Robust Decentralized Formation Tracking Control of Complex Multiagent Systems

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Introduction
Multiagent systems have gained popularity as a result of their ability to perform distributed sensing and actuating operations.In comparison with a single agent, they are more fexible, cost-efective, reliable, and robust while operating in an uncertain environment and are capable of handling complex tasks.Owing to these benefts, they are being adopted in a variety of applications, such as rescue and retrieval operations, intelligence, surveillance, and reconnaissance (ISR) missions, precision agriculture, space and planetary exploration, cooperative transportation, communication networking.[1,2].Typically, in these applications, the agent can be a robotic manipulator, an unmanned aerial vehicle, an underwater vehicle, a ground vehicle, etc.
In order to accomplish the aforementioned tasks, a cooperative controller is required such that the multiagent system attains the anticipated group behavior based on individual dynamic models and the information sharing among these agents [3].In literature, several linear, nonlinear, fuzzy, and observer-based techniques are employed to design cooperative controllers such as [4][5][6][7][8][9][10].Cooperative control can either be centralized or decentralized [11].Te centralized approach entails the control of all agents by a central station; consequently, each agent shall perpetually have knowledge of the central station's states and inputs.Te most fatal weakness of the centralized approach is its reliance on a central station, resulting in poor robustness, high computational requirements, and susceptibility to disruptions.In order to address these problems, the decentralized approach was introduced in which the central station is not present to manage the multiagent system [12].A controller is assigned to each agent in accordance with its local interaction in order to execute a global group mission [13].
In the formation tracking problem, convergence time is of signifcant importance in complex scenarios where there are constraints on the time available for accomplishing a particular task.For example, in some industrial applications, the operations are constrained by the production schedule; in ISR missions that often take place in hostile or challenging environments, such as battlefelds or disaster zones, the task execution time becomes crucial for the success of the mission.One of the viable solutions in such circumstances is to design a controller that ensures fnitetime stability, as discussed in [25,[32][33][34][35][36].In fnite-time control schemes, systems' states converge to a fxed value or a limit set within a fnite time interval.Consequently, fnitetime stability ofers high control precision and improved disturbance rejection attributes [37].Te existing fnite time control approaches employ the terminal sliding mode that sufers from singularity.Tis paper employs a linear sliding manifold to establish fnite time formation tracking control, and as a result, the singularity problem is mitigated.
In practice, the complexity of control design arises when the multiagent systems are subjected to particular constraints.Typical examples include the current through a branch of the synchronous circuit, keeping a self-driving car platoon inside the road boundary, or the fights of an unmanned aerial system may be restricted by speed, angle of attack, or position limits.Moreover, it is common practice to deploy multiagent systems in a confned space, where it is necessary to impose constraints on the agents' states [38].To handle this complexity, these constraints are fulflled by introducing the barrier Lyapunov functions (BLFs) in nonlinear controllers [39].In another article, the distributed consensus of the multiagent system under position constraints is studied using BLF in conjunction with adaptive backstepping [40].In another research, the consensus problem was addressed under input saturation constraints [41].BLF-based constrained nonlinear control along with a disturbance observer is presented for surface ships [9].In another study, an output synchronization problem with prescribed constraints is incorporated for nonlinear strict-feedback multiagent systems by utilizing a high-gain observer and adaptive controller [8].In another study, BLF is used in conjunction with sliding mode control (SMC) for fnite-time tracking of a single quadrotor under output constraints [42].Te linear sliding manifold is used in this work to achieve a fnite settling time of states.A similar approach is used in References [43][44][45][46][47] for fnite and fxed settling time control of single agents under constraints.It is essential to note that recent work also includes fxed-time or predefned-time controllers derived from fnite-time controllers [48][49][50].However, these approaches do not constitute the consideration of constraints.In general, existing research focuses on either fnite-time cooperative control or the performance of cooperative control under constraints.However, singularity-free fnite-time cooperative control of multiagent systems with output constraints and the efect of external disturbances is still an open problem.In this paper, we present the singularity-free formation tracking of multiagents in fnite settling time under output constraints.
From the preceding discussion, it is evident that in the literature, robustness, settling time, and output constraints are the primary performance matrices that are considered when dealing with multiagent systems' control.Comparing the existing robust cooperative control methods, it is to be noted that the majority of the research is carried out without considering the fnite convergence time or output constraints, as in references [15-19, 21-24, 27].Only works of references [25,26,[32][33][34][35][36]51] investigated the coordinated control with convergence time consideration.References [8,38,40,41] present the cooperative control under output constraints but only establish asymptotic convergence.A brief comparison of existing cooperative control methods is presented in Table 1.It demonstrates that none of the available techniques provide robust formation tracking in fnite time while maintaining output constraints, i.e., the literature lacks to present a controller that considers three performance matrices of robustness, fnite time, and output constraints, simultaneously.In this research, we investigate the formation tracking problem for nonlinear multiagent systems with output constraints under disturbances in fnite settling time.
Te subsequent points outline the main contributions of this paper: (1) A decentralized formation tracking control topology is introduced for nonlinear multiagent systems that is robust against the matched disturbances.
(2) Using the proposed control, the output of all the agents is guaranteed to stay within the user-specifed constraints under the directed communica tion graph.
(3) Singularity-free convergence of outputs within a fnite time interval is guaranteed, and the upper bound on the settling time is predetermined.
In a nutshell, this article presents a new formation tracking control framework for multiagent systems that simultaneously considers the three performance metrics of robustness, fnite-time convergence, and output constraints 2 Complexity while mitigating the singularity problem.To the best of the authors' knowledge, such a cooperative controller that considers these performance matrices simultaneously has not been reported in the literature.Tis article is organized as follows: Section 2 presents the essential defnitions and lemmas to be used in this work.Te problem statements with considered assumptions are described in Section 3, whereas Section 4 presents the proposed fnite settling time formation tracking control scheme.In Section 5, the MATLAB simulation-based results are discussed.Te conclusion of this article is given in Section 6.

Preliminaries
In this section, the preliminaries are defned which will be used throughout this article.Moreover, a concise summary of previously published fndings is also provided in this section of the paper.

Graph Teory.
In multiagent systems consider that there are N agents.Te communication graph G comprises a pair is the node set which denotes the vertex of the graph, and is the edge set that defnes the fow of information from the node ] j to node ] i .For an undirected graph, the set E(G) must satisfy the condition for any edge, then it is said to be a directed graph.In the case of an undirected graph, the degree of a node is defned as the number of edges that are incident to a node [30].In graph theory, a communication path is defned to be the sequence of the edges among the nodes.Whenever there exists at least one communication path among all nodes of the graph, the graph G is connected.For communication graph analytics, some of the matrices are very signifcant.In a diagonal matrix D, each diagonal element characterizes the degree of the respective node whereas in an adjacency matrix A, the nondiagonal elements describe the information fows from the neighboring nodes.Te communication graph is entirely defned by a semipositive defnite Laplacian matrix which is defned by

Notations.
In the course of this article, the following notations will be used: A n × m dimensional set of real matrices is denoted by R n×m ; sets of negative, nonnegative, and positive real numbers are denoted by R − , R 0+ , and R + , respectively; the N-dimensional identity matrix is represented by I N ; the construction of a diagonal matrix is defned by diag • { }; and l N denotes an N-dimensional column matrix with all entries equal to 1. Te Kronecker product operator is denoted by ⊗ and sgn(x) defnes the discontinuous scalar function, which is given by sgn Consider an autonomous single agent with a globally asymptotically stable origin defned as follows: where f: where x 0 � x(0) denotes the initial condition.Te defnitions below are taken from the literature and are provided here for the reader's convenience.Defnition 1 (fnite-time stability) [53,54].For the agent (2), the origin is said to achieve global fnite-time stability, if after time T(x 0 ) its solution converges to the origin, i.e., x(t, x 0 ) � 0, ∀t ≥ T(x 0 ).T(•) denotes the settling time function.
Lemma 2 (see [54,55]).Let V(t, x) be a radially unbounded and positive defnite function that is continuously defned on the open neighborhood Q around the origin, i.e., V(t, x): R + × Q ⟶ R n , and satisfes Ten, the agent ( 2) is said to be fnite-time stable.[45]: For the agent (2), the origin is said to achieve globally fnite settling time stability if the solution x(t, x 0 ) satisfes x(t, x 0 ) ≤ ϵ ∀ t ≥ T(x 0 ) where x 0 ∈ R − ∪ R + , ϵ is a bound such that ϵ ≈ 0, and fnite settling time is denoted by T(x 0 ).
Remark 4. According to Defnition 3, an agent is considered to achieve fnite settling time stability if the states of the agent approach the small bound ϵ ≈ 0 after some fnite-time T, rather than being exactly 0. Tis assumption takes the practical aspect into account since in tracking control problems, it is typically acceptable to have a small userdefned error margin ϵ (e.g., order of 10 − 2 ).In recent literature, this type of stability is also referred to as practical fnite-time stability [56].

Problem Formulation
We consider N agents in a multiagent system, and each agent has single-input single-output second-order dynamics, as considered in [3].Te multiagent system is defned by where X ≔ [x i , v i ] T ∈ Q⊆R 2×1 defnes the state vector; the initial conditions of each agent in the multiagent system.
Te main aim of this article is to propose a robust fnite settling time decentralized formation tracking control for the multiagent system (3) under output constraints.For this purpose, we defne the error variables as p i � x i − r and q i � v i − _ r for reference trajectory r and i � 1, 2, . . ., N. We where the vectors f ⋆ and g ⋆ are defned by . ., g(x N , v N )}, respectively.Consider the vector sliding surface s defned by where L � L + I N is a positive defnite matrix, β ∈ R + , and F � [F 1 , F 2 , . . ., F N ] T denotes the desired formation.Furthermore, we defne few variables that will be employed to construct the main results of this article, such as, p � η + F, q � _ p, i � 1, 2, . . ., N, and w ⋆ � [w 1 , w 2 , . . ., w N ] T .It is pertinent to mention here that to avoid singularities in the nominal control and disturbance compensator, all diagonal entries of g ⋆ must be diferent from zero.Te block diagram to explain the notations and signal fow of the proposed cooperative controller is given in Figure 1.
Te error bound is defned by h � B − (A 0 + F max ) and initially, all agents satisfy the symmetric constraint |p i (0) − F i | < h.Te pictorial illustration is given in Figure 2. Assumption 8. Te uniform upper bound W(t) on the matched perturbation is known, i.e., Remark 9. Tis paper deals with consensus-based formation tracking.Terefore, virtually, the consensus point of multiagent is assumed to track the reference r.

Problem Statement. Under directed communication,
propose a decentralized formation tracking control for a multiagent system (3) to (i) Achieve the desired formation within fnite settling time (ii) Constraint the output of all agents, i.e., − B < x i < B, ∀t ∈ R 0+ and i � 1, 2, . . ., N Provide the predetermined upper bound on the settling time, depending upon the initial conditions of the agents.

Finite-Time Formation Tracking Control
In this section, the main contribution of this article is presented.Te fnite-time formation tracking control is presented in Teorem 10. Figure 3 presents the overview of the proposed formation tracking controller.Te settling time function is derived in Corollary 11.

Theorem 10. Under Assumptions 5-8 the multiagent system (3) is guaranteed to achieve the desired decentralized formation tracking within a fnite settling time using the following control protocol:
where nominal control u n and disturbance compensator u c are, respectively, given by

Complexity
where s denotes the sliding surfaces defned in ( 5) and κ ∈ R + .Moreover, the outputs of all agents will satisfy the constraint |χ| < B ∀ t ≥ 0.
Proof.Consider a Lyapunov function Te derivative of the Lyapunov function results in Plugging _ ] and _ η from ( 4) and hence, Plugging the control law [7], Note that the second term in ( 14) is scalar and therefore sgn(s T )sgn(s) � N.Moreover, the Assumption 8 results in sgn(s T )w ⋆ − sgn(s T )sW(t) ≤ 0 and hence, Since, ln(h 2 /(h 2 − (Lp) 2 )) ≤ p T L T Lp/(h 2 − p T L T Lp) (Rens' inequality [57]), therefore, Using inequality, 6 Complexity Terefore, by virtue of Lemma 2, the sliding mode is attained in fnite time.Since the error is bounded by an upper bound h and exponential convergence of states is guaranteed after attaining sliding mode, therefore, it is straightforward to determine the maximum settling time that states require after attaining sliding mode.Tis results in fnite settling time stability.Moreover, by the construction of the Lyapunov function, it is evident that |p| < h, which eventually leads to |χ| < B based on Assumption 7. Te mathematical expressions for fnite settling time are provided in Corollary 11.

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Corollary 11.Te settling time function T of the closed-loop multiagent system using control law ( 6) is upper bounded by where V 0 denotes the value of Lyapunov function V at time t � 0 and is dependent on the initial conditions of agents, Laplacian matrix, and initial value of the reference.ϵ denotes the error margin as explained in Defnition 3.
Proof.Simplify the inequality (19) to solve the reaching time T r of states.By straightforward evaluation of ordinary diferential inequality where V 0 denotes the initial values of the Lyapunov function.By virtue of Defnition 1 and Lemma 2, V � 0; ∀ t ≥ T r (x 0 ), which implies that Once the states of each agent reach the respective sliding manifold, i.e., s � 0, it is guaranteed by the construction of the sliding surface that the output converges exponentially to the origin, i.e., p � p ⋆ 10 exp(− βLt), where p ⋆ 10 denotes the value of p when s � 0 is achieved.Subsequently, p ⋆ 10 < h ⊗ l N implies that the maximum settling time T s to establish p 1 < ϵ becomes

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Remark 12.It is noticeable that the settling time function T, of Corollary 11, can be predetermined, depending upon the initial conditions, communication graph, disturbance bound, and the control parameters.Te settling time function of Corollary 11 exhibits relatively more overestimation of the settling time, and it can be improved by knowing the exact knowledge of states at the time of attaining the sliding mode.
Remark 13.Table 2 elaborates on the selection criteria as well as the essence of all the formation tracking controller parameters: Remark 14. Excitation in the unmodeled closed loop dynamics, known as chattering, is caused by the highfrequency switching that occurs during the sliding phase (using the signum function).Actuators in the system can experience wear and tear due to chattering.While it is impossible to completely eliminate the chattering, the techniques outlined in [58] can help reduce it to a manageable level.Te interesting fact about these methods is that the controller is initially designed using the signum function, and then its approximations are used to diminish chattering.In this article, however, the chattering suppression is not taken into account, and the focus is kept on the design of a decentralized fnite settling time formation tracking controller with output constrained in the presence of disturbances.
Remark 15.By systematic extension of Teorem 10 and Corollary 11, the validation of the results of this paper can be employed to the square MIMO multiagent system; with each agent having n th order dynamics which are characterized by n/2 block diagonals, each block having a relative degree 2.

Simulations and Results
In order to validate the theoretical developments, two numerical examples are presented here.Four frictionless carts (agents) under external force and disturbances are considered, as illustrated in Figure 4. Assuming M � 1, the dynamics can be written as follows: Te directed communication graph is given in Figure 5. Te resulting Laplacian matrix L is as follows: In Example 1, a formation controller is devised for multiagent cart dynamics while the solution is extended to time-varying formation tracking control in Example 2.

Example 1 (Formation of Cart-Multiagent Dynamics).
While moving on a single axis under the efect of external force, the cart dynamics are in the form of double integrator dynamics (i.e., in (3), f(x i , v i ) � 0; g(x i , v i ) � 1).First, the simulations are carried out for formation control only (Example-2 is presented for evaluation of performance for the formation tracking control), and the initial positions of four carts are considered as x 0 � [− 2, 2, − 0.9, − 0.1] T and speeds as v 0 � [1, − 1, 0.5, − 0.5] T , respectively.
Te desired formation is F � [1, − 1, 3, − 3] T while reference is set to r � 0 ⇒ A 0 � 0.Moreover, for simulations, the disturbance is assumed to be w i � [0.25, 0.1, 0.5, 0.1] T .Te controller parameters are set to β � 2; κ � 1; W � 5; h � 3. From (18), the settling time function yields a maximum time of 14.11 s.Te resulting plots are given in Figures 6 and 7; it is evident from these Complexity  Complexity results that the settling time of the multiagent is 1.6 s, therefore, the upper bound evaluated using the settling time function is over-estimated.Te plot in Figure 6 shows the position and speed state of each cart starting from diferent initial conditions.Te position of each cart attains the desired formation and then sustains it for all future time.Furthermore, the position state satisfes the output constraints for all time.Figure 7 displays the control input for all agents.Te chattering is observed with the presented controller but is of a very small amplitude, and it is inevitable when working with sliding mode control as discussed in Remark 14.

Example 2 (Time-Varying Formation Tracking Control).
In this section, the time-varying formation tracking control is presented for cart-multiagent dynamics of Example-1.Here, the desired formation is time-dependent and given as F � 0.1 sin(2πt) × [1, − 1, 3, − 3] T while reference is set to r � cos(0.2πt)⇒ A 0 � 1. Te perturbation is assumed to be w i � sin(t) × [0.25, 0.1, 0.5, 0.1] T .Te controller parameters are set to β � 2; κ � 1; W � 5; h � 3. From (18), the upper bound on settling time shall be 14.30s.Te results (see Figures 8 and 9) show that there is no degradation in the performance of the proposed control for varying formations.Te constraint on the output is also satisfed for all time.
Te time-varying reference formation F is kept similar with just bias involved in the provided simulations for clear visualization; nonetheless, the controller performs equivalently well for alternative formations as well.Moreover, the reference r is exactly in the center of the solid-magenta and the dashed-black agent and the multiagent system attains the formation around that reference according to desired F. Figure 9 represents the control efort for the proposed time varying formation controller.From the position plot of Figure 8, it is to be noted that the cart multiagent system is not only tracking the lowfrequency reference but also has attained the highfrequency time varying formation.

Conclusion
A novel control topology with robustness and fnite-time formation tracking for complex multiagent systems, subjected to output constraints, is designed in this paper.It is established using BLF that the controller ensures that the output state never leaves that preassigned bound, provided it starts from that bound.As a result, the output bound during reaching phase of SMC is known and consequently, the fnite convergence time is achieved using linear sliding surface.Terefore, no singularity occurs in the proposed formation tracking controller unlike in the existing literature.Contrarily, nonlinear sliding surfaces are utilized in existing literature to achieve fnite time stability of the agents that causes the singularity in the control input.Te simulations are conducted for cart-multiagent dynamics, and the results show that the proposed decentralized controller performs well for the formation tracking of constrained agents in the presence of unwanted disturbances.Tus the strength of this work lies in providing a unique solution to the formation tracking control of multiagent systems that ofers output constraints, robustness, and fnite time convergence simultaneously without singularity issues.However, the settling time function exhibits relatively more overestimation and is the weakness of this cooperative control framework.Te future directions for this study include exploring the nonsymmetric constraints, predefned settling time, and higher-order multiagent systems.Moreover, a parallel study will be carried out to design a fnite-time distributed cooperative control under output constraints.

Figure 1 :Figure 2 :
Figure 1: Block diagram of the proposed formation tracking controller.

Figure 9 :
Figure 9: Control inputs for time-varying formation tracking of cart-multiagent system.

Table 1 :
Comparison of existing cooperative control methods.