Autonomous and non-autonomous fixed-time leader-follower consensus for second-order multi-agent systems

Autonomous and non-autonomous fixed-time leader-follow er consensus for second-order m ulti-agen t systems a M. A. T rujillo b R. Aldana-L´ op ez c D. G´ omez Guti ´ errez d M. Defo ort e J. Ruiz Le´ on a H. M. Becerra f Abstract This pap er addresses the problem o f consensus tracking with fixed- time conv ergence, for leader-follower m ulti-agent systems with double-integrator dynamics, where only a subset of followers has access to the state o f the leader. The co ntrol scheme is divided in to tw o steps. The first o ne is dedicated to the estimation of the leader state by ea c h follow er in a distributed wa y and in a fix ed-time. Then, based on the estimate o f the lea der state, each follow er computes its control law to track the leader in a fixe d- time. In this pap er, t wo control strategies are inv estigated and co mpared to solve the tw o mentioned steps. The fir s t one is an autonomous proto col which ensures a fixed-time conv ergence for the obser v er and for the controller parts where the Upp er B o und of the Settling-Time ( UBST ) is set a prior y by the user. Then, the previous stra tegy is redesigned using time-v arying gains to obtain a non-autonomo us proto col. This enables to obtain less conserv ativ e estima tes of the UBST while guara n teeing tha t the time-v arying g a ins rema in b ounded. Some numerical examples show the effectiveness of the pr opo sed consensus proto cols. 1 In tro duction In the last years, the problems of co ordination and con trol of Multi-Agen t System (MAS) ha v e b een widely studied (see for ins tance [1, 2, 3, 4, 5]), due mainly to the abilit y of a MAS to face complex tasks that a single agen t is not able to h an d le. Distributed cont rol approac hes applied to a MAS requir e a comm unication netw ork allo wing to sh are information with a s ubset of agen ts (neigh b ors). In this con text, sev eral int eresting p roblems and applications ha v e b een in v estiga ted in the literature, for instance, sync hronization of complex net w orks [6], d istributed resource allo catio n [7], consensus [8] and form ation con trol of m ultiple agen ts [9]. Among all the mentioned p roblems, an in teresting one is the leader-follo wer consensu s pr oblem where a set of agen ts, through lo cal in teractio n, conv erge to the state of a leader, ev en though the leader ma y not b e accessible for all agen ts. The consensus problem consists in r eac hing a common agreemen t state b y exc hanging only lo cal inform ation [8, 10]. Linear a verage consen s us proto cols with asymptotic conv ergence w ere prop osed in [8, 10]. It h as b een d emonstrated th at the second smallest eigen v alue of the Laplacian graph (i.e. the algebraic connectivit y) determines the con v ergence rate of the MAS. a This is the accepted version of th e manuscri pt: T rujillo, M.A., Aldana-L´ opez, R., G´ omez-Guti´ errez, D. et al. Autonomous and non-autonomous fixed-time leader–follo wer consensus for second-ord er m ulti-agent sys- tems. Nonlinear Dy namics 102 , 2669–268 6 (2020). DOI: 10.1007/s1 1071-020-06075-7. Please cite the p u blisher’s versi on. F or the publisher’s version and full citation details see: https://doi.org /10.100 7/s11071-020-06075-7 . b CINVEST A V Unidad Guadala jara, Av. d el Bosque 1145, Zap opan, 45019, Jalisco, Mexico c Department of Computer Science and Systems Engineering, Univers ity of Zaragoza, Zaragoza, 500 09, Espa˜ na d Intel T ecnolog ´ ıa de M´ exico, Intel Labs, Multi agen t autonomous systems lab, Jalisco, M´ exico. T ecnol´ ogi co de Mon terrey , Escuela de Ingenier ´ ıa y Ciencias, Jalisco, M´ exico. Email: D a vid.Gomez.G@ieee .org e LAMIH, UMR CNR S 8201, IN SA, Pol ytechnic U niv ersit y of Hauts de F rance, V alenciennes, 59313 F ran ce f Cen tro de In v estigaci´ o n en Matem´ aticas (CIMA T), Jalisco S N, Col. V alenciana, 36023, Guana juato, Mexico 1 F urthermore, the problem of trac king a reference by a MAS (i.e. leader-follo wer consensus p rob- lem) has b een inv estigated where the common agreemen t to reac h is the state of a r eference imp osed by a leader wh ic h ev olv es ind ep enden tly of the MAS [11, 12, 13, 14]. In [14], th e con- sensus pr oblem has b een add r essed w here the agen ts reac h a time-v arying reference. Ho wev er, the con trol proto col has b een derived f or first-order MAS. The problem for s eco nd-order MAS has b een studied in [12] an d extended to high-order MAS in [11, 13]. F urthermore, [15] has considered the consensus tracki ng con trol problem of u ncertain n onlinear MAS with predefined accuracy . Nev ertheless, in these w orks, the conv ergence is only asymptotic. T o improv e the con v ergence rate of a MAS, finite-time consensus p rotocols hav e b een in v es- tigated in [16 ]. Fin ite-time stability h as b een studied in [17, 18, 19]. Ho w ever, the settling time is an u n b ounded fu nction of the initial conditions of the system. T herefore, the concept of fixed- time stabilit y h as b een introdu ced and applied to systems with time constraints [20, 21, 22]. In this case, the settling time is b ounded b y a constant whic h is in dep enden t of the initial conditions of the system. In the literature, there are several con tributions on algorithms with fixed-time con v ergence pr op ert y , su c h as stabiliz ing controlle rs [21, 23], state observ ers [24], m ulti-agen t co ordination [25, 26], online d ifferen tiation algorithms [27, 28], etc. Nev ertheless, one can mentio n that th e fixed-time stabilization problem of second-order s y s tems is not an easy task since u sually the settling time is not p ro vid ed or is o v erestimate d. Indeed, there are seve ral w orks for second-order systems stabilizatio n b ased on b lock-c on trol tec h niques ([29, 21, 30, 31]) or on the homogeneit y in the bi-limit ([32]). Ho w ev er, the h omoge neit y-based algorithms do not pr o vide an estimate of the settling time and many blo c k-con trol-based algorithms neglect some transien t when the system tra jectories sta y on a region around a manifold. Moreo ver, the w orks [33, 34, 35, 36] deal with the problem of leader-follo wer consens u s. Nev ertheless, these algorithms requir e that eac h follo we r kno w the inpu ts of its n eigh b ors simultaneously , whic h causes comm unication lo op p r oblems. In th is pap er, w e address th e leader-follo wer consensus problem of a MAS, where eac h agen t of the MAS estimates and trac ks the tra j ecto ry of the leader using lo cal a v ailable inf ormation eve n wh en ju s t a sub s et of MAS has access to the leader state, and we p ro vid e the necessary conditions to ac hiev e th e con v ergence in a fixed -time. A Lyapuno v differen tial inequalit y for an autonomous system to exhibit fi xed-time stabilit y w as presente d in [21]. Based on this metho dology usin g autonomous s ystems, the consensus problem w ith fixed-time con v ergence prop erty has b een derived for fir st-order MAS in [25, 37, 38]. Nev ertheless, in [38], the UBST has b een estimated from design p arameters, algebraic connectivit y and group order. Thus, it cannot b e easily tuned . In [25], the UB ST was a design parameter whic h wa s established a p riory by the u s er. How eve r, the settling time b ecomes o v er- estimated and the slac k b et w een the settling time and the UBST is conserv ativ e. F urthermore, the works [39, 40, 41] ha v e addressed the consensus trac king problem, i.e., the MAS follo w s a tra jectory imp osed by the leader. The scheme presen ted in [40] has introd u ced a fixed-time algorithm considering inherent dynamics for the agen ts. Ho wev er, disturb an ces w ere not tak en in to accoun t. The leader-follo w er consensus pr ob lem for agents with second-order an d high order in tegrator dynamics has b een add ressed in [42, 41], resp ectiv ely . Th e appr oac h was based on a fixed-time observer to estimate the leader state and a fixed -time con troller to drive the s tate of the agent to the estimated leader state. Unf ortunately , although the observ er can b e designed to conv erge at a desired UBST (with a conserv ativ e estimate of the UBST ), the con troller is based on the homogeneit y theory [43] and no metho dology has b een pro vided to estimate an UBST . Thus, although th e algorithm is fixed -time con v ergen t, the desired con v ergence time cannot b e set a priory by the user. T o address this issu e, autonomous algorithms were prop osed in [44, 45, 46] with an estimation of the UBST . Unf ortunately , such estimate of th e U BST results v ery conserv ativ e leading to o v er-engineered consensus p rotocols. Therefore, the design of fixed- time leader-follo w er consensu s algorithms where th e UBST is set explicitly as a parameter of the system, as w ell as the reduction of the conserv ativ eness of th e estimate of the UBST is of a great in terest. 2 An appr oac h to deriv e predefin ed-time consensu s algorithms has b een addressed via a linear function of the su m of the errors b et w een neigh b oring no des together with a time-v arying gain, using time base generators [47], see e.g., [48, 49, 50, 51, 52, 53, 54, 55]. This approac h ensu res that the conv ergence is obtained exactly at a p r edefined time. Ho w ever, such time-v arying gain b ecomes singular at the p redefined time, either b ecause th e gain go es to infi nite as the time tends to the predefin ed time [53, 54, 55] or b ecause it p ro duces Zeno b eh a vior (infinite num b er of switc hing in a fin ite-ti me in terv al) as the time tend s to the predefin ed time [49]. In this pap er, we p resen t a metho dology to ac hiev e leader-follo wer consensus with fi xed-time con v ergence. It consists in t w o steps. The first one estimates the leader state (p osition and v elocit y) u sing a fixed-time observer that only requir es information of the neigh b ors. Then, the second step compu tes the co n trol la w to dr iv e the follo w ers to the observ er states in a fixed-time. Moreo v er, w e in v estigate t w o p rotocols to solv e th e considered p roblems. In one proto col, called autonomous proto col, the con v ergence for th e ob s erv er and for the con troller is in fixed-time, wh ere the UBST is established a pr iory by the user. In the second proto col, called non-autonomous p rotocol, w e red esign the pr evious one b y add ing time-v arying gains to obtain a less conserv ativ e estimate of th e U BST while guaran teeing that the time-v arying gains remain b ounded. The con tribution lies in the follo wing. A no v el proto col is deriv ed for second-order MAS w ith fixed-time stabilit y where the UBST is a design parameter. Moreo ver, a n on-autonomous pr otocol is present ed to achiev e the con v ergence in a p redefined-time with less conserv ativ e estimates of the UBST compared to existing results in th e literature. In fact, the r esulting U BST can b e made arbitrarily tight. A t last, our algorithm yields a b ounded time- v arying gain, thus w e a void the d ra wb ac ks p resen t in the existing algorithms with time-v arying gains where the gain go es to in fi nit y . This w ork is structured as follo ws. Section 2 recalls some definitions and r esults from graph theory , and pr eliminaries on finite-time and fi xed-time con v ergence are p resen ted. In Section 3, the problem of consensus trac kin g with fixed-time con v ergence is form ulated. Section 4 in tro duces tw o metho dologie s to solve the consensu s trac king pr oblem. The first (resp. second) one is based on algorithms to obtain a fixed-time stable autonomous (resp. non autonomous) system with an U B ST fun ction indep end en t of the in itia l conditions of the system. Numerical results using b oth metho dologies are shown in Section 6. Finally , the conclusions are presen ted in Section 7. 2 Preliminaries 2.1 Graph Theory In this section, some n ota tions and pr eliminaries ab out graph and consensus theory are pr e- sen ted. One can r efer to [56, 57] for a deep er insigh t in these fields. This pap er is only f ocused on undirected graphs f or the follo w er agen ts. Definition 1. A graph consists of a set of v ertices V ( X ) and a set of agen ts E ( X ) wh er e an edge is an unordered p air of d istinct v ertices of X . ij denotes an edge, if v ertex i and v ertex j are adjacent or neigh b ors. The set of neigh b ors of i in graph X is expressed by N i ( X ) = { j ∈ X : j i ∈ E ( X ) } . Definition 2. A p ath f rom i to j in a graph is a sequence of distinct vertic es starting with i and endin g w ith j such th at consecutiv e ve rtices are adjacen t. If there is a path b et ween an y t w o ve rtices of graph X , th en X is s aid to b e conn ecte d. Definition 3. L et X b e a weigh ted graph suc h that ij ∈ E h as we igh t a ij and let N = |V ( X ) | . Then, the adjacency matrix A ( X ) (or simply A when the graph is clear fr om th e con text) is an N × N matrix where A = [ a ij ] and, the Laplacian is denoted by Q ( X ) (or simp ly Q ) and is defined as Q ( X ) = ∆( X ) − A ( X ) where ∆( X ) = diag( d 1 , ..., d N ) with d i = P j ∈N i a ij . 3 Through this w ork, it is assumed that a ij = a j i , i.e. on ly und irected and b alanced graphs are considered. Definition 4. Let ˆ X b e a we igh ted graph among all the agen ts (i.e. the leader and the follo wers). Then, the communicatio n matrix b et w een all the agen ts is represent ed by M ( ˆ X ) = Q ( X ) + B w h ere B = d iag( b 1 . . . b N ) ∈ R N × N with b i > 0 when there is an edge from the leader to the i -agen t and Q ( X ) is the w eigh ted graph asso ciate d to the comm unication top ology of the follo w ers. Lemma 1. [58, 13] L et ˆ X b e the c ommunic ation gr aph among al l the agents with the le ader as the r o ot. Then, matrix M ( ˆ X ) is symmetric p ositive definite. 2.2 On finite-time and fixed-time Consider the system ˙ x ( t ) = f ( x ( t ) , t ; ρ ) , x (0) = x 0 (1) where x ∈ R n is the system state, the v ecto r ρ ∈ R b stands for more parameters of system (1) whic h are assumed to be constan t, i.e., ˙ ρ = 0. F urthermore, there is no limit for the n um b er of p arameters, so b can take an y v alue in the natural n umber set N . Th e fu nction f : R n × R + → R n is nonlinear and the origin is assumed to b e an equilibrium p oint of system (1), so that f (0 , t ; ρ ) = 0. Besides, when function f do es not dep end explicitly on t , the system is said to b e autonomous or time-in v arian t. Otherwise, it is called n on-autonomous or time-v arying [59]. Definition 5. [60] T he origin of (1) is globally finite-time stable if it is globally asymp totically stable and any solution x ( t ; x 0 ) of (1) reac h es the equilibrium p oin t at some fi nite time moment i.e. x ( t ; x 0 ) = 0 , ∀ t ≥ T ( x 0 ) where T : R n → R + ∪ { 0 } is called th e s ettli ng-time function. Definition 6. [61] Th e origin of (1) is fixed-time stable if it is globally finite-time stable an d the settling fun ctio n is b ounded, i.e., ∃ T max > 0 : T ( x 0 ) ≤ T max , ∀ x 0 ∈ R n . Theorem 1. [62] Consider the system ˙ x = − ( α | x | p + β | x | q ) k sign ( x ) , x (0) = x 0 (2) with x ∈ R . The p ar ameters of the system ar e the r e al numb ers α, β , p, q , k > 0 which satisfy the c onstr aints k p < 1 and k q > 1 . L et ρ = [ α, β , p, q , k ] T ∈ R 5 . Then, the origin x = 0 of system (2) is fixe d-time stable and the settling time function satisfies T ( x 0 ) ≤ T f = γ ( ρ ) , wher e γ ( ρ ) = Γ  1 − k p q − p  Γ  q k − 1 q − p  α k Γ( k )( q − p )  α β  1 − kp q − p , (3) and Γ( · ) is the Gamma function define d as Γ( z ) = R ∞ 0 e − t t z − 1 dt (se e [63] for details on the Gamma function). Theorem 2. [62] Consider the system ˙ x ( t ) = f ( x ( t ) , t ; ρ ) , x (0) = x 0 (4) wher e x ∈ R n is the system state, the ve ctor ρ ∈ R b stands for the system p ar ameters which ar e assume d to b e c onsta nt. The function f : R n × R + → R n is su ch that f (0 , t ; ρ ) = 0 . Assume that ther e exists a c ontinuous r adial ly unb ounde d f unction V : R n → R such that: V (0) = 0 V ( x ) > 0 , ∀ x ∈ R n \{ 0 } 4 and the derivative of V alon g the tr aje ctories of (4) satisfies ˙ V ( x ) ≤ − γ ( ρ ) T c ( αV ( x ) p + β V ( x ) q ) k , ∀ x ∈ R n \{ 0 } wher e α, β , p, q , k > 0 , k p < 1 , k q > 1 , γ is given in (3) and ˙ V is the upp er right-hand time- derivative of V . Then, the origin of (4) is pr e define d-time stable with pr e define d-time T c . Definition 7. F or any real num b er r , the function x 7→ ⌊ x ⌉ r is defined as ⌊ x ⌉ r = | x | r sign ( x ) for an y x ∈ R if r > 0, and for an y x ∈ R \ 0 if r ≤ 0. Moreo ver, if r > 0, ⌊ 0 ⌉ r = 0 . 3 Problem statemen t Let us consider a group of N + 1 agen ts with one leader and N follo w ers lab eled 0 and i ∈ { 1 , . . . , N } , resp ectiv ely . The dynamics of the leader is describ ed b y ˙ x 0 ( t ) = v 0 ( t ) ˙ v 0 ( t ) = u 0 ( t ) where X 0 = [ x 0 , v 0 ] T ∈ R 2 is the state of the leader and u 0 ∈ R is the con trol in put of the leader, whic h is assumed to satisfy | u 0 ( t ) | ≤ u max 0 , ∀ t ≥ 0 with u max 0 a kn own constant. The dynamics of the i − th f ollo w er agent is giv en by: ˙ x i ( t ) = v i ( t ) ˙ v i ( t ) = u i ( t ) + ∆ i ( t ) (5) where X i = [ x i , v i ] T ∈ R 2 is th e state of agen t i , u i ∈ R is the control input of agen t i and ∆ i is an un kno w n external d isturbance wh ic h is assumed to s atisfy | ∆ i ( t ) | ≤ δ i , ∀ t ≥ 0 with δ i a kno wn constant. Besides, eac h agen t estimates the leader states, r epresen ted by ˆ x i (p osition) and ˆ v i (v elocit y). T he comm unication top ology is represente d by an undirected graph, whic h is assumed to cont ain a spanning tree with the leader agen t as the ro ot. T h e i − th agen t shares its estimated states of the leader with its n eigh b ors, d efined by the neigh b or s et N i . The con trol ob jectiv e is to design a d istributed con trol u i suc h that the consensus is achiev ed in a fixed-time T c , i.e.  lim t → T c k X i ( t ) − X 0 ( t ) k = 0 X i ( t ) = X 0 ( t ) , ∀ t > T c . This goal is ac hiev ed into tw o s tages. An “observe r”, based on consensu s algorithms, allo ws eac h agen t to obtain an estimate of the leader state in a distribu ted m an n er in a fixed-time. Then, after the obs er ver conv erges, a con troller d riv es the state of the agen t to w ards the s tate tra jectory of the leader. Two proto cols are inv estigated hereafter. In the fir st one, known as an autonomous p rotocol, we guarant ee that eac h agen t is driven to w ards th e leader state in a fixed- time, where the Upp er Bound of th e Settling-Time (UBST) is sp ecified a priory by the us er. In the second on e, known as a non-autonomous pr otocol, we redesign the previous one b y adding time-v arying gains to obtain a less conserv ative estimate of the UB ST while guarantee ing that the time-v arying gains remain b ounded. 4 Fixed-time leader-follo w er consensus using autonomous p ro- to cols 4.1 Distributed fixed-time observer Since the leader state is n ot a v ailable to all follo w ers, for eac h agent, an observer is d esigned to estimate the state of the leader in a fixed-time. T h e observer has the follo wing structur e: ˙ ˆ x i = ˆ v i − κ i,x  ( α | e 1 ,i | p + β | e 1 ,i | q ) k + ζ x  sign ( e 1 ,i ) ˙ ˆ v i = − κ i,v  ( α | e 2 ,i | p + β | e 2 ,i | q ) k + ζ v  sign ( e 2 ,i ) (6) 5 with e 1 ,i = P j ∈N i a ij ( ˆ x j ( t ) − ˆ x i ( t )) + b i ( x 0 ( t ) − ˆ x i ( t )) and e 2 ,i = P j ∈N i a ij ( ˆ v j ( t ) − ˆ v i ( t )) + b i ( v 0 ( t ) − ˆ v i ( t )), ˆ x i (resp. ˆ v i ) is the estimate of the leader p osition (resp. v elo cit y) for th e i -th follo wer. κ i,x , κ i,v , α , β , k , p , q , ζ x and ζ v are p ositiv e constants to b e defined later. F or eac h agen t, let u s denote the observ er errors as ˜ x i = ˆ x i − x 0 ˜ v i = ˆ v i − v 0 . (7) Therefore, the observ atio n err or d y n amics can b e expressed as: ˙ ˜ x i = ˜ v i − κ i,x  ( α | e 1 ,i | p + β | e 1 ,i | q ) k + ζ x  sign ( e 1 ,i ) ˙ ˜ v i = − κ i,v  ( α | e 2 ,i | p + β | e 2 ,i | q ) k + ζ v  sign ( e 2 ,i ) − u 0 (8) with e 1 ,i = P j ∈N i a ij ( ˜ x j ( t ) − ˜ x i ( t )) − b i ˜ x i ( t )) and e 2 ,i = P j ∈N i a ij ( ˜ v j ( t ) − ˜ v i ( t )) − b i ˜ v i ( t )). In a compact f orm , with ˜ x = [ ˜ x 1 · · · ˜ x N ] T ∈ R N and ˜ v = [ ˜ v 1 · · · ˜ v N ] T ∈ R N , system (8) can b e written as: ˙ ˜ x = ˜ v − Φ x  M ( ˆ X ) ˜ x  ˙ ˜ v = − Φ v  M ( ˆ X ) ˜ v  − 1 u 0 (9) where M ( ˆ X ) repr esen ts the connection m atrix of the graph describ ing the net w ork b et ween the follo wers and the leader, and for z = [ z 1 · · · z N ] T ∈ R N , the fun ctions Φ x : R N → R N and Φ v : R N → R N are defined as Φ x ( z ) =    κ 1 ,x  ( α | z 1 | p + β | z 1 | q ) k + ζ x  sign ( z 1 ) . . . κ N ,x  ( α | z N | p + β | z N | q ) k + ζ x  sign ( z N )    , Φ v ( z ) =    κ 1 ,v  ( α | z 1 | p + β | z 1 | q ) k + ζ v  sign ( z 1 ) . . . κ N ,v  ( α | z N | p + β | z N | q ) k + ζ v  sign ( z N )    . Theorem 3. If the observer p ar ameters ar e sele cte d as α, β , p, q , k > 0 , k p < 1 , k q > 1 , ζ x ≥ 0 , κ x ≥ N γ ( ρ ) λ min ( M ( ˆ X )) T c 2 , κ v ≥ N γ ( ρ ) λ min ( M ( ˆ X )) T c 1 and κ v ζ v ≥ u max 0 wher e κ x = min i ∈{ 1 ...N } κ i,x and κ v = min i ∈{ 1 ...N } κ i,v and γ ( ρ ) is define d in Equation (3) , then under the distribute d observer (6) , the observer err or dynamics (8) is fixe d-time stable with a pr e define d-time T o = T c 1 + T c 2 . Pr o of. Consid er the r ad ially un b ounded Ly apuno v f unction cand id ate V 1 ( ˜ v ) = 1 N r λ min  M ( ˆ X )  ˜ v T M ( ˆ X ) ˜ v . Its time-deriv ativ e along the tra jectories of system (9) is ˙ V 1 = r λ min  M ( ˆ X )  N ˜ v T M ( ˆ X ) ˙ ˜ v q ˜ v T M ( ˆ X ) ˜ v (10) 6 Let us d enote e 2 = M ( ˆ X ) ˜ v = [ e 2 , 1 · · · e 2 ,N ] T . Th en, Equation (10) can b e written as follo w s ˙ V 1 = r λ min  M ( ˆ X )  N e T 2 q ˜ v T M ( ˆ X ) ˜ v  − Φ v  M ( ˆ X ) ˜ v  − 1 u 0  = r λ min  M ( ˆ X )  N   − 1 q ˜ v T M ( ˆ X ) ˜ v N X i =1 κ i,v e 2 ,i h ( α | e 2 ,i | p + β | e 2 ,i | q ) k + · · · + ζ v ] sign ( e 2 ,i ) − e T 2 1 u 0 q ˜ v T M ( ˆ X ) ˜ v   = r λ min  M ( ˆ X )  N   − 1 q ˜ v T M ( ˆ X ) ˜ v N X i =1 κ i,v | e 2 ,i | ( α | e 2 ,i | p + β | e 2 ,i | q ) k − ζ v q ˜ v T M ( ˆ X ) ˜ v N X i =1 κ i,v | e 2 ,i | − e T 2 1 u 0 q ˜ v T M ( ˆ X ) ˜ v   (11) No w , u sing the inequ ality (28) of Lemma 3 in App endix, the fi rst term of Equation (11) can b e written as 1 q ˜ v T M ( ˆ X ) ˜ v N X i =1 κ i,v | e 2 ,i | ( α | e 2 ,i | p + β | e 2 ,i | q ) k ≥ N κ v q ˜ v T M ( ˆ X ) ˜ v 1 N n X i =1 | e 2 ,i | ! α 1 N N X i =1 | e 2 ,i | ! p + β 1 N N X i =1 | e 2 ,i | ! q ! k with κ v = min { κ 1 ,v , . . . , κ N ,v } . Since k e 2 k 1 = P N i | e 2 ,i | the last expression can b e written as 1 q ˜ v T M ( ˆ X ) ˜ v N X i =1 κ i,v | e 2 ,i | ( α | e 2 ,i | p + β | e 2 ,i | q ) k ≥ N κ v q ˜ v T M ( ˆ X ) ˜ v  1 N k e 2 k 1   α  1 N k e 2 k 1  p + β  1 N k e 2 k 1  q  k . F urthermore, f rom Lemma 4 in App endix, one gets k e 2 k 1 ≥ k e 2 k 2 = q e T 2 e 2 = q ˜ v T M ( ˆ X ) 2 ˜ v ≥ r λ min  M ( ˆ X )  ˜ v T M ( ˆ X ) ˜ v . Hence, 1 q ˜ v T M ( ˆ X ) ˜ v N X i =1 κ i,v | e 2 ,i | ( α | e 2 ,i | p + β | e 2 ,i | q ) k ≥ κ v N q ˜ v T M ( ˆ X ) ˜ v V ( αV p + β V q ) k = κ v r λ min  M ( ˆ X )  ( αV p 1 + β V q 1 ) k . 7 No w , for the last tw o terms of Equation (11), one can obtain − ζ v q ˜ v T M ( ˆ X ) ˜ v N X i =1 k i,v | e 2 ,i | − e T 2 1 u 0 q ˜ v T M ( ˆ X ) ˜ v ≤ − k e 2 k 1 q ˜ v T M ( ˆ X ) ˜ v ( κ v ζ v − u max 0 ) ≤ 0 . Therefore, the follo wing in equalit y can b e obtained ˙ V 1 ≤ − κ v λ min  M ( ˆ X )  N ( αV p 1 + β V q 1 ) k . F rom Theorem 2, the observ atio n error in v elocity ˜ v con v erges to the origin in a fixed-time b efore the predefi ned-time T c 1 where γ ( ρ ) is giv en by Eq. (3). Once the observ atio n error in velocit y ˜ v conv erges to zero (i.e. after time T c 1 ), the observ ation error dynamics in p osition can b e r ed uced to ˙ ˜ x i = − κ i,x h ( α | e 1 ,i | p + β | e 1 ,i | q ) k + ζ x i sign ( e 1 ,i ) . Similarly to the previous analysis, one can easily sh o w that V 2 ( ˜ x ) = 1 N r λ min  M ( ˆ X )  ˜ x T M ( ˆ X ) ˜ x satisfies ˙ V 2 ( ˜ x ) ≤ − κ x λ min  M ( ˆ X )  N ( αV p 2 + β V q 2 ) k , ∀ t ≥ T c 2 . F rom Theorem 2 , the observ ation err or in p osition ˜ x con v erges to th e origin in a fixed-time b efore the predefi ned-time T c 2 . Therefore, the prop osed d istributed observ er guarante es th e estimation of the leader states in a fixed-time b efore th e p redefined-time T o = T c 1 + T c 2 . 4.2 A Fixed-time track ing con troller After time T o , eac h agen t has an accurate estimation of the leader s tate. F or eac h agent, the trac kin g error is defi n ed as e x,i = x i − ˆ x i e v,i = v i − ˆ v i , (12) or, equiv alent ly , after the con v ergence of the observ ation error: e x,i = x i − x 0 e v,i = v i − v 0 . (13) Its dynamics can b e expr essed as: ˙ e x,i = e v,i ˙ e v,i = u i + ∆ i − u 0 . Here, the ob j ecti v e is to design the con tr ol input u i suc h that the origin ( e x,i , e v,i ) = (0 , 0) is fixed-time stable wh ere the Up p er Bound of th e S ettling-Time ( UBST ) is set a pr iory by the user, in spite of the unknown b ut b ounded p ertur bation term ∆ i − u 0 . He refater, we pr esent the follo wing results motiv ated b y the work [62]. 8 Theorem 4. If for e ach agent, the c ontr ol ler is sele cte d as u i = υ ( e x,i , e v,i ) = − " γ 2 ˆ T c 2  α 2 | σ i | p ′ + β 2 | σ i | q ′  k ′ + γ 2 1 2 ˆ T 2 c 1  α 1 + 3 β 1 e 2 x,i  + ζ i ( t ) # sign ( σ i ) (14) with the fol lowing sliding variable σ i = e v,i + $ ⌊ e v,i ⌉ 2 + γ 2 1 ˆ T 2 c 1  α 1 ⌊ e x,i ⌉ 1 + β 1 ⌊ e x,i ⌉ 3  ' 1 / 2 , (15) wher e p ar ameters ar e sele cte d as α 1 , α 2 , β 1 , β 2 , T ′ o , ˆ T c 1 , ˆ T c 2 > 0 , p ′ , q ′ , k ′ > 0 , k ′ p ′ < 1 , k ′ q ′ > 1 , ζ i ≥ u max 0 + δ i , γ 1 = Γ ( 1 4 ) 2 2 α 1 / 2 1 Γ ( 1 2 )  α 1 β 1  1 / 4 and γ 2 = Γ( m p )Γ( m q ) α k ′ 2 Γ( k ′ )( q ′ − p ′ )  α 2 β 2  m p with m p = 1 − k ′ p ′ q ′ − p ′ and m q = k ′ q ′ − 1 q ′ − p ′ , then the le ader-fol lower c onsensus is achieve d in a pr e define d-time ˆ T c = T ′ o + ˆ T c 1 + ˆ T c 2 . Pr o of. First, the time deriv ativ e of σ i along the tra jectory of the s y s tem solution is giv en by ˙ σ i = u i + ∆ i − u 0 + | e v,i | ( u i + ∆ i − u 0 ) + γ 2 1 2 ˆ T 2 c 1  α 1 + 3 β 1 e 2 x,i  e v,i     ⌊ e v,i ⌉ 2 + γ 2 1 ˆ T 2 c 1  α 1 ⌊ e x,i ⌉ 1 + β 1 ⌊ e x,i ⌉ 3      1 / 2 . Using the con trol inp ut u i giv en by (14), one obtains ˙ σ i = − γ 2 ˆ T c 2  α 2 | σ i | p ′ + β 2 | σ i | q ′  k ′ sign ( σ i )      1 + | e v,i |     ⌊ e v,i ⌉ 2 + γ 2 1 ˆ T 2 c 1  α 1 ⌊ e x,i ⌉ 1 + β 1 ⌊ e x,i ⌉ 3      1 / 2      − γ 1 2 ˆ T c 1  α 1 + 3 β 1 e 2 x,i       sign ( σ i ) + | e v,i | sign ( σ i ) − e v,i     ⌊ e v,i ⌉ 2 + γ 2 1 ˆ T 2 c 1  α 1 ⌊ e x,i ⌉ 1 + β 1 ⌊ e x,i ⌉ 3      1 / 2      − ( ζ i sign ( σ i ) − ∆ i + u 0 )      1 + | e v,i |     ⌊ e v,i ⌉ 2 + γ 2 1 ˆ T 2 c 1  α 1 ⌊ e x,i ⌉ 1 + β 1 ⌊ e x,i ⌉ 3      1 / 2      . (16) Let us consider th e candid ate Ly apuno v function V 1 ( σ i ) = | σ i | with its time d eriv ativ e as ˙ V 1 = sign ( σ i ) ˙ σ i . Since | e v,i |     ⌊ e v,i ⌉ 2 + γ 2 1 ˆ T 2 c 1  α 1 ⌊ e x,i ⌉ 1 + β 1 ⌊ e x,i ⌉ 3      1 / 2 ≥ 0 , | e v,i | − e v,i sign ( σ i ) ≥ 0 , and ζ i ≥ u max 0 + δ i 9 one can easily rewr ite the Lyapuno v fu nction deriv ativ e, us in g (16), in th e follo wing in equ alit y ˙ V 1 ( σ ) ≤ − γ 2 ˆ T c 2  α 2 V 1 ( σ ) p ′ + β 2 V 1 ( σ ) q ′  k ′ . F rom Th eorem 2, on e can deduce that σ i con v erges to zero in a fixed-time ˆ T c 2 . After time ˆ T c 2 , one obtains 0 = e v,i + $ ⌊ e v,i ⌉ 2 + γ 2 1 ˆ T 2 c 1  α 1 ⌊ e x,i ⌉ 1 + β 1 ⌊ e x,i ⌉ 3  ' 1 / 2 , whic h in turn implies, ˙ e x,i = e v,i = − γ 1 ˆ T c 1  α 1 | e x,i | + β 1 | e x,i | 3  1 / 2 sign ( e x,i ) . F rom Theorem 1 , it is clear that e x,i con v erges to ze ro in a fixed-time b efore the settl ing time ˆ T c 1 . Moreo ver, from (15), since σ i = 0 and e x,i = 0, then e v,i = 0. Hence, we can conclude that system (5) with (14) as the con trol inpu t, is fixed-time stable with predefin ed - time ˆ T c 1 + ˆ T c 2 . Moreo ver, d ue to Theorem 3, where the leader states are estimated in a fixed -time with the p redefined settling time T o . Hence, if T ′ o = T o , one can ded u ce th at leader-follo w er consensus is ac h iev ed in fixed-time b efore the pr edefined-time ˆ T c = T ′ o + ˆ T c 1 + ˆ T c 2 . A t last, if T ′ o = 0 an d T o < ˆ T c 1 + ˆ T c 2 , the leader-follo we r consensus is ac h iev ed b efore th e pr ed efined-time ˆ T c = ˆ T c 1 + ˆ T c 2 . 5 Fixed-time leader-follo w er consensus with impro v ed estimate for the UBS T using non-autonomous pr oto col The autonomous leader-follo w er proto col p resen ted in Section 4 allo ws a fi xed-time conv ergence. Ho wev er, the estimate of the UBST for th e observer and cont roller are b oth to o conserv ativ e. This is a common drawbac k on existing fi xed-time consensus proto cols, see e.g., [26, 40] for the leader -follo w er problem f or agen ts with first-order in tegrat or dynamics and [64, 45] for agent s with second-order integrato r dynamics. T o address this issue, w e p resen t new proto cols, based on the class of time-v arying gains prop osed in [20 ], to significan tly redu ce such conserv atism. Con trary to some existing proto cols suc h as [52, 47, 54], w here the time-v arying gains b ecome singular when consens us is reac hed, in our approac h the con v ergence is ac hiev ed with b ound ed time-v arying gains in a user-defined time. Before d esigning th e prop osed fixed-time leader-follo we r consensu s proto col, let us define the follo wing functions: Definition 8. Let us define the follo wing • Φ : R + → R + ∪ { + ∞} \ { 0 } is a con tin u ous function on R + \ { 0 } that satisfies – R + ∞ 0 Φ( z ) dz = 1, – Φ( τ ) < + ∞ , ∀ τ ∈ R + \ { 0 } , – is either non-increasing or lo cally Lipschitz on R + \ { 0 } . • ψ : R + → R + satisfies ψ ( τ ; T c ) = T c Z τ 0 Φ( ξ ) dξ with T c a p ositiv e constan t, 10 • η is suc h that η ( T ) = lim τ → T 1 T c ψ ( τ ; T c ) ≤ 1 with T a p ositiv e parameter, • ρ : R + → R + satisfies ρ ( τ ; T c ) = 1 T c Φ( τ ) − 1 . Definition 9. (Definition in [65]). A regular p arametrized cu r v e, with parameter t , is a C 1 ( I ) inmersion c : I 7→ R , defined on a real in terv al I ⊆ R . This means that dc dt 6 = 0 holds ev erywhere Definition 10. (Pg. 8 in [65]). A r egular curve is an equiv alence class of regular parametrized curv es, where the equiv alence r elat ion is giv en by regular (orien tarion preserving) parameter transformation ψ , where ψ : I → I ′ is C 1 ( I ), b ij ective , and dψ dt > 0. Therefore, if c : I → R is a regular parametrized curv e and ψ : I → I ′ is a regular parameter transform atio n, then c and c ◦ ψ : I ′ → R are considered to b e equiv alent. Lemma 2. [66] L et t 0 b e the i nitial time. The function t = ψ ( τ ) + t 0 , defines a p ar ameter tr ansformatio n with τ = ψ − 1 ( t − t 0 ) as its inverse mapping. Pr o of. It follo ws from Definition 10 T o der ive the fixed-time n on-autonomous scheme, we define the f ollo wing time-v arying gain for eac h p r edefined settling time T c i as follo ws usin g the previously d efined fu nctions: ˆ κ i ( t ; t 0 , T c i , T ) =  ρ i ( ψ − 1 i ( t − t 0 ; T c i ); T c i ) if t ∈ [ t 0 , t 0 + η i ( T ) T c i ) 1 otherwise . R emark 1 . Notice th at if T < + ∞ , then ˆ κ i ( t ; t 0 , T c i , T ) is b ounded. Suc h b ound can b e user- defined by tuning T . No w , w e are ready to p resen t our main result. Theorem 5. L et us c onsider the same observer p ar ameters as in The or em 3 (i.e. α, β , p, q , k > 0 , ζ x , ζ v , κ i,x , κ i,v , T c 1 , T c 2 ). L et  1 ( t ) = ˆ κ 1 ( t ; t 0 , T c 1 , T α ) and  2 ( t ) = ˆ κ 2 ( t ; t ′ 0 , T c 2 , T β ) b e time- varying gains. T α and T β ar e p ositive p ar ameters and t ′ 0 = t 0 + η 1 ( T α ) T c 1 . Using the fol lowing distribute d observer ˙ ˆ x i = ˆ v i −  2 ( t ) κ i,x h ( α | e 1 ,i | p + β | e 1 ,i | q ) k + ˆ ζ x ( t ) i sign ( e 1 ,i ) (17) ˙ ˆ v i = −  1 ( t ) κ i,v h ( α | e 2 ,i | p + β | e 2 ,i | q ) k + ˆ ζ v ( t ) i sign ( e 2 ,i ) , with ˆ ζ x ( t ) =  − 1 2 ( t ) ζ x and ˆ ζ v ( t ) =  − 1 1 ( t ) ζ v , the observer err or dynamics is fixe d-time stable with the UBST given by T o = t 0 + η 1 ( T α ) T c 1 + η 2 ( T β ) T c 2 , with T α , T β > 0 . L et us c onsider the same c ontr ol p ar ameters as in The or em 4 (i.e. α 1 , α 2 , β 1 , β 2 , ˆ T c 1 , ˆ T c 2 > 0 , p ′ , q ′ , k ′ > 0 , ζ i , γ 1 , γ 2 ) and set T c 3 = ˆ T c 1 + ˆ T c 2 . L et  3 ( t ) = ˆ κ 3 ( t ; T o , T c 3 , T γ ) b e a time- varying gain with T γ a p ositive p ar ameter. Using the fol lowing c ontr ol ler u i =   3 ( t ) 2 υ ( e x,i ,  3 ( t ) − 1 e v,i ) + ˙  3 ( t )  3 ( t ) − 1 e v,i if t ∈ [ T ′ o , T ′ o + η 3 ( T γ ) T c 3 ) υ ( e x,i , e v,i ) if t ∈ [ T ′ o + η 3 ( T γ ) T c 3 , + ∞ ) (18) wher e υ ( e x,i ,  − 1 3 ( t ) e v,i ) is given by (14) , the le ader-fol lower c onsensus is achieve d in fixe d time with the UBST given by ˆ T = T ′ o + η ( T γ ) T c 3 . 11 The pr oof of T heorem 5 will b e divided into t w o p arts. The firs t part f o cuses on the ob s erv er stabilit y wh er eas the second one fo cuses on the cont roller s tabilit y . Pr o of. First, let us stud y th e observ er error d ynamics usin g the non-autonomous observe r (17). Let u s consider the obs er ver errors as in (7). Using (17), the observ ation error dynamics is expressed as follo ws ˙ ˜ x i = ˜ v i −  2 κ i,x h ( α | e 1 ,i | p + β | e 1 ,i | q ) k + ˆ ζ x i sign ( e 1 ,i ) ˙ ˜ v i = −  1 κ i,v h ( α | e 2 ,i | p + β | e 2 ,i | q ) k + ˆ ζ v i sign ( e 2 ,i ) − u 0 . No w , considering the observ er err or d ynamics of th e v elocit y ˜ v i in the n ew τ -time v ariable as follo ws d ˜ v i dτ = d ˜ v i dt dt dτ , (19) and according to the parameter transformation giv en in L emma 2, dt dτ = d dτ ( ψ i ( τ ) − t 0 )     τ = ψ − 1 i ( t − t 0 ; T c i ) = ρ i ( ψ − 1 i ( t − t 0 ); T c i ) − 1 (20) = ˆ κ i ( t ; t 0 , T c i , T ) − 1 . Th us, the observ ation error dynamics of the vel o cit y giv en by (19) is rewr itten, using (20), as follo ws d ˜ v i dτ = − ˆ κ 1 ( t ; t 0 , T c 1 , T α ) − 1  1 κ i,v h ( α | e 2 ,i | p + β | e 2 ,i | q ) k + ˆ ζ v i sign ( e 2 ,i ) −  ( τ ) , where  ( τ ) = ˆ κ 1 ( t ; t 0 , T c 1 , T α ) − 1 u 0 with | u 0 | < u max 0 is the d isturbance term and ˆ κ 1 ( t ; t 0 , T c 1 , T α ) − 1 =  − 1 1 ( t ). Th en, the last expr ession is wr itten as d ˜ v i dτ = − κ i,v h ( α | e 2 ,i | p + β | e 2 ,i | q ) k + ˆ ζ v i sign ( e 2 ,i ) −  ( τ ) . (21) Note that b y th e d efinition of the fun ctio n Φ 1 ( τ ), the time-v arying gain ˆ κ 1 ( t ; t 0 , T c 1 , T α ) − 1 for t ∈ [ t 0 , t 0 + η 1 ( T α ) T c 1 ), ca n b e written as ρ 1 ( τ ; T c 1 ) − 1 ∀ τ , and function ρ 1 ( τ ; T c 1 ) − 1 is non- increasing. Besides, ρ 1 ( τ ; T c 1 ) − 1 is b ound ed and ρ 1 ( τ ; T c 1 ) − 1 → 0 as τ → + ∞ . Th en, the disturbance  ( τ ) = ρ 1 ( τ ; T c 1 ) − 1 u o is v anish ing. F u rthermore, notice that ˆ ζ v ( t ) =  − 1 1 ( t ) ζ v and |  ( τ ) | < ˆ ζ v , ∀ τ since u max 0 ≤ κ v ζ v . Then, similarly to (9) , the compact form of (21) is d ˜ v dτ = − Φ v  M ( ˆ X ) ˜ v  − 1  ( τ ) (22) with ˜ v = [ ˜ v 1 · · · ˜ v N ] T ∈ R N . F urthermore, from T heorem 3, the observ atio n error d ynamics of v elo cit y (22) is fixed-time stable in the time-v ariable τ and , con v erges to the origin with a settling time T ′ c 1 < + ∞ . Then, the observ ation error in v elocit y reac hes th e origin at T ( ˜ v 0 ) = lim τ → T ′ c 1 ( ψ 1 ( τ ) + t 0 ) ≤ t 0 + η 1 ( T α ) T c 1 ; ∀ ˜ v 0 ∈ R N as the initial conditions. In a similar wa y , the observ ation error dynamics of the p osition ˜ x i in the time-v ariable τ is written as follo ws d ˜ x i dτ = υ i ( τ ) − κ i,x h ( α | e 2 ,i | p + β | e 2 ,i | q ) k + ˆ ζ x i sign ( e 2 ,i ) (23) where υ i ( τ ) = ˆ κ 2 ( t ; t 0 + η 1 ( T α ) T c 1 , T c 2 , T β ) − 1 ˜ v i . The compact form of (23 ) is d ˜ x dτ = υ − Φ x  M ( ˆ X ) ˜ x  (24) 12 with υ = [ υ 1 · · · υ N ] ∈ R N . Then, due to ˜ v = 0 for t ≥ t 0 + η 1 ( T α ) T c 1 , the observ ation error dynamics of p osition (24) is fi xed-time stable in the time-v ariable τ , and con verge s to the origin with a settling time T ′ c 2 < + ∞ . Hence, the observ ation error in p osition reac hes the origin at T ( ˜ x 0 ) = lim τ → T ′ c 2 ( ψ 2 ( τ ) + t ′ 0 ) ≤ t ′ 0 + η 2 ( T β ) T c 2 ; ∀ ˜ x 0 ∈ R N , with t ′ 0 = t 0 + η 1 ( T α ) T c 1 . Therefore, the observer error dynamics is fixed-time stable and con v er ges to the origin b efore the pr edefined-time T o = t 0 + η 1 ( T α ) T c 1 + η 2 ( T β ) T c 2 . Notice that T ′ c 1 and T ′ c 2 are the settling time for the sys tem in the time-v ariable τ . Then, let us study the trac kin g error d y n amics u sing the non-autonomous con troller (18). Consider the follo wing co ordinate change e x,i = ˜ e x,i (25) e v,i =  3 ( t ) ˜ e v,i where e x,i and e v,i are the trac king errors for eac h agen t defined in (13) or, its equiv alen t in (1 2) after th e con v ergence of th e observ ation error. T hen, the dynamics of the v ariable ˜ e i = [ ˜ e x,i , ˜ e v,i ] T is the follo wing ˙ ˜ e x,i =  3 ˜ e v,i ˙ ˜ e v,i =  − 1 3 u i −  − 1 3 ˙  3 ˜ e v,i +  − 1 3 (∆ i − u 0 ) . No w , let us consider th e parameter transformation giv en by Lemma 2 to get the d ynamics in τ -v ariable. Then, the dynamics of (25) expressed in the time-v ariable τ is d ˜ e x,i dτ = ˜ e v,i (26) d ˜ e v,i dτ =  − 2 3 u i −  − 2 3 ˙  3 ˜ e v,i +  − 2 3 (∆ i − u 0 ) . Using the con troller (18 ) for t ∈ [ T ′ o , T ′ o + η 3 ( T γ ) T c 3 ), system (26) is written as d ˜ e x,i dτ = ˜ e v,i (27) d ˜ e v,i dτ = υ ( ˜ e x,i , ˜ e v,i ) + π i ( τ ) , with π i ( τ ) =  − 2 3 (∆ i ( t ) − u 0 )   t = ψ 3 ( τ )+ T o . Notice that ∆ i ( t ) satisfies | ∆ i ( t ) | ≤ δ i and u 0 is unknown but b oun ded b y | u o | ≤ u max 0 . By Definition 8, the function  3 ( t ) − 2 is non incr easing. Then, π i ( τ ) is b oun ded and π i ( τ ) → 0 as τ → + ∞ . Thus, from Th eorem 4, system (27) is fi xed-time stable in the time-v ariable τ with T ′ c 3 < + ∞ as its settling time. He nce, the trac kin g errors ( ˜ e x,i and ˜ e v,i ), w ith e 0 = [ ˜ e x,i ( T o ) , ˜ e v,i ( T o )] as initial conditions, reac h the origin at T ( e 0 ) = lim τ → T ′ c 3 ( ψ 3 ( τ ) + T o ) ≤ T o + η 3 ( T γ ) T c 3 ; ∀ e 0 ∈ R 2 . Th us, the trac king error dynamics is fixed-time stable with η 3 ( T γ ) T c 3 as the predefined UBST . Hence, if T ′ o = T o and from the fact that observer (17) estimates the leader state in predefined -time and con troller (18 ) drive s the agen ts to w ard s the leader state tra jectory , one can conclude that the leader-follo wer consensus is ac h iev ed in fi x ed -time b efore the predefi n ed- time ˆ T = T o + η ( T γ ) T c 3 . A t last, if T ′ o = t o and T o < η ( T γ ) T c 3 the leader-follo w er consensus is ac h iev ed b efore the p redefined time ˆ T = t 0 + η ( T γ ) T c 3 . R emark 2 . It is worth noting that the n on-autonomous proto col is deriv ed from the autonomous one. As discussed in the next section, this scheme based on b ounded time-v arying gains, h as b een introd uced to impro v e the con v er gence time estimation, i.e., the slac k b et w een the UBST and the con v ergence time is reduced. 13 6 Sim ulation results In this section, we illustrate our main resu lts with the autonomous and non-autonomous p ro- to cols for the leader-follo wer co nsensus p roblem. In order to co mpare the autonomous and non-autonomous sc hemes pr op osed in this w ork, w e will also compare the slac k b etw een the UBST and the real conv ergence time of th e system for eac h con trol sc h eme. F or all sc hemes, 0 1 2 3 5 4 Figure 1: Communicatio n top ology w ith 5 follo wers. w e consider the same scenario for comparison pur p oses. W e consid er a m ulti-agen t system com- p osed of N = 5 agen ts where the d ynamics of eac h agen t is giv en by Eq. (5) with an external p erturbation ∆ i ( t ) = s in(4 0 t + 0 . 1 i ), with i = 1 . . . N . The comm unication top ology , giv en in Figure 1 is u ndirected and con tains a spann ing tree with the leader agen t as the r oot. F or th e leader, its control inpu t is giv en by u 0 = 4 cos (2 t ) with the initial cond itions [ x 0 , v 0 ] = [ − 1 , 0]. F rom Figure 1, one gets λ min ( M ( ˆ X )) = 0 . 2907. The initial conditions of the ag en ts are as follo ws x (0) = [ − 10 , − 5 , 0 , 5 , 10] v (0) = [0 , 0 , 0 , 0 , 0] and the initial conditions for the observer are r andomly set as ˆ x (0) = [ − 5 . 81 , − 7 . 82 , 4 . 57 , 9 . 22 , 5 . 94] ˆ v (0) = [5 . 57 , − 6 . 42 , 4 . 91 , 8 . 39 , − 7 . 87] . 6.1 Fixed-time leader-follo wer consensus using autonomous proto cols In this subsection, w e present the results of the autonomous sc heme present ed in Section 4. According to Theorem 3, the d istributed fixed-time obs er ver (6) w ith p = 1 . 5, q = 3 . 0, k = 0 . 5, α = 1, β = 2, T c 1 = 0 . 9 s , T c 2 = 0 . 1, κ x = 3 . 53, κ v = 31 . 82 and ζ v = 0 . 0678 guaran tees that the observer err or con v er ges to zero b efore the predefined-time T c 1 + T c 2 = 1 s . Figure 2 shows the leader state estimation for eac h agen t, wh ile the left column of Figure 6 sho ws the ob s erv er errors for eac h agen t. One can see in more details in the left column of Figure 7 the settling time of the observ atio n errors and the UBST as the d otted line, where the settling time for th e v elocit y error ( ˜ v ) is T 1 ≈ 0 . 013 s , and for the p osition err or ( ˜ x ) is T 2 ≈ 0 . 143 s . I t is p ossible to see that th e settling time for the observ atio n error o ccurs b efore the predefin ed time, i.e., T 1 < T c 1 for the v elocit y error and T 1 + T 2 < T o for the observ ation error. In the simulation the controlle r is activ ated at the same time that the observer, i.e., T ′ o = 0 and T o < ˆ T c 1 + ˆ T c 2 . Then, the control ler (14) is applied in ord er to follo w the tra jectory of the leader with p ′ = 1 . 5, q ′ = 3 . 0, k ′ = 0 . 5, α 1 = α 2 = 1 /β 1 = 1 /β 2 = 1 / 4, ˆ T c 1 = 1 s and ˆ T c 2 = 1 s . The left column of 14 Figure 8 shows the trac kin g err or, wh er e the tr acking errors e x and e v con v erge to zero b efore ˆ T c = ˆ T c 1 + ˆ T c 2 = 2 s with T ′ o = 0. One can see in more details in Figure 9 that the conv ergence time of the trac king error o ccurs at T 3 ≈ 1 . 228 s < ˆ T c where the UBST is plotted in dotted line. The states of the agen ts are shown in Figur e 3, wh ere it can b e seen that the leader-follo w er consensus is successfully ac hiev ed. 0 0.5 1 1.5 2 2.5 3 -2 0 2 0 0.5 1 1.5 2 2.5 3 -4 -2 0 2 4 Figure 2: Autonomous proto col. Leader s tates estimation for eac h agent . 0 0.5 1 1.5 2 2.5 3 -2 0 2 0 0.5 1 1.5 2 2.5 3 -4 -2 0 2 4 Figure 3: Autonomous proto col. T ra jectories of eac h agen t. 6.2 Fixed-time leader-follo wer consensus using non-autonomous prot o cols The results of the fixed-time observ er/con troller scheme present ed in Section 5 , u sing time- v arying gains, are sh o wn in this subsection. Consider th e same example as in the pr evious 15 subsection with the same external p erturbation. No w , consider the observ er/con troller sc heme p r op osed in Theorem 5, w h ere fu nction Φ i is defined as Φ i ( τ ) = ˆ α i η − 1 i e − ˆ α i τ for i = 1 , 2 , 3 with η i ( T ) = 1 − e − ˆ α i T and ˆ α 1 = 220, ˆ α 2 = 90 and ˆ α 3 = 1 . 8. This fun ctio n is used to compu te the time-v arying gains  i . Then, the gain ˆ κ i ( t ; t o , T c i , T ) used f or the observer (17) and the con troller (18 ), is defined as ˆ κ i ( t ; t 0 , T c i , T ) = ( η i ˆ α i ( T c i − η i ( t − t 0 )) if t ∈ [ t 0 , t 0 + η i ( T ) T c i ) 1 otherwise . The u s er-defined p aramete rs are s et as follo ws T c 1 = 0 . 1 s , T c 2 = 0 . 9 s and T c 3 = ˆ T c 1 + ˆ T c 2 with ˆ T c 1 = 1 s and ˆ T c 2 = 1 s , the parameters for η i are set as T α = 0 . 016, T β = 0 . 055 and T γ = 1 . 5 for i = 1 , 2 , 3 resp ectiv ely , t o = 0 s for the observe r and T ′ o = t 0 for the control ler. In ord er to co mpare the cont rol sc heme prop osed in Theorems 3-4 with Theorem 5, the p arameters α 1 , α 2 , β 1 , β 2 , p ′ , q ′ , k ′ for the con troller and the p arameters α, β , p, q , k , ζ x , ζ v , κ x , κ v for the observ er were tak en from the result presented in S ectio n 6.1. Fig ure 4 sh o ws the leader state estimation for eac h ag en t, while the righ t column of Figure 6 shows the observ er errors for eac h agen t. One can see in more details in the righ t column of Figure 7 the settling time of the observ ation errors ( ˜ x and ˜ v ) and the UBST as the d otted line, where the settling time for th e v elocity error ˜ v is T 1 ≈ 0 . 0947 8 s , and for the p osition error ˜ x is T 2 ≈ 0 . 9891 . It is p ossible to s ee that the settling time for the observ ation error o ccurs before the predefined time T o = η 1 ( T α ) T c 1 + η 2 ( T β ) T c 2 . Besides, the cont roller (18) is applied in order to follo w the tra jectory of the leader. The righ t column of Figure 8 sho ws th e fixed-time con v ergence of the trac kin g error ( e x and e v ). One can see in more details in the r igh t column of Figure 9 that the con v ergence time of the trac king err or o ccurs at T 3 ≈ 1 . 952 s . The states of th e agen ts are sho wn in Figure 5, wh ere it can b e seen that the leader-follo wer consen s us is successfully ac hiev ed. Unlik e th e con trol sc heme pr op osed in Th eorem 3 (for the observer) and T heorem 4 (for the con troller), the slac k b et w een the predefin ed UB ST giv en by the u s er and the real conv ergence time is less conserv ativ e. Figure 7 sh o ws the conv ergence of b oth p rotocols for the observer and Figur e 9 s h o ws the conv ergence of b oth proto cols for the con troller, wh ere one can see the UBST as the dotted line and, the difference b et w een the slac k of the autonomous an d non- autonomous p rotocols. Moreo v er, the slac k of the n on-autonomous proto col can b e adjusted b y the parameters of the time-v aryin g gain ˆ κ i ( t ; t 0 , T c i , T ). Ho w ev er, in this ca se, this gain increases. Th us, one needs to establish a trade-off b et ween th e size of the upp er b oun d for ˆ κ i ( t ; t 0 , T c i , T ) and how small the slac k is. In ord er to compare the t w o pr otocols present ed in th is w ork (autonomous and non-autonomous ones), we will define the s lac k f unction s ( x ) as the error b etw een the predefined UBST and th e con v ergence time of the v ariable x , i.e., s ( x ) = T c − T where T c is the predefined UBST and T is th e actual con v ergence time. This index for the observ ation error and the trac king error v ariables is sho wn in T able 1. It can b e seen that the slac k for the non-autonomous p rotocol is lo wer than for the autonomous proto col. s ( ˜ v ) = T c 1 − T 1 s ( ˜ x ) = T o − T 2 s ( ˜ e ) = ˆ T − T 3 Autonomous pr otocol 0.0870 9 0.8744 0.743 Non-autonomous p rotocol 0.0023 0.0016 0.1154 T able 1: Slac k of con vergence time. These numerica l examples sh o w th e effectiv eness of the pr op osed consensus proto cols. 16 0 0.5 1 1.5 2 2.5 3 -2 0 2 0 0.5 1 1.5 2 2.5 3 -4 -2 0 2 4 Figure 4: Non-autonomous proto col. Lea der states estimation for eac h agen t. 0 0.5 1 1.5 2 2.5 3 -2 0 2 0 0.5 1 1.5 2 2.5 3 -4 -2 0 2 4 Figure 5: Non-autonomous proto col. T r a jectories of eac h agen t. 7 Conclusions In this work, we presente d no v el p rotocols for the problem of consensus tracking with fixed-time con v ergence, for leader-follo wer multi-ag en t systems with d ouble-in tegrator dynamics, w here only a su bset of follo w ers h as access to the state of the leader. A d istributed observ er is prop osed for eac h agen t to estimate the leader state, and a lo cal con troller dr iv es the agen ts to wards th e estimated state, b oth with fixed-time conv ergence. Two con trol str ategies hav e b een in v estiga ted and compared for the observe r and con troller parts. The fi rst one is an autonomous proto col w hic h ensu res th at the UBST is set a priory b y the user. T hen, the pr evious strategy is redesigned using time-v arying gains to obtain a non-autonomous proto col. T his enables to 17 Figure 6: Obser v ation error of eac h agen t. Figure 7: Conv ergence of the observ atio n err or for eac h agent . obtain less conserv ativ e estimates of th e UBST w hile guaran teeing that th e time-v arying gains remain b ou n ded. F u tu re wo rk in clud es the extension of the algorithm to c hained form systems or high order MAS, the robustness against faults in the communicatio n links and the extension of the proto col to d irected graphs. 18 0 0.5 1 1.5 2 2.5 -50 0 50 0 0.5 1 1.5 2 2.5 -400 -200 0 200 0 0.5 1 1.5 2 2.5 -50 0 50 0 0.5 1 1.5 2 2.5 -100 0 100 Figure 8: T racking err or f or eac h agen t. 0 0.5 1 1.5 2 2.5 -1 0 1 0 0.5 1 1.5 2 2.5 -1 0 1 0 0.5 1 1.5 2 2.5 -1 0 1 0 0.5 1 1.5 2 2.5 -1 0 1 Figure 9: Conv ergence of the trac king error for eac h agen t. Appp endix Lemma 3. [25] L et n ∈ N . If a = ( a 1 , . . . , a n ) is a se quenc e of p ositive numb ers, then the fol lowing i ne quality is satisfie d 1 n n X i =1 a i ( αa p i + β a q i ) k ≥ 1 n n X i =1 a i ! α 1 n n X i =1 a i ! p + β 1 n n X i =1 a i ! q ! k . (28) 19 Lemma 4. [67] L et z = [ z 1 . . . z n ] T ∈ R n and k z k p = n X i =1 | z i | p ! 1 p then, k z k l ≤ k z k r for l > r > 0 . 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