Fuzzy Adaptive Finite-Time Cooperative Control With Input Saturation for Nonlinear Multiagent Systems and its Application

In this paper, the fuzzy adaptive finite-time cooperative control with input saturation (FAFTCCIS) is designed to quickly accomplish the cooperation of nonlinear multiagent systems (MASs) without the risk of bumping among agents. At least one agent must communicate with the leader and the information of neighborhood agents is required to accomplish the assigned task. Each agent, including the leader and the followers, is first approximated by <inline-formula> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> fuzzy-based linear subsystems. To accomplish the null cooperation error in finite time, the proposed adaptive control possesses the switching surface with an integration of fraction order, a time-varying switching gain, and an on-line learning of the upper bound of the uncertainties in each fuzzy subsystem of agent <inline-formula> <tex-math notation="LaTeX">$j$ </tex-math></inline-formula>. The stability of all the cooperative uncertain systems is then verified by the Lyapunov stability theory. Finally, the application to the cooperative control of intelligent chef is presented to confirm the effectiveness, robustness and feasibility of the proposed FAFTCCIS.


I. INTRODUCTION
Recently, many consensus, containment or cooperative controls of multiagent systems (MASs) are published [1]- [33]. These typical studies are briefly discussed in the next three paragraphs. Afterwards, their features are summarized to lead to the motivations, objectives, and contributions of this paper.
Based on local information that is measured or received from its neighbors and itself, a distributed consensus controller for each follower agent is designed by the fuzzy approximations of unknown nonlinear functions and one adaptive parameter to decay the effect of external disturbances [1]. Under a weighted undirected topology, the robust fixed-time consensus control for nonlinear MASs with uncertainties is investigated [2]. In [3], a multiagent collision avoidance problem is formulated by differential game to steer each agent from its initial position to a desired goal while avoiding collisions with obstacles and other agents.
The associate editor coordinating the review of this manuscript and approving it for publication was Venanzio Cichella .
Despite unknown nonaffine dynamics and mismatched uncertainties, a distributed neural adaptive control for containment achieves the dynamic containment in finite time with sufficient accuracy [4]. In [5], a distributed adaptive containment control for nonlinear MASs with input quantization is developed by employing a matrix factorization and normalization technique. In [6], the output consensus problem of the heterogeneous stochastic nonlinear MASs with directed communication topologies is developed by the fuzzy approximation of the unknown nonlinear functions of agents. Under the directed communication topology, the distributed adaptive output feedback consensus problem for linear MASs with the matched nonlinear functions and actuator bias faults is designed [7]. In [8], a joint objective of the distributed formation tracking control and learning/identification of the nonlinear uncertain autonomous underwater vehicle dynamics is designed. In [9], the cooperative global robust practical output regulation problem for a class of second-order uncertain nonlinear MASs is achieved by a distributed event-triggered state feedback VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ control strategy. In [10], the swarming behavior of multiple Euler-Lagrange uncertain systems with cooperationcompetition interactions is investigated. In [11], a neural network-based distributed adaptive finite-time control for unknown mismatched nonlinear dynamics is tackled by the modified command filtered backstepping with high order sliding mode differentiator. In [12], a sufficient condition for the local consensus in terms of linear matrix inequalities associated with the dynamics of the agents, the eigenvalues of Laplacian matrix, and the time-varying delay is derived. In [13], practical time-varying formation tracking problems for second-order nonlinear MASs with multiple leaders are addressed by adaptive neural networks. In [14], the formation-containment control for multiple multirotor unmanned aerial vehicle systems with the directed topologies are studied, where the states of leaders form the desired formation and the states of followers converge to the convex hull spanned by those of the leaders. Based on measurement error with a fast reaching law, an integral sliding mode-based robust finite-time event-triggered control is proposed to improve the convergence [15]. In [16], a group of nonholonomic unicycle-type agents with nonidentical constant speeds steers their orientations such that the circular motion center is adopted as a virtual position for each agent to achieve the desired formation. Based on neuro-adaptive approximation and a non-smooth feedback, the cooperative robust containment control of the multileader MASs subject to unknown nonlinear dynamics and external disturbances is designed [17]. In [18], the formation tracking control for a multi-agent system in constrained space with the communication limitations is designed. In [19], the leaderfollowing consensus problem of MASs using a distributed event-triggered impulsive control method is designed such that the controller of each agent is updated only when some state-dependent errors exceed a tolerable bound. In [20], the robust consensus tracking problem for a class of uncertain fractional-order MASs with a leader, whose input is unknown and bounded, is investigated. In [21], a cooperative fuzzy TS based dynamical sliding-mode controller is constructed. In [22], the consensus tracking problem of second-order nonlinear MASs with disturbance and actuator fault is tackled by the sliding mode control method.
In [23], an iterative learning control algorithm for the leader-follower formation tracking problem of a class of nonlinear MASs with actuator faults is designed. In [24], an adaptive asymptotic cooperative control scheme for nonlinear time-varying MASs is designed to simultaneously tolerate unknown actuator failures and unknown control directions. In [25], a tracking control with the switching formation in the nonomniscient constrained space for multi-agent system is designed in the restricted path. In [26], the event-based leader-following consensus problem for high-order nonlinear MASs, whose dynamics are in the strict feedback forms and satisfy Lipschitz condition, is tackled by the self-triggered dynamic output feedback control. In [27], a self-tuning adaptive observer based distributed control is proposed to preserve the connectivity of the initially connected rendezvous network, as well as to achieve the asymptotic tracking of leader's signal. An event-triggered rendezvous control method for the multiple two-wheeled mobile robots with the first order dynamics subject to time-varying communication delays is designed in [28]. In [29], the fuzzy adaptive distributed eventbased control scheme for a class of uncertain nonlinear MASs in the strict feedback form is designed. In [30], a kinematic model describing the interactions and evolutions of a swarm of mobile agents is proposed to create different forms of agent aggregation. In [31], the distributed adaptive output feedback leader-following consensus control for high-order nonlinear MASs with the unknown parameters and nonlinear terms is developed. Besides each follower is modeled by a nonlinear nonstrict feedback system, the unknown functions are approximated by RBFN to design an adaptive formation tracking control [32]. The large-scale unknown industrial processes are tackled by the reinforcement learning method and the multiagent game theory plant-wide performance optimization [33].
Since most of the above papers only adopt a simple dynamic system for each agent (e.g., the first or second order system with a strict feedback form), it is difficult to represent a practical system (e.g., the dynamics of robot manipulator, mobile robot, unmanned autonomous vehicle, industry complex process [33]). Moreover, these studies generally assume that the leader is the desired trajectory for the followers to track, which seems unpractical. Simply put, these papers are for the regulation problem or the single-input-single-output system or the finite-time bounded control, or without the saturated input's compensation or virtual leader which is not suitable for on-line sensor based path planning. To overcome the above shortcomings, each agent in this paper is approximated by N fuzzy-based linear subsystems to integrate the local behaviors of each agent. The same fuzzy sets of the system rule are applied to design the fuzzy adaptive finite-time cooperative control with input saturation. It is well-known that (adaptive) finite-time possesses the following properties: fast tracking and excess robustness but possibly large transient [34]- [38]. Under these circumstances, a fuzzy finite-time adaptive cooperative control with the saturated input, including a switching surface with fraction order, time-varying switching gains [38]- [40], and a learning law for two unknown coefficients of the relatively bounded above uncertainties in each fuzzy subsystem, is designed such that under suitable conditions the null cooperative error is achieved in finite time.
In this work, the lumped uncertainties are caused by the approximation error of fuzzy-model, the interaction dynamics resulting from the other subsystems, the saturated input of practical consideration, and the external disturbance from the other agents due to communication limitation. It is also assumed that these lumped uncertainties in each subsystem are bounded above by two coefficients multiplied by the switching surface's norm of agent j with exponents of one and positive fractional number. Subsequently, an adaptive law with e-modification and projection guarantees the boundedness of these learning coefficients if the value of switching surface is bounded. Since each subsystem is linear, the corresponding adaptive law can achieve a better convergence rate. After the effective learning of these coefficients, the on-line compensation of these lumped uncertainties in each subsystem of agent j is added into the proposed FAFTCCIS to accomplish the assigned cooperation of the multiple agents with nonlinear dynamics, which are described by fuzzy T-S models. Under appropriate scenarios, the finite-time convergence of switching surface and then cooperation error are achieved.
The main contributions of this paper are as follows. (i) Each agent, including the leader and the followers, is approximated by N fuzzy-based linear subsystems to design the proposed FAFTCCIS. It possesses the switching surface with fraction order, the time-varying switching gains, and the learning upper bound compensation of the uncertainties in each subsystem, so that under appropriate conditions the null cooperation pose error in finite time is achieved. (ii) The time-varying switching gain possesses a larger (smaller) value for the smaller (larger) switching surface value such that the saturated input can be postponed or avoided. (iii) Besides the application to intelligent chef, the stability of all the cooperative uncertain systems is verified by the Lyapunov stability theory.

II. MATHEMATICAL PRELIMINARIES
A fuzzy set A in Z is characterized by a membership function f A (z) which associates with each point in Z a real number in the interval [0, 1], with the value of f A (z) at z representing the ''grade of membership'' of z in A. The notation n×m denotes the sets of real matrices with dimension n × m. The symbol N j=1 f j = f 1 f 2 . . . f N represents a scalar multiplication. The symbols z and z 1 are a Euclidean norm and L1 norm of vector z, respectively; in addition, z 1 ≥ z . The notation · F is the Frobenius norm, i.e., W 2 F = tr W T W = tr WW T , where W ∈ n×m . The I m stands for a unit matrix of dimension m.
Relationship among agents is introduced as follows. A directed graph G consisting of M agents includes, A = {1, 2, . . . , M } denoting the node set of agents, E = {(k, j), where k, j ∈ A, k = j} standing for the set of undirected edges, and C = (c kj ) ∈ M ×M representing the relevant connected matrix. If the agent k and agent j can receive the information from each other, the edge (k, j) exists, i.e., (k, j) ∈ E. The agent k and agent j are the neighbors when the edge (k, j) exists. All the elements in the connected matrix C are nonnegative. If (k, j) ∈ E, then c kj > 0; if (k, j) / ∈ E or k = j, then c kj = 0. The directed graph G is connected if and only if there is at least one path between every pair of distinct agents in C, i.e., c kj = c jk . On the other hand, in the undirected graph c kj = c jk [1], [22]. In this work, agent 1 is called the leader, and agents 2, 3, . . . , M are called the followers. The leader cannot be affected by any of the followers; therefore, the leader is a special agent in the MASs. Moreover, only a few followers can receive the information from the leader. Define a nonnegative element c j1 standing for the communication from the follower j to the leader; then c j1 = 0, j = 2, 3, . . . , M . In this paper, it is supposed that at least one follower can receive the information of the leader to track and maintain the reference pose c 1j = 1, j = 2, 3, . . . , or M .

III. PROBLEM FORMULATION
Consider a class of nonlinear dynamic MASs: where z j (t) ∈ n denotes the system state of agent j which is available, u j (t) ∈ m represents the control input of agent j, A j (z j ) : n → n and B(z j ) : VOLUME 8, 2020 n → n×m are the known continuous mappings of agent j, and C j (z j , t) : n × + → n is the unknown but relatively bounded mapping of the uncertainties caused by uncertain system function of agent j or external disturbance from the other agents. Moreover, the saturated input is defined as sat (u j At the outset, a fuzzy dynamic model of agent j is employed to represent local linear input/output relations of nonlinear dynamic systems (1). That is, the i th rule of this fuzzy model for the nonlinear dynamic system of agent j is expressed as follows: System rule i: . . , M are also known and controllable. The saturated input is not included into fuzzy model but contained in the lumped uncertainties (10) such that the controller design is simplified. The output of the overall fuzzy system of agent j is inferred as the following form: Based on the approximation of fuzzy model of agent j, its approximation errors A j (z j ) and B j (z j ) are described as follows: To obtain the finite-time tracking ability, the following switching surface with fraction order is defined.
In fact, c kj = 1 or 0 denotes agent j will or won't track agent k, respectively; y kj d (t) is to maintain the desired cooperation avoiding any sort of collision between agent j and agent k. Moreover, c jj = 0, j = 2, 3, .., M , except c 11 = 1 and y k (t) = 0 forỹ 1 (t), i.e., the leader is only tracking the desired pose y 11 d (t). In addition, the values of c kj are not always fixed but at least one neighbor agent must communicate with the leader and other agents to complete a cooperation control. Here, P j and Q j are selected such that the linear dynamics is stable and its frequency response is shaped; in contrast, R j is chosen such that the finite-time stability to s j (t) = 0 is achieved. Furthermore, the integration of fraction order in (8) can make the steady-state control input smooth.
Suppose that the proposed fuzzy adaptive finite-time cooperative control with input saturation (FAFTCCIS) of agent j shares the same fuzzy sets with the fuzzy system (5).
Controller Rule i: Before discussing the controller design, the lumped uncertainties, resulting from the approximation error of fuzzymodel, the interaction dynamics resulting from the other subsystems, the saturated input of practical consideration, and the external disturbance from the other agents, are expressed as follows: wherez j (t) = (z j ) T (u j ) T T . Since the fact (5) and C j (z j , t) is relatively bounded, the upper bound of (10) is assumed as follows: . . , M are unknown, which are on-line learned. As the operating point is in the vicinity of switching surface, e.g., s j (t) 1, s j (t) β j i s j (t) , the right hand side of equation (11) covers all the value of s j (t) for the upper bound of uncertainties. Although the other agents can be connected with agent j, the corresponding uncertainties are distributively tackled. Finally, the overall control law of agent j is described as follows: The objectives of this paper are expressed as follows (cf. Fig. 1). (i) The system output of agent j, y j (t), is set to track the output of the leader or other agent with assigned pose to achieve zero cooperative error in finite time by avoiding collision with other agents and possessing smoothness in the steady-state control input. To achieve this objective, the proposed FAFTCCIS (9) and (12) includes the switching surface with the integration of fraction order (8), the time-varying switching gains (17) and (18), and the on-line  [43] based on the three-link planar robot [44] is presented to verify the effectiveness and robustness of the proposed control. (11) are unknown, the following adaptive laws are designed for the on-line compensation of the uncertainties for each subsystem i of agent j.

IV. FUZZY ADAPTIVE FINITE-TIME COOPERATIVE CONTROL WITH INPUT SATRATION
Hereσ Then, the proposed FAFTCCIS is designed as follows: u j i,eq (t) Here, P j D j B j i ∈ m×m is nonsingular for allỹ j (t), F j i ≥ λ f I m ≥ I m , ζ j i > 0, and λ j i satisfies the inequality: ∀i, j, z j (t). (16) Moreover, the time-varying switching gains j i,1 ( s j ), j i,2 ( s j ) > 0 ∈ m×m are the diagonal matrices and designed as  (17) and (18), as the operating point is near or away from the switching surface, a higher or smaller switching gain is produced, respectively, so that a saturation of control input can be postponed or avoided. Then, the excess robustness of the closed-loop system is achieved [38]- [40]. Before describing the system analysis, properties of the adaptive law (13) are discussed in the following lemma.
Lemma 2: The following relation is first achieved.
The following lemma discusses the computational complexity of (11) as compared with radial basis function neural network.
Lemma 3: Computational complexity of the upper bound of uncertainties in the right hand side of (11) is much smaller than that of radial basis function neural network.
Proof: Since s j (t) β i and s j (t) are already computed by the control law of (14) or (15), the multiplication and addition of the overall fuzzy system of all agents for the right-hand side of (11) are O (2NM ) and O (NM ) , respectively. In contrast, the RBFNN approximation of the lumped uncertainties includes [36]. Here, is the bipolar distributive number of each variable, which is an odd integer and at least 3; ''+1'' is for the dc biasing. Their corresponding multiplication and addition of the overall RBFNN approximation of all agents, are, respectively, O (hmNM ) and O ((h − 1)mNM ) , which are more complex than those of (11) since h = (n + m) + 1 2.
Q.E.D. Remark 2: As given in [45], the computation time using MLPN for the inverse kinematics of 6 DOF serial robot is over 7.5 times of that using RBFN. Even the successful implementation of adaptive MLPN based control [46], most of them need a larger computation time. Since the proposed method has less computation complexity than that in RBFNN, it can be generalized as having less computational complexity as compared with MLPN. Moreover, the designed adaptive compensation (13) is based on the implementation viewpoint such that it is effective and improved as compared with robust or fuzzy logic controls.
Subsequently, the key relationship of time-derivative of the switching surface for the controller design is derived. Using (4)- (12) and u j i,eq (t) in (14), the time derivative of the switching surface (8) is given as follows: where j i (z j , t) is described in (10). To deal with these uncertainties (11), the adaptive law (13) is applied to learn these coefficients. This adaptive law is simple as compared with the adaptive fuzzy control using the learning of a whole nonlinear function [1], [4]- [7], [13], [15], [27], [29], [32]. Then, the properties for the MAS with uncertainties using the proposed adaptive control are addressed as follows.
Theorem 1: Consider the nonlinear dynamic system (4) in the presence of the lumped uncertainties (10) satisfying the upper bound (11), the FAFTCCIS (14)-(15) with the adaptive law (13) and the time-varying switching gains (17) and (18). Then, the operating point of all agents converges to the switching surface of (8) in finite time, i.e., D d = s j (t), j = 1, 2, . . . , M s j (t) = 0 ∀j as t ≥ t 1 (22) where Here, t 0 is an initial time and V s = Proof: First, the following Lyapunov function is defined.
The nonlinear dynamics of robot j is given as follows: where N · m) are the disturbance torques. Then, each robot arm is approximated by the following 27 fuzzy models using (39) with the membership functions centered at −0.5π, 0, 0.5π as shown in Fig. 3.
2 ) 2 (±a for j = 1, 2). Once q j 2 is known, the solution for q j 1 is achieved by The desired elliptical trajectory for the leader is generated by the following equation: As shown in Fig. 4, the end effector of the leader (x 1 e , y 1 e , θ 1 e ) is planned to track the desired 2D pose (x d , y d , θ d ). On the other hand, the end effector of the follower (x 2 e , y 2 e , θ 2 e ) is planned to track the (x 1 e , y 1 e , θ 1 e ) with an assigned pose (x s , y s , θ s ). Hence, the required task given to the intelligent chef such as preparing or cooking foods is performed successfully. The desired trajectory is depicted in the red solid line of Fig. 5(c). Using the inverse kinematics equations of (44)- (46), the desired joint angles for the leader are plotted in the magenta solid line of Fig. 5(e). Similar result for the follower can be achieved (cf. the solid lines in Figs. 5(d) and 5(f)). To achieve the task of the intelligent chef for the system (39)-(42) with j = 1, 2 respectively for the leader and the follower, the assigned pose 0.2m between them is set as The assigned pose (x s , y s , θ s ) is required to avoid the collision between the agents when performing the assigned task and should be maintained throughout the cooperation period (cf. Figs. 5(a) and 5(b)).   Table 1, is shown in Fig. 5.
Based on the results of Fig. 5, the important observations are discussed as follows: (i) Since each agent (three-link planar robot) is effectively approximated by 27 fuzzy subsystems (39)- (41) with the advantageous feature of the finite-time convergence of cooperation, the task of intelligent chef (48)- (49) in the uncertain environment (50) and the saturated input u s = 30Nm is satisfactorily achieved (cf. Figs. 5(a)-(d)). Moreover, the responses of the joint angle (43)-(45) for robots 1 and 2 (cf. Figs. 5(e) and 5(f)) are adequately accomplished. (ii) Although the initial cooperative error and uncertainties exist, the control inputs in Fig. 5(g) possess a larger value and are saturated by u s = 30Nm due to the practical consideration. Since the switching surface (8) is designed as the integration of nonlinear fraction order, after transient the control inputs become smooth and the maximum steady-state control response is smaller than 55% of its saturated value. It implies that the further reduction of the saturated control input still can work but it will cause the slightly slow or poor tracking response. (iii) Even in the face of the larger uncertainties, the finite time (0.25s) convergence to the neighborhood of the switching surfaces in Fig. 5(h) is achieved. Since the output of robot 2 is designed to track the output of robot 1 with an assigned pose (49), the dynamics of robot 2 is slightly slower than that of robot 1. (iv) From Fig. 5(i), the upper bounds of the first coefficient of all fuzzy subsystems are larger than those in the second coefficient. It indicates that the operating points for all fuzzy systems of robots 1 and 2 indeed converge to the vicinity of their switching surfaces (cf. Fig. 5(h) and (11)) such that the desired cooperative task of intelligent chef is accomplished. (v) The cooperated task of intelligent chef must satisfy the constraint of workspace (42). More followers can be extended without difficulty. For simplicity, it is omitted. (vi) Since our leader is not virtual, it can track the on-line sensor based planning path (e.g., visual signal [43]). Simply put, the proposed method is practical and effective. Based on the developed theory and the successful simulation result, the proposed method can apply to many engineering problems, such as the satellite formation control for the development of 5G communication [47], [48], VOLUME 8, 2020  the swarm control for UAV with switching topology and obstacle avoidance ability [49]- [52].

VI. CONCLUSION
The cooperation of nonlinear multiagent dynamic systems is accomplished by the proposed fuzzy adaptive finite-time cooperative control with input saturation (FAFTCCIS). At least one agent must communicate with the leader for following the assigned pose and the information of neighborhood agents is also required to accomplish the cooperative task. As compared with most of previous studies, the leader is not necessary to be the desired trajectory to be tracked by all the followers. In contrast, each agent, including the leader and followers, is approximated by N fuzzy-based linear subsystems to design the proposed FAFTCCIS. To obtain the zero cooperation pose error in finite time and the smoothness of steady-state control response, the FAFTCCIS possesses the following advantageous features: the switching surface with the integration of fraction order, the time-varying switching gains, and the learning compensation for the upper bound of uncertainties in each subsystem. The successful application to intelligent chef confirms the effectiveness, robustness, and feasibility for many cooperation or formation problems of nonlinear multi-agent systems. Future studies include the integration of visual system, different mechanism design, and the required cooperation for cooking various foods [41]- [43], the formation control of satellite for the development of 5G communication [47], [48], and the swarm control of UAV with obstacle avoidance ability [49]- [52]. He was an Assistant Professor with the Computer Science Department, Tuskegee University, from 2009 to 2014. He was an Associate Professor with the Computer Science Department, Tuskegee University, from 2014 to 2017. Since 2017, he has been a Professor with the Computer Science Department, Tuskegee University, where he has been the Head of the Department, since 2019. His research interests are in the areas of mobile graphics, high-performance computing with graphics processing units, mobile security, robotics, and mobile computing.