Adaptive neural network control for course-keeping of ships with input constraints

Abstract

In this paper, an adaptive neural network (NN) control method is proposed for the problem of nonlinear course control of ships with input constraints and unknown direction control gains. Specifically, dynamic surface control is used to overcome the problem of explosion of complexity inherent in the backstepping technique, and the Nussbaum function is employed to deal with the unknown signs of control gains. It is proved that the proposed adaptive NN control method, which is composed of dynamic surface control and a backstepping technique with the Nussbaum gain function, is able to guarantee uniform ultimate boundedness of all the signals in the controlled system. In addition, the tracking error between the output of the controlled system and a desired trajectory is shown to converge to a small neighbourhood of the origin. Finally, one example is introduced to illustrate the proposed theoretical results.

Keywords

Neural network input saturation backstepping dynamic surface control course-keeping

Introduction

Recently, many researchers have focused their attention on the course control problem of ships (Fossen, 2000), which is known to be a difficult problem. Much work has been done with regard to this topic, and some modern control methods have been proposed (Mei et al., 2015; Sun et al., 2016, 2017a,b); see for example adaptive fuzzy control (Yang et al., 2003), backstepping control (Du et al., 2007), dynamic surface control (Du et al., 2014; Hou and Duan, 2011) and adaptive neural network (NN) control (Luo et al., 2009).

It has been observed that the adaptive NN control method has proved to be particularly effective for nonlinear uncertain systems of ships with unknown nonlinear functions. By employing the approximation property of NNs, it is very accurately estimate the unknown nonlinear part of systems, which will not make much effort on system modelling. There exist some real applications. For example, a back propagation NN model was proposed in Wang et al. (2017) by using solar radiation as the input parameter to establish the relationship between solar radiation and air temperature. In Gu et al. (2015), two finite mixture models to capture the structural information of the data from binary classification were used with a nonlinear model by exploiting kernelization techniques. A self-adaptive algorithm which is based on the global best candidate for global optimization was proposed in Xue et al. (2017) for some complex strategies. Different from earlier NN control methods using optimization techniques, where analytical results may be not obtained, several adaptive NN control methods have been proposed with Lyapunov stability (Lewis et al., 1996; Sanner and Slotine, 1992; Zhang et al., 1999). Specifically, the NN controller was proposed in Zhang et al. (2000b) for a class of unknown and minimum phase nonlinear systems. In Zhang et al. (2000a), an adaptive NN controller with backstepping technology was proposed by using an integral-type Lyapunov function, which overcomes the controller singularity problem. A direct adaptive NN controller was presented in Ge and Wang (2002) for a class of affine nonlinear systems with unknown nonlinearities, which avoids the singularity problem of the controller using a special property of the affine term.

Note that in the above-mentioned works input constraints are not considered, which a crucial issue for energy balance. For example, in the work of Zhang et al. (2017), an optimal cluster-based mechanism was proposed to load balancing with multiple mobile sinks for these problems by using a modified multi-hop layered model. In Zhang et al. (2016), the two problems of the relocation of sensors with the minimum number of mobile sensors and formation with minimum energy cost were studied. In fact, input constraints have been extensively studied due to their universal existence in real applications (Chen et al., 2010). For example, the controller design has been given for ocean surface vessels subject to input saturation in Chen et al. (2009) with an adaptive NN control method. Considering the tracking error dynamic systems of underactuated ships with input saturation, two finite-time controllers were designed in Huang et al. (2015) with the backstepping technique. By using a gain scheduled control method, the rudder roll stabilization problem of ships with input rate saturation has been studied in Lauvdal and Fossen (1998) in a computationally effective way. It is known that the uncertainty usually causes performance degradation (Xiao and Yin, 2016; Xiao et al., 2016, 2017). In Li et al. (2011b), an adaptive backstepping control algorithm is proposed for the course autopilot of ships with parameter uncertainty and input saturation. In Li et al. (2011a), a new adaptive NN controller was proposed for the course autopilot of ships with input saturation, where an auxiliary system is introduced to deal with the input constraints of ships. However, the signs of control gains in Li et al. (2011a) are known a priori, and, in general, the directions of control gains of ships are unknown.

In this paper, we will propose a new adaptive NN controller to solve the course-keeping problem of ships with input constraints. Specifically, the dynamic surface control is used to overcome the problem of explosion of complexity inherent in the backstepping technique, and the Nussbaum function is employed to deal with the unknown signs of control gains. To deal with the input constraints, a smooth function is introduced to approximate the saturation function, which is motivated by Wen et al. (2011). It is shown that the proposed adaptive NN control method, which is composed of dynamic surface control and a backstepping technique with the Nussbaum gain function, is able to guarantee uniform ultimate boundedness of all the signals in the controlled system. In addition, the tracking error between the output of the controlled system and a desired trajectory is shown to converge to a small neighbourhood of the origin. The main challenge of this work is how to deal with input constraints to make use of the backstepping technique.

The rest of this paper is organized as follows. In ‘Problem formulation’, the course-keeping problem of ships is formulated. ‘Main results’ presents the design procedure and its stability analysis. In ‘Simulation’, a simulation example is given to show the performance of the controller. The final section concludes this paper.

Problem formulation

In this study, we consider the model of ship steering introduced by Jia and Yang (1999), whose physical steering model is shown in Figure 1, and whose mathematical model is in the following form:

\overset{\cdot}{r} + \frac{K}{T} H (r) = \frac{K}{T} δ

(1)

where $r$ is the rate of the heading angle $φ$ of the ship, that is, $r = \overset{\cdot}{φ}$ , and $δ$ is the rudder angle. The parameters $K$ and $T$ represent the gain constant and the time constant, respectively. $H (r)$ represents an unknown nonlinear function, which is given as $H (r) = a_{1} r + a_{2} r^{3} + a_{3} r^{5} + \dots,$ where $a_{i}, i = 1, 2, \dots$ , are all constants. In this paper, we assume that the parameters $K$ , $T$ and $a_{i}$ are unknown constants.

Defining $x_{1} = φ,$ $x_{2} = r$ and $u (v) = δ$ , the nonlinear model of (1) can be transformed into

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = f_{2} (x_{2}) + g_{2} (x_{2}) u (v) + d_{2} (t) \end{matrix}

(2)

where $f_{2} (x_{2}) = - (K / T) H (r)$ , $g_{2} (x_{2}) = K / T$ , $d_{2} (t)$ is the bounded disturbance, and $u (v)$ is the output of the control input with saturation constraints, and is defined as follows:

u (v) = {sat}_{M} (v) = {\begin{matrix} M sign (v), | v | \geq M \\ v, | v | < M \end{matrix}

(3)

where $M$ is a known bound of $u (v)$ , $sign (\cdot)$ and $sat (\cdot)$ are the standard sign and saturation functions, and $v$ is the designed control input.

The control objective is to design an adaptive NN control law for system (2) with input constraints such that all the signals in the closed-loop systems remain uniformly ultimately bounded, and the tracking error between the output $y = x_{1}$ and a desired trajectory $y_{d}$ converges to a small neighbourhood of the origin.

To design the control law of (2) with input saturation, we consider a general class of second-order nonlinear systems

{\begin{matrix} {\overset{\cdot}{x}}_{1} = f_{1} (x_{1}) + g_{1} (x_{1}) x_{2} + d_{1} (t) \\ {\overset{\cdot}{x}}_{2} = f_{2} ({\bar{x}}_{2}) + g_{2} ({\bar{x}}_{2}) u (v) + d_{2} (t) \\ y = x_{1} \end{matrix}

(4)

where ${\bar{x}}_{2} = {[x_{1}, x_{2}]}^{T}$ and $y$ are the states and output of the system, respectively; $f_{i} (\cdot)$ and $g_{i} (\cdot)$ , $i = 1, 2$ , are unknown smooth functions; and $d_{i} (\cdot)$ , $i = 1, 2$ , are the bounded disturbances.

In what follows, some useful definitions and lemmas will be introduced.

A radial basis function NN (RBFNN) is usually adopted to approximate any continuous nonlinear functions, and can be regarded as a two-layer network including the hidden layer and the output layer. In this study, we select the following RBFNN (Haykin, 1999) to approximate a continuous function $h (Z) : R^{q} \to R$ :

h_{n n} (Z) = W^{T} S (Z)

(5)

where $Z \in Ω_{Z} \subset R^{q}$ is the input vector, $W \in R^{l}$ is the weight vector, $l \geq 1$ is the NN node number, and $S (Z) = {[s_{1} (Z), s_{2} (Z), \dots, s_{l} (Z)]}^{T}$ , with $s_{i} (Z), i = 1, 2, \dots, l$ , being chosen as the Gaussian functions:

s_{i} (Z) = \exp [\frac{- {(Z - {\bar{μ}}_{i})}^{T} (Z - {\bar{μ}}_{i})}{η_{i}^{2}}]

(6)

where ${\bar{μ}}_{i} = {[μ_{i 1}, μ_{i 2}, \dots, μ_{i l}]}^{T}$ is the centre of the receptive field and $η_{i}$ is the width of the Gaussian function. It has been proved (Sanner and Slotine, 1992) that if $l$ is chosen to be sufficiently large the network (5) can approximate any continuous nonlinear functions over a compact set $Ω_{Z} \subset R^{q}$ to any arbitrary accuracy as

h (Z) = W^{* T} S (Z) + ϵ, \forall Z \in Ω_{Z}

(7)

where $W^{*}$ are the ideal constant weights and $ϵ$ is the approximation error.

Assumption 1. There exist ideal constant weights $W^{*}$ such that $| ϵ | \leq ϵ^{*}$ with constant $ϵ^{*} > 0$ for all $Z \in Ω_{Z}$ .

Remark 1. The ideal constant weights $W^{*}$ can be regarded as an artificial quantity required for analytical purposes, which is defined as the value of $W$ that minimizes $| ϵ |$ for all $Z \in Ω_{Z}$ , that is,

W^{*} \overset{Δ}{=} \arg \min_{W \in R^{l}} {\sup_{Z \in Ω_{Z}} | h (Z) - W^{T} S (Z) |}

(8)

Furthermore, we let $\hat{W}$ be the estimate of $W^{*}$ , and the weight estimation error is $\tilde{W} = \hat{W} - W^{*}$ .

Definition 1. ( Ye and Jiang, 1998 ) The function $N (s) : R \to R$ is of Nussbaum type if it has the following properties:

\lim_{s \to + \infty} \sup \frac{1}{s} \int_{0}^{s} N (ζ) d ζ = + \infty

(8)

and

\lim_{s \to + \infty} \inf \frac{1}{s} \int_{0}^{s} N (ζ) d ζ = - \infty

(9)

Lemma 1. ( Deng and Krstić, 1997 ) For $\forall (x, y) \in R^{2},$ we have the following results:

x y \leq \frac{ε^{p}}{p} {| x |}^{p} + \frac{1}{q ε^{p}} {| y |}^{q}

(10)

where $ε > 0,$ $p > 1,$ $q > 1,$ and $(p - 1) (q - 1) = 1 .$

Lemma 2. ( Ge et al., 2004 ) Let $V (\cdot)$ and $ζ (\cdot)$ be smooth functions defined on $[0, t_{f})$ with $V (t) \geq 0, \forall t \in [0, t_{f})$ , and $N (\cdot)$ be an even smooth Nussbaum-type function. If the following inequality holds:

V (t) \leq c_{0} + e^{- c_{1} t} \int_{0}^{t} [g (\cdot) N (ζ) + 1] \overset{\cdot}{ζ} e^{c_{1} τ} d τ,

(11)

where $c_{0}$ and $c_{1} > 0$ are some suitable constants, $g (\cdot)$ is a time-varying parameter with values in the unknown closed intervals $I_{l} : = [l^{-}, l^{+}]$ and $0 \notin I_{l}$ , then $V (t)$ and $\int_{0}^{t} g (\cdot) N (ζ) \overset{\cdot}{ζ} e^{c_{1} τ} d τ$ must be bounded on $[0, t_{f})$ .

It is noted that the saturation model (3) is non-smooth, that is, compared with $u (v)$ there is a sharp corner in control input $v$ . Therefore, the backstepping method cannot be directly used. In order to apply this method, we define a smooth function to approximate the saturation function as follows:

g (v) = M \tanh (\frac{v}{M}) = M \frac{e^{v / M} - e^{- v / M}}{e^{v / M} + e^{- v / M}}

(12)

Then we can obtain

u (v) = b v (t) + Δ (v)

(13)

where $Δ (v) = g (v) - b v (t)$ and $b > 0$ is a constant. Substituting the approximated saturation function (13) into (4), we can obtain

{\begin{matrix} {\overset{\cdot}{x}}_{1} = f_{1} (x_{1}) + g_{1} (x_{1}) x_{2} + d_{1} (t) \\ {\overset{\cdot}{x}}_{2} = f_{2} ({\bar{x}}_{2}) + g_{2} ({\bar{x}}_{2}) [b v (t) + Δ (v)] + d_{2} (t) \\ y = x_{1} \end{matrix}

(14)

In order to design the control laws of (14), the following assumptions are reasonable and needed.

Assumption 2. The parameters $Δ (v)$ , $b$ and $d_{i} (t)$ , $i = 1, 2$ , are all bounded such that $Δ (v) \leq \bar{Δ},$ $0 < \underline{b} \leq b \leq \bar{b}$ and $d_{i} (t) \leq {\bar{d}}_{i},$ $i = 1, 2$ , respectively.

Assumption 3. The desired trajectory $y_{d}$ is a smooth function, and $y_{d},$ ${\overset{\cdot}{y}}_{d}$ and ${\overset{\cdot\cdot}{y}}_{d}$ are bounded.

Assumption 4. The signs of the unknown control gains $g_{1} (x_{1})$ and $g_{2} ({\bar{x}}_{2})$ are unknown, but the control gains are bounded such that $g_{1} \leq | g_{1} (x_{1}) | \leq {\bar{g}}_{1}$ and $g_{2} \leq | g_{2} ({\bar{x}}_{2}) | \leq {\bar{g}}_{2}$ , where $0 < g_{1} \leq {\bar{g}}_{1}$ and $0 < g_{2} \leq {\bar{g}}_{2}$ .

Main results

In this section, we will design an adaptive NN control law which is composed of dynamic surface control and the backstepping method. The design procedure can be divided into two steps, and in each step we construct an appropriate Lyapunov function for the stabilization problem of the system. The detailed design procedure is given as follows.

Step 1

Define $ω_{1} = y_{d}$ , and the error variables $z_{1} = x_{1} - ω_{1}$ and $z_{2} = x_{2} - ω_{2}$ , where $ω_{2}$ is the output of a first-order filter with $α_{1}$ as a virtual control law for the first subsystem. We also define the notation $y_{2} = ω_{2} - α_{1}$ . In view of (14), the derivative of $z_{1}$ is

{\overset{\cdot}{z}}_{1} = f_{1} (x_{1}) + g_{1} (x_{1}) x_{2} + d_{1} (t) - {\overset{\cdot}{ω}}_{1}

(15)

Since the condition is that $f_{1} (x_{1})$ is unknown, the feedback control law cannot be implemented in practice. As we see, the unknown part $h_{1} (Z_{1}) = f_{1} (x_{1}) - {\overset{\cdot}{ω}}_{1}$ is a smooth function of $x_{1}$ and ${\overset{\cdot}{ω}}_{1}$ , where $Z_{1} = {[x_{1}, {\overset{\cdot}{ω}}_{1}]}^{T} \subset R^{2}$ . Therefore, we should use an RBFNN $W_{1}^{T} S_{1} (Z_{1})$ to approximate $h_{1} (Z_{1})$ , such that

h_{1} (Z_{1}) = {\hat{W}}_{1}^{T} S_{1} (Z_{1}) - {\tilde{W}}_{1}^{T} S_{1} (Z_{1}) + ϵ_{1}

(16)

It follows from (15) that

{\overset{\cdot}{z}}_{1} = g_{1} (x_{1}) x_{2} + h_{1} (Z_{1}) + d_{1} (t)

(17)

Since we have defined $z_{2} = x_{2} - y_{2} - α_{1}$ , there exists a desired feedback control law

α_{1} = N (ζ_{1}) [k_{1} z_{1} + {\hat{W}}_{1}^{T} S_{1} (Z_{1})]

(18)

{\overset{\cdot}{ζ}}_{1} = k_{1} z_{1}^{2} + z_{1} {\hat{W}}_{1}^{T} S_{1} (Z_{1})

(19)

{\overset{\cdot}{\hat{W}}}_{1} = {\overset{\cdot}{\tilde{W}}}_{1} = Π_{1} [S_{1} (Z_{1}) z_{1} - δ_{1} {\hat{W}}_{1}]

(20)

where $k_{1}$ is the design parameter, $Π_{1} = Π_{1}^{T}$ is the adaptive gain matrix, $δ_{1}$ is a small constant, and the term $δ_{1} {\hat{W}}_{1}$ is introduced to improve the robustness with the NN approximation error $ϵ_{1}$ . It has been proved in Ge and Wang (2002) that without such a term, the NN weight estimate ${\hat{W}}_{1}$ may lead to very large values and result in a variation of high-gain control laws.

Consider the Lyapunov function

V_{z_{1}} = \frac{1}{2} z_{1}^{2}

(21)

Then the time derivative of $V_{z_{1}}$ is

\begin{array}{l} {\overset{\cdot}{V}}_{z_{1}} = z_{1} {\overset{\cdot}{z}}_{1} \\ = z_{1} [g_{1} (x_{1}) x_{2} + h_{1} (Z_{1}) + d_{1} (t)] \\ = z_{1} [g_{1} (x_{1}) (z_{2} + y_{2} + α_{1}) \\ + {\hat{W}}_{1}^{T} S_{1} (Z_{1}) - {\tilde{W}}_{1}^{T} S_{1} (Z_{1}) + ϵ_{1} + d_{1} (t) \\ = g_{1} (x_{1}) z_{1} z_{2} + z_{1} (ϵ_{1} + d_{1} (t)) - z {\tilde{W}}_{1}^{T} S_{1} (Z_{1}) \\ + [g_{1} (x_{1}) N (ζ_{1}) + 1] {\overset{\cdot}{ζ}}_{1} - k_{1} z_{1}^{2} + g_{1} (x_{1}) z_{1} y_{2} \\ \leq - k_{1} z_{1}^{2} + [g_{1} (x_{1}) N (ζ_{1}) + 1] {\overset{\cdot}{ζ}}_{1} + g_{1} (x_{1}) z_{1} z_{2} \\ + g_{1} (x_{1}) z_{1} y_{2} + | z_{1} | | ε_{1} | - z_{1} {\tilde{W}}_{1}^{T} S_{1} (Z_{1}) \end{array}

(22)

where we have defined $ε_{1} = ϵ_{1}^{*} + {\bar{d}}_{1}$ . According to Lemma 1, we have

g_{1} (x_{1}) z_{1} z_{2} \leq z_{1}^{2} + \frac{1}{4} g_{1}^{2} (x_{1}) z_{2}^{2}

(23)

| z_{1} | | ε_{1} | \leq z_{1}^{2} + \frac{1}{4} ε_{1}^{2}

(24)

g_{1} (x_{1}) z_{1} y_{2} \leq z_{1}^{2} + \frac{1}{4} g_{1}^{2} (x_{1}) y_{2}^{2}

(25)

It follows from (22) that

\begin{matrix} {\overset{\cdot}{V}}_{z_{1}} \leq - (k_{1} - 3) z_{1}^{2} + [g_{1} (x_{1}) N (ζ_{1}) + 1] {\overset{\cdot}{ζ}}_{1} + \frac{1}{4} g_{1}^{2} (x_{1}) z_{2}^{2} \\ + \frac{1}{4} g_{1}^{2} (x_{1}) y_{2}^{2} + \frac{1}{4} ε_{1}^{2} - z_{1} {\tilde{W}}_{1}^{T} S_{1} (Z_{1}) \end{matrix}

(26)

Since $ω_{2}$ is the output of a first-order filter with $α_{1}$ as a virtual control law for the first subsystem, we can obtain

κ_{2} {\overset{\cdot}{ω}}_{2} + ω_{2} = α_{1}

(27)

where $κ_{2}$ is the time constant of the first-order filter, and $ω_{2} (0) = α_{1} (0)$ . Since $y_{2} = ω_{2} - α_{1}$ , we have

{\overset{\cdot}{ω}}_{2} = \frac{α_{1} - ω_{2}}{κ_{2}} = - \frac{y_{2}}{κ_{2}}

(28)

Therefore, we obtain

{\overset{\cdot}{y}}_{2} = - \frac{y_{2}}{κ_{2}} + N (ζ_{1}) [k_{1} {\overset{\cdot}{z}}_{1} + {\overset{\cdot}{\hat{W}}}_{1}^{T} S_{1} (Z_{1}) + {\hat{W}}_{1}^{T} {\overset{\cdot}{S}}_{1} (Z_{1})]

(29)

which implies

| {\overset{\cdot}{y}}_{2} + \frac{y_{2}}{κ_{2}} | \leq Ψ_{2}

(30)

where $Ψ_{2}$ is a positive continuous function associated with parameters $z_{1},$ $ζ_{1},$ ${\tilde{W}}_{1}$ and $ω_{1}$ . Again, with Lemma 1 we have

\begin{matrix} {\overset{\cdot}{y}}_{2} y_{2} \leq - \frac{y_{2}^{2}}{κ_{2}} + | y_{2} | Ψ_{2} \\ \leq - \frac{y_{2}^{2}}{κ_{2}} + y_{2}^{2} + \frac{1}{4} Ψ_{2}^{2} \end{matrix}

Consider the following Lyapunov function:

V_{1} = V_{z_{1}} + \frac{1}{2} y_{2}^{2} + \frac{1}{2} {\tilde{W}}_{1}^{T} Π_{1}^{- 1} {\tilde{W}}_{1}

(31)

Its time derivative is

\begin{matrix} {\overset{\cdot}{V}}_{1} = {\overset{\cdot}{V}}_{z_{1}} + {\tilde{W}}_{1}^{T} Π_{1}^{- 1} {\overset{\cdot}{\hat{W}}}_{1} + {\overset{\cdot}{y}}_{2} y_{2} \\ \leq - (k_{1} - 3) z_{1}^{2} + [g_{1} (x_{1}) N (ζ_{1}) + 1] {\overset{\cdot}{ζ}}_{1} + \frac{1}{4} g_{1}^{2} (x_{1}) z_{2}^{2} \\ + \frac{1}{4} ϑ_{1}^{2} - δ_{1} {\tilde{W}}_{1}^{T} {\hat{W}}_{1} - \frac{y_{2}^{2}}{κ_{2}} + y_{2}^{2} + \frac{1}{4} g_{1}^{2} (x_{1}) y_{2}^{2} \end{matrix}

(32)

where $ϑ_{1}^{2} = ε_{1}^{2} + Ψ_{2}^{2}$ . By using completion of squares, we obtain

\begin{matrix} - δ_{1} {\tilde{W}}_{1}^{T} {\hat{W}}_{1} = - δ_{1} {\tilde{W}}_{1}^{T} ({\tilde{W}}_{1} + W_{1}^{*}) \\ \leq - δ_{1} {‖ {\tilde{W}}_{1} ‖}^{2} + δ_{1} ‖ {\tilde{W}}_{1} ‖ ‖ W_{1}^{*} ‖ \\ \leq - \frac{1}{2} δ_{1} {‖ {\tilde{W}}_{1} ‖}^{2} + \frac{1}{2} δ_{1} {‖ W_{1}^{*} ‖}^{2} \end{matrix}

(33)

It follows from (32) that

\begin{matrix} {\overset{\cdot}{V}}_{1} \leq - (k_{1} - 3) z_{1}^{2} + [g_{1} (x_{1}) N (ζ_{1}) + 1] {\overset{\cdot}{ζ}}_{1} + \frac{1}{4} g_{1}^{2} (x_{1}) z_{2}^{2} + \frac{1}{4} ϑ_{1}^{2} \\ - \frac{1}{2} δ_{1} {‖ {\tilde{W}}_{1} ‖}^{2} + \frac{1}{2} δ_{1} {‖ W_{1}^{*} ‖}^{2} - (\frac{1}{4} g_{1}^{2} (x_{1}) - \frac{1}{κ_{2}} - 1) y_{2}^{2} \end{matrix}

(34)

Step 2

In this step, we will design the control input $v (t)$ in (14). Since we have defined $z_{2} = x_{2} - ω_{2}$ , the time derivative of $z_{2}$ is

{\overset{\cdot}{z}}_{2} = f_{2} ({\bar{x}}_{2}) + g_{2} ({\bar{x}}_{2}) [b v (t) + Δ (v)] + d_{2} (t) - {\overset{\cdot}{ω}}_{2}

(35)

Similarly, we construct a Lyapunov function

V_{z_{2}} = \frac{1}{2} z_{2}^{2}

(36)

and its derivative is

\begin{matrix} {\overset{\cdot}{V}}_{z_{2}} = z_{2} {\overset{\cdot}{z}}_{2} \\ = z_{2} [f_{2} ({\bar{x}}_{2}) + g_{2} ({\bar{x}}_{2}) [b v (t) + Δ (v)] + d_{2} (t) - {\overset{\cdot}{ω}}_{2}] \\ = z_{2} [h_{2} (Z_{2}) + g_{2} ({\bar{x}}_{2}) (b v (t) + Δ (v)) + d_{2} (t)] \end{matrix}

(37)

where $Z_{2} = [{\bar{x}}_{2}, {\overset{\cdot}{ω}}_{2}] \subset R^{3}$ , and we have used an RBFNN $W_{2}^{T} S_{2} (Z_{2})$ to approximate $h_{2} (Z_{2}) = f_{2} ({\bar{x}}_{2}) - {\overset{\cdot}{ω}}_{2}$ , such that

h_{2} (Z_{2}) = {\hat{W}}_{2}^{T} S_{2} (Z_{2}) - {\tilde{W}}_{2}^{T} S_{2} (Z_{2}) + ϵ_{2}

(38)

The following feedback control laws are proposed for the control input $v (t)$ :

v (t) = N (ζ_{2}) [k_{2} z_{2} + {\hat{W}}_{2}^{T} S_{2} (Z_{2})]

(39)

{\overset{\cdot}{ζ}}_{2} = k_{2} z_{2}^{2} + z_{2} {\hat{W}}_{2}^{T} S_{2} (Z_{2})

(40)

{\overset{\cdot}{\hat{W}}}_{2} = {\overset{\cdot}{\tilde{W}}}_{2} = Π_{2} [S_{2} (Z_{2}) z_{2} - δ_{2} {\hat{W}}_{2}]

(41)

where $k_{2}$ is the design parameter, $Π_{2} = Π_{2}^{T}$ is the adaptive gain, $δ_{2}$ is a small constant, and the term $δ_{2} {\hat{W}}_{2}$ is introduced to improve the robustness with the NN approximation error $ϵ_{2}$ .

It follows from (37) that

{\overset{\cdot}{V}}_{z_{2}} \leq - k_{2} z_{2}^{2} + [g_{2} ({\bar{x}}_{2}) b N (ζ_{2}) + 1] {\overset{\cdot}{ζ}}_{2} + z_{2} ε_{2} - z_{2} {\tilde{W}}_{2}^{T} S_{2} (Z_{2})

(42)

where we have defined $ϑ_{2} = {\bar{g}}_{2} \bar{Δ} + {\bar{d}}_{2} + ϵ_{2}^{*}$ . Using Young’s inequality, we have

{\overset{\cdot}{V}}_{z_{2}} \leq - (k_{2} - 1) z_{2}^{2} + [g_{2} ({\bar{x}}_{2}) b N (ζ_{2}) + 1] {\overset{\cdot}{ζ}}_{2} + \frac{1}{4} ϑ_{2}^{2} - z_{2} {\tilde{W}}_{2}^{T} S_{2} (Z_{2})

(43)

Consider the following Lyapunov function:

V_{2} = V_{z_{2}} + \frac{1}{2} {\tilde{W}}_{2}^{T} Π_{2}^{- 1} {\tilde{W}}_{2}

(44)

Then we have

\begin{matrix} {\overset{\cdot}{V}}_{2} = {\overset{\cdot}{V}}_{z_{2}} + \frac{1}{2} {\tilde{W}}_{2}^{T} [S_{2} (Z_{2}) z_{2} - δ_{2} {\hat{W}}_{2}] \\ \leq - (k_{2} - 1) z_{2}^{2} + [g_{2} ({\bar{x}}_{2}) b N (ζ_{2}) + 1] {\overset{\cdot}{ζ}}_{2} \\ + \frac{1}{4} ϑ_{2}^{2} - \frac{1}{2} δ_{2} {‖ {\tilde{W}}_{2} ‖}^{2} + \frac{1}{2} δ_{2} {‖ W_{2}^{*} ‖}^{2} \end{matrix}

(45)

Based on the above design procedure, the main result can be summarized as follows.

Theorem 1. Consider the system (4) with input saturation (13) under Assumptions 1 to 4. If the initial conditions of the system are bounded, all signals of the closed-loop system remain bounded with the designed control laws (18) and (39) and the updating laws (19), (20), (40) and (41). Meanwhile, by properly choosing design parameters, the tracking error between the output $y = x_{1}$ and the desired trajectory $y_{d}$ converges to a small neighbourhood of the origin.

Proof. To prove the signals of the closed-loop system are bounded with the designed control laws, we first consider the Lyapunov function (31). We have

\begin{matrix} {\overset{\cdot}{V}}_{1} \leq - (k_{1} - 3) z_{1}^{2} + [g_{1} (x_{1}) N (ζ_{1}) + 1] {\overset{\cdot}{ζ}}_{1} + \frac{1}{4} g_{1}^{2} (x_{1}) z_{2}^{2} + \frac{1}{4} ϑ_{1}^{2} \\ - \frac{1}{2} δ_{1} {‖ {\tilde{W}}_{1} ‖}^{2} + \frac{1}{2} δ_{1} {‖ W_{1}^{*} ‖}^{2} + (\frac{1}{4} g_{1}^{2} (x_{1}) - \frac{1}{κ_{2}} - 1) y_{2}^{2} \\ \leq - γ_{1} V_{1} + [g_{1} (x_{1}) N (ζ_{1}) + 1] {\overset{\cdot}{ζ}}_{1} + \frac{1}{4} g_{1}^{2} (x_{1}) z_{2}^{2} + m_{1} \end{matrix}

(46)

where the parameters $γ_{1}$ and $m_{1}$ are chosen as

γ_{1} \leq 2 \min {(k_{1} - 3), \frac{δ_{1}}{2 λ_{\max} (Π_{1}^{- 1})}, (\frac{1}{κ_{2}} + 1 - \frac{1}{4} {\bar{g}}_{1}^{2})}

(47)

m_{1} = \frac{1}{4} ϑ_{1}^{2} + \frac{1}{2} δ_{1} ‖ W_{1}^{*} ‖^{2}

(48)

Then, multiplying $e^{γ_{1} t}$ on both sides of (46) and integrating it on the interval $[0, t]$ , we have

\begin{matrix} V_{1} \leq V_{1} (0) + e^{- γ_{1} t} \int_{0}^{t} [(g_{1} (x_{1}) N (ζ_{1}) + 1) {\overset{\cdot}{ζ}}_{1} e^{γ_{1} τ}] d τ \\ + e^{- γ_{1} t} \int_{0}^{t} [\frac{1}{4} g_{1}^{2} (x_{1}) z_{2}^{2} e^{γ_{1} τ}] d τ + \frac{m_{1}}{γ_{1}} \end{matrix}

(49)

According to Assumption 4, we have

e^{- γ_{1} t} \int_{0}^{t} [\frac{1}{4} g_{1}^{2} (x_{1}) z_{2}^{2} e^{γ_{1} τ}] d τ \leq \frac{1}{4 γ_{1}} {\bar{g}}_{1}^{2} \sup_{τ \in [0, t]} [z_{2}^{2} (τ)]

Consequently, if $z_{2}$ is bounded on the interval $[0, t_{f})$ , we find that the term $e^{- γ_{1} t} \int_{0}^{t} [\frac{1}{4} g_{1}^{2} (x_{1}) z_{2}^{2} e^{γ_{1} τ}] d τ$ is bounded, and (49) can be written as

V_{1} \leq e^{- γ_{1} t} \int_{0}^{t} [(g_{1} (x_{1}) N (ζ_{1}) + 1) {\overset{\cdot}{ζ}}_{1} e^{γ_{1} τ}] d τ + m_{1}^{f}

(50)

where we have defined

m_{1}^{f} = V_{1} (0) + e^{- γ_{1} t} \int_{0}^{t} [\frac{1}{4} g_{1}^{2} (x_{1}) z_{2}^{2} e^{γ_{1} τ}] d τ + \frac{m_{1}}{γ_{1}}

(51)

In view of Lemma 2, it is shown that $V_{1},$ $ζ_{1},$ ${\hat{W}}_{1}$ and $\int_{0}^{t} [(g_{1} (x_{1}) N (ζ_{1}) + 1) {\overset{\cdot}{ζ}}_{1} e^{γ_{1} τ}] d τ$ are all bounded on the interval $[0, t_{f})$ . Moreover, since $V_{1}$ is bounded, it is easy to know that $z_{1}$ is bounded.

In what follows, we consider the Lyapunov function (44). We have

\begin{matrix} {\overset{\cdot}{V}}_{2} \leq - (k_{2} - 1) z_{2}^{2} + [g_{2} ({\bar{x}}_{2}) b N (ζ_{2}) + 1] {\overset{\cdot}{ζ}}_{2} \\ + \frac{1}{4} ϑ_{2}^{2} - \frac{1}{2} δ_{2} {‖ {\tilde{W}}_{2} ‖}^{2} + \frac{1}{2} δ_{2} {‖ W_{2}^{*} ‖}^{2} \\ \leq - γ_{2} V_{2} + [g_{2} ({\bar{x}}_{2}) b N (ζ_{2}) + 1] {\overset{\cdot}{ζ}}_{2} + m_{2} \end{matrix}

(52)

where we have defined

γ_{2} \leq 2 \min {(k_{2} - 1), \frac{δ_{2}}{2 λ_{\max} (Π_{2}^{- 1})}}

(53)

m_{2} = \frac{1}{4} ϑ_{2}^{2} + \frac{1}{2} δ_{2} ‖ W_{2}^{*} ‖^{2}

(54)

Similarly, multiplying $e^{γ_{2} t}$ on both sides of (52) and integrating it on the interval $[0, t]$ , we have

\begin{matrix} V_{2} \leq V_{2} (0) + e^{- γ_{2} t} \int_{0}^{t} [(g_{2} ({\bar{x}}_{2}) b N (ζ_{2}) + 1) {\overset{\cdot}{ζ}}_{2} e^{γ_{2} τ}] d τ + \frac{m_{2}}{γ_{2}} \\ = m_{2}^{f} + e^{- γ_{2} t} \int_{0}^{t} [(g_{2} ({\bar{x}}_{2}) b N (ζ_{2}) + 1) {\overset{\cdot}{ζ}}_{2} e^{γ_{2} τ}] d τ \end{matrix}

(55)

where

m_{2}^{f} = V_{2} (0) + \frac{m_{2}}{γ_{2}}

(56)

In view of Lemma 2 and Assumption 2, it can be proved that $V_{2},$ $ζ_{2},$ ${\hat{W}}_{2}$ and $\int_{0}^{t} [(g_{2} ({\bar{x}}_{2}) b N (ζ_{2}) + 1) {\overset{\cdot}{ζ}}_{2} e^{γ_{2} τ}] d τ$ are all bounded on the interval $[0, t_{f})$ . Moreover, since $V_{2}$ is bounded, it is easy to know that $z_{2}$ is bounded.

Finally, we consider the following Lyapunov function:

V = V_{1} + V_{2}

(57)

Substituting (46) and (52) into the time derivative of (57), we can obtain that

\begin{matrix} \overset{\cdot}{V} \leq - (k_{1} - 3) z_{1}^{2} + [g_{1} (x_{1}) N (ζ_{1}) + 1] {\overset{\cdot}{ζ}}_{1} + \frac{1}{4} g_{1}^{2} (x_{1}) z_{2}^{2} + \frac{1}{4} ϑ_{1}^{2} \\ - \frac{1}{2} δ_{1} {‖ {\tilde{W}}_{1} ‖}^{2} + \frac{1}{2} δ_{1} {‖ W_{1}^{*} ‖}^{2} + (\frac{1}{4} g_{1}^{2} (x_{1}) - \frac{1}{κ_{2}} - 1) y_{2}^{2} \\ - (k_{2} - 1) z_{2}^{2} + [g_{2} ({\bar{x}}_{2}) b N (ζ_{2}) + 1] {\overset{\cdot}{ζ}}_{2} \\ + \frac{1}{4} ϑ_{2}^{2} - \frac{1}{2} δ_{2} {‖ {\tilde{W}}_{2} ‖}^{2} + \frac{1}{2} δ_{2} {‖ W_{2}^{*} ‖}^{2} \\ \leq - γ V + \sum_{i = 1}^{2} [(\hat{g} N (ζ_{i}) + 1) {\overset{\cdot}{ζ}}_{i}] + m \end{matrix}

(58)

where we have defined $\hat{g} = \max {{\bar{g}}_{1}, {\bar{g}}_{2} \bar{b}}$ ,

\begin{matrix} γ = 2 \min {(k_{1} - 3), \frac{δ_{1}}{2 λ_{\max} (Π_{1}^{- 1})}, (\frac{1}{κ_{2}} + 1 - \frac{1}{4} {\bar{g}}_{1}^{2}), \\ (k_{2} - 1), \frac{δ_{2}}{2 λ_{\max} (Π_{2}^{- 1})}} \end{matrix}

(59)

and

m = \frac{1}{4} ϑ_{1}^{2} + \frac{1}{2} δ_{1} ‖ W_{1}^{*} ‖^{2} + \frac{1}{4} ϑ_{2}^{2} + \frac{1}{2} δ_{2} ‖ W_{2}^{*} ‖^{2}

(60)

Multiplying $e^{γ t}$ on both sides of (58) and integrating it on the interval $[0, t]$ , we have

V \leq m^{f} + e^{- γ t} \sum_{i = 1}^{2} \int_{0}^{t} [(\hat{g} N (ζ_{i}) + 1) {\overset{\cdot}{ζ}}_{i}] e^{γ τ} d τ

(61)

where

m^{f} = V (0) + \frac{m^{f}}{γ}

(62)

By using Lemma 2, it is easy to see that $V,$ $ζ_{i},$ ${\hat{W}}_{i}$ and $\int_{0}^{t} [(\hat{g} N (ζ_{i}) + 1) {\overset{\cdot}{ζ}}_{i}] e^{γ τ} d τ$ are all bounded on the interval $[0, t_{f})$ . Therefore, all the signals of the controlled system are bounded. In addition, the tracking error between the output and the desired trajectory converges to a small neighbourhood of the origin. This completes the proof. □

Simulation

In this section, an example is presented to demonstrate the effectiveness of the proposed adaptive NN control for the course-keeping of ships with input constraints. Consider the following model of ships:

{\begin{matrix} {\overset{\cdot}{x}}_{1} = x_{2} \\ {\overset{\cdot}{x}}_{2} = f_{2} (x_{2}) + g_{2} (x_{2}) u (v) + d_{2} (t) \\ y = x_{1} \end{matrix}

(63)

where $d_{2} (t) = 0.01 \sin (2 π t)$ , $f_{2} (x_{2}) = - \frac{1}{T} x_{2} - \frac{α}{T} x_{2}^{3},$ $g_{2} (x_{2}) = \frac{K}{T}$ and the input saturation is $u (v) = b v + Δ (v)$ with $b = 8$ and $Δ (v) = 10 \tanh (\frac{v}{10}) - b v$ , where $v$ is the designed control input. The control objective is to guarantee that all the signals of the system (63) with the proposed controllers $v$ are bounded, and that the output $y$ can track the following desired trajectory:

y_{d} = - 0.2 \cos (t) + 0.2

(64)

In this simulation, we choose $T = 15,$ $α = 0.3$ and $K = 0.23$ , and the simulation time is $t = 80$ s. The NN term ${\hat{W}}_{1}^{T} S_{1} (Z_{1})$ with input vector $Z_{1} = {[x_{1}, {\overset{\cdot}{ω}}_{1}]}^{T}$ contains $21$ nodes with centres $μ_{1 l}$ , $l = 1, 2, \dots, 21$ , evenly spaced in $[- 5, 5] \times [- 5, 5]$ , and widths $η_{1 l} = 2,$ $l = 1, 2, \dots, 21$ . The NN term ${\hat{W}}_{2}^{T} S_{2} (Z_{2})$ with input vector $Z_{2} = {[x_{1}, x_{2}, {\overset{\cdot}{ω}}_{2}]}^{T}$ contains $121$ nodes with centres $μ_{2 l}$ , $l = 1, 2, \dots, 121$ , evenly spaced in $[- 5, 5] \times [- 5, 5] \times [- 5, 5]$ , and widths $η_{2 l} = 5,$ $l = 1, 2, \dots, 121$ . The initial conditions are chosen as ${\hat{W}}_{1} (0) = {\hat{W}}_{2} (0) = 0, x_{1} (0) = x_{2} (0) = 0$ , and the control parameters are $Π_{1} = 0.1 I_{21}, Π_{2} = 0.1 I_{121}, δ_{1} = 0.1, δ_{2} = 0.1, k_{1} = 4, k_{2} = 6$ , and the function $N (s) = s^{2} \cos (s)$ . The simulation results are shown in Figures 2 to 6. It is seen from Figures 2 and 3 that the tracking performance is achieved in less than $40$ s’. That is, the output $y$ can track the desired trajectory in less than $40$ s’. Furthermore, due to the unknown control directions, some large control inputs are shown from $t = 2$ s to $t = 4$ s; see Figure 4. Figures 5 and 6 show the process of Nussbaum gains $N (ζ)$ and $ζ$ .

Conclusions

In this paper, we have studied the course-keeping problem of ships with input constraints, and an adaptive NN control method has been proposed based on dynamic surface control and the backstepping technique with Nussbaum functions. To deal with the input constraints, a smooth function has been introduced to approximate the saturation function. The proposed adaptive NN control method has been proved able to guarantee uniform ultimate boundedness of all the signals in the controlled system. In addition, the tracking error between the output of the controlled system and a desired trajectory can converge to a small neighbourhood of the origin.

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 partially supported by the National Natural Science Foundation of China (61503079, 61520106009, 61533008 and 61673106), by the Jiangsu Natural Science Foundation (BK20150625 and BK20171362), by the Fundamental Research Funds for the Central Universities, and was a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions.

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