A novel active disturbance rejection control with hyperbolic tangent function for path following of underactuated marine surface ships

Abstract

This paper presents a TADRC method via active disturbance rejection control (ADRC) with hyperbolic tangent function for path following of underactuated surface ships with input constraint, heading rate constraint, parameters uncertainties, as well as environment disturbances. The line of sight (LOS) guidance scheme that computes the desired heading angle on basis of cross tracking error and a look ahead distance, converts path following into heading control, and also renders good helmsman behavior. Moreover, hyperbolic tangent function is introduced to modify the linear extended state observer (LESO) to design a nonlinear observer (TESO) for promoting the estimation performance of the heading, heading rate and total disturbances including parameters uncertainties and environmental disturbances. Then, the linear error feedback control is modified by a nonlinear sliding mode control scheme with hyperbolic tangent function to handle the heading rate constraint and to obtain the better control action. Furthermore, the feedback control law is embedded in a standard Quadratic Programming (QP) cost function to handle the input constraints including rudder saturation and rudder rate limit. Finally, the comparison simulation demonstrates the effectiveness of the proposed method for the underactuated ship’s path following.

Keywords

Path following underactuated ship active disturbance rejection control line of sight extended state observer

Introduction

Path following problem of marine surface ships has attracted more attention from the control and ship engineering community for many years. Path following is one of the typical control scenarios for underactuated ships, which can be described as following a predefined path without temporal constraint.¹ The underactuated nature of these problems, together with the matter of uncertain parameters, external disturbances as well as input constraints, renders the control problem both challenging and interesting.^2,3

Some techniques have been proposed for the control design of path following. The line of sight (LOS) guidance was also widely used in path following.^4,5 The LOS was used by Fossen for path following, which could convert path following into heading control.⁶ And it was extended in⁷ where a dynamic LOS was proposed to improve the rate of the convergence. After that, a time-varying look ahead distance was exploited in,⁸ which could promote the maneuvering behavior and track the desired path faster.

The problems of the rudder constraints, parameters uncertainties, environment disturbances even heading rate constraint, caused path following to be challenging. To cope with the disturbances, a nonlinear disturbance observer was employed in.⁹ And the adaptive method was proposed in¹⁰ and¹¹ to solve the parameters uncertainties and environmental disturbances. Furthermore, the adaptive technique was used in¹² where both the curvature of the path and the disturbance of ocean current were addressed. In,^13,14 radial basis function (RBF) neural network and adaptive law were used to deal with the model uncertainties and disturbances, respectively.

However, the aforesaid methods are so complex relatively that they are hard to be implemented in practice presently. Active disturbance rejection control (ADRC) approach proposed by Han¹⁵ is proved a novel approach to handle the uncertainties and disturbance. Extended state observer (ESO), as the key part of ADRC, realize the system a significant robustness by estimating the total unknown including internal uncertainties and external disturbances in real time.¹⁶ Gao¹⁷ proposed a linear ADRC (LESO) scheme, which adopts a linear ESO (LESO) and proportional-derivative (PD) error feedback control law for easy parameter tuning. In¹⁸ and,¹⁹ the LESO was designed to estimate the sideslip angles. In²⁰ and,²¹ the LESO was designed to tackle both the uncertain dynamics and external disturbances. It is noted that a rapid disturbance estimation requires large LESO gain that is probably exceed the bandwidth of engineering systems, and even high gain observers could cause the peaking phenomenon. The estimation may be less accurate when the error is small, or easily causes large oscillatory when the error is large. An adaptive ESO (AESO) with time-varying gains was design in²² and,²³ which overcomes the drawbacks of LESO and nonlinear ESO. The hyperbolic tangent function was used in²⁴ where the derivative peaking phenomenon in conventional LESO was suppressed. Literature²¹ presented a nonlinear sliding mode with hyperbolic tangent function to modify the PD control law for the better control action.

In addition to the aforementioned problems, it is also significant to obtain a constrained and smoother control input. In recent years, the MPC algorithm has become the standard optimization method for constrained systems.²⁵

Considering aforementioned investigations, this paper proposes a modified ADRC (TADRC) via both modified ESO (TESO) and nonlinear sliding mode error feedback control law with hyperbolic tangent function for the ship’s path following to promote the performance of control, simplify the design of the ESO parameters and improve estimation accuracy of LESO. The LOS algorithm is employed to design the reference heading angle, so that path following is converted into heading control. Then, a nonlinear sliding mode approach is presented via a cost function, by which the rudder amplitude and rate constraints are handled. Finally, the comparison simulation results illustrate the effectiveness of the designed controller.

The remainder of this paper is organized as follows: the modeling of ship path following is presented in Section 2. Section 3 proposes a TADRC control design method for ship path following on the basis of LOS guidance. The simulation results are conducted and explained in Section 4 followed the conclusions.

Modeling of ship path following

Modeling of ship and disturbance

The ship position in the horizontal plane and the motion parameters are shown in Figure 1.

Figure 1.

The ship position and parameters.

where, φ is the heading, r is the heading rate, u is the surge velocity, v is the sway velocity, V = (u²+v²)^1/2 is the ground velocity, and β = arctan(v/u) is the drift angle. The ship maneuvering Mathematical Model Group (MMG) was proposed in 1970s, which took the revolution of propeller and the rudder angle as the control input. Consider the environment disturbances, MMG model can be expressed²⁶

{\begin{matrix} x' = u \cos φ - v \sin φ = \sqrt{u^{2} + v^{2}} \cos (φ + β) \\ y' = u \sin φ + vcos φ = \sqrt{u^{2} + v^{2}} \sin (φ + β) \\ φ' = r \\ u' = [(m + m_{y}) vr + X_{H} + X_{P} + X_{R} + X_{W} + X_{Wave} \\ + (m_{x} - m_{y}) V_{c} \sin (φ_{c} - φ) r] / (m + m_{x}) \\ v' = [(m + m_{x}) ur + Y_{H} + Y_{P} + Y_{R} + Y_{W} + Y_{Wave} \\ - (m_{x} - m_{y}) V_{c} \cos (φ_{c} - φ) r] / (m + m_{y}) \\ r' = (N_{H} + N_{P} + N_{R} + N_{W} + N_{Wave}) / (I_{zz} + J_{zz}) \end{matrix}

(1)

where, m is the ship mass, m_x and m_y are the additional masses, X_H, Y_H and N_H are the bare hull forces (moment), X_P, Y_P and N_P are the propeller forces (moment), X_W, Y_W and N_W are the wind forces (moment), X_Wave, Y_Wave and N_Wave are the wave forces (moment). It is noted that only the slowly varying forces are counteracted by steering system. The oscillatory motion caused by the first-order wave-induced forces should be prevented from entering the feedback loop. φ_c and V_c are the current set and speed, respectively. I_ZZ is the moment of inertia of the ship around the vertical axis, J_ZZ the additional moment of inertia, and X_R, Y_R and N_R are the rudder forces (moment) given by

{\begin{matrix} X_{R} = (1 - t_{R}) F_{N} \sin δ \\ Y_{R} = (1 + a_{H}) F_{N} \cos δ \\ N_{R} = - (x_{R} + a_{H} x_{H}) F_{N} \cos δ \end{matrix}

(2)

where, t_R is the coefficient of rudder resistance deductions, α_H is the ratio of the hull’s additional lateral force to the rudder lateral force caused by steering, x_H is the distance from the steering induced hull lateral force center to the ship’s center of gravity, and F_N is the rudder positive pressure. δ is the control rudder angle. |δ| ≤ 35° and |Δδ| ≤ 3–6°/s are the rudder amplitude saturation and rudder rate limit, respectively.

The winds (moments) $X_{W}$ , $Y_{W}$ and $N_{W}$ in the dynamics (1) are given²⁷by

{\begin{matrix} X_{W} = \frac{1}{2} ρ_{a} A_{f} U_{R}^{2} C_{wx} (α_{R}) \\ Y_{W} = \frac{1}{2} ρ_{a} A_{s} U_{R}^{2} C_{wy} (α_{R}) \\ N_{W} = \frac{1}{2} ρ_{a} A_{s} L_{oa} U_{R}^{2} C_{wn} (α_{R}) \end{matrix}

(3)

where, $ρ_{a}$ is the air density, $α_{R}$ is the wind angle of attack, $U_{R}$ is the relative wind speed, A_f and A_s are the ship frontal and lateral projected areas, respectively. L_oα is the ship length overall, C_wx(αR), C_wy(αR) and C_wn(αR) are the nondimensional wind coefﬁcients, respectively.

The waves (moments) X_Wave, Y_Wave and N_Wave are given²⁸ by

{\begin{matrix} X_{Wave} = \frac{1}{2} ρ L α^{2} \cos χ C_{Xw} (λ) \\ Y_{Wave} = \frac{1}{2} ρ L α^{2} \sin χ C_{Yw} (λ) \\ N_{Wave} = \frac{1}{2} ρ L α^{2} \sin χ C_{Nw} (λ) \end{matrix}

(4)

where, λ is the wavelength, χ is the encounter angle, ρ is the seawater density, and α is the wave amplitude, L is the ship length, C_Xw(λ), C_Yw(λ) and C_Nw(λ) are the coefficients related to wave length λ, respectively.

The assumptions are as follows

The ship states x, y, φ, u, v and r can be measured.

The generalized disturbance f including parameters uncertainties and environment disturbances is bounded and slowly time-varying,²⁹ namely | f | ≤f_max and $\overset{\cdot}{f}$ ≈ 0.

The first and second order derivatives of x and y are bounded, namely $| \overset{\cdot}{x} | \leq {\overset{\cdot}{x}}_{max}, | \overset{\cdot}{y} | \leq {\overset{\cdot}{y}}_{max}, | \overset{\cdot\cdot}{x} | \leq {\overset{\cdot\cdot}{x}}_{max} and | \overset{\cdot\cdot}{y} | \leq {\overset{\cdot\cdot}{y}}_{max},$

Control of path following

In this paper, the revolution of propeller is set as a constant. The control objective is that, the rudder angle δ will be designed to force ship to follow the reference path. And the ship motion model could be simplified by the kinematics and heading model with disturbances for the controller design

{\begin{matrix} y' = u \sin φ + v \cos φ = \sqrt{u^{2} + v^{2}} \sin (φ + β) \\ x' = u \cos φ - v \sin φ = \sqrt{u^{2} + v^{2}} \cos (φ + β) \\ φ' = r \\ r' = - \frac{r}{T} + \frac{K}{T} δ + d (t) \end{matrix}

(5)

where, K and T are the manipulability index of ship, $d (t)$ is the total unknowns including the parameters uncertainties and environment disturbances.

Path following controller consist of the reference heading design based on the LOS algorithm and heading control by the nonlinear sliding mode method in the paper. The framework of the control design is shown in Figure 2.

Figure 2.

The framework of the control design.

Control design for path following

The LOS guidance system

Fossen⁷ applied LOS schemes to ship path following. LOS guidance principle is used to generate the desired heading angle that is calculated from cross track error $y_{e}$ and a look ahead distance Δ. The reference heading φ_d to make the control objects from (x, y, φ) to only φ, so that is to say, the matter of path following is converted into the heading control.⁴ The LOS guidance principle is shown in Figure 3.

Figure 3.

The LOS guidance principle.

where, y_e is the cross-track error, θ_k is the current straight line angle, α is the angle between the adjacent straight paths, R_a is the radius of the circle of acceptance for the current waypoint, namely the acceptable radius of tacking the next waypoint, Δ is the look ahead distance, R is the distance of ship position P_{(x, y)} from LOS position P_los. And in order to obtain a variable look ahead distance Δ, a time-varying radius R here is introduced as

R = c_{1} (c_{2} | y_{e} | + L)

(6)

where, c₁ and c₂ are positive parameters. A time-varying look ahead distance Δ would be achieved using a time-varying R. The smaller LOS angle could reduce overshoot when the ship is nearer to the desired path. Meanwhile, the larger LOS angle could improve convergence rate as the ship is far from the desired path. Thus, the LOS angle will be offered

φ_{los} = θ_{k} + \arcsin (y_{e} / R)

(7)

Then, the next waypoints P_k₊₂ should be chosen when the ship position P_{(x, y)} satisfies

(x - x_{k + 1})^{2} + (y - y_{k + 1})^{2} \leq R_{a}^{2}

(8)

Meanwhile, k will be incremented to k=k+1. This varying radius R_α will be determined by the angle α. Here, a varying radius R_α is utilized to improve the behavior of path tracking at waypoints like the action of a good helmsman.

Theorem 1., Define a reference heading φ_d as follows

{\begin{matrix} φ_{d} = φ_{los} - β \\ β = \arctan (v / u) \end{matrix}

(9)

and $φ_{e} = φ - φ_{d}$ , while $t \to + \propto$ , if $φ_{e} \to 0$ , then $y_{e} \to 0$

Proof:

According to the point-line distance formula and Figure 3, the cross tracking error is given

y_{e} = \frac{a_{k} x + b_{k} y + C_{k}}{\sqrt{a_{k}^{2} + b_{k}^{2}}} = \frac{\tan (θ_{k}) x - y + C_{k}}{\sqrt{\tan^{2} (θ_{k}) + 1}}

(10)

where, a_k, b_k, C_k are the current straight line parameters. Derivative equation (10) with respect to time

\begin{matrix} y'_{e} = \frac{\tan (θ_{k}) x' - y'}{\sqrt{\tan^{2} (θ_{k}) + 1}} \\ = \frac{\sqrt{u^{2} + v^{2}}}{\sqrt{\tan^{2} (θ_{k}) + 1}} (\tan (θ_{k}) \cos (φ + β) - \sin (φ + β)) \\ = \sqrt{u^{2} + v^{2}} \sin (θ_{k} - φ - β) \end{matrix}

(11)

Substitute equation (9) into (11)

\begin{matrix} y'_{e} = \sqrt{u^{2} + v^{2}} \sin (θ_{k} - (θ_{k} + \arcsin (\frac{y_{e}}{R}) - β) - β) \\ = - \frac{\sqrt{u^{2} + v^{2}}}{c_{1} (c_{2} | y_{e} | + L)} y_{e} \end{matrix}

(12)

The Lyapunov function candidate V₁ = (1/2)y_e² is chosen, derivative V₁ with respect to time

V'_{1} = y_{e} y'_{e} = - \frac{\sqrt{u^{2} + v^{2}} y_{e}^{2}}{c_{1} (c_{2} | y_{e} | + L)} \leq 0

(13)

Consequently, while $t \to + \propto$ , if ship heading φ tracks the designed φ_d, namely $φ_{e} \to 0$ , then $y_{e} \to 0$ .

The φ_d is also taken as the reference heading for path following control design in the next section.

Design of the conventional LADRC

Design of the conventional LESO

On the basis of the items 3 and 4 in equation (5), a nonlinear second ship heading dynamic system can be given

{\begin{matrix} φ' = r \\ r' = f (φ, r, d) + b δ \\ φ = φ_{d} \end{matrix}

(14)

where, $f (φ, r, d)$ denotes the nonlinear dynamics, b is the unknown control gain. Equation (14) can be rewritten as

φ ″ = f (φ, r, d) + (b - b_{0}) δ + b_{0} δ = f + b_{0} δ

(15)

where, $f = f (y' (t), y (t)), d (t)) + (b - b_{0}) δ$ represents the generalized disturbance. b₀ is the estimation of the unknown b, which may be a parameter to be tuned. If $f$ is differentiable, the LESO can be given as follows¹⁷

{\begin{matrix} \hat{φ}' = \hat{r} - l_{11} (\hat{φ} - φ) \\ \hat{r}' = \hat{f} - l_{12} (\hat{φ} - φ) + b_{0} δ \\ \hat{f}' = - l_{13} (\hat{φ} - φ) \end{matrix}

(16)

where, the positive parameters. $\hat{φ}$ , and $\hat{f}$ are the observer states of φ, r and f, respectively. l₁₁, l₁₂ and l₁₃ are the observer bandwidth $ω_{o}$ , normally, $l_{11} = 3 ω_{o}$ , $l_{12} = 3 ω_{o}^{2}$ , $l_{13} = ω_{o}^{3}$ . The state $\hat{f}$ will converge to the extended state f if the observer bandwidth are well tuned.

The linear state error feedback law

Once the observer is designed and bandwidth is well tuned, its outputs will track $φ$ , $r$ and $f$ , respectively. By canceling the effect of $f$ by $\hat{f}$ , the ADRC compensates actively for $f$ in real time.¹⁷ The ADRC control law is given as follows

δ = \frac{- k_{p} (\hat{φ} - φ_{d}) - k_{d} (r - φ'_{d}) + {φ ″}_{d} - \hat{f}}{b_{0}}

(17)

where $k_{p}$ and $k_{d}$ are the controller gains with $k_{p} = ω_{c}^{2}$ and $k_{d} = 2 ω_{c}$ , and $ω_{c}$ is the closed loop natural frequency. Consequently, parameter tuning burden can be reduced as bandwidth regulation, for example, $ω_{c}$ and $ω_{o}$ . Let

u_{0} = - k_{p} (\hat{φ} - φ_{d}) - k_{d} (\hat{r} - φ'_{d})

(18)

Note that with a LESO well-designed, if the estimation errors of $\hat{φ}$ , and $\hat{f}$ are ignored, then $\hat{φ} \approx φ$ , $\hat{r} \approx r$ and $\hat{f} \approx f$ . Let $φ_{e} = \hat{φ} - φ_{d}$ , then equation (18) becomes

u_{0} = - k_{p} φ_{e} - k_{d} φ'_{e}

(19)

which is a linear state error feedback law.

Design of TADRC

ESO with hyperbolic tangent function

The drawback of the LESO as equation (16) may be that, the large parameters of l₁₁, l₁₂ and l₁₃ may introduce the large oscillatory when the estimation error ${\hat{φ}}_{e} = \hat{φ} - φ$ is large. On the other hand, the small parameters of l₁₁, l₁₂ and l₁₃ may introduce the inaccuracy when the estimation error ${\hat{φ}}_{e}$ is small. To solve the aforesaid problems, the smooth monotone bounded hyperbolic tangent function is provided to design a nonlinear ESO, is named TESO here.

{\begin{matrix} \hat{φ}' = \hat{r} - l_{1} \tanh l_{2} (\hat{φ} - φ) \\ \hat{r}' = - \frac{1}{T} \hat{r} + \frac{K}{T} δ + \hat{f} - l_{3} \tanh l_{4} (\hat{φ} - φ) \\ \hat{f}' = - l_{5} \tanh l_{6} (\hat{φ} - φ) \end{matrix}

(20)

where, l₁, l₂, l₃, l₄, l₅ and l₆ are positive parameters. l₁, l₃, and l₅ used to restrict the feedback range of the estimation error, which can prevent the oscillatory as ${\hat{φ}}_{e}$ is large. l₂, l₄ and l₆ are used to improve the estimated accuracy by tuning the proportionality coefficient of ${\hat{φ}}_{e}$ as ${\hat{φ}}_{e}$ is small.

proof: The observer estimation errors could be expressed

{\begin{matrix} \hat{φ}'_{e} = {\hat{r}}_{e} - l_{1} \tanh l_{2} {\hat{φ}}_{e} \\ \hat{r}'_{e} = - \frac{1}{T} {\hat{r}}_{e} + {\hat{f}}_{e} - l_{3} \tanh l_{4} {\hat{φ}}_{e} \\ \hat{f}'_{e} = - l_{5} \tanh l_{6} {\hat{φ}}_{e} - f' \end{matrix}

(21)

The Lyapunov function candidate is chosen

V_{4} = \frac{1}{2} {\hat{φ}}_{e}^{2} + \frac{1}{2} μ {\hat{r}}_{e}^{2} + \frac{1}{2} η {\hat{f}}_{e}^{2}

(22)

where, μ and η are positive coefficients. Derivative V₄ with respect to time, yields

\begin{matrix} V'_{4} = {\hat{φ}}_{e} \hat{φ}'_{e} + μ {\hat{r}}_{e} \hat{r}'_{e} + η {\hat{f}}_{e} \hat{f}'_{e} \\ = {\hat{φ}}_{e} ({\hat{r}}_{e} - l_{1} \tanh l_{2} {\hat{φ}}_{e}) + μ {\hat{r}}_{e} (- \frac{1}{T} {\hat{r}}_{e} + {\hat{f}}_{e} - l_{3} \tanh l_{4} {\hat{φ}}_{e}) \\ + η {\hat{f}}_{e} (- l_{5} \tanh l_{6} {\hat{φ}}_{e} - f') \\ = - l_{1} {\hat{φ}}_{e} \tanh l_{2} {\hat{φ}}_{e} - \frac{1}{T} μ {\hat{r}}_{e}^{2} + {\hat{r}}_{e} ({\hat{φ}}_{e} - μ l_{3} \tanh l_{4} {\hat{φ}}_{e}) \\ + {\hat{f}}_{e} (μ {\hat{r}}_{e} - η l_{5} \tanh l_{6} {\hat{φ}}_{e}) - η \hat{f} f' \end{matrix}

(23)

The item 1 in equation (21) is rewritten

{\hat{r}}_{e} = \hat{φ}'_{e} + l_{1} \tanh l_{2} {\hat{φ}}_{e}

(24)

Substitute equation (24) into (23), we have

\begin{matrix} V'_{4} = - l_{1} {\hat{φ}}_{e} \tanh l_{2} {\hat{φ}}_{e} - \frac{1}{T} μ {\hat{r}}_{e}^{2} + {\hat{r}}_{e} ({\hat{φ}}_{e} - μ l_{3} \tanh l_{4} {\hat{φ}}_{e}) \\ + {\hat{f}}_{e} (μ {\hat{r}}_{e} - η l_{5} \tanh l_{6} {\hat{φ}}_{e}) - η {\hat{f}}_{e} f' \\ = - l_{1} {\hat{φ}}_{e} \tanh l_{2} {\hat{φ}}_{e} - \frac{1}{T} μ {\hat{r}}_{e}^{2} + {\hat{r}}_{e} ({\hat{φ}}_{e} - μ l_{3} \tanh l_{4} {\hat{φ}}_{e}) \\ + μ {\hat{f}}_{e} \hat{φ}'_{e} + {\hat{f}}_{e} (μ l_{1} \tanh l_{2} {\hat{φ}}_{e} - η l_{5} \tanh l_{6} {\hat{φ}}_{e}) - η {\hat{f}}_{e} f' \\ = z_{1} + z_{2} + z_{3} + z_{4} + z_{5} \end{matrix}

(25)

where,

z_{1} = - l_{1} {\hat{φ}}_{e} \tanh l_{2} {\hat{φ}}_{e} - \frac{1}{T} μ {\hat{r}}_{e}^{2},

z_{2} = {\hat{r}}_{e} ({\hat{φ}}_{e} - μ l_{3} \tanh l_{4} {\hat{φ}}_{e}),

z_{3} = μ {\hat{f}}_{e} \hat{φ}'_{e},

z_{4} = {\hat{f}}_{e} (μ l_{1} \tanh l_{2} {\hat{φ}}_{e} - η l_{5} \tanh l_{6} {\hat{φ}}_{e}),

z_{5} = - η {\hat{f}}_{e} f' .

Obviously, z₁ < 0. Consider the assumption ãÀ and the saturation characteristic of the hyperbolic tangent function, z₅ could be ignored, and the suitable parameters l₁, l₃ and l₅ could be chosen for leading the signs in brackets of z₂ and z₄ are the same as ${\hat{φ}}_{e}$ . Hence, if $\hat{φ}'_{e} > 0$ , ${\hat{φ}}_{e}$ will increase to ${\hat{φ}}_{e} > 0$ , and $\hat{f}'_{e} < 0$ , then, ${\hat{f}}_{e}$ will decrease to ${\hat{f}}_{e} < 0$ , thus, there are z₃ < 0 and z₄ < 0, and $\hat{r}'_{e}$ will decrease to $\hat{r}'_{e} < 0$ , then, ${\hat{r}}_{e}$ will decrease to ${\hat{r}}_{e} < 0$ , and z₂<0, so, $V'_{4} < 0$ can be satisfied. Meanwhile, if $\hat{φ}'_{e} < 0$ , ${\hat{φ}}_{e}$ will decrease to ${\hat{φ}}_{e} < 0$ , and $\hat{f}'_{e} > 0$ , then, ${\hat{f}}_{e}$ will increase to ${\hat{f}}_{e} > 0$ , thus, there are z₃<0 and z₄<0, and $\hat{r}'_{e}$ will increase to $\hat{r}'_{e} > 0$ , then, ${\hat{r}}_{e}$ will increase to ${\hat{r}}_{e} > 0$ , and z₂ < 0, so, $V'_{4} < 0$ can also be satisfied.

The control law with hyperbolic tangent function

For the equation (19), let $k_{1} = k_{p} / k_{d}$ , then

u_{0} = - k_{d} (k_{1} φ_{e} + φ'_{e})

(26)

Defined

σ = k_{1} φ_{e} + φ'_{e}

(27)

where, σ is the phrase locus of $φ_{e}$ and $φ'_{e}$ on the phrase plane, and σ may be also taken as a linear sliding mode function.

Linear sliding mode requires faster convergence rate as the larger track error of the system states through the larger and faster control input. However, for the better control action when the track error is large, normally, convergence rate caused by the control input constraint is hard to obtain in the practical engineering system. The system would be stable slowly during the track error is small. Therefore, a nonlinear sliding mode is urgently needed to deal with the drawback of the linear sliding mode, whose characteristic is that the convergence rate increases nonlinearly until it tends to a constant with the increase of tracking error. The nonlinear sliding mode could lead to the better control action in a large scale. So we can choise hyperbolic tangent function that is monotone and bounded as the nonlinear sliding mode function. Defined

σ = k_{1} \tanh (k_{0} φ_{e}) + φ'_{e}

(28)

The feedback control law of system (26) becomes

u_{0} = - k_{d} k_{1} \tanh (k_{0} φ_{e}) + φ'_{e})

(29)

where k₀ and k₁ are positive parameters. If $σ$ → 0, then $φ'_{e} \to - k_{1} \tanh (k_{0} φ_{e})$ and $φ'_{e} \in (- k_{1}, k_{1})$ . The maximum convergence rate of the system is limited within $(- k_{1}, k_{1})$ . $φ_{e}$ will be convergence as almost fixed rate $k_{1}$ when $φ_{e}$ is large enough. On the other hand, $φ_{e}$ will be convergence as index law when $φ_{e}$ is small enough. Currently, the problem of the heading rate constraint was solved and the control action is improved.

The final feedback control law of ADRC with hyperbolic tangent function (TADRC) is

δ_{1} = \frac{- k_{d} (k_{1} \tanh (k_{0} φ_{e}) + φ'_{e}) + {φ ″}_{d} - \hat{f}}{b_{0}}

(30)

Input constraints handled by a cost function

δ ₁ always addresses the rudder amplitude and rudder rate constraints poorly, which even may introduce chattering phenomenon. Hence, δ₁ incorporates in a cost function to promote performance of the rudder angle as follows

min_{\binom{δ_{min} \leq δ \leq δ_{max}}{Δ δ_{min} \leq Δ δ \leq Δ δ_{max}}} J = \sum_{i = 0}^{n} Q (δ - δ_{1} (t - i))^{2} + G δ^{2}

(31)

where, n is the historical horizon, Q and G are the weightings, which could adjust the balance between current rudder angle and historical rudder angle. The stability could be still satisfied by adjusting the suitable Q and G. And δ₁(t−i) is the current information of rudder angle, namely, current control law (30) if i = 0. δ₁(t−i) is the historical information of rudder angle if i > 0. The optimal rudder angle δ will be obtained by solving the Quadratic Programming (QP)²⁵ problem with the control input constraints including δ_min≤δ≤δ_max and Δδ_min≤Δδ≤Δδ_max.

Simulation analysis

The parameters of the ship

The effectiveness of the designed controller is demonstrated by the comparison simulations. Motor vessel named “Yulong” with single propeller and rudder is taken as the example in the simulation, and the ship parameters concerned are shown in Table 1.

Table 1.

The parameters of the ship.

The parameter	Values	Units
Draft	8	m
Displacement	14635	t
Ship length	126	m
Ship breadth	20.8	m
Front wind projection area	369.9	m²
Side wind projection area	1031.94	m²
Propeller diameter	4.6	m
Propeller pitch	3.66	m
The area of rudder blade	18.8	m²
Revolutions of the propeller	100	r/min
Following coefficient	216	\
Turning coefficient	0.478	\

Simulation results

Simulation 1: In order to illustrate that the NESO can promote the estimated accuracy or performance, the compared simulations on the basis of the NESO and conventional LESO are conducted. The initial state variables: (x, y, φ, u, v, r) = (0, 250 m, 0, 7.2 m/s, 0, 0). The environment disturbances: wind speed and time-varying wind direction 10 m/s and 50°sin(0.025t)+45°, current velocity and time-varying direction-going 1 m/s and 10°sin(0.001t) + 45°, wavelength 83 m, wave encounter angle φ+ 135°− 50°sin(0.025t), and wave height 3 m.

Figure 4 shows the two controllers could force ship to follow the reference path because the range of oscillations from the time-varying disturbances are less than 2 m or 10% of ship breadth. Meanwhile, the oscillation range of y_TESO is smaller than y_LESO. Figure 5 shows the control input of rudder angle with TESO is smoother and less oscillations than the one via LESO. Figure 6 decribes that the estimation errors of both heading angle and heading rate based on the NESO are smaller. Figure 7 shows that the generalized disturbances f could be estimated accurately by TESO, and the estimation errors of f via TESO is also smaller. The aforesaid results demonstrate that the proposed TESO promoted the estimated accuracy and performance, and outperformed the conventional LESO.

Figure 4.

Path following for straight line.

Figure 5.

Rudder angle for following straight line.

Figure 6.

The estimation errors of the heading angle and yaw rate for following straight line.

Figure 7.

The total unknowns and estimation errors of total unknowns for following straight line.

Simulation 2: The compared simulations using the proposed TADRC and LADRC, on the basis of LOS guidance method, are used to demonstrate the effectiveness of path following action with the rudder constraints, environment disturbances, as well as parameters uncertainties. The reference paths are computed by connecting the adjacent predefined waypoints as follows: P₁(0, 0), P₂(4000, 0), P₃(5500, 1000), P₄(8500, 1000), P₅(8500, 7500), P₆(11000, 10000), P₇(13000, 10000). The initial states: (x, y, φ, u, v, r) = (0, 0 m, 0, 7.2 m/s, 0, 0). The environment disturbances: wind speed and time-varying direction are 10m/s and 30°sin(0.01t)+45°, current velocity and direction-going are 1 m/s and 5°sin(0.005t) + 45°, wavelength 83m, wave encounter angle φ+ 135°− 30°sin(0.05t), wave height 3 m.

Figure 8 shows that the two control schemes of TADRC and LADRC could force ship to follow the desired path in the case of uncertain parameters and environment disturbances. Moreover, the actual path via TADRC is the same smooth as good helmsman behavior at large angle heading-altering. Figure 9 shows the control input rudder angle via TADRC is smoother and less oscillations than the one via LADRC. These results illustrate that the TADRC method could address the rudder amplitude and rate constraints successfully, and handle the parameters uncertainties as well as environmental disturbances effectively.

Figure 8.

Path following for curve.

Figure 9.

The heading angle, rudder angle and rudder rate for following curve.

Conclusions

A novel TADRC design method with the LOS guidance scheme was presented for path following of underactuated ships in this paper. The TESO was designed by modifying LESO via hyperbolic tangent function to improve the estimated performance for the generalized disturbances including parameters uncertainties and environment disturbances. The hyperbolic tangent function was introduced to design a nonlinear sliding mode control law instead of the conventional PD control to improve the control action. The nonlinear sliding mode control law was embedded in a standard QP cost function to handle the input constraints. The LOS guidance generated the reference heading for shaping desired paths like a good ship’s helmsman behavior. The simulation results showed that the proposed controller could force ship to follow the desired path effectively in case of the time-varying disturbances including wind, current and wave. Meanwhile, the comparison results indicate that the TADRC controllers with the LOS guidance scheme outperformed the conventional LADRC method. Moreover, the TESO method can improve the estimation accuracy for the heading, the heading rate and the generalized disturbance. More theoretical research about TADRC is needed in the future.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported in part by the special projects of key fields (Artificial Intelligence) of Universities in Guangdong Province (2019KZDZX1035), the National Natural Science Foundation of China (Nos. 51979045, 51939001, 61976033), and program for scientific research start-up funds of Guangdong Ocean University.

ORCID iD

Ronghui Li

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