Fixed-Time Cooperative Tracking Control for Double-Integrator Multi-Agent Systems: A Time-Based Generator Approach
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Fixed-Time Cooperative Tracking Control forDouble-Integrator Multi-Agent Systems: ATime-Based Generator Approach
Qiang Chen, Yu Zhao, Guanghui Wen, Guoqing Shi and Xinghuo Yu
Abstract —In this paper, both the fixed-time distributed con-sensus tracking and the fixed-time distributed average trackingproblems for double-integrator-type multi-agent systems withbounded input disturbances are studied, respectively. Firstly, anew practical robust fixed-time sliding mode control methodbased on the time-based generator is proposed. Secondly, afixed-time distributed consensus tracking observer for double-integrator-type multi-agent systems is designed to estimate thestate disagreements between the leader and the followers underundirected and directed communication, respectively. Thirdly,a fixed-time distributed average tracking observer for double-integrator-type multi-agent systems is designed to measure theaverage value of reference signals under undirected communica-tion. Note that both the observers for the distributed consensustracking and the distributed average tracking are devised basedon time-based generators and can be extended to that of high-order multi-agent systems trivially. Furthermore, by combing thefixed-time sliding mode control with the fixed-time observers, thefixed-time controllers are designed to solve the distributed con-sensus tracking and the distributed average tracking problems.Finally, a few numerical simulations are shown to verify theresults.
Index Terms —Fixed-time, sliding mode control, time-basedgenerator, distributed observer, consensus tracking, distributedaverage tracking.
I. I
NTRODUCTION
Distributed cooperation control has been a popular scientificresearch issue over the past decades owing to its significantvalue in reality such as distributed optimization [1], [2],tracking control [3]–[5], flocking and containment control [6]–[8].In the distributed cooperation control of a flock of agentswith local interactions, a premier task is to design an algo-rithm which makes each agent achieve consensus in position,velocity and so on. The consensus algorithm for single-integrator multi-agent systems was first developed in [9],and then some sufficient and necessary conditions for theconsensus of double-integrator multi-agent systems were gen-eralized in [10]. Distributed tracking control can be regarded
Q. Chen is with the School of Electronics and Information and the schoolof Automation, Northwestern Polytechnical University, Xi’an 710129, China(e-mail: [email protected]).Y. Zhao is with the School of Automation, Northwestern PolytechnicalUniversity, Xi’an 710129, China (Corresponding author).G. Wen is with the School of Mathematics, Southeast University, Nanjing210096, China.G. Shi is with the School of Electronics and Information, NorthwesternPolytechnical University, Xi’an 710129, China.X. Yu is with the School of Engineering, RMIT University, Melbourne,VIC 3001, Australia. as an extension of generalized consensus control, in whichthe followers not only have to reach consensus, but alsoto follow with the specified trajectory. For example, in thedistributed consensus tracking and distributed average track-ing, the target trajectories are the states of the leader andthe average value of multiple reference signals, respectively.However, in distributed algorithms, only a few or none of theagents can acquire the target information directly. Therefore, afrequently-used method to measure the target information is toestablish a distributed observer. An observer-based algorithmfor nonlinear agents was proposed in [11], [12] to achievedistributed consensus tracking. In [13]–[15], the distributedobservers were designed to measure the average value ofreference signals. However, these protocols are asymptoticallystable, which implies that the upper-bounded convergence timeis not guaranteed. So as to estimate the precise upper-boundedconvergence time, the finite-time observers relying on initialconditions were proposed in [16]–[18]. Unfortunately, in someengineering practices, the initial states are not available orthe convergent rate has to be faster. Therefore developing thefast converging algorithms without dependence on initial statesis quite necessary. The fixed-time stability strategy was firstinvestigated in [19], in which the prerequisite of initial con-ditions was eliminated. Some novel fixed-time algorithms forsingle-integrator multi-agent systems were developed in [20].A fixed-time observer for double-integrator-type multi-agentsystems to estimate the states of the leader was designed in[21] under undirected communication topology. Then A fixed-time directed edition for high-order integrator-type multi-agentsystems was developed in [22]; Although, after the systemstates converging into the unit circle in the last step it isasymptotically stable, this method provides some inspirationsfor the protocol in this paper.After estimating the task trajectory in a fixed-time, the nextstep is to devise the fixed-time controller for the agent totrack the target trajectory. For double-integrator-type systems,sliding mode control is a type of classic nonlinear controlprotocol [23], which has the advantages of fast response,parameter change, insensitive to disturbance and simple phys-ical implementation. Some finite-time sliding mode controlmethods were proposed in [24], [25]. However, the processof extending it to that of double-integrator-type systems isnontrivial due to the singular problem. An attempt to designthe fixed-time sliding mode control protocol was made in [26]by utilizing a sinusoid function to offset the singularity in theneighborhood of zero, but it leads to a little uncertainty of the
EEE TRANSACTIONS ON 2 convergence time. The price of reducing the uncertainty is asharp rise of the control input.Besides the conventional fixed-time protocol that usuallyuse two feedback terms, another popular technique in terminalcontrol is the time-based generator technique, which wasinduced in [27] to induce the attraction of force fields. In [28],an finite-time sliding mode surface is designed, but it can’taccurately track the trajectory and has no robustness. In [29], apredefined-time control method for single-input single-outputcontrollable linear systems was proposed. A novel fixed-timeconsensus strategy for single-integrator multi-agent systemswas developed in [30]. Furthermore, it proved that the fixed-time protocol based on the time-based generators had a lessmagnitude of control inputs.As for the time synchronization between different agents,the clock synchronization device has been proposed in theexisting paper [31], [32] to ensure the time synchronization.Therefore, it is not repeated here.Motivated by the above results, by utilizing the time-basedgenerator technique, five main contributions are made in thispaper. Firstly, a new fixed-time nonsingular sliding modecontrol method is developed, which can precisely predesignthe upper-bounded convergence time without dependence oninitial states and has a less magnitude of control inputscompared with the conventional ones [21], [26]. Secondly, anew fixed-time distributed observer under undirected topologyis proposed to evaluate the state disagreements between theleader and the followers. Thirdly, inspired by [22], the observerfor undirected communication systems is extended to thesystems with directed communication, but what’s different isthat the observer in this paper is a fully fixed-time protocolwith the precise upper-bounded convergence time. Note that,all the observers proposed in this paper can be extended tothat of high-order multi-agent systems trivially. Moreover,by combining the sliding mode control protocol with thedistributed consensus tracking observers, two fixed-time con-trollers are developed which successfully extended the fixed-time distributed consensus tracking algorithms based on time-based generators for single-integrator-type multi-agent systemsin [30] to the double ones. More importantly, the disturbanceis considered in this paper, which is of great significance inpractice. Finally, a controller is given to solve the fixed-timedistributed average tracking problems for double-integrator-type multi-agent systems. As far as I am concerned, there isno other fixed-time distributed average tracking algorithm fordouble-integrator-type multi-agent systems.The rest of this paper is given as below. In section II,some mathematical preliminaries were given. In section III,the fixed-time sliding mode control protocol is investigated.Next, the observers for distributed consensus tracking underboth undirected and directed graph are designed. Then theobserver for distributed average tracking under undirectedgraph is designed. Furthermore, the distributed consensustracking and the distributed average tracking algorithms aregiven. In section IV, several simulations are given. In sectionV, a few conclusions are made. II. M
ATHEMATICAL PRELIMINARIES
A. Notations
The real number set and the N-dimensional real vectorspace are denoted by R and R n , respectively. The signumfunction is represented by sgn( · ) and its vector form can bewritten as sgn( z ) = [sgn( z ) , sgn( z ) , ..., sgn( z n )] T , where z = [ z , z , ..., z n ] T . Let | · | stand for the absolute valueof a scalar. The vector q-norm can be written as k z k q =( | z | q + | z | q + ... + | z n | q ) q . Let λ ( Q ) and λ ( Q ) representthe smallest and the second smallest eigenvalues of the matrix Q , respectively. B. Graph Theory
The communication topology of a group of n + 1 agentscan be represented by a graph G . If there is a leader in them,the other n agents can be expressed as a subgraph G s . Theweighted graph G = ( V , E ) is constructed with a set of nodes V = { v , v , ..., v n +1 } and a set of edges E = { e , e , ..., e m } .A directed edge from v j to v i can be denoted as ( v i , v j ) ,which means v i can receive information from v j . An directedpath from v j to v i consists of a sequence of edges in theform of E ij = { ( v i , v i +1 ) , ..., ( v j − , v j ) } , which means theinformation can flow from v j to v i . When replacing thedirected edges by the undirected, it becomes undirected pathand the information flow is bidirectional. It is said to contain aspanning tree if at leat there exists a node which has directedpaths to all other nodes. The undirected graph is connectedif and only if there at least exists an undirected path betweenany two notes. Let A = [ a ij ] ∈ R n × n and D ∈ R n × m denotethe adjacency matrix and the incidence matrix of the graphrespectively, and a ij = 1 if there exists a directed edge from v j to v i , else a ij = 0 . With regard to undirected graphs, a ij = a ji . let O = [ o ij ] ∈ R n × n denote the degree matrixand o ii = P nj =1 a ij , else p ij = 0 . Then the Laplacian matrixis written as L ∈ R n × n = O − A . Set a i = 0 and a i = 1 if the agent i can acquire information from the leader, else a i = 0 and then set B = diag { a i , a i , ..., a in } . C. Time-Based Generator
The time-based generator ξ ( t ) is a kind of time dependentfunction that can be seen as a termination function. Its generalproperties can be generalized as follows.1) ξ ( t ) is a non-decreasing and continuous function.2) With time going by, ξ ( t ) increases from the initial state ξ (0) = 0 to ξ ( t s ) = 1 , and when t > t s , ξ ( t ) ≡ , where t s can be predesigned arbitrarily.3) ˙ ξ (0) = 0 and when t ≥ t s , ˙ ξ ( t ) ≡ . Remark 1.
A typical time-based generator function ξ ( t ) ispresented as follows [30]. ( ξ ( t ) = t s t − t s t + t s t , ≤ t ≤ t s , , t > t s , Think about the differential equation as below. ˙ z = − h ( t ) z, z (0) = z , (1) EEE TRANSACTIONS ON 3 where h ( t ) is constructed as h ( t ) = k ˙ ξ − ξ + δ , (2)where k ∈ R and δ ∈ R are two positive constants whichsatisfy k > and < δ << .Solving the differential equation (1) one has z = ( 1 − ξ + δ δ ) k z . (3)With t growing from to t s , ξ grows from to smoothly.Therefore, when t ∈ [0 , t s ) , z gradually approaches z ( δ δ ) k .When t ≥ t s , the result will remain the same. If let δ = 0 . and k = 3 , at the terminal moment t s , the solution of (1) willbe z = 10 − z . Thus we can nearly think that z reaches zeroat t s and the initial state z has no effect on the convergencetime. D. Problem Description1) Fixed-Time Distributed Consensus Tracking:
Supposethat there is a double-integrator-type multi-agent system witha leader and n agents. The leader can be represented by ( ˙ x = v , ˙ v = u , (4)where x ∈ R and v ∈ R represent the position and velocityof the leader, respectively. u ∈ R represents the control inputbounded by a positive constant u max .Then the followers can be modeled by ( ˙ x i = v i , ˙ v i = u i + d i , i = 1 , , ..., n, (5)where x i ∈ R and v i ∈ R denote the position and the velocityof the agent i , respectively. u i ∈ R and d i ∈ R denote thecontrol input and the uncertainty, where d i takes the positiveconstant d max as the boundary.The objective of fixed-time distributed consensus trackingis to devise the control input only using local information foreach follower, which enable the followers to achieve consensuswith the leader in a fixed time independent of initial states. Definition 1. (Fixed-time distributed consensus tracking) Forthe system described by (4) and (5) , with the given observerand control input u i , it is said to achieve fixed-time distributedconsensus tracking if all the followers can achieve consensuswith the leader in a fixed-time T max independent of initialconditions, i.e., ( lim t → T max | x i − x | + | v i − v | ≤ c lim t →∞ | x i − x | + | v i − v | = 0 , (6) where T max can be predesignated arbitrarily independent ofinitial conditions and c can be limited to the desired level. 2) Fixed-Time Distributed Average Tracking: Consider adouble-integrator-type multi-agent system with n agents repre-sented by (5) , and each agent i has a reference signal r i ∈ R described as follows. ( ˙ r i = f i , ˙ f i = a ri , i = 1 , , ..., n, (7)where f i and a ri are the velocity and acceleration of referencesignal r i , respectively. Note that a ri is bounded by a positiveconstant a max . Let r = n P ni =1 r i , f = n P ni =1 f i and a = n P ni =1 f i be the average value of the reference signals.The objective of fixed-time distributed average tracking isto devise control inputs only using local information for theagents, which enable them to achieve consensus with theaverage value of multiple reference signals in a fixed timewithout dependence on initial states. Definition 2. (Fixed-time distributed average tracking) For thesystem described by (5) and (7) , with the given observer andcontrol input u i , it is said to achieve fixed-time distributedaverage tracking if all the agents can achieve consensuswith the average value of the multiple reference signals ina fixed-time T max which can be predesigned arbitrarily andindependent of initial states, i.e., ( lim t → T max | x i − r | + | v i − f | ≤ c lim t →∞ | x i − r | + | v i − r | = 0 , (8)III. M AIN R ESULTS
A. Fixed-Time Sliding Mode Control
Lemma 1. [22] Suppose that z (0) = z and V ( z ) isa positive definite Lyapunov candidate which satisfies theinequality as below. ˙ V ( z ) + µV ν ( z ) ≤ , (9) where µ ≥ and ν ∈ (0 , . Then z will converge to zero ina finite time T ( z ) such that T ( z ) ≤ µ (1 − ν ) V − ν ( z ) . (10)A typical double-integrator-type control system is given asfollows. ( ˙ z = z , ˙ z = u + ̺, (11)where z ∈ R and z ∈ R are the system states. δ ∈ R is adisturbance bounded by a positive constant ̺ max .The objective of fixed-time sliding mode control is todevise a control input u which drives the system (11) to theequilibrium point in a fixed time, i.e., [ z , z ] = [0 , . Theprocess of fixed-time double-integrator sliding mode controlis generally divided into two sections. In the first section,the control input forces the system to arrive at the prescribedsurface in a fixed time t a ; In the second section, the systemwill slide along the surface to the equilibrium point in afixed time t a . Therefore, The whole convergence time isbounded by T a = t a + t a . In order to converge in thefixed time in each stage, two time-based generators ξ a and EEE TRANSACTIONS ON 4 ξ a are used sequentially. ξ a ensures the system to arrive atthe prescribed surface in t a and then invalid. ξ a guaranteesthe fixed convergence time t a . Let h a ( t ) = k ˙ ξ a − ξ a + δ and h a ( t ) = k ˙ ξ a − ξ a + δ . Then we have h ( t ) = h a ( t ) , t ∈ [0 , t a ) ,h a ( t ) , t ∈ [ t a , t a + t a ) , , t ∈ [ t a + t a , + ∞ ) . (12) Remark 2.
Since ˙ ξ a (0) = ˙ ξ a ( t a ) = ˙ ξ a ( t a ) = ˙ ξ a ( t a + t a ) = 0 , one obtains h a (0) = h a ( t a ) = h a ( t a ) = h a ( t a + t a ) = 0 , which shows the connectivity of h ( t ) .Furthermore, owing to the nonnegativity of ˙ ξ a and ˙ ξ a , h ( t ) is also nonnegative. Remark 3.
In order to clarify the idea, let ξ a , h a ( t ) and ξ a , h a ( t ) have the same structure. But, in simulation, owingto ξ a ( t a ) = 1 and ξ a ( t a ) = 0 , which leads to a sharpdecrease of the derivative and causes problems. By resetting ˆ ξ a ( t ) = ξ a + 1 and ˆ h a ( t ) = k ˙ ξ a − ξ a + δ , the problem causedby discontinuity is solved. Moreover, in the different steps, k and δ can be selected as different constants respectively. In this paper, the fixed-time sliding mode surface is selectedas s = ( 12 h ( t ) + 1) z + z . (13)If s = 0 , the system arrives at the sliding mode surface andhas the form as below. z = ˙ z = − ( 12 h ( t ) + 1) z . (14)The control input is devised as follows. u = −
12 ˙ h ( t ) z − ( 12 h ( t ) + 1) z − h ( t ) s − ρ sgn( s ) , (15)where ρ is a positive constant satisfying ρ ≥ | ̺ max | + 1 . Theorem 1.
With the given control input (15), the system (11)will arrive at the sliding mode surface ( s = 0 ) in a fixed-time t a , and then slide along the surface ( s = 0 ) to the equilibriumpoint [ z , z ] = [0 , in a fixed-time t a . Thus the final upper-bounded convergence time is T a = t a + t a .Proof. The Lyapunov candidate is constructed as V = s .Differentiating (13) against time one has ˙ s = 12 ˙ h ( t ) z + ( 12 h ( t ) + 1) z + u + ̺. (16)Substituting the control input (15) in to (16) one has ˙ s = − h ( t ) s − ρ sgn( s ) + ̺. (17) Differentiating the Lyapunov candidate V against time andthen substituting (17) into it one has ˙ V = s ˙ s = − h ( t ) s − ρ | s | + ̺s ≤ − h ( t ) s − ( ρ − ̺ max ) | s |≤ − h ( t ) s = − h ( t ) V . (18)When t ∈ [0 , t a ) , h ( t ) = h a ( t ) . According to (1) oneobtains lim t → t a V ≤ ( 1 − ξ a + δ δ ) k V (0) = ( δ δ ) k V (0) , (19)where, according to (3), ( δ δ ) k V (0) is in the near region ofzero.When t ≥ t a , h ( t ) = h a ( t ) and one obtains ˙ V = − h a ( t ) s − ρ | s | + δs ≤ − ( ρ − δ max ) | s |≤ − | s | = − p V (20)According to Lemma 1 one has that V will converge to zeroafter t a within a finite time ˆ t a , i.e., ˆ t a ≤ p V ( t a ) ≤ q V (0)( δ δ ) k . Although V doesn’t converge to zero per-fectly at t a , which means that the system states are in thenear region of the sliding mode surface ( s = 0 ), the systemstates will still converge to zero along the sliding surface inthe fixed-time. In order to clarify the idea clearly, at first, it isassumed that V converges to zero at t a , which means s = 0 as well. Then the case that there is a small error between thesliding surface and the system states at t a is investigated.A Lyapunov candidate is constructed as V = z . Differ-entiating V along (14) one has ˙ V = z ˙ z = − h ( t ) z − z ≤ − h ( t ) z = − h ( t ) V (21)From (1) one obtains that lim t → t a + t a V = ( δ δ ) k V ( t a ) .When t ≥ t a + t a , owing to h ( t ) = 0 and ˙ V = − z = − V one concludes that V will converge to zeroexponentially. Since z = − ( h ( t ) + 1) z = − z , z willconverge to zero with the same rate of z as well.In the following proof, the influence of V ( t a ) = 0 isanalysed. According to the relationship between V and s ,suppose that the system states converge to the adjacent regionof the sliding surface at t a and there exists an error e , i.e., s = { e | t ≥ t a } . When t ≥ t a , ˙ V ≤ −| s | , which means V as well as | e | are non-increasing functions and bounded by EEE TRANSACTIONS ON 5 ˆ V = ( δ δ ) k V (0) and ˆ | e | = q δ δ ) k V (0) , respectively.Then (13) can be rewritten as e = ( 12 h ( t ) + 1) z + z . (22)The derivative of z can be obtained as ˙ z = − ( 12 h ( t ) + 1) z + e. (23)Substituting (23) into ˙ V one has ˙ V = − h ( t ) z − z + ez = − h ( t ) V − z + ez . (24)Note that | e | is very small and non-increasing, and the conver-gence time is bounded by ˆ t a . If | z | < | e | , which means that | z | has been in the near region of zero. Meanwhile, | z | isbounded by ˆ | e | . After t a +ˆ t a , due to | e | = 0 , | z | will at leastconverge exponentially. If | z | ≥ | e | , one has ˙ V ≤ − h ( t ) V .From (1), V will nearly converge to zero within a fixed time t a or converge to | ˆ e | in ˆ t a , Whatever the case may be, V will converge nearly to zero in a fixed time T a = t a + t a .Then the proof has been completed. B. Fixed-Time Distributed Consensus Tracking Observer Un-der Undirected Communication
Assumption 1.
The topology subgraph G s for the followersis undirected and connected; There at least exists a followerwhich can acquire information from the leader. Lemma 2. [21] If L ∈ R n × n is the Laplacian matrix ofa undirected connected graph, and the nonnegative diagonalmatrix B = diag { a , ..., a n } with at least one elementgreater than zero, then Q = L + B is a positive definite matrix. In the subsection, a fixed-time distributed observer basedon time-based generators is designed for each follower tomeasure the relative position and velocity disagreements be-tween the leader and itself under undirected communication.set two time-based generators as ξ b and ξ b , and then h b ( t ) = k ˙ ξ b − ξ b + δ and h b ( t ) = k ˙ ξ b − ξ b + δ . Let h ( t ) has thesame structure as h ( t ) . Denote the real tracking errors as ˜ x i = x i − x and ˜ v i = v i − v . Then fixed-time distributedthe observers α i and β i of estimating ˜ x i and ˜ v i are proposedas below. ˙ α i = β i − b h ( t ) (cid:26) n X j =0 a ij [( α i − α j ) − ( x i − x j )] (cid:27) − b sgn (cid:26) n X j =0 a ij [( α i − α j ) − ( x i − x j )] (cid:27) , ˙ β i = u i − c h ( t ) (cid:26) n X j =0 a ij [( β i − β j ) − ( v i − v j )] (cid:27) − c sgn (cid:26) n X j =0 a ij [( β i − β j ) − ( v i − v j )] (cid:27) , (25)where i = 1 , ..., n , α = 0 , β = 0 . b , b , c and c arepositive constants satisfying b = c ≥ λ ( Q ) , b ≥ and c > u max + d max . Let ˜ α i = α i − ˜ x i and ˜ β i = β i − ˜ v i be the errors between theobserving disagreements and the real disagreements. If all theerrors converge to zero in t b + t b , the observer is designedsuccessfully. Theorem 2.
With the given dynamics (4), (5) and observer(25), under Assumption 1, α i and β i converges to ˜ x i and ˜ v i within a fixed-time T b = t b + t b .Proof. Following from (25), ˙˜ α i and ˙˜ β i can be written as ˙˜ α i = ˜ β i − b h ( t ) n X j =0 a ij (˜ α i − ˜ α j ) − b sgn (cid:20) n X j =0 a ij (˜ α i − ˜ α j ) (cid:21) , ˙˜ β i = − c h ( t ) n X j =0 a ij ( ˜ β i − ˜ β j ) − c sgn (cid:20) n X j =0 a ij ( ˜ β i − ˜ β j ) (cid:21) − d i + u . (26)Let ˜ α = [˜ α , ..., ˜ α n ] T , ˜ β = [ ˜ β , ..., ˜ β n ] T , d = [ d , ..., d n ] T and u ∈ R n = [ u , ..., u ] T . The vector form of (26) can bewritten as ˙˜ α = ˜ β − b h ( t ) Q ˜ α − b sgn( Q ˜ α ) , ˙˜ β = − c h ( t ) Q ˜ β − c sgn( Q ˜ β ) − d + u, (27)where Q = L + B is a positive definite matrix according toLemma 2.Construct a Lyapunov candidate as V = ˜ β T Q ˜ β . Because Q is a positive definite matrix, V is well defined. Differentiate V against time such that ˙ V = ˜ β T Q [ − c h ( t ) Q ˜ β − c sgn( Q ˜ β ) − d + u ] ≤ − c h ( t )( Q ˜ β ) T Q ( Q ˜ β ) − ( c − u max − d max ) || Q ˜ β || ≤ − c λ ( Q ) h ( t ) ˜ β T Q ˜ β ≤ − h ( t ) V . (28)When t ∈ [0 , t b ) , h ( t ) = h b ( t ) . According to (1) oneconcludes that lim t → t b V ≤ ( δ δ ) k V (0) << . Then dueto V = 12 ˜ β T ( L + B ) ˜ β = 14 n X i =1 n X j =1 a ij ( ˜ β i − ˜ β j ) + 12 n X i =1 a i ˜ β i , (29)one has that lim t → t b | ˜ β i | ≤ q ( δ δ ) k V (0) << .When t ≥ t b , one has ˙ V = − ( θ − u max − d max ) || Q ˜ β || . (30)Owing to || Q ˜ β || ≥ || Q ˜ β || = q ( Q ˜ β ) T Q ˜ β ≥ q λ ( Q ) ˜ β T Q ˜ β, (31)one obtains ˙ V ≤ − ( θ − u max − d max ) p λ ( Q ) V ≤ . (32) EEE TRANSACTIONS ON 6
Therefore, V will keep decreasing and | ˜ β i | << is ensured.Following from Lemma 1, when t ≥ t b , V will convergeto zero within a finite time ˆ t b , i.e., ˆ t b ≤ ( θ − u max − d max ) q ( δ δ ) k V (0) λ ( Q ) .Construct a Lyapunov candidate as V = ˜ α T Q ˜ α . Differ-entiate it against time and then one has ˙ V = ˜ α T Q ˙˜ α = ˜ α T Q ˜ β − b h ( t )˜ α T QQ ˜ α − b ˜ α T Q sgn( Q ˜ α )= − b h ( t )˜ α T QQ ˜ α − b || Q ˜ α || + ( Q ˜ α ) T ˜ β. (33)Since when t ≥ t b , | ˜ β i | << , one has ˙ V ≤ − b λ ( Q ) h ( t )˜ α T Q ˜ α − ( b − || Q ˜ α || ≤ − h ( t ) V . (34)When t ∈ [ t b , t b + t b ) , h ( t ) = h b ( t ) . According to (1), oneconcludes that lim t → t b + t b V ≤ ( δ δ ) k V ( t b ) ≈ . When t ≥ t b + t b , due to h ( t ) = 0 and ˜ β = 0 , from (33) onehas ˙ V = − b || Q ˜ α || . Compared with (32), one concludesthat V will converge to zero within finite time after t b + t b .That also means that the observer successfully complete theobserving task in a fixed time T b = t b + t b . Thus the wholeproof has been finished. C. Fixed-Time Distributed Consensus Tracking Observer Un-der Directed Communication
Assumption 2.
There is a spanning tree in the directed graph G , where the leader is set as the root node. Note that, thesubgraph G s doesn’t need to be strongly connected or containa spanning tree. Lemma 3. [33] Under Assumption 2, define H = L + B , p = [ p , ..., p n ] T = H − T n , P = diag { p i } , Q = H T P + P H .Then we have that P and Q are both positive definite. Let d = | max i { P nj =0 a ij ( d i − d j ) }| and p max = max i { p i } ,where d = 0 . Then the observer is given as below. ˙ α i = β i − b [ h ( t ) + 2] (cid:26) n X j =0 a ij [( α i − α j ) − ( x i − x j )] (cid:27) − b [ h ( t ) + 2]sgn (cid:26) n X j =0 a ij [( α i − α j ) − ( x i − x j )] (cid:27) , ˙ β i = u i − c [ h ( t ) + 2] (cid:26) n X j =0 a ij [( β i − β j ) − ( v i − v j )] (cid:27) − c [ h ( t ) + 2]sgn (cid:26) n X j =0 a ij [( β i − β j ) − ( v i − v j )] (cid:27) , (35)where α = β = 0 , and b , b , c , c are positive constantssatisfying b = c ≥ p max λ ( Q ) , b ≥ p max λ ( Q ) , c ≥ p max ( d + u max ) λ ( Q ) Theorem 3.
With the given dynamics (4), (5) and observer(35), under Assumption 2, α i and β i converges to ˜ x i and ˜ v i within a fixed-time T b = t b + t b . Proof. Let ˜ x i = x i − x , ˜ v i = v i − v and ˜ α i = α i − ˜ x i , ˜ β i = β i − ˜ v i . Then we have ˙˜ α i = ˜ β i − b [ h ( t ) + 2] n X j =0 a ij (˜ α i − ˜ α j ) − b [ h ( t ) + 2]sgn (cid:20) n X j =0 a ij (˜ α i − ˜ α j ) (cid:21) , ˙˜ β i = − c [ h ( t ) + 2] n X j =0 a ij ( ˜ β i − ˜ β j ) − c [ h ( t ) + 2]sgn (cid:20) n X j =0 a ij ( ˜ β i − ˜ β j ) (cid:21) − d i + u . (36)Let z i = P nj =0 a ij ( ˜ β i − ˜ β j ) and then one obtains ˙˜ β i = − c [ h ( t ) + 2] z i − c [ h ( t ) + 2]sgn( z i ) − d i + u . (37)Differentiating z i against time one has ˙ z i = − c [ h ( t ) + 2] n X j =0 a ij ( z i − z j ) − c [ h ( t ) + 2] (cid:26) n X j =0 a ij [sgn( z i ) − sgn( z j )] (cid:27) − n X j =0 a ij ( d i − d j ) + a i u . (38)According to z = H ˜ β , where H is a nonsingular matrix.Thus if z converges to zero, ˜ β converges as well. Construct aLyapunov candidate as V = n X i =1 p i [ c z i + c | z i | ] . (39)Then, one has ˙ V = n X i =1 p i [2 c z i + c sgn( z i )] × (cid:26) − c [ h ( t ) + 2] n X j =0 a ij ( z i − z j ) − c [ h ( t ) + 2] n X j =0 a ij [sgn( z i ) − sgn( z j )] − n X j =0 a ij ( d i − d j ) + a i u (cid:27) = − [ h ( t ) + 2] n X i =1 p i [2 c z i + c sgn( z i )] × (cid:26) c n X j =0 a ij ( z i − z j ) + c n X j =0 a ij [sgn( z i ) − sgn( z j )] (cid:27) − n X i =1 p i [2 c z i + c sgn( z i )] (cid:20) n X j =0 a ij ( d i − d j ) − a i u (cid:21) = − [ h ( t ) + 2][2 c z + c sgn( z )] T P H [2 c z + c sgn( z )] EEE TRANSACTIONS ON 7 − n X i =1 p i [2 c z i + c sgn( z i )] (cid:20) n X j =0 a ij ( d i − d j ) − a i u (cid:21) = − [ h ( t ) + 2][2 c z + c sgn( z )] T Q [2 c z + c sgn( z )] − n X i =1 p i [2 c z i + c sgn( z i )] (cid:20) n X j =0 a ij ( d i − d j ) − a i u (cid:21) ≤ − λ ( Q )[ h ( t ) + 2][2 c z + c sgn( z )] T [2 c z + c sgn( z )] − n X i =1 p i [2 c z i + c sgn( z i )] (cid:20) n X j =0 a ij ( d i − d j ) − a i u (cid:21) = − λ ( Q )[ h ( t ) + 1] (cid:26) n X i =1 [4 c z i + 4 c c | z i | + c ] (cid:27) − λ ( Q ) (cid:26) n X i =1 [4 c z i + 4 c c | z i | + c ] (cid:27) − n X i =1 p i [2 c z i + c sgn( z i )] (cid:20) n X j =0 a ij ( d i − d j ) − a i u (cid:21) ≤ − c λ ( Q )[ h ( t ) + 1] (cid:26) n X i =1 [ c z i + c | z i | ] (cid:27) − c λ ( Q ) (cid:26) n X i =1 [4 c | z i | + c ] (cid:27) + p max ( d + u max ) (cid:26) n X i =1 [2 c | z i | + c ] (cid:27) ≤ − c λ ( Q ) p max [ h ( t ) + 1] (cid:26) n X i =1 p max [ c z i + c | z i | ] (cid:27) ≤ − h ( t ) V − V ≤ − h ( t ) V . (40)When t ∈ [0 , t b ) , h ( t ) = h b ( t ) . Using differential equation(1) one has lim t → t b V ≤ ( δ δ ) k V (0) ≈ . When t ≥ t b , ˙ V ≤ − V . Then V will converge exponentially and | z i | << is ensured. Therefore, the observer can successfully estimatethe velocity disagreements between the leader and the follow-ers in t b .In the following proof, let w i = P nj =0 a ij ( ˜ α i − ˜ α j ) . Thensubstituting w i into (36) one obtains ˙˜ α i = ˜ β i − b [ h ( t ) + 2] w i − b [ h ( t ) + 2]sgn( w i ) . (41)Differentiating w i against time one has ˙ w i = z i − b [ h ( t ) + 2] n X j =0 a ij ( w i − w j ) − b [ h ( t ) + 2] (cid:26) n X j =0 a ij [sgn( w i ) − sgn( w j )] (cid:27) . (42)When t ∈ [ t b , t b + t b ) , due to | z i | << , z i in (42) can beseen as a bounded disturbance in (38). The following proof isthe same as before and not restated. Until now, the proof ofTheorem 3 has been finished. D. Fixed-Time Distributed Averaging Tracking Observer
Assumption 3.
The topology graph for the n agents isundirected and connected. Each agent can only receive theinformation form one reference signal. Lemma 4. [34] If L ∈ R n × n is a Laplacian matrix of aconnected undirected graph and D ∈ R n × m is its relativeincidence matrix. Then for any vector z ∈ R n one has z T LD sgn( D T z ) ≥ λ ( L ) z T D sgn( D T z ) . (43)The distributed observer is given as below. ˙ α i = − b h ( t ) n X j =1 a ij ( α i − α j ) − b n X j =1 a ij sgn( α i − α j ) + β i , ˙ β i = − c h ( t ) n X j =1 a ij ( β i − β j ) − c n X j =1 a ij sgn( β i − β j ) + a ri , (44)where b , b , c , c are positive constants satisfying b = c ≥ λ ( L ) , b ≥ , c > a max ; Moreover the initial states satisfy P ni =1 α i (0) = P ni =1 x i (0) and P ni =1 β i (0) = P ni =1 v i (0) .Note that P ni =1 ˙ α i = P ni =1 β i and P ni =1 ˙ β i = P ni =1 a ri .Therefore under the given initial states, we have P ni =1 β i = P ni =1 v i and P ni =1 α i = P ni =1 x i all the time. Following fromthis, if all the observers achieve consensus in a fixed time, theaverage value of the multiple reference signals is obtainedsuccessfully. Theorem 4.
With the given dynamics (5), (7) and observer(44), under Assumption 3, α i and β i converges to r and f within a fixed-time T b = t b + t b .Proof. Construct a Lyapunov candidate as V = β T Lβ and(44) can be written in the vector form as ˙ α = − b h ( t ) Lα − b D sgn( D T α )˙ β = − c h ( t ) Lβ − c D sgn( D T β ) + a. (45)Then we have ˙ V = β T L ˙ β = β T L [ − c h ( t ) Lβ − c D sgn( D T β ) + a ] ≤ − c h ( t ) β T LLβ − c β T LD sgn( D T β ) + β T La ≤ − c h ( t ) λ ( L ) β T Lβ − c β T D sgn( D T β ) + β T La ≤ − h ( t ) V − c || D T β || + ( D T β ) T D T a ≤ − h ( t ) V − ( c − a max ) || D T β || ≤ − h ( t ) V . (46)When t ∈ [0 , t b ) , h ( t ) = h b ( t ) . Using differential equation(1) one has lim t → t b V ≤ ( δ δ ) k V (0) ≈ . When t ≥ t b , ˙ V ≤ − ( c − a max ) || D T β || . Then compared with (32) one EEE TRANSACTIONS ON 8 has V will converge in finite time after t b and β i << isensured. Construct the Lyapunov candidate as V = α T Lα ˙ V = α T L ˙ α = α T L [ − b h ( t ) Lα − c D sgn( D T α ) + β ]= − b h ( t ) α T LLα − c α T LD sgn( D T α ) + αLβ ] ≤ − b λ ( L ) h ( t ) α T Lα − c || D T α || + ( D T α ) T D T β ≤ − h ( t ) V. (47)Due to β i << , | β i − β j | << is ensured. Then one has c || D T α || > ( D T α ) T D T β . The following is the same asbefore and hence omitted. Until now, the proof of Theorem 4has been finished. Remark 4.
All the observers proposed in this paper can beextended to that of high-order multi-agent systems by usingmore time-based generators and more integrators.E. Distributed Consensus Tracking and Distributed AverageTracking Control
After designing the observers, the next step is to design thecontrollers by using the information provided by the observers.In this subsection, the control inputs for the distributed con-sensus tracking and the distributed average tracking will begiven.
Theorem 5.
Under dynamics (4), (5) and Assumption 1(Assumption 2), with the observer (25) (observer (35)) andthe control input u i = = 0 , t ∈ [0 , T c ) , = − ˙ h ( t ) α i − ( h ( t ) + 1) β i − h ( t ) s i − ρ sgn( s i ) , t ≥ T b , (48) where T c ≥ T b , s i = ( h ( t ) + 1) α i + β i , ρ ≥ d max + u max + 1 , the fixed-time distributed consensus tracking fordouble-integrator-type multi-agent systems is solved. Furthermore, the upper-bounded convergence time is T a + T c .Proof. When t ≥ T b , one has α i = x i − x and β i = v i − v .Then we have ˙ s i = 12 ˙ h ( t ) α i + ( 12 h ( t ) + 1) ˙ α i + ˙ β i = 12 ˙ h ( t ) α i + ( 12 h ( t ) + 1) β i + ˙ v i − ˙ v = 12 ˙ h ( t ) α i + ( 12 h ( t ) + 1) β i + u i + d i − u . (49)Then substitute (48) into (49), one has ˙ s i = − h ( t ) s i − ρ sgn( s i ) + d i − u . (50)The other part of the proof is the same as Theorem 1 andomitted. Thus the whole proof has been finished. Theorem 6.
Under dynamics (5), (7) and Assumption 3, withthe observer (44) and the control input u i = = 0 , ∈ [0 , T c ) , = − ˙ h ( t )( x i − α i ) − ( h ( t ) + 1)( v i − β i ) − h ( t ) s i − ρ sgn( s i ) , t ≥ T b , (51) where s i = ( h ( t ) + 1)( x i − α i ) + ( v i − β i ) , ρ ≥ d max + a max + 1 , the fixed-time distributed average trackingproblems for double-integrator-type multi-agent systems issolved. Further more, the upper-bounded convergence time is T a + T c .Proof. When t ≥ T b , one has α i = r and β i = f . Then wehave ˙ s i = 12 ˙ h ( t )( x i − α i ) + ( 12 h ( t ) + 1)( v i − β i )+ u i + d i − ˙ β i . (52)Then substitute (51) into (52), one has ˙ s i = − h ( t ) s i − ρ sgn( s i ) + d i − a. (53)The other part of the proof is the same as Theorem 1 andomitted. Thus the whole proof has been completed.IV. N UMERICAL SIMULATIONS
Example 1.
A simulation for Theorem 1 is given as follows.Set k = 2 , δ = 0 . , ̺ = sin( t ) , ρ = 2 , t a = t a = 3 , z (0) = 200 and z (0) = 100 . The results with upper-boundedconvergence time T a = 6 s are shown in Fig. 1. time(s) -200-1000100200300 z , z . convergence time = 6s z z Fig. 1. The results of the sliding mode control in Example 1.
Example 2.
A simulation for Theorem 5 under Assumption1 is given as follows. Consider a multi-agent system describedby (4) and (5) with the undirected communication topology inFig. 2. Set k = 2 , δ = 0 . , ρ = 8 , b = c = 4 , b = 1 , c =8 , t a = t a = 3 , t b = t b = 1 . , T b = T c , u = 1 + 5sin( t ) and x i (0) , v i (0) with random states. The results with upper-bounded convergence time T a + T c = 9 s are shown in Fig. 3and Fig. 4. Fig. 2. The communication topology in Example 2.
EEE TRANSACTIONS ON 9 time(s) x i , i = , ,..., . convergence time = 9s x x x x x x Fig. 3. The positions of the agents in Example 2. time(s) -300-200-1000100 v i , i = , ,..., . convergence time = 9s v v v v v v Fig. 4. The velocities of the agents in Example 2.
Example 3.
A simulation for Theorem 5 under Assumption2 is given as follows. Consider a multi-agent system describedby (4) and (5) with the directed communication topology inFig. 5. Set k = 2 , δ = 0 . , ρ = 21 , b = c = 2 , b = 7 , c = 34 , t a = t a = 2 , t b = t b = 1 , T b = T c , u =2 + 18sin( t ) and x i (0) , v i (0) with random states. The resultswith upper-bounded convergence time T a + T c = 6 s are shownin Fig. 6 and Fig. 7. Fig. 5. The communication topology in Example 3.
Example 4.
A simulation for Theorem 6 under Assumption3 is given as follows. Consider a multi-agent system describedby (5) and (7) with the undirected communication topology inFig. 8. Set k = 2 , δ = 0 . , ρ = 63 , b = c = 0 . , b = 1 , c = 123 , t a = t a = 4 , t b = t b = 2 , T b = T c , a r = 41+20sin(5 t ) , a r = 51+10sin(5 t ) , a r = 30+30sin(5 t ) , a r = 40 + 20sin(5 t ) and x i (0) , v i (0) with random states. Theresults with upper-bounded convergence time T a + T c = 12 s are shown in Fig. 9 and Fig. 10. time(s) -150-100-50050100150 x i , i = , ,..., . convergence time = 6s x x x x x Fig. 6. The positions of the agents in Example 3. time(s) -100-50050100150 v i , i = , ,..., . convergence time = 6s v v v v v Fig. 7. The velocities of the agents in Example 3.Fig. 8. The communication topology in Example 4. time(s) -1000010002000300040005000 x i , i = ,..., . convergence time = 12s Fig. 9. The positions of the agents in Example 4.
EEE TRANSACTIONS ON 10 time(s) -2000200400600800 v i , i = ,..., . convergence time = 12s Fig. 10. The velocities of the agents in Example 4.
V. C
ONCLUSIONS
In this paper, both the fixed-time distributed consensustracking and the fixed-time distributed average tracking prob-lems for double-integrator-type multi-agent systems are solvedby using time-based generators. Different from traditionalfixed-time methods, the time-based generator approach candirectly predesign the fixed time, which is of great significancein reality. But the tradeoff is the introduce of time dependentfunction. Moreover, it is trivial to extend the fixed-time slidingmode control method in this article to Euler-Lagrange systems.By combining the fixed-time sliding mode control methodof Euler-Lagrange systems and the observers in this article,the fixed-time distributed consensus tracking and distributedaverage tracking for multiple Euler-Lagrange systems can beachieved. Also, the fixed-time distributed consensus trackingproblem for single-integrator multi-agent systems under di-rected graph can be solved by devising a controller similar tothe velocity observer in (44).R
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