# Fixed-Time Trajectory Tracking Control of Autonomous Surface Vehicle with Model Uncertainties and Disturbances

Authors: Jiawen Cui; Haibin Sun
Journal: Complexity
 (2020)

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## Abstract

The issue of fixed-time trajectory tracking control for the autonomous surface vehicles (ASVs) system with model uncertainties and external disturbances is investigated in this paper. Particularly, convergence time does not depend on initial conditions. The major contributions include the following: (1) An integral sliding mode controller (ISMC) via integral sliding mode surface is first proposed, which can ensure that the system states can follow the desired trajectory within a fixed time. (2) Unknown external disturbances are absolutely estimated by means of designing a fixed-time disturbance observer (FTDO). By combining the FTDO and ISMC techniques, a new control scheme (FTDO-ISMC) is developed, which can achieve both disturbance compensation and chattering-free condition. (3) Aiming at reconstructing the unknown nonlinear dynamics and external disturbances, a fixed-time unknown observer (FTUO) is proposed, thus providing the FTUO-ISMC scheme that finally achieves trajectory tracking of ASVs with unknown parameters. Finally, simulation tests and detailed comparisons indicate the effectiveness of the proposed control scheme.


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## Body

## 1. Introduction

With marine engineering operations developing and progressing, autonomous surface vehicles (ASVs) are instrumental in river and oil pipeline inspection, hull inspection, ocean survey, levee inspection, underwater archaeology, and underwater wreck inspection [1–5]. ASVs are usually perceived as a class of nonlinear dynamic systems equipped with complex external disturbances and model uncertainties [6]. It is an overwhelming matter to design a highly efficient controller for the ASV system.Trajectory tracking is a basic problem for ASVs; however, as the system dynamics of ASV are highly nonlinear and there are unpredictable external disturbances in the marine environment, designing an effective controller for ASVs is a challenging issue. Many classical control algorithms such as feedback linearization [7], backstepping control [8], PID control [9], adaptive control [10], fuzzy control [11], and neural networks control [12] have been implemented to the trajectory tracking control of ASVs. The system states generally realize either asymptotic convergence or exponential convergence. In addition, many scholars have proposed many compound control methods in combination with above different control theories that apply to actual control systems [13–18], especially ASV systems [5, 6, 9], etc.The finite-time control scheme profits from short divergence span and strong robustness, which is applied to various nonlinear control systems. In this respect, continuous finite-time control schemes for robotic manipulators have already been designed utilizing terminal sliding mode control theory in [19], and a nonsingular terminal sliding mode control scheme for a marine vehicle with complex unknowns has been presented in [20]. Moreover, a finite-time integral sliding controller in [21] has been designed to realize the path following of the ASVs.Although the finite-time tracking control problems are achieved in the above references, the convergence time is to depend on the original states. When the initial time tends to infinity, the convergence time also tends to infinity. However, compared with the finite-time control schemes, the converge speed of the fixed-time control schemes is quite insensitive to initial condition [22, 23]. Recently, many fixed-time sliding mode control schemes have been proposed. For nonlinear systems with matched uncertainties and disturbances, a fixed-time nonsingular terminal sliding mode controller has been proposed in [24]. In [25], for the trajectory tracking control, a fixed-time nonsingular terminal sliding mode controller for a warship-launched submarine with multiple disturbances has been presented. In addition, a fixed-time sliding mode control for fault-tolerant trajectory tracking of an ASV has been designed in [26].As known, chattering is an inherent phenomenon in sliding mode control. To reduce the chattering, a feasible solution is that disturbance observer-based control (DOBC) schemes are used to estimate external disturbances and model uncertainties and the disturbance estimation values are introduced into sliding mode control law. In order to improve the convergence speed and robustness, some finite-time disturbance observers were designed to estimate external disturbances and model uncertainties [27–32].In this brief, the fixed-time trajectory tracking control scheme for ASVs with external disturbances and model uncertainties is explored. An integral sliding mode controller (ISMC) is firstly intended by using ISM surface for the ASV system without external disturbances, so that system states can attain the expected value in fixed time. Next, a fixed-time disturbance observer (FTDO) is designed to estimate the external disturbances and a new control scheme (FTDO-ISMC) is constructed to enable that the system states can accurately track the expected trajectory within fixed time even if there exist unexpected external disturbances. Furthermore, to guarantee good tracking performance against both disturbances and unknown system dynamics, a corresponding fixed-time unknown observer based on ISMC (FTUO-ISMC) control scheme is proposed. As a consequence, simulation results imply that the proposed control schemes can guarantee the system states to track the desired trajectory in a fixed time in spite of the ASV system subject to unknown disturbances and model uncertainties and the convergence speed is regardless of the origin states of the ASV system.The remainder of this paper is structured as follows: In Section2, some definitions and lemmas related to the trajectory tracking problem are formulated. The problem of the paper is described in Section 3. Section 4 describes the design of fixed-time controller and its stability analysis. Simulation results and discussion are mentioned in Section 5. And, Section 6 summarizes the main conclusions of this paper.


## 2. Preliminaries

Lemma 1 (see [33]).
Consider the following double-integrator system:(1)x˙1=x2,x˙2=u,x0=x0,with the control law(2)ut=−k1x1ϱ1+k1′x1+k2″x1ϱ1′−k2x2ϱ2+k2′x2+k2″x2ϱ2′,where parameters ki>0, ki′>0, ki′′>0 (i=1,2), and ϱi, ϱi′, (i=1,2) are chosen by(3)ϱ1=ϱ2−ϱ,ϱ2=ϱ,ϱ1′=4−3ϱ2−ϱ,ϱ2′=4−3ϱ3−2ϱ,with ϱ∈0,1 and xα=xαsignx, α≥0. Next, the state of the double-integrator system is fixed-time stability with convergence time tf.Lemma 2 (see [34]).
Consider the following nonlinear system:(4)x˙t=fx,t,f0,t=0,x0=x0,where x∈ℝn and f:ℝ+×ℝn⟶ℝn is a nonlinear function. For the above system, suppose that there is a continuous radially unbounded function V:ℝn⟶ℝ+U0 which satisfies(1)
Vx=0, when x=0(2)
V˙x≤−αVpx−βVqx for some α,β,p,q>0, with 0<p<1 and q>1
The considered nonlinear system is globally fixed-time stable within the settling timeT satisfying(5)T≤Tmax≔1α1−p+1βq−1.


## 3. System Modeling and Problem Formulation

The kinematics and dynamics of the ASV system regarded as rigid body with three degrees of freedom (3-DOF) are represented by [6](6)η˙=Rφυ,(7)Mυ˙=−Cυυ−Dυυ−gη+τ+MRTdlt,where η=x,y,φT is the location x,y and course angle φ of ASVs in an earth-fixed inertial frame, υ=u,ν,rT is the linear velocities u,ν and angular rate r in the body-fixed frame, τ=Fu,Fν,FrT stands for the actual control thrust, and dlt denotes unsuspected external disturbances owing to complex surface environment including wind, waves, and ocean current. The rotation matrix Rφ is defined by(8)Rφ=cosφ−sinφ0sinφcosφ0001,being provided with the following characters: RTφRφ=I, R˙φ=RφSr, ∀φ⊆0,2π, and RTφSrRφ=RφSrRTφ=Sr, where(9)Sr=0−r0r00000is the inertia matrix M=MT>0 and(10)M=m−Xu˙000m−Yν˙mxg−Yr˙0mxg−Nν˙Iz−Nr˙,where m is the mass of the system, Iz is the inertia matrix concerned with the yaw angle, Yr˙=Nν˙, and X∗,Y∗,andZ∗ denote the corresponding hydrodynamic derivatives. Coriolis and centripetal matrix Cυ=−CυT have the following form:(11)Cv=00c13v00c23v−c13v−c23v0,and the damping matrix Dv is described by(12)Dv=d11v000d22vd23v0d32vd33v,where c13v=−mxgr+ν+Xu˙ν+Yr˙r, c23v=mu−Xu˙u, and d11v=−Xu−Xuuu−Xuuuu2, d22v=−Yν−Yννν, d23v=−Yr−Yνrν−Yrrr, d32v=−Nν−Nννν−Nrνr, and d33v=−Nr−Nνrν−Nrrr. And gη is the gravity and buoyancy forces and moments, which is usually used as a constant in ASV.Consider the desired trajectory as follows:(13)η˙d=Rφdυd,(14)Mυ˙d=−Cυdυd−Dυdυd+τd,where ηd=xd,yd,φdT and υd=ud,νd,rdT denote its desired position and velocity vectors, and the model contains no unknown nonlinear dynamics including external disturbances and model uncertainties.Assumption 1.
For disturbance vectordlt, it is given that constants κi satisfies d˙lit≤κi, where κi is an unknown nonnegative bounded constant.
The control purpose, in this paper, is to design fixed-time trajectory tracking control schemes so that the practical position and velocity (6)-(7) can precisely pursuit the expected ones (13)-(14), respectively.


## 4. Controller Design and Stability Analysis

### 4.1. Coordinate Transformation

Consider coordinate transformations as follows:(15)σ=Rυ,σ∈σ,σd,υ∈υ,υd,R∈R,Rd,where σ=σ1,σ2,σ3T, σd=σd1,σd2,σd3T, R=Rφ, and Rd=Rφd.By combining (6)-(7) and (15), we can obtain(16)η˙=σ,σ˙=RM−1τ+Θη,σ+dlt,where(17)Θη,σ=Sσσ−RM−1CRTσ+DRTσRTσ−RM−1gη.Similarly, together with (13), (14), and (15), we obtain(18)η˙d=σd,σ˙d=RdM−1τd+Θdηd,σd,where(19)Θdηd,σd=Sσdσd−RdM−1CRdTσd+DRdTσdRdTσd.Define the position and velocity errorηe=ηe1,ηe2,ηe3T and σe=σe1,σe2,σe3T. Then, we have(20)η˙e=σe,σ˙e=RM−1τ−RdM−1τd+dlt+Θeη,σ,ηd,σd,where(21)Θeη,σ,ηd,σd=Θη,σ−Θdηd,σd.


### 4.2. Design of the ISMC without External Disturbances

In this section, an ISMC is firstly proposed for tracking error systems (20) and (21) without external disturbances, and the fixed-time stability is verified.The ISM manifold is designed as follows:(22)sσet=σet+∫t0tunσeθdθ,where(23)un=k1ηeϱ1+k1′ηe+k1″ηeϱ1′+k2σeϱ2+k2′σe+k2″σeϱ2′.and parameters are provided in Lemma 1.The ISMC can be designed as(24)τISMC=MR−1RdM−1τd−Θe−un−ξ1sα−ξ2sβ,where 0<α<1, β>1, ξ1>0, and ξ2>0.Theorem 1 (ISMC).
Consider tracking error systems (20)-(21) without external disturbances, and an ISMC is designed by (24). Systems (6) and (7) can converge to desired trajectories (13) and (14) within a fixed time, i.e., η≡ηd and υ≡υd, when t>Tf1.Proof.
There are two processes in the whole verification: the reaching and the sliding phases:(i)
Step 1: taking the derivative of ISM surfaces (22) and (23) along error systems (20) and (21) without external disturbances dlt and combining with the control law (24), we can obtain(25)s˙=σ˙e+un=−ξ1sα−ξ2sβ.Take the candidate Lyapunov function as follows:(26)Vs=12s2.Differentiating it along the dynamics (25), we obtain(27)V˙s=ss˙=s−ξ1sα−ξ2sβ=−ξ1sα+1−ξ2sβ+1=−2α+1/2ξ1Vα+1/2−2β+1/2ξ2Vβ+1.According to Lemma2, it is claimed that the ISM control law will let the system states reach the ISM surface s=0 within a fixed time.(ii)
Step 2: at that moment, error system (20) will reduce to the following system:(28)η˙e=σe,σ˙e=−unηe,σe.
By applying Lemma1, we have that system (28) is globally fixed-time stable. Eventually, errors ηe and σe are converging to zero within a fixed time tf, so this completes the proof. Under the ISMC scheme, the convergence time of the ASV system is Tf1=T0+tf, where T0=1/2α−1/2ξ11−α+1/2β−1/2ξ2β−1.Remark 1.
External disturbances are not considered in ISMC excogitation, but they are actually exist in the actual environment. Therefore, controller design for the ASV system with external disturbances is essential.


### 4.3. Design of FTDO-ISMC

To achieve accurate tracking performance, in this section, a FTDO is built to estimate the external disturbances. Inspired by [35–37], a FTDO algorithm is established as(29)χ0=σ−χ,χ˙=RM−1τ−RdM−1τd+Θeη,σ,ηd,σd+d^lt+ψi,ψi=λ1iχ0i1/2signχ0i+λ2iχ0iγisignχ0i,d^˙lt=βisignχ0i,i=1,2,3,where χ0=χ01,χ02,χ03T is an auxiliary variable, ψ=ψ1,ψ2,ψ3T, d^lt=d^l1t,d^l2t,d^l3tT denotes the estimation of dlt, λ1i and λ2i are constants greater than zero, βi>κi,andγi>1.The error dynamic of the observer is described as(30)χ˙0i=χ1i−λ1iχ0i1/2signχ0i−λ2iχ0iγisignχ0i,χ1i.=−βisignχ0i+d˙lit,where χ1i=dlit−d^lit is the estimation error of external disturbance. It can be derived from [36, 37] that observer error system (18) is fixed-time stable according to Assumption 1, i.e., when t>t1, χ0i=χ1i=0. And the t1 satisfies t1≤1/λ2γ−1εγ−1+2ε1/2/λ11+1/m1/M−hλ1/λ1 with ε>0, M=α−κ,m=α+κ, and hλ1=1/λ1+2exp1/mλ11/3.Theorem 2 (FTDO-ISMC).
Consider tracking error systems (20)-(21) with external disturbances satisfying Assumption 1; then, a FTDO-ISMC is designed as(31)τFTDO−ISMC=MR−1RdM−1τd−Θe−un−d^lt−ξ1sα−ξ2sβ,with d^l estimated by FTDO (30). Systems (6) and (7) can converge to desired trajectories (13) and (14) within a fixed time, i.e., d^lt≡dlt, η≡ηd, and υ≡υd, when t>Tf2.Proof.
The derivative of ISM surface (13) is rewritten as(32)s˙=χ1−ξ1sα−ξ2sβ.
And derivative Lyapunov functionV based on (32) obtains(33)V˙=sχ1−ξ1sα+1−ξ2sβ+1.
When the timet>t1, d^lt equals dlt, that is to say that the disturbance estimation error χ1i converges to zero within a fixed time. (33) is rewritten as(34)V˙=−ξ1sα+1−ξ2sβ+1.
Based on Lemma2, (34) is described by V˙=−2α+1/2ξ1Vα+1/2−2β+1/2ξ2Vβ+1/2 and we obtain that the ISM manifold s=0 will be arrived within a fixed time T1=1/2α−1/2ξ11−α+1/2β−1/2ξ2β−1. The system is still the same as (28) while s=0. In other words, the tracking error system states ηe and υe are still sufficient to ensure convergence along the sliding surface in a fixed time tf. Under the FTDO-ISMC scheme, the convergence time of the ASV system is Tf2≤t1+T1+tf. This concludes the proof.Remark 2.
Note that we do not testify that the proposed controller (31) can ensure the boundedness of errors ηe and σe in the time quantum 0,t1 since the analysis of the dynamics of tracking error system is a difficult subject due to the complex nonlinear terms. In view of this reason, we have done a large number of simulations for ASV systems (6)-(7) under observer (29) and control laws (31), in which any divergence phenomenon is not observed. Practically, for the purpose of guaranteeing the boundedness of error system states in engineering, a bounded control rule can be applied in 0,t1. Therefore, it can assume that all states will not be divergent in fixed time in advance of the observer error dynamics converging.Remark 3.
Notice that it is difficult forΘe to obtain the real value because Cv and Dv are unknown because of existing uncertainties in ASV uncertainties. The FTDO only estimates the external disturbances and the influence of nonlinear terms is not considered, which will be solved in this context.


### 4.4. Design of FTUO-ISMC

In this section, the unknown nonlinear term consisting ofCυ, Dυ, and gη together with the external disturbances dlt are regarded as the lumped disturbances. Therefore, rewriting tracking error systems (20) and (21) yields(35)η˙e=σe,σ˙e=RM−1τ−RdM−1τd−Θd+Sσσ+Φ,where(36)Φ=−RM−1CRTσ+DRTσRTσ−RM−1gη+dlt,is an unknown nonlinearity regarded as the lumped disturbances. According to [29], an assumption is presented as follows.Assumption 2.
There exists a bounded constanth0 such that Φ˙≤h0.
A FTUO is presented as(37)e0=σ−e,e˙=RM−1τ−RdM−1τd−Θdηd,σd+Sσσ+Φ^+ψi,ψi=λ1ie0i1/2signe0i+λ2ie0iγisigne0i,Φ^.i=βisigne0i,i=1,2,3,where e0=e01,e02,e03T is an auxiliary variable, ψ=ψ1,ψ2,ψ3T, Φ^=Φ^1,Φ^2,Φ^3T is the estimation of the unknown lumped disturbances Φ, λ1i and λ2i are positive constants, βi>h0,andγi>1.
The error dynamic of the observer is described as(38)e˙0i=e1i−λ1ie0i1/2signe0i−λ2ie0iγisigne0i,e˙1i=−βisigne0i+Φ˙it,where the lumped disturbance estimation error is described as e1i=Φi−Φ^i. With the help of Assumption 2, observer error system (38) is found to be fixed-time stable according to [36, 37], i.e., when t>t2, e0i=e1i=0. And, the t2≤1/λ2γ−1εγ−1+2ε1/2/λ11+1/m1/M−hλ1/λ1 with M=α−h0andm=α+h0 is different from the FTDO.Theorem 3 (FTUO-ISMC).
Considering tracking error systems (35)-(36) with the unknown lumped disturbances satisfying Assumption 2, a FTUO-ISMC is designed as(39)τFTUO−ISMC=MR−1RdM−1τd+Θd−Sσσ,with Φ^ estimated by FTUO (40). Systems (6) and (7) can converge to desired ones (13) and (14) within a fixed time, i.e., Φ^≡Φ, η≡ηd, and υ≡υd, when t>Tf3.Proof.
The derivative of ISM surface (22) is redescribed as follows:(40)s˙=e1−ξ1sα−ξ2sβ.
And the derivation of Lyapunov function (25) will be changed as(41)V˙=se1−ξ1sα+1−ξ2sβ+1.
When the timet>t2, Φ^ equals Φ; to put it differently, the unknown lumped disturbance estimation error e1 converges to zero in a fixed time, which makes equation (41) become (27) such that system states arrive at s=0 within a fixed time T2=1/2α−1/2ξ11−α+1/2β−1/2ξ2β−1 based on Lemma 2. The tracking error system is the same as (28) after reaching the sliding surface, and it converges to origin along the sliding surface in a fixed time based on Lemma 1. Under the FTUO-ISMC scheme, the convergence time of the ASV system is Tf3≤t2+T2+tf. The proof is absolutely accomplished.Remark 4.
Although the states of tracking error systems (35)-(36) under control law (40) are not proved to be bounded within 0,t2 analogous to Remark 2, sufficient simulation results have been shown that any state does not diverge in 0,t2. Hence, we still suppose that the states of (35) are bounded within the time period 0,t2.Remark 5.
The ISMC scheme is designed based on integrated sliding mode surface for ASV systems without external disturbances. The FTDO-ISMC scheme is designed when there are external disturbances in the ASV system. The FTUO-ISMC scheme is designed considering the model uncertainties and external disturbances of the ASV system. The similarity lies in that ISMC, FTDO-ISMC, and FTUO-ISMC schemes are all based on the state feedback to design the integral sliding mode controllers.


## 4.1. Coordinate Transformation

Consider coordinate transformations as follows:(15)σ=Rυ,σ∈σ,σd,υ∈υ,υd,R∈R,Rd,where σ=σ1,σ2,σ3T, σd=σd1,σd2,σd3T, R=Rφ, and Rd=Rφd.By combining (6)-(7) and (15), we can obtain(16)η˙=σ,σ˙=RM−1τ+Θη,σ+dlt,where(17)Θη,σ=Sσσ−RM−1CRTσ+DRTσRTσ−RM−1gη.Similarly, together with (13), (14), and (15), we obtain(18)η˙d=σd,σ˙d=RdM−1τd+Θdηd,σd,where(19)Θdηd,σd=Sσdσd−RdM−1CRdTσd+DRdTσdRdTσd.Define the position and velocity errorηe=ηe1,ηe2,ηe3T and σe=σe1,σe2,σe3T. Then, we have(20)η˙e=σe,σ˙e=RM−1τ−RdM−1τd+dlt+Θeη,σ,ηd,σd,where(21)Θeη,σ,ηd,σd=Θη,σ−Θdηd,σd.


## 4.2. Design of the ISMC without External Disturbances

In this section, an ISMC is firstly proposed for tracking error systems (20) and (21) without external disturbances, and the fixed-time stability is verified.The ISM manifold is designed as follows:(22)sσet=σet+∫t0tunσeθdθ,where(23)un=k1ηeϱ1+k1′ηe+k1″ηeϱ1′+k2σeϱ2+k2′σe+k2″σeϱ2′.and parameters are provided in Lemma 1.The ISMC can be designed as(24)τISMC=MR−1RdM−1τd−Θe−un−ξ1sα−ξ2sβ,where 0<α<1, β>1, ξ1>0, and ξ2>0.Theorem 1 (ISMC).
Consider tracking error systems (20)-(21) without external disturbances, and an ISMC is designed by (24). Systems (6) and (7) can converge to desired trajectories (13) and (14) within a fixed time, i.e., η≡ηd and υ≡υd, when t>Tf1.Proof.
There are two processes in the whole verification: the reaching and the sliding phases:(i)
Step 1: taking the derivative of ISM surfaces (22) and (23) along error systems (20) and (21) without external disturbances dlt and combining with the control law (24), we can obtain(25)s˙=σ˙e+un=−ξ1sα−ξ2sβ.Take the candidate Lyapunov function as follows:(26)Vs=12s2.Differentiating it along the dynamics (25), we obtain(27)V˙s=ss˙=s−ξ1sα−ξ2sβ=−ξ1sα+1−ξ2sβ+1=−2α+1/2ξ1Vα+1/2−2β+1/2ξ2Vβ+1.According to Lemma2, it is claimed that the ISM control law will let the system states reach the ISM surface s=0 within a fixed time.(ii)
Step 2: at that moment, error system (20) will reduce to the following system:(28)η˙e=σe,σ˙e=−unηe,σe.
By applying Lemma1, we have that system (28) is globally fixed-time stable. Eventually, errors ηe and σe are converging to zero within a fixed time tf, so this completes the proof. Under the ISMC scheme, the convergence time of the ASV system is Tf1=T0+tf, where T0=1/2α−1/2ξ11−α+1/2β−1/2ξ2β−1.Remark 1.
External disturbances are not considered in ISMC excogitation, but they are actually exist in the actual environment. Therefore, controller design for the ASV system with external disturbances is essential.


## 4.3. Design of FTDO-ISMC

To achieve accurate tracking performance, in this section, a FTDO is built to estimate the external disturbances. Inspired by [35–37], a FTDO algorithm is established as(29)χ0=σ−χ,χ˙=RM−1τ−RdM−1τd+Θeη,σ,ηd,σd+d^lt+ψi,ψi=λ1iχ0i1/2signχ0i+λ2iχ0iγisignχ0i,d^˙lt=βisignχ0i,i=1,2,3,where χ0=χ01,χ02,χ03T is an auxiliary variable, ψ=ψ1,ψ2,ψ3T, d^lt=d^l1t,d^l2t,d^l3tT denotes the estimation of dlt, λ1i and λ2i are constants greater than zero, βi>κi,andγi>1.The error dynamic of the observer is described as(30)χ˙0i=χ1i−λ1iχ0i1/2signχ0i−λ2iχ0iγisignχ0i,χ1i.=−βisignχ0i+d˙lit,where χ1i=dlit−d^lit is the estimation error of external disturbance. It can be derived from [36, 37] that observer error system (18) is fixed-time stable according to Assumption 1, i.e., when t>t1, χ0i=χ1i=0. And the t1 satisfies t1≤1/λ2γ−1εγ−1+2ε1/2/λ11+1/m1/M−hλ1/λ1 with ε>0, M=α−κ,m=α+κ, and hλ1=1/λ1+2exp1/mλ11/3.Theorem 2 (FTDO-ISMC).
Consider tracking error systems (20)-(21) with external disturbances satisfying Assumption 1; then, a FTDO-ISMC is designed as(31)τFTDO−ISMC=MR−1RdM−1τd−Θe−un−d^lt−ξ1sα−ξ2sβ,with d^l estimated by FTDO (30). Systems (6) and (7) can converge to desired trajectories (13) and (14) within a fixed time, i.e., d^lt≡dlt, η≡ηd, and υ≡υd, when t>Tf2.Proof.
The derivative of ISM surface (13) is rewritten as(32)s˙=χ1−ξ1sα−ξ2sβ.
And derivative Lyapunov functionV based on (32) obtains(33)V˙=sχ1−ξ1sα+1−ξ2sβ+1.
When the timet>t1, d^lt equals dlt, that is to say that the disturbance estimation error χ1i converges to zero within a fixed time. (33) is rewritten as(34)V˙=−ξ1sα+1−ξ2sβ+1.
Based on Lemma2, (34) is described by V˙=−2α+1/2ξ1Vα+1/2−2β+1/2ξ2Vβ+1/2 and we obtain that the ISM manifold s=0 will be arrived within a fixed time T1=1/2α−1/2ξ11−α+1/2β−1/2ξ2β−1. The system is still the same as (28) while s=0. In other words, the tracking error system states ηe and υe are still sufficient to ensure convergence along the sliding surface in a fixed time tf. Under the FTDO-ISMC scheme, the convergence time of the ASV system is Tf2≤t1+T1+tf. This concludes the proof.Remark 2.
Note that we do not testify that the proposed controller (31) can ensure the boundedness of errors ηe and σe in the time quantum 0,t1 since the analysis of the dynamics of tracking error system is a difficult subject due to the complex nonlinear terms. In view of this reason, we have done a large number of simulations for ASV systems (6)-(7) under observer (29) and control laws (31), in which any divergence phenomenon is not observed. Practically, for the purpose of guaranteeing the boundedness of error system states in engineering, a bounded control rule can be applied in 0,t1. Therefore, it can assume that all states will not be divergent in fixed time in advance of the observer error dynamics converging.Remark 3.
Notice that it is difficult forΘe to obtain the real value because Cv and Dv are unknown because of existing uncertainties in ASV uncertainties. The FTDO only estimates the external disturbances and the influence of nonlinear terms is not considered, which will be solved in this context.


## 4.4. Design of FTUO-ISMC

In this section, the unknown nonlinear term consisting ofCυ, Dυ, and gη together with the external disturbances dlt are regarded as the lumped disturbances. Therefore, rewriting tracking error systems (20) and (21) yields(35)η˙e=σe,σ˙e=RM−1τ−RdM−1τd−Θd+Sσσ+Φ,where(36)Φ=−RM−1CRTσ+DRTσRTσ−RM−1gη+dlt,is an unknown nonlinearity regarded as the lumped disturbances. According to [29], an assumption is presented as follows.Assumption 2.
There exists a bounded constanth0 such that Φ˙≤h0.
A FTUO is presented as(37)e0=σ−e,e˙=RM−1τ−RdM−1τd−Θdηd,σd+Sσσ+Φ^+ψi,ψi=λ1ie0i1/2signe0i+λ2ie0iγisigne0i,Φ^.i=βisigne0i,i=1,2,3,where e0=e01,e02,e03T is an auxiliary variable, ψ=ψ1,ψ2,ψ3T, Φ^=Φ^1,Φ^2,Φ^3T is the estimation of the unknown lumped disturbances Φ, λ1i and λ2i are positive constants, βi>h0,andγi>1.
The error dynamic of the observer is described as(38)e˙0i=e1i−λ1ie0i1/2signe0i−λ2ie0iγisigne0i,e˙1i=−βisigne0i+Φ˙it,where the lumped disturbance estimation error is described as e1i=Φi−Φ^i. With the help of Assumption 2, observer error system (38) is found to be fixed-time stable according to [36, 37], i.e., when t>t2, e0i=e1i=0. And, the t2≤1/λ2γ−1εγ−1+2ε1/2/λ11+1/m1/M−hλ1/λ1 with M=α−h0andm=α+h0 is different from the FTDO.Theorem 3 (FTUO-ISMC).
Considering tracking error systems (35)-(36) with the unknown lumped disturbances satisfying Assumption 2, a FTUO-ISMC is designed as(39)τFTUO−ISMC=MR−1RdM−1τd+Θd−Sσσ,with Φ^ estimated by FTUO (40). Systems (6) and (7) can converge to desired ones (13) and (14) within a fixed time, i.e., Φ^≡Φ, η≡ηd, and υ≡υd, when t>Tf3.Proof.
The derivative of ISM surface (22) is redescribed as follows:(40)s˙=e1−ξ1sα−ξ2sβ.
And the derivation of Lyapunov function (25) will be changed as(41)V˙=se1−ξ1sα+1−ξ2sβ+1.
When the timet>t2, Φ^ equals Φ; to put it differently, the unknown lumped disturbance estimation error e1 converges to zero in a fixed time, which makes equation (41) become (27) such that system states arrive at s=0 within a fixed time T2=1/2α−1/2ξ11−α+1/2β−1/2ξ2β−1 based on Lemma 2. The tracking error system is the same as (28) after reaching the sliding surface, and it converges to origin along the sliding surface in a fixed time based on Lemma 1. Under the FTUO-ISMC scheme, the convergence time of the ASV system is Tf3≤t2+T2+tf. The proof is absolutely accomplished.Remark 4.
Although the states of tracking error systems (35)-(36) under control law (40) are not proved to be bounded within 0,t2 analogous to Remark 2, sufficient simulation results have been shown that any state does not diverge in 0,t2. Hence, we still suppose that the states of (35) are bounded within the time period 0,t2.Remark 5.
The ISMC scheme is designed based on integrated sliding mode surface for ASV systems without external disturbances. The FTDO-ISMC scheme is designed when there are external disturbances in the ASV system. The FTUO-ISMC scheme is designed considering the model uncertainties and external disturbances of the ASV system. The similarity lies in that ISMC, FTDO-ISMC, and FTUO-ISMC schemes are all based on the state feedback to design the integral sliding mode controllers.


## 5. Simulation

For the sake of illustrating the efficiency and superiority of the developed ISMC, FTDO-ISMC, and FTUO-ISMC points at trajectory following of an ASV, simulation machine adopted the surface vehicle CyberShipII [38] whose main parameters can be found in [27].In this section, simulation proposed three control laws verifying that the system states can track the trajectory given by (13) and (14) within a fixed time, where desired control input τd is conducted by τd=5,2cos20.1πt,sin20.1πtT. The origin condition is given by η0=2,1,π/2T, υ0=0,0,0T, ηd0=1,2,π/4T, and υd0=0,0,0T.


### 5.1. Simulation on the ISMC

The ISMC scheme is simulated without considering external disturbances, and the parameters of this scheme are chosen as follows:ϱ=0.7, ki=ki′=ki′′=5, ξ1=ξ2=3, α=0.5, and β=2. It is clearly expressed from Figures 1–3 that the desired system state is fully tracked by the system states within a fixed time.Figure 1
Planar trajectory tracking on ISMC.Figure 2
Position tracking on ISMC.Figure 3
Velocity tracking on ISMC.In Figures4 and 5, curves of tracking error are shown under two sets of initial conditions. The case one is as above and the case two is η0=1,1,π/3T, υ0=1,0,0T, ηd0=1,0,π/2T, and υd0=0,1,0T. The convergence time can be demonstrated to be similar in distinct initial states. The control input of ISMC is shown in Figure 6.Figure 4
Position tracking errors on ISMC with different conditions.Figure 5
Velocity tracking errors on ISMC with different conditions.Figure 6
Control input on ISMC.


### 5.2. Simulation on the FTDO-ISMC

In general, the external disturbances always exist in the ASV dynamics. The unknown external disturbancesdlt are chosen as(42)dlt=9sin0.1πt−π56sin0.3πt+π63sin0.2πt+π3.And the parameters of the proposed FTDO and corresponding controllerτFTDO−ISMC are chosen as follows:λ1i=λ2i=6, β1=9, β2=6, β3=3; γ1=γ2=0.3, γ3=0.8; ϱ=0.7, ki=ki′=ki′′=5, ξ1=ξ2=4, α=0.5, and β=2.Accordingly, simulation results of FTDO-ISMC scheme are illustrated in Figures7–13. From Figures 7–12, compared to the performance with ISMC, the FTDO-ISMC scheme can also accurately control for tracking control of the ASV system among brief time in the presence of external disturbances. As can be seen from Figure 12, the FTDO can accurately estimate the external disturbances. The control input of the ASV system under FTDO-ISMC scheme is shown in Figure 13. Therefore, it can be concluded that the FTDO-ISMC scheme can achieve good tracking performance while external disturbances are existing.Figure 7
Planar trajectory tracking on ISMC and FTDO-ISMC.Figure 8
Position tracking on ISMC and FTDO-ISMC.Figure 9
Velocity tracking on ISMC and FTDO-ISMC.Figure 10
Position tracking errors on ISMC and FTDO-ISMC.Figure 11
Velocity tracking errors on ISMC and FTDO-ISMC.Figure 12
External disturbances and their estimations on FTDO-ISMC.Figure 13
Control input under FTDO-ISMC.


### 5.3. Simulation on the FTUO-ISMC

The unknown lumped disturbances composed of the external disturbances and the model uncertainties of the ASV system are estimated together by the FTUO. The parameters of the proposed FTUO and corresponding controllerτFTUO−ISMC are chosen as follows: λ1i=λ2i=6, β1=9, β2=7, and β3=4; γ1=γ2=0.3 and γ3=0.6; ϱ=0.3, ki=ki′=ki′′=5, ξ1=4, ξ2=4, α=0.1, and β=2.The simulation results of the ASV system under FTUO-ISMC are all provided in Figures14–17. It could be described from Figures 14–16 that the vehicle can follow the expected trajectory accurately with the fixed time while the unknown lumped disturbances are presenting. In Figure 17, curves of unknown lumped disturbances and estimations are presented. From this figure, we can see that the proposed FTUO can effectively estimate the lumped disturbances. As a conclusion, the tracking performance of the FTUO-ISMC is satisfied for the ASV system with external disturbances and parameter uncertainties.Figure 14
Planar trajectory tracking.Figure 15
Position tracking on FTUO-ISMC.Figure 16
Velocity tracking on FTUO-ISMC.Figure 17
Unknown disturbance and its estimation on FTUO-ISMC.


## 5.1. Simulation on the ISMC

The ISMC scheme is simulated without considering external disturbances, and the parameters of this scheme are chosen as follows:ϱ=0.7, ki=ki′=ki′′=5, ξ1=ξ2=3, α=0.5, and β=2. It is clearly expressed from Figures 1–3 that the desired system state is fully tracked by the system states within a fixed time.Figure 1
Planar trajectory tracking on ISMC.Figure 2
Position tracking on ISMC.Figure 3
Velocity tracking on ISMC.In Figures4 and 5, curves of tracking error are shown under two sets of initial conditions. The case one is as above and the case two is η0=1,1,π/3T, υ0=1,0,0T, ηd0=1,0,π/2T, and υd0=0,1,0T. The convergence time can be demonstrated to be similar in distinct initial states. The control input of ISMC is shown in Figure 6.Figure 4
Position tracking errors on ISMC with different conditions.Figure 5
Velocity tracking errors on ISMC with different conditions.Figure 6
Control input on ISMC.


## 5.2. Simulation on the FTDO-ISMC

In general, the external disturbances always exist in the ASV dynamics. The unknown external disturbancesdlt are chosen as(42)dlt=9sin0.1πt−π56sin0.3πt+π63sin0.2πt+π3.And the parameters of the proposed FTDO and corresponding controllerτFTDO−ISMC are chosen as follows:λ1i=λ2i=6, β1=9, β2=6, β3=3; γ1=γ2=0.3, γ3=0.8; ϱ=0.7, ki=ki′=ki′′=5, ξ1=ξ2=4, α=0.5, and β=2.Accordingly, simulation results of FTDO-ISMC scheme are illustrated in Figures7–13. From Figures 7–12, compared to the performance with ISMC, the FTDO-ISMC scheme can also accurately control for tracking control of the ASV system among brief time in the presence of external disturbances. As can be seen from Figure 12, the FTDO can accurately estimate the external disturbances. The control input of the ASV system under FTDO-ISMC scheme is shown in Figure 13. Therefore, it can be concluded that the FTDO-ISMC scheme can achieve good tracking performance while external disturbances are existing.Figure 7
Planar trajectory tracking on ISMC and FTDO-ISMC.Figure 8
Position tracking on ISMC and FTDO-ISMC.Figure 9
Velocity tracking on ISMC and FTDO-ISMC.Figure 10
Position tracking errors on ISMC and FTDO-ISMC.Figure 11
Velocity tracking errors on ISMC and FTDO-ISMC.Figure 12
External disturbances and their estimations on FTDO-ISMC.Figure 13
Control input under FTDO-ISMC.


## 5.3. Simulation on the FTUO-ISMC

The unknown lumped disturbances composed of the external disturbances and the model uncertainties of the ASV system are estimated together by the FTUO. The parameters of the proposed FTUO and corresponding controllerτFTUO−ISMC are chosen as follows: λ1i=λ2i=6, β1=9, β2=7, and β3=4; γ1=γ2=0.3 and γ3=0.6; ϱ=0.3, ki=ki′=ki′′=5, ξ1=4, ξ2=4, α=0.1, and β=2.The simulation results of the ASV system under FTUO-ISMC are all provided in Figures14–17. It could be described from Figures 14–16 that the vehicle can follow the expected trajectory accurately with the fixed time while the unknown lumped disturbances are presenting. In Figure 17, curves of unknown lumped disturbances and estimations are presented. From this figure, we can see that the proposed FTUO can effectively estimate the lumped disturbances. As a conclusion, the tracking performance of the FTUO-ISMC is satisfied for the ASV system with external disturbances and parameter uncertainties.Figure 14
Planar trajectory tracking.Figure 15
Position tracking on FTUO-ISMC.Figure 16
Velocity tracking on FTUO-ISMC.Figure 17
Unknown disturbance and its estimation on FTUO-ISMC.


## 6. Conclusion

In this paper, the problem of fixed-time trajectory tracking control has been investigated for ASVs with external disturbances and model uncertainties. By introducing integral sliding mode surface, an ISMC scheme has been proposed for ASVs, which can achieve position and velocity tracking in fixed time. Then, a fixed-time disturbance observer (FTDO) has been designed to estimate the external disturbances. The control scheme (FTDO-ISMC) proposed by combining the ISM surface and the output of FTDO can accurately track the desired trajectory for ASVs with the external disturbances. A fixed-time unknown observer (FTUO) has been developed to calculate the unknown lumped disturbances, and a control law in the light of fixed-time unknown observer (FTUO-ISMC) has been proposed to achieve accurate disturbances attenuation and trajectory tracking for the ASVs. Results on simulation have been used to illustrate the superiority and efficiency of the control schemes proposed.



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*Source: 3281368-2020-10-15.xml*