Distributed adaptive formation control for underactuated quadrotors with guaranteed performances

This paper investigates a distributed adaptive formation control problem for underactuated quadrotors with guaranteed performances. To ensure a robust and stable formation pattern with predefined behavior bounds, by transforming the original constrained formation synchronization error dynamics into an equivalent unconstrained one, a prescribed performance mechanism is introduced in the translational loop to render the formation regulation as a prior. An adaptive consensus strategy is developed according to undirected graph theory and Lyapunov stability rules for follower quadrotors to achieve a distributed cooperative formation with prescribed tracking abilities via exchanging local information with neighbors. The presented control scheme has the following salient merits: (1) the formation synchronization errors can be guaranteed within pre-assigned bounds with desired transient behaviors despite of uncertain disturbances; (2) by using a state estimation error to update neural network (NN) parameters, rather than the tracking error that widely applied in traditional NN approximators, and with the help of MLP technique, the proposed SE-MLP observer capable of decreasing the computational complexity can achieve a fast identification of lumped disturbances without causing high-frequency oscillations even using a large adaptive gain, and the transient solutions of L2 norm of the differential of neural weights are established to illustrate the mechanism of SE-MLP observer in reducing chattering behaviors. The merits of presented algorithm are confirmed by sufficient simulations.


Introduction
Recently, there have been a surge of interests in formation control for multiple quadrotors, owing to their practical applications in collectively fulfilling difficult missions across civilian and military fields, such as distributed environmental monitoring with portable chemical and biological sensor arrays to detect toxic pollutants, disaster relief in an inaccessible area, aerial transportation of large objects, etc. [1][2][3][4][5][6]. However, the coordinated control problem is rather challenging, due to the strong nonlinear coupling in quadrotor modeling and various sources of uncertainties arising from unmodeled dynamics and time-varying disturbances acting on each quadrotor dynamics [7][8][9]. In addition, some time-urgent tasks always entail the design of swarm control strategy equipped with a preset convergence rate to ensure a high-efficiency mission outcome. Therefore, complicated model dynamics [10,11], uncertain circumstances, and rigid performance constraints [12,13] pose great difficulties on guaranteeing the success of formation flight mission.
To realize a distributed cooperative control of multi-agents, consensus is a fundamental issue where each agent reaches an agreement on the interested states utilizing only local communication flow [14][15][16][17]. As described in [15][16][17], a consensus topic for strict-feedback multi-agent systems is investigated in a single-input single-output (SISO) form. Consensus strategies for second-order nonlinear multi-agent systems are discussed in [18][19][20]. Consider a formation system with a union of unmanned surface vehicles; a consistent tracking result can be found in [21]. However, the aforementioned consensus schemes only concentrate on system dynamics obeying SISO or fully actuated features, which are hard to be generalized to formation control problems of multiple quadrotors exposed to multiple-input multiple-output (MIMO) and underactuated characteristics [22]. Recently, some formation control approaches of multiple quadrotors have been reported. For example, a leader-follower control approach based on a suboptimal H ? is developed in [23] for quadrotors suffering from external disturbances and model parameter uncertainties, contributing to an enhanced control performance. In [24], a robust formation problem is studied, and with the aid of a robust compensation strategy, a distributed controller is developed to accomplish a desired formation of networked uncertain quadrotors. By employing a finite-time stability and optimal control principle, a non-smooth cooperative formation control algorithm is discussed in [25] under a leader-follower structure. Unfortunately, most of the previous formation control methods can merely provide ultimately uniformly bounded (UUB) tracking via various anti-disturbance avenues, where the final upper bound of synchronization error primarily relies on some design scalars and unknown auxiliary terms, i.e., the preset performance indices cannot be easily met even tuning parameters tediously, inevitably leading to a severe design conservatism and further cause unpredictable tracking errors in the presence of large uncertainties. Additionally, little consideration has been focused on transient performances, while transient performances dominate an important role in ensuring an autonomous and effective execution of cooperative flight mission. For instance, in the situation of performing a formation task in an uncertain environment with randomly distributed obstacles, to avoid collision between neighbor quadrotors and communication interruption, practical conditions typically require high-performance formation control algorithms to hold certain transient and steady-state specifications [26][27][28]. If the preselected performances cannot be specified as a prior, the unexpected consequences, including a relatively large overshoot, a slow convergence rate, or an enlarged steady-state deviation, may deteriorate system performances and even cause the failure of missions. Thus, it is imperative to design an enhanced anti-disturbance formation control for multiple quadrotors with preselected characteristics. Unfortunately, up to now, few results have been available for the guaranteed performance formation control design of multiple quadrotors.
Neural network (NN) is an efficient self-learning tool to counteract the adverse effect of unmodeled dynamics [29][30][31][32]. During the past decades, NN-based adaptive estimation algorithms have drawn much attentions, and a great deal of research advancements have been made in [33][34][35][36][37][38]. For instance, in [33], a radial basis function (RBF) neural network is employed for multiagent systems to neutralize unknown system dynamics and external disturbances. A constrained backstepping adaptive NN control scheme is presented in [35] for MIMO aeroelastic systems to approximate the system uncertainties. A RBF neural network is developed in [36] to identify unknown dynamics existing in multi-agent systems. In [37], by integrating with a neural adaptive observer, a consensus coordination controller is interviewed for nonlinear multi-agent systems considering unknown external disturbances. In [38], by synthesizing a hybrid rescheduling policy, a fuzzy NN observer is studied for spacecrafts to neutralize the uncertainties with fuzzy features. It is worthwhile stating that unknown time-varying uncertainties can be straightforwadly recovered with a favorable convergence speed and estimation accuracy by utilizing a fast learning solution, which corresponds to a high adaptive gain in NN [33][34][35][36][37][38]. However, a sizeable adaptive gain may deteriorate transient performances and even make control inputs suffer from high-frequency oscillations during the initial stage [34][35][36][37], which will directly lead to collisions and communication interruptions among quadrotors. Meanwhile, another crucial issue inherent in implementing the calculation of NN is the computational burden owing to the involved huge number of NN hidden nodes. It is acknowledged that a large amount of NN nodes are inevitably needed to realize an excellent system identification capability against a vast range of uncertainties [39][40][41]. Nevertheless, the increase in the computation load results in a remarkable controlling error, especially for the maneuver formation control scenario that requires a real time computational capability. Consequently, it is paramount to explore an improved NN approximator for uncertainty identification with less chattering transient behaviors and reduced computational complexity even using a large adaptive gain.
Based on the above observations, a distributed adaptive consensus policy comprising of a low cost approximator is proposed to achieve a rapid and precise formation maintenance for multiple quadrotors with performance requirements and uncertainties. By introducing a prescribed performance control (PPC) technique, communication interruption and every possible collision between quadrotors under aggressive maneuvers can be circumvented. To handle unknown uncertainties, SE-MLP observers are effectively designed in the translational and rotational loop, respectively, greatly alleviating the computational complexity and avoiding high-frequency oscillations even using a large adaptive gain. The specific contributions of presented control algorithm are twofold: 1. Differing from the available cooperative issues concentrating on multi-agents formulated as single integrator [42], high-order nonlinear uncertain form [43,44], fractional-order systems [45,46], herein a nonlinear collective control policy is devised for quadrotor robust formation suffering from nonlinearity, underactuated MIMO properties, and uncertainties. Distinguished with the existing distributed consensus solutions [23][24][25], where synchronization errors can only converge to an unknown and conservative performance bound, a PPC technique consisting of an exponential decaying behavior bound and an error transformation function is enforced, allowing for a satisfaction of preset time-varying formation constraint rather than a constant limitation executed by the reported output constraint approach [46], indicating that the tracking precision, the maximum overshoot, and convergence time can be prescribed as a priori despite of uncertainties.
And it is worth stressing that the proposed design can be suitable for a general type of intelligent unmanned systems considering nonlinear uncertain dynamics, including flocking and swarming control of autonomous surface vehicles [21,47], circular formation of ground vehicles [48]. 2. Different from the published approximation-based methods [39][40][41]49] where transient oscillations are frequently encountered due to nonzero initial tracking errors and the usage of large adaptive gain, by using a state estimation error instead of tracking error to update NN parameters, a SE-MLP observer is proposed to decouple the control loop and learning loop via appropriately selecting the bandwidth of state estimator, which enables a fast and smooth transient behavior even using a sizeable adaptive gain. And transient solutions of L 2 norm of the differential of neural weights are established to illustrate the mechanism in reducing chattering behaviors. In addition, unlike the traditional NN schemes subject to explosion of learning [36,50], herein an artificial quantity rather than each elements in NN weight vector is entailed to be updated online; thus, a competitive observation performance can be realized by virtue of only regulating one parameter in each subsystem, effectively reducing the computational complexity and ensuring an affordable neuro-adaptive cooperative control framework to robust formation configuration under unsigned network, and comparative simulations are conducted to show the merits and demerits of developed SE-MLP observer.
The rest of this paper is arranged in the following order. Preliminaries and problem formulation are illustrated in Sect. 2. Formation controller design together with stability analysis is presented in Sect. 3. Section 4 provides simulation results, and conclusions are drawn in Sect. 5.

Graph theory
Given the multiple quadrotors consisting of one virtual leader and N followers, the data communication among N followers is described by an undirected graph G ¼ V; E; A f g, with V ¼ v i f g being the set of nodes and v i ði ¼ 1; 2; . . .; NÞ being the i-th quadrotor.
indicates the information exchange between i-th quadrotor and its neighbor, i.e., the j-th quadrotor. The adjacency matrix A ¼ a ij Â Ã 2 R NÂN denotes topology of undirected graph, given by a ij ¼ a ji ¼ 1 if ðv i ; v j Þ 2 E, and a ij ¼ a ji ¼ 0, otherwise. The set of neighbors of quadrotor i is denoted by Specify the adjacency matrix between the follower quadrotors and the virtual leader Assumption 1 [20,21,25,26] Consider the involved multiple nodes, the unsigned graph for N follower quadrotors, and one virtual leader preserve connected if and only if there exists at least one path from the virtual leader to each follower, demonstrating that all the eigenvalues of matrix H have positive real parts.

Quadrotor model
Define the inertial frame O e X e Y e Z e and body frame o b x b y b z b , as shown in Fig. 1, where F i;n ðn ¼ 1; . . .; 4Þ represents the rotor thrusts of quadrotors. According to [51,52], the kinematics and kinetics of i-th quadrotor conform to the following dynamics: where X i;p ¼ ½X i;p1 ; X i;p2 ; X i;p3 T and X i;v ¼ ½X i;v1 ; X i;v2 ; X i;v3 T stand for the translational motions including position and linear velocity vectors expressed in O e X e Y e Z e , respectively. X i;X ¼ ½X i;X1 ; X i;X2 ; X i;X3 T and X i;x ¼ ½X i;x1 ; X i;x2 ; X i;x3 T represent rotational angles and angular velocity vector. F i;v ,ðg i;1 u i;1 À G i Þ m i denotes the equivalent control action, with m i representing the i-th quadrotor mass. G i ¼ ½0; 0; m i g T with g being the gravity acceleration.g i; 1 ¼ ½cosðX i;X3 Þ sinðX i;X2 Þ cosðX i;X1 Þþ sinðX i;X3 Þ sinðX i;X1 Þ; sinðX i;X3 Þ sinðX i;X2 Þ cosðX i;X1 ÞÀ cosðX i;X3 Þ sinðX i;X1 Þ; cosðX i;X2 Þ cosðX i;X1 Þ T indicates the interaction matrix between translation and rotational motions. u i;1 and U i;x ¼ ½u i;2 ; u i;3 ; u i;4 T express the actual lifting force and torque vector, which are produced by a linear combination of rotor thrusts as where c i denotes force-tomoment factor and l i represents the distance between each rotor of quadrotor and the center of mass. Obviously, the quadrotor is underactuated, since the number of actual control inputs is less than that of flight states. Notations f i;v ðX i;v Þ ¼ ÀP i;1 X i;v m i and f i;x ðX i;x Þ ¼ ÀJ À1 i P i;2 X i;x are model perturbations induced by inaccurate aerodynamic damping matrices ; D i;x3 T stand for unknown bounded environmental disturbances.

Radial basis function NN
For any given scalar continuous function f ðÁÞ, one can use a NN with a sufficient degree of accuracy as follows: where eðx p Þ stands for the unavoidable approximation Fig. 1 Schematic diagram of ith quadrotor error with its upper bound being e. w Ã ¼ ½w Ã 1 ; w Ã 2 ; . . .; w Ã L T 2 R L denotes the ideal weight with L being the hidden node number. x p ¼ ½x 1 ; x 2 ; . . .; x p T is the neural input with p being the input number. hðx p Þ ¼ ½h 1 ðx p Þ; h 2 ðx p Þ; . . .; h L ðx p Þ T represents the commonly used Gaussian basis function. Referring to [34], each component h j is chosen as Remark 2 It is well known that when traditional NNs are designed to identify unknown functions with a high precision, explosion of learning parameters unavoidably appears due to the involved numerous hidden units; especially, as argument vector dimension of the observed function increases, the number of parameters to be regulated will drastically grow, leading to a heavy computational complexity, which is not appropriate for the considered quadrotor formation control system that can only tolerate a slight time delay during the computational process.
Control Objectives The purpose of this paper is to design a distributed neural adaptive formation consensus for follower quadrotors with guaranteed performances, such that (i) The follower individual can be steered to tend to the predefined trajectory of virtual leader, i.e., Ã T , and the anticipated formation configuration is described as where q i ¼ q i;1 ; q i;2 ; q i;3 Â Ã T is the expected pattern expressed in the inertial frame among networked quadrotors and q i;1 ; q i;2 ; q i;3 are the expected distance deviation between the i-th individual and formation geometry center. q ij ¼ q i À q j denotes the anticipated relative position discrepancy between i-th and j-th individual.
(ii) The formation synchronization deviation e i;pk ðk ¼ 1; 2; 3Þ obeys the following preselected condition: Àr i;pk q i;pk ðtÞ\e i;pk ðtÞ\r i;pk q i;pk ðtÞ ð 5Þ where 0\r i;pk 1, 0\r i;pk 1 are design scalars, q i;pk ðtÞ represents the prescribed performance function, and its definition is given in Sect. 3.

Remark 3
The control objectives imply that not only the desired formation pattern can be maintained through application of the distributed consensus protocol, but also the pre-given time-varying trajectory determined by virtual leader can be tracked accurately for followers. Equation (4) implies that the location of each individual can synchronize to the leader with a designed deviation. Besides, it should be pointed out that since only a portion of follower individuals can have access to the data of virtual leader, the developed strategy is distributed essentially; the effectiveness relies on the network topology to be designed and consensus parameters, irrespective of the number of agents. In addition, note that the anticipated relative position discrepancy q ij can be specified according to requirements of different tasks, for example, when q ij is designated as a time-related signal, the proposed technique can be readily expanded to a time-varying formation pattern case.

Formation consensus design and stability analysis
Here, a distributed neural adaptive consensus protocol with guaranteed performances for multiple quadrotors is proposed to acquire the previous formation control objectives. The control block diagram is given in Fig. 2.

SE-MLP observer
In the section, to facilitate the subsequent demonstration and calculation, and referring to the design principle of MLP [41], take the translational dynamics of i-th quadrotor into consideration and the lumped disturbances can be approximated by the following MLP: Gaussian basis function with L being the number of hidden nodes in NN. X i;v ¼ ½X T i;p ;X T i;v T , and e i;v ¼ ½e i;v1 ; e i;v2 ; e i;v3 T represents the approximation error. Note that v i;v is an unknown and continuous function, indicating that the precise information of v i;v is not obtained. [39,50], the number of learning parameters to be regulated is mainly determined by the dimension of nodes in hidden layer; thus, h i;v ðX i;v Þ 2 R L inevitably causes a substantial time-consuming computational process, especially when a high accuracy identification result is required. Here, only one adaptive learning parameter is needed to be updated online for each subsystem, regardless of the specific number of hidden nodes L, which can eliminate the issue of learning explosion inherent in the existing adaptive NN algorithms [39,40].

Remark 4 For classical NN approximators
To approximate the unknown disturbance v i;v , the following SE-MLP observer is designed as À Á is the positive estimator bandwidth. c i;v [ 0 represents the adaptive gain, and r i;v ¼ diag r i;v1 ; r i;v2 ; r i;v3 À Á is the modification parameter matrix with r i;v1 ; r i;v2 ; r i;v3 being positive constants.Ŵ i;v is an estimate of ideal weight vector W Ã i;v . By invoking (1), the total disturbances (6), and observer (7), the estimation error dynamics can be formulated in the subsequent form: Next, a rigorous mathematical derivation and theoretical analysis will be conducted to show the improved transient performances of presented SE-MLP observer by using series of algebraic operations. It is noteworthy that the truncated L 2 norm of the differential of NN weights can reflect their frequency features, and generally, a larger L 2 norm of the differential of NN weight corresponds to more oscillations included in system response. Here, Thus, Theorem 1 is established.
Theorem 1 Take the estimation error dynamics (8) into consideration, and then, L 2 norm of the differential of weight vectorŴ i;v during the time interval where k min ðÁÞ and k max ðÁÞ denote the minimum and maximum eigenvalues of a matrix, respectively.
Proof Construct the following Lyapunov function candidate: Differentiating V with respect to time yields Resorting to Young's inequality produces Substituting (12) into (11), _ V further satisfies with Þ being a positive constant. (13) can be rewritten as Solving the inequality (14) results in Combining the Lyapunov function candidate (10) and utilizing the inequality it further generates the following bound: To obtain the transient bound of estimation error dynamicsX i;v , by recollecting (13), we can deduce the following inequality: Afterwards, through integrating (17) over ½0; t Ã , we get Following (18), one has To make a quantitative evaluation for the transient performance, one concern is to derive the upper bound of _ W i;v , and aiming at the purpose, it follows from the time derivative ofŴ i;v in (7) that for ease of presentation. Then, we can obtain Squaring both sides of (21) gives Similarly, via using integration of (22) over ½0; t Ã , we have which further produces Noticing the bound ofX i;v in (19), the truncated L 2 norm ofŴ i;v can be rewritten as The proof of Theorem 1 is completed.
To facilitate comparison, the following lemma is given to show the truncated L 2 norm of differential of weightQ i;v 2 R LÂ3 used in traditional RBF neural networks.
Lemma 1 [53] Consider the classical radial basis function NNs given in Sect. 2.3, where the adaptive law of the weight is described as Then, L 2 norm of the differential of the weightQ i;v during the time interval ½0; t Ã satisfies where k i;v ¼ diag k i;v1 ; k i;v2 ; k i;v3 À Á denotes the positive control gain matrix in the velocity loop. (25) and (26), it is noteworthy that as the adaptive gain increases, the truncated L 2 norms of differential of the weights for both classical NN and presented SE-MLP gradually become larger, contributing to a fact that the transient oscillations will aggravate. But compared with the current NN approaches [39,50], the proposed SE-MLP observer provides two extra design freedoms to alleviate oscillations with the precondition of the same adaptive gain. One is that by employing an observation error X i;v , instead of tracking error e i;v , to learn the adaptive parameters, the developed SE-MLP observer renders a time-scale separation between controlling and leaning loop with the selection of observer bandwidth g i;v satisfying g i;vk ! 2k i;vk , and a larger observer bandwidth will lead to a less fluctuation transient, implying that the presented SE-MLP can tolerate a relatively higher adaptive gain compared with classical NNs. Another is that by setting X i;v ð0Þ ¼X i;v ð0Þ, the proposed SE-MLP observer effectively removes the undesired transient learning process induced by nonzero initial tracking errors. Obviously, the above two avenues cannot be provided by utilizing the current NN methods.

Formation synchronization error transformation
To facilitate the formation controller design, based on graph theory, define the formation synchronization error e i;pk as follows: where N i denotes a set consisting of quadrotors excluding quadrotor i, while the subscript k denotes k-th component of position vector. Note that only a team of agents can have access to the leader in (27), leading to a reduced communication cost compared to centralized formation framework [54]. Differentiating e i;pk along (1) gives Note that it is difficult to establish a control law with respect to inequality (5). To tackle this problem, based on prescribed performance control principle [55][56][57][58], a formation error transformation function S i;pk ðZ i;pk Þ is introduced to transform the limited formation synchronization deviation to an equivalent one free of constraints: e i;pk ðtÞ ¼ q i;pk ðtÞS i;pk ðZ i;pk Þ ð 29Þ with S i;pk ðZ i;pk Þ ¼ r i;pk expðZ i;pk Þ À r i;pk expðÀZ i;pk Þ expðZ i;pk Þ þ expðÀZ i;pk Þ ð30Þ where Z i;pk represents the transformed formation error, and the performance function is selected as q i;pk ðtÞ ¼ ðq 0 i;pk À q 1 i;pk Þe Àk i;pk t þ q 1 i;pk . The decreasing rate k i;pk is applied to regulate the decaying rate of e i;pk . The positive constant q 1 i;pk is the upper bound of allowable steady-state error, while q 0 i;pk is tuned to maintain Àr i;pk q i;pk ð0Þ\e i;pk ð0Þ\r i;pk q i;pk ð0Þ.
The inverse transformation of (29) is given as Z i;pk ¼ 1 2 ln r i;pk þ e i;pk q i;pk r i;pk À e i;pk q i;pk ! ð31Þ with its time derivative _ Z i;pk ¼ n i;pk ð _ e i;pk À _ q i;pk e i;pk q i;pk Þ ¼ n i;pk ðd i þ b i ÞX i;vk À X j2Ni a ij X j;vk À b i _ X d k À _ q i;pk e i;pk q i;pk " # ð32Þ where n i;pk ¼ 1 2q i;pk 1 e i;pk =qi;pkþr i;pk À 1 e i;pk =qi;pkÀri;pk . Lemma 2 [55,59,60] Considering the formation synchronization error e i;pk and transformed error Z i;pk , the prescribed limitation imposed on e i;pk will be always achieved as long as Z i;pk is bounded, i.e., (5) is satisfied.

Neural adaptive control design
Step 1 Assuming a i;vk as a virtual control input of position dynamics, to make the transformed error Z i;pk tend to zero, a virtual control policy is derived: For the sake of facilitating the analysis and design, (33) can be reformulated as the following compact form: where C ¼ diag _ q i;p1 ðtÞ q i;p1 ðtÞ; _ q i;p2 ðtÞ À :q i;p2 ðtÞ; _ q i;p3 ðtÞ q i;p3 ðtÞÞ, a i;v ¼ ½a i;v1 ; a i;v2 ; a i;v3 T ,Z i;p ¼ ½Z i;p1 ; Z i;p2 ; Z i;p3 T , k i;p ¼ diag k i;p1 ; k i;p2 ; k i;p3 À Á with k i;p1 ; k i;p2 ; k i;p3 being nonnegative control gains.
To avoid analytically differentiating a i;v , inevitably causing complexity explosion problem due to the recursive analytic computation of backstepping, a dynamic surface control (DSC) principle is embedded here. Let a i;v inject into the following filter to yield an estimate of a i;v , denoted as a i;v ¼ ½a i;v1 ; a i;v2 ; a i;v3 T : where B i;vk ðZ i;pk ; _ Z i;pk ; _ X j;vk ; € X d ; _ e i;pk Þ is a continuous function.
Step 2 In this step, F i;v is treated as a virtual control signal to stabilize velocity X i;v . The tracking error of translational velocity is defined as whose time derivative along (1) is where e i;v ¼ ½e i;v1 ; e i;v2 ; e i;v3 T . Considering system (38), based on (7), the velocity subsystem control law is designed as follows: where k i;v ¼ diag k i;v1 ; k i;v2 ; k i;v3 À Á with k i;v1 ; k i;v2 ; k i;v3 being control gains.
In fact, to implement the velocity adjustment for follower quadrotors, the magnitude of F i;v is mainly determined by thrust force u i;1 , and the desired body attitude (u d i , h d i , w d i ) depends on its orientation, with u d i , h d i , and w d i being the anticipated roll, pitch, and yaw angles. F i;v ¼ ½F i;v1 ; F i;v2 ; F i;v3 T corresponds to the expected position control input meeting Subsequently, the lifting force u i;1 and the anticipated angles u d i , h d i can be derived as  Considering the second filtering error as Step 3 In this step, to construct attitude control law for individual quadrotor to track attitude commands, the tracking error in attitude loop is defined as with its derivative along (1) being computed as Thus, the virtual angular velocity law is calculated as where k i;X ¼ diag k i;X1 ; k i;X2 ; k i;X3 À Á with k i;X1 ; k i;X2 ; k i;X3 denoting nonnegative values.
Following the same line with Step 1, let a i;x pass through the following filter: where s i;x ¼ diag s i;x1 ; s i;x2 ; s i;x3 À Á is a time constant to be regulated.
Define the third filtering error y i;x ¼ a i;x À a i;x , and combining with (46), we have Step 4 Consider the controlling deviation of rotational level as The eventual practical controller consisting of a neural approximator is given by where U i;x is the rotational control vector; other notations discussed in (50) are similar with (7).

Convergence analysis
To promote the convergence analysis, definẽ For the formation deviation, state estimation error, weight updating error, and filtering error discussed aforementioned in controller design, the resulting distributed dynamics of formation deviation can be derived as Theorem 2 Considering the multiple quadrotors (1) and overall error dynamics comprising of (51), (52), given that Assumption 1 satisfies and controllers are devised as (34), (39), (46), (50) along with adaptive laws (7), (50), all error variables are steered to convergence within a small vicinity of zero. In particular, for the initial condition of formation synchronization error fulfilling Àr i;pk q i;pk ð0Þ\e i;pk ð0Þ\r i;pk q i;pk ð0Þ, formation synchronization errors of all quadrotors will be constrained within the prescribed bounds, i.e., Àr i;pk q i;pk ðtÞ\e i;pk ðtÞ\r i;pk q i;pk ðtÞ.
Proof Consider the positive definite Lyapunov function as below: Computing the time differential of (53) and using (36), (43), (48), (51) as well as (52) produces Define the sets X 0 , Recall the filtering error represented by (36), (43) and (48) with f i;v , f i;X and f i;x being nonnegative parameters. By virtue of the aforementioned procedures, one has Next, choose the design parameters in (56) to satisfy Obviously, where l ¼ min i¼1;...;N 2l i1 ; 2l i2 ; 2l i3 ; 2l i4 ; 2l i5 ; f 2l i6 ; 2l i7 ; 2l i8 ; 2l i9 ; 2c i;v l i10 ; 2c i;x l i11 g. The inequality (58) means _ V\0 on V ¼ 1 as long as l [ -=1. Therefore, V 1 belongs to an invariant set, i.e., if Vð0Þ 1, then V 1, 8t ! 0. Calculating the inequality (58) yields Therefore, all error states of overall dynamics are UUB, that is, when t ! 1, one has It is obvious from (59) that convergence rate of error states in the overall dynamics depends on parameter l. Theoretically, increasing l will result in a faster convergence speed. Moreover, the eventual size of error relies on -. In addition, the formation synchronization deviation can be forced to decay to a predefined arbitrary small neighborhood around the origin by improving adaptive learning parameter, control gains, and decreasing time constants in filtering as t ! 1. Simultaneously, the converted formation synchronization deviation Z i;pk converges to an arbitrary small neighborhood of zero. Furthermore, by appropriately tuning the constants q 0 i;pk , q 1 i;pk , and k i;pk in the behavior function, and we can easily derive that the formation synchronization error e i;pk can always evolve within the prescribed performance bound based on Lemma 2. Let e Ã;pk ¼ e 1;pk ; . . .; e N;pk Â Ã T , then e Ã;pk ¼ H X Ã;pk À 1X d k þ 1 P N i¼1 q i;k 0 N , where X Ã;pk ¼ X 1;pk À q 1;k ; . . .; X N;pk À q N;k Â Ã T , it yields X Ã;pk À 1X d k þ 1 P N i¼1 q i;k 0 N e Ã;pk r min ðHÞ, where r min ðHÞ represents the minimum singular value of H, showing that the formation controlling deviation The proof of Theorem 2 is completed.
Remark 6 A parameter tuning guideline for the proposed controller is summarized as follows: 1. For PPC, the transient performance of formation error can be determined by the initial value q 0 i;pk , and generally, a larger q 0 i;pk will result in a wider range and a smoother transient. And q 1 i;pk corresponds to the eventual boundary of controlling deviation, which should be adjusted larger than the equipped sensor resolution; k i;pk is a decaying factor of performance boundary. A faster convergence speed can be accomplished by increasing k i;pk . 2. For the presented SE-MLP, g i;v and g i;x should be chosen 2 to 3 times greater than the controlling parameters in the corresponding level to allow for a decoupling between tracking and learning channels. In addition, a huge adaptive learning parameter is usually required to permit a rapid learning of high-frequency uncertainties.

Simulation results
In the section, all the simulation results are carried out under MATLAB /SIMULINK environment with a sampling frequency being 50 Hz. The suggested algorithms, as shown in Fig. 2, are basically formulated as algebraic equations and differential dynamics, which are prone to be programed and performed via resorting to Interpreted MATLAB Fcn or S-Functions. For the purpose of validating the effectiveness and advantages of the derived results, we provide simulations on a networked system constituted by one virtual leader and three follower quadrotors, while pseudocode for suggested formation control policy is provided in Table 1 to illustrate the proposed control scheme clearly. Model parameters and external perturbations are collected in Table 2, where model parameters are borrowed from [51], which is well recognized and extensively adopted in academia as a benchmark data to testify the efficacy of quadrotor control design. And regarding external perturbations, to validate the proposed SE-MLP in terms of enhanced adaptiveness against fast time-varying uncertainties, we choose external perturbations as a compound formulation involving sine and cosine functions with different frequencies and amplitudes. And according to Theorem 2, the control gains of formation controller are given in Table 3 via a trial-and-error manner. It is worth mentioning that for PPC, select q 0 i;pk , q 1 i;pk , initial value of X i;p , and ensure initial formation synchronization error e i;pk ð0Þ satisfying Àr i;pk q i;pk ð0Þ\e i;pk ð0Þ\r i;pk q i;pk ð0Þ, then transform the constrained error dynamics e i;pk into an equivalent unconstrained one Z i;pk utilizing (31), and finally, formation synchronization error e i;pk ðtÞ can be constrained by Àr i;pk q i;pk ðtÞ\e i;pk ðtÞ\r i;pk q i;pk ðtÞ. For SE-MLP observer, the detailed training procedure is stated as follows: Firstly, setting the inputs and neurons of SE-MLP and configuring the center and width of neurons (see line 5 in Table 1). Then,Ŵ i;v and W i;x are online updated according to adaptive laws (7) and (50). As a consequence, the learning results are According to Assumption 1, to assure the coordinated control design and stability derivation, the designed communication topology should fulfill the subsequent property, i.e., there exists at least one path from the virtual leader to each follower; herein we just give a representative example shown in Fig. 3, where the weights of information exchange among quadrotors are selected as: b 1 ¼ 1, a 12 ¼ a 21 ¼ 1, and a 23 ¼ a 32 ¼ 1. Note that recalling the graph theory, one can know that b 1 ¼ 1 indicates that 1-th quadrotor has access to the leader, a 12 ¼ a 21 ¼ 1 denotes that these exists information exchange between 1-th and 2th quadrotors, and similarly, a 23 ¼ a 32 ¼ 1 declares that the information of 2-th and 3-th quadrotors can be obtained from each other. The desired relative position deviation is given by q 12 ¼ Àq 21 ¼ q 1 À q 2 ¼ ð½2; 0; 0 T À ½2 sinðÀp=6Þ; 2 cosðÀp=6Þ; 0 T Þ (m), q 13 ¼ Àq 31 ¼ q 1 À q 3 ¼ ð½2; 0; 0 T À ½2 sinðÀp=6Þ; 2 cos ð5p=6Þ; 0 T Þ m),q 23 ¼ q 2 À q 3 ¼ ð½2 sinðÀp=6Þ; 2 cos ðÀp=6Þ; 0 T À ½2 sinðÀp=6Þ; 2 cosð5p=6Þ; 0 T Þ (m). The initial conditions of three quadrotors are chosen as: X 1;p ð0Þ ¼ ½0; À5:5; 5:5 T (m), X 2;p ð0Þ ¼ ½À4:5; 0; 3:3 T (m), X 3;p ð0Þ ¼ ½À4; 0; À5:5 T (m), X 1;X ð0Þ ¼ ½0; 0; 0:2 T (rad), X 2;X ð0Þ ¼ ½0; 0; 0:2 T (rad), and X 3;X ð0Þ ¼ ½0; 0; 0:2 T (rad). And based on undirected communication topology devised in Fig. 3, the Laplacian matrix is  In the following, the desired leader's trajectory for follower quadrotors to track is X d ¼ ½0; 0; 9ð1 À e À0:3t Þ T ; t 9 ½10ð1 À cosð2pðt À 9Þ=23ÞÞ; 5 sinð4pðt À 9Þ=23Þ; 9ð1 À e À0:3t Þ T ; t [ 9: . . .; L. In the comparative simulation, modification parameter matrix r 1;x is set as diag 0:3; 0:3; 0:3 ð Þ , the center vector d is set as ½0; 0 T , the width / j is given as 70, the node number of hidden layer, i.e., L is selected as 9. In addition, to make the fairness of comparisons, the values of adaptive gains are respectively adjusted for Firstly, to validate the estimation performances against low-frequency disturbances, the external disturbance of roll channel of the first follower quadrotor is formulated as D 1;x1 ¼ 0:2ðsinðtÞ þ sinð0:5tÞÞ. Estimation performances of low-frequency disturbances using SE-MLP and MLP are described in Fig. 9. It is demonstrated that the presented SE-MLP observer can obtain a slightly better estimation for disturbances with smooth transients.
Next, to illustrate the necessity of employing big adaptive gains to acquire a rapid approximation ability, we modify the disturbance of roll channel of first follower quadrotor as a high-frequency one, i.e., D 1;x1 ¼ 0:2ðsinð8tÞ þ sinð4tÞÞ. As shown in Fig. 10, as the adaptive gain increases, although the steadystate estimation ability tends to be better for MLP, the transient behaviors aggravate with significant oscillations, whereas for the proposed SE-MLP observer, a smooth and rapid identification result can be obtained using a large adaptive gain without incurring highfrequency vibrations, implying that the proposed SE-MLP observer can further relax the constraints on the maximum learning rate of MLP and permit a relatively wider range of adaptive gain, which is more suitable for solving coordinated formation consensus considering fast time-varying uncertainties.
In addition, to demonstrate the effect of observer bandwidth on improving transient estimation performances, different bandwidths of state observer are used for the developed SE-MLP, and Fig. 11 shows that when the observer bandwidth is selected at least two times larger than that of controller gain, as depicted in the right column of Fig. 11, in this sense, the time-scale separation between control and learning loops satisfies, leading to smooth and fast learning profiles in disturbance estimates and control actions. Thus, one can increase the observer bandwidth to achieve a smooth transient performance, which is consistent with the results in Theorem 1.
Finally, to quantitatively evaluate the merits and demerits of proposed SE-MLP observer, Table 4 gives observation comparisons among RBF [39], MLP [41], and SE-MLP for disturbances with different frequencies; we can see that for the involved two cases, SE-MLP observer performs the best in reducing transient chattering without influencing the steady-state     where DT is the time interval and N is the number of recorded digital signals precision, which can be treated as the consequence of using observation errors, instead of tracking deviations to implement the weight learning, and in contrast with RBF [39] suffering from learning explosion, the computational burden of SE-MLP is obviously eased via using the artificial quantity rather than the weight vector, and it is acceptable that with the introduction of state observer, the design complexity is a bit more than that of MLP, which can be easily circumvented by the existing digital processors. Moreover, Table 5 collects transient performance comparisons using SE-MLP with different bandwidths, which further demonstrates that a large bandwidth can facilitate a smooth and fast learning, avoiding the undesired oscillations arising from strong coupling between tracking and learning loops. Consequently, the proposed SE-MLP observer can enable a rapid and smooth learning with a reduced computational complexity and chattering degree.

Conclusion
This paper deals with an adaptive formation consensus scheme for a group of quadrotors with prescribed performances subject to unknown uncertainties. Prescribed performance constraints acting on formation synchronization errors are explicitly considered in the controller design. By introducing the technique of MLP and constructing state estimators, the proposed SE-MLP observer not only can offer a smooth disturbance estimate even with a high adaptive learning rate, but also can avoid the issue of heavy calculation burden encountered in the available NN approximators. Finally, a distributed formation consensus approach is achieved to provide a robust formation configuration despite of unknown system perturbations. Comparative studies are illustrated to reveal the superiority and efficacy of the presented algorithm. Future direction will include an eventtriggered control design for multiple quadrotors to pursue a resource efficient tracking result.