Controller design
Firstly, the trajectory tracking error is defined as follows
$$e = \eta – \eta_{d} ,$$
(16)
where \(\eta = [x,y,\psi ]^{T}\) is the actual trajectory; \(\eta_{d} = [x_{d} ,y_{d} ,\psi_{d} ]^{T}\) is the reference trajectory.
To achieve the prescribed-time prescribed performance, the subsequent auxiliary error vector is formulated:
$$z = \left( {k_{e} + c_{e} \frac{{\dot{\gamma }}}{\gamma }} \right)e$$
(17)
where \(k_{e}\) and \(c_{e}\) are constants. By taking the derivative of \(z\), we get
$$\dot{z} = \frac{{c_{e} (\gamma \ddot{\gamma } – \dot{\gamma }^{2} )}}{{\gamma^{2} }}e + \left( {k_{e} + c_{e} \frac{{\dot{\gamma }}}{\gamma }} \right)\dot{e}$$
(18)
Owing to the challenges in directly formulating the controller under constraint (10), the following error transformation is proposed.
$$z = \lambda_{2} \varphi (t)\Upsilon (s_{1} ,\bar{\lambda})$$
(19)
where \(\bar{\lambda} = \frac{{\lambda_{1} }}{{\lambda_{2} }}\) and \(\Upsilon (s_{1} ,\bar{\lambda}) = \frac{{e^{{s_{1} }} – e^{{ – s_{1} }} }}{{e^{{s_{1} }} + \bar{\lambda}^{ – 1} e^{{ – s_{1} }} }}\) is strictly monotonically increasing.
Consequently, the transformation error \(s_{1}\) can be derived by resolving (19)
$$s_{1} = \frac{1}{2}\ln \left( {\lambda_{1} \lambda_{2} \varphi + \lambda_{2} z} \right) – \frac{1}{2}\left( {\lambda_{1} \lambda_{2} \varphi – \lambda_{1} z} \right)$$
(20)
Subsequently, taking the time derivative of \(s_{1}\) yields
$$\dot{s}_{1} = \partial (\dot{z} – z\dot{\varphi }\varphi^{ – 1} )$$
(21)
where \(\partial = \left( {\frac{1}{{2(z + \lambda_{1} \varphi )}} – \frac{1}{{2(z – \lambda_{2} \varphi )}}} \right)\).
We establish the subsequent Lyapunov function as
$$V_{1} = \frac{1}{2}s_{1}^{T} s_{1}$$
(22)
The time derivative of \(V_{1}\) along with (20) is
$$\begin{aligned} \dot{V}_{1} & = s_{1}^{T} \left( {\partial (\dot{z} – z\dot{\varphi }\varphi^{ – 1} )} \right) \\ & = s_{1}^{T} \left( {\partial \left( {\frac{{c_{e} (\gamma \ddot{\gamma } – \dot{\gamma }^{2} )}}{{\gamma^{2} }}e + \left( {k_{e} + c_{e} \frac{{\dot{\gamma }}}{\gamma }} \right)(\xi – \dot{\eta }_{d} ) – z\dot{\varphi }\varphi^{ – 1} } \right)} \right) \\ \end{aligned}$$
(23)
Then, the virtual control law \(\alpha\) is designed as:
$$\alpha = \left( {k_{e} + c_{e} \frac{{\dot{\gamma }}}{\gamma }} \right)^{ – 1} \left[ {z\dot{\varphi }\varphi^{ – 1} – \partial^{ – 1} \kappa_{1} s_{1} – \frac{{c_{e} (\gamma \ddot{\gamma } – \dot{\gamma }^{2} )}}{{\gamma^{2} }}e} \right] + \dot{\eta }_{d}$$
(24)
where \(\kappa_{1}\) is a positive control gain.
Subsequently, the velocity tracking error and the filtering error are defined as follows
$$s_{2} = \xi – \alpha_{f}$$
(25)
$$\phi = \alpha_{f} – \alpha$$
(26)
where \(\phi\) is the filtering error and \(\alpha_{f}\) is the filter output.
$$\iota \dot{\alpha }_{f} + \alpha_{f} = \alpha , \, \alpha_{f} (0) = \alpha (0)$$
(27)
here, \(\iota\) is the filter time constant.
According to (26) and (27), one obtains
$$\dot{\phi } = – \frac{1}{\iota }\phi + \Delta$$
(28)
where \(\Delta = – \dot{\alpha }\).
By differentiating \(s_{2}\), we have
$$\begin{aligned} \dot{s}_{2} & = J(\psi )M^{ – 1} \tau_{p} + \Gamma – \dot{\alpha }_{f} \\ & = J(\psi )M^{ – 1} (\tau_{c} + \Delta \tau ) + \Gamma – \dot{\alpha }_{f} \\ \end{aligned}$$
(29)
To mitigate the thruster saturation, an anti-windup compensator is introduced
$$\dot{\Xi } = – K_{\Xi } \Xi + \Delta \tau$$
(30)
where \(K_{\Xi }\) is a positive-definite matrix, \(\Xi \in R^{3}\) is the state vector of the compensator.
Following that, the prescribed-time controller is developed in accordance with the PTPPF constraint
$$\tau_{p} = MJ(\psi )^{T} \left( { – \kappa_{2} s_{2} + \dot{\alpha }_{f} – \hat{\Gamma }} \right) + \kappa_{3} \Xi$$
(31)
where \(\kappa_{2} > 0\), and \(\hat{\Gamma }\) is the estimation of \(\Gamma\).
To obtain the total perturbations, a prescribed-time lumped disturbance observer (PTLDO) is constructed:
$$\dot{\hat{s}}_{2} = J(\psi )M^{ – 1} \tau_{p} + \hat{\Gamma } – \dot{\alpha }_{f}$$
(32)
$$\hat{\Gamma } = \left( {k_{1} + c_{1} \frac{{\dot{\gamma }}}{\gamma }} \right)\tilde{s}_{2} + \int_{0}^{t} {\Phi \tanh \left( {\frac{{\tilde{s}_{2} (\varpi )}}{\varepsilon }} \right)d\varpi }$$
(33)
where \(\tilde{s}_{2} = s_{2} – \hat{s}_{2}\); \(\varepsilon > 0\) is a small design constant; \(k_{1}\), \(c_{1}\), and \(\Phi\) are control gains.
Theorem 1
Considering system (4) and Assumption 1, the designed PTLDO (32, 33) can enable the estimation error \(\tilde{\Gamma }\) to converge to zero within the prescribed time \(T_{o}\).
Proof of Theorem 1
From (29) and (32), one can deduce.
$$\dot{\tilde{s}}_{2} = \dot{s}_{2} – \dot{\hat{s}}_{2} = \Gamma – \hat{\Gamma } = \tilde{\Gamma }$$
(34)
Choose the Lyapunov function as follows
$$V_{{\tilde{s}_{2} }} = \frac{1}{2}\tilde{s}_{2}^{T} \tilde{s}_{2}$$
(35)
Afterwards, calculating the derivative of (35) results in
$$\begin{aligned} \dot{V}_{{\tilde{s}_{2} }} & = \tilde{s}_{2}^{T} \dot{\tilde{s}}_{2} = \tilde{s}_{2}^{T} (\dot{s}_{2} – \dot{\hat{s}}_{2} ) = \tilde{s}_{2}^{T} (\Gamma – \hat{\Gamma }) \\ & = – \tilde{s}_{2}^{T} \left( {k_{1} + c_{1} \frac{{\dot{\gamma }}}{\gamma }} \right)\tilde{s}_{2} + \tilde{s}_{2}^{T} \left( {\int_{0}^{t} {\dot{\Gamma }(\varpi )d\varpi – \int_{0}^{t} {\Phi \tanh \left( {\frac{{\tilde{s}_{2} (\varpi )}}{\varepsilon }} \right)d\varpi } } } \right) \\ & \le – \tilde{s}_{2}^{T} \left( {k_{1} + c_{1} \frac{{\dot{\gamma }}}{\gamma }} \right)\tilde{s}_{2} \\ & = – k_{1} V_{{\tilde{s}_{2} }} – c_{1} \frac{{\dot{\gamma }}}{\gamma }V_{{\tilde{s}_{2} }} \\ \end{aligned}$$
(36)
The estimation error can converge to zero within the prescribed time \(T_{o}\), as indicated by Lemma 1. The proof of Theorem 1 is thus established.
Subsequently, to lessen the frequency of controller signal updates and conserve communication resources, an event-triggered strategy is implemented41
$$\tau_{j} (t) = \chi_{j} (t_{k}^{j} ),\forall t \in [t_{k}^{j} ,t_{k + 1}^{j} )$$
(37)
$$t_{k + 1}^{j} = \inf \left\{ {t \in R|\left| {e_{\tau j} (t)} \right| \ge \theta_{i} \left| {\tau_{j} (t)} \right| + \delta_{i} } \right\}$$
(38)
where \(e_{\tau j} (t) = \chi_{j} (t) – \tau_{j}\) is the measurement error; \(0 < \theta_{i} < 1\) and \(\delta_{i} > 0\) are user-defined constants; and \(t_{k}^{j}\) denotes the input update time of the \(j\) th actuator. At time \(t_{k + 1}^{j}\), the control signal \(\tau_{j} (t_{k + 1}^{j} )\) is applied to the actuator; during the interval \([t_{k}^{j} ,t_{k + 1}^{j} )\), the control signal remains unchanged, namely \(\chi_{j} (t_{k}^{j} )\).
For interval \([t_{k}^{j} ,t_{k + 1}^{j} )\), by combining (37) and (38), we obtain
$$\chi_{j} (t) = (1 + \mu_{1} (t)\theta_{i} )\tau_{j} (t) + \mu_{2} (t)\delta_{i}$$
(39)
where \(\mu_{1} (t)\) and \(\mu_{2} (t)\) are designed parameters satisfying \(\left| {\mu_{1} (t)} \right| \le 1\) and \(\left| {\mu_{2} (t)} \right| \le 1\). In addition, (37) is further equivalent to
$$\tau_{j} (t) = \frac{{\chi_{j} (t)}}{{1 + \theta_{i} \mu_{1} (t)}} – \frac{{\mu_{2} (t)\delta_{i} }}{{1 + \theta_{i} \mu_{1} (t)}}$$
(40)
We define the subsequent Lyapunov function
$$V_{2} (t) = \frac{1}{2}s_{1}^{T} s_{1} + \frac{1}{2}s_{2}^{T} s_{2}$$
(41)
By differentiating \(V_{2} (t)\) and utilizing (29), one obtains
$$\dot{V}_{2} (t) = s_{1}^{T} \dot{s}_{1} + s_{2}^{T} \left( {J(\psi )M^{ – 1} \left( {\frac{{\chi_{j} (t)}}{{1 + \theta_{i} \mu_{1} (t)}} – \frac{{\mu_{2} (t)\delta_{i} }}{{1 + \theta_{i} \mu_{1} (t)}} + \Delta \tau } \right) + \Gamma – \dot{\alpha }_{f} } \right)$$
(42)
Thus, an event-triggered based prescribed-time controller is expressed as follows
$$\chi_{j} (t) = – (1 + \theta_{i} )\left[ {\tanh \left( {\frac{{s_{2}^{T} \tau_{p} }}{\omega }} \right)\tau_{p} + \tanh \left( {\frac{{s_{2}^{T} \overline{\delta }}}{\omega }} \right)\overline{\delta }} \right]$$
(43)
where \(\omega > 0\), \(\overline{\delta } > \delta_{i} /(1 – \theta_{i} )\).
Stability analysis
The major findings of this research are provided in Theorem 2.
Theorem 2
Considering the controlled USV system (4), combined with the PTLDO (32, 33), the virtual control law (24), and the dynamic controller with an event-triggered mechanism (43), the USV trajectory tracking control system satisfies the prescribed-time stability. In addition, through rigorous theoretical deduction, it has been proven that the introduced event-triggered mechanism can avoid Zeno behavior.
Proof of Theorem 2
Select the Lyapunov function as follows.
$$V = \frac{1}{2}s_{1}^{T} s_{1} + \frac{1}{2}s_{2}^{T} s_{2} + \frac{1}{2}\Xi^{T} \Xi + \frac{1}{2}\phi^{T} \phi$$
(44)
Differentiating (44) yields
$$\dot{V} = s_{1}^{T} \dot{s}_{1} + s_{2}^{T} \dot{s}_{2} + \Xi^{T} \dot{\Xi } + \phi^{T} \dot{\phi }$$
(45)
It follows from (25) and (26) that \(\xi = s_{2} + \phi + \alpha\). Then, by combining this with (23), we obtain
$$\begin{aligned} s_{1}^{T} \dot{s}_{1} & = s_{1}^{T} \left( {\partial (\dot{z} – z\dot{\varphi }\varphi^{ – 1} )} \right) \\ & = s_{1}^{T} \left( {\partial \left( {\frac{{c_{e} (\gamma \ddot{\gamma } – \dot{\gamma }^{2} )}}{{\gamma^{2} }}e + \left( {k_{e} + c_{e} \frac{{\dot{\gamma }}}{\gamma }} \right)(\xi – \dot{\eta }_{d} ) – z\dot{\varphi }\varphi^{ – 1} } \right)} \right) \\ & = s_{1}^{T} \left( {\partial \left( {\frac{{c_{e} (\gamma \ddot{\gamma } – \dot{\gamma }^{2} )}}{{\gamma^{2} }}e + \left( {k_{e} + c_{e} \frac{{\dot{\gamma }}}{\gamma }} \right)(s_{2} + \phi + \alpha – \dot{\eta }_{d} ) – z\dot{\varphi }\varphi^{ – 1} } \right)} \right) \\ \end{aligned}$$
(46)
Substituting (24) into (46) yields
$$\begin{aligned} s_{1}^{T} \dot{s}_{1} & = s_{1}^{T} \left( {\partial \left( {k_{e} + c_{e} \frac{{\dot{\gamma }}}{\gamma }} \right)\left( {s_{2} + \phi } \right) – \kappa_{1} s_{1} } \right) \\ & = \partial \left( {k_{e} + c_{e} \frac{{\dot{\gamma }}}{\gamma }} \right)s_{1}^{T} s_{2} + \partial \left( {k_{e} + c_{e} \frac{{\dot{\gamma }}}{\gamma }} \right)s_{1}^{T} \phi – \kappa_{1} s_{1}^{T} s_{1} \\ \end{aligned}$$
(47)
By using Young’s inequality, we obtain
$$\begin{aligned} s_{1}^{T} \dot{s}_{1} & = \partial \left( {k_{e} + c_{e} \frac{{\dot{\gamma }}}{\gamma }} \right)s_{1}^{T} s_{2} + \partial \left( {k_{e} + c_{e} \frac{{\dot{\gamma }}}{\gamma }} \right)s_{1}^{T} \phi – \kappa_{1} s_{1}^{T} s_{1} \\ & \le \partial \left( {k_{e} + c_{e} \frac{{\dot{\gamma }}}{\gamma }} \right)\left( {\frac{1}{2}s_{1}^{T} s_{1} + \frac{1}{2}s_{2}^{T} s_{2} } \right) + \partial \left( {k_{e} + c_{e} \frac{{\dot{\gamma }}}{\gamma }} \right)\left( {\frac{1}{2}s_{1}^{T} s_{1} + \frac{1}{2}\phi^{T} \phi } \right) – \kappa_{1} s_{1}^{T} s_{1} \\ & \le – \left[ {\kappa_{1} – \partial \left( {k_{e} + c_{e} \frac{{\dot{\gamma }}}{\gamma }} \right)} \right]s_{1}^{T} s_{1} + \frac{1}{2}\partial \left( {k_{e} + c_{e} \frac{{\dot{\gamma }}}{\gamma }} \right)s_{2}^{T} s_{2} + \frac{1}{2}\partial \left( {k_{e} + c_{e} \frac{{\dot{\gamma }}}{\gamma }} \right)\phi^{T} \phi \\ \end{aligned}$$
(48)
Subsequently, based on (42), we obtain
$$s_{2}^{T} \dot{s}_{2} = s_{2}^{T} \left( {J(\psi )M^{ – 1} \left( {\frac{\chi (t)}{{1 + \theta \mu_{1} (t)}} – \frac{{\mu_{2} (t)\delta }}{{1 + \theta \mu_{1} (t)}} + \Delta \tau } \right) + \Gamma – \dot{\alpha }_{f} } \right)$$
(49)
Since \(\rho_{1} \in R\) and \(\mu_{2} > 0\), \(- \rho_{1} \tanh (\rho_{1} /\mu_{2} ) \le 0\). By combining this with (43), we obtain \(s_{2}^{T} \chi (t) \le 0\). Therefore, the following inequality can be obtained:
$$\frac{{s_{2}^{T} \chi (t)}}{{1 + \theta \mu_{1} }} \le \frac{{s_{2}^{T} \chi (t)}}{1 + \theta },\left| {\frac{{s_{2}^{T} \mu_{2} \delta }}{{1 + \theta \mu_{1} }}} \right| \le \left| {\frac{{s_{2}^{T} \delta }}{1 – \theta }} \right|$$
(50)
Given (50) and Lemma 2, we can deduce that
$$\begin{aligned} s_{2}^{T} \tau & = s_{2}^{T} \left( {\frac{\chi (t)}{{1 + \theta \mu_{1} (t)}} – \frac{{\mu_{2} (t)\delta }}{{1 + \theta \mu_{1} (t)}}} \right) \\ & \le s_{2}^{T} \left[ { – \tanh \left( {\frac{{s_{2}^{T} \tau_{p} }}{\omega }} \right)\tau_{p} – \tanh \left( {\frac{{s_{2}^{T} \overline{\delta }}}{\omega }} \right)\overline{\delta }} \right] + \left| {\frac{{s_{2}^{T} \delta }}{1 – \theta }} \right| \\ & \le 1.671\omega + s_{2}^{T} \tau_{p} \\ \end{aligned}$$
(51)
According to (31), (49) and (51), we obtain
$$\begin{aligned} s_{2}^{T} \dot{s}_{2} & \le JM^{ – 1} \left( {1.671\omega + s_{2}^{T} \tau_{p} } \right) + s_{2}^{T} \Gamma – s_{2}^{T} \dot{\alpha }_{f} \\ & \le \left\| {M^{ – 1} } \right\|1.671\omega – s_{2}^{T} \kappa_{2} s_{2} + \left\| {M^{ – 1} } \right\|s_{2}^{T} \kappa_{3} \Xi + s_{2}^{T} JM^{ – 1} \Delta \tau \\ \end{aligned}$$
(52)
From (28), we obtain
$$\phi^{T} \dot{\phi } = \phi^{T} \left( { – \frac{1}{\iota }\phi + \Delta } \right)$$
(53)
where \(\Delta = – \dot{\alpha }\).
Next, by combining (30), (48), (52) and (53), (45) can be rewritten as
$$\begin{aligned} \dot{V} & \le – \left[ {\kappa_{1} – \partial \left( {k_{e} + c_{e} \frac{{\dot{\gamma }}}{\gamma }} \right)} \right]s_{1}^{T} s_{1} + \frac{1}{2}\partial \left( {k_{e} + c_{e} \frac{{\dot{\gamma }}}{\gamma }} \right)s_{2}^{T} s_{2} + \frac{1}{2}\partial \left( {k_{e} + c_{e} \frac{{\dot{\gamma }}}{\gamma }} \right)\phi^{T} \phi \\ & \quad + \left| {M^{ – 1} } \right|1.671\omega – s_{2}^{T} \kappa_{2} s_{2} + \left| {M^{ – 1} } \right|s_{2}^{T} \kappa_{3} \Xi + s_{2}^{T} JM^{ – 1} \Delta \tau \\ & \quad – \frac{1}{\iota }\phi^{T} \phi + \phi^{T} \Delta – \Xi^{T} K_{\Xi } \Xi + \Xi^{T} \Delta \tau \\ \end{aligned}$$
(54)
Considering the compact set \(\Omega_{d} = \left\{ {\left[ {\eta_{d}^{T} ,\dot{\eta }_{d}^{T} ,\ddot{\eta }_{d}^{T} } \right]^{T} :\left\| {\eta_{d} } \right\|^{2} + \left\| {\dot{\eta }_{d} } \right\|^{2} + \left\| {\ddot{\eta }_{d} } \right\|^{2} \le B_{0} ,B_{0} > 0} \right\}\) and \(\Omega_{1} = \left\{ {\left[ {s_{1}^{T} ,s_{2}^{T} ,\phi^{T} } \right]^{T} :V \le \hbar_{0} ,\hbar_{0} > 0} \right\}\), \(\Omega_{d} \times \Omega_{1}\) is likewise a compact set. Then, there is a non-negative continuous \(\beta ( \cdot )\) such that \(\left\| \Delta \right\| \le \beta ( \cdot )\) and \(\beta ( \cdot )\) has maximum \(N\) on \(\Omega_{d} \times \Omega_{1}\)41.
The subsequent inequality is established in alignment with Young’s inequality:
$$\phi^{T} \Delta \le a_{1} \phi^{T} \phi + \frac{{N^{2} }}{{4a_{1} }}$$
(55)
Substituting (55) into (54) yields
$$\begin{aligned} \dot{V} & \le – \left[ {\kappa_{1} – \partial \left( {k_{e} + c_{e} \frac{{\dot{\gamma }}}{\gamma }} \right)} \right]s_{1}^{T} s_{1} + \frac{1}{2}\partial \left( {k_{e} + c_{e} \frac{{\dot{\gamma }}}{\gamma }} \right)s_{2}^{T} s_{2} + \frac{1}{2}\partial \left( {k_{e} + c_{e} \frac{{\dot{\gamma }}}{\gamma }} \right)\phi^{T} \phi \\ & \quad + \left\| {M^{ – 1} } \right\|1.671\omega – s_{2}^{T} \kappa_{2} s_{2} + \left\| {M^{ – 1} } \right\|s_{2}^{T} \kappa_{3} \Xi + s_{2}^{T} JM^{ – 1} \Delta \tau \\ & \quad – \frac{1}{\iota }\phi^{T} \phi + a_{1} \phi^{T} \phi + \frac{{N^{2} }}{{4a_{1} }} – \Xi^{T} K_{\Xi } \Xi + \Xi^{T} \Delta \tau \\ \end{aligned}$$
(56)
Let \(k_{1} = \lambda_{\max } (JM^{ – 1} )\), \(k_{2} = \lambda_{\min } (\kappa_{2} )\), \(k_{3} = \left\| {M^{ – 1} } \right\|\lambda_{\max } (\kappa_{3} )\), and \(k_{\Xi } = \lambda_{\min } (\kappa_{\Xi } )\); we thus obtain
$$\begin{aligned} \dot{V} & \le – \left[ {\kappa_{1} – \partial \left( {k_{e} + c_{e} \frac{{\dot{\gamma }}}{\gamma }} \right)} \right]s_{1}^{T} s_{1} + \frac{1}{2}\partial \left( {k_{e} + c_{e} \frac{{\dot{\gamma }}}{\gamma }} \right)s_{2}^{T} s_{2} + \frac{1}{2}\partial \left( {k_{e} + c_{e} \frac{{\dot{\gamma }}}{\gamma }} \right)\phi^{T} \phi \\ & \quad + \left\| {M^{ – 1} } \right\|1.671\omega – k_{2} \left\| {s_{2} } \right\|^{2} + k_{3} \left\| {s_{2}^{T} } \right\|\left\| \Xi \right\| + k_{1} \left\| {s_{2}^{T} } \right\|\left\| {\Delta \tau } \right\| \\ & \quad – \frac{1}{\iota }\phi^{T} \phi + a_{1} \phi^{T} \phi + \frac{{N^{2} }}{{4a_{1} }} – k_{\Xi } \left\| \Xi \right\|^{2} + \left\| \Xi \right\|\left\| {\Delta \tau } \right\| \\ & \le – \left[ {\kappa_{1} – \partial \left( {k_{e} + c_{e} \frac{{\dot{\gamma }}}{\gamma }} \right)} \right]s_{1}^{T} s_{1} + \frac{1}{2}\partial \left( {k_{e} + c_{e} \frac{{\dot{\gamma }}}{\gamma }} \right)s_{2}^{T} s_{2} + \frac{1}{2}\partial \left( {k_{e} + c_{e} \frac{{\dot{\gamma }}}{\gamma }} \right)\phi^{T} \phi \\ & \quad + \left\| {M^{ – 1} } \right\|1.671\omega – \frac{{k_{2} }}{3}\left( {\left\| {s_{2} } \right\| – \frac{{3k_{3} }}{{2k_{2} }}\left\| \Xi \right\|} \right)^{2} – \frac{{k_{\Xi } }}{3}\left( {\left\| \Xi \right\| – \frac{3}{{2k_{\Xi } }}\left\| {\Delta \tau } \right\|} \right)^{2} \\ & \quad – \left( {\frac{{2k_{\Xi } }}{3} – \frac{{3k_{3}^{2} }}{{4k_{2} }}} \right)\left\| \Xi \right\|^{2} – \left( {\frac{{2k_{2} }}{3} – \frac{{k_{1} }}{2}} \right)\left\| {s_{2} } \right\|^{2} \, – \frac{1}{\iota }\phi^{T} \phi + a_{1} \phi^{T} \phi + \frac{{N^{2} }}{{4a_{1} }} \\ & \quad + \left( {\frac{3}{{4k_{\Xi } }} + \frac{1}{{2k_{1} }}} \right)\left\| {\Delta \tau } \right\|^{2} \\ \end{aligned}$$
(57)
We can derive additional results by manipulating and reorganizing specific terms in the equation provided above
$$\begin{aligned} \dot{V} & \le – \left[ {\kappa_{1} – \partial \left( {k_{e} + c_{e} \frac{{\dot{\gamma }}}{\gamma }} \right)} \right]s_{1}^{T} s_{1} – \left( {\left( {\frac{{2k_{2} }}{3} – \frac{{k_{1} }}{2}} \right) – \frac{1}{2}\partial \left( {k_{e} + c_{e} \frac{{\dot{\gamma }}}{\gamma }} \right)} \right)s_{2}^{T} s_{2} \\ & \quad – \left( {\frac{{2k_{\Xi } }}{3} – \frac{{3k_{3}^{2} }}{{4k_{2} }}} \right)\left\| \Xi \right\|^{2} – \left( {\frac{1}{\iota } – a_{1} – \frac{1}{2}\partial \left( {k_{e} + c_{e} \frac{{\dot{\gamma }}}{\gamma }} \right)} \right)\phi^{T} \phi \\ & \quad + \frac{{N^{2} }}{{4a_{1} }} + \left( {\frac{3}{{4k_{\Xi } }} + \frac{1}{{2k_{1} }}} \right)\left\| {\Delta \tau } \right\|^{2} + \left\| {M^{ – 1} } \right\|1.671\omega \\ & \le – \mu V + \Lambda \\ \end{aligned}$$
(58)
here, \(\mu = \min \left\{ {2\left( {\kappa_{1} – \partial \varsigma } \right),2\left( {\left( {\frac{{2k_{2} }}{3} – \frac{{k_{1} }}{2}} \right) – \frac{1}{2}\partial \varsigma } \right),2\left( {\frac{{2k_{\Xi } }}{3} – \frac{{3k_{3}^{2} }}{{4k_{2} }}} \right),2\left( {\frac{1}{\iota } – a_{1} – \frac{1}{2}\partial \varsigma } \right)} \right\}\), \(\varsigma = k_{e} + c_{e} \frac{{\dot{\gamma }}}{\gamma }\), \(\Lambda = \frac{{N^{2} }}{{4a_{1} }} + \left( {\frac{3}{{4k_{\Xi } }} + \frac{1}{{2k_{1} }}} \right)\left\| {\Delta \tau } \right\|^{2} + \left\| {M^{ – 1} } \right\|1.671\omega\). With proper adjustment of the designed parameters such that \(\mu > \Lambda /p\), we have \(\dot{V} < 0\) on \(V = p\). Consequently,\(V \le p\) represents an invariable set; that is , if \(V(0) \le p\), then \(V(t) \le p\) for \(t \ge 0\). Then, by solving (58), we obtain
$$V(t) \le e^{ – \mu t} V(0) + \frac{\Lambda }{\mu }(1 – e^{ – \mu t} )$$
(59)
Therefore, when \(V(t) \in {\mathcal{L}}_{\infty }\), \(\left\| {s_{1} } \right\|\), \(\left\| {s_{2} } \right\|\), and \(\left\| \Xi \right\|\) are bounded. From the definition of \(s_{1}\), \(\left\| e \right\|\), \(\left\| z \right\|\), and \(\left\| \alpha \right\|\) are also bounded. Furthermore,\(\left\| e \right\|\) and \(\left\| {s_{2} } \right\|\) are bounded, which means \(\left\| \eta \right\|\) and \(\left\| \nu \right\|\) are bounded. Since the control input is a continuous function about \(s_{2}\), \(\alpha\), \(\Xi\) and \(\hat{\Gamma }\),the control law \(\tau_{c}\) are also bounded. From the analysis provided, it is evident that all signals are ultimately uniformly bounded. In the meantime, according to inequality (10), it can be inferred that for \(\forall t \ge 0\), \(z \in ( – \lambda_{1} \varphi ,\lambda_{2} \varphi )\). By selecting \(\lambda_{1} = \lambda_{2} = 1\), one obtains \(\left\| z \right\| < \varphi\). By combining with (5), (6), and (17), it can be seen that
$$\left\| e \right\| = \left( {k_{e} + c_{e} \frac{{\dot{\gamma }}}{\gamma }} \right)^{ – 1} z < \left\{ {\begin{array}{*{20}l} {\frac{{\varphi (T + t_{0} – t)}}{{k_{e} (T + t_{0} – t) + c_{e} h}},} \hfill & {t \in [t_{0} ,T)} \hfill \\ {\frac{\varphi }{{k_{e} }},} \hfill & {t \in [T,\infty )} \hfill \\ \end{array} } \right.$$
(60)
Obviously, according to (60), the trajectory tracking errors can converge to the prescribed performance constraints within the prescribed time \(T\), namely \(\left\| e \right\| < \frac{\varphi }{{k_{e} }}\).
In addition, the Zeno behavior is an issue that cannot be ignored in event triggering control systems, which refers to the infinite number of samples occurring within a finite time, i.e. the sampling time interval tends to zero. Subsequently, we will demonstrate that the introduced event-triggered mechanism can avoid Zeno behavior. Presumably, there is a \(t^{ * } > 0\) such that \(t_{k + 1} – t_{k} \ge t_{k}^{*} \forall k \in z^{ + }\). In the light of the event triggering error \(e_{\tau } (t) = \chi (t) – \tau (t)\), one can deduce that
$$\frac{d}{dt}\left| {e_{\tau } (t)} \right| = sign(e_{\tau } (t))\dot{e}_{\tau } (t) \le \left| {\dot{\chi }(t)} \right|$$
(61)
From (43), we can deduce that \(\chi (t)\) is differentiable and satisfies \(\left| {\dot{\chi }(t)} \right| \le \hbar\). Due to \(e_{\tau } (t_{k} ) = 0\) and \(\lim_{{t \to t_{k + 1} }} e_{\tau } (t) = K\), the minimum sampling interval \(t_{k}^{ * }\) satisfies \(t_{k}^{ * } \ge K/\hbar\). Hence, Zeno behavior can be prevented. The proof of Theorem 2 is complete.
Remark 5
Although the prescribed-time control algorithm has many advantages, the contradiction between the convergence time of the prescribed-control and the control energy consumption of the USV tracking system seems to be inevitable. The specific reasons are as follows: (1) The prescribed-time control usually adopts the time-varying gain technology. As the system approaches the equilibrium point, the control gain may increase indefinitely. This rapidly growing control gain requires the thruster to provide greater torque or thrust to achieve rapid tracking, resulting in a significant increase in energy consumption. (2) To achieve rapid tracking in a prescribed short time, the system is required to respond quickly to the control input. This may require sufficient flexibility and strength of system dynamics, and large systems such as ships often have large inertia and delay, so greater energy input is needed to overcome these limitations and achieve rapid tracking. (3) In practical applications, there are physical limitations on the thrust and response speed of the thruster. To meet the requirements of rapid tracking, the propeller may need to operate under extreme conditions, which not only increases energy consumption, but also may affect the service life of the thruster. (4) It is difficult to establish an accurate mathematical model of the USV tracking system, which limits the design and optimization of the control strategy. The inaccuracy of the model may lead to over-adjustment or insufficient control input, which in turn affects energy consumption and tracking accuracy. In summary, the contradiction between the convergence time of the ship’s prescribed-time trajectory tracking control and the control energy consumption is mainly due to the accuracy of system model, external disturbances, the physical limitations of the thrusters, and the complexity of fault-tolerant control. To solve this contradiction, it is necessary to comprehensively consider the innovation of control theory, the accuracy of system model, the progress of sensor technique and the improvement of thruster technology. More importantly, we must make a trade-off between control performance (accuracy or efficiency) and energy consumption.
