4.1. Neural network approximator

According to the universal approximation property, radial basis function neural networks (RBFNNs) are widely applied to compensate structured uncertainties due to the fact that RBFNNs are capable to approximate smooth uncertain functions in many nonlinear systems such as autonomous surface vessels. To this end, a nonlinear function is approximated as follows:

(17) \begin{equation} \hat{f}_{i}(\upsilon _{i})\;:\!=\;{\hat{\theta }_{i}^T}\Phi _{i}(\upsilon _{i}), \end{equation}

where $\upsilon _{i} \in{\Omega _{\upsilon i}} \subset{R^q}$ is the input vector and $q$ is the NN input dimension, $\hat \theta _{i} ={[{\hat \theta _{1i}},{\hat \theta _{2i}},\ldots,{\hat \theta _{li}}]^T} \in{R^l}$ is the NN weights vector and $l \gt 1$ is the NN number. The regression vector $\Phi _{i} (\upsilon _{i}) ={[{\varphi _{1i}}(\upsilon _{i}),\ldots,{\varphi _{li}}(\upsilon _{li})]^T}$ is the basis function vector where ${\varphi _i}(\upsilon _{i})$ is defined as ${\varphi _i}(\upsilon _{i}) = \exp\![{{\|\upsilon _{i} -{\mu _i}}\|^2}/{c _i}]$ , $i = 1,\ldots,l$ ; where, ${\mu _i} ={[{\mu _{i1}},\ldots,{\mu _{iq}}]^T}$ is the center of Gaussian functions and ${c _i} \gt 0$ is the width of them. For any unknown nonlinear function $f_{i}(\upsilon _{i})$ that contains $C_{i}(\upsilon _{i})$ and $D_{i}(\upsilon _{i})$ , there exists the NN function ${{\theta _{i}}^*{^T}}\Phi _{i}(\upsilon _{i})$ in a sense that

(18) \begin{equation} f_{i}(\upsilon _{i})\;:\!=\;{\theta _{i}^{{*^T}}}\Phi _{i}(\upsilon _{i}) +{\delta _{i}^*}(\upsilon _{i}),\quad \forall \upsilon _{i} \in{\Omega _{\upsilon i} } \in{R^q}, \end{equation}

where ${\delta _{i}^*}(\upsilon _{i})$ is the minimum approximation error and $\theta _{i}^*$ is the ideal constant vector defined as:

(19) \begin{equation} {\theta _{i}^*}\;:\!=\; \arg \;{\min _{\hat \theta _{i} \in{R^l}}}\ {{{{{\sup }_{\upsilon _{i} \in{\Omega _{\upsilon i}}}}} f(\upsilon _{i}) -{{\hat \theta _{i} }^T}\Phi _{i} (\upsilon _{i})} }.\ \end{equation}

4.2. HGO

In some practical systems, the limitations and environmental disturbances together with technological limitations make accurate velocity measurements of surface vessels difficult [Reference Sarrafan44]. A HGO is applied to estimate these unmeasurable states, referred to as $\upsilon _i$ for $i = 1,\ldots,N$ . The dynamics of HGO are expressed as follows:

(20) \begin{equation} \left \{ \begin{array}{l}{m_i}{{\dot \lambda }_{i1}} ={\lambda _{i2}}\\[5pt]{m_i}{{\dot \lambda }_{i2}} = -{m_{1i}}{\lambda _{i2}} -{\lambda _{i1}} +{\eta _i} \end{array} \right ., \end{equation}

where ${\lambda _{i1}},{\lambda _{i2}} \in{R^n}$ are the state vectors of HGO, $m_{1i}$ and $m_i$ are the small positive constants. To prevent peaking, the saturation functions are applied to a given observer signal when it is outside the preferable range $\Omega$ as [Reference Khalil46]:

(21) \begin{equation} {\lambda _{ij}} ={\textrm{B}_{ij}}\Delta \!\left(\frac{{{\lambda _{ij}}}}{{{\textrm{B}_{ij}}}}\right),\quad{\textrm{B}_{ij}} \ge \mathop{\max\! ({\lambda _{ij}})}\limits _{\upsilon \in{\Omega _{\upsilon i}} }, \ \end{equation}

where

\begin{align*} \Delta (a) = \left \{ \begin{array}{l} - 1\quad \quad for\;a \lt - 1\\[5pt] a\quad \quad \;for\;\left | a \right | \le + 1\\[5pt] + 1\quad \quad for\;a \gt + 1 \end{array} \right. \end{align*}

Consequently, the following equation can be proved [Reference Tee45]:

(22) \begin{equation} {\Lambda _{i2}} = \frac{{{\lambda _{2i}}}}{{{m_i}}} -{\upsilon _i} ={m_i}\textrm{X}_i^{(2)} \end{equation}

where ${\textrm{X}_i} ={\lambda _{i2}} +{m_i}{\lambda _{i1}}$ and there exist positive constants $t^*$ and $g_{i2}$ in a sense that $\forall \;t \gt{t^*}$ , $\left \|{{\Lambda _{i2}}} \right \| \le{\kappa _i}{g_{i2}}$ holds true.

4.3. Output feedback adaptive-neural formation controller design

To design DSC for the formation control of surface vessels, the formation tracking errors are defined for the first and second steps through the state variables $\eta _i$ and $\upsilon _i$ as:

(23) \begin{equation} \left \{ \begin{array}{l}{z_{1i}} ={\eta _i} -{\eta _0}\\[5pt]{z_{2i}} ={\upsilon _i} -{\textrm{P}_{iF}} \end{array} \right .\quad i = 1,\ldots,N. \end{equation}

Let ${z_{1i}} ={[{z_{11i}},{z_{12i}},{z_{13i}}]^T}$ and ${z_{2i}} ={[{z_{21i}},{z_{22i}},{z_{23i}}]^T}$ . With respect to the suggested HGO, the estimation of second surface errors in (23) is rewritten as:

(24) \begin{equation} {\hat z_{2i}} ={\hat \upsilon _i} -{\textrm{P}_{iF}} ={J^{ - 1}}({\psi _i})\frac{{{\lambda _{2i}}}}{{{m_i}}} -{\textrm{P}_{iF}}, \end{equation}

where $\textrm{P}_{iF}$ is the virtual control signal that is passed through the following first-order filter:

(25) \begin{equation} {\alpha _i}{\dot{\textrm{P}}_{iF}} +{\textrm{P}_{iF}} ={\textrm{P}_i}\quad \quad{\textrm{P}_{iF}}(0) ={\textrm{P}_i}(0), \end{equation}

where $\textrm{P}_i$ is the virtual controller and $\alpha _i$ is a design parameter. Consequently, the following error is defined for each surface vessel in the formation control:

(26) \begin{equation} {\textrm{P}_{ie}} ={\textrm{P}_{iF}} -{\textrm{P}_i}. \end{equation}

Next, one has the following equation through (25) and (26):

(27) \begin{equation} {\dot{\textrm{P}}_{ie}} ={\dot{\textrm{P}}_{iF}} -{\dot{\textrm{P}}_i} = \frac{1}{{{\alpha _i}}}{\textrm{P}_{ie}} -{\dot{\textrm{P}}_i}. \end{equation}

Consequently, one may find that

(28) \begin{equation} \left \|{{{\dot{\textrm{P}}}_{ie}} + \frac{{{\textrm{P}_{ie}}}}{{{\alpha _i}}}} \right \| \le \varpi, \end{equation}

where $\varpi$ is the continuous function of closed-loop variables and their derivatives that is used to derive the following inequality, where will be used in the stability analysis in the sequel:

(29) \begin{equation} \textrm{P}_{ie}^T{\dot{\textrm{P}}_{ie}} \le - \frac{1}{{{\alpha _i}}}{\left \|{{\textrm{P}_{ie}}} \right \|^2} +{\left \|{{\textrm{P}_{ie}}} \right \|^2} + \frac{1}{4}{\varpi ^2}. \end{equation}

In the second step, the virtual surface error of each surface vessel is obtained by

(30) \begin{equation} {\tilde z_{2i}} ={\hat z_{2i}} -{z_{2i}} ={J^{ - 1}}({\psi _i})\frac{{{\lambda _{2i}}}}{{{m_i}}} -{\textrm{P}_{iF}} -{\upsilon _i} +{\textrm{P}_{iF}} ={\Lambda _{i2}}. \end{equation}

With respect to the conditions of HGO and $\left \|{{J^{ - 1}}} \right \| \le{S_i}$ where ${S_i} \in{\Re ^ + }$ is an unknown constant, the following inequalities are of concern:

(31) \begin{equation} \left \|{{{\tilde z}_2}} \right \| = \left \|{{J^{ - 1}}(\eta )\left(\frac{{{\lambda _{i2}}}}{{{m_i}}} - \dot \eta \right)} \right \| = \left \|{{J^{ - 1}}(\eta ){m_i}\ddot \chi } \right \| \le{m_i}{P_2}\!\left \|{{J^{ - 1}}(\eta )} \right \| \le{m_i}{P_2}{S_i}\;:\!=\;{S_2}, \end{equation}

(32) \begin{equation} {\left \|{{{\tilde z}_2}} \right \|^2} ={\left \|{{J^{ - 1}}(\eta )\left(\frac{{{\lambda _{i2}}}}{{{m_i}}} - \dot \eta \right)} \right \|^2} ={\left \|{{J^{ - 1}}(\eta ){m_i}\ddot \chi } \right \|^2} \le m_i^2P_2^2{\left \|{{J^{ - 1}}(\eta )} \right \|^2} \le m_i^2P_2^2S_i^2\;:\!=\; S_2^2. \end{equation}

By replacing ${\hat \upsilon _i} ={\hat z_{2i}} +{\textrm{P}_{iF}}$ and derivation from the calculated tracking errors, the following equations are obtained:

(33) \begin{equation} \begin{array}{l}{\dot z_{1i}} ={J_i}({\hat z_{2i}} +{\textrm{P}_{iF}}) -{\dot \eta _0},\\\\ \quad \quad{\dot z_{2i}} ={\dot{ \hat{ \upsilon }}}_i -{\dot{\textrm{P}}_{iF}}. \end{array} \end{equation}

At this step, to facilitate discussion on the limited FOV constraint requirements of the formation tracking errors, the symmetric BLF is introduced for each surface vessel [Reference Huang, Zhang and Zhang22] as follows:

(34) \begin{equation} {\Upsilon _{1i}} = \frac{1}{2}\log \frac{{\beta _{1i}^2}}{{\beta _{1i}^2 - z_{1i}^T{z_{1i}}}}, \quad \left \|{{z_{1i}}(0)} \right \| \lt{\beta _{1i}}. \end{equation}

The derivative of the BLF is calculated based on the formation tracking errors of each surface vessel. By inserting the calculated dynamic error in the derivative of the BLF, the following equation is obtained:

(35) \begin{equation} {\dot \Upsilon _{1i}} = \frac{{z_{1i}^T{{\dot z}_{1i}}}}{{\beta _{1i}^2 - z_{1i}^T{z_{1i}}}}= \frac{{z_{1i}^T({J_i}({\psi _i})({{\hat z}_{2i}} +{\textrm{P}_i}) -{{\dot \eta }_0})}}{{\beta _{1i}^2 - z_{1i}^T{z_{1i}}}}. \end{equation}

Consequently, the virtual control signals are designed according to the BLF derivative and finite-time approach on the formation control of autonomous surface vessels as

(36) \begin{equation} {\textrm{P}_i} ={J_i^{-1}}({\psi _i})({\dot \eta _0} -{c_{1i}}{z_{1i}}(\beta _{1i}^2 - z_{1i}^T{z_{1i}}) -{c_{2i}}{{\textrm{sgn}} ^\alpha }({z_{1i}})), \end{equation}

where $c_{1i}$ and $c_{2i}$ are the designed positive parameters, ${{\textrm{sgn}} ^\alpha }({z_{1i}}) ={\left |{{z_{1i}}} \right |^\alpha }sign({z_{1i}})$ which $0.5 \lt \alpha \lt 1$ . With substituting the virtual control (36) into (35), the following equation is obtained:

(37) \begin{equation} {\dot \Upsilon _{1i}} = \frac{{{J_i}{z_{1i}}{{\hat z}_{2i}} -{c_{1i}}z_{1i}^2(\beta _{1i}^2 - z_{1i}^T{z_{1i}}) -{c_{2i}}{z_{1i}}{{{\textrm{sgn}} }^\alpha }({z_{1i}})}}{{\beta _{1i}^2 - z_{1i}^T{z_{1i}}}}. \end{equation}

With respect to equation ${z_{1i}}sign({z_{1i}}) = \left |{{z_{1i}}} \right |$ , term ${c_{2i}}{\left |{{z_{1i}}} \right |^{\alpha + 1}}$ is replaced into ${c_{2i}}{z_{1i}}{{\textrm{sgn}} ^\alpha }({z_{1i}})$ ; thus, the following equation is achieved:

(38) \begin{equation} {\dot \Upsilon _{1i}} = -{c_{1i}}z_{1i}^2 + \frac{{{J_i}{z_{1i}}{{\hat z}_{2i}}}}{{\beta _{1i}^2 - z_{1i}^T{z_{1i}}}} - \frac{{{c_{2i}}{{\left |{{z_{1i}}} \right |}^{\alpha + 1}}}}{{\beta _{1i}^2 - z_{1i}^T{z_{1i}}}}. \end{equation}

By applying a simple math calculations, the following equation is obtained:

(39) \begin{equation} {\dot \Upsilon _{1i}} = -{c_{1i}}z_{1i}^2 + \frac{{{J_i}{z_{1i}}{{\hat z}_{2i}}}}{{\beta _{1i}^2 - z_{1i}^T{z_{1i}}}} -{c_{2i}}{(\beta _{1i}^2 - z_{1i}^T{z_{1i}})^{\frac{{\alpha - 1}}{2}}}{\left(\frac{{z_{1i}^2}}{{\beta _{1i}^2 - z_{1i}^T{z_{1i}}}}\right)^{\frac{{\alpha + 1}}{2}}}. \end{equation}

According to Lemma 1, we have $\dfrac{1}{2}\log \dfrac{{\beta _{1i}^2}}{{\beta _{1i}^2 - z_{1i}^T{z_{1i}}}} \lt \dfrac{{z_{1i}^2}}{{2(\beta _{1i}^2 - z_{1i}^T{z_{1i}})}}$ . Let ${\rho _{1i}} = \beta _{1i}^2 - z_{1i}^T{z_{1i}}$ which is positive, then, the following equation is obtained:

(40) \begin{equation} {\dot \Upsilon }_{1i}=- 2{c_{1i}}{\rho _{1i}}{\Upsilon _{1i}} -{c_{2i}}{\rho _{1i}}^{\frac{{\alpha - 1}}{2}}{(2{\Upsilon _{1i}})^{\frac{{\alpha + 1}}{2}}} + \frac{{{J_i}{z_{1i}}{{\hat z}_{2i}}}}{{\beta _{1i}^2 - z_{1i}^T{z_{1i}}}}. \end{equation}

After designing the virtual control singals for each surface vessel in step 1, the quadratic Lyapunov function is proposed in step 2 as follows:

(41) \begin{equation} {\Upsilon _{2i}} ={\Upsilon _{1i}} + \frac{1}{2}z_{2i}^T{M_i}\,{z_{2i}} + \frac{1}{{2{\gamma _i}}}\tilde \theta _i^T{\tilde \theta _i} + \frac{1}{2}\textrm{P}_{ie}^T\textrm{P}_{ie}, \end{equation}

where ${\gamma _i} \gt 0$ is a design parameter and ${\tilde \theta _i} = \theta _i^* -{\hat \theta _i}$ . By calculating the derivative of Lyapunov function in step 2, the following equation is obtained:

(42) \begin{equation} {\dot \Upsilon _{2i}} ={\dot \Upsilon _{1i}} + z_{2i}^T{M_i}\,{\dot z_{2i}} - \frac{1}{{{\gamma _i}}}\tilde \theta _i^T{\dot{ \hat{\theta }} _i} + \textrm{P}_{ie}^T{\dot{\textrm{P}}_{ie}}. \end{equation}

By inserting the error dynamic into this derivative, the following result is achieved:

(43) \begin{align} {{\dot \Upsilon }_{2i}} & = - 2{c_{1i}}{\rho _{1i}}{\Upsilon _{1i}} -{c_{2i}}{\rho _{1i}}^{\frac{{\alpha - 1}}{2}}{(2{\Upsilon _{1i}})^{\frac{{\alpha + 1}}{2}}} + \frac{{{J_i}z_{1i}^T{{\hat z}_{2i}}}}{{\beta _{1i}^2 - z_{1i}^T{z_{1i}}}}\nonumber \\[5pt] & \quad + z_{2i}^T({ -{C_i}({{\upsilon _i}}){z_{2i}} -{D_i}({{\upsilon _i}}){z_{2i}} +{\tau _i} + f_i^*({\upsilon _i})}) - \frac{1}{{{\gamma _i}}}\tilde \theta _i^T{{\dot{\hat{\theta }}}_i} + \textrm{P}_{ie}^T{{\dot{\textrm{P}}}_{ie}}, \end{align}

where $f_i^*({\upsilon _i}) = -{C_i}({\upsilon _i}){\textrm{P}_{iF}} -{D_i}({{\upsilon _i}}){\textrm{P}_{iF}} -{G_i}\!\left ({{\eta _i}} \right ) -{M_i}{\dot{\textrm{P}}_{iF}}.$ By inserting $f_i^*({\upsilon _i}) = \theta _i^{{*^T}}{\varphi _i} +{\varepsilon _i}$ into Eq. (43), where $\varepsilon _i$ is the approximator error in the derivative of Lyapunov function in step 2, the following is yield:

(44) \begin{align} {{\dot{\Upsilon }}_{2i}} & = - 2{c_{1i}}{\rho _{1i}}{\Upsilon _{1i}} -{c_{2i}}{\rho _{1i}}^{\frac{{\alpha - 1}}{2}}{(2{\Upsilon _{1i}})^{\frac{{\alpha + 1}}{2}}} + \frac{{{J_i}z_{1i}^T{{\hat{z}}_{2i}}}}{{\beta _{1i}^2 - z_{1i}^T{z_{1i}}}}\nonumber \\[5pt] & \quad + z_{2i}^T\!\left ({ -{C_i}\!\left ({{\upsilon _i}} \right ){z_{2i}} -{D_i}({{\upsilon _i}}){z_{2i}} +{\tau _i} + \tilde \theta _i^T{\varphi _i} + \hat{\theta }_i^T{\varphi _i} +{\varepsilon _i}} \right ) - \frac{1}{{{\gamma _i}}}\tilde \theta _i^T{{\dot{\hat{\theta }}}_i} \nonumber \\[5pt] & \quad - \left ({\frac{1}{{{\alpha _i}}} - 1 - 0.5{\lambda _{\max }}\{{J_i}({\psi _i})\} } \right ){\left \|{{\textrm{P}_{ie}}} \right \|^2} + \frac{1}{4}{\varpi ^2}. \end{align}

The sigma-modification structure is utilized as a robust adaptive law until it makes the estimated parameters to become fixed at specific points. The adaptive laws are chosen for each surface vessel as follows:

(45) \begin{equation} \dot{\hat{\theta } } _i={\gamma _i}z_{2i}^T{\varphi _i} -{\gamma _i}{\delta _i}({\hat \theta _i} -{\theta _{i0}}), \end{equation}

where ${\gamma _i} \gt 0$ and ${\delta _i} \gt 0$ are the design parameters. The parameters ${\theta _{i0}} \in{R^3}$ are priori estimates of the parameters $\theta _i$ . Consequently, the formation control signal for each surface vessel is designed as:

(46) \begin{equation} {\tau _i} = -{c_{3i}}{\hat z_{2i}} -{c_{4i}}si{g^\alpha }({\hat z_{2i}}) - \hat \theta _i^T{\varphi _i} - \frac{{{J_i}{z_{1i}}}}{{\beta _{1i}^2 - z_{1i}^T{z_{1i}}}} -{\hat \tau _{di}}, \end{equation}

where parameters $c_{3i}$ and $c_{4i}$ are the designed positive constants and ${\hat \tau _{di}}$ will be designed as the finite-time disturbance observer in the next step. With respect to $z_{2i}^T\!\left ({{{\dot M}_i} - 2{C_i}\!\left ({{\upsilon _i}} \right )} \right ){z_{2i}} = 0$ and by substituting the control signals into the derivative of Lyapunov function, the following equation is obtained:

(47) \begin{equation} \begin{array}{l}{{\dot \Upsilon }_{2i}} = - 2{c_{1i}}{\rho _{1i}}{\Upsilon _{1i}} -{c_{2i}}{\rho _{1i}}^{\frac{{\alpha - 1}}{2}}{(2{\Upsilon _{1i}})^{\frac{{\alpha + 1}}{2}}} - z_{2i}^T{c_{3i}}{z_{2i}} - z_{2i}^T{c_{3i}}{{\tilde z}_{2i}}\\\\ \,\quad \quad \; -{c_{4i}}{\left ({{{\left \|{{z_{2i}}} \right \|}^2}} \right )^{\frac{{\alpha + 1}}{2}}}\; -{c_{4i}}{\left ({{{\left \|{{{\tilde z}_{2i}}} \right \|}^2}} \right )^{\frac{{\alpha + 1}}{2}}} - z_{2i}^T{D_i}{z_{2i}} + z_{2i}^T{\varepsilon _i} + \tilde \theta _i^T{\delta _i}({{\hat \theta }_i} -{\theta _{i0}})\\\\ \,\quad \quad \; -\left({\frac{1}{{{\alpha _i}}} - 1 - 0.5{\lambda _{\max }}\{{J_i}({\psi _i})\} }\right){{{\textrm{P}_{ie}}}^2} + \frac{1}{4}{\varpi ^2}. \end{array} \end{equation}

With respect to inequalities (29) and (32) and through assumption ${{{\varepsilon _i}}^2} \lt \varepsilon _i^{{*^2}}$ and Lemma 2, the following inequalities can be obtained by applying the Young’s inequality as follows:

(48) \begin{equation} z_{2i}^T{\varepsilon _i} \le \frac{1}{2}z_{2i}^T{z_{2i}} + \frac{1}{2}\varepsilon _i^{{*^2}}, \end{equation}

(49) \begin{equation} \tilde \theta _i^T{\delta _i}({\hat \theta _i} -{\theta _{i0}}) \le - (1 - 0.5/{k^2}){\delta _i}{\left \|{{{\tilde \theta }_i}} \right \|^2} + 0.5{\delta _i}{\left \|{{\theta _i} -{\theta _{i0}}} \right \|^2}, \end{equation}

(50) \begin{equation} - z_{2i}^T{c_{3i}}{\tilde z_{2i}} \le 0.5{\lambda _{\max }}\!\left \{{{c_{3i}}} \right \}{\left \|{{z_{2i}}} \right \|^2} + 0.5{\lambda _{\max }}\!\left \{{{c_{3i}}} \right \}{\left \|{{{\tilde z}_{2i}}} \right \|^2}. \end{equation}

Consequently, with replacing the inequalities (48)–(50) into Eq. (47), the following inequality is achieved:

(51) \begin{equation} \begin{array}{l}{{\dot \Upsilon }_{2i}} \le - 2{c_{1i}}{\rho _{1i}}{\Upsilon _{1i}} -{c_{2i}}{\rho _{1i}}^{\frac{{\alpha - 1}}{2}}{(2{\Upsilon _{1i}})^{\frac{{\alpha + 1}}{2}}} -{\rho _{2i}}{\left \|{{\textrm{P}_{ie}}} \right \|^2} -{\rho _{5i}}{\left \|{{z_{2i}}} \right \|^2}\\\\ \quad \quad -{c_{4i}}{\left ({{{\left \|{{z_{2i}}} \right \|}^2}} \right )^{\frac{{\alpha + 1}}{2}}} -{\rho _{3i}}{\left \|{{{\tilde \theta }_i}} \right \|^2} + \frac{1}{2}\varepsilon _i^{{*^2}} +{\rho _{4i}} +{\rho _{6i}}, \end{array} \end{equation}

where parameters ${\rho _{2i}} = \left ({\frac{1}{{{\alpha _i}}} - 1 -{\lambda _{\max }}\!\left ({{J_i}\!\left ({{\psi _i}} \right )} \right )} \right )$ , ${\rho _{3i}} = (1 - 0.5/{k^2}){\delta _i}$ , ${\rho _{4i}} = 0.5{\delta _i}{\left \|{{\theta _i} -{\theta _{i0}}} \right \|^2}$ , ${\rho _{5i}} ={\lambda _{\min }}\!\left \{{{D_i} +{c_{3i}} +{c_{4i}}} \right \} - 0.5{\lambda _{\max }}\!\left \{{{c_{3i}}} \right \}$ and ${\rho _{6i}} = \frac{{S_2^2}}{2} + \frac{1}{4}{\varpi ^2}$ are positive constants with conditions ${\lambda _{\min }}\!\left \{{{D_i} +{c_{3i}} +{c_{4i}}} \right \} \gt 0.5{\lambda _{\max }}\!\left \{{{c_{3i}}} \right \}$ and $\frac{1}{{{\alpha _i}}} \gt 1 +{\lambda _{\max }}\!\left ({{J_i}\!\left ({{\psi _i}} \right )} \right )$ .

Remark 5. Since in the constrained formation control problem, each ASV can only acquire the position information from its immediate predecessor, such position states of ASVs are unmeasurable and the tracking errors can be considered in the predefined and constrained bounds which are time-varying by recalling the limited FOV. To satisfy these constraints on the formation tracking errors, the following constraints are considered:

(52) \begin{equation} {\underline \beta _i}(t) \le{z_i}(t) \le{\bar \beta _i}(t) \end{equation}

with

(53) \begin{equation} \begin{array}{l}{{\bar \beta }_i}(t) ={{\bar \beta }_{i,\infty }} + ({{\bar \beta }_{i,0}} -{{\bar \beta }_{i,\infty }}){e^{ -{k_i}t}},\\\\{\underline \beta _i}(t) ={\underline \beta _{i,\infty }} + ({\underline \beta _{i,0}} -{\underline \beta _{i,\infty }}){e^{ -{k_i}t}}, \end{array} \end{equation}

where ${\bar \beta _{i,0}},$ ${\bar \beta _{i,\infty }},$ ${\underline \beta _{i,0}},$ $\underline \beta _{i,\infty }$ and $k_i$ are the positive design parameters, ${\bar \beta _i}(t)$ and ${\underline \beta _i}(t)$ are the maximum and minimum allowable tracking errors with ${\bar \beta _{i,\infty }} \le{\bar \beta _{i,0}}$ and ${\underline \beta _{i,\infty }} \le{\underline \beta _{i,0}}$ , respectively. These time-varying boundaries of the tracking errors are utilized in the conversion of the error transformation.

The asymmetric BLFs are proposed to assure the time-varying limited FOV errors constrained within certain bounds such that leaders and followers can preserve the predefined structure and ensure the desired steady-state bounds as follow:

(54) \begin{equation} {\Upsilon _i} = \frac{{{q_z}({z_i})}}{2}\log \frac{{\overline \beta _i^2}}{{\overline \beta _i^2 - z_i^2}} + \frac{{1 -{q_z}({z_i})}}{2}\log \frac{{\underline \beta _i^2}}{{\underline \beta _i^2 - z_i^2}}, \end{equation}

where $q_z$ is the logic variable which is equal to one if the kinematic error $z_i$ is positive and the otherwise is equal to zero; $\underline \beta _i$ and $\overline \beta _i$ are the minimum and maximum constraints on the time-varying error $z_i$ , respectively. To dominate the time dependency, the following conversion of variables is proposed:

(55) \begin{equation} {\Omega _{z1}} = \frac{{{z_i}}}{{{\beta _i}}},\quad{\Omega _{z2}} = \frac{{{z_i}}}{{{\beta _i}}},\quad{\Omega _z} ={q_z}{\Omega _{z2}} + \left ({1 -{q_z}} \right )\!{\Omega _{z1}}. \end{equation}

With the above conversion of coordinates, the BLF $\Upsilon _i$ is rewritten without the time dependency as follows:

(56) \begin{equation} {\Upsilon _i} = \frac{1}{2}\log \frac{1}{{1 - \Omega _z^2}}. \end{equation}

It is obvious that the BLF $\Upsilon _i$ is positive-definite and continuously differentiable in the set $\left |{{\Omega _z}} \right | \lt 1$ . This suggested BLF based on the error transformation can be substituted into the proposed BLF in Eq. (34) with respect to the time-varying predefined and limited FOV constraints.

4.4. Terminal sliding-mode disturbance observer design

The environmental disturbances vector $\tau _{di}$ is bounded and not measurable precisely in practice, indicating that the designed terminal sliding-mode disturbance observer has more practical value. The physical meanings of these imposed environmental disturbances are mostly due to the sea and ocean currents, storms, waves and winds. However, its interpretation may generally cover strong couplings, uncertain nonlinearities, modeling errors and unmodeled dynamics of the system. In this step, a terminal sliding-mode observer-based estimation scheme will be developed more to assure the actual external disturbance for each surface vessel to allow high-accuracy estimation after finite-time. To design an estimation law for disturbance $\tau _{di}$ , the new error variable ${\Xi _i} ={\upsilon _i} -{\Phi _i}$ is introduced firstly for each surface vessel where ${\Phi _i} \in{R^3}$ is updated through the following equation:

(57) \begin{equation} {\Phi _i} = M_i^{ - 1}({K_{i1}}{\Xi _i} +{K_{i2}}{\textrm{sgn}} ({\Xi _i}) +{\tau _i} -{C_i}({\upsilon _i}){\upsilon _i} -{D_i}({\upsilon _i}){\upsilon _i} -{g_i}({\eta _i})). \end{equation}

This nonlinear terminal sliding-mode disturbance observer estimation law is implemented in the control design to obtain in a high performance control against external disturbances through finite-time convergence of tracking errors vector for each surface vessel into a compact set as:

(58) \begin{equation} {\hat \tau _{di}} ={K_{1i}}{\Xi _i} +{K_{2i}}{\textrm{sgn}} ({\Xi _i}) \end{equation}

where $K_{1i}$ and $K_{2i}$ are positive constants. The actual $\tau _{di}$ can be estimated precisely through $\hat \tau _{di}$ . Moreover, the estimated disturbance error ${e_{di}} ={\hat \tau _{di}}-{\tau _{di}}$ for each surface vessel is assured to converge to zero in an asymptotic manner. In practice, to reduce the chattering in the finite-time convergence, a continuous function ${{\Xi _i}} \mathord{\left/ \right. }{(\!\left \|{{\Xi _i}} \right \| +{\varepsilon _i})}$ must be adopted to approximate the ${\textrm{sgn}} ({\Xi _i})$ function. This designed terminal sliding-mode disturbance observer is implemented without any assumption on the upper bound of disturbances in the control design to increase the control system performance.

To prove the convergence of this terminal sliding-mode disturbance observer law, the following Lyapunov function is proposed for each surface vessel:

(59) \begin{equation} {\Upsilon _{3i}} = 0.5\,\Xi _i^T{M_i}{\Xi _i}. \end{equation}

With respect to Eq. (4), control signal (46) and terminal sliding-mode law (53), the following equation is yielded:

(60) \begin{equation} \begin{array}{l}{M_i}{{\dot \Xi }_i} ={M_i}{{\dot \upsilon }_i} -{M_i}{{\dot \Phi }_i}\\[5pt] \quad \quad \;\, = -{K_{i1}}{\Xi _i} -{K_{i2}}{\textrm{sgn}} ({\Xi _i}) +{\tau _{id}}. \end{array} \end{equation}

The time derivative of Lyapunov function (59) is calculated along with disturbance observer error ${\Xi }_i$ as:

(61) \begin{equation} \begin{array}{l}{{\dot \Upsilon }_{3i}} = \Xi _i^T({\tau _{di}} -{K_{i1}}{\Xi _i} -{K_{i2}}{\textrm{sgn}} ({\Xi _i}) -{K_{i3}}{\left \lfloor{{\Xi _i}} \right \rfloor ^{{{{p_i}} \mathord{\left/ \right. }{{q_i}}}}})\\[5pt] \quad \; \le -{K_{i1}}{\left \|{{\Xi _i}} \right \|^2} -{K_{i2}}\!\left \|{{\Xi _i}} \right \| +{\Sigma _i}\!\left \|{{\Xi _i}} \right \| -{K_{i3}}\Xi _i^T{\left \lfloor{{\Xi _i}} \right \rfloor ^{{{{p_i}} \mathord{\left/ \right. }{{q_i}}}}}\\[5pt] \quad \; \le -{K_{i1}}{\left \|{{\Xi _i}} \right \|^2} -{K_{i3}}\Xi _i^T{\left \lfloor{{\Xi _i}} \right \rfloor ^{{{{p_i}} \mathord{\left/ \right. }{{q_i}}}}} \end{array} \end{equation}

With applying Lemma 1, the inequality (61) will be bounded more through:

(62) \begin{equation} {\dot \Upsilon _{3i}} \le - \frac{{2{K_{i1}}}}{{{\lambda _{\max }}({M_i})}}{\Upsilon _{3i}} -{K_{3i}}{\left(\frac{2}{{{\lambda _{\max }}({M_i})}}\right)^\sigma }{({\Upsilon _{3i}})^\sigma } \end{equation}

where $\sigma = \dfrac{{{p_i} +{q_i}}}{{2{q_i}}}$ . Consequently, ${\dot \Upsilon _{3i}}(t) \equiv 0$ for all convergence rate $t \ge{t_{ix}}$ , which, $t_{ix}$ is obtained through the following inequality at finite-time [Reference Xiao, Yang and Huo43].

(63) \begin{equation} {t_{ix}} \le \frac{1}{{c(1 + \alpha )}}\ln \frac{{c\Upsilon _{3i}^{1 - \alpha }(0) +{\iota _i}}}{{{\iota _i}}} \end{equation}

where ${\iota _i} = \dfrac{{2{K_{i1}}}}{{{\lambda _{\max\! ({M_i})}}}}$ and ${c_i} ={K_{i3}}{\left(\dfrac{2}{{{\lambda _{\max\! ({M_i})}}}}\right)^{{\sigma _i}}}$ .

Consequently, it can be deduced that ${\Upsilon _{3i}} \equiv 0$ , where ${\dot \Xi _i} = 0,\;t \ge{t_{ix}}$ and the disturbance estimation error $e_{di}$ meets

(64) \begin{align} {e_{di}} & ={K_{i1}}{\Xi _i} +{K_{i2}}{\textrm{sgn}} ({\Xi _i}) +{K_{i3}}{\left \lfloor{{\Xi _i}} \right \rfloor ^{{{{p_i}} \mathord{\left/ \right. }{{q_i}}}}} - ({M_i}{\upsilon _i} +{C_i}({\upsilon _i}){\upsilon _i} -{\tau _i} +{D_i}({\upsilon _i}){\upsilon _i} + G({\eta _i}))\nonumber \\[5pt] &={M_i}({-}{{\dot \upsilon }_i} +{{\dot \Phi }_i}). \end{align}

Figure 2. Suggested block diagram for the leader-follower formation control of autonomous surface vessels.

As a result, ${e_{di}} = -{M_i}{\dot \Xi _i}$ , which is finite-time stable, ${e_{di}} \equiv 0,\;t \ge{t_{ix}}$ . Thus, the states of the closed-loop control system converge to the origin in finite-time and the estimation parameters are bounded. This suggested control structure is illustrated for the leader-follower formation control of autonomous surface vessels in Fig. 2.

Proof. Consider the following Lyapunov function for the overall formation control system:

(65) \begin{equation} {\Upsilon _T}(t) = \sum \limits _{i = 1}^N{\left ({\frac{1}{2}\log \frac{{\beta _{1i}^2}}{{\beta _{1i}^2 - z_{1i}^T{z_{1i}}}} + \frac{1}{2}z_{2i}^T{M_i}\,{z_{2i}} + \frac{1}{{2{\gamma _i}}}\tilde \theta _i^T{{\tilde \theta }_i} + \frac{1}{2}\textrm{P}_{ie}^T\textrm{P}_{ie} + \frac{1}{2}\,\Xi _i^T{M_i}\,{\Xi _i}} \right )}. \end{equation}

The following inequality is obtained for the entire formation control system:

(66) \begin{equation} \begin{array}{l}{{\dot \Upsilon }_T}(t) \le \sum \limits _{i = 1}^N{\left ({ - 2{c_{1i}}{\rho _{1i}}{\Upsilon _{1i}} -{c_{2i}}{\rho _{1i}}^{\frac{{\alpha - 1}}{2}}{{(2{\Upsilon _{1i}})}^{\frac{{\alpha + 1}}{2}}} -{\rho _{2i}}{{\left \|{{\textrm{P}_{ie}}} \right \|}^2}\; -{\rho _{3i}}{{\left \|{{{\tilde \theta }_i}} \right \|}^2} -{\rho _{5i}}{{\left \|{{z_{2i}}} \right \|}^2}} \right .} \\\\ \left .{\quad \quad \quad -{c_{4i}}{{\left ({{{\left \|{{z_{2i}}} \right \|}^2}} \right )}^{\frac{{\alpha + 1}}{2}}} + \frac{1}{2}\varepsilon _i^{{*^2}} +{\rho _{4i}} +{\rho _{6i}} - \dfrac{{2{K_{i1}}}}{{{\lambda _{\max }}({M_i})}}{\Upsilon _{3i}} -{K_{3i}}{{\left(\dfrac{2}{{{\lambda _{\max }}({M_i})}}\right)}^\sigma }{{({\Upsilon _{3i}})}^\sigma }} \right ). \end{array} \end{equation}

Consequently, inequality (66) can be represented for the overall formation control of autonomous surface vessels as follows:

(67) \begin{equation} {\dot \Upsilon _T}(t) = \sum \limits _{i = 1}^N{\left ({ -{s_i}{{\left \|{{\chi _{ti}}(t)} \right \|}^2} +{\gamma _i}(t)} \right )} \le -{s_{\min }}{\left \|{{\chi _t}(t)} \right \|^2} + \gamma (t), \end{equation}

where ${\chi _{ti}} = [z_{1i}^T,z_{2i}^T,\tilde \theta _i^T,\,{\textrm{P}_{ie}},\Xi _i^T]$ , ${\chi _t} ={[\chi _{t1}^T,\chi _{t2}^T,\ldots,\chi _{tN}^T]^T}$ , ${s_{\min }} = \min \!\left \{{{s_i}} \right \}_{i = 1}^N$ and $\gamma (t) = \sum \limits _{i = 1}^N{{\gamma _i}(t)}$ . In this context, $\xi _i$ and $\gamma _i$ are defined as follows:

(68) \begin{equation} {\xi } = \min \!\left \{{2{c_{1i}}{\rho _{1i}},{c_{2i}}{\rho _{1i}}^{\frac{{\alpha - 1}}{2}},{\rho _{2i}},{\rho _{5i}},{\rho _{3i}},{c_{4i}},\frac{{2{K_{i1}}}}{{{\lambda _{\max }}({M_i})}},{K_{3i}}{{\left(\frac{2}{{{\lambda _{\max }}({M_i})}}\right)}^\sigma }} \right \}. \end{equation}

(69) \begin{equation} {\gamma } ={\rho _{4i}} +{\rho _{6i}} + \frac{1}{2}\varepsilon _i^{{*^2}}. \end{equation}

Consequently, with respect to definitions (68) and (69), the following inequality is obtained for the overall formation control of autonomous surface vessels:

(70) \begin{equation} {\dot \Upsilon _T}(t) \le - \xi \,{\Upsilon _T}(t) + \gamma, \end{equation}

which assures that the closed-loop control system is SGUUB. According to Lemma 1, inequality $\left \|{{z_{1i}}(t)} \right \| \lt{\beta _{1i}}$ $\forall t \gt 0$ yields that the initial conditions $\left \|{{z_{1i}}(0)} \right \| \lt{\beta _{1i}}$ are satisfied. Hence, it is known that ${\eta _i}(t) ={z_{1i}}(t) +{\eta _0}(t)$ , $\left \|{{z_{1i}}(t)} \right \| \lt{\beta _{1i}}$ and $\left \|{{\eta _0}(t)} \right \| \le{A_0}$ , consequently, $\left \|{{\eta _i}(t)} \right \| \lt{\beta _{1i}} +{A_0}$ and the position output constraints will never be violated. The signals ${z_i}(t) ={[{z_{1i}}(t),{z_{2i}}(t),{z_{3i}}(t)]^T}$ remain in a defined bounded compact set through:

(71) \begin{equation} {\Omega _z} = \left \{{{z_i} \in{R^n}\;:\left |{{z_{1i}}} \right | \le \beta _{1i}^2\sqrt{1 -{e^{ - 2{\Upsilon _{1i}}}}},\left \|{{z_{2i}}} \right \| \le \sqrt{\frac{{\max \!\left \{{\Upsilon ({t_0}),\dfrac{{{\gamma _i}}}{{{s_i}}}} \right \}}}{{{\lambda _{{z_{\min }}}}}}} } \right \}. \end{equation}

This equation indicates that $\left \|{{z_i}(t)} \right \|$ decreases whenever ${z_i}(t)$ is outside the set $\Omega _z$ , revealing that ${z_i}(t)$ is SGUUB and converges exponentially to a neighborhood of the origin which assures that the leader-follower formation control system for the surface vessels under output constraints is SGUUB. By the function approximation capability of NNs, an adaptive-neural formation controller is designed for each surface vessel in a sense that all of them can synchronize the leader with tracking errors by being SGUUB in an asymptotic manner. Thus, all the signals in the closed-loop system are bounded and the tracking errors converge to a small neighborhood of the origin by choosing the design parameters appropriately. Consequently, the obtained proof process is analogous to theorem. The proposed control method not only can guarantee the formation objective achieved within finite settling time but also can estimate the external disturbances in the finite-time. It is easy to draw the conclusion that the formation control problem of surface vessels can be solved with constrained states.

5.1. A comparative experiment

To verify the performance of the suggested controller for the formation control of surface vessels, four scenarios are considered in this study. In the first scenario, the simulation is carried out for the DSC approach. In the second scenario, the BLF-based design is implemented in the suggested controller. In the third scenario, the finite-time technique is applied to the DSC approach, and in the fourth scenario, both BLF and finite-time strategies are utilized in the proposed control system. To compare between these approaches, the following performance indices are defined:

(72) \begin{equation} \left \{ \begin{array}{l} ISE = \int _0^T{{e_{i}^2}(t)dt}, \\\\ IAE = \int _0^T{\left |{e_{i}(t)} \right |dt}, \\\\ ITAE = \int _0^T{t\!\left |{e_{i}(t)} \right |dt}, \end{array} \right. \end{equation}

where $e_{i}(t)$ is the tracking error of the formation control, $T$ is the constant time in which the transient response settles and Integral Square Error (ISE), Integral Absolute Error (IAE) and Integral Time Absolute Error (ITAE) are the ISE, IAE and ITAE, respectively. The overshoot, settling time, the tracking errors $z_{xi}$ , $z_{yi}$ and $z_{\psi i}$ in terms of three well-known criteria including ISE, IAE and ITAE are reported in Tables I and II, respectively, between four scenarios without considering disturbances and with applying disturbances. Environmental disturbances as currents and waves effects cause the desired performance not to be achieved and the tracking errors of the formation control are transgressed. The forced external disturbances to each autonomous surface vessel are considered as follows:

(73) \begin{equation} {\tau _{di}} = \left \{ \begin{array}{l}{\tau _{d1}} = 5\sin\! (2t) + 5\cos\! (t),\\\\{\tau _{d2}} = 5\cos\! (3t) + 5\cos\! (2t),\\\\{\tau _{d3}} = 5\sin\! (2t) + 5\sin\! (t), \end{array} \right. \end{equation}

By a comparison between different scenarios, in Scenario 1, the control structure only assures the tracking of all ASVs without any novelty. In Scenario 2, the BLF approach decreases the formation tracking errors for all ASVs in the defined and limited bounds. In Scenario 3, the finite-time approach decreases the settling time and increases the convergence rate in an effective way. In Scenario 4, the control performance is significantly increased via a combination of the finite-time control and BLF-based design in the control structure. In Fig. 4, the yaw moment $\tau _r$ is presented to be much more noisy than $\tau _u$ and $\tau _v$ . This is because the relative effect of the noise in the yaw dynamics is larger than that one in the surge motion.

5.2. The efficacy and robustness of the proposed controller

To verify the efficacy and robustness of this proposed controller, the following disturbances are applied to the dynamic models of ASVs:

(74) \begin{equation} {\tau _{di}} = \left \{ \begin{array}{l}{\tau _{d1}} = + \frac{1}{2}u(t) + \frac{1}{2}u(t - 6) + \frac{1}{2}u(t - 14) \\\\{\tau _{d2}} = - \frac{1}{2}u(t) - \frac{1}{2}u(t - 6) - \frac{1}{2}u(t - 14). \\\\{\tau _{d3}} = + 1u(t) - 2u(t - 6) - \frac{5}{2}u(t - 14) \end{array} \right. \end{equation}

As can be seen from Figs. 12 and 13, despite the imposed disturbances to the ASVs dynamics, the desired trajectory tracking of ASVs in the formation control problem is performed well. Although this methodology is an effective approach to handle these disturbances, unwanted peaks are appeared in the disturbance observer errors, while the consecutive step disturbances in Eq. (74) are applied to the ASVs dynamics. In spite of such a peaking, which may be removed by employing the saturation function in the disturbance observer design, all ASVs successfully track the desired trajectories in the formation control problem and the boundary of tracking errors are never transgressed. Moreover, we can see that the finite-time terminal sliding-mode disturbance observer estimates the disturbance $\tau _{di}$ quite well for a finite-time presentation of $t _{ix}$ .

It is clear that from Fig. 14, in spite of peaking at $t=6$ and $t=8$ s due to the consecutive step disturbances calculated from Eq. (74), all the vehicles construct the desired formation very well. In comparison with the other disturbance observers in the literature, the proposed approach is illustrated as a suitable solution to the problem of disturbance observer design with the finite-time convergence $t _{ix}$ . Moreover, the proposed methodology can eliminate the well-known assumptions on the existence of the derivative of external disturbances in previous works. It is clear that this method does not require any knowledge about the amplitude and frequency of imposed environmental disturbances, and such parameters are not implemented in the controller design.

Table I. Comparison results between four scenarios without disturbance effects on ASVs.

Table II. Comparison results between four scenarios with disturbance effects on ASVs.