1 Introduction

Motion control for an MSV plays a increasingly pivotal and irreplaceable behavior for realizing and accomplishing various arduous missions and modern ocean engineering. Autonomous trajectory tracking control of an MSV is one of the most important roles in the motion control. Achieving and acquiring superior high-precision and fast tracking performance of an MSV has been a challenging topic due to some adverse effects including strong coupling, high nonlinearity, unknown ocean environment, etc. As a consequence, considerable attentions have been attracted from control theoreticians and engineers [1, 2].

Autonomous trajectory tracking control means that an MSV autonomously tracks a spatial and temporal trajectory under unmanned operations. Its core and key point is the design of control algorithms. Among the published results, some remarkable control methods have been presented, such as model predictive control [3, 4], adaptive fuzzy control [5, 6], adaptive neural network control [7, 8], event-triggered control [9, 10], H\( \infty \) control [11, 12], and so on. As we all known, as a strong robust control, sliding mode control has been widely applied to various uncertain nonlinear systems due to its remarkable characteristic including implemented simplicity, fast globally convergence, order reduction and insensitivity to model uncertainties and external disturbances [13,14,15,16,17]. In terms of the trajectory tracking control of an MSV, a sliding mode trajectory tracking control scheme is proposed for an MSV in [18], where a disturbance observation method is introduced to reduce the inherent oscillation, and the tracking errors are only with asymptotic convergence in infinite time. As a negation, finite-time control with high-accuracy and fast convergence is proposed [19, 20]. However, the setting time of finite-time control is highly related to initial conditions. It prevents predicting the setting time in advance when initial conditions are unavailable or unmeasurable. In response, fixed-time control is proposed to deal with the above issue [21]. What’s more, fixed-time control has the inherent capacities in faster transient and higher steady-state accuracy than finite-time one [22]. In [23], based on an FXDO, a fixed-time sliding mode trajectory tracking control scheme is developed, and the tracking errors can converge to the origin within a setting time, where the FXDO is employed to alleviate the chattering phenomenon in a direct feedforward compensation manner. In addition to the disturbance observation method, the boundary layer method and the adaptive method can also alleviate the chattering phenomenon. Nevertheless, the tracking performance is perhaps deteriorated. Accordingly, it would be a promising scheme for improving the tracking performance of an MSV by virtue of fixed-time sliding mode control and a disturbance observation method.

Although time optimization control can greatly improve and enhance the tracking performance of an MSV, the transient performance including convergence rate and overshoot cannot be ignored [22, 24]. If the output tracking errors are constrained in a known manner, the transient and steady-state performance can be guaranteed. Moreover, the constrained output tracking errors are helpful to make collisions avoided and retain an MSV in a safe channel during the navigation. It is noteworthy that input saturation is a universal and widespread physical constraint in most nonlinear systems [1, 25,26,27]. It can weaken system performance, and even cause instability. Hence, considering input saturation and prescribed performance constraints is crucial for the trajectory tracking control of an MSV. In [27], considering input saturation and prescribed performance constraints, an anti-saturation trajectory tracking control scheme for an MSV is proposed based on an auxiliary dynamic system. Subsequently, fixed-time auxiliary dynamic systems in [28] and [26] are respectively proposed to deal with the saturated nonlinearity. In [29], considering input saturation, a trajectory tracking control scheme for an MSV is developed, where a gaussian error function-based continuous differentiable asymmetric saturation model is introduced to solve the problem that the non-smooth input saturation function matrix is not differentiable. In addition to the two methods mentioned above, the disturbance observation method in [30] and [24] is also used to deal with the saturated nonlinearity. However, the premise assumes that lumped disturbances including the input saturation function matrix is differentiable. Therefore, it is necessary to design a novel disturbance observer to solve this problem.

As a consequence, motivated by the aforementioned analyses and statements, this paper develops a fixed-time nonsingular terminal sliding mode trajectory tracking control scheme for an MSV in the presence of model uncertainties, external disturbances, input saturation and prescribed performance constraints. More specifically, the main contributions of this paper are outlined below.

  1. (1)

    An FXDO is designed. In comparison with [31], the designed FXDO not only realizes fixed-time stability, but also solves the design method problem. In comparison with [32, 33], the designed FXDO can get rid of choosing relative complex parameters. In comparison with [33,34,35,36,37,38], the disturbance estimation errors can converge to the origin within a setting time. Moreover, the designed FXDO can eradicate the assumption that the derivatives of external disturbances or lumped disturbances exist in most disturbance observer designs.

  2. (2)

    An FXNTSMM with simple structures is designed. In comparison with [13, 39], the designed FXNTSMM reduces the calculation burden of the system. In comparison with [22, 26, 40, 41], the designed FXNTSMM ensures that the system states converge to the origin within a setting time in the sliding phase.

  3. (3)

    A fixed-time trajectory tracking control scheme in the presence of model uncertainties, external disturbances, input saturation and prescribed performance constraints is proposed based on the FXDO, the FXNTSMM and a prescribed performance function. In comparison with [24, 30] by disturbance observation methods dealing with input saturation, the proposed FXDO-based anti-saturation trajectory tracking control scheme is more reasonable, which avoids an unreasonable assumption that lumped disturbances including the input saturation function matrix are differentiable. In comparison with [7, 25,26,27,28, 39, 42] by auxiliary dynamic systems dealing with input saturation, the proposed FXDO-based anti-saturation trajectory tracking control scheme not only improves the system robustness, but also simplifies the one structures without introducing an auxiliary dynamic system. In comparison with [2, 14], in addition to considering external disturbances and model uncertainties, the proposed scheme also considers input saturation and prescribed performance constraints, which is more in line with practical scenarios during the navigation.

In what follows, preliminaries are described in Sect. 2. In Sect. 3, an MSV model and a control problem are provided. Section 4 shows main results including an FXDO, an FXNTSMM and a trajectory tracking control scheme. The effectiveness of the proposed control scheme is demonstrated in Sect. 5. Conclusions are drawn in Sect. 6.

2 Preliminaries

In terms of a vector, \( {\left\| \bullet \right\| } \) denotes an Euclidean 2-norm. In terms of a scalar, \( \left| \bullet \right| \) denotes an absolute value. \( si{g^*}(\bullet ) = {\left| \bullet \right| ^*}sign(\bullet )\).

Lemma 1

([43]): The system is given below

$$\begin{aligned} {\dot{z}}(t) = f(z(t)),\mathrm{{ }}z(0) = {z_0} \end{aligned}$$
(1)

where \( z \in {\Re ^n} \), \( f(z):{\Re ^n} \rightarrow {\Re ^n} \) is continuous on \( {\Re ^n} \), \( f(0) = 0 \). A continuous positive-definite function \( V(z):{\Re ^n} \rightarrow \Re _0^+ \) and constants \( a'>0, b'>0, c' > 0 \), \( \alpha ' > 1 \) and \(0< \beta ' <1 \) with

$$\begin{aligned} {\dot{V}}(z) \le - a'V(z) - b'{V^{\alpha '} }(z) - c'{V^{\beta '} }(z),z \in {\Re ^n}\backslash \{ 0\} \end{aligned}$$
(2)

Then, the system (1) is with globally fast fixed-time stability, and the setting time is expressed as follows

$$\begin{aligned} T \le \frac{1}{{a'(1 - \beta ' )}}\ln (\frac{{a' + c'}}{c'}) + \frac{1}{{b'(\alpha ' - 1)}} \end{aligned}$$
(3)

Lemma 2

([44]): If \( {x_i} \in \Re (i = 1,2,3,...,n) \) and \( p>0 \), \( {(\left| {{x_1}} \right| + \left| {{x_2}} \right| + \cdots + \left| {{x_n}} \right| )^p} \le \max ({n^{p - 1}},1)({\left| {{x_1}} \right| ^p} + {\left| {{x_2}} \right| ^p} + \cdots + {\left| {{x_n}} \right| ^p}) \).

Definition

([45]): A smooth function \( \vartheta :{\Re ^+} \rightarrow {\Re ^+} \) is called as a performance function if:

  1. (1)

    \( \vartheta (t) \) is positive and decreasing.

  2. (2)

    \( \mathop {\lim }\limits _{t \rightarrow \infty } \vartheta (t) = {\vartheta _\infty } > 0 \).

3 System modelling and problem formulation

In this section, a three-degree-of-freedom MSV is described as follows [2]

$$\begin{aligned} \left\{ \begin{aligned}&{\dot{\eta }} \mathrm{{ = }}R(\psi )v\mathrm{{ }}\\&M{\dot{v}} + C(v)v + D(v)v = \tau + d\mathrm{{ }} \end{aligned} \right. \end{aligned}$$
(4)

where \( \eta = {[x,y,\psi ]^T} \) and \( v = {[u,\upsilon ,r]^T} \) are measurable state vectors of the system. (x, y) and \( \psi \) are respectively called as positions and a yaw angle, and v is called as a velocity vector. \( \tau = {[{\tau _1},{\tau _2},{\tau _3}]^T} \) is called as a control input vector. \( d(t) = {[{d_1}(t),{d_2}(t),{d_3}(t)]^T} \) is called as an external disturbance vector. M is called as an inertia matrix. C(v) is called as a coriolis and centripetal matrix. D(v) is called as a damping matrix. The following \( R(\psi ) \) is called as a rotation matrix.

$$\begin{aligned} R(\psi ) = \left[ {\begin{array}{*{20}{c}} {\cos (\psi )}&{}{ - \sin (\psi )}&{}0\\ {\sin (\psi )}&{}{\cos (\psi )}&{}0\\ 0&{}0&{}1 \end{array}} \right] \end{aligned}$$
(5)

The properties pertaining to \( R(\psi ) \) are presented below [2]

$$\begin{aligned}&{\dot{R}}(\psi ) = R(\psi )S(r) \end{aligned}$$
(6)
$$\begin{aligned}&{R^T}(\psi )S(r)R(\psi ) = R(\psi )S(r){R^T}(\psi ) = S(r)\mathrm{{ }} \end{aligned}$$
(7)
$$\begin{aligned}&S(r) = \left[ {\begin{array}{*{20}{c}} 0&{}{ - r}&{}0\\ r&{}0&{}0\\ 0&{}0&{}0 \end{array}} \right] \mathrm{{ }} \end{aligned}$$
(8)
$$\begin{aligned}&{R^T}(\psi )R(\psi ) = I\quad and \quad \left\| {R(\psi )} \right\| = 1 \end{aligned}$$
(9)

The MSV (4) is converted below by coordinate transformation [2]

$$\begin{aligned} \left\{ {\begin{array}{*{20}{l}} {{\dot{\eta }} = w}\\ {{\dot{w}} = R(\psi ){M^{ - 1}}\tau + \chi + R(\psi ){M^{ - 1}}d }\\ {{{{\dot{\eta }} }_d} = {w_d}} \end{array}} \right. \end{aligned}$$
(10)

where \( \chi = R(\psi )S(r)v - R(\psi ){M^{ - 1}}C(v)v - R(\psi ){M^{ - 1}}D(v)v \). \( {\eta _d} = {[{x_d},{y_d},{\psi _d}]^T} \) is called as a desired trajectory vector. \( w = {[{w_1},{w_2},{w_3}]^T} \) is called as a velocity vector under an earth-fixed coordinate frame. \( {w_d} = {[{w_{d1}},{w_{d2}},{w_{d3}}]^T} \) is called as a desired velocity vector under an earth-fixed coordinate frame. \( {\eta _d} = {[{\eta _{d1}},{\eta _{d2}},{\eta _{d3}}]^T} = {[{x_d},{y_d},{\psi _d}]^T} \) is called as desired positions and a desired yaw angle.

The MSV (10) with input saturation is presented below

$$\begin{aligned} \left\{ {\begin{array}{*{20}{l}} {{\dot{\eta }} = w}\\ {{\dot{w}} = R(\psi ){M^{ - 1}}{\tau _c} + R(\psi ){M^{ - 1}}\Delta \tau + \chi + R(\psi ){M^{ - 1}}d}\\ {{{{\dot{\eta }} }_d} = {w_d}} \end{array}} \right. \end{aligned}$$
(11)

where \( \Delta \tau = \tau - {\tau _c} \). \( {\tau _c} = {[{\tau _{c1}},{\tau _{c2}},{\tau _{c3}}]^T} \) is called as a command control input vector to be designed. The relationship between \( \tau \) and \( {\tau _c} \) is listed below [27]

$$\begin{aligned} \tau = \left\{ {\begin{array}{*{20}{l}} {{\tau _{\max }},\mathrm{{~~~if }}~~~{\tau _c} > {\tau _{\max }}}\\ {{\tau _c},~~~~~~\mathrm{{if }}~~~{\tau _{\min }} \le {\tau _c} \le {\tau _{\max }}}\\ {{\tau _{\min }},\mathrm{{~~~if}}~~~{\tau _c} < {\tau _{\min }}} \end{array}} \right. \end{aligned}$$
(12)

where \( {{\tau _{\max }}} \) and \( {{\tau _{\min }}} \) are respectively called as maximum and minimum control input vectors.

Defining the tracking error vectors is presented below

$$\begin{aligned} \left\{ {\begin{array}{*{20}{l}} {{\eta _e} = \eta - {\eta _d}}\\ {{w_e} = w - {w_d}} \end{array}} \right. \end{aligned}$$
(13)

where \( {\eta _e} = {[{\eta _{e1}},{\eta _{e2}},{\eta _{e3}}]^T} = {[{x_e},{y_e},{\psi _e}]^T} \), \( {w_e} = {[{w_{e1}},{w_{e2}},{w_{e3}}]^T} \).

In view of (11) and (13), the error dynamics with input saturation is developed below

$$\begin{aligned} \left\{ {\begin{array}{*{20}{l}} {{{{\dot{\eta }} }_e} = {w_e}}\\ {{{{\dot{w}}}_e} = R(\psi ){M^{ - 1}}{\tau _c} + R(\psi ){M^{ - 1}}\Delta \tau + \chi + R(\psi ){M^{ - 1}}d - {{{\dot{w}}}_d}} \end{array}} \right. \end{aligned}$$
(14)

A prescribed performance function is introduced as follows [45]

$$\begin{aligned} {\vartheta _i}(t) = ({\vartheta _{i0}} - {\vartheta _{i\infty }}){e^{ - \omega t}} + {\vartheta _{i\infty }}(i = 1,2,3) \end{aligned}$$
(15)

where \( \omega \), \( {\vartheta _{i0}} \) and \( {\vartheta _{i\infty }} \) are called as positive scale parameters.

And then, in the light of (15), the following tracking error is constrained as follows

$$\begin{aligned} - {\varepsilon _i}{\vartheta _i}(t)< {\eta _{ei}}< {\varepsilon _i}{\vartheta _i}(t),0 < {\varepsilon _i} \le 1 \end{aligned}$$
(16)

A transformation function is presented as follows

$$\begin{aligned} {J_{1i}} = \frac{1}{2}\ln \left( \frac{{{\eta _{ei}} + {\varepsilon _i}{\vartheta _i}(t)}}{{{\varepsilon _i}{\vartheta _i}(t) - {\eta _{ei}}}}\right) \end{aligned}$$
(17)

The time derivative of (17) is obtained as follows

$$\begin{aligned} {{\dot{J}}_{1i}} = \frac{{{\varepsilon _i}({{{\dot{\eta }} }_{ei}}{\vartheta _i}(t) - {\eta _{ei}}{{{\dot{\vartheta }} }_i}(t))}}{{({\eta _{ei}} + {\varepsilon _i}{\vartheta _i}(t))({\varepsilon _i}{\vartheta _i}(t) - {\eta _{ei}})}} \end{aligned}$$
(18)

where \( {{\dot{J}}_{1i}} = {J_{2i}} \).

Therefore, combining with (14), we have

$$\begin{aligned} \left\{ {\begin{array}{*{20}{l}} {{{\dot{J}}_1} = {J_2}}\\ {{{\dot{J}}_2} = H\Theta R(\psi ){M^{ - 1}}{\tau _c} - H\Theta {{{\dot{w}}}_d} - H\ddot{\Theta }{\eta _e} + L} \end{array}} \right. \end{aligned}$$
(19)

where \( L = H\Theta R(\psi ){M^{ - 1}}\Delta \tau + H\Theta \chi + \dot{H}(\Theta {{{\dot{\eta }} }_e} - {\dot{\Theta }} {\eta _e}) \). \( {J_1} = {[{J_{11}},{J_{12}},{J_{13}}]^T} \), \( {J_2} = {[{J_{21}},{J_{22}},{J_{23}}]^T} \), \( \Theta = \mathrm{{diag}}({\vartheta _1}(t),{\vartheta _2}(t),{\vartheta _3}(t)) \), \( H = \mathrm{{diag}}({h_1},{h_2},{h_3}) \), \( {h_i} = \frac{{{\varepsilon _i}}}{{({\eta _{ei}} + {\varepsilon _i}{\vartheta _i}(t))({\varepsilon _i}{\vartheta _i}(t) - {\eta _{ei}})}} \).

Assumption

  1. (1)

    The desired trajectory vector \( {\eta _d} \) is bounded, and its first and second order derivatives are also bounded.

  2. (2)

    \( L = {[{L_1},{L_2},{L_3}]^T} \) is regarded as a lumped disturbance vector with \( \left\| L \right\| \le \delta \).

  3. (3)

    The matrices C(v) and D(v) are unknown.

In this paper, the main objective is to design a fixed-time trajectory tracking control scheme based on an FXDO, an FXNTSMM and a prescribed performance function for the MSV (11) in the presence of model uncertainties, external disturbances, input saturation and prescribed performance constraints, which not only ensures the system is with fixed-time stability, but also guarantees the tracking error vector \( {\eta _e} \) is within prescribed constraints. The control objectives can be stated mathematically as follows

$$\begin{aligned} \mathop {\lim }\limits _{t \rightarrow T} {\eta _e} = 0 \end{aligned}$$
(20)

and

$$\begin{aligned} - {\varepsilon _i}{\vartheta _i}(t)< {\eta _{ei}}< {\varepsilon _i}{\vartheta _i}(t),0 < {\varepsilon _i} \le 1, (i = 1,2,3) \end{aligned}$$
(21)

4 Main results

4.1 Design of FXDO

In this subsection, in the light of (19), an FXDO is designed as follows

$$\begin{aligned} \left\{ {\begin{array}{*{20}{l}} {e' = {J_2} - \xi }\\ {{\bar{s}} = e' + \int _0^t {gd\tau } }\\ {g = {c_\alpha }e' + {c_\beta }si{g^{{\sigma _1}}}(e') + {c_\gamma }si{g^{{\sigma _2}}}(e')}\\ {{\dot{\xi }} = H\Theta R(\psi ){M^{ - 1}}{\tau _c} - H\Theta {{{\dot{w}}}_d} - H\ddot{\Theta }{\eta _e} + {\hat{L}}}\\ {{\hat{L}} = lsign({\bar{s}}) + {k_1}{\bar{s}} + {k_2}si{g^{{p_1}}}({\bar{s}}) + {k_3}si{g^{{p_2}}}({\bar{s}}) + g} \end{array}} \right. \end{aligned}$$
(22)

where \( {k_{1}} > 0 \), \( {k_{2}} > 0 \), \( {k_{3}} > 0 \), \( {c_\alpha } > 0 \), \( {c_\beta } > 0 \), \( {c_\gamma } > 0 \), \( {p_1} > 1 \), \( 0< {p_2} < 1 \), \( {\sigma _1} > 1 \), \( 0< {\sigma _2} < 1 \) and \( \left\| L \right\| \le \delta <l \).

Theorem 1

Based on the (19), if an FXDO is designed as (22), the lumped disturbance estimation error vector \( {{\tilde{L}}} \) can converge to the origin within a setting time \( {T_0} \) by choosing appropriate parameters, and the setting time \( {T_0} \) is determined as follows

$$\begin{aligned} {T_0} \le {T_1} + {T_2} \end{aligned}$$
(23)

where \( {T_1} \le \frac{1}{{{k_1}(1 - {p_2})}}\ln ({2^{\frac{{1 - {p_2}}}{2}}}\frac{{{k_1}}}{{{k_3}}} + 1) + {3^{\frac{{{p_1} - 1}}{2}}}{2^{\frac{{1 - {p_1}}}{2}}}\frac{1}{{{k_2}({p_1} - 1)}} \), \( {T_2} \le \frac{1}{{{c_\alpha }(1 - {\sigma _2})}}\ln ({2^{\frac{{1 - {\sigma _2}}}{2}}}\frac{{{c_\alpha }}}{{{c_\gamma }}} + 1) + {3^{\frac{{{\sigma _1} - 1}}{2}}}{2^{\frac{{1 - {\sigma _1}}}{2}}}\frac{1}{{{c_\beta }({\sigma _1} - 1)}} \).

Proof

A Lyapunov function is selected as follows

$$\begin{aligned} {V_{{\bar{s}}}} = \frac{1}{2}{{{\bar{s}}}^T}{\bar{s}} \end{aligned}$$
(24)

The time derivative of (24) is obtained as follows

$$\begin{aligned} {\dot{V}_{{\bar{s}}}} = {{{\bar{s}}}^T}\dot{{\bar{s}}} \end{aligned}$$
(25)

Combining with (19) and (22), we have

$$\begin{aligned} \dot{{\bar{s}}}&= \dot{e}' + g\nonumber \\&= {{\dot{J}}_2} - {\dot{\xi }} + g\nonumber \\&= H\Theta R(\psi ){M^{ - 1}}{\tau _c} - H\Theta {{{\dot{w}}}_d} - H\ddot{\Theta }{\eta _e} + L\nonumber \\&- H\Theta R(\psi ){M^{ - 1}}{\tau _c} + H\Theta {{{\dot{w}}}_d} + H\ddot{\Theta }{\eta _e} - {\hat{L}} + g\nonumber \\&= L - {\hat{L}} + g\nonumber \\&= L - lsign({\bar{s}}) - {k_1}{\bar{s}} - {k_2}si{g^{{p_1}}}({\bar{s}}) - {k_3}si{g^{{p_2}}}({\bar{s}}) \end{aligned}$$
(26)

In view of \( \left\| L \right\| \le \delta <l \), combining with (25) and (26), we have

$$\begin{aligned} {\dot{V}_{{\bar{s}}}} = {{{\bar{s}}}^T}\dot{{\bar{s}}} \le {{{\bar{s}}}^T}( - {k_1}{\bar{s}} - {k_2}si{g^{{p_1}}}({\bar{s}}) - {k_3}si{g^{{p_2}}}({\bar{s}})) \end{aligned}$$
(27)

Based on Lemma 2, (27) can be written as follows

$$\begin{aligned} {\dot{V}_{{\bar{s}}}}&\le {{{\bar{s}}}^T}( - {k_1}{\bar{s}} - {k_2}si{g^{{p_1}}}({\bar{s}}) - {k_3}si{g^{{p_2}}}({\bar{s}}))\nonumber \\&\le - {k_1}{{{\bar{s}}}^T}{\bar{s}} - {k_2}{{{\bar{s}}}^T}si{g^{{p_1}}}({\bar{s}}) - {k_3}{{{\bar{s}}}^T}si{g^{{p_2}}}({\bar{s}})\nonumber \\&\le - 2{k_1}{V_{{\bar{s}}}} - {2^{\frac{{{p_1} + 1}}{2}}}{3^{\frac{{1 - {p_1}}}{2}}}{k_2}V_{{\bar{s}}}^{\frac{{{p_1} + 1}}{2}} - {2^{\frac{{{p_2} + 1}}{2}}}{k_3}V_{{\bar{s}}}^{\frac{{{p_2} + 1}}{2}} \end{aligned}$$
(28)

Due to \( {p_1} > 1 \) and \( 0< {p_2} < 1 \), we have \( \frac{{{p_1} + 1}}{2} > 1 \) and \( 0< \frac{{{p_2} + 1}}{2} < 1 \). Based on Lemma 1, combining with (28), \( {V_{{\bar{s}}}} \) can converge to the origin within a setting time \( {T_1} \), and the setting time \( {T_1} \) is determined as follows

$$\begin{aligned} {T_1}&\le \frac{1}{{{k_1}(1 - {p_2})}}\ln ({2^{\frac{{1 - {p_2}}}{2}}}\frac{{{k_1}}}{{{k_3}}} + 1) \nonumber \\&\quad + {3^{\frac{{{p_1} - 1}}{2}}}{2^{\frac{{1 - {p_1}}}{2}}}\frac{1}{{{k_2}({p_1} - 1)}} \end{aligned}$$
(29)

Therefore, when \( t > {T_1 } \), it yields

$$\begin{aligned} {V_{{\bar{s}}}} = {\dot{V}_{{\bar{s}}}} = 0 \end{aligned}$$
(30)

Further, according to the definition of \( {V_{{\bar{s}}}} \), we have

$$\begin{aligned} {\bar{s}} = \dot{{\bar{s}}} = 0 \end{aligned}$$
(31)

When \( {\bar{s}} = 0 \), we have

$$\begin{aligned} \dot{e}' = - {c_\alpha }e' - {c_\beta }si{g^{{\sigma _1}}}(e') - {c_\gamma }si{g^{{\sigma _2}}}(e') \end{aligned}$$
(32)

Similarly, \( e' \) and \( {\dot{e}'} \) can converge to the origin within a setting time \( {T_2} \), and the setting time \( {T_2} \) is determined as follows

$$\begin{aligned} {T_2}&\le \frac{1}{{{c_\alpha }(1 - {\sigma _2})}}\ln ({2^{\frac{{1 - {\sigma _2}}}{2}}}\frac{{{c_\alpha }}}{{{c_\gamma }}} + 1) \nonumber \\&\quad + {3^{\frac{{{\sigma _1} - 1}}{2}}}{2^{\frac{{1 - {\sigma _1}}}{2}}}\frac{1}{{{c_\beta }({\sigma _1} - 1)}} \end{aligned}$$
(33)

Combining with (26), we have \( {\tilde{L}} = L - {\hat{L}} = \dot{e}'=0 \), i.e., \( {\tilde{L}} = 0 \). According to [46], when \( {\bar{s}} = 0 \) and \( e' = 0 \), the equivalent value \( {[lsign(0)]_{eq}} \) will compensate for the lumped disturbance vector L. This completes the proof. \(\square \)

Remark 1

In terms of disturbance observers, in comparison with [31], the designed FXDO not only realizes fixed-time stability, but also solves the design method problem. Additionally, the following FXDO for the system in [31] is developed as (34). In comparison with [32, 33], the designed FXDO can get rid of choosing relative complex parameters. In comparison with [33,34,35,36,37,38], the disturbance estimation errors can converge to the origin within a setting time. Moreover, the designed FXDO can eradicate the assumption that the derivatives of external disturbances or lumped disturbances exist in most disturbance observer designs.

$$\begin{aligned} \left\{ {\begin{array}{*{20}{l}} {{e_x} = x - {\hat{x}}}\\ {\dot{{\hat{x}}} = A{\hat{x}} + Du + C{\hat{f}}}\\ {s = {e_x} + \int _0^t {{c_\alpha }{e_x} + {c_\beta }si{g^{{\sigma _1}}}({e_x}) + {c_\gamma }si{g^{{\sigma _2}}}({e_x})d\tau } }\\ {G = A{e_x} + {c_\alpha }{e_x} + {c_\beta }si{g^{{\sigma _1}}}({e_x}) + {c_\gamma }si{g^{{\sigma _2}}}({e_x})}\\ {{\hat{f}} = {C^{ - 1}}(lsign(s) + {k_1}s + {k_2}si{g^{{p_1}}}(s) + {k_3}si{g^{{p_2}}}(s)) + {C^{ - 1}}G} \end{array}} \right. \end{aligned}$$
(34)

4.2 FXDO-based fixed-time sliding mode trajectory tracking control

The control frame diagram of the fixed-time control system for the MSV (11) is presented in Fig. 1.

Fig. 1
figure 1

The control frame diagram of the fixed-time control system for the MSV (11)

Full size image

In this subsection, an FXNTSMM with simple structures is designed as follows

$$\begin{aligned} S&= si{g^{{q_1}}}({J_2}) + {a_1}si{g^{{q_2}}}({J_1}) \nonumber \\&\quad + {b_1}si{g^{{q_3}}}({J_1}) + {c_1}si{g^{{q_1}}}({J_1}) \end{aligned}$$
(35)

where \( {a_{1}}>0, {b_{1}}>0 \), \( {c_{1}}>0 \), \( 1< {q_{2}}< {q_{1}}< {q_{3}} < 2 \), \( {q_{1}} = \frac{{{m_1}}}{{{n_1}}} \) is with \( {m_1} \) and \( {n_1} \) being positive odd integers. \( S = {[{S_1},{S_2},{S_3}]^T} \).

Differentiating (35) with respect to time, one has

$$\begin{aligned}&{\dot{S}}\mathrm{{ }} = {q_1}\mathrm{{diag}}({\left| {{J_{2i}}} \right| ^{{q_1} - 1}}){{\dot{J}}_2} + {a_1}{q_2}\mathrm{{diag}}({\left| {{J_{1i}}} \right| ^{{q_2} - 1}}){J_2}\mathrm{{ }} \nonumber \\&\qquad + {b_1}{q_3}\mathrm{{diag}}({\left| {{J_{1i}}} \right| ^{{q_3} - 1}}){J_2} + {c_1}{q_1}\mathrm{{diag}}({\left| {{J_{1i}}} \right| ^{{q_1} - 1}}){J_2} \end{aligned}$$
(36)

Combining with (35) and the FXDO (22), a fixed-time nonsingular terminal sliding mode trajectory tracking control law is presented as follows

$$\begin{aligned} {\tau _c}\mathrm{{ }}&= M{R^{ - 1}}(\psi ){\Theta ^{ - 1}}{H^{ - 1}}\nonumber \\&\quad {}\left[ \frac{1}{{{q_1}}}\mathrm{{diag}}({\left| {{J_{2i}}} \right| ^{1 - {q_1}}})\frac{{2\theta }}{\pi }\mathrm{{diag}}(\rho ({\left| {{J_{2i}}} \right| ^{{q_1} - 1}}))( - {\lambda _0}si{g^{{q_4}}}(S) \right. \nonumber \\&\quad \left. - {\lambda _1}si{g^{{q_5}}}(S) - {\lambda _2}S)\right] + M{R^{ - 1}}(\psi ){\Theta ^{ - 1}}{H^{-1}}\nonumber \\&\quad {}\left[ \frac{1}{{{q_1}}}( - {a_1}{q_2}\mathrm{{diag}}({\left| {{J_{1i}}} \right| ^{{q_2} - 1}}) - {b_1}{q_3}\mathrm{{diag}}({\left| {{J_{1i}}} \right| ^{{q_3} - 1}})\right. \nonumber \\&\quad \left. - {c_1}{q_1}\mathrm{{diag}}({\left| {{J_{1i}}} \right| ^{{q_1} - 1}}))si{g^{2 - {q_1}}}({J_2})\right] \nonumber \\&\quad + M{R^{ - 1}}(\psi ){\Theta ^{ - 1}}{H^{ - 1}}(H\Theta {{{\dot{w}}}_d} + H\ddot{\Theta }{\eta _e} - {\hat{L}}) \end{aligned}$$
(37)

where \( \rho (x) = \left\{ {\begin{array}{*{20}{l}} {\sin (\frac{\pi }{2}\frac{x}{\theta })\mathrm{{ ~~~if }}\left| x \right| \le \theta }\\ {1\mathrm{{ ~~~~~~~~~~~if }}\left| x \right| > \theta } \end{array}} \right. \), \( {\lambda _{0}}>0, {\lambda _{1}}>0\), \( {\lambda _{2}}>0 \), \( {q_{4}}>1 \), \( 0<{q_{5}}<1 \). It is worth noting that \( \frac{{2\theta \rho (x)}}{{\pi x}} \rightarrow 1 \) holds when \( \frac{{\pi x}}{{2\theta }} \rightarrow 0 \). Therefore, the control input signals do not tend to infinity. In this paper, \( {\left| x \right| > \theta } \) and \( {\left| x \right| \le \theta } \) are respectively called as the case 1 and the case 2 to facilitate the following proof.

Theorem 2

Considering the MSV (11) subject to model uncertainties, external disturbances, input saturation and prescribed performance constraints, based on the FXNTSMM (35) and the FXDO (22) with appropriate parameters, if the fixed-time trajectory tracking control law is designed as (37), the MSV (11) can accurately track a desired trajectory with the prescribed performance (15), and the tracking error vector \( {\eta _e} \) can converge to the origin within a setting time.

Proof

A Lyapunov function is selected as follows

$$\begin{aligned} {V_1} = \left| {{S_{1}}} \right| + \left| {{S_{2}}} \right| + \left| {{S_{3}}} \right| \end{aligned}$$
(38)

The time derivative of (38) is presented as follows

$$\begin{aligned} {\dot{V}_1} = {sign}({S_{1}}){{{\dot{S}}}_{1}} + {sign}({S_{2}}){{{\dot{S}}}_{2}} + {sign}({S_{3}}){{{\dot{S}}}_{3}} \end{aligned}$$
(39)

Combining with (19), (36) is written as follows

$$\begin{aligned} {\dot{S}}&= {q_1}\mathrm{{diag}}({\left| {{J_{2i}}} \right| ^{{q_1} - 1}})(H\Theta R(\psi ){M^{ - 1}}{\tau _c} \nonumber \\&\quad - H\Theta {{{\dot{w}}}_d} - H\ddot{\Theta }{\eta _e} + L) \nonumber \\&\quad + {a_1}{q_2}\mathrm{{diag}}({\left| {{J_{1i}}} \right| ^{{q_2} - 1}}){J_2}\mathrm{{ }} + {b_1}{q_3}\mathrm{{diag}}({\left| {{J_{1i}}} \right| ^{{q_3} - 1}}){J_2} \nonumber \\&\quad + {c_1}{q_1}\mathrm{{diag}}({\left| {{J_{1i}}} \right| ^{{q_1} - 1}}){J_2} \end{aligned}$$
(40)

Substituting (37) into (40), when the disturbance estimation error vector \( {{\tilde{L}}} \) converge to the origin, we have

$$\begin{aligned} {\dot{S}}&= \frac{{2\theta }}{\pi }\mathrm{{diag}}(\rho ({\left| {{J_{2i}}} \right| ^{{q_1} - 1}}))\nonumber \\&\quad ( - {\lambda _0}si{g^{{q_4}}}(S) - {\lambda _1}si{g^{{q_5}}}(S) - {\lambda _2}S) \end{aligned}$$
(41)

In view of the case 1, (41) is written as follows

$$\begin{aligned} {\dot{S}} = \frac{{2\theta }}{\pi }( - {\lambda _0}si{g^{{q_4}}}(S) - {\lambda _1}si{g^{{q_5}}}(S) - {\lambda _2}S) \end{aligned}$$
(42)

Accordingly, combining with (42), (39) is rewritten as follows

$$\begin{aligned} {\dot{V}_1}&= {{sign}}({S_{1}}){{{\dot{S}}}_{1}} + {{sign}}({S_{2}}){{{\dot{S}}}_{2}} + {{sign}}({S_{3}}){{{\dot{S}}}_{3}}\nonumber \\&= \frac{{2\theta }}{\pi }{{sign}}({S_{1}}) ( - {\lambda _{0}}si{g^{{q_{4}}}}({S_{1}}) - {\lambda _{1}}si{g^{{q_{5}}}}({S_{1}})- {\lambda _{2}}{S_{1}})\nonumber \\&+ \frac{{2\theta }}{\pi }{{sign}}({S_{2}})( - {\lambda _{0}}si{g^{{q_{4}}}}({S_{2}}) - {\lambda _{1}}si{g^{{q_{5}}}}({S_{2}})- {\lambda _{2}}{S_{2}})\nonumber \\&+ \frac{{2\theta }}{\pi }{{sign}}({S_{3}})( - {\lambda _{0}}si{g^{{q_{4}}}}({S_{3}}) - {\lambda _{1}}si{g^{{q_{5}}}}({S_{3}})- {\lambda _{2}}{S_{3}}) \end{aligned}$$
(43)

Further, one has

$$\begin{aligned} {\dot{V}_1}&= - \frac{{2\theta }}{\pi }{\lambda _{0}}(\left| {{S_{1}}} \right| ^{q_{4}} + \left| {{S_{2}}} \right| ^{q_{4}} + \left| {{S_{3}}} \right| ^{q_{4}}) \nonumber \\&- \frac{{2\theta }}{\pi }{\lambda _{1}}({{\left| {{S_{1}}} \right| }^{q_{5}}} + {{\left| {{S_{2}}} \right| }^{q_{5}}} + {{\left| {{S_{3}}} \right| }^{q_{5}}}) \nonumber \\&- \frac{{2\theta }}{\pi }{\lambda _{2}}({{\left| {{S_{1}}} \right| }} + {{\left| {{S_{2}}} \right| }} + {{\left| {{S_{3}}} \right| }}) \end{aligned}$$
(44)

Combining with (44) and in the light of Lemma 2, one has

$$\begin{aligned} {\dot{V}_1} \le - \frac{{2\theta }}{\pi }{\lambda _{0}}{3^{1- {q_{4}}}}V_1^{q_{4}} - \frac{{2\theta }}{\pi }{\lambda _{1}}V_1^{q_{5}}- \frac{{2\theta }}{\pi }{\lambda _{2}}{V_1} \end{aligned}$$
(45)

In the light of Lemma 1, the sliding mode manifold S (35) can converge to \(S= 0\) within a setting time or reach the case 2. In view of the case 2, \( S = 0 \) is still an attractor due to \(0< \rho ({\left| {{J_{2i}}} \right| ^{{q_{1}} - 1}}) < 1 \) when \( {J_{2i}} \ne 0 \). Subsequently, we will exclude that \( {J_{2}}=0 \) is not attractive except for the origin. Substituting (37) into (19) when \( {J_{2}} = 0 \) yields

$$\begin{aligned} {{\dot{J}}_{2}} = \frac{1}{{{q_{1}}}}( - {\lambda _{0}}si{g^{{q_{4}}}}({S}) - {\lambda _{1}}si{g^{{q_{5}}}}({S})- {\lambda _{2}}{S}) \end{aligned}$$
(46)

In terms of (46), it is concluded that \( {{\dot{J}}_{2}} < 0 \) when \( S > 0 \) and \( {{\dot{J}}_{2}} > 0 \) when \( S < 0 \). Hence, according to [44], the system states transgress the case 2 into the case 1 within a finite time. Consequently, the sliding mode manifold S (35) can converge to \( S= 0 \) from anywhere.

When \( S=0 \), (35) is written as follows

$$\begin{aligned} si{g^{{q_{1}}}}({J_{2}}) =- {a_{1}}si{g^{{q_{2}}}}({J_{1}}) - {b_{1}}si{g^{{q_{3}}}}({J_{1}}) - {c_{1}}si{g^{{q_{1}}}}({J_{1}}) \end{aligned}$$
(47)

Further, one has

$$\begin{aligned} {J_{2}} = {(- {a_{1}}si{g^{{q_{2}}}}({J_{1}}) - {b_{1}}si{g^{{q_{3}}}}({J_{1}}) - {c_{1}}si{g^{{q_{1}}}}({J_{1}}))^{\frac{1}{{{q_{1}}}}}} \end{aligned}$$
(48)

Considering a function \( z = \left| {{J_{1i}}} \right| \), and its derivative with respect to time is developed as follows

$$\begin{aligned} {\dot{z}} = {\mathop { sign}} ({J_{1i}}){J_{2i}} \end{aligned}$$
(49)

Combining with (48), (49) is rewritten as follows

$$\begin{aligned} {{\dot{z}}}&= {( - {a_{1}}{\left| {{J_{1i}}} \right| ^{q_{2}}} - {b_{1}}{\left| {{J_{1i}}} \right| ^{q_{3}}} - {c_{1}}{\left| {{J_{1i}}} \right| ^{q_{1}}})^{\frac{{{1}}}{{{q_{1}}}}}}\nonumber \\&= {( - {a_{1}}{z}^{q_{2}} - {b_{1}}{z}^{q_{3}} - {c_{1}}{z}^{q_{1}})^{\frac{{{1}}}{{{q_{1}}}}}}\nonumber \\&=-{( {a_{1}}{z}^{q_{2}} + {b_{1}}{z}^{q_{3}} + {c_{1}}{z}^{q_{1}})^{\frac{{{1}}}{{{q_{1}}}}}} \end{aligned}$$
(50)

Constructing the following inequality holds

$$\begin{aligned} \left\{ \begin{array}{l} {({a_{1}}{z^{{q_{2}}}} + {b_{1}}{z^{{q_{3}}}} + {c_{1}}{z^{{q_{1}}}})^{\frac{1}{{{q_{1}}}}}} \ge {({a_{1}}{z^{{q_{2}}}})^{\frac{1}{{{q_{1}}}}}}\\ {({a_{1}}{z^{{q_{2}}}} + {b_{1}}{z^{{q_{3}}}} + {c_{1}}{z^{{q_{1}}}})^{\frac{1}{{{q_{1}}}}}} \ge {({b_{1}}{z^{{q_{3}}}})^{\frac{1}{{{q_{1}}}}}}\\ {({a_{1}}{z^{{q_{2}}}} + {b_{1}}{z^{{q_{3}}}} + {c_{1}}{z^{{q_{1}}}})^{\frac{1}{{{q_{1}}}}}} \ge {({c_{1}}{z^{q_{1}}})^{\frac{1}{{{q_{1}}}}}} \end{array} \right. \end{aligned}$$
(51)

Subsequently, (51) is rewritten as follows

$$\begin{aligned} {({a_{1}}{z^{{q_{2}}}} + {b_{1}}{z^{{q_{3}}}} + {c_{1}}{z^{{q_{1}}}})^{\frac{1}{{{q_{1}}}}}}&\ge \frac{1}{3}a_{1}^{\frac{1}{{{q_{1}}}}}{z^{\frac{{{q_{2}}}}{{{q_{1}}}}}} \nonumber \\&\quad + \frac{1}{3}b_{1}^{\frac{1}{{{q_{1}}}}}{z^{\frac{{{q_{3}}}}{{{q_{1}}}}}} + \frac{1}{3}c_{1}^{\frac{1}{{{q_{1}}}}}z \end{aligned}$$
(52)

Further, we have

$$\begin{aligned} {\dot{z}}&= - {({a_{1}}{z^{{q_{2}}}} + {b_{1}}{z^{{q_{3}}}} + {c_{1}}{z^{{q_{1}}}})^{\frac{1}{{{q_{1}}}}}} \nonumber \\&\le - \frac{1}{3}a_{1}^{\frac{1}{{{q_{1}}}}}{z^{\frac{{{q_{2}}}}{{{q_{1}}}}}} - \frac{1}{3}b_{1}^{\frac{1}{{{q_{1}}}}}{z^{\frac{{{q_{3}}}}{{{q_{1}}}}}} - \frac{1}{3}c_{1}^{\frac{1}{{{q_{1}}}}}z \end{aligned}$$
(53)

where \( 0< \frac{{{q_{2}}}}{{{q_{1}}}} < 1 \) and \( \frac{{{q_{3}}}}{{{q_{1}}}} > 1 \) due to \( 1< {q_{2}}< {q_{1}}< {q_{3}} < 2 \).

In the light of Lemma 1 and (53), the state z can converge to the origin within a setting time. Obviously, the tracking error vector \( {\eta _e} \) can also converge to the origin within a setting time.

Further, the inverse function of (17) is expressed as follows

$$\begin{aligned} {\rho _i} = \frac{{{\varepsilon _i}({e^{2{J_{1i}}}} - 1)}}{{{e^{2{J_{1i}}}} + 1}} \end{aligned}$$
(54)

where \( {\rho _i} \) is a strictly monotonically increasing function with respect to \( {J_{1i}} \) with \( {\rho _i}(0) = 0 \). \( {\rho _i} = \frac{{{\eta _{ei}}}}{{{\vartheta _i}(t)}} \), \( \mathop {\lim }\limits _{{J_{1i}} \rightarrow + \infty } {\rho _i} = {\varepsilon _i} \), \( \mathop {\lim }\limits _{{J_{1i}} \rightarrow - \infty } {\rho _i} = - {\varepsilon _i} \).

According to the above theoretical derivations, all signals in the system are bounded. Consequently, one has

$$\begin{aligned} - {\varepsilon _i}< {\rho _i} < {\varepsilon _i} \end{aligned}$$
(55)

Accordingly, we have

$$\begin{aligned} - {\varepsilon _i}{\vartheta _i}(t)< {\eta _{ei}} < {\varepsilon _i}{\vartheta _i}(t) \end{aligned}$$
(56)

As a consequence, it is concluded that the designed scheme can not only make the tracking error vector \( {\eta _e} \) converge to the origin within a setting time, but also keep the one within prescribed constraints. This completes the proof. \(\square \)

Remark 2

In terms of the FXNTSMM, in comparison with [13, 39], the designed FXNTSMM reduces the calculation burden of the system. In comparison with [22, 26, 40, 41], the designed FXNTSMM ensures that the system states converge to the origin within a setting time in the sliding phase.

Table 1 Design parameters and their values in the control law (37) and the observer (22)
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Remark 3

In terms of the proposed scheme, in comparison with [24, 30] by disturbance observation methods dealing with input saturation, the proposed FXDO-based anti-saturation trajectory tracking control scheme is more reasonable, which avoids an unreasonable assumption that lumped disturbances including the input saturation function matrix are differentiable. In comparison with [7, 25,26,27,28, 39, 42] by auxiliary dynamic systems dealing with input saturation, the proposed FXDO-based anti-saturation trajectory tracking control scheme not only improves the system robustness, but also simplifies the one structures without introducing an auxiliary dynamic system. In comparison with [2, 14], in addition to considering external disturbances and model uncertainties, the proposed scheme also considers input saturation and prescribed performance constraints, which is more in line with practical scenarios during the navigation.

5 Simulation and comparison studies

In this section, a Matlab/Simulink test is carried out on a scale model named CyberShip II [27] to verify the effectiveness of the designed trajectory tracking control scheme. The parameters of the prescribed performance function are selected as \( {\vartheta _{10}} = 2,{\vartheta _{20}} = 1,{\vartheta _{30}} = 2,{\vartheta _{1\infty }} = 0.1,{\vartheta _{2\infty }} = 0.1,{\vartheta _{3\infty }} = 0.1,\omega = 2 \). The ranges of the control input signals are determined as \( {\tau _1} \in [-100,100]\textrm{N} \), \( {\tau _2} \in [ -500,500]\textrm{N} \) and \( {\tau _3} \in [-50,50]\mathrm{N \cdot m} \). The desired trajectory is selected as \( {\eta _d} = {[(0.3t)\textrm{m},(5\sin (0.03t))\textrm{m},(0.02t)\textrm{rad}]^T} \). The initial state vectors of the MSV are respectively selected as \( {\eta }(0) = {[-1\,\textrm{m},0.5 \mathrm{\,m},\pi /3 \mathrm{\,rad}]^T} \) and \( {v}(0) = {[0\,\mathrm{\,m/s},0\,\mathrm{\,m/s},0 \mathrm{\,rad/s}]^T} \). The control parameters are listed in Table 1.

The structure of an external disturbance vector is selected as follows [47]

$$\begin{aligned} \dot{d} = - {T^{ - 1}}d + \gamma {\bar{\omega }} \end{aligned}$$
(57)

where \( {T^{ - 1}} = diag(0.03,0.05,0.03) \), \( \gamma = diag(0.6,0.5,0.4) \).

Simulation results are depicted in Figs. 2, 3, 4, 5, 6, 7, 8, 9 and 10. Figure 2 shows actual positions (x, y) and a yaw angle \( \psi \) track a desired trajectory \( {\eta _d} \) with satisfying tracking performance. Figure 3 shows the tracking errors of the two schemes, where the scheme with lumped disturbances and prescribed performance constraints is called as FXPPC and the other scheme without lumped disturbances and prescribed performance constraints is called as FX. As can be seen from Fig. 3, the FXPPC scheme can ensure the tracking errors are within constraints, and can obtain fast tracking performance, while the FX scheme can only ensure system stability. The velocity tracking and their tracking errors relative to the desired velocities are respectively shown in Figs. 4 and 5, where the tracking errors can converge to the origin in an accurate manner. Figure 6 shows that the control input signals are bounded. Figure 7 shows the constrained control input signals. The trajectories with \( t=300s \) on the horizontal plane are shown in Fig. 8. Figure 9 shows the sliding mode manifold. From Fig. 10, it can be seen that the proposed FXDO can estimate lumped disturbances.

Fig. 2
figure 2

The tracking of positions and a yaw angle under the control law (37)

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Fig. 3
figure 3

The comparative tracking errors of positions and a yaw angle

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Fig. 4
figure 4

The velocity tracking under the control law (37)

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Fig. 5
figure 5

The velocity tracking errors under the control law (37)

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Fig. 6
figure 6

The control inputs without the constraints

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Fig. 7
figure 7

The control inputs with the constraints

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Fig. 8
figure 8

The horizontal trajectories under the control law (37)

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Fig. 9
figure 9

The sliding mode manifold under the control law (37)

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Fig. 10
figure 10

The estimation of lumped disturbances by the FXDO (22)

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To verify the impact of input saturation on system tracking performance, two schemes are designed, where the scheme using FXDO for estimating lumped disturbances including the input saturation function matrix is called as FXWIS and the scheme using FXDO for estimating lumped disturbances excluding the one is called as FXNIS. The two sets of data are shown below, and the transient and steady-state performance of the scheme FXWIS is significantly better than the scheme FXNIS. Therefore, the designed scheme can not only improve the robustness of the system, but also handle the adverse impact of input saturation on system tracking performance.

$$\begin{aligned} \left\{ \begin{array}{l} \mathrm{{IA}}{\mathrm{{E}}_{\mathrm{{FXWIS}}}} = \int _0^{{t_{final}}} {\left\| {{\eta _e}} \right\| dt = 1.0140} \\ \mathrm{{ITA}}{\mathrm{{E}}_{\mathrm{{FXWIS}}}} = \int _0^{{t_{final}}} {t\left\| {{\eta _e}} \right\| dt = 1.3906} \end{array} \right. \\ \left\{ \begin{array}{l} \mathrm{{IA}}{\mathrm{{E}}_{\mathrm{{FXNIS}}}} = \int _0^{{t_{final}}} {\left\| {{\eta _e}} \right\| dt = 1.5436} \\ \mathrm{{ITA}}{\mathrm{{E}}_{\mathrm{{FXNIS}}}} = \int _0^{{t_{final}}} {t\left\| {{\eta _e}} \right\| dt = 3.5274} \end{array} \right. \end{aligned}$$

Remark 4

In the light of [2], the transient and steady-state performance can be evaluated by the integrated absolute error (IAE) \( \int _0^{{t_{final}}} {\left| x \right| dt} \) and the integrated time absolute error (ITAE) \( \int _0^{{t_{final}}} {t\left| x \right| dt} \), where \( {t_{final}}=30\,s \) is the running time of the simulation.

6 Conclusions

In this paper, an FXDO is firstly designed, and it can realize fixed-time stability and solve the design method problem in the existing disturbance observer. Subsequently, an FXNTSMM with simple structures is designed, and it can reduce the calculation burden of the system. And then, considering model uncertainties, external disturbances, input saturation and prescribed performance constraints, a fixed-time nonsingular terminal sliding mode trajectory tracking control scheme for an MSV is proposed based on an FXDO, an FXNTSMM and a prescribed performance function. In terms of the future work and research, the actual experiment will be carried out on the control scheme mentioned above. Additionally, an underactuated trajectory traking control scheme for an MSV with input saturation and prescribed performance constraints will be proposed.