence of input saturation constraints on 𝝉, we have 𝜏_𝑖min≤𝜏_𝑖≤𝜏_𝑖max(𝑖=𝑥,𝑦,𝜓), where 𝜏_𝑖min and 𝜏_𝑖max are the known lower limit and upper limit of input saturation constraints. Thus, the control input is defined as 𝜏_𝑖=𝜏_𝑖max,𝜏_0𝑖>𝜏_𝑖max,𝜏_0𝑖,𝜏_𝑖min≤𝜏_0𝑖≤𝜏_𝑖max,𝜏_𝑖min,𝜏_0𝑖<𝜏_𝑖min, (5) where 𝜏_0𝑖 is the 𝑖th control input of the designed control law 𝝉₀.

Assume that the system parameters are unknown and bounded and the following properties are satisfied.(i)𝐌_𝜂(𝜼) is a positive definite matrix, and it is bounded, which means that there exists a constant 𝜎₀>0, such that 𝟎<𝐌_𝜂(𝜼)≤𝜎₀𝐈, where 𝐈 is a 3rd-order identity matrix.(ii)The matrix 𝐌_𝜂(𝜼) and 𝐂_𝜂(𝝂,𝜼) have the following property: the matrix 𝐌̇_𝜂(𝜼)−2𝐂_𝜂(𝝂,𝜼) is skew-symmetric and for all 𝐱∈ℝ³ the following relation is satisfied: 𝐱^Τ𝐌̇_𝜂(𝜼)−2𝐂_𝜂(𝝂,𝜼)𝐱=0,∀𝝂,𝜼∈ℝ³. (6)

2.2. Model for Controller Design

In order to discuss conveniently and simplify the controller design, state transformation is introduced here to obtain a new form of the system model.

Let 𝐱₁=𝜼 and 𝐱₂=𝜼̇ then the system model (1) can be written as; 𝐱̇₁=𝐱₂,𝐱̇₂=𝐌_𝜂⁻¹(𝜼)𝐮−𝐌_𝜂⁻¹(𝜼)𝐂_𝜂(𝝂,𝜼)𝐱₂−𝐌_𝜂⁻¹(𝜼)𝐃_𝜂(𝝂,𝜼)𝐱₂+𝐌_𝜂⁻¹(𝜼)𝐛, (7)(8) where 𝐮=𝐉^−Τ(𝜼)𝝉=𝐉(𝜼)𝝉.

3. Control Strategy

An adaptive fuzzy logic control using backstepping technique is designed for fully actuated surface vessels with dead-zone in actuator. To compensate the dead-zone nonlinearity, a fuzzy logic system is introduced.

3.1. Backstepping Design without Dead-Zone

Step 1. Define the desired position and heading vector as 𝜼_𝑑, and a perfect guidance system is introduced to generate the tracking target 𝜼_𝑟; and its first- and second-order derivation 𝜼̇_𝑟 and 𝜼̈_𝑟 are also generated. Then the control objective is to design a controller to make the ship track the target 𝜼_𝑟. Define the tracking error as 𝐳₁=𝜼−𝜼_𝑟. (9)

Taking the time derivation of (9) yields 𝐳̇₁=𝜼̇−𝜼̇_𝑟=𝐱̇₁−𝜼̇_𝑟=𝐱₂−𝜼̇_𝑟. (10)

Choose 𝐱₂ as virtual control and defined as 𝝃₁≜𝐱₂=𝐳₂+𝜶₁, (11) where 𝐳₂ is the second error vector, and 𝜶₁ is the stable function which is defined later.

Substitute (11) into (10), we obtain 𝐳̇₁=𝐱₂−𝜼̇_𝑟=𝐳₂+𝜶₁−𝜼̇_𝑟. (12)

Choose the stable function 𝜶₁ as 𝜶₁=−𝝀₁𝐳₁+𝜼̇_𝑟, (13) where 𝝀₁ is a positive definite diagonal matrix.

The Lyapunov function is chosen for the 𝐳₁-subsystem as 𝑉₁=12𝐳^Τ₁𝐳₁. (14)

Then the time derivation of (14) is 𝑉̇₁=𝐳^Τ₁𝐳̇₁=𝐳^Τ₁𝐳₂+𝜶₁−𝜼̇_𝑟=−𝐳^Τ₁𝝀₁𝐳₁+𝐳^Τ₁𝐳₂. (15)

If 𝐳₂=0, then 𝑉̇₁≤0 which implies that the 𝐳₁-subsystem is stable.

Step 2. From (11) we can get 𝐳₂=𝐱₂−𝜶₁. (16)

Differencing (16) with respect to time yields 𝐳̇₂=𝐱̇₂−𝜶̇₁. (17)

Substitute (8) into (17), then (17) becomes 𝐳̇₂=𝐌_𝜂⁻¹𝐮−𝐌_𝜂⁻¹𝐂_𝜂𝐱₂−𝐌_𝜂⁻¹𝐃_𝜂𝐱₂+𝐌_𝜂⁻¹𝐛−𝜶̇₁. (18)

Note that, we omit the variables of coefficient matrices for denoting conveniently.

For convenience of constraint effect analysis of the input saturation, the following auxiliary design system is given by 𝜻̇=−𝐊₁𝐳𝜻−^Τ2Δ𝐮+0.5Δ𝐮^ΤΔ𝐮‖𝜻‖²‖𝜻‖‖𝜻‖𝜻+Δ𝐮,≥𝜇,𝟎,<𝜇, (19) where Δ𝐮=𝐮−𝐮₀=𝐉(𝜼)(𝝉−𝝉₀),  𝐊₁=𝐊^Τ₁>0, 𝜇 is a small positive design parameter, and 𝜻∈ℝ³ is the state of the auxiliary design system. Control command 𝝉₀ will be designed later.

For 𝐳₂-subsystem, the Lyapunov function is chosen as 𝑉₂=𝑉₁+12𝜻^Τ1𝜻+2𝐳^Τ₂𝐌_𝜂𝐳₂. (20)

Take the time derivation of (20), which yields 𝑉̇₂=𝑉̇₁+𝐳^Τ₂𝐌_𝜂𝐳̇₂+𝜻^Τ𝜻̇+12𝐳^Τ₂𝐌̇_𝜂𝐳₂=−𝐳^Τ₁𝝀₁𝐳₁+𝐳^Τ₁𝐳₂+𝐳^Τ₂𝐌_𝜂𝐱̇₂−𝜶̇₁+12𝐳^Τ₂𝐌̇_𝜂𝐳₂+𝜻^Τ𝜻̇=−𝐳^Τ₁𝝀₁𝐳₁+𝐳^Τ₁𝐳₂+𝐳^Τ₂𝐌_𝜂𝐌_𝜂⁻¹𝐮−𝐌_𝜂⁻¹𝐂_𝜂𝐱₂−𝐌_𝜂⁻¹𝐃_𝜂𝐱₂+𝐌_𝜂⁻¹𝐛−𝜶̇₁+12𝐳^Τ₂𝐌̇_𝜂𝐳₂+𝜻^Τ𝜻̇≤−𝐳^Τ₁𝝀₁𝐳₁+𝐳^Τ₁𝐳₂+𝐳^Τ₂𝐮−𝐂_𝜂𝐱₂−𝐃_𝜂𝐱₂+𝐛−𝐌_𝜂𝜶̇₁+12𝐳^Τ₂𝐌̇_𝜂𝐳₂−𝜻^Τ𝐊₁−12𝐈₃𝜻−𝐳^Τ₂Δ𝝉. (21)

According to the skew-symmetric property of the matrix 𝐌̇_𝜂(𝜼)−2𝐂_𝜂(𝝂,𝜼) in (6), we have (1/2)𝐳^Τ₂𝐌̇_𝜂𝐳₂=𝐳^Τ₂𝐂_𝜂(𝝂,𝜼)𝐳₂. Substituting it into (21) yields 𝑉̇₂≤−𝐳^Τ₁𝝀₁𝐳₁+𝐳^Τ₁𝐳₂+𝐳^Τ₂×𝐮+𝐂_𝜂𝐳₂−𝐂_𝜂𝐱₂−𝐃_𝜂𝐱₂+𝐛−𝐌_𝜂𝜶̇₁−𝜻^Τ𝐊₁−12𝐈₃𝜻−𝐳^Τ₂Δ𝝉. (22)

Invoking (11) and (22) becomes 𝑉̇₂≤−𝐳^Τ₁𝝀₁𝐳₁+𝐳^Τ₁𝐳₂+𝐳^Τ₂𝐮−𝐂_𝜂𝜶₁−𝐃_𝜂𝐱₂−𝐌_𝜂𝜶̇₁+𝐳^Τ₂𝐛−𝜻^Τ𝐊₁−12𝐈₃𝜻−𝐳^Τ₂Δ𝝉. (23)

Let 𝐟=−𝐂_𝜂𝜶₁−𝐃_𝜂𝐱₂−𝐌_𝜂𝜶̇₁, and (23) can be written as 𝑉̇₂≤−𝐳^Τ₁𝝀₁𝐳₁+𝐳^Τ₁𝐳₂+𝐳^Τ₂(𝐮+𝐟+𝐛)−𝜻^Τ𝐊₁−12𝐈₃𝜻−𝐳^Τ₂Δ𝝉. (24)

Then the ideal control law can be chosen as 𝐮^∗₀=−𝝀₂𝐳₂−𝜻−𝐳₁−𝐟̂+𝐯, (25) where 𝝀₂ is a positive definite diagonal matrix, 𝐟̂ is a nonlinear function designed later, and 𝐯 is a robust term for disturbance and estimated error.

3.2. Dead-Zone Compensator Design

The unsymmetrical dead-zone nonlinearity is shown as Figure 2, and it can be described as 𝜏=𝐷_𝑢(𝑢)=𝑢+𝑑₋,𝑢<−𝑑₋,0,−𝑑₋≤𝑢<𝑑₊,𝑢−𝑑₊,𝑢≥𝑑₊, (26) where 𝑢 is the control input before the dead-zone, 𝜏 is the output of dead-zone model, and 𝑑₋,  𝑑₊ are unknown positive constant. Then the dead-zone model can be denoted as 𝜏=𝑢−sat_𝑑(𝑢),sat_𝑑(𝑢)=−𝑑₋,𝑢<−𝑑₋,𝑢,−𝑑₋≤𝑢<𝑑₊,𝑑₊,𝑢≥𝑑₊. (27)

Figure 2 

Unsymmetrical dead-zone nonlinearity.

For multiple input multiple output system such as dynamics positioning ships, control input of the 𝑖th control loop is 𝜏_𝑖=𝐷𝑢_𝑖=𝑢_𝑖−sat_𝑑𝑖(𝑢), (28) where 𝑑_𝑖=[𝑑_𝑖+,𝑑_𝑖−]^Τ, and let 𝐃=diag{𝑑₁,…,𝑑_𝑛}, it becomes 𝝉=𝐮−sat_𝐷(𝐮). (29)

According to the characteristic of dead-zone, the rules for compensation are designed as if𝑤_𝑖ispositive,then𝑢_𝑖=𝑤_𝑖+𝑑_𝑖+,if𝑤_𝑖isnegative,then𝑢_𝑖=𝑤_𝑖+𝑑_𝑖−, (30) and 𝐝_𝑖=[𝑑_𝑖+,𝑑_𝑖−]^Τ is estimate of dead-zone 𝐝_𝑖=[𝑑_𝑖+,𝑑_𝑖−]^Τ.

After compensating the dead-zone, the control input 𝑢_𝑖 is 𝑢_𝑖=𝑤_𝑖+𝑤_𝐹𝑖, (31) where 𝑤_𝐹𝑖 is a compensation term which is determined by if𝑤_𝑖∈𝑋₊𝑤_𝑖,then𝑤_𝐹𝑖=𝑑_𝑖+if𝑤_𝑖∈𝑋₋𝑤_𝑖,then𝑤_𝐹𝑖=−𝑑_𝑖−, (32) where 𝑋₊(⋅),  𝑋₋(⋅) are the membership degree of 𝑤_𝑖 and 𝐝_𝑖=[𝑑_𝑖+,𝑑_𝑖−]^Τ. Let 𝐰_𝐹=[𝑤_𝐹1,…,𝑤_𝐹𝑛]^Τ; then the control action after compensation is 𝐮=𝐰+𝐰_𝐹=𝐰+𝐃^Τ𝑋(𝐰), (33) where 𝐃=diag{𝐝₁,…,𝐝_𝑛} and 𝑋(𝐰) is described as 𝑋(𝐰)=𝑋₊𝑤₁,−𝑋₋𝑤₁,…,𝑋₊𝑤_𝑛,−𝑋₋𝑤_𝑛^Τ. (34)

Then membership of 𝑤_𝑖 is designed as 𝑋₊𝑤_𝑖=0,𝑤_𝑖<0,1,𝑤_𝑖𝑋≥0,₋𝑤_𝑖=1,𝑤_𝑖<0,0,𝑤_𝑖≥0. (35)

According to Theorem 1 in [17], the control input after the dead-zone is 𝝉=𝐰−𝐃^Τ𝑋(𝐰)+𝐃^Τ𝜹, (36) where 𝐃=𝐃−𝐃=diag{𝑑₁,…,𝑑₁} and ‖𝜹‖≤√𝑛.

With the control input chosen as (25) and 𝐰=𝐽⁻¹(𝜂)𝐮^∗₀, the time derivation of 𝑉₂ becomes 𝑉̇₂≤−𝐳^Τ₁𝝀₁𝐳₁+𝐳^Τ₁𝐳₂+𝐳^Τ₂−𝝀₂𝐳₂−𝜻−𝐳₁−𝐟̂+𝐟+𝐛+𝐯−𝐉⁻¹𝐃^Τ𝑋(𝐰)+𝐉⁻¹𝐃^Τ𝜹−𝜻^Τ𝐊₁−12𝐈₃𝜻−𝐳^Τ₁Δ𝜏≤−𝐳^Τ₁𝝀₁𝐳₁−𝐳^Τ₂𝝀₂𝐳₂−𝜻+𝐳^Τ₂𝐟−𝐟̂+𝐯+𝐛−𝐉⁻¹𝐃^Τ𝑋(𝐰)+𝐉⁻¹𝐃^Τ𝜹−𝜻^Τ𝐊₁−12𝐈₃𝜻. (37)

We can choose the robust term and adaptive law of dead-zone width as 𝐯(𝑡)=−𝐛_𝑀sgn𝐳₂,𝐃̇=−Γ𝑋(𝐰)𝐳^Τ₂𝐉⁻¹−12𝝌𝐃, (38)(39) where 𝝌>0,Γ>0 are diagonal matrices and 0≤‖𝐛‖≤𝐛_𝑀.

3.3. Adaptive RBFNN Controller Design

From the description of 𝐟 mentioned above, we can see that it contains the knowledge about the system model. In order to design a control law without explicit knowledge about the ship motion model, an adaptive RBFNN is introduced to approximate the nonlinear function 𝐟. And here 𝐟̂ is the adaptive neural network system used to approximate 𝐟.

The structure of RBFNN is shown in Figure 3. RBFNN is a forward network with three layers: input layer, implicit layer, and output layer. The mapping from input to output is nonlinear, while it is linear from implicit layer to output. RBFNN can approximate nonlinear function locally, so its learning rate is fast and the local minimization problem can be avoided.

Figure 3 

The structure of RBFNN.

Assume there are 𝑛 inputs and 𝑚 implicits nodes in the RBF network. Let 𝐗=[𝑥₁,𝑥₂,…,𝑥_𝑛]^Τ be the input vector; then denote the radial basis vector as 𝐇=[ℎ₁,ℎ₂,…,ℎ_𝑚]^Τ, where ℎ_𝑖 is Gaussian function defined as ℎ_𝑖=exp−𝐗−𝐜_𝑖²2𝑏²_𝑖,𝑖=1,2,…,𝑚, (40) where 𝐜_𝑖=[𝑐_𝑖1,𝑐_𝑖2,…,𝑐_𝑖𝑛]^Τ is the central vector of the ith node and 𝑏_𝑖>0 is the basis width parameter of the 𝑖th node.

Define the weight vector of the RBF neural network as 𝐖=[𝑤₁,𝑤₂,…,𝑤_𝑚]^Τ; then the output of the RBFNN can be expressed as follows: 𝑦=𝐇^Τ𝐖. (41)

In order to approximate the nonlinear function 𝐟 which is relative to the ship modeled parameters and the system states, the elements of 𝐟 are, respectively, estimated by corresponding RBFNN as follows: 𝑓_𝑘(𝐱)=𝐇_𝑘(𝐱)^Τ𝐖_𝑘,(𝑘=1,2,3). (42)

Then define the Gaussian basis function vector 𝝋 as 𝐟̂()=𝑓𝐱,𝜽𝑓¹₂𝑓₃=𝐇^Τ₁𝟎𝟎𝟎𝐇^Τ₂𝟎𝟎𝟎𝐇^Τ₃𝐖₁𝐖₂𝐖₃=𝝃(𝐱)^Τ𝜽, (43) where 𝝃^Τ(𝐱)=diag{𝐇^Τ₁,𝐇^Τ₂,𝐇^Τ₃} and  𝜽=[𝐖^Τ₁,𝐖^Τ₂,𝐖^Τ₃]^Τ.

Assumption 1. The output of RBFNN 𝐟(̂𝐱,𝜽) is continuous.

Assumption 2. There exist an ideal approximation 𝐟(̂𝐱,𝜽^∗), such that for any small positive constant 𝜀₀max𝐟(𝐱)−𝐟̂𝐱,𝜽^∗≤𝜀₀, (44) where 𝜽^∗ is the optimal weight vector of the RBFNN for approximating 𝐟(𝐱).

Denote 𝐞_𝑎 as the approximate error of the ideal RBFNN: 𝐞_𝑎=𝐟(𝐱)−𝐟̂𝐱,𝜽^∗. (45)

According to the approximate capability of RBFNN, it is easy to obtain that 𝐞_𝑎 is bounded. Denote the bound as 𝐞_𝑎0=sup𝐟(𝐱)−𝐟̂𝐱,𝜽^∗, (46) where 𝐟(̂𝐱,𝜽^∗)=𝝃(𝐱)^Τ𝜽^∗.

𝐟(𝐱) is bounded, so 𝜽^∗ is bounded and ‖𝜽^∗‖_𝐹≤𝜽_max. Let 𝜽=𝜽^∗−𝜽, and the adaptive law is chosen as 𝜽̇_𝑖=𝑟_𝑖𝑧_2𝑖𝝃_𝑖^Τ(𝐱)^Τ−2𝑘_𝑖𝜽_𝑖,(𝑖=1,2,3), (47) where 𝑟_𝑖>0 and 𝑘_𝑖>0 are designed constant and 𝑧_2𝑖 is the 𝑖th element of 𝐳₂.

Let 𝜸∈ℜ^3𝑚×3𝑚, 𝜿∈ℜ^3𝑚×3𝑚, and define matrices as 𝜸=diag𝑟₁𝐈_𝑚,𝑟₂𝐈_𝑚,𝑟₃𝐈_𝑚,𝜿=diag𝑘₁𝐈_𝑚,𝑘₂𝐈_𝑚,𝑘₃𝐈_𝑚, (48) where 𝐈_𝑛 is a 𝑛th order identity matrix. Then the adaptive law (47) becomes 𝜽̇=𝜸𝐳^Τ₂𝝃^Τ(𝐱)^Τ−2𝜿𝜽. (49)

3.4. Stability Analysis

Now we can choose the total Lyapunov function as 𝑉=𝑉₂+12𝜽^Τ𝜸⁻¹1𝜽+2tr𝐃^ΤΓ⁻¹𝐃. (50)

Differentiate (50) with respect to time as 𝑉̇≤𝑉̇₂+𝜽^Τ𝜸⁻¹𝜽̇+tr𝐃^ΤΓ𝐃̇≤−𝐳^Τ₁𝝀₁𝐳₁−𝐳^Τ₂𝝀₂𝐳₂−𝜻+𝐳^Τ₂𝐟−𝝃(𝐱)^Τ𝜽−𝜽^Τ𝜸⁻¹𝜽̇−𝐳^Τ₂𝐉⁻¹𝐃^Τ𝑋(𝐰)−𝐉⁻¹𝐃^Τ𝜹+tr𝐃^ΤΓ̇𝐃+𝐳^Τ₂(𝐛+𝐯)−𝜻^Τ𝐊₁−12𝐈₃𝜻≤−𝐳^Τ₁𝝀₁𝐳₁−𝐳^Τ₂𝝀₂𝐳₂−𝜻+𝐳^Τ₂𝐟−𝝃(𝐱)^Τ𝜽^∗−𝜽+𝐛+𝐯_Τ𝜸⁻¹𝜽̇+𝐳^Τ₂𝝃(𝐱)^Τ𝜽^∗−𝜽−𝐳^Τ₂𝐉⁻¹𝐃^Τ(𝑋(𝐰))𝐃−𝜹+tr^ΤΓ̇𝐃−𝜻^Τ𝐊₁−12𝐈₃𝜻. (51)

It is obviously that 𝐳^Τ₂𝝀₂𝜻≤𝝅𝐳^Τ₂𝐳₂+𝝅⁻¹𝜻^Τ𝝀^Τ₂𝝀₂𝜻, (52) where 𝝅∈ℝ^3×3 and 𝝅>0.

According to 𝐳^Τ₂𝐉⁻¹𝐃^Τ𝑋(𝐰)−𝐉⁻¹𝐃^Τ𝜹=𝐳^Τ₂𝐉⁻¹𝐃^Τ(𝑋(𝐰))𝐃−𝜹=tr^Τ(𝑋(𝐰)−𝜹)𝐳^Τ₂𝐉⁻¹, (53) we can obtain 𝜗≜−𝐳^Τ₂𝐉⁻¹𝐃^Τ𝑋(𝐰)−𝐉⁻¹𝐃^Τ𝜹+tr𝐃^ΤΓ⁻¹̇𝐃=tr𝐃^Τ(−𝑋(𝐰)+𝜹)𝐳^Τ₂𝐉⁻¹+tr𝐃^ΤΓ⁻¹̇𝐃=tr𝐃^Τ−𝑋(𝐰)𝐳^Τ₂𝐉⁻¹+𝜹𝐳^Τ₂𝐉⁻¹+Γ⁻¹̇𝐃. (54)

From 𝐃=𝐃−𝐃 we have =−𝐃̇; then 𝜗=tr𝐃^Τ−𝑋(𝐰)𝐳^Τ₂𝐉⁻¹+𝜹𝐳^Τ₂𝐉⁻¹−Γ⁻¹𝐃̇=tr𝐃^Τ𝛿𝐳^Τ₂𝐉⁻¹+12Γ⁻¹𝐃≤√𝐃−𝑛𝐳₂𝐃+12𝑔_𝑚𝐷_𝑀𝐃−𝜒_𝑚2tr𝐃^ΤΓ⁻¹𝐃, (55) where 𝜒_𝑚=𝜎_min(𝝌),𝑔_𝑚=𝜎_min(Γ) and 𝐷_𝑀>‖𝐃‖,‖𝜹‖≤√𝑛, so (51) yields 𝑉̇≤−𝐳^Τ₁𝝀₁𝐳₁−𝐳^Τ₂𝝀₂−𝝅𝐳₂+𝐳^Τ₂⋅𝐟−𝝃(𝐱)^Τ𝜽^∗+𝜗−𝜽^Τ𝜸⁻¹𝜽̇+𝐳^Τ₂𝝃(𝐱)^Τ𝜃+𝐳^Τ₂𝐛−𝐛_𝑀sgn𝐳₂−𝜻^Τ𝐊₁−12𝐈₃−𝝅⁻¹𝝀^Τ₂𝝀₂𝜻≤−𝐳^Τ₁𝝀₁𝐳₁−𝐳^Τ₂𝝀₂−𝝅𝐳₂+12𝐳^Τ₂²+12𝜀₀²+𝜽+𝜗_Τ𝐳^Τ₂𝝃^Τ(𝐱)^Τ−𝜸⁻¹𝜽̇−𝜻^Τ𝐊₁−12𝐈₃−𝝅⁻¹𝝀^Τ₂𝝀₂𝜻. (56)

Substitute adaptive law (39) into (56) and we can get 𝑉̇≤−𝐳^Τ₁𝝀₁𝐳₁−𝐳^Τ₂𝝀₂−𝝅𝐳₂−𝜻^Τ𝐊₁−12𝐈₃−𝝅⁻¹𝝀^Τ₂𝝀₂𝜻+12𝜀²−𝜽^Τ𝐳^Τ₂𝝃^Τ(𝐱)^Τ−𝜸⁻¹𝜸𝐙₂𝝃^Τ(𝐱)^Τ1−2𝜿𝜽+𝜗+2𝐳^Τ₂²=−𝐳^Τ₁𝝀₁𝐳₁−𝐳^Τ₂𝝀₂−𝝅𝐳₂+𝐳^Τ₂𝐳₂+12𝜀²+2𝜽^Τ𝜸⁻¹𝜿𝜽+𝜗−𝜻^Τ𝐊₁−12𝐈₃−𝝅⁻¹𝝀^Τ₂𝝀₂𝜻=−𝐳^Τ₁𝝀₁𝐳₁−𝐳^Τ₂𝝀₂−12𝐈₃𝐳₂+12𝜀²+2𝜽^∗Τ𝜸⁻¹𝜿𝜽−2𝜽^Τ𝜸⁻¹𝜿𝜽+𝜗−𝜻^Τ𝐊₁−12𝐈₃−𝝅⁻¹𝝀^Τ₂𝝀₂𝜻. (57)

If 𝜿 and 𝜸 are positive definite diagonal matrices, then (𝜽^∗−𝜽)^Τ𝜸⁻¹𝜿(𝜽^∗−𝜽)≥0. So the following inequality is satisfied: 2𝜽^∗Τ𝜸⁻¹𝜿𝜽−2𝜽^Τ𝜸⁻¹𝜿𝜽≤−𝜽^Τ𝜸⁻¹𝜿𝜽+𝜽^∗Τ𝜸⁻¹𝜿𝜽^∗. (58)

Substituting (58) into (57) yields 𝑉̇≤−𝐳^Τ₁𝝀₁𝐳₁−𝐳^Τ₂𝝀₂−12𝐈₃−𝝅𝐳₂+12𝜀²+𝜽^∗Τ𝜸⁻¹𝜿𝜽^∗−𝜽^Τ𝜸⁻¹𝜿𝜽+𝜗−𝜻^Τ𝐊₁−12𝐈₃−𝝅⁻¹𝝀^Τ₂𝝀₂𝜻=−𝐳^Τ₁𝝀₁𝐳₁−𝐳^Τ₂𝝀₂−12𝐈₃−𝝅𝐳₂−𝜽^Τ𝜸⁻¹𝜿𝜽−𝜽^∗Τ𝜸⁻¹𝜿𝜽^∗+12𝜀²+2𝜽^∗Τ𝜸⁻¹𝜿𝜽^∗+𝜗−𝜻^Τ𝐊₁−12𝐈₃−𝝅⁻¹𝝀^Τ₂𝝀₂𝜻. (59)

Due to (𝜽^∗+𝜽)^Τ𝛾⁻¹𝜿(𝜽^∗+𝜽)≥0, we obtain −𝜽^∗Τ𝜸⁻¹𝜿𝜽−𝜽^Τ𝜸⁻¹𝜿𝜽^∗≤𝜽^Τ𝜸⁻¹𝜿𝜽+𝜽^∗Τ𝜸⁻¹𝜿𝜽^∗. (60)

Then 𝜽^Τ𝜸⁻¹𝜿𝜽=𝜽^∗−𝜽^Τ𝜸⁻¹𝜿𝜽^∗−𝜽=𝜽^∗Τ𝜸⁻¹𝜿𝜽^∗+𝜽^Τ𝜸⁻¹𝜿𝜽−2𝜽^∗Τ𝜸⁻¹𝜿𝜽≤2𝜽^∗Τ𝜸⁻¹𝜿𝜽^∗+2𝜽^Τ𝜸⁻¹𝜿𝜽, (61) which implies that −𝜽^∗Τ𝜸⁻¹𝜿𝜽^∗−𝜽^Τ𝜸⁻¹1𝜿𝜽≤−2𝜽^Τ𝜸⁻¹𝜿𝜽. (62)

Now (59) becomes 𝑉̇≤−𝐳^Τ₁𝝀₁𝐳₁−𝐳^Τ₂𝝀₂−12𝐈₃−𝝅𝐳₂−12𝜽^Τ𝜸⁻¹𝜿1𝜽+2𝜀²+2𝜽^∗Τ𝜸⁻¹𝜿𝜽^∗−𝜻^Τ𝐊₁−12𝐈₃−𝝅⁻¹𝝀^Τ₂𝝀₂𝜻+𝜗. (63)

Choose appropriate 𝝀₂,  𝝅,  𝐊₁ such that 𝝀₂−(1/2)𝐈₃−𝝅 and 𝐊₁−(1/2)𝐈₃−𝝅⁻¹𝝀^Τ₂𝝀₂ are positive definite, and 𝜆_1𝑚, 𝜆_2𝑚, 𝜅_𝑚 and 𝑘_𝑚 are, respectively, the minimum eigenvalue of 𝝀₁, 𝝀₂−(1/2)𝐈₃−𝝅, 𝜿 and 𝐊₁−(1/2)𝐈₃−𝝅⁻¹𝝀^Τ₂𝝀₂. Then (63) becomes 𝑉̇≤−𝜆_1𝑚𝐳^Τ₁𝐳₁−𝜆_2𝑚𝐳^Τ₂𝐳₂−12𝜅_𝑚𝜽^Τ𝜸⁻¹𝜽−𝑘_𝑚𝜻^Τ𝜻+2𝜽^∗Τ𝜸⁻¹𝜿𝜽^∗+12𝜀²+𝜗=−𝜆_1𝑚𝐳^Τ₁𝐳₁−𝜆_2𝑚𝐳^Τ₂𝐌_𝜂⁻¹𝐌_𝜂𝐳₂−𝑘_𝑚𝜻^Τ1𝜻+2𝜽^Τ𝜸⁻¹𝜿𝜽+2𝜽^∗Τ𝜸⁻¹𝜿𝜽^∗+12𝜀²+𝜗. (64)

According to 𝟎<𝐌_𝜂(𝜼)≤𝜎₀𝐈, that is, −𝐌_𝜂⁻¹≤−(1/𝜎₀); then 𝑉̇≤−𝜆_1𝑚𝐳^Τ₁𝐳₁−1𝜎₀𝜆_2𝑚𝐳^Τ₂𝐌_𝜂𝐳₂−𝑘_𝑚𝜻^Τ1𝜻−2𝜅_𝑚𝜽^Τ𝜸⁻¹𝜽+2𝜽^∗Τ𝜸⁻¹𝜿𝜽^∗+12𝜀²+𝜗. (65)

Define 𝑐₀=min{2𝜆_1𝑚,(2/𝜎₀)𝜆_2𝑚,𝜅_𝑚,𝑘_𝑚,𝜒_𝑚}, then 𝑉̇≤−12𝑐₀𝐳^Τ₁𝐳₁+𝐳^Τ₂𝐌_𝜂𝐳₂+𝜻^Τ𝜻+𝜽^Τ𝜸⁻¹𝜽+tr𝐃^ΤΓ⁻¹𝐃+12𝜀²+√𝑛𝐳₂𝐃+𝜒_𝑀𝐷_𝑀𝐃−𝜒_𝑀𝐃²+2𝜽^∗Τ𝜸⁻¹𝜿𝜽^∗=−𝑐₀𝑉+2𝜽^∗Τ𝜸⁻¹𝜿𝜽^∗+12𝜀²+√𝑛𝐳₂𝐃+12𝑔_𝑚𝐷_𝑀𝐃=−𝑐₀𝑉+𝑐_𝑉𝑀, (66) where 𝑐_𝑉𝑀=2𝜽^∗Τ𝜸⁻¹𝜿𝜽^∗+(1/2)𝜀²+√𝑛‖𝐳₂‖‖𝐃‖+(1/2𝑔_𝑚)𝐷_𝑀‖𝐃‖>0.

So we can obtain that 𝑉 and all the signals of the close-loop system are bounded [18].

4. Case Study

A computer simulation has been used to evaluate the performance of the adaptive neural network control with backstepping for fully actuated surface ships with dead-zone in actuator. The ship system used for simulation is described as follows: 𝐌=9.1948×10⁷0009.1948×10⁷1.2607×10⁹01.2607×10⁹1.0724×10¹¹,𝐃_𝑙=1.5073×10⁶0008.1687×10⁶−1.3180×10⁸0−1.3180×10⁸1.2568×10¹¹,𝐃_𝑛(𝝂)=−diag𝑋_|𝑢|𝑢|𝑢|,𝑌_|𝑣|𝑣|𝑣|,𝑁_|𝑟|𝑟|𝑟|, (67) where 𝑋_|𝑢|𝑢|𝑢|, 𝑌_|𝑣|𝑣|𝑣|, 𝑁_|𝑟|𝑟|𝑟| are hydrodynamic coefficients, 𝑋_|𝑢|𝑢=−2.9766⋅10⁴, 𝑌_|𝑣|𝑣=−8.0922⋅10⁴, and 𝑁_|𝑟|𝑟=−1.2228⋅10¹².

Note that the elements of matrices are the nominal values of the ship plant.

The bound of dead-zone is set as 𝑑₋=0.5×10⁵,𝑑₊=0.6×10⁵. (68)

To verify the robustness of the controller and the control effect, the unpredictable disturbances and parameter uncertainties are introduced. The environmental disturbances acting on the ship can be treated together as 𝐛=0.25×10⁵sin(0.1𝑡),0.25×10⁵sin(0.1𝑡),0.25×10⁶sin(0.1𝑡)^Τ. (69)

And parameters of the ship model vary from 0.5 to 1.5 times the size of its nominal value. The limit of control input is set as |𝝉|≤0.2×10⁶,0.15×10⁶,0.15×10^8Τ. (70)

The initial position and heading of the vessel is (0 m, 0 m, 0 deg), and the initial velocity is (0 m/s, 0 m/s, 0 rad/s) and the desired position is set as (2 m, 1 m, 5 deg). A smooth reference path is generated by a guidance system. The simulation results with proposed controller are shown in Figures 4 and 5, and the ones with PID controller are listed in Figures 6 and 7. The blue lines in Figures 4 and 5 are values without using dead-zone compensator, and the red ones are the results using it. Figure 4 shows that the dead-zone compensator improves the control performance and reduces the tracking errors. Figure 5 is the control inputs for both simulations with and without dead-zone compensator. From Figure 5, the control input is zero when the control command is in the dead-zone, so its control effect is not excellent unless the dead-zone is compensated well. The blue lines in Figures 6 and 7 are results without dead-zone in actuators, and the red ones are values with dead-zone characteristic. Comparing Figures 4 and 5 to Figures 6 and 7, the method proposed in this work ameliorates the control effect, and the tracking errors are 5–10 times smaller than the results using PID controller. Meanwhile, the forces and moments acting on the ship become smaller and smoother than the ones with PID method. So the control strategy designed in this work can protect the actuator from wear and reduce fuel consumption.

Figure 4 

Tracking error of the position and heading.

Figure 5 

Control inputs in surge, sway, and yaw.

Figure 6 

Tracking error of the position and heading with PID controller.

Figure 7 

Control inputs in surge, sway, and yaw with PID controller.

5. Conclusion

In this paper, an adaptive neural network control using backstepping is derived for fully actuated surface ships with dead-zone characteristics in actuator. A three degree of freedom model including disturbances has been established for ships. In order to overcome the difficulties brought by unknown model parameters, the adaptive RBFNN is introduced to approximate the unknown nonlinear functions needed in controller design. The dead-zone character is quite common in actuator for ships, and the existence of dead-zone affects the control effect of traditional controllers, especially when the dead-zone parameters are unknown. The fuzzy logic system is utilized here to handle this problem. Fuzzy logic technique can estimate the dead-zone parameters which are used for compensator design. It has been shown that the adaptive fuzzy control with the dead-zone compensator can drive the ship to the desired position with certain heading angles. The input saturation is overcome by adopting the auxiliary design system. Meanwhile, through the Lyapunov stable theory it is proved that the system is bounded for all states. The simulation results showed that the controller proposed in this work performs excellently for dynamic positioning ships.