Nonlinear disturbance observer-based backstepping finite-time sliding mode tracking control of underwater vehicles with system uncertainties and external disturbances

Abstract

In this paper, a nonlinear disturbance observer-based backstepping finite-time sliding mode control scheme for trajectory tracking of underwater vehicles subject to unknown system uncertainties and time-varying external disturbances is proposed. To reduce the influence of the uncertainties and external disturbances, a nonlinear disturbance observer is developed without any acceleration measurements to identify the lumped disturbance term. Additionally, the finite-time trajectory tracking controller is designed by combining second-order sliding mode control and backstepping design technique with the nonlinear disturbance observer. The finite-time convergence of motion tracking errors and the stability of the overall closed-loop control system are guaranteed by the Lyapunov approach. Besides, comprehensive simulation studies on trajectory tracking control of underwater vehicles are provided to demonstrate the effectiveness and performance of the proposed control scheme.

1 Introduction

Underwater vehicles are self-energizing and self-propelled intelligent equipments with autonomous controllers and various sensors, where the efficient and reliable autonomous controllers are highly desired for different types of underwater missions. Thus, a great deal of attentions have been paid to the accurate trajectory tracking (TT) control of underwater vehicles, which provides the fundamental technical support for various dynamic and complex missions, mainly because of the great practical significance for marine traffic safety. Commonly in practical situation, underwater vehicles are always coupled and highly nonlinear subject to diverse external disturbances that adversely affect performance, and the system parameters of underwater vehicles may not be known exactly; therefore, the design of TT controller of underwater vehicles is quite challenging and difficult.

To achieve high-precision tracking performance, a large variety of motion tracking control schemes of underwater vehicles have been proposed in the literatures for solving the TT control problem of underwater vehicles, which includes adaptive control [1,2,3], backstepping design technique [4,5,6], sliding mode control (SMC) [7,8,9,10,11], LMI-based control [12], fuzzy logic system [13,14,15,16], neural network control [17,18,19,20] and fuzzy neural network [21,22,23,24,25]. Among these methods, SMC has been shown to be a potentially useful approach when applied to a system subject to disturbances which satisfies the matched uncertainty condition. Moreover, SMC can offer many good properties, such as insensitivity to system uncertainties, disturbance rejection and fast dynamic response. However, the SMC-based motion controllers of underwater vehicles proposed in [7,8,9,10,11] are designed based on an asymptotic stability analysis which implies that the system trajectories converge to the equilibrium with infinite settling time. Therefore, a faster convergence to the required orientation in finite time is highly desired. Fortunately, the finite-time stabilization of dynamic systems may provide not only faster convergence but also better disturbance attenuation. The terminal sliding mode control (TSMC) [26, 27] is a representative method used to design a robust controller which can guarantee a finite-time convergence to the origin. Another advanced sliding mode, namely the second-order SMC (SOSMC), which preserves the robustness ability of SMC and also yields improved accuracy and performance, has been paid attention in many literatures [28,29,30]. Although SOSMC controller is less sensitive to parameter uncertainties and external disturbances, its robustness is normally obtained by a large switching gain which causes the undesired chattering phenomenon. To alleviate above drawbacks, recently, the backstepping design technique is combined with sliding mode control to relax the matching condition at the expense of a high feedback gain required for robustness [31].

On the other hand, the system uncertainties and external disturbance are always unknown and nonlinear; thus, it is necessary to employ some disturbance suppression or attenuation methods to achieve the desired robust performance. Despite of above methods, an alternative emerged in recent years is the use of so-called disturbance observers control (DOBC). Roughly stated, the essence of the DOBC is to lump all internal uncertainties and external unknown disturbances acting on the underwater vehicles into a single term and then identify the unknown lumped term with the disturbance observer. It is found that the disturbance observers can provide fast, excellent tracking performance and smooth control actions without the use of large feedback gains such that the robustness of controller can be improved [32,33,34,35,36,37]. A nonlinear disturbance observer was proposed in [32] to handle nonlinear systems with disturbances, which is applied to tracking control of pneumatic artificial muscle actuator using dynamic surface control (DSC) scheme [33]. A new type of composite control scheme for the uncertain structural systems was proposed in [34], which combines the disturbance observer-based control and the TSMC. Yang et al. [35] investigated a sliding mode control for nonlinear systems with mismatched uncertainties by the use of a disturbance observer. In the design process of above disturbance observers, it is assumed to vary slowly relative to the observer dynamics, namely, the derivative of disturbance is approximately equal to zero. However, in practice, the external disturbance usually varies in a complex way due to the ocean currents and waves, and its change rate is always bounded in the specific mission environment. Bu et al. [36] proposed a new nonlinear disturbance observer (NDO) based on tracking differentiator, which can be used to estimate many types of uncertain disturbances, and can overcome the disadvantages of existing NDOs that need the priori information concerning the upper and lower bounds of the disturbance and its ith derivatives Lipschitz upper bound. In [37], a novel NDO is constructed using a new TD designed based on hyperbolic sine function to estimate the model uncertainties and varying disturbances.

In this paper, a novel nonlinear disturbance observer-based backstepping finite-time sliding mode control (NDOB-BFTSMC) scheme is proposed for trajectory tracking of underwater vehicles with system uncertainties and external disturbances. In the proposed NDOB-BFTSMC scheme, backstepping design technique and the second-order sliding mode control are integrated to preserve the advantages and alleviate the drawbacks of each method. Particularly, the finite-time stability of the closed-loop control system is guaranteed by this way. Under the framework of BFTSMC, to compensate the unknown system uncertainties and external disturbance and further relax the matching condition of exact model information, a nonlinear disturbance observer (NDOB) is developed for online identifying the lumped disturbance term accurately. The main contributions of the proposed control scheme are summarized as follows.

(1) A backstepping second-order sliding mode control framework is constructed for trajectory tracking control of underwater vehicles. To relax the matching condition at the expense of a larger switching gain of SOSMC, the backstepping design technique is combined with SOSMC such that a second-order sliding surface is defined in backstepping procedures, and the second-order sliding mode control law is designed to guarantee the finite-time stability of the overall control system.

(2) Note that there exist unmodeled dynamics, parameter variations and unknown external disturbances in the underwater vehicle systems, then these uncertainties and disturbances are lumped into a single disturbance term. To online identify the lumped disturbance, a nonlinear disturbance observer is proposed without any acceleration measurements, which requires no knowledge of full dynamics such that further relax the matching condition at the expense of a larger switching gain on the basis of backstepping technique.

The remainder of this paper is organized as follows. The problem formulations and preliminaries are described in Sect. 2. The design of nonlinear disturbance observer and the proposed NDOB-BFTSMC are presented in Sect. 3. Moreover, the stability of the disturbance observer and overall closed-loop control system are also addressed in Sect. 3. The simulation studies are given in Sect. 4. Finally, the conclusion is drawn in Sect. 5.

2 Problem formulations and preliminaries

Consider the dynamics of an 5-DOF underwater vehicles in following form [38, 39]:

$$\begin{aligned}&{\dot{\pmb {\eta }}}={\pmb J(\pmb \eta ) \pmb v} \end{aligned}$$

(1a)

$$\begin{aligned}&\pmb M{\dot{\pmb v}}+\pmb C(\pmb v)\pmb v+\pmb D(\pmb v)\pmb v+\pmb g(\pmb \eta )=\pmb \tau +\pmb \tau _{ex} \end{aligned}$$

(1b)

where $\pmb \eta =[x,y,z,\theta ,\psi ]^T$ is the positions and attitudes vector, $\pmb v=[u,v,w,q,r]^T$ is the linear and angle velocities vector, $\pmb M\in \pmb {\mathbb {R}}^{5\times 5}$ is the inertia matrix, $\pmb C\in \pmb {\mathbb {R}}^{5\times 5}$ is the Coriolis and centrifugal matrix, $\pmb D\in \pmb {\mathbb {R}}^{5\times 5}$ is the drag forces (gravity) matrix, $\pmb g\in \pmb {\mathbb {R}}^{5\times 1}$ is the restoring forces (buoyancy) vector, $\pmb \tau =[\tau _1,\tau _2,\tau _3,\tau _4,\tau _5]^T$ is the control torques vector applied to the underwater vehicles, $\pmb \tau _{ex}=[\tau _{ex1},\tau _{ex2},\tau _{ex3},\tau _{ex4},\tau _{ex5}]^T$ is the time-varying unknown external disturbances vector due to ocean currents and waves, and $\pmb J(\pmb \eta )$ is the rotation matrix defined as

$$\begin{aligned} \pmb J(\pmb {\eta })=\left[ \begin{matrix}{\cos {\psi }\cos {\theta }}&{}{-\sin {\psi }}&{}{\cos {\psi }\sin {\theta }}&{}0&{}0\\ {\sin {\psi }\cos {\theta }}&{}{\cos {\psi }}&{}{\sin {\psi }\sin {\theta }}&{}0&{}0\\ {-\sin {\theta }}&{}0&{}{\cos {\theta }}&{}0&{}0\\ 0&{}0&{}0&{}1&{}0\\ 0&{}0&{}0&{}0&{}{\frac{1}{\cos {\theta }}}\end{matrix}\right] \nonumber \\ \end{aligned}$$

(2)

Underwater vehicles have several inherent dynamic properties, which are listed below and will be used for designing the extend disturbance observer later.

Property 1

The inertia matrix $\pmb M$ is symmetric positive definite and bounded, i.e., $\pmb M=\pmb M^T>0$, $\varpi _{M_0}\le \Vert \pmb M\Vert \le \varpi _{M_1}$. By using body symmetry conditions, i.e., $xz-$, $yz-$ and xy-planes of symmetry, the detailed expression of $\pmb M$ is simplified as [39]

$$\begin{aligned} \pmb M=\text {diag}\{m_{11},m_{22},m_{33},m_{44},m_{55}\} \end{aligned}$$

(3)

where $\varpi _{M_0}$ and $\varpi _{M_1}$ are unknown constants, $m_{11}=m-X_{\dot{u}}$, $m_{22}=m-Y_{\dot{v}}$, $m_{33}=m-Z_{\dot{w}}$, $m_{44}=I_y-M_{\dot{q}}$, $m_{55}=I_z-N_{\dot{r}}$. Here, m is the mass of the underwater vehicle, $I_{y}$ and $I_{z}$ are the moments of inertia about the pitch and yaw rotation, respectively, and $X_{*}$, $Y_{*}$, $Z_{*}$, $M_{*}$ and $N_{*}$ are the corresponding hydrodynamic derivatives.

Property 2

The matrix $\dot{\pmb M}-2\pmb C(\pmb v)$ is skew-symmetric [39], i.e.,

$$\begin{aligned}{}[\dot{\pmb M}-2\pmb C(\pmb v)]^T= & {} -[\dot{\pmb M}-2\pmb C(\pmb v)]\nonumber \\\Rightarrow & {} \dot{\pmb M}=\pmb C(\pmb v)+\pmb C^T(\pmb v) \end{aligned}$$

(4)

it follows that

$$\begin{aligned} \pmb v^T[\dot{\pmb M}-2\pmb C(\pmb v)]\pmb v\equiv 0,~ \forall \pmb v\in \pmb {\mathbb {R}}^{5\times 1} \end{aligned}$$

(5)

Here, the Coriolis and centrifugal matrix $\pmb C(\pmb v)$ is given by

$$\begin{aligned} \pmb C(\pmb v)=\left[ \begin{matrix}{0}&{}\quad {0}&{}\quad {0}&{}\quad {c_{14}}&{}\quad {c_{15}}\\ {0}&{}\quad {0}&{}\quad {0}&{}\quad {0}&{}\quad {c_{25}}\\ {0}&{}\quad {0}&{}\quad {0}&{}\quad {c_{34}}&{}\quad {0}\\ {c_{41}}&{}\quad {0}&{}\quad {c_{43}}&{}\quad {0}&{}\quad {0}\\ {c_{51}}&{}\quad {c_{52}}&{}\quad {0}&{}\quad {0}&{}\quad {0}\end{matrix}\right] \end{aligned}$$

(6)

where $c_{14}=(m-Z_{\dot{w}})w$, $c_{15}=-(m-Y_{\dot{v}})v$, $c_{25}=(m-X_{\dot{u}})u$, $c_{34}=-(m-X_{\dot{u}})u$, $c_{41}=-(m-Z_{\dot{w}})w$, $c_{43}=(m-X_{\dot{u}})u$, $c_{51}=(m-Y_{\dot{v}})v$, $c_{52}=-(m-X_{\dot{u}})u$. Moreover, the drag force matrix and restoring force vector are given by

$$\begin{aligned} {\pmb D(\pmb v)}= & {} \mathrm{diag}(d_{11},d_{22},d_{33},d_{44},d_{55}) \end{aligned}$$

(7)

$$\begin{aligned} {\pmb g(\pmb {\eta })}= & {} [g_{1},g_{2},g_{3},g_{4},g_{5}]^{T} \end{aligned}$$

(8)

where $d_{11}=-X_{u}-X_{|u|u}|u|$, $d_{22}=-Y_{v}-Y_{|v|v}|v|$, $d_{33}=-Z_{w}-Z_{|w|w}|w|$, $d_{44}=-M_{q}-M_{|q|q}|q|$, $d_{55}=-N_{r}-N_{|r|r}|r|$, $g_{1}=(P-B)\sin {\theta }$, $g_{2}=0$, $g_{3}=-(P-B)\cos {\theta }$, $g_{4}=-z_{B}B\sin {\theta }$, $g_{5}=0$. Here, P and B are the gravity and the buoyancy of underwater vehicles with center $\pmb r_{P}$ and $\pmb r_B$, respectively.

Now, by considering the additive uncertainties $\Delta \pmb *$, the parameters in (1b) can be defined as $\pmb C=\pmb C_0+\Delta \pmb C$, $\pmb D=\pmb D_0+\Delta \pmb D$, $\pmb g=\pmb g_0+\Delta \pmb g$. Then, the effect of unknown parameter uncertainties and external disturbances can be lumped into a disturbance vector $\pmb \tau _d$. Therefore, the underwater vehicles dynamics (1) is rewritten as

$$\begin{aligned} \dot{\pmb \eta }= & {} \pmb J(\pmb \eta )\pmb v \end{aligned}$$

(9a)

$$\begin{aligned} \dot{\pmb v}= & {} \pmb F(\pmb v, \pmb \eta )+\pmb G(\pmb \tau +\pmb \tau _d) \end{aligned}$$

(9b)

where $\pmb F(\pmb v, \pmb \eta )=-\pmb M^{-1}(\pmb C_0(\pmb v)\pmb v+\pmb D_0(\pmb v)\pmb v+\pmb g_0(\pmb \eta ))$, $\pmb G=\pmb M^{-1}$, $\pmb \tau _d=\pmb \tau _{ex}-\Delta \pmb C(\pmb v)\pmb v-\Delta \pmb D(\pmb v)-\Delta \pmb g(\pmb \eta )$.

Lemma 1

[40] Suppose that $V(\pmb x)$ is a smooth positive definite function (defined on $U\subset \mathbb {R}^n$) and $\dot{V}(\pmb x)+{\varrho V}^\varsigma (\pmb x)$ is negative semi-definite on $U\subset \mathbb {R}^n$ for $\varsigma \in (0,1)$ and $\varrho \in \mathbb {R}^+$, then there exists an area $U_0\subset \mathbb {R}^n$ such that any $V(\pmb x)$ which starts from $U_0\subset \mathbb {R}^n$ can reach $V(\pmb x)\equiv 0$ in finite time. Moreover, if $t_r$ is the time required to reach $V(\pmb x)\equiv 0$ then

$$\begin{aligned} t_r\le \frac{V^{1-\varsigma }(\pmb x_0)}{\varrho (1-\varsigma )} \end{aligned}$$

(10)

where $V(\pmb x_0)$ is the initial value of $V(\pmb x)$.

Lemma 2

[41] For $\nu \in \mathbb {R}^+$ and $\varepsilon >0$ it holds that

$$\begin{aligned} 0<|v|-v\tanh (\varepsilon v)\le \frac{0.2785}{\varepsilon } \end{aligned}$$

(11)

The control objective of this paper is to design a nonlinear disturbance observer-based backstepping second-order sliding mode control scheme such that all signals in the overall closed-loop system are bounded, and the trajectory states can track the given reference trajectories in the presence of time-varying disturbances and system uncertainties. Moreover, the tracking errors and sliding surface converge to a invariant region in finite time.

3 Nonlinear disturbances observer-based backstepping finite-time sliding mode control

In this section, a nonlinear disturbance observer-based backstepping finite-time sliding mode controller (NDOB-BFTSMC) will be developed, where the nonlinear disturbance observer is proposed without any acceleration measurement to online identify the unknown time-varying disturbances, and then the control bandwidth and tracking precision are improved.

3.1 Nonlinear disturbance observer design

Acceleration measurements are usually available for underwater vehicles system, so the disturbances observer can take following form

$$\begin{aligned} \dot{\hat{\pmb \tau }}_d=-\pmb L\hat{\pmb \tau }_d+\pmb L(\pmb M\dot{\pmb v}+\pmb N({{\pmb \eta }},{{\pmb v}})-\pmb \tau ) \end{aligned}$$

(12)

where

$$\begin{aligned} \pmb N({{\pmb \eta }},{{\pmb v}})=\pmb C_0\pmb v+\pmb D_0\pmb v+\pmb g_0 \end{aligned}$$

(13)

and $\pmb L=\text {diag}(l_{1},\ldots ,l_{5})$ is the observer gain matrix, which is positive definite, i.e., $l_{i}>0$, $i=1,\ldots ,5$.

Note that the precision of above observer relies on that of acceleration measurements; however, high-accuracy accelerometers are not available for many underwater vehicles; thus, a nonlinear disturbance observer without any acceleration measurement is highly desired. Now, we introduce an auxiliary variable given as

$$\begin{aligned} \pmb \epsilon =\hat{\pmb \tau }_d-\pmb p(\pmb v) \end{aligned}$$

(14)

where the auxiliary vector $\pmb p(\pmb v)$ can be determined from the observer gain matrix as follows

$$\begin{aligned} \frac{{\hbox {d}}}{{\hbox {d}}t}{\pmb p}(\pmb v)=\pmb L\pmb M{\dot{\pmb v}} \end{aligned}$$

(15)

By taking the time derivative of (14) together with (12) yields

$$\begin{aligned} \dot{\pmb \epsilon }&=\dot{\hat{\pmb \tau }}_d-\dot{\pmb p}({\dot{\pmb v}})=\dot{\hat{\pmb \tau }}_d-\pmb L\pmb M{\dot{\pmb v}}\nonumber \\&=-\pmb L(\pmb \epsilon +\pmb p)+\pmb L(\pmb M{\dot{\pmb v}}+\pmb N(\pmb \eta ,\pmb v)-\pmb \tau -\pmb M{\dot{\pmb v}})\nonumber \\&=-\pmb L\pmb \epsilon +\pmb L(\pmb N(\pmb \eta ,\pmb v)-\pmb \tau -\pmb p) \end{aligned}$$

(16)

Combining (14)–(16) together, the nonlinear disturbance observer takes following form

$$\begin{aligned}&\dot{\pmb \epsilon }=-\pmb L\pmb \epsilon +\pmb L(\pmb N(\pmb \eta ,\pmb v)-\pmb \tau -\pmb p)\nonumber \\&\dot{\pmb p}=\pmb L\pmb M{\dot{\pmb v}}\nonumber \\&\hat{\pmb \tau }_d=\pmb \epsilon +\pmb p(\pmb v) \end{aligned}$$

(17)

From (15), note that the proposed nonlinear disturbance observer designed in (15) has no requirement of acceleration measurement due to cancellation of the term $\pmb M\dot{\pmb v}$. Then, with (9) and (12)–(14), the error dynamics of observation $\dot{\tilde{\pmb \tau }}_d$ can be obtained as

$$\begin{aligned} \dot{\tilde{\pmb \tau }}_d&=\dot{\pmb \tau }_d-\dot{\hat{\pmb \tau }}_d=\dot{\pmb \tau }_d-\dot{\pmb \epsilon }-\frac{d}{dt}{\pmb p}(\pmb v)\nonumber \\&=\dot{\pmb \tau }_d+\pmb L(\hat{\pmb \tau }_d-\pmb p)-\pmb L(\pmb \tau _d-\pmb M\dot{\pmb v}-\pmb p)-\pmb L\pmb M{\dot{\pmb v}}\nonumber \\&=\dot{\pmb \tau }_d-\pmb L\tilde{\pmb \tau }_d \end{aligned}$$

(18)

The following theorem addresses the case when the underwater vehicles are subject to complex disturbances with bounded change rate.

Theorem 1

Consider the underwater vehicles subject to lumped disturbances described by (9), the disturbance observer is designed as (15) with the observation gain matrix $\pmb L$ and the auxiliary vector $\pmb p(\pmb v)$. Under the condition that the change rate of the lumped disturbance is bounded, i.e., there exists an unknown constant $\varpi _d>0$ such that $\Vert \dot{\pmb \tau }_d\Vert \le \varpi _d$ for all $t>0$. The disturbance tracking error $\tilde{\pmb \tau }_d$ is globally uniformly ultimately bounded (GUUB) and can be made arbitrarily small.

Proof

Consider the following Lyapunov candidate function:

$$\begin{aligned} V_0(\tilde{\pmb \tau }_d)=\frac{1}{2}\tilde{\pmb \tau }_d^T\pmb Q\tilde{\pmb \tau }_d \end{aligned}$$

(19)

where $\pmb Q$ is a positive definite matrix satisfying $\Vert \pmb Q\Vert \le \varpi _Q$. From (16), taking the derivative of $V_0$, it is obtained that

$$\begin{aligned} \dot{V}_0&= \tilde{\pmb \tau }_d^T\pmb Q\dot{\tilde{\pmb \tau }}_d=\tilde{\pmb \tau }_d^T\pmb Q(\dot{\pmb \tau }_d-\pmb L\tilde{\pmb \tau }_d)\nonumber \\&=-\pmb Q\pmb L\Vert \tilde{\pmb \tau }_d\Vert ^2+\tilde{\pmb \tau }_d^T\pmb Q\dot{\pmb \tau }_d \end{aligned}$$

(20)

Since $\Vert {\dot{\tau }}_d\Vert \le \varpi _d$, one can obtain following inequality

$$\begin{aligned} \tilde{\pmb \tau }_d^T\pmb Q\dot{\pmb \tau }_d\le \Vert \tilde{\pmb \tau }_d\Vert ^2+\Vert \pmb Q\Vert ^2\Vert \dot{\pmb \tau }_d\Vert ^2\le \Vert \tilde{\pmb \tau }_d\Vert ^2+\varpi _Q^2\varpi _d^2 \end{aligned}$$

(21)

By the inequality (19) and (41), it is observed that

$$\begin{aligned} \dot{V}_0\le&-(\lambda _{\min }(\pmb L)-1)\Vert \tilde{\pmb \tau }_d\Vert ^2+\varpi _Q^2\varpi _d^2\nonumber \\ \le&-\alpha _0 V_0+\beta _0 \end{aligned}$$

(22)

where $\alpha _0=\lambda _{\min }(\pmb L)-1$, $\beta _0=\varpi _Q^2\varpi _d^2$. From (20), it can be seen that

$$\begin{aligned} 0\le V_0(t)\le \frac{\beta _0}{\alpha _0}+\left( V_0(0)-\frac{\beta _0}{\alpha _0}\right) \exp ({-\alpha _0 t}) \end{aligned}$$

(23)

According to the uniform ultimate boundedness theorems [51] and (21), it implies that $V_0(t)$ and the disturbance tracking error $\tilde{\pmb \tau }_d$ are GUUB, namely,

$$\begin{aligned}&\Vert \tilde{\pmb \tau }_d\Vert ^2\le \frac{2\beta _0}{\alpha _0}+2\left( V_0(0)-\frac{\beta _0}{\alpha _0}\right) \exp ({-\alpha _0 t})\nonumber \\&\quad \Rightarrow \Vert \tilde{\pmb \tau }_d\Vert \le \sqrt{\frac{2\beta _0}{\alpha _0}+2\left( V_0(0)-\frac{\beta _0}{\alpha _0}\right) \exp ({-\alpha _0 t})} \end{aligned}$$

(24)

Thus, the convergence rate of the disturbance tracking error is $\alpha _0$, and there exists a time constant $T_0>0$, for any $\mu _0>\sqrt{{2\beta _0}/{\alpha _0}}$ such that $\Vert \tilde{\pmb \tau }_d\Vert \le \mu _0$, where the bound $\mu _0$ can be made arbitrarily small since $\beta _0/\alpha _0$ can be rendered arbitrarily small if parameter $\pmb L$ is selected appropriately, namely, a large enough $\pmb L$ can guarantee a arbitrarily small disturbance tracking error even in the case of fast-varying disturbance. From (18) and (24), it is concluded that $\dot{\tilde{\pmb \tau }}_d$ is bounded, i.e., $\Vert \dot{\tilde{\pmb \tau }}_d\Vert \le \varpi _{c}$, where $\varpi _{c}>0$ is an unknown constant. This concludes the proof. $\square$

3.2 Design of NDOB-BFTSMC

Define the tracking error $\pmb e_1=[e_{11},\ldots ,e_{15}]^T$ as

$$\begin{aligned} \pmb e_1(t)=\pmb \eta (t)-\pmb \eta _r(t) \end{aligned}$$

(25)

where $\pmb \eta _r(t)$ is the reference trajectory. From (9), let $\pmb v$ be a virtual input given by

$$\begin{aligned} \pmb v=\pmb J^T\left( \dot{\pmb \eta }_r-\pmb \phi (\pmb \kappa _\eta ,\pmb e_1)\right) =\pmb \alpha (\pmb e_1) \end{aligned}$$

(26)

where $\pmb \phi (\pmb \kappa _\eta ,\pmb e_1)=\pmb \kappa _\eta \tanh \big (\frac{\pmb e_1}{\mu _\eta }\big )$. Here, $\pmb \kappa _\eta =\text {diag}({\kappa _{\eta 1},\ldots ,\kappa _{\eta 5}})>0$, $\mu _\eta$ is a small positive scalar and $\tanh \big (\frac{\pmb e_1}{\mu _\eta }\big )=[\tanh \big (\frac{e_{11}}{\mu _\eta }\big )~\tanh \big (\frac{e_{12}}{\mu _\eta }\big ) \tanh \big (\frac{e_{13}}{\mu _\eta }\big ) \tanh \big (\frac{e_{14}}{\mu _\eta }\big )~ \tanh \big (\frac{e_{15}}{\mu _\eta }\big )]^T$.

Define the tracking error $\pmb e_2=[e_{21},\ldots ,e_{25}]^T$ as $\pmb e_2=\pmb v-\pmb \alpha$. Then, (9) can be rewritten as follows

$$\begin{aligned} {\left\{ \begin{array}{ll} \dot{\pmb e}_1&{}=\pmb J\pmb e_2-\pmb \kappa _\eta \tanh \big (\frac{\pmb e_1}{\mu _\eta }\big )\\ \dot{\pmb e}_2&{}=\pmb F+\pmb G(\pmb \tau +\pmb \tau _d)-\dot{\pmb \alpha }(\pmb e_1) \end{array}\right. } \end{aligned}$$

(27)

where $\dot{\pmb \phi }(\pmb e_1)$ in $\dot{\pmb \alpha }(\pmb e_1)$ is determined by

$$\begin{aligned} \dot{\pmb \phi }(\pmb e_1)=\mu _\eta ^{-1}\text {diag}\left( \kappa _{\eta i}\text {sech}^2\left( \frac{e_{1i}}{\mu _\eta }\right) \right) \dot{\pmb e}_1 \end{aligned}$$

(28)

Then, select the tracking error $\pmb s_1=\pmb e_2$ and its derivative $\pmb s_2=\dot{\pmb e}_2$ as the sliding variables, such that the sliding surface $\pmb \sigma$ can be defined in following form

$$\begin{aligned} \pmb \sigma =\pmb s_2+\pmb \lambda \pmb s_1+\pmb \gamma \int _0^t(\pmb s_2+\pmb \lambda \pmb s_1)d\tau \end{aligned}$$

(29)

where $\pmb \lambda =\text {diag}(\lambda _1,\ldots ,\lambda _5)$ and $\pmb \gamma =\text {diag}(\gamma _1,\ldots ,\gamma _5)$ are positive definite matrices.

Therefore, the NDOB-BFTSMC law is designed to drive the system trajectory onto the sliding surface as

$$\begin{aligned} \pmb \tau =\pmb G^{-1}(-\pmb F+\dot{\pmb \alpha }+\pmb \tau _s)-\hat{\pmb \tau }_d \end{aligned}$$

(30)

where $\pmb \tau _s$ is determined as

$$\begin{aligned} \dot{\pmb \tau }_s=-\pmb \lambda \pmb s_2-\pmb \rho \text {sgn}(\pmb \sigma )-\pmb \gamma (\pmb s_2+\pmb \lambda \pmb s_1)-\pmb {K\sigma } \end{aligned}$$

(31)

Here, $\pmb \rho =\text {diag}(\rho _1,\ldots ,\rho _5)$ and $\pmb K=\text {diag}(k_1,\ldots ,k_5)$ are positive definite matrices.

From (27), and according to the above definition of sliding variables, the derivatives of $\pmb s_1$ and $\pmb s_2$ can be obtained as

$$\begin{aligned} {\left\{ \begin{array}{ll} \dot{\pmb s}_1=\pmb s_2\\ \dot{\pmb s}_2=-\pmb \lambda \pmb s_2-\pmb \rho \text {sgn}(\pmb \sigma )-\pmb \gamma (\pmb s_2+\pmb \lambda \pmb s_1)-\pmb K\pmb \sigma +\pmb G\dot{\tilde{\pmb \tau }}_d \end{array}\right. } \end{aligned}$$

(32)

and then the time derivative of the sliding surface is obtained as

$$\begin{aligned} \dot{\pmb \sigma }=-\pmb \rho \text {sgn}(\pmb \sigma )-\pmb K\pmb \sigma +\pmb G\dot{\tilde{\pmb \tau }}_d \end{aligned}$$

(33)

Theorem 2

Consider the system dynamics in (9), the virtual input $\pmb \alpha$ in (26) and the control law $\pmb \tau$ in (30) and (31), in finite time, the tracking errors $\pmb e_1$ and $\pmb e_2$ and the sliding surface $\pmb \sigma$ of the underwater vehicle system converge to the region

$$\begin{aligned} \Vert \pmb \chi \Vert \le \frac{\varpi }{\overline{\phi }} \end{aligned}$$

(34)

where $\pmb \chi =[\frac{\pmb e_1^T}{\sqrt{2}}~\frac{\pmb \sigma ^T}{\sqrt{2}}]^T$, $\varpi =1.9325\mu _\eta$ and $\overline{\phi }=\min (\sqrt{2}\overline{\kappa }_\eta , \sqrt{2}(\overline{\rho }-\varpi _c))$ with $\overline{\kappa }_\eta =\min ({\kappa _{\eta i}})$ and $\overline{\rho }=\min ({\rho _i})~(i=1,\ldots ,5)$.

Proof

Substituting $\pmb v$ defined in (26) into the first equation of (27), one can obtain

$$\begin{aligned} \dot{\pmb e}_1=-\pmb \kappa _\eta \tanh \big (\frac{\pmb e_1}{\mu _\eta }\big ) \end{aligned}$$

(35)

Consider the Lyapunov candidate function as

$$\begin{aligned} V_1=\frac{1}{2}\pmb e_1^T\pmb e_1=\frac{1}{2}\sum _{i=1}^5 e_{1i}^2 \end{aligned}$$

(36)

Then from (35), differentiating (36) with respect to time as follows

$$\begin{aligned} \dot{V}_1&=\pmb e_1^T\dot{\pmb e}_1\nonumber \\&=-\pmb e_1^T\pmb \kappa _\eta \tanh \big (\frac{\pmb e_1}{\mu _\eta }\big )\nonumber \\&=-\sum _{i=1}^5 \kappa _{\eta i}e_{1i}\tanh \big (\frac{e_{1i}}{\mu _\eta }\big ) \end{aligned}$$

(37)

According to Lemma 2, it is obtained that $0<|e_{1i}|\le e_{1i}\tanh (\frac{e_{1i}}{\mu _\eta })+0.2785\mu _\eta$ such that (37) can be rearranged as

$$\begin{aligned} \dot{V}_1&\le -\sum _{i=1}^5 \kappa _{\eta i}\big (|e_{1i}|-0.2785\mu _\eta \big )\nonumber \\&\le -\sum _{i=1}^5 \kappa _{\eta i}|e_{1i}|+0.2785\sum _{i=1}^5 \mu _\eta \nonumber \\&\le -\sqrt{2}\overline{\kappa }_{\eta }\Vert \pmb e_1\Vert +\varpi \end{aligned}$$

(38)

where $\overline{\kappa }_\eta =\min ({\kappa _{\eta i}}),~i=1,\ldots ,5$, and $\varpi =1.3925\mu _\eta$.

From Lemma 1, if $\Vert \pmb e_1\Vert >\frac{\varpi }{\sqrt{2}\overline{\kappa }_{\eta }}$, then the finite-time stability is still guaranteed, namely, when the state moves outside the region

$$\begin{aligned} \Vert \pmb e_1\Vert \le \frac{\varpi }{\sqrt{2}\overline{\kappa }_{\eta }} \end{aligned}$$

(39)

$\dot{V}_1<0$ is guaranteed. Hence, the states $e_{1i}$ of equilibrium of the differential equation $\dot{e}_{1i}=\kappa _{\eta i}\tanh \big (\frac{e_{1i}}{\mu _\eta }\big )$ will converge to the region (38) in finite time.

Furthermore, consider following Lyapunov candidate function

$$\begin{aligned} V_2=\pmb \chi ^T\pmb \chi =V_1+\frac{1}{2}{\pmb \sigma }^T{\pmb \sigma } \end{aligned}$$

(40)

From (33), the time derivative of $V_2$ is obtained as

$$\begin{aligned} \dot{V}_2=&\,\dot{V}_1+{\pmb \sigma }^T\dot{\pmb \sigma }\nonumber \\ =&\,\dot{V}_1+{\pmb \sigma }^T\big (-\pmb \rho \text {sgn}(\pmb \sigma )-\pmb K\pmb \sigma +\pmb G\dot{\tilde{\pmb \tau }}_d\big )\nonumber \\ \le&-\sum _{i=1}^5 \kappa _{\eta i}|e_{1i}|-\sum _{i=1}^5\rho _i|\sigma _i|+0.2785\sum _{i=1}^5 \mu _\eta \nonumber \\&+\sum _{i=1}^5|\sigma _i||G_i||\dot{\tilde{\tau }}_{di}| \end{aligned}$$

(41)

From Theorem 1 and Property 1, it is obtained that $\Vert \dot{\tilde{\pmb \tau }}_d\Vert \le \varpi _c$ and $\Vert \pmb G\Vert \le \varpi _{M_0}^{-1}$. Thus, (40) can be rewritten as

$$\begin{aligned} \dot{V}_2\le&-\overline{\kappa }_{\eta }\sum _{i=1}^5 |e_{1i}|\!-\!\overline{\rho }\sum _{i=1}^5|\sigma _i|+\varpi _{M_0}^{-1}\sum _{i=1}^5\varpi _c|\sigma _i|\!+\!\varpi \nonumber \\ \le&-\overline{\kappa }_{\eta }\sum _{i=1}^5 |e_{1i}|-\big (\overline{\rho }-\varpi _c\big )\sum _{i=1}^5|\sigma _i|+\varpi \nonumber \\ \le&-\min \big (\sqrt{2}\overline{\kappa }_{\eta },\sqrt{2}(\overline{\rho }-\varpi _c)\big )\nonumber \\&\quad \left( \sum _{i=1}^5\left( \frac{|\sigma _i|}{\sqrt{2}}\right) ^2+\sum _{i=1}^5\left( \frac{|e_{1i}|}{\sqrt{2}}\right) ^2\right) ^{1/2}+\varpi \nonumber \\ =&-\overline{\phi }\Vert \pmb \chi \Vert +\varpi \end{aligned}$$

(42)

where $\overline{\phi }=\min \big (\sqrt{2}\overline{\kappa }_{\eta },\sqrt{2}(\overline{\rho }-\varpi _c)\big )$. Note that $\dot{V}_2<0$ if $\Vert \pmb \chi \Vert >{\varpi }/{\overline{\phi }}$, namely, the decrease of $V_2$ finally forces the trajectories of the closed-loop system into $\Vert \pmb \chi \Vert >{\varpi }/{\overline{\phi }}$. Thus, the trajectories of the closed-loop system are ultimately bounded as

$$\begin{aligned} \lim _{t\rightarrow \infty }\pmb \chi \in \left( \Vert \pmb \chi \Vert >\frac{\varpi }{\overline{\phi }}\right) \end{aligned}$$

(43)

which is a small set containing the origin of the closed-loop system. Therefore, from Lemma 1, the state $\pmb \chi$ will converge to the region

$$\begin{aligned} \Vert \pmb \chi \Vert \le \frac{\varpi }{\overline{\phi }} \end{aligned}$$

(44)

in finite time. This implies that $\sigma _i=0~(i=1,2,\ldots ,5)$, and it follows that $s_i$ and $\dot{s}_i$ are both quite close to zero. This completes the proof. $\square$

It is worth noting that the accuracy of tracking errors and the convergence time are determined by the values of the parameters $\pmb \kappa _\eta$, $\pmb \rho$ and $\mu _\eta$, which can be made arbitrarily small by increasing the value of $\pmb \kappa _\eta$ and $\pmb \rho$ and decreasing the value of $\mu _\eta$, namely, large enough $\pmb \kappa _\eta$ and $\pmb \rho$ as well small enough $\mu _\eta$ can make the tracking errors arbitrarily small such that the convergence performance is improved.

4 Simulation studies

In order to verify the effectiveness and superiority of the proposed NDOB-BFTSMC scheme, several simulations are conduct on the underwater vehicle developed by the research group at Tokyo University of Marine Science and Technology [42]. The nominal parameter values of (1) are employed as follows: the mass $m=390 kg$; the moments of inertia $I_{y}=305.67 kgm^{2}$ and $I_{z}=305.67 kgm^{2}$; the derivatives of the added mass $X_{\dot{u}}=-49.12, Y_{\dot{v}}=-311.52, Z_{\dot{w}}=-311.52$ and $N_{r}=-200$; the coefficients of the linear skin friction $X_{u}=-20, Y_{v}=-200, Z_{w}=-200, M_{q}=-200$ and $N_{r}=-200$; the coefficients of the quadratic drag $X_{|u|u}=-30, Y_{|v|v}=-300, Z_{|w|w}=-300, M_{|q|q}=-300$ and $N_{|r|r}=-300$; the center of the underwater vehicle $x_{B}=0, y_{B}=0, z_{B}=-0.15$. Moreover, the actual plant model in (9) is constructed under the following conditions: for parameter uncertainties, $\triangle \pmb C=-0.2 \pmb C$, $\triangle \pmb D=0.2 \pmb D$, $\triangle \pmb g=0.1 \pmb g$; for external disturbances $\pmb \tau _{ex}(t)$, the complex environment involving ocean currents and waves are governed by

$$\begin{aligned} \pmb \tau _{ex}(t)=\left[ \begin{aligned} 3+5\sin (4t-\pi /3)\\ 2+3\cos (2t-\pi /6)\\ 4+7\sin (3t-\pi /4)\\ 5+2\cos ({0.5t-{\pi }/{4}})\\ 2+2\sin ({0.5t+{\pi }/{6}}) \end{aligned}\right] \end{aligned}$$

(45)

For achieving the control objective of high-accuracy tracking, the reference trajectory $\pmb \eta _r$ governed by

$$\begin{aligned} \pmb \eta _r(t)=\left[ \begin{matrix} \sin (0.02t)\\ \cos (0.01t)\\ \sin (0.01t)+2\cos (0.01t)\\ -0.1\cos (0.01t)+0.2\sin (0.01t)\\ 0.1\cos (0.01t)-0.1\sin (0.01t) \end{matrix}\right] \end{aligned}$$

(46)

Moreover, the initial conditions are set as follows: $\pmb \eta (0)=[0(m), 1(m), 1.5(m),0(rad), 0(rad)]^{T}$, and the design parameters associated with the BFTSMC and NDOB are selected as follows: $\pmb \lambda =\text {diag}(3,3,3, 3,3)$, $\pmb \gamma =\text {diag}(30,30,30,30,30)$, $\pmb K=\text {diag}(30,30,30,30,30)$, $\pmb \kappa _\eta =\text {diag}(2,2,2,1.1,1.1)$, $\pmb \rho =\text {diag}(5,5,5,5,5)$, $\pmb L=\text {diag}(80,80,80,80,80)$, $\mu _\eta =0.05$.

Fig. 1

Reference and actual trajectories in 3D presentation

Fig. 2

Reference and actual trajectories in separate-dimension presentation

Fig. 3

Observation of the unknown lumped disturbance

Fig. 4

Observation errors of unknown lumped disturbance

Fig. 5

Response of sliding surface

Fig. 6

Response of sliding variable $\pmb s_1$

Fig. 7

Response of sliding variable $\pmb s_2$

Fig. 8

Comparisons on the tracking error $\pmb e_1$ among different schemes

The simulation results of reference and actual trajectories of tracking underwater vehicles are shown in Figs. 1 and 2, including the time history of the reference and actual trajectories of underwater vehicles in 3D and separate-dimension presentations, from which one can clearly observe that underwater vehicles are able to track the reference trajectory with high accuracy using the proposed NDOB-BFTSMC scheme, and a remarkable transient response of trajectory tracking can be obtained by the proposed control laws. The output of disturbance observer, namely, the estimate of the lumped disturbance, and the corresponding observation error of unknown lumped disturbance is shown in Figs. 3 and 4, respectively, which show that a remarkable estimation of the unknown lumped disturbance is achieved by using NDOB-BFTSMC. Figures 5, 6 and 7 show the responses of the sliding surface and the sliding variables, from which one can observe that the sliding surface is forced into a region of origin, namely, the system trajectories can be forced into the region of sliding surface $\pmb \sigma$, and the validity is demonstrated by the convergence of sliding variables $\pmb s_1$ and $\pmb s_2$ in Figs. 6 and 7, respectively.

Table 1 Performance comparison

To verify the superiority of the proposed NDOB-BFTSMC scheme in terms of tracking accuracy, the comparisons with other schemes, i.e., SMC, BFTSMC, AFSMC, on the tracking error $\pmb e_1$ are exhibited in Fig. 8, from which one can observe higher initial errors in the first 20 runs for the NDOB-BFTSMC scheme due to the high frequency disturbances, and then the robustness of the controller is enhanced along with the finite-time convergence stability. The responses of tracking errors can immediately reflect the transient responses of each control schemes such that the proposed control scheme can obtain better transient and steady-state responses by comparing with other control schemes in Fig. 8. Moreover, the comprehensive performance comparisons for tracking errors of the position and attitude of underwater vehicles are summarized in Table 1, where the integral time square error (IAE), i.e., IAE $=\int _0^t|e(\tau )|d\tau$, the integral square error (ISE), i.e., ISE $=\int _0^te^2(\tau )d\tau$ and the root-mean-square error (RMSE), i.e., RMSE $=\left( {\frac{1}{t}\int _0^t e^2(\tau )d\tau }\right) ^{1/2}$, are utilized to assess the transient and steady-state performance in trajectory tracking of underwater vehicles. For computational time of each scheme, we run each scheme ten times and take the average value as the computation time of the corresponding scheme. According to [43], theoretically, the computational time of the proposed NDOB-BFTSMC is higher than that of SMC and BFTSMC since there exists more online parameters in the proposed scheme. The theoretical analysis is demonstrated by computational time in Table 1, but there is only slight gaps with other schemes which is acceptable for practical application. Furthermore, the lines of code are used to assess the program complexity of each control scheme since the program complexity is an important fact for controller design in practice. From Table 1, it can be seen that the lines of code for the proposed NDOB-BFTSMC are larger than these of SMC and BFTSMC, but only slightly. In addition, the control inputs are shown in Fig. 9, from which one can see that the control inputs of the proposed NDOB-BFTSMC scheme are much smaller than these of BFTSMC and the chattering phenomenon is suppressed, which demonstrates that the effectiveness of the proposed NDO. From above simulation results, it is evident that the NDOB-BFTSMC scheme can provide improved robust performance with regard to system uncertainties and external disturbances.

Fig. 9

Comparisons on control inputs $\pmb \tau$ among different schemes

5 Conclusion

In this paper, a novel backstepping finite-time sliding mode control scheme using nonlinear disturbance observer for trajectory tracking of the underwater vehicle system which suffers from system uncertainties and external disturbances. A robust output feedback control framework has been constructed by integrating backstepping design technique into the second-order sliding mode control such that the finite-time stability can be guaranteed. Under the control framework, by combining the system uncertainties and external disturbances into a single disturbance term, a nonlinear disturbance observer is employed to online identify the unknown lumped disturbance term. The stability of the overall closed-loop control system and the boundedness of the signals have been proved using the Lyapunov stability theorem, and the position and attitude tracking errors can be arbitrarily small by adjusting the designed parameters. From the comprehensive simulation results, the effectiveness and superiority of the proposed NDOB-BFTSMC scheme are indicated, such that the robust finite-time tracking performance of underwater vehicle systems with high-accuracy and rapid transient responses is obtained.