Composite anti-disturbance control for ship dynamic positioning systems with thruster faults

Abstract

Anti-disturbance control problem is studied for ship dynamic positioning systems with model uncertainties and ocean disturbances under thruster faults. For the ocean disturbance, a stochastic disturbance observer (SDO) is established to give the online estimation. For thruster faults, an adaptive law is used to evaluate, which is obtain from Lyapunov function. For model uncertainties, a robust control term with adaptive technology is used to attenuate it. Then, a composite anti-disturbance control (CADC) strategy is raised by combining disturbance observer-based control (DOBC), adaptive technology, and robust control term, which makes the position and yaw angle of ship reach the desired values. Finally, the simulation example proves the validity of the controller.

Introduction

The dynamic positioning (DP) can keep the ship in a predetermined trajectory in the presence of external environmental disturbance.^1–3. For unknown constant disturbance, a nonlinear set-point-regulation controller was proposed using a port-Hamiltonian framework. ⁴ Considering the second-order wave forces, a proportion-integration-differentiation controller was given based on fuzzy rules. ⁵ An observer is designed to estimate the ocean disturbances influencing in systems DP ships. ⁶ As we all known that, in many results only the influence of external disturbance are considered on DP ships.

In fact, there are various disturbances at sea level, thus it is necessary to study the anti-disturabnce control for DP systems with multiple disturbances.^7,8 Yet conventional anti-disturbance control is conservative in the attenuation of multiple disturbances, in this cases, the composite hierarchical anti-disturbance control (CHADC) strategy was presented, which has advantages of high control precision and strong robustness.^9–12 Considering modeling uncertainty and marine environmental disturbance, a composite control method is raised by combining DOBC and $H_{\infty }$ control. ¹³ In Du et al., ¹⁴ a robust adaptive controller was given for systems with multiple disturbance. Furthermore, the DOBC and stochastic control theory are combined to deal with environmental disturbances such as wind, second-order wave force, and current. ¹⁵

Thruster faults will appear inevitably when ships sailing on the sea for a long time, which will degrade system performance, reduce the DP accuracy, and even make the DP system unstable, thus, it is essential to study the fault diagnosis of ships to improve the reliability of the DP systems.^16–18 Under the condition of the fault type and upper bound are known, the dynamic positioning was realized for ships with disturbances. ¹⁹ In Su et al., ²⁰ the asymptotic stability of DP ships with actuator constraints was addressed, whereas the external environmental disturbances is not considered. Considering the dynamic positioning ship with thruster faults, a dynamic surface control method based on fault state observer is proposed, which lacks the ability to deal with disturbances. ²¹

In this paper, the problem of anti-disturbance control for DP ship systems with model uncertainties and ocean disturbances under thruster faults is researched. A composite anti-disturbance control strategy is proposed to make all states of DP ships asymptotically bounded in mean square, such that the desired position and heading can be achieved. The main contributions are as follows:

1. The DP system of ships with ocean disturbances, model uncertainties, and thruster faults are considered concurrently, the model uncertainties and thruster faults are unknown and norm bounded. In addition, the ocean disturbance is generated by an external system with stochastic term.

2. A stochastic disturbance observer is established to compensate the ocean disturbance, an adaptive algorithm-based robust control term is designed to suppress the model uncertainties, as well as the thruster faults are estimated by using adaptive technology. On these basis, a composite anti-disturbance controller is constructed to ensure that the states of system are asymptotically bounded in mean square.

Mathematical modeling of ships

Two coordinate frames are defined to describe the ship motion, as represented in the Figure 1. The dynamic positioning of ship’s mathematical model is given as follow ²²

$$\stackrel{·}{\eta }(t)=J(\psi )\mathbf{v}(t),$$

(1)

$$M\stackrel{·}{\mathbf{v}}(t)=-D\mathbf{v}(t)+\tau (t)+b(t),$$

(2)

where $\eta =[x,y,\psi {]}^{\mathrm{T}}$ is the vector consisting of the north-east-down position $(x,y)$ and heading angle $\psi$ , $\mathbf{v}=[u,v,r{]}^{\mathrm{T}}$ is the vector consisting of the ship-fixed surge velocity $u$ , sway velocity $v$ , and yaw rate $r$ , and

$$J(\psi )=\left[\begin{array}{ccc}\cos (\psi )\hfill & -\sin (\psi )\hfill & 0\hfill \\ \multicolumn{1}{c}{\sin (\psi )}& \cos (\psi )\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill \end{array}\right],$$

(3)

which satisfies $J^{-1}(\psi )=J^{\mathrm{T}}(\psi )$ . $M\in {\Re }^{3\times 3}$ is a inertia matrix, $D\in {\Re }^{3\times 3}$ is a damping matrix. $b=[b_1,b_2,b_3{]}^{\mathrm{T}}$ is slowly-varying environmental disturbances. $\tau =[{\tau }_1,{\tau }_2,{\tau }_3{]}^{\mathrm{T}}$ is a three-dimensional column vector, which is formed from forces and moments that generated by the propulsion device with thruster faults, where ${\tau }_1$ is the surge force, ${\tau }_2$ is the sway force, and ${\tau }_3$ is the yaw moment. The thruster faults can be describe as

$${\tau }_i={\Gamma }_i{\tau }_{\mathrm{ci}},i=1,2,3,$$

(4)

where ${\tau }_{\mathrm{ci}}$ is actual control input, ${\Gamma }_i$ indicates a loss-of-effectiveness (LOE) fault.

Figure 1.

North-east-down and ship-fixed coordinate frames. ²²

Based on Fossen and Strand, ²³ $b(t)$ is described as

$$\stackrel{·}{b}(t)=-T^{-1}b(t)+\Psi \xi (t),$$

(5)

where $T\in {\Re }^{3\times 3}$ is the known positive definite diagonal matrices, $\Psi \in {\Re }^{3\times 3}$ is the bounded positive definite diagonal matrices, $\xi (t)\in {\Re }^3$ is the zero-mean Gauss white noise, and $\parallel \xi (t){\parallel }^2\le c^{*}$ , $c^{*}$ is a positive constant.

The ship has uncertainty in dynamic modeling, which is shown as follows

$$M=M_0+\Delta M,D=D_0+\Delta D,$$

(6)

where $M_0,D_0$ are known, and $\Delta M,\Delta D$ represent modeling uncertainty.

Let

$$\delta =-\Delta M\stackrel{·}{\mathbf{v}}-\Delta D\mathbf{v}.$$

(7)

Based on Wang and Han, ²⁴ the yaw angle $\psi$ satisfies

$$J(\psi )\approx I.$$

(8)

The DP system is expressed as

$$\begin{array}{c}\stackrel{·}{x}(t)=\mathrm{Gx}(t)+H({\Gamma }^{\mathrm{T}}{\tau }_c(t)+b(t)+\delta (t)),\hfill \\ x=\left[\begin{array}{c}\eta \hfill \\ \multicolumn{1}{c}{\mathbf{v}}\end{array}\right],G=\left[\begin{array}{cc}{O}_{3\times 3}\hfill & I\hfill \\ \multicolumn{1}{c}{O_{3\times 3}}& -M_0^{-1}{D}_0\hfill \end{array}\right],H=\left[\begin{array}{c}{O}_{3\times 3}\hfill \\ \multicolumn{1}{c}{M_0^{-1}}\end{array}\right],\hfill \end{array}$$

(9)

where $x(t)\in {\Re }^6$ , $G\in {\Re }^{6\times 6}$ , $H\in {\Re }^{6\times 3}$ , and ${\tau }_c(t)=[{\tau }_{c1},{\tau }_{c2},{\tau }_{c3}{]}^{\mathrm{T}}\in {\Re }^3$ is the state vector, the coefficient matrix and the control input vector. $b(t)$ is slowly-varying environmental disturbances. $\delta (t)$ is the modeling uncertainty. $\Gamma =\mathrm{diag}\{{\Gamma }_1,{\Gamma }_2,{\Gamma }_3\}$ represents the LOE fault caused by the thruster system.

• Assumption 1: The LOE fault satisfies ${\Gamma }^{*}<{\Gamma }_i\le 1,i=1,2,3$ , where ${\Gamma }^{*}$ is an unknown positive constant.

• Assumption 2: The modeling uncertainty term is bounded, which satisfies $\parallel \delta \parallel <\rho$ and $\rho$ is unknown positive constant.

• Assumption 3: The pair $(T^{-1},H)$ is observable and the pair $(G,H)$ is controllable.

• Lemma 1 ²⁵ : For

$$\mathrm{dx}(t)=f(x(t),t)\mathrm{dt}+g(x(t),t)d\omega (t),t\ge t_0\ge 0.$$

(10)

If $\exists V\in C^{2.1}({\Re }^n\times {\Re }_{+})$ , $\kappa \in {\mathcal{K}}_{\nu }\subset {\mathcal{K}}_{\infty }$ , and $\psi ,s,\theta >0$ , satisfy

$$\kappa (|x{|}^{\psi })\le V(x,t),$$

(11)

$$\mathcal{L}V(x,t)\le -\mathrm{sV}(x,t)+\theta ,$$

(12)

for $\forall (x,t)\in {\Re }^n\times {\Re }_{+}$ , that is

$$\underset{t\to \infty }{lim}supE|x(t;t_0,x_0){|}^{\psi }\le {\kappa }^{-1}\left(\frac{\theta }{s}\right)$$

(13)

for $\forall (x,t)\in {\Re }^n\times {\Re }_{+}$ , the system (10) is asymptotically bounded in $p$ th moment.

According to Øksendal, ²⁶ by replacing $\xi (t)$ with $\frac{d\omega (t)}{\mathrm{dt}}$ , the DP systems can be depicted by:

$$\mathrm{dx}(t)=\mathrm{Gx}(t)\mathrm{dt}+H({\Gamma }^{\mathrm{T}}{\tau }_c(t)+b(t)+\delta (t))\mathrm{dt},$$

(14)

$$\mathrm{db}(t)=-T^{-1}b(t)\mathrm{dt}+\Psi d\omega (t),$$

(15)

based on Hu, ²⁷ Hu et al., ²⁸ $\omega (t)$ is an independent standard Wiener process.

• Lemma 2 ²⁹ : If the characteristic value of the matrix $\mathcal{X}$ belong to the LMI region $\mathcal{D}(\alpha ,\beta )$ , the following conditions are satisfied when and only when there exists a symmetric positive definite matrix $\mathcal{P}$

$$\mathcal{X}\mathcal{P}+\mathcal{P}{\mathcal{X}}^{\mathrm{T}}+2\alpha \mathcal{P}<0$$

(16)

$$\left[\begin{array}{cc}-\beta \mathcal{P}\hfill & \mathcal{X}\mathcal{P}\hfill \\ \multicolumn{1}{c}{*}& -\beta \mathcal{P}\hfill \end{array}\right]<0$$

(17)

Design of observers and controller

Supposing $x(t)$ is available. The stochastic disturbance observer (SDO) is devised to estimate $b(t)$ . After that, the CADC strategy is proposed.

Stochastic disturbance observer (SDO)

The SDO is given as

$$\{\begin{array}{c}\hat{b}(t)=q(t)+\mathrm{Lx}(t),\hfill \\ \multicolumn{1}{c}{\mathrm{dq}(t)=(-T^{-1}-\mathrm{LH})(q(t)+\mathrm{Lx}(t))\mathrm{dt}-\mathrm{LGx}(t)\mathrm{dt}}\end{array}-\mathrm{LH}{\hat{\Gamma }}^{\mathrm{T}}{\tau }_c(t)\mathrm{dt},$$

(18)

where $\hat{b}(t)$ is the evaluation of $b(t)$ , $q(t)$ is the auxiliary variable of SDO (18), and $L$ is the observer gain matrix. $\hat{\Gamma }$ is the evaluation of $\Gamma$ , which will be determined later.

Define the disturbance estimation error as $\tilde{b}(t)$ and the fault estimation error as $\tilde{\Gamma }$ , and make it satisfy $\tilde{b}(t)=b(t)-\hat{b}(t)$ and $\tilde{\Gamma }=\Gamma -\hat{\Gamma }$ . The disturbance estimation error systems is

$$\begin{array}{c}d\tilde{b}(t)=(-T^{-1}-\mathrm{LH})\tilde{b}(t)\mathrm{dt}-\mathrm{LH}{\tilde{\Gamma }}^{\mathrm{T}}{\tau }_c(t)\mathrm{dt}\hfill \\ -\mathrm{LH}\delta (t)\mathrm{dt}+\Psi d\omega (t).\hfill \end{array}$$

(19)

Composite anti-disturbance control (CADC)

The CADC strategy is put forward in order that the system (14) is asymptotically mean-square bounded.

To repress the modeling uncertainty (7), the adaptive law-based robust control term $h_r(t)$ is given as follows:

$$h_r(t)=\frac{{\hat{\rho }}^2{H}^{\mathrm{T}}{P}_{1}x(t)}{\hat{\rho }\parallel H^{\mathrm{T}}{P}_{1}x(t)\parallel +\sigma },$$

(20)

$$d\hat{\rho }=2{\gamma }_1(\parallel H^{\mathrm{T}}{P}_{1}x(t)\parallel -{\gamma }_2\hat{\rho })\mathrm{dt}.$$

(21)

Here, $\hat{\rho }$ is the modeling uncertainty bound estimation of $\rho$ , ${\gamma }_1$ , and ${\gamma }_2$ are positive design constants, $\sigma$ is a small positive design constant, and $P_1$ is the matrix to be designed, which will be determined later.

Considering the SDO (18) and robust control term (20), the composite controller is devised as follows

$${\tau }_c(t)={\hat{\Gamma }}^{-1}(\mathrm{Kx}(t)-\hat{b}(t)-h_r(t)),$$

(22)

where $K$ is coefficient matrix, which can be derived using the LMI.

On the basis of (22), one has

$$\begin{array}{c}\mathrm{dx}(t)=(G+\mathrm{HK})x(t)\mathrm{dt}+H\tilde{b}(t)\mathrm{dt}+H{\tilde{\Gamma }}^{\mathrm{T}}{\tau }_c(t)\mathrm{dt}\hfill \\ +H(\delta (t)-h_r(t))\mathrm{dt}.\hfill \end{array}$$

(23)

Combining (19) and (23), then

$$\begin{array}{c}d\overline{x}(t)=\overline{G}\overline{x}(t)\mathrm{dt}+{\overline{H}}_1{\tilde{\Gamma }}^{\mathrm{T}}{\tau }_c(t)\mathrm{dt}+{\overline{H}}_1\delta (t)\mathrm{dt}\hfill \\ -{\overline{H}}_2{h}_r(t)\mathrm{dt}+\underset{¯}{\Psi }d\omega (t),\hfill \end{array}$$

(24)

where

$$\begin{array}{c}\overline{G}=\left[\begin{array}{cc}G+\mathrm{HK}\hfill & H\hfill \\ \multicolumn{1}{c}{O_{3\times 6}}& -T^{-1}-\mathrm{LH}\hfill \end{array}\right],{\overline{H}}_1=\left[\begin{array}{c}H\hfill \\ \multicolumn{1}{c}{-\mathrm{LH}}\end{array}\right],\hfill \\ {\overline{H}}_2=\left[\begin{array}{c}H\hfill \\ \multicolumn{1}{c}{O_{3\times 3}}\end{array}\right],\underset{¯}{\Psi }=\left[\begin{array}{c}{O}_{6\times 3}\hfill \\ \multicolumn{1}{c}{\Psi }\end{array}\right],\overline{x}(t)=\left[\begin{array}{c}x(t)\hfill \\ \multicolumn{1}{c}{\tilde{b}(t)}\end{array}\right].\hfill \end{array}$$

The following results can be obtained through the stability analysis for the system (24).

• Theorem 1: Under Assumption 1–3, consider system (14) with disturbances and thruster faults, for given constants $\lambda >0$ , $\alpha >0$ , $\beta >0$ , if $\exists Q_1>0$ , $P_2>0$ , constant ${\gamma }_3>0$ , and matrix $R_1$ and $R_2$ , satisfying

$$\Xi =\left[\begin{array}{c}\begin{array}{cccc}{\Pi }_1\hfill & H\hfill & O\hfill & O\hfill \\ \multicolumn{1}{c}{*}& {\Pi }_2\hfill & O\hfill & -R_{2}H\hfill \\ \multicolumn{1}{c}{*}& *\hfill & -{\gamma }_3^{2}I\hfill & O\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & -{\gamma }_3^{2}I\hfill \end{array}\hfill \end{array}\right]<0,$$

(25)

$$\Sigma +{\Sigma }^{\mathrm{T}}+2\alpha P_2<0$$

(26)

$$\left[\begin{array}{cc}-\beta P_2\hfill & \Sigma \hfill \\ \multicolumn{1}{c}{*}& -\beta P_2\hfill \end{array}\right]<0$$

(27)

where

$$\begin{array}{c}{\Pi }_1=G{Q}_1+Q_1{G}^{\mathrm{T}}+H{R}_1+R_1^{\mathrm{T}}{H}^{\mathrm{T}},\hfill \\ {\Pi }_2=\Sigma +{\Sigma }^{\mathrm{T}},\hfill \\ \Sigma =-P_2{T}^{-1}-R_{2}H,\hfill \end{array}$$

and the adaptive law satisfies

$$\stackrel{·}{\hat{\Gamma }}=2\lambda {\tau }_c{\overline{x}}^{\mathrm{T}}P{H}_1,$$

(28)

then all states of system (24) are asymptotically mean-square bounded by adjusting the gain $L$ of SDO (18) with $L=P_2^{-1}{R}_2$ and solving the gain $K$ of the controller (20) with $K=R_1{Q}_1^{-1}$ , where the characteristic values of $-T^{-1}-\mathrm{LH}$ belong to the LMI region $\mathcal{D}(\alpha ,\beta )$ .

Proof: Select the following Lyapunov function

$$V(\overline{x}(t),t)={\overline{x}}^{\mathrm{T}}(t)P\overline{x}(t)+\mathrm{Tr}\left(\frac{{\tilde{\Gamma }}^{\mathrm{T}}\tilde{\Gamma }}{2\lambda }\right)+\frac{1}{2{\gamma }_1}{\tilde{\rho }}^2,$$

(29)

and

$$P=\left[\begin{array}{cc}{P}_1\hfill & O\hfill \\ \multicolumn{1}{c}{O}& P_2\hfill \end{array}\right]=\left[\begin{array}{cc}{Q}_1^{-1}\hfill & O\hfill \\ \multicolumn{1}{c}{O}& P_2\hfill \end{array}\right]>0.$$

(30)

the derivative of (29) is

$$\begin{array}{c}\mathcal{L}V(\overline{x}(t),t)=\frac{\partial V}{\partial \overline{x}}(\overline{G}\overline{x}(t)+{\overline{H}}_1{\tilde{\Gamma }}^{\mathrm{T}}{\tau }_c(t)+{\overline{H}}_1\delta (t)\hfill \\ -{\overline{H}}_2{h}_r(t))+\mathrm{Tr}({\underset{¯}{\Psi }}^{\mathrm{T}}P\underset{¯}{\Psi })\hfill \\ -\mathrm{Tr}\left(\frac{{\tilde{\Gamma }}^{\mathrm{T}}\stackrel{·}{\hat{\Gamma }}}{\lambda }\right)-\frac{1}{{\gamma }_1}\tilde{\rho }\stackrel{·}{\hat{\rho }}\hfill \\ ={\overline{x}}^{\mathrm{T}}(t)(P\overline{G}+{\overline{G}}^{\mathrm{T}}P)\overline{x}(t)+2{\overline{x}}^{\mathrm{T}}(t)P{\overline{H}}_1\delta (t)\hfill \\ -2{\overline{x}}^{\mathrm{T}}(t)P{\overline{H}}_2{h}_r(t)+\mathrm{Tr}(2{\overline{x}}^{\mathrm{T}}(t)P{\overline{H}}_1{\tilde{\Gamma }}^{\mathrm{T}}{\tau }_c(t))\hfill \\ -\mathrm{Tr}\left(\frac{{\tilde{\Gamma }}^{\mathrm{T}}\stackrel{·}{\hat{\Gamma }}}{\lambda }\right)-\frac{1}{{\gamma }_1}\tilde{\rho }\stackrel{·}{\hat{\rho }}+\mathrm{Tr}({\underset{¯}{\Psi }}^{\mathrm{T}}P\underset{¯}{\Psi })\hfill \\ ={\overline{x}}^{\mathrm{T}}(t)(P\overline{G}+{\overline{G}}^{\mathrm{T}}P)\overline{x}(t)+2{\overline{x}}^{\mathrm{T}}(t)P({\overline{H}}_1-{\overline{H}}_2)\delta (t)\hfill \\ +2x^{\mathrm{T}}(t)P_{1}H\delta (t)+\mathrm{Tr}({\underset{¯}{\Psi }}^{\mathrm{T}}P\underset{¯}{\Psi })\hfill \\ -2x^{\mathrm{T}}(t)P_{1}H\frac{{\hat{\rho }}^2{H}^{\mathrm{T}}{P}_{1}x(t)}{\hat{\rho }\parallel H^{\mathrm{T}}{P}_{1}x(t)\parallel +\sigma }\hfill \\ -\frac{1}{{\gamma }_1}\tilde{\rho }\stackrel{·}{\hat{\rho }}+\mathrm{Tr}(2{\overline{x}}^{\mathrm{T}}(t)P{\overline{H}}_1{\tilde{\Gamma }}^{\mathrm{T}}{\tau }_c(t))-\mathrm{Tr}\left(\frac{{\tilde{\Gamma }}^{\mathrm{T}}\stackrel{·}{\hat{\Gamma }}}{\lambda }\right)\hfill \end{array}$$

(31)

It can be obtained from Young’s inequality

$$\begin{array}{c}\mathcal{L}V(\overline{x}(t),t)\le {\overline{x}}^{\mathrm{T}}(t)\hfill \\ (P\overline{G}+{\overline{G}}^{\mathrm{T}}P+\frac{1}{{\gamma }_3^2}P({\overline{H}}_1-{\overline{H}}_2)({\overline{H}}_1-{\overline{H}}_2{)}^{\mathrm{T}}P)\overline{x}(t)\hfill \\ +{\gamma }_3^2\parallel \delta (t){\parallel }^2+\mathrm{Tr}({\underset{¯}{\Psi }}^{\mathrm{T}}P\underset{¯}{\Psi })\hfill \\ +2x^{\mathrm{T}}(t)P_{1}H\delta (t)-2x^{\mathrm{T}}(t)P_{1}H\frac{{\hat{\rho }}^2{H}^{\mathrm{T}}{P}_{1}x(t)}{\hat{\rho }\parallel H^{\mathrm{T}}{P}_{1}x(t)\parallel +\sigma }\hfill \\ -\frac{1}{{\gamma }_1}\tilde{\rho }\stackrel{·}{\hat{\rho }}+\mathrm{Tr}(2{\overline{x}}^{\mathrm{T}}(t)P{\overline{H}}_1{\tilde{\Gamma }}^{\mathrm{T}}{\tau }_c(t))-\mathrm{Tr}\left(\frac{{\tilde{\Gamma }}^{\mathrm{T}}\stackrel{·}{\hat{\Gamma }}}{\lambda }\right)\hfill \end{array}$$

(32)

According to $\tilde{\rho }=\rho -\hat{\rho }$ , and the square inequality, we have

$$\begin{array}{c}2x^{\mathrm{T}}(t)P_{1}H\delta (t)-2x^{\mathrm{T}}(t)P_{1}H\frac{{\hat{\rho }}^2{H}^{\mathrm{T}}{P}_{1}x(t)}{\hat{\rho }\parallel H^{\mathrm{T}}{P}_{1}x(t)\parallel +\sigma }-\frac{1}{{\gamma }_1}\tilde{\rho }\stackrel{·}{\hat{\rho }}\hfill \\ \le 2\rho \parallel H^{\mathrm{T}}{P}_{1}x(t)\parallel -\frac{2{\hat{\rho }}^2\parallel H^{\mathrm{T}}{P}_{1}x(t){\parallel }^2}{\hat{\rho }\parallel H^{\mathrm{T}}{P}_{1}x(t)\parallel +\sigma }-\frac{1}{{\gamma }_1}\tilde{\rho }\stackrel{·}{\hat{\rho }}\hfill \\ =2\tilde{\rho }\parallel H^{\mathrm{T}}{P}_{1}x(t)\parallel +\frac{2\sigma \hat{\rho }\parallel H^{\mathrm{T}}{P}_{1}x(t)\parallel }{\hat{\rho }\parallel H^{\mathrm{T}}{P}_{1}x(t)\parallel +\sigma }\hfill \\ -2\tilde{\rho }(\parallel H^{\mathrm{T}}{P}_{1}x(t)\parallel -{\gamma }_2\hat{\rho })\le 2\sigma +2{\gamma }_2\tilde{\rho }\hat{\rho }\hfill \\ =2\sigma +{\gamma }_2({\rho }^2-{\tilde{\rho }}^2-{\hat{\rho }}^2)\le -{\gamma }_2{\tilde{\rho }}^2+2\sigma +{\gamma }_2{\rho }^2\hfill \end{array}$$

(33)

According to (28), we get

$$\begin{array}{c}2{\overline{x}}^{\mathrm{T}}(t)P{\overline{H}}_1{\tilde{\Gamma }}^{\mathrm{T}}{\tau }_c(t)-\mathrm{Tr}\left(\frac{{\tilde{\Gamma }}^{\mathrm{T}}\stackrel{·}{\hat{\Gamma }}}{\lambda }\right)\hfill \\ =\mathrm{Tr}(2{\overline{x}}^{\mathrm{T}}(t)P{\overline{H}}_1{\tilde{\Gamma }}^{\mathrm{T}}{\tau }_c(t))-\mathrm{Tr}\left(\frac{{\tilde{\Gamma }}^{\mathrm{T}}\stackrel{·}{\hat{\Gamma }}}{\lambda }\right)\hfill \\ =\mathrm{Tr}(2{\tilde{\Gamma }}^{\mathrm{T}}{\tau }_c(t){\overline{x}}^{\mathrm{T}}(t)P{\overline{H}}_1)-\mathrm{Tr}(2{\tilde{\Gamma }}^{\mathrm{T}}{\tau }_c(t){\overline{x}}^{\mathrm{T}}(t)P{\overline{H}}_1)=0\hfill \end{array}$$

(34)

From (32)-(34), it is obvious that

$$\mathcal{L}V\le {\overline{x}}^{\mathrm{T}}(t){\Xi }_1\overline{x}(t)-{\gamma }_2{\tilde{\rho }}^2+C$$

(35)

where

$$\begin{array}{c}{\Xi }_1=P\overline{G}+{\overline{G}}^{\mathrm{T}}P+\frac{1}{{\gamma }_3^2}P({\overline{H}}_1-{\overline{H}}_2)({\overline{H}}_1-{\overline{H}}_2{)}^{\mathrm{T}}P,\hfill \\ C={\gamma }_3^2\parallel \delta (t){\parallel }^2+\mathrm{Tr}({\underset{¯}{\Psi }}^{\mathrm{T}}P\underset{¯}{\Psi })+2\sigma +{\gamma }_2{\rho }^2.\hfill \end{array}$$

Our main results are as follows:

• (1):  ${\Xi }_1<0\iff {\Xi }_2<0$ . Considering (24), (35), and Schur complement lemma, ${\Xi }_1<0\iff {\Xi }_2<0$ , where

$${\Xi }_2=\left[\begin{array}{c}\begin{array}{cccc}{\Lambda }_1\hfill & P_{1}H\hfill & O\hfill & O\hfill \\ \multicolumn{1}{c}{*}& {\Lambda }_2\hfill & O\hfill & -P_2\mathrm{LH}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & -{\gamma }_3^{2}I\hfill & O\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & -{\gamma }_3^{2}I\hfill \end{array}\hfill \end{array}\right],$$

(36)

with

$$\begin{array}{c}{\Lambda }_1=P_{1}G+G^{\mathrm{T}}{P}_1+P_1\mathrm{HK}+K^{\mathrm{T}}{H}^{\mathrm{T}}{P}_1,\hfill \\ {\Lambda }_2=-P_2{T}^{-1}-T^{-1}{P}_2-P_2\mathrm{LH}-H^{\mathrm{T}}{L}^{\mathrm{T}}{P}_2,\hfill \end{array}$$

• (2):  ${\Xi }_2<0\iff {\Xi }_3<0$ . Multiplied by $\mathrm{diag}\{Q_1,I,I,I\}$ on both sides of the matrix ${\Xi }_2$ , yields

$${\Xi }_3=\left[\begin{array}{c}\begin{array}{cccc}{\Pi }_1\hfill & H\hfill & O\hfill & O\hfill \\ \multicolumn{1}{c}{*}& {\Pi }_2\hfill & O\hfill & -P_2\mathrm{LH}\hfill \\ \multicolumn{1}{c}{*}& *\hfill & -{\gamma }_3^{2}I\hfill & O\hfill \\ \multicolumn{1}{c}{*}& *\hfill & *\hfill & -{\gamma }_3^{2}I\hfill \end{array}\hfill \end{array}\right]<0,$$

(37)

here

$${\Pi }_1=G{Q}_1+Q_1{G}^{\mathrm{T}}+\mathrm{HK}{Q}_1+Q_1{K}^{\mathrm{T}}{H}^{\mathrm{T}},$$

$${\Pi }_2=-P_2{T}^{-1}-T^{-1}{P}_2-P_2\mathrm{LH}-H^{\mathrm{T}}{L}^{\mathrm{T}}{P}_2,$$

(3):  ${\Xi }_3<0\iff \Xi <0$ . For ${\Xi }_3<0$ , choosing $K=R_1{Q}_1^{-1}$ and $L=P_2^{-1}{R}_2$ , we have $\Xi <0$ .

So, one has $\Xi <0\iff {\Xi }_3<0\iff {\Xi }_2<0\iff {\Xi }_1<0$ .

When $\Xi <0$ , $\exists \theta >0$ , one has

$$\Xi <0\iff {\Xi }_1<0\Rightarrow {\Xi }_1+\theta I<0.$$

(38)

Based on (29), (35), and (38), choosing $\kappa ={\lambda }_{\min }(P)|\overline{x}{|}^p$ , $\vartheta =min\{\frac{\theta }{{\lambda }_{\max }(P)},{\gamma }_2\}$ , and $p=2$ , the following inequalities hold

$$\kappa (|\overline{x}{|}^p)={\lambda }_{min}(P)|\overline{x}{|}^2\le {\overline{x}}^T(t)P\overline{x}(t)+Tr\left(\frac{{\tilde{\Gamma }}^T\tilde{\Gamma }}{2\lambda }\right)+\frac{1}{2{\gamma }_1}{\tilde{\rho }}^2=V(\overline{x}(t),t),$$

(39)

$$\mathcal{L}V(\overline{x}(t),t)\le -\vartheta V(\overline{x}(t))+C.$$

(40)

That is

$$\mathrm{EV}(\overline{x}(t),t)\le V(\overline{x}(0))e^{-\vartheta t}+\frac{C}{\vartheta }$$

(41)

and

$$\underset{t\to \infty }{lim}supE|\overline{x}(t;t_0,{\overline{x}}_0){|}^p\le \frac{C}{\vartheta }.$$

(42)

In the light of Lemma 1, system (24) is asymptotically mean-square bounded.

The next step is considered the problem of regional pole constraints.

On account of Lemma 2, by selecting the symmetric positive definite matrix $\mathcal{P}$ as $P_2^{-1}$ , matrix $\mathcal{X}$ as $-T^{-1}-\mathrm{LH}$ , the characteristic values of $-T^{-1}-\mathrm{LH}$ belong to the region $\mathcal{D}(\alpha ,\beta )$ , when

$$\overline{\Sigma }+{\overline{\Sigma }}^{\mathrm{T}}+2\alpha P_2^{-1}<0$$

(43)

$$\left[\begin{array}{cc}-\beta P_2^{-1}\hfill & \overline{\Sigma }\hfill \\ \multicolumn{1}{c}{*}& -\beta P_2^{-1}\hfill \end{array}\right]<0$$

(44)

where $\overline{\Sigma }=(-T^{-1}-\mathrm{LH})P_2^{-1}$ .

Pre-multiplying and post-multiplying by $P_2$ in (43), by $\mathrm{diag}\{P_2,P_2\}$ in (44), yields

$$\Sigma +{\Sigma }^{\mathrm{T}}+2\alpha P_2<0$$

(45)

$$\left[\begin{array}{cc}-\beta P_2\hfill & \Sigma \hfill \\ \multicolumn{1}{c}{*}& -\beta P_2\hfill \end{array}\right]<0$$

(46)

where $\Sigma =-P_2{T}^{-1}-R_{2}H$ .

That is, if (26) and (27) hold, the characteristic values of $-T^{-1}-\mathrm{LH}$ belong to the region $\mathcal{D}(\alpha ,\beta )$ .

That is to say, the proof is done.

Simulation examples

Taking Northern Clipper with mass of $4.591\times {10}^6$  kg and length of $76.2$  m as an example to simulate different situations. ²² Based on Fossen, ²² the coefficient matrices in (6) are taken as follows:

$$M_0=\left[\begin{array}{ccc}5.3122\times {10}^6\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 8.2831\times {10}^6\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 3.7454\times {10}^6\hfill \end{array}\right],$$

$$D_0=\left[\begin{array}{ccc}5.0242\times {10}^6\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 2.7229\times {10}^5\hfill & -4.3933\times {10}^6\hfill \\ \multicolumn{1}{c}{0}& -4.3933\times {10}^6\hfill & 4.1894\times {10}^8\hfill \end{array}\right].$$

• Case 1:

$\mathrm{disturbance}:T=\mathrm{diag}\{1000,1000,1000\}\mathrm{and}\Psi =\mathrm{diag}\{130,100,80\}$

$$\mathrm{LOE}\mathrm{fault}:{\Gamma }_i=\{\begin{array}{c}1,0\le t<50\hfill \\ \multicolumn{1}{c}{0.8,t\ge 50}\end{array},i=1,2,3$$

$$uncertainterm:\Delta M=0\%M_{0}and\Delta D=0\%D_0$$

Selecting ${\eta }_0=[20\mathrm{m},20\mathrm{m},10\mathrm{rad}{]}^{\mathrm{T}}$ and ${\mathbf{v}}_0=[0\mathrm{m}/\mathrm{s},0\mathrm{m}/\mathrm{s},0\mathrm{rad}/\mathrm{s}{]}^{\mathrm{T}}$ , then $X(0)=[1,1,\pi /4,0,0,0{]}^{\mathrm{T}}$ . Selecting the designed parameters as $\alpha =10$ , $\beta =20$ , ${\gamma }_1=2\times {10}^4$ , ${\gamma }_2=1\times {10}^{-9}$ , and $\sigma =1\times {10}^3$ .

Based on Theorem 1, one has

$$Q_1=\left[\begin{array}{cccccc}0.6272\hfill & 0.0000\hfill & 0.0000\hfill & -0.2091\hfill & 0.0000\hfill & 0.0000\hfill \\ \multicolumn{1}{c}{0.0000}& 0.6272\hfill & 0.0000\hfill & 0.0000\hfill & -0.2091\hfill & 0.0001\hfill \\ \multicolumn{1}{c}{0.0000}& 0.0000\hfill & 0.6386\hfill & 0.0000\hfill & -0.0002\hfill & -0.1854\hfill \\ \multicolumn{1}{c}{-0.2091}& 0.0000\hfill & 0.0000\hfill & 0.6272\hfill & 0.0000\hfill & 0.0000\hfill \\ \multicolumn{1}{c}{0.0000}& -0.2091\hfill & -0.0002\hfill & 0.0000\hfill & 0.6272\hfill & 0.0000\hfill \\ \multicolumn{1}{c}{0.0000}& 0.0001\hfill & -0.1854\hfill & 0.0000\hfill & 0.0000\hfill & 0.2338\hfill \end{array}\right],$$

$$P_2=\left[\begin{array}{ccc}0.0217\hfill & 0.0000\hfill & 0.0000\hfill \\ \multicolumn{1}{c}{0.0000}& 0.0217\hfill & 0.0000\hfill \\ \multicolumn{1}{c}{0.0000}& 0.0000\hfill & 0.0116\hfill \end{array}\right],$$

$$R_1={\left[\begin{array}{ccc}-3.3423\times {10}^6\hfill & 0.0000\hfill & 0.0000\hfill \\ \multicolumn{1}{c}{0.0000}& -5.2525\times {10}^6\hfill & 4.5118\times {10}^5\hfill \\ \multicolumn{1}{c}{0.0000}& 8.1464\times {10}^5\hfill & -3.5183\times {10}^8\hfill \\ \multicolumn{1}{c}{-1.4459\times {10}^6}& 0.0000\hfill & 0.0000\hfill \\ \multicolumn{1}{c}{0.0000}& -2.1327\times {10}^6\hfill & -5.9087\times {10}^3\hfill \\ \multicolumn{1}{c}{0.0000}& -1.0331\times {10}^6\hfill & -4.5593\times {10}^8\hfill \end{array}\right]}^{\mathrm{T}},$$

$$R_2={\left[\begin{array}{ccc}0.0000\hfill & 0.0000\hfill & 0.0000\hfill \\ \multicolumn{1}{c}{0.0000}& 0.0000\hfill & 0.0000\hfill \\ \multicolumn{1}{c}{0.0000}& 0.0000\hfill & 0.0000\hfill \\ \multicolumn{1}{c}{1.6279\times {10}^6}& 0.0000\hfill & 0.0000\hfill \\ \multicolumn{1}{c}{0.0000}& 2.5382\times {10}^6\hfill & 0.0000\hfill \\ \multicolumn{1}{c}{0.0000}& 0.0000\hfill & 6.2247\times {10}^8\hfill \end{array}\right]}^{\mathrm{T}},$$

$$P=\left[\begin{array}{ccccccccc}1.7936\hfill & 0.0000\hfill & 0.0000\hfill & 0.5979\hfill & 0.0000\hfill & 0.0000\hfill & 0.0000\hfill & 0.0000\hfill & 0.0000\hfill \\ \multicolumn{1}{c}{0.0000}& 1.7936\hfill & -0.0002\hfill & 0.0000\hfill & 0.5979\hfill & -0.0011\hfill & 0.0000\hfill & 0.0000\hfill & 0.0000\hfill \\ \multicolumn{1}{c}{0.0000}& -0.0002\hfill & 2.0340\hfill & 0.0000\hfill & 0.0007\hfill & 1.6127\hfill & 0.0000\hfill & 0.0000\hfill & 0.0000\hfill \\ \multicolumn{1}{c}{0.5979}& 0.0000\hfill & 0.0000\hfill & 1.7937\hfill & 0.0000\hfill & 0.0000\hfill & 0.0000\hfill & 0.0000\hfill & 0.0000\hfill \\ \multicolumn{1}{c}{0.0000}& 0.5979\hfill & 0.0007\hfill & 0.0000\hfill & 1.7938\hfill & 0.0007\hfill & 0.0000\hfill & 0.0000\hfill & 0.0000\hfill \\ \multicolumn{1}{c}{0.0000}& -0.0011\hfill & 1.6127\hfill & 0.0000\hfill & 0.0007\hfill & 5.5563\hfill & 0.0000\hfill & 0.0000\hfill & 0.0000\hfill \\ \multicolumn{1}{c}{0.0000}& 0.0000\hfill & 0.0000\hfill & 0.0000\hfill & 0.0000\hfill & 0.0000\hfill & 0.0217\hfill & 0.0000\hfill & 0.0000\hfill \\ \multicolumn{1}{c}{0.0000}& 0.0000\hfill & 0.0000\hfill & 0.0000\hfill & 0.0000\hfill & 0.0000\hfill & 0.0000\hfill & 0.0217\hfill & 0.0000\hfill \\ \multicolumn{1}{c}{0.0000}& 0.0000\hfill & 0.0000\hfill & 0.0000\hfill & 0.0000\hfill & 0.0000\hfill & 0.0000\hfill & 0.0000\hfill & 0.0116\hfill \end{array}\right],$$

$$K={\left[\begin{array}{ccc}-6.8592\times {10}^6\hfill & 0.0000\hfill & 0.0000\hfill \\ \multicolumn{1}{c}{0}& -1.0695\times {10}^7\hfill & 1.3776\times {10}^6\hfill \\ \multicolumn{1}{c}{0}& -9.5523\times {10}^3\hfill & -1.4509\times {10}^9\hfill \\ \multicolumn{1}{c}{-4.5919\times {10}^6}& 0.0000\hfill & 0.0000\hfill \\ \multicolumn{1}{c}{0}& -6.9662\times {10}^6\hfill & -3.0627\times {10}^5\hfill \\ \multicolumn{1}{c}{0}& -4.4222\times {10}^6\hfill & -3.1007\times {10}^9\hfill \end{array}\right]}^{\mathrm{T}},$$

$$L={\left[\begin{array}{ccc}0.0000\hfill & 0.0000\hfill & 0.0000\hfill \\ \multicolumn{1}{c}{0.0000}& 0.0000\hfill & 0.0000\hfill \\ \multicolumn{1}{c}{0.0000}& 0.0000\hfill & 0.0000\hfill \\ \multicolumn{1}{c}{7.5017\times {10}^7}& 0.0000\hfill & 0.0000\hfill \\ \multicolumn{1}{c}{0.0000}& 1.1697\times {10}^8\hfill & 0.0000\hfill \\ \multicolumn{1}{c}{0.0000a}& 0.0000\hfill & 5.3661\times {10}^{10}\hfill \end{array}\right]}^{\mathrm{T}},$$

• Case 2:

$$\begin{array}{c}\mathrm{disturbance}:T=\mathrm{diag}\{1000,1000,1000\}\mathrm{and}\Psi \hfill \\ ={10}^6\times \mathrm{diag}\{130,100,80\}\hfill \end{array}$$

$$\mathrm{LOE}\mathrm{fault}:{\Gamma }_i=\{\begin{array}{c}1,0\le t<50\hfill \\ \multicolumn{1}{c}{0.3,t\ge 50}\end{array},i=1,2,3$$

$$uncertainterm:\Delta M=10\%M_{0}and\Delta D=10\%D_0$$

• Case 3:

$$\begin{array}{c}\mathrm{disturbance}:T=\mathrm{diag}\{1000,1000,1000\}\mathrm{and}\Psi \hfill \\ ={10}^6\times \mathrm{diag}\{150,130,4000\}\hfill \end{array}$$

$$\mathrm{LOE}\mathrm{fault}:{\Gamma }_i=\{\begin{array}{c}1,0\le t<50\hfill \\ \multicolumn{1}{c}{0.25,t\ge 50}\end{array},i=1,2,3$$

$$uncertainterm:\Delta M=15\%M_{0}and\Delta D=15\%D_0$$

The initial values and design control parameters are the same as case 1. In Figures 2 to 4, Figures 2 to 4(a) is the ship’s trajectory from initial position $(20\mathrm{m},20\mathrm{m})$ . The curves of ship position are illustrated in Figures 2 to 4(b). Figures 2 to 4(c) represents the responses of ship velocity. The control input can be seen in Figures 2 to 4(d). Figures 2 to 4(e) and (f) show the curves of estimation for the disturbances and faults. Furthermore, Table 1 indicates the performance of the controlled ship at $100\mathrm{s}$ .

Figure 2.

Simulation results under CADC and CHADC for case 1: (a) trajectory of the ship in plan- $\mathrm{xy}$ , (b) responses of ship position, (c) responses of ship velocity, (d) curves of the control input, (e) curves of the disturbance estimation, and (f) curves of the fault estimation.

Figure 3.

Simulation results under CADC and CHADC for case 2: (a) trajectory of the ship in plan- $\mathrm{xy}$ , (b) responses of ship position, (c) responses of ship velocity, (d) curves of the control input, (e) curves of the disturbance estimation, and (f) curves of the fault estimation.

Figure 4.

Simulation results under CADC and CHADC for case 3: (a) trajectory of the ship in plan- $\mathrm{xy}$ , (b) responses of ship position, (c) responses of ship velocity, (d) curves of the control input, (e) curves of the disturbance estimation, and (f) curves of the fault estimation.

Table 1.

Performance indices under ${\tau }_{\mathrm{CADC}}$ and ${\tau }_{\mathrm{CHADC}}$ .

Case	Case 1		Case 2		Case 3
Controller	${\tau }_{\mathrm{CADC}}$	${\tau }_{\mathrm{CHADC}}$	${\tau }_{\mathrm{CADC}}$	${\tau }_{\mathrm{CHADC}}$	${\tau }_{\mathrm{CADC}}$	${\tau }_{\mathrm{CHADC}}$
Settling time $t(s)$	36.2	36.1	36.3	46.6	36.5	47.2
${\int }_0^{100}|x(s)|\mathrm{ds}$	34.1531	35.0623	38.3493	68.5296	40.5499	81.5761
${\int }_0^{100}|y(s)|\mathrm{ds}$	34.1539	35.0657	36.3498	51.5139	37.4709	60.8935
${\int }_0^{100}|\psi (s)|\mathrm{ds}$	28.9219	33.5340	28.9369	33.5930	29.3148	36.8310

In Figures 2(a) and 3(a), it is shown that the CADC can make the ship stay in desired position $(0\mathrm{m},0\mathrm{m})$ and maintain a fixed attitude steadily. In of Figures 2 to 4(b) and (c), we know that the CADC scheme presented in this paper can make the states of DP ship asymptotically mean-square bounded. As can be seen from Figures 2(e) to (f), 3(e) to (f), and 4(e) to (f), the estimations of disturbances and faults in this paper are satisfactory. It can be concluded from Figures 2(a) to (f), 3(a) to (f), and 4(a) to (f) that the proposed method can effectively compensate the disturbances and faults at the same time. Therefore, under the condition of thruster faults and environmental disturbances at the same time, the DP ship can still complete the marine task excellently by using the proposed composite controller.

Conclusions

For the dynamic positioning ship systems with ocean disturbances and modeling uncertainties under thruster faults, an adaptive law-based SDO is raised to evaluate the disturbances and faults simultaneously. Based on the estimation, the CADC strategy is put forward by using disturbance observer-based control, robust control term and adaptive technology to make the closed-loop system asymptotically mean-square bounded at the same time. The controlled ship can stay in desired position and maintain a fixed attitude. The next work in the future is the anti-disturbance control for the dynamic positioning ship systems with multiple disturbances and thruster faults under input saturation.