Distributed Finite-time Prescribed Performance Security Control for Unmanned Ships Utilizing the Novel Disturbance Estimator

Authors: yuhui  song
huanqing  wang
siwen  liu
tieshan  li
Venue: IEEE_org_ICIST_2024_Conference
Published: 2024-07-28 08:25:41

Abstract: The article developed a distributed security control protocol of  unmanned ships (USs) with the disturbances. To constraint the tracking errors within a predefined boundary, an infinite-time performance function and a logarithmic Lyapunov function are simultaneously introduced.Using the novel disturbance estimator, the disturbances are estimated. With the aid of the backstepping method, a distributed security controller is obtained and it is given that all the states of the system are bounded in finite time and the tracking errors can also meet preset performance requirements. Eventually, unmanned ships are given to show the utilizability of the obtained protocol.

Main paper: 
 i. introduction: 
 U Ss are primarily preferred for tasks characterized by being dangerous or unsuitable for manned vessels. In the future, they will be developed for ocean surveying, meteorological monitoring, maritime search and rescue, and more. Due to its excellent characteristics, USs have been listed as a key research object in the field of marine armaments by countries around the world. In order to deal with the US control issue effectively, some control strategies were proposed [1] - [3] . Recently, the cooperative control for USs has attracted more and more attention due to that multiple USs can achieve the cooperative tasks efficiently [4] - [6] . In general, the cooperative control problem includes the tracking issue and the regulation issue. With regard to the control issue, all the USs will eventually incline to a non-specified value that is not previously known. Then, the follower USs can incline to the reference formed by the leader US or the convex hull generated by the trajectories of the leader USs [7] , [8] . When there exist multiple leader USs, that is called the containment control. As a matter of fact, it is invulnerable for multiple USs if the follower USs incline to the convex hull generated by the leader USs [9] .Although the backstepping control technique has become one of the most common methods to overcome the control issue of nonlinear systems [10] - [12] , it is not difficult to obtain that, utilizing the classical backstepping technique to dispose the control issue of nonlinear systems, there exists the complexity explosion problem that is produced by repeated differentiation of virtual control variables. This may lead to the rapid multiplication of the systems and further makes the system performance unsatisfactory or even unstable [13] . Even if the proposed controller makes the performance of the studied system satisfy our design requirements, it is clear that the actual controller will become more complex [14] . It is obvious that the complex controller structure limits the practicability of the proposed strategy. To solve the above difficulty, the dynamic surface control method (DSCM) was presented [15] , such that, by the conventional backstepping control technique, the designed first-order filter can be applied to work out the intermediate control variable's derivative at each step. After that, the method was extended to handle the control issue of different nonlinear systems, especially, nonlinear multi-agent systems [16] , [17] .However, in the above articles, the infinite time boundedness of the states can be obtained. From the view of the convergence time, this may restrict the use of the developed controllers. Therefore, in the article, the finite-time first-order filter is applied to work out the intermediate control variable's derivative and, employing the finite-time control way, the convergence of all the variables of USs will also be assured in finite time. In reality, many real systems, such as temperature of chemical reactor and physical stoppages, are with the constraints of the output and such constraints may come from safety consideration. Thus, it is very necessary to consider the problem of output constraints during the engineering system analysis. In order to handle such issue, the prescribed performance control (PPC) was developed [18] - [22] . By using a PPC method, a tracking controller was given for USs [23] , and the PPC issue was discussed for USs in [24] .Consequently, the article discuss the distributed security control issue of USs with the disturbances. The major innovations of this work are as follows:(1) By utilizing the proposed finite-time disturbance estimator, the disturbances existing in USs can be approximated.(2) In this paper, by introducing the infinite-time performance function, a finite-time distributed security control scheme is obtained for USs with multiple leader USs.The following paper structures are shown. Section II shows the problem formulation and preliminaries. Main results are shown in Section III. In Section IV, the simulation examples will prove the utilizability of the obtained method. The conclusion is shown in Section V.Consider M USs including N follower USs and M -N leader USs. The model of ith follower US is [25] EQUATIONwhere η i = [x i , y i , ψ i ] T denote the position and direction.υ i = [u i , v i , r i ]T is the velocity. M is the inertia matrix, D is the damping matrix, and τ i = [τ iu , τ iv , τ ir ] T is the control vector. d i (t) is the disturbance. It is assumed that the external disturbances and the derivatives of them satisfy|τ iu | ≤ τiu , |τ iv | ≤ τiv , |τ ir | ≤ τir , | τiu | ≤ τiu , | τiv | ≤ τiv , | τir | ≤τir , where τiu , τiv , τir , τiu , τiv and τir represent positive constants. J i (ψ i ) is the rotation matrix described asJ i (ψ i ) = [cos(ψ i ), -sin(ψ i ), 0; sin(ψ i ), cos(ψ i ), 0; 0, 0, 1].Then, define the reference of the jth leader US as η jd = {x jd , y jd , ψ jd }, where j = N + 1, N + 2, . . . , M .Lemma 1 [26] :For the system ẋ = f (x), if there is a continuous function V (x) such that V (x) ≤ -λ 1 V (x) - λ 2 V γ (x) + η, where λ 1 > 0, λ 2 > 0, 0 < γ < 1, 0 < η < ∞,then the trajectory of the system ẋ = f (x) is practical finite-time stable, and the resid-ual set is lim t-→Tr V (x) ≤ min{ η (1-θ0)λ1 , ( η (1-θ0)λ2 ) 1 γ } where θ 0 satisfies 0 < θ 0 < 1. The setting time is bounded as T r ≤ max{t 0 + 1 θ0λ1(1-γ) ln θ0λ1V 1-γ (t0)+λ2 λ2 , t 0 + 1 λ1(1-γ) ln λ1V 1-γ (t0)+θ0λ2 θ0λ2 }. Lemma 2 [27]: Considering χ i ∈ R(i = 1, . . . , n) and κ ∈ [0, 1], one gets (|χ 1 | + • • • + |χ n |) κ ≤ |χ 1 | κ + • • • + |χ n | κ .Lemma 3 [28] :For Ξ > 0 and x ∈ R, it gets 0 ≤ |x| -x • tanh( x Ξ ) ≤ κΞ with κ = sup t>0 1 1+e t = 0.2785.Lemma 4 [29] : Define Ω e = {e |e| < 0.8814v} where v is a positive constant. If any e /∈ Ω e , we can obtain 1 -2 tanh 2 ( e v ) ≤ 0.The information flow among M USs including N follower USs and M -N leader USs is described by G = (Υ, E, A), where Υ = {r 1 , r 2 , . . . , r m } is the set of USs, E ⊆ Υ × Υ is the set of edges, and A = [a ij ] ∈ R M ×M is the weighted adjacency matrix. r i is the ith US and a directed edge (i, j) means that US♯i can obtain information from US♯j, i.e., a ij > 0, otherwise, a ij = 0 and define a ii = 0. The set of neighbors of US♯i is denoted as N i = {j|(j, i) ∈ E}. The Laplacian matrix is defined asL a = D -A ∈ R M ×M where D = diag[D 1 , D 2 , . . . , D M ], D i = Σ M j=1,j̸ =i a ij , i = 1, 2, .. . , M . Since each follower US has at least one neighbor, and the leader USs have no neighbors, L a can be represented as [30] L a = [L 1a , L 2a ; 0 M ×N , 0 M ×M ] where L 1a ∈ R N ×N with N denoting the number of follower USs and L 2a ∈ R N ×M with M denoting the number of leader USs.The containment control scheme is designed such that the position and direction of all follower USs converge into the convex hulls Co(X d ), Co(Y d ) and Co(ψ d ), respectively, which are given by Co(X d ) = {Σ M N +1 c i x id |c i ≥ 0, Σ M N +1 c i = 1}, Co(Y d ) = {Σ M N +1 c i y id |c i ≥ 0, Σ M N +1 c i = 1}, Co(ψ d ) = {Σ M N +1 c i ψ id |c i ≥ 0, Σ M N +1 c i = 1} where X d = {x (N +1)d , x (N +2)d , . . . , x M d }, y d = {y (N +1)d , y (N +2)d , . . . , y M d } and ψ d = {ψ (N +1)d , ψ (N +2)d , . . . , ψ M d }.Then the output synchronization error of the ith follower US isEQUATIONThe PPC design is that the tracking error is included in the preset residual set, such that the tracking error e i1j , i = 1, 2, . . . , N, j = 1, 2, 3, is included in a predefined range, which is shown asEQUATIONwhere i = 1, 2, . . . , N , j = x, y, ψ, k = 1, 2, 3, v ij (t) satisfies lim t→∞ v ij (t) = v tf ij and v tf ij is the steady-state value of e i1k . Choose v ij (t) = (v 0ij -v tf ij )e -βij t + v tf ijas the performance function, where v 0ij , v tf ij and β ij are positive constants andv 0ij > v tf ij . v 0ij = v ij (0) is chosen such that -v ij (0) < e i1k (0) < v ij (0). β ijrepresents the convergence rate of e i1k (t). Hence, by choosing v 0ij , v tf ij and β ij to make the tracking error be included in the predefined range (3), the PPC can be obtained.For the PPC design, the following lemma is shown.                                    m 11 ui = -d 11 u i + τ iu + diu +k iu1 tanh( ui-ûi Ξũ i ) + k iu2 (u i -ûi ) ḋiu = β iu (e i21 + u i -ûi ) -δ iu diu m 22 vi = -d 22 v i -d 23 r i + τ iv + div +k iv1 tanh( vi-vi Ξṽ i ) + k iv2 (v i -vi ) ḋiv = β iv (e i22 + v i -vi ) -δ iv div m 33 ṙi = -d 33 r i -d 32 v i + τ ir + dir +k ir1 tanh( ri-ri Ξr i ) + k ir2 (r i -ri ) ḋir = β ir (e i23 + r i -ri ) -δ ir dir (4)wherek iu1 , Ξ ũi , k iu2 , β iu , δ iu , k iv1 , Ξ ṽi , k iv2 , β iv , δ iv , k ir1 , Ξ ri , k ir2 ,β ir and δ ir are positive constants; e i21 , e i22 and e i23 will be defined later.By defining ũi = u i -ûi , ṽi = v i -vi and ri = r i -ri , we can obtainEQUATIONWe introduce the changes of coordinates as follows:EQUATIONwhere υ d idj , the first-order filter output signal, is given asEQUATIONwhere υ d idj (0) = υ idj (0), j = u, v, r. υ idj is the input. k idj1 , Ξ y idj and k idj2 are the design constants. The filter errors arey idj = υ d idj -υ idj , j = u, v, r. Step 1. Firstly, considering the first equation in (2), we can get ėi11 = Σ M j=1 a ij (e i21 cos(ψ i ) -e i22 sin(ψ i )) + Σ M j=1 a ij (υ idu cos(ψ i ) -υ idv sin(ψ i )) -Σ N j=1 a ij ẋj - Σ M j=N +1 a ij ẋid + Σ M j=1 a ij (y idu cos(ψ i ) -y idv sin(φ i )). Next, the Lyapunov function is V ei11 = 1 2 ln v 2 ix v 2 ix -e 2 i11. Subsequently, the dynamics of V ei11 areVei11 = e i21 e i11 M j=1 a ij cos(ψ i ) v 2 ix -e 2 i11 - e i22 e i11 M j=1 a ij sin(ψ i ) v 2 ix -e 2 i11 + e i11 v 2 ix -e 2 i11 ( M j=1 a ij (υ idu cos(ψ i ) -υ idv sin(ψ i ) - N j=1 a ij ẋj ) - M j=N +1 a ij ẋid + 2(v 2 ix -e 2 i11 ) e i11 tanh 2 ( e i11 vix )(- v ix vix v 2 ix -e 2 i11)+ e i11 v 2 ix -e 2 i11 ) + vix v ix + (1 -2 tanh 2 ( e i11 vix ))(- v ix vix v 2 ix -e 2 i11 ) + ( M j=1 a ij ) 2 2 y 2 idu + ( M j=1 a ij ) 2 2 y 2 idv (8)Then, we design the following expression.EQUATIONwhere T ix = υ idu cos(ψ i ) -υ div sin(ψ i ) and k i1x > 0. Substituting ( 9) into (8), we haveEQUATION)Secondly, considering the second equation in (2), we can getEQUATIONThe following Lyapunov function candidate is defined asV ei12 = 1 2 ln v 2 iy v 2 iy -e 2 i12. Subsequently, the dynamics ofV ei12 are Vei12 = e i21 e i12 M j=1 a ij sin(ψ i ) v 2 iy -e 2 i12 + e i22 e i12 M j=1 a ij cos(ψ i ) v 2 iy -e 2 i12 + e i12 v 2 iy -e 2 i12 ( M j=1 a ij (υ idu sin(ψ i ) + υ idv cos(ψ i )) - N j=1 a ij ẏj - M j=N +1 a ij ẏid + 2(v 2 iy -e 2 i12 ) e i12 tanh 2 ( e i12 viy )(- v iy viy v 2 iy -e 2 i12)EQUATION)Then, we give the equation as follows:T iy = 1 M j=1 a ij (-k i1y e i12 + N j=1 a ij ẏj + M j=N +1 a ij ẏid - 2(v 2 iy -e 2 i12 ) e i12 tanh 2 ( e i12 viy )(- v iy viy v 2 iy -e 2 i12)EQUATIONwhere T iy = υ idu sin(ψ i ) + υ div cos(ψ i ) and k i1y > 0. Substituting ( 13) into ( 12), we haveEQUATION)Finally, considering the third equation in (2), we canget ėi13 = Σ M j=1 a ij (e i23 + υ idr + y idr ) -Σ N j=1 a ij ψj - Σ M j=N +1 a ij ψid . The Lyapunov function candidate is defined as follows: V ei13 = 1 2 ln v 2 iψ v 2 iψ -e 2 i13Subsequently, the dynamics ofEQUATION)Then, we design the following equation.EQUATIONwhere T iψ = υ idr and k i1ψ > 0. Substituting ( 16) into (15), we haveEQUATION)Define T i = [T ix , T iy , T iψ ] T . Then, according to ( 9), ( 13) and ( 16), we can obtainυ id = J -1 i (ψ i )T i where υ id = [υ idu , υ idv , υ idr ] T .Step 2. Firstly, considering the first equation in ( 6), we can get ėi21 = ui -υd idu . The following Lyapunov function candidate is defined asV ei21 = 1 2 m 11 e 2 i21 + 1 2 m 11 ũ2 i + 1 2βiu d2 iu + 1 2 y 2idu where diu = d iu -diu and β iu > 0. Using ( 5) and ( 7), the dynamics of V ei21 areEQUATIONSubsequently, the force τ iu in surge is designed as follows:EQUATION)where k i21u > 0 and k i22u > 0.Substituting ( 4) and ( 19) into (18) , we haveEQUATION)Utilizing Lemma 3 and Young's inequality, we getEQUATIONCombining ( 20) and ( 21), we haveEQUATIONwhereT ei21 = k i21u κΞ ei21 + k iu1 κΞ ũi + k idu1 κΞ y idu + 1 2 ( υidu ) 2 + 1 2βiu ( ḋiu ) 2 + δiu 2βiu d 2 iu + 1 2 .Secondly, considering the second equation in (6) , we can get ėi22 = vi -υd idv . The Lyapunov function candidate is defined as follows:V ei22 = 1 2 m 22 e 2 i22 + 1 2 m 22 ṽ2 i + 1 2βiv d2 iv + 1 2 y 2 idvwhere diu = d iu -diu and β iu > 0. Using ( 5) and ( 7), the dynamics of V ei22 areVei22 ≤ e i22 (-EQUATIONSubsequently, the force τ iv in sway is designed as follows:EQUATION)where k i21v > 0 and k i22v > 0. Substituting ( 4) and ( 24) into ( 23), we haveEQUATION)Utilizing Lemma 3 and Young's inequality, we getEQUATIONCombining ( 25) and ( 26), we haveEQUATIONwhereT ei22 = k i21v κΞ ei22 + k iv1 κΞ ṽi + k idv1 κΞ y idv + 1 2 ( υidv ) 2 + 1 2βiv ( ḋiv ) 2 + δiv 2βiv d 2 iv + 1 2 .Finally, considering the third equation in (6) , we can get ėi23 = ṙi -υd idr . The following Lyapunov function candidate is defined asV ei23 = 1 2 m 33 e 2 i23 + 1 2 m 33 r2 i + 1 2βir d2 ir + 1 2 y 2 idrwhere dir = d ir -dir and β ir > 0. Using ( 5) and ( 7), the dynamics of V ei23 areEQUATION)y 2 ide -k idr1 y idr tanh(EQUATIONSubsequently, the force τ ir in yaw is designed as follows:EQUATIONwhere k i21r > 0 and k i22r > 0. Substituting ( 4) and ( 29) into (28), we haveVei23 ≤ -k i21r e i23 tanh( e i23 Ξ ei23 ) -k i22r e 2 i23 -k ir1 ri tanh( ri Ξ ri ) -k ir2 r2 i + 1 β ir dir ḋir + δ ir β ir dir dir -k idr1 y idr tanh( y idr Ξ y idr ) -(k idr2 - 1 2 )y 2 idr + 1 2 ( υidr ) 2 -e i23 • M j=1a ij e i13 (30) Utilizing Lemma 3 and Young's inequality, we getEQUATIONCombining ( 30) and (31), we haveEQUATIONwhereT ei23 = k i21r κΞ ei23 + k ir1 κΞ ri + k idr1 κΞ y idr + 1 2 ( υidr ) 2 + 1 2βir ( ḋir ) 2 + δir 2βir d 2 ir + 1 2 .Theorem 1: Consider the networked USs (1) under the distributed controllers ( 19), ( 24), ( 29), the first-order filters (7) and the disturbance estimators (4). If the initial conditions are bounded with e i11 (0) < v ix (0), e i12 (0) < v iy (0) and e i13 (0) < v iψ (0)(i = 1, 2, . . . , N ), the following properties are true: i) All the state signals and disturbance estimation errors are bounded in finite time. ii) The synchronization tracking errors can be included in the specific decaying ranges.Proof: The Lyapunov function is given asV = N i=1 ( 1 2 ln v 2 ix (t) v 2 ix (t) -e 2 i11 + 1 2 ln v 2 iy (t) v 2 iy (t) -e 2 i12 + 1 2 ln v 2 iψ (t) v 2 iψ (t) -e 2 i13 + 1 2 m 11 e 2 i21 + 1 2 m 11 ũ2 i + 1 2β iu d2 iu + 1 2 y 2 idu + 1 2 m 22 e 2 i22 + 1 2 m 22 ṽ2 i + 1 2β iv d2 iv + 1 2 y 2 idv + 1 2 m 33 e 2 i23 + 1 2 m 33 r2 i + 1 2β ir d2 ir + 1 2 y 2 idr )(33)Taking the time derivative of V yields )EQUATIONEQUATIONBy utilizing Lemma 5, we haveEQUATIONFurthermore, by the definition of v ix (t), v iy (t) and v iψ (t), v ix (t), vix (t), v iy (t), viy (t), v iψ (t) and viψ (t) are bounded. Then, the following results are true.EQUATION)whereK 1i > 0, K 2i > 0 and K 3i > 0.Subsequently, substituting ( 35) and ( 36) into (34) gets following result:EQUATION))DefiningEQUATION))In (38), according to Lemma 4, e i11 , e i12 and e i13 respectively determine the signs of 1 -2 tanh 2 ( ei11 vix ), 1 -2 tanh 2 ( ei12 viy ) and 1 -2 tanh 2 ( ei13 viψ ). Therefore, the following three cases will be given.Case 1: For vix , viy and viψ , we havee i11 ∈ Ω ei11 = {e i11 |e i11 | < 0.8814v ix }, e i12 ∈ Ω ei12 = {e i12 |e i12 | < 0.8814v iy } and e i13 ∈ Ω ei13 = {e i13 |e i13 | < 0.8814v iψ }. Because e i11 , v ix (t), vix (t), e i12 , v iy (t), viy (t), e i13 , v iψ (t) and viψ (t) are bounded, 1 -2 tanh 2 ( ei11 vix ) -vix(t) vix(t) v 2 ix (t)-e 2 i11 + 1 -2 tanh 2 ( ei12 viy ) - viy(t) viy(t) v 2 iy (t)-e 2 i12 + 1 -2 tanh 2 ( ei13 viψ ) - v iψ (t) viψ (t) v 2 iψ (t)-e 2 i13is also bounded and it's supposed that c i (> 0) is the upper boundedness. Then, for (38), we can obtainV ≤ -λ 1 V -λ 2 V 1 2 + d where d = Σ N i=1 ( 3 2 + K 1i + K 2i + K 3i + T ei21 + T ei22 + T ei23 + c i ).Case 2: For vix , viy and viψ , we havee i11 / ∈ Ω ei11 = {e i11 |e i11 | < 0.8814v ix }, e i12 / ∈ Ω ei12 = {e i12 |e i12 | < 0.8814v iy } and e i13 / ∈ Ω ei13 = {e i13 |e i13 | < 0.8814v iψ }. According to Lemma 4, based on the fact that -vix(t) vix(t) v 2 ix (t)-e 2 i11 ≥ 0, - viy(t) viy(t) v 2 iy (t)-e 2 i12 ≥ 0 and - v iψ (t) viψ (t) v 2 iψ (t)-e 2 i13 ≥ 0, we can obtain (1 -2 tanh 2 ( ei11 vix ))(-vix(t) vix(t) v 2 ix (t)-e 2 i11 ) + (1 - 2 tanh 2 ( ei12 viy ))(- viy(t) viy(t) v 2 iy (t)-e 2 i12 ) + (1 - 2 tanh 2 ( ei13 viψ ))(- v iψ (t) viψ (t) v 2 iψ (t)-e 2 i13 ) ≤ 0. Then, for (38), we can obtain V ≤ -λ 1 V -λ 2 V 1 2 + d where d = Σ N i=1 ( 3 2 + K 1i + K 2i + K 3i + T ei21 + T ei22 + T ei23 + c i ).Case 3: For e i11 , e i12 and e i13 , some are inside the set and some are outside the set. Based on Case 1 and Case 2, we can obtainV ≤ -λ 1 V -λ 2 V 1 2 + d where d = Σ N i=1 ( 3 2 + K 1i + K 2i + K 3i + T ei21 + T ei22 + T ei23 + c i ).According to the above three cases, (38) can be rewritten asEQUATIONAccording to Lemma 1, we can get0 ≤ V ≤ min{ d (1 -θ 0 )λ 1 , ( d (1 -θ 0 )λ 2 ) 1 2 }, t > T rwhere 0 < θ 0 < 1 and Then, we can obtain that, according to Lemma 4, all the state signals and disturbance estimation errors are bounded when t > T r .EQUATIONBy (39), it can be concluded that lnv 2 ix (t) v 2 ix (t)-e 2 i11 ≤ 2 min{ d (1-θ0)λ1 , ( d (1-θ0)λ2 ) 2 }, ln v 2 iy (t) v 2 iy (t)-e 2 i12 ≤ 2 min{ d (1-θ0)λ1 , ( d (1-θ0)λ2 ) 2 } and ln v 2 iψ (t) v 2 iψ (t)-e 2 i13 ≤ 2 min{ d (1-θ0)λ1 , ( d (1-θ0)λ2 ) 2 }. Subsequently, we know that v 2 ix (t) v 2 ix (t)-e 2 i11 ≤ e 2 min{ d (1-θ 0 )λ 1 ,( d (1-θ 0 )λ 2 ) 2 } , v 2 iy (t) v 2 iy (t)-e 2 i12 ≤ e 2 min{ d (1-θ 0 )λ 1 ,( d (1-θ 0 )λ 2 ) 2 } and v 2 iψ (t) v 2 iψ (t)-e 2 i13 ≤ e 2 min{ d (1-θ 0 )λ 1 ,( d (1-θ 0 )λ 2 ) 2 } . Because v 2 ix (t) -e 2 i11 > 0, v 2 iy (t) -e 2 i12 > 0 and v 2 iψ (t) -e 2 i13 > 0, we have |e i11 | ≤ 1 - 1 e 2 min{ d (1-θ 0 )λ 1 ,( d (1-θ 0 )λ 2 ) 2 } |v ix (t)|, |e i12 | ≤ 1 - 1 e 2 min{ d (1-θ 0 )λ 1 ,( d (1-θ 0 )λ 2 ) 2 } |v iy (t)| and |e i13 | ≤ 1 - 1 e 2 min{ d (1-θ 0 )λ 1 ,( d (1-θ 0 )λ 2 ) 2 } |v iψ (t)| which imply that |e i11 | ≤ |v ix (t)|, |e i12 | ≤ |v iy (t)|, |e i13 | ≤ |v iψ (t)| where min{ d (1-θ0)λ1 , ( d (1-θ0)λ2 ) 2 } > 0.The proof is thus completed.For the utilizability of the obtained method, 5 USs including 3 follower USs (US♯1-US♯3) and 2 leader USs (US♯4-US♯5) are given. The topological structure is given in Fig. 1 . The adjacency matrix is selected as A = [0, 0.4, 0.5, 0.5, 0; 0.4, 0, 0.5, 0, 0; 0, 0.5, 0, 0, 0.4; 0, 0, 0, 0, 0; 0, 0, 0, 0, 0].The disturbance is The controller gains are chosen asd i = [k i1x = k i1y = k i1ψ = 5, vix = viy = viψ = 1, k i21u = k i21v = k i21r = 2, k i22u = k i22v = k i22r = 5 and Ξ ei21 = Ξ ei22 = Ξ ei23 = 0.1.The disturbance estimators' initial states are υi (0) = [0, 0, 0] T and the parameters are chosen ask iu1 = k iu2 = k iv1 = k iv2 = k ir1 = k ir2 = 1 and Ξ ũi = Ξ ṽi = Ξ ri = 0.1.The design parameters of first-order filters are k idu1 = k idv1 = k idr1 = 2, k idu2 = k idv2 = k idr2 = 15 and Ξ y idu = Ξ y idv = Ξ y idr = 0.1. The performance functions for the PPC are v ix (t) = v iy (t) = v iψ (t) = (10 -2)e -20t + 2.Fig. 2 shows the curves of US♯1-US♯5. It is obtained that the curves of US♯1-US♯3 are included in the convex hull generated by the trajectories of US♯4-US♯5. From Fig. 3 , the synchronization errors of US♯1-US♯3 are included in the specified performance bounds.The velocity curves of the follower US♯1-US♯3 are shown in Fig. 4 . Fig. 5 gives the controller responses of US♯1-US♯3. From Fig. 4 and Fig. 5 , the velocities and control inputs are bounded. Fig. 6 shows the disturbance estimation errors, which are finite-time bounded.The article has discussed the distributed security control issue for USs with the disturbances. To constraint the tracking errors within predefined boundaries, infinite-time performance functions have been introduced and the tracking errors were successfully controlled within the prespecified ranges. The disturbances were approximated by the disturbance estimators with finite-time estimation errors. Compared with the cooperative control of USs with one leader US, the obtained controller can solve the distributed security control issue with multiple leader USs and make further tracking errors satisfy the transient and steady-state performances. Numerical simulation results have revealed that the synchronization tracking errors satisfied the pre-specified prescribed performance.However, in practice, the considered system states are normally unknown or unmeasurable. Therefore, the issue of how to propose the observer-based containment control scheme can be discussed in our future research. 
 
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Reviews: review: title: title: comment
main_review: main_review: This paper investigates a distributed security control problem for unmanned ships with the disturbances. However, in the reviewer’s opinion, there are some comments in the paper which should be addressed by the authors:
1.	Introduction about the prescribed performance control is limited, it is recommended to add some literature.
2.	The label format of formulas 8 and 9 are not consistent with other formulas and the alignment in the formulas are not standardized, please correct it.
3.	The arrangement of some simulation results is not quite reasonable, please re-typeset.
title: title: The topic under consideration is interesting. This paper can be accepted after minor modifications.
main_review: main_review: This paper developed a distributed security control protocol of unmanned ships with the disturbances. The topic under consideration is interesting. Detailed comments and suggestions are listed as follows.

1.	In (1), the model of $i$th follower has been defined, but the model of leaders is not be found.
2.	The derivation of (15) cannot be easily followed. Please give a detailed explanation in the content.
3.	The English writing of the paper needs to be further polished, and some typos should be fixed, such as “Take into account the networked USVs (1) under the distributed controllers (19), (24), (29), the first-order filters (7) and the disturbance estimators (4)” in Page 6.  Moreover, the manuscript format should be revised.
title: title: minor repair
main_review: main_review: 1. Notations in this paper are easily confused, especially the symbols of damping matrix and D = diag[d1, d2, . . . , dM].
2. When using abbreviations for the first time, the full name should be provided, such as "USVs".
3. The simulation section should provide more selection parameters to facilitate readers' understanding.