Abstract
This paper solves a fixedtime trajectory tracking control problem for a marine surface vessel in the presence of model uncertainties, external disturbances, input saturation and prescribed performance constraints. Firstly, a fixedtime disturbance observer (FXDO) is designed, which not only realizes fixedtime stability, but also solves the design method problem in the existing disturbance observer. Secondly, a fixedtime nonsingular terminal sliding mode manifold (FXNTSMM) with simple structures is designed, whereby the designed FXNTSMM reduces the calculation burden of the system. Thirdly, a fixedtime 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, and the entire constrained closedloop control system is with fixedtime stability, simplicity and nonsingularity. Finally, several numerical simulation results are presented to demonstrate the effectiveness of the proposed trajectory tracking control scheme.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
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 highprecision 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], eventtriggered 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, finitetime control with highaccuracy and fast convergence is proposed [19, 20]. However, the setting time of finitetime control is highly related to initial conditions. It prevents predicting the setting time in advance when initial conditions are unavailable or unmeasurable. In response, fixedtime control is proposed to deal with the above issue [21]. What’s more, fixedtime control has the inherent capacities in faster transient and higher steadystate accuracy than finitetime one [22]. In [23], based on an FXDO, a fixedtime 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 fixedtime 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 steadystate 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 antisaturation trajectory tracking control scheme for an MSV is proposed based on an auxiliary dynamic system. Subsequently, fixedtime 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 functionbased continuous differentiable asymmetric saturation model is introduced to solve the problem that the nonsmooth 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 fixedtime 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)
An FXDO is designed. In comparison with [31], the designed FXDO not only realizes fixedtime 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)
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)
A fixedtime 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 FXDObased antisaturation 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 FXDObased antisaturation 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 2norm. 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
where \( z \in {\Re ^n} \), \( f(z):{\Re ^n} \rightarrow {\Re ^n} \) is continuous on \( {\Re ^n} \), \( f(0) = 0 \). A continuous positivedefinite function \( V(z):{\Re ^n} \rightarrow \Re _0^+ \) and constants \( a'>0, b'>0, c' > 0 \), \( \alpha ' > 1 \) and \(0< \beta ' <1 \) with
Then, the system (1) is with globally fast fixedtime stability, and the setting time is expressed as follows
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)
\( \vartheta (t) \) is positive and decreasing.

(2)
\( \mathop {\lim }\limits _{t \rightarrow \infty } \vartheta (t) = {\vartheta _\infty } > 0 \).
3 System modelling and problem formulation
In this section, a threedegreeoffreedom MSV is described as follows [2]
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.
The properties pertaining to \( R(\psi ) \) are presented below [2]
The MSV (4) is converted below by coordinate transformation [2]
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 earthfixed coordinate frame. \( {w_d} = {[{w_{d1}},{w_{d2}},{w_{d3}}]^T} \) is called as a desired velocity vector under an earthfixed 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
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]
where \( {{\tau _{\max }}} \) and \( {{\tau _{\min }}} \) are respectively called as maximum and minimum control input vectors.
Defining the tracking error vectors is presented below
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
A prescribed performance function is introduced as follows [45]
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
A transformation function is presented as follows
The time derivative of (17) is obtained as follows
where \( {{\dot{J}}_{1i}} = {J_{2i}} \).
Therefore, combining with (14), we have
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)
The desired trajectory vector \( {\eta _d} \) is bounded, and its first and second order derivatives are also bounded.

(2)
\( L = {[{L_1},{L_2},{L_3}]^T} \) is regarded as a lumped disturbance vector with \( \left\ L \right\ \le \delta \).

(3)
The matrices C(v) and D(v) are unknown.
In this paper, the main objective is to design a fixedtime 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 fixedtime stability, but also guarantees the tracking error vector \( {\eta _e} \) is within prescribed constraints. The control objectives can be stated mathematically as follows
and
4 Main results
4.1 Design of FXDO
In this subsection, in the light of (19), an FXDO is designed as follows
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
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
The time derivative of (24) is obtained as follows
Combining with (19) and (22), we have
In view of \( \left\ L \right\ \le \delta <l \), combining with (25) and (26), we have
Based on Lemma 2, (27) can be written as follows
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
Therefore, when \( t > {T_1 } \), it yields
Further, according to the definition of \( {V_{{\bar{s}}}} \), we have
When \( {\bar{s}} = 0 \), we have
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
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 fixedtime 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.
4.2 FXDObased fixedtime sliding mode trajectory tracking control
The control frame diagram of the fixedtime control system for the MSV (11) is presented in Fig. 1.
In this subsection, an FXNTSMM with simple structures is designed as follows
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
Combining with (35) and the FXDO (22), a fixedtime nonsingular terminal sliding mode trajectory tracking control law is presented as follows
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 fixedtime 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
The time derivative of (38) is presented as follows
Combining with (19), (36) is written as follows
Substituting (37) into (40), when the disturbance estimation error vector \( {{\tilde{L}}} \) converge to the origin, we have
In view of the case 1, (41) is written as follows
Accordingly, combining with (42), (39) is rewritten as follows
Further, one has
Combining with (44) and in the light of Lemma 2, one has
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
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
Further, one has
Considering a function \( z = \left {{J_{1i}}} \right \), and its derivative with respect to time is developed as follows
Combining with (48), (49) is rewritten as follows
Constructing the following inequality holds
Subsequently, (51) is rewritten as follows
Further, we have
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
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
Accordingly, we have
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.
Remark 3
In terms of the proposed scheme, in comparison with [24, 30] by disturbance observation methods dealing with input saturation, the proposed FXDObased antisaturation 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 FXDObased antisaturation 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]
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.
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 steadystate 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.
Remark 4
In the light of [2], the transient and steadystate 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 fixedtime 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 fixedtime 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.
Data availability
Research data are not shared.
Code availability
Not applicable.
References
Liu, H., Chen, G.: Robust trajectory tracking control of marine surface vessels with uncertain disturbances and input saturations. Nonlinear Dyn. 100(4), 3513–3528 (2020)
Zhang, J., Yu, S., Yan, Y.: Fixedtime output feedback trajectory tracking control of marine surface vessels subject to unknown external disturbances and uncertainties. ISA Trans. 93, 145–155 (2019)
Grandia, R., Jenelten, F., Yang, S., Farshidian, F., Hutter, M.: Perceptive locomotion through nonlinear modelpredictive control. IEEE Trans. Rob. 39(5), 3402–3421 (2023)
Feng, N., Wu, D., Yu, H., Yamashita, A.S., Huang, Y.: Predictive compensator based eventtriggered model predictive control with nonlinear disturbance observer for unmanned surface vehicle under cyberattacks. Ocean Eng. 259, 111868 (2022)
Wang, W., Li, Y.: Distributed fuzzy optimal consensus control of stateconstrained nonlinear strictfeedback systems. IEEE Trans. Cybern. 53(5), 2914–2929 (2022)
Singhal, K., Kumar, V., Rana, K.: Robust trajectory tracking control of nonholonomic wheeled mobile robots using an adaptive fractional order parallel fuzzy PID controller. J. Franklin Inst. 359(9), 4160–4215 (2022)
Wang, T., Li, Y.: Neuralnetwork adaptive outputfeedback saturation control for uncertain active suspension systems. IEEE Trans. Cybern. 52(3), 1881–1890 (2022)
ElSousy, F.F.M., Amin, M.M., Mohammed, O.A.: Robust adaptive neural network tracking control with optimized supertwisting slidingmode technique for induction motor drive system. IEEE Trans. Ind. Appl. 58(3), 4134–4157 (2022)
Zhu, W., Du, H., Li, S.: On eventtriggered nonsmooth attitude tracking controller for a rigid spacecraft. Int. J. Robust Nonlinear Control 32(2), 900–916 (2022)
Dou, W., Ding, S., Yu, X.: Eventtriggered secondorder slidingmode control of uncertain nonlinear systems. IEEE Trans. Syst. Man Cybern. Syst. 53(11), 7269–7279 (2023)
Schwerdtner, P., Voigt, M.: Fixedorder \({H_\infty }\) controller design for portHamiltonian systems. Automatica 152, 110918 (2023)
Huang, M., Ding, L., Li, W., Chen, C.Y., Liu, Z.: Distributed observerbased \({H_\infty }\) faulttolerant control for DC microgrids with sensor fault. IEEE Trans. Circuits Syst. I Regul. Pap. 68(4), 1659–1670 (2021)
Huang, Y., Jia, Y.: Adaptive fixedtime sixDOF tracking control for noncooperative spacecraft flyaround mission. IEEE Trans. Control Syst. Technol. 27(4), 1796–1804 (2018)
Zhang, J., Yu, S., Wu, D., Yan, Y.: Nonsingular fixedtime terminal sliding mode trajectory tracking control for marine surface vessels with antidisturbances. Ocean Eng. 217, 108158 (2020)
Ding, S., Hou, Q., Wang, H.: Disturbanceobserverbased secondorder sliding mode controller for speed control of PMSM drives. IEEE Trans. Energy Convers. 38(1), 100–110 (2023)
Cong, Y., Du, H., Zhu, W., Chen, C.C.: Position regulation of industrial robots via bounded integral terminal sliding mode control algorithm. IEEE Trans. Ind. Electron. (2023). https://doi.org/10.1109/TIE.2023.3310016
Sai, H., Xu, Z., Xia, C., Sun, X.: Approximate continuous fixedtime terminal sliding mode control with prescribed performance for uncertain robotic manipulators. Nonlinear Dyn. 110(1), 431–448 (2022)
Van, M.: An enhanced tracking control of marine surface vessels based on adaptive integral sliding mode control and disturbance observer. ISA Trans. 90, 30–40 (2019)
Bhat, S.P., Bernstein, D.S.: Finitetime stability of continuous autonomous systems. SIAM J. Control. Optim. 38(3), 751–766 (2000)
Chen, Y., Liu, Z.W., Wen, G., Wang, Y.W.: Distributed finitetime observer for multiple line outages detection in power systems. IEEE Trans. Syst. Man Cybern. Syst. 54(3), 1768–1778 (2024)
Polyakov, A.: Nonlinear feedback design for fixedtime stabilization of linear control systems. IEEE Trans. Autom. Control 57(8), 2106–2110 (2012)
Yang, P., Su, Y.: Proximate fixedtime prescribed performance tracking control of uncertain robot manipulators. IEEE/ASME Trans. Mechatron. 27(5), 3275–3285 (2021)
Wang, T., Wang, C., Yan, S., Liu, Y.: Disturbance observerbased fixedtime sliding mode trajectory tracking control for marine surface vehicles with uncertain dynamics. Trans. Inst. Meas. Control. (2023). https://doi.org/10.1177/01423312231193188
Sun, H., Zong, G., Cui, J., Shi, K.: Fixedtime sliding mode output feedback tracking control for autonomous underwater vehicle with prescribed performance constraint. Ocean Eng. 247, 110673 (2022)
Sadek, B.A., Badreddine, E.H., Noreddine, C., El Houssaine, T.: Achieving prescribedtime stabilization in nonlinear TCP/AQM systems with input saturation. IEEE Trans. Netw. Sci. Eng. 11(1), 1363–1373 (2024)
Sun, Y., Kuang, J., Gao, Y., Chen, W., Wang, J., Liu, J., Wu, L.: Fixedtime prescribed performance tracking control for manipulators against input saturation. Nonlinear Dyn. (2023). https://doi.org/10.1007/s11071023084993
Zhu, G., Du, J., Kao, Y.: Robust adaptive neural trajectory tracking control of surface vessels under input and output constraints. J. Franklin Inst. 357(13), 8591–8610 (2020)
Sai, H., Xu, Z., He, S., Zhang, E., Zhu, L.: Adaptive nonsingular fixedtime sliding mode control for uncertain robotic manipulators under actuator saturation. ISA Trans. 123, 46–60 (2022)
Zheng, Z., Jin, C., Zhu, M., Sun, K.: Trajectory tracking control for a marine surface vessel with asymmetric saturation actuators. Robot. Auton. Syst. 97, 83–91 (2017)
Hu, Q., Li, B., Qi, J.: Disturbance observer based finitetime attitude control for rigid spacecraft under input saturation. Aerosp. Sci. Technol. 39, 13–21 (2014)
Wang, F., He, L.: FPGAbased predictive speed control for PMSM system using integral slidingmode disturbance observer. IEEE Trans. Industr. Electron. 68(2), 972–981 (2020)
Ni, J., Liu, L., Chen, M., Liu, C.: Fixedtime disturbance observer design for Brunovsky systems. IEEE Trans. Circuits Syst. II Express Briefs 65(3), 341–345 (2018)
Sun, J., Pu, Z., Yi, J., Liu, Z.: Fixedtime control with uncertainty and measurement noise suppression for hypersonic vehicles via augmented sliding mode observers. IEEE Trans. Ind. Inf. 16(2), 1192–1203 (2020)
Xu, H., Cui, G., Ma, Q., Li, Z., Hao, W.: Fixedtime disturbance observerbased distributed formation control for multiple QUAVs. IEEE Trans. Circuits Syst. II Express Briefs 70(6), 2181–2185 (2023)
Wang, X., Wang, X., Wang, Z., Xiao, X., Li, S.: Composite finitetime control for PMSM with prescribed performance using disturbance compensation technique. Control. Eng. Pract. 141, 105677 (2023)
Sun, J., Yang, J., Zeng, Z.: Safetycritical control with control barrier function based on disturbance observer. IEEE Trans. Autom. Control (2024). https://doi.org/10.1109/TAC.2024.3352707
Zhou, C., Yang, J., Li, S., Chen, W.H.: Robust temporal logic motion control via disturbance observers. IEEE Trans. Ind. Electron. 70(8), 8286–8295 (2022)
Wang, X., Zheng, W.X., Wang, G.: Distributed finitetime optimization of secondorder multiagent systems with unknown velocities and disturbances. IEEE Trans. Neural Netw. Learn. Syst. 34(9), 6042–6054 (2023)
Liu, K., Wang, R.: Antisaturation adaptive fixedtime sliding mode controller design to achieve faster convergence rate and its application. IEEE Trans. Circuits Syst. II Express Briefs 69(8), 3555–3559 (2022)
Liu, Z., Lin, X., Gao, Y., Xu, R., Wang, J., Wang, Y., Liu, J.: Fixedtime sliding mode control for DC/DC buck converters with mismatched uncertainties. IEEE Trans. Circuits Syst. I Regul. Pap. 70(1), 472–480 (2023)
Yan, Y., Yu, S., Gao, X., Wu, D., Li, T.: Continuous and periodic eventtriggered slidingmode control for path following of underactuated surface vehicles. IEEE Trans. Cybern. 54(1), 449–461 (2024)
Neisarian, S., Arefi, M.M., Abooee, A., Yin, S.: Fast finitetime observerbased sliding mode controller design for a class of uncertain nonlinear systems with input saturation. Inf. Sci. 630, 599–622 (2023)
Zhang, H., Duan, J., Wang, Y., Gao, Z.: Bipartite fixedtime output consensus of heterogeneous linear multiagent systems. IEEE Trans. Cybern. 51(2), 548–557 (2021)
Zuo, Z.: Nonsingular fixedtime consensus tracking for secondorder multiagent networks. Automatica 54, 305–309 (2015)
Bechlioulis, C.P., Rovithakis, G.A.: Robust adaptive control of feedback linearizable MIMO nonlinear systems with prescribed performance. IEEE Trans. Autom. Control 53(9), 2090–2099 (2008)
Yu, S., Long, X.: Finitetime consensus for secondorder multiagent systems with disturbances by integral sliding mode. Automatica 54, 158–165 (2015)
Du, J., Hu, X., Krstić, M., Sun, Y.: Robust dynamic positioning of ships with disturbances under input saturation. Automatica 73, 207–214 (2016)
Funding
This work was supported by the National Natural Science Foundation of China under Grant Numbers: 62073054, 62173054, 62373072.
Author information
Authors and Affiliations
Contributions
J.Z. and S.Y. wrote the main manuscript text. Y.Y. and Y.Z. prepared Figs. 19. All authors reviewed the manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors have no relevant financial or nonfinancial interests to disclose.
Ethical approval and consent to participate
Not applicable.
Consent for publication
When the article is accepted, authors consent for publication.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author selfarchiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Zhang, J., Yu, S., Yan, Y. et al. Fixedtime sliding mode trajectory tracking control for marine surface vessels with input saturation and prescribed performance constraints. Nonlinear Dyn 112, 17169–17181 (2024). https://doi.org/10.1007/s11071024099189
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071024099189