Fast Fixed-Time Output Multi-Formation Tracking of Networked Autonomous Surface Vehicles: A Mathematical Induction Method

In this paper, we aim to exploit an effective way to solve the output multi-formation tracking problem of the networked autonomous surface vehicles (ASVs) in a fast fixed time manner. Specifically, addressing the output multi-formation tracking problem implies that 1) the networked ASVs are divided into multiple interconnected subnetworks with respect to multiple targets; 2) for each subnetwork, the outputs of the networked ASVs form a desired geometric formation with exchanging the local interactions. Besides, solving the fast fixed-time tracking problem in this paper implies that 1) the convergence time is independent of the initial conditions; 2) the system states are forced to reach the employed nonsingular fixed-time sliding surface in a prescribed time, which thus called fast fixed-time control. Then, based on a time-related function and a nonsingular fixed-time sliding surface, a hierarchical fast fixed-time control algorithm is proposed to solve the aforementioned problem within a fast fixed time being independent of the initial conditions. Furthermore, by employing the Lyapunov argument and mathematical induction, we present the sufficient conditions for fast fixed-time convergence of the tracking errors with respect to multiple targets. Finally, numerous simulation examples are presented to demonstrate the effectiveness of the proposed control algorithm.


I. INTRODUCTION
A UTONOMOUS surface vehicles (ASVs) are a class of intelligent surface vessels with self-navigation capability, and have been widely applied to accomplish different practical tasks, including water quality measurement, target detection, military reconnaissance and other tasks [1], [2], [3]. Compared with a single ASV, the networked ASVs admits information exchange, cooperation, task assignments, design of decentralized algorithms among different individuals, thus possessing the characteristics of higher working efficiency, higher fault-tolerant capability, stronger robustness and more flexibility [4]. This then motivats people to develop different control strategies for networked ASVs to achieve different control goals, including containment control [5], [6], formation control [7], [8], trajectory tracking [9], [10], [11], path following [12], [13] and target tracking [11], [14]. Among the above-mentioned collective goals of the networked ASVs, formation control has been recognized as one of the most fundamental objectives, aiming to drive the networked ASVs to form a desired formation configuration for some user-assigned missions.
The formation control has been researched extensively owing to its wide applications. On the one hand, from the perspective of control objectives, formation control can be classified into single-formation and multi-formation, where the controlled systems are requested to perform single or multiple missions respectively. The existing researches are mainly concentrate on single-formation control, which aims to urge multiple ASVs into a single formation pattern [15], [16], [17], [18]. Besides, in the multi-formation control [19], [20], [21], the overall interaction networks are generally segmented into several subnetworks and each subnetwork form a formation pattern respectively. On the other hand, from the perspective of control theory, formation control can be categorized into state formation and output formation, where the formation maneuvering is utilized by state coordination or output coordination. Different from the state formation control studied in [22], [23], the problem of output formation is more significance from the aspect of engineering application [24], [25]. Despite the rich literature in the area of formation control, there are less references on the combination of multi-formation control and output formation control (i.e., output multi-formation control) for the networked ASVs due to its highly complicated nonlinear controlled system and the complex interaction behaviors.
In addition, the convergence speed is also a significant concern in the coordinated control area owing to its vital index in assessing the quality of the designed control protocols. The finite-time control [26], [27], [28] generally performs faster convergence rate than asymptotic convergence [29], [30], [31]. However, the settling time of the finite-time control is related to the initial conditions, which weakens its practicability in some scenarios. To loosen this restriction, the fixed-time control has been proposed in [11], [32], [33], in which the settling time can be prescribed arbitrarily by adjusting several constrained parameters. Recently, some user-assignable time control methods have already arisen to alleviate the uncertainty of the settling time. The relevant references can be mainly classified into prescribed-time stability [34], [35] and predefined-time stability [36], [37], [38], in which the settling time can be viewed as a parameter in the designed algorithm and thus allowed to be user-assigned arbitrarily. Motivated by the existing fixed-time control and prescribed-time control, we aim to exploit a new control algorithm to combine the both two control technologies for achieving a newly designed fast fixed-time control by simultaneously considering the issues of output multi-formation, uncertainty and nonlinear models.
Consequently, inspired by the above discussions, we propose a novel fast fixed-time control algorithm for the networked ASVs with uncertain disturbances and devote to achieve the output multi-formation tracking in the fast fixed time. The designed control algorithm includes the distributed prescribed-time estimator layer and the local control layer. Based on a time-related function, the states of the leader are estimated in a prescribed time at the distributed estimator layer. Then, based on a nonsingular fixed-time sliding surface and the same time-related function, the controlled system is forced to reach the sliding surface in a prescribed time. This can be further derived that the output multi-formation tracking problem is solved in a fast fixed time. Further, both the Lyapunov stability theory and mathematical induction method are employed to obtain the sufficient conditions. The main highlights are summarized as follows.
1) Different from the finite-time control algorithms [26], [27], [28], where the convergence time is related to the initial conditions. The proposed fast fixed-time control algorithm reduces this constraint. Different from the fixedtime sliding mode control [39], in which the reaching time is relevant to the multiple control gains and can not be determined easily. The reaching time of the proposed algorithm can be prescribed arbitrary by choosing a parameter, even to an extremely small value. 2) Different from the researches on multi-target tracking problem in asymptotic convergence [19] and finite-time convergence [20], [21], we solve the fast fixed-time output multi-formation tracking problem of the networked ASVs successfully for the first time. 3) Different from the conventional analytical method of the dynamic stability [34], [35], we provide a method combining mathematical induction and Lyapunov function to analyze the prescribed stability of the controlled system. The remainder of this paper is organized as follows. The preliminaries and problem formulation are given in Section II. The main results are given in Section III, in which the designed algorithm is presented and analyzed completely. The simulation results are presented and discussed in Section VI. Finally, conclusions are summarized in Section V.

A. Graph Theory
Spontaneously, B = diag(℘ 10 , . . . , ℘ N 0 ) is defined to depict the interaction between the ith ASV and its corresponding tracking target. ℘ i0 > 0 if the ith ASV can receive information from the target directly, otherwise ℘ i0 = 0.
For the interaction of the networked ASVs in multi-formation tracking control, consider that the networked ASVs are divided into J(1 ≤ J ≤ N ) groups, each of which has the same tracking target and the related interaction can be depicted by subgraph Similarly, B l is the pinning matrix of subgraph G l .
Assumption 1: For each subgraph G l , l ∈ {1, 2, . . . , J}, the information of the sub-leaders is globally accessible to all its follower vehicles.
Assumption 2: The directed graph G with acyclic partition satisfy the in-degree balanced condition, i.e., each row sum of L lm is zero.
According to [40,Lemma 1], the Laplacian matrix can be rewritten into the following form for the directed graph G with acyclic partition.
where L l is the laplacian matrix of G l , L lm denotes the interaction between subgroup G l and G m , ∀l = m, l, m ∈ {1, . . . , J}.

B. Definitions and Lemmas
Definition 1: is globally finite-time stable with settling-time function.
Definition 2: The systemẋ = f (x, t) is said to be prescribedtime stable if there are two constants δ and T such that x(t) ≤ δ for any initial conditions when t ≥ T , where T can be prescribed freely.
Lemma 1: [35] Consider that there exists a continuous differ- where the time-related function μ(t) is proposed as in which ρ > 1 is a real number, t 0 > 0 and T u > 0 are the initial time and the user-prescribed time. Beside, it holds that . Lemma 2: [41] Under Assumption 1, for each subgraph G l , l ∈ {1, 2, . . . , J}, there exist two positive-defined matrices P and Q such that where Lemma 4: [43]There exists a positive constant such that where x, y are vectors arbitrarily.

C. System Formulation
Consider that the networked ASVs include N individuals. The dynamics and kinematics of the ith ASV with port-starboard symmetric model are established as follows [44]. where is the velocity and r i is the angular rate in the body-fixed frame (AXY ). More details refer to Fig. 1.
The positive-definite and symmetric inertia parameter matrix M i , the Coriolis and centripetal terms C i (v i ) and damping matrix D i (v i ) are the dynamical terms of the vehicle, whose specific mathematical formulation are proposed in Appendix C. Besides, the rotation matrix R(ψ i ) is defined as with the following relevant properties where S( Additionally, define the velocity of the networked ASVs in the earth-fixed frame as where Combing with the properties (7), (6) can be redefined as where The leaders in multi-formation tracking control are modeled asη 0 a l 0 are the position, velocity and acceleration in the earth-fixed frame.
Assumption 3: The derivatives of all the leaders' accelerations are bounded, namely, Assumption 4: The external disturbances are bounded,

D. Control Objective
Definition 3: For the system (6), the fast fixed-time output multi-formation tracking control problem could be solved if there exists a fixed time T > 0 such that

A. The Fast Fixed-Time Tracking Algorithm
The hierarchical fast fixed-time tracking algorithm is designed to solve the output multi-formation tracking problem in the fast fixed time. To this end, a distributed prescribed-time estimator is designed to estimate the leaders' states, and then a nonsingular fixed-time sliding surface is introduced to constitute the local control layer, as well as a time-related function applied to force the system states reaching the sliding surface in the prescribed time. For the distributed prescribed-time estimator, the algorithm is designed as whereη i ,ŵ i andâ i are the estimator of η 0 l , w 0 l and a 0 l in the earthfixed frame, α 1 , α 2 , α 3 , β 1 , β 2 , β 3 ≥ d a are positive parameters, ϕ(t) has been defined in (1). Before move on, define the tracking errors as where o i is the formation offset. Then for the local control layer, the algorithm is designed as where is non-negative with the property of δ ξ (x)/x → 0 when x → 0, which guarantees the control input is well-defined even when tracking errors converge to orgin. Π i is a diagonal matrix with positive designed gains. According to [39,Lemma 4], once the nonsingular sliding surface s i = 0 is satisfied, e i andė i converge to the origin in the fixed time T κ , which is bounded with Remark 1: Noting that the implement of sliding mode control (SMC) generally contains two-stage process. Firstly, the controlled system is drived to reach the sliding surface. Secondly, the system states remains on the sliding surface and then reach to the origin within infinite time. Compared with the conversional SMC, the control strategy in this paper has the following advantages. 1) the controlled systems are drived to reach the sliding surface in the prescribed time, which can be predesigned arbitrary; 2) the system states reach to the origin within fixed time without the existence of the singularity problem. Superiorly, compared with the conversional fixed-time sliding mode control, the designed algorithm has a faster settling time due to the existence of the time-related function in some sense.
Remark 2: Compared with the existing fixed-time algorithms, the designed fast fixed-time controller shows faster convergence gate in the following aspects. 1) In [15], [45], the fixed-time observers are employed to estimate the leaders states in a fixed time, where the settling time is a bounded constant that related to multiple control gains and thus can not be prescribed easily. The designed estimator (11) can estimate the states of the leader in the prescribed time, which can be predetermined arbitrarily to an extremely small value by selecting only one parameter. This is one reason for achieving the fast convergence rate. 2) Compared with [15], [46], where the settling time of reaching the fixed-time sliding surface is always in connection with several control parameters. The prescribedtime reachability of the designed algorithm is guaranteed by selecting an arbitrary small constant. This is another reason for achieving the fast convergence rate. 3) Different from [18], [47], where the fast convergence rate is achieved by using the fixed-time sliding mode control approach. We present a novel framework for achieving the fast fixed-time stability by combining the prescribedtime control with the fixed-time sliding mode control for accelerating the reaching time to the sliding surface.

B. Analysis of the Distributed Prescribed-Time Estimator
Theorem 1: Suppose that Assumptions 1-3 hold. By using (11), the states of the leaders can be estimated in the prescribedtime manner and the prescribed time T 1 can be set as . . , J}, t 0 > 0 and T u > 0 are the initial time and the user-prescribed time. Besides, the necessary conditions for the designed control gains are presented as: Proof: The proof includes two main steps. In the first step, it will be proved that the third equation of (11) can estimate the acceleration a 0 l of the leaders in the prescribed time in which h ll = L l + B l , l ∈ {1, 2, . . . , J}. Taking the derivative ofẽ along the acceleration estimator (i.e., the third equation of (11)), and then combing with the specific form of H, we can obtain the expansion ofė as the following forṁ The Lyapunov function candidate is expressed as The remaining proof is based on the mathematical induction.
Step 1: Suppose l = 1. It follows that For the time interval [t 0 , t 0 + T u ), taking the derivative of V 11 , it can be obtained thaṫ Based on Lemma 2 and Lemma 3, the following inequalities hold where p i is the ith element of P 1 . It then follows thaṫ Noting the fact that It further obtains Combing (23) with (25), it derives thaṫ If (16) holds, it follows thaṫ It thus can be concluded that lim t→t 0 +T u ẽ 1 = 0. Noting that H is a nonsingular matrix, it thus comes to the conclusion that lim t→t 0 +T u â i − a 0 1 = 0, ∀i ∈ V 1 . For the time interval [t 0 + T u , +∞), with the similar analysis of (20)-(27), we can derived thatV 11 ≤ −(β 1 + β 2 ϕ(t))V 11 . Then we can derive that V 11 = 0 for t ∈ [t 0 + T u , +∞).
Construct the following Lyapunov function candidate where P l is the diagonal matrix with respect to h ll , see Lemma 2 for more details.
Step 2: Suppose l = 2. According to the analyse above, when t ≥ T 11 + T u ,Ṡ 2 can be rewritten aṡ Similar to the steps (37)- (46), it can be derived thatη i ,w i converge to zero as t → T 11 + 2T u .
Step 3: For l = J, when t ≥ T 11 + J−1 l=1 T u , it can be concluded thaṫ Based on the mathematical induction, it obtains thatη i ,ŵ i will approach to zero within prescribed time T 11 + J l=1 T u . i.e., t 0 + 2 J l=1 T u , ∀i ∈ V, l ∈ {1, 2, . . . , J}. Above all, the prescribed-time convergence ofη i ,ŵ i andâ i can be achieved in . . , J}. This ends the proof.

C. Analysis of the Fast Fixed-Time Tracking Algorithm
Theorem 2: Suppose that Assumptions 1-4 hold. By using (11) and (13), the fast fixed-time output multi-formation tracking problem of the networked ASVs can be solved and the fast fixed time can be set as T = T 1 + T u + T κ ,∀i ∈ V, l ∈ {1, 2, . . . , J} if the following conditions hold: Proof: This part aims to prove the fast fixed-time convergence of the tracking errors defined in (12). For t ≥ T 1 , the tracking error (12) equals to the following equation The Lyapunov function is chosen as Taking the derivative of V 3 obtains thaṫ Then, it follows from λ max (Π) ≥ d m thaṫ Then similar to the analysis in [39], it derives that the system (9) will be driven on s i = 0 in settling time t 0 + T u + T ι , in which T ι = (ξ 1/(γ 2 −1) )/(λ min (Π i ) − d m ). Noting that ξ is a sufficiently small parameter, coupled with the limited condition 1 < γ 2 ≤ 2 (further illustrates that 0 < ξ 1/(γ 2 −1) << 1), it obtains T ι is infinitesimal and thus can be omitted here. This leads to the inescapable conclusion that the system (9) will be driven on s i = 0 in settling time t 0 + T u . Moreover, once s i = 0, e i ,ė i converge to zero less than T κ . Further, it gets that system states achieve fast fixed-time stability within T = T 1 + T u + T κ . When t > T , the position tracking error of the networked ASVs is It further obtains that lim t→T w i − w 0 l = 0, ∀i ∈ V, l ∈ {1, 2, . . . , J}. This ends the proof.
Remark 3: To make it more clear, the framework of (11) and (13) is provided in Fig. 2 to illustrate the information flow of the Step 1: Set the interaction graph and obtain the gain λ max (Q l ), λ min (P l ); Step 2: Set the prescribed time T u and the related appropriate gain t 0 , ρ, b, c according to lemma 1; Step 3: Set an appropriate gain α 1 /α 2 according to the first and second inequalities in conditions (49), and then set the appropriate gains α 1 , α 2 respectively; Step 4: Set the gains α 3 , according to the known λ max (Q l ), λ min (P l ), and set the gains Π, β 3 to satisfy the conditions in (49).
states. It shows that the fast fixed-time control algorithm includes distributed prescribed-time estimator layer and local control layer. The states of the leader is estimated in the prescribed time T 1 in the distributed estimator layer. Then, there are two involved user-designed time in the local control layer, one is that the states of the closed-loop system reach to the sliding surface s i in the prescribed time t 0 + T u (where t 0 = T 1 holds), the another is that the error states converge to the origin in the fixed time T κ . Apparently, the whole convergence time is bounded with T = T 1 + T u + T κ . Furthermore, the Algorithm 1 is introduced to illustrate the design procedure of the proposed controller. Remark 4: Compared with the existing researches of the formation control for the networked ASVs, the designed algorithm in this paper shows several advantages in the following aspects.
1) The multi-formation tracking control shows more efficient in practical application, in which every subnetwork can be employed to implement the respective control operations. 2) The multi-formation tracking can be achieved in a fast fixed time, in which the settling time is independent of the initial conditions. 3) The fast fixed-time stability analysis of the multiformation tracking has certain difficulties in technical implementation, since the analysis methods of the singleformation tracking problem can not be directly applied to the multi-formation one. Remark 5: The term ϕ(t) in (1) is essential for realizing the prescribed-time stability of the controlled system. The prescribed-time controller (11) includes both the ϕ(t) and ϕ 2 (t). This will inevitably place strain on the analysis of the prescribedtime stability when employing the Lyapunov function. Referring to [31], a state transformation (30) is applied to analyze the effectiveness of (11). Correspondingly, (32) can be obtained according to the specific form of ϕ(t) and (30).
Remark 6: The uncertainties regarding model and external have no impact on the recursive feasibility, since we design the hierarchical fast fixed-time tracking algorithm, where the stability of the distributed prescribed-time estimator and the local control layer is analyzed, respectively. Besides, the uncertainties we considered only exist in the local control layer, where the sliding mode control method is introduced to eliminate the influence of the uncertainties as long as Assumption 4 holds. In addition, the uncertainty of the model parameters are not considered in this paper. However, based on the existing work [48], an extend state observer is employed to reduce the negative effect of the unknown model parameters. Please refer to the model-free approach in [48] for more details.

IV. SIMULATION RESULTS
To link some practical operation, consider that there is a detection scenario that the networked ASVs are divided into several groups to arrive at the corresponding desired areas respectively. In Fig. 3, there are 3 groups including 11 follower vehicles and 3 sub-leaders. The sub-leaders can be virtual or physical, which play a role in providing a trajectory for the follower vehicles of the related group. Spontaneously, the follower vehicles track their corresponding trajectory of the sub-leaders to accomplish appointed task, such as maritime rescuemilitary detection and underwater surveillance. As for the following simulation for illustrating the validity of the designed algorithm, the model parameters of ASV are selected as the same the well-known CyberShip II in [49].         converge to the origin in T , even though the initial values vary enormously under the two simulations. Further, it can safely come to the conclusion that the fast fixed time T is independent of the initial conditions. Example 3: (The comparison with the fixed-time and finitetime control law) Based on the discussion in Remark 2, several comparisons with the existing fixed-time and finite-time control algorithms are carried out to show the fast convergence rate of the proposed algorithm. To this end, the initial conditions and the time-varying targets, as well as the interaction topology are chosen the same (i.e., the second network in Fig. 4). Firstly, the finite-time algorithm in [20] is employed to replace the estimator layer (11) and is presented as (55) Although the settling time of (55) are set as small as possible by adjusting the control parameters and initial conditions, it still needs lager convergence time than the controller we designed. Secondly, we compared the fixed-time controller in [39], where the fixed-time sliding mode control method are utilized in the controller design and the reaching time to the sliding surface are associated with the several control parameters. Refer to [39] for more details about the convergence time and the form of the fixed-time algorithm is given as follows.
(56) The specific conditions of the control parameters can be found in [39] and is omitted for space limitation.
The simulation results can be observed in Figs. [13][14][15]. It demonstrates that the designed algorithm can achieve fast convergence performance than the general finite-time and fixedtime control laws. Specifically, it can be observed from Fig. 13 that the settling time of the designed algorithm is set at 0.02 s. This displays the superiority of the designed algorithm for realizing the control objective within any arbitrarily convergence time. Besides, Fig. 14 shows the evolution of the tracking errors  under the finite-time control and the settling time can not be set as small as the designed algorithm. From Fig. 15, the settling time of the general fixed-time control is also longer than the designed algorithm, although the fixed-time sliding surface in (56) is the same as that in (13). This is because the reaching time to the sliding surface of (56) is related to several control parameters, which can be further seen in [39]. However, the reaching time to the sliding surface of (13) can be set artificially and arbitrarily, even to 0.02 s. Above all, the faster convergence rate could be guaranteed under the designed fast fixed-time control algorithm.

V. CONCLUSION
In this paper, a novel fast fixed-time tracking control algorithm has been proposed to solve output multi-formation tracking problem of the networked ASVs with external disturbances under directed graph. The proposed algorithm is capable of guaranteeing satisfactory tracking performances in a fast fixed-time manner. Accordingly, an appropriate mathematical inductionbased method has been introduced into stability analysis for dealing with the multi-formation tracking problem. The sufficient conditions for achieving fast fixed-time multi-formation tracking of the networked ASVs have been obtained by employing the mathematical induction method and the Lyapunov stability theory. Finally, several illustrative simulation examples have been presented to demonstrate the superiorities of the fast fixed-time control algorithm. Future work will be focused on the prescribed-time multi-formation tracking control of the networked ASVs with input constraint.