RECENTLY, multiagent systems have attracted considerable

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1 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART B: CYBERNETICS, VOL 42, NO 3, JUNE Leader-Following Formation of Switching Multirobot Systems via Internal Model Xiaoli Wang, Wei Ni, and Xinsheng Wang Abstract In this paper, the leader-following formation problem of multirobot systems with switching interconnection topologies is considered The robots are required to move in a formation with formation constrains described in terms of relative distances of the robots and the formation as whole entity is required to track the trajectory generated by an exosystem The exosystem of the considered multirobot systems provides driving forces or environmental disturbance, whose dynamics is different from the dynamics of the robots A systematic distributed design approach for the leader-following formation problem is proposed via dynamic output feedback with the help of canonical internal model Index Terms Canonical internal model, formation, multirobot systems, switching topology I INTRODUCTION RECENTLY, multiagent systems have attracted considerable attention Many coordination algorithms for multiple agents have been reported, including consensus, formation, flocking, and so on [1], [3], [6], [1] [14], [18], [21], [22], [25], [27] Among them, the formation stabilization of a group of autonomous wheeled vehicles problem was studied by many authors [1], [3], [6], [7], [21] It requires that agents collectively maintain a prescribed geometric shape, ie, prescribed relative positions and orientations with respect to each other The formation phenomenon is observed in nature in many forms, such as flocking birds, schooling fish, and swarming bees [18] Experts in artificial intelligence, control theory, robotics, systems engineering, and biology have worked on the formation control problem with the goal of understanding these phenomena Many significant results were obtained Formation control of point-mass-type robots was discussed in [1], [3], and [15] Dimarogonas and Johansson [1] examined the relation between the cycle space of the formation graph and the resulting equilibria of cyclic formations The agents motion obeyed the single integrator model Fax and Murray Manuscript received June 23, 211; revised September 22, 211; accepted November 22, 211 Date of publication January 12, 212; date of current version May 16, 212 This work was supported in part by the National Natural Science Foundation of China under Grant and in part by the Natural Science Foundation of Shandong Province under Grants ZR211FQ14 and HITNSRIF212 This paper was recommended by Associate Editor W E Dixon X Wang and X Wang are with the School of Information and Electrical Engineering, Harbin Institute of Technology at Weihai, Weihai 26429, China xiaoliwang@amssaccn; wangxswh@126com W Ni is with the School of Science, Nanchang University, Nanchang 3331, China niw@amssaccn Color versions of one or more of the figures in this paper are available online at Digital Object Identifier 1119/TSMCB [3] and Lafferriere et al [15] used graph theory to model a formation of a group of vehicle Regarding a group of wheeled vehicles with nonholonomic constraints, the formation control problem was investigated in [6], [7], [21], and [24] Dong and Farrell [6] discussed the design of cooperative control laws such that a group of nonholonomic mobile agents can converge to some stationary point or some desired trajectory under various communication scenarios using σ-processes and graph theory Egerstedt and Hu [7] developed a formation control strategy based on formation functions and virtual leaders In [21], multirobot formation based on decentralized output regulation was studied, where the exogenous information leader is known to each agent and topology is fixed Lin et al [24] studied the feasibility of the formation stabilization to a point and a line problem among a group of autonomous unicycles using graph theory Here, we consider the formation control of a group of wheeled vehicles with nonholonomic constraints Although basic multiagent formation problems have been studied with many publications in recent years, new frameworks are still needed for investigation this problem It is significant to use the idea of the results in distributed output regulation dealing with formation control Consensus and formation problem can be formulated as distributed output regulation problem which has strong theoretical and practical background [21], [23] Distributed output regulation is mainly about designing distributed feedback controller for the considered multiagent systems such that all agents can track an active leader, and/or distributed rejection with disturbance signals generated by some external system, usually called exosystem In this problem, the exogenous information is not available to all the agents, and the topology is time varying It can be viewed as the extension of classical output regulation problem, which mainly deals with designing a control law for a plant such that the closed-loop system asymptotically tracks a class of reference inputs [2], [8], [9], [16], [2] Wang et al [23] formulated the distributed output regulation of multiagent systems Its background includes active leader following model and multiagent consensus with environmental inputs Distributed feedback controller based on internal model can solve the distributed output regulation problem with fixed topology Gazi [21] formulated the formation control problem of multiagent systems as a output regulation servomechanism problem They solved the problem by decentralized output regulation for multiagent systems assuming that the exogenous information is known to each agent and interconnection topology is fixed Here, we remove these strong assumptions and concern with the case that the exogenous information is not available to all the agents and the topology is time varying /$ IEEE

2 818 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART B: CYBERNETICS, VOL 42, NO 3, JUNE 212 In this paper, we consider the leader-following formation problem of a group of nonholonomic mobile robots with switching topology from distributed output regulation viewpoint The idea of using the results in distributed output regulation dealing with formation control is a novel method that has few results Applying distributed output regulation to the formation problem is significant, which opens up a new research avenue Here, a group of nonholonomic mobile robots keeping a required formation determined by a set of formation constraints are required to follow a leader, determined by a set of tracking constraints In this paper, the leader dynamics is modeled as an exosystem that has a relative general form To solve this problem, we appeal to canonical internal model [8], [9], [17] Under certain assumptions on the system matrix of the leader dynamics, there always exists an observable steady-state generator and a canonical internal model candidate with output for the considered system Then, coupling the multirobot systems with the canonical internal model dynamics, we get an augmented system By aid of the augmented system, a distributed dynamic feedback controller is proposed using only relative output measurements It is shown that there exists an observer-based controller solving the leaderfollowing formation problem under some suitable assumptions on the system matrices and the interconnection topology A constructive three-step procedure of dynamic control design is further proposed In the first step, an observable steady-state generator and a canonical internal model candidate are given with suitable assumptions on the leader dynamics, producing an augmented system The second step deals with the design of feedback gain matrices of the formation protocol with the augmented system While in the third step, the observer-based controller is obtained in a general frame Here, using the designed distributed dynamic feedback law, we can solve the leader-following formation problem with switching topology This paper is organized as follows In Section II, preliminaries are introduced In Section III, we formulate the leader-following formation problem of multirobot systems The leader-following formation problem is formulated as the distributed output regulation problem in Section IV Distributed controller design based on canonical internal model is constructed for multirobot systems in Section V Finally, concluding remarks are given in Section VI II PRELIMINARIES In this section, preliminary knowledge is introduced First, we introduce some basic concepts and notations from graph theory [4] First, we consider the N robots Regarding the N robots as the nodes, the relationships between N robots can be conveniently described by an undirected graph G There is an edge i, j leaving from node i and entering into node j if node i can get information from node j In this case, node j is said to be a neighbor of node i N i i =1,,N is called the neighbor set of agent i The weighted adjacency matrix of G is denoted as A =a ij N N R N N, where a ii = and a ij a ij > if there is an edge from agent i to agent j For the undirected graph G, a ij = a ji Its degree matrix D = diagā 1,,ā N } RN N is a diagonal matrix, where diagonal elements ā i = N j=1 a ij for i =1,,N Then, the Laplacian of the weighted graph is defined as L = D A Furthermore, let us consider the digraph Ḡ containing N robots and a leader denoted as node with directed edges from some robots to the leader by the connection weights a i > if robot i can get information from the leader, otherwise, a i = note that Ḡ is directed though G is undirected A path in digraph Ḡ is an alternating sequence i 1e 1 i 2 e 2,,e k i k of nodes i j and edges e j =i j,i j+1 for j =1, 2,,k 1 If there exists a path from node i to node j, then node j is said to be reachable from node i A node that is reachable from every other node of Ḡ is called a globally reachable node of Ḡ Set A = diaga 1,,a N }, which is an N N diagonal matrix Define H = L + A, which describes the connectivity of the whole graph Ḡ Obviously, we have H1 = A 1 The following lemma is about the matrix H [25] Lemma 1: H is positive definite if and only if Ḡ is connected node is globally reachable in Ḡ In this paper, the leader-following formation problem of multirobot systems with switching interaction topologies is considered Strictly, suppose that there is an infinite sequence of bounded nonoverlapping contiguous time intervals [t i,t i+1, i =, 1,, starting at t = To avoid infinite switching within a finite time interval and related nonsmooth description, as usual, assume that there is a constant τ >, often called dwell time, with t i+1 t i τ i Denote S = Ḡ1, Ḡ2,,Ḡμ} as the set of all possible graphs satisfying that node the leader is globally reachable in Ḡ TakeP = 1, 2,,μ} as its index set To describe the time-varying interconnection topology with a given dwell time, we define a switching signal σ :[, P, which is piecewise constant Therefore, Laplacian L σ associated with the switching graph G σ and A,σ corresponding to the connections between agents and the leader are time varying switched at t i,i=, 1, Obviously, H σ = L σ + A,σ is also time varying However, L p, A,p and H p are time-invariant matrices noting that Ḡp p P is the graph during some time interval Clearly, under switching topology, the multirobot formation problem becomes much harder to be solved For later use, we introduce a lemma [19] Lemma 2: Consider the system ẋ = Ãx + Bu R n, y = Cx, with C,Ã being detectable For any positive definite matrices ˆM, M, there is a unique positive definite matrix P satisfying the Riccati equation, ie, P ÃT + ÃP P C T ˆM CP + M = Furthermore, Ã T C T ˆM CP is stable The next lemma was given in [5] to check the positive definiteness of a matrix Lemma 3: Suppose that a symmetric matrix is partitioned as R1 R R = 2 R2 T R 3 where R 1 and R 3 are square R is positive definite if and only if both R 1 and R 3 R T 2 R 1 R 2 are positive definite

3 WANG et al: SWITCHING MULTIROBOT SYSTEMS VIA INTERNAL MODEL 819 III LEADER-FOLLOWING FORMATION OF ROBOTS In this section, we consider the leader-following formation of robots from an output regulation viewpoint Consider a system of N mobile robots with motion equations, ie, ẋ i = ν i cosθ i ẏ i = ν i sinθ i θ i = ω i i =1,,N 1 ν i = 1 M i F i ω i = 1 J i T i where x i y i T is the location of the ith robot in the plane, θ i is the steering angle, ν i is the linear speed, and ω i is the angular speed The quantities M i and J i are positive constants and represent the mass and the moment of inertia of agent i, respectively The control inputs to the system are the force input F i and the torque input T i Here, we are not concerned with the position and orientation of the center of the robot We are interested in the formation point z i, which is defined as z i =x i + d i cosθ i y i + d i sinθ i T 2 which is a point in front of the robot at a distance d i from the center of the robot This point may represent a gripper at the end of a hand of length d i or a sensor positioned in front of the robot We treat z i as the output of the system Assume that the robots are required to move in a formation with formation constrains described in terms of relative distances of the outputs ie, points of interest of the robots given by z i z j = d ij, i,j =1,,N 3 where d ij, i,j =1,,N, are constants In addition to the above constraints, we assume that the formation as whole entity is required to track the output of the following exosystem: v =Γv, v R q z = Fv, z R 2 4 where F and Γ are system matrices of exosystem, z is the output, and v is the exogenous signal representing the disturbance input and/or the driving reference signal, which can be viewed as the leader denoted as node To ensure tracking, we define a set of tracking constraints as the relative distances of the outputs of the robots with respect to the output position of the leader z i z = d il, i =1,,N 5 where d il, i =1,,N, are constants With this formulation, the considered leader-following formation problem is stated as follows Problem 1: Design a distributed feedback F i T i T such that the N robots achieve the desired formation, ie, 3 and 5 are asymptotically achieved In the following, we force the output of ith subsystem z i to asymptotically follow or track: q i v =z + μ i 1R μ i 2 where μ i 1, μ i 2 are constants to be determined later, R : R R 2 is the rotation vector, and Rμ i 2= cosμ i 2 sinμ i 2 With this formation, μ i 1 determines the relative distance of robot i to the leader, whereas μ i 2 determines the relative angle between its position and the position of the leader in some global coordinates located for example on the leader Set μ i 1 = d il Dueto Rμ i 2 =1,wehave q i v z = d il Similarly, to satisfy the formation constraints, we need q i v q j v = μ i 1R μ i 2 μ j 1 μ 2 R j = dij As a result d il R μ i 2 djl R μ2 j = d 2 il 2d ild jl cos where μ i 2 μ j 2 can be expressed as μ i 2 μ j 2 =cos d 2 il + d 2 jl d2 ij 2d il d jl μ i 2 μj 2 + d 2 jl = d ij If the constraints are feasible, then there exist constants μ i 2, μ j 2 such that the above conditions are satisfied Since the cosine inverse has two possible solutions we need the relative angle and the constraints of at least two robots j 1 and j 2 to be able to uniquely determine the desired relative angle of robot i From here on, we assume that there exist constants μ i 1, μ i 2 such that q i v, q j v satisfying the following formation and tracking requirements: q i v q j v = d ij, q i v z = d il, i,j =1,,N 6 Then, the considered leader-following formation problem, ie, Problem 1 becomes Problem 2 Problem 2: Design a distributed feedback F i T i T such that 6 is asymptotically achieved IV FORMULATED AS OUTPUT REGULATION PROBLEM Define the regulated error as e i = z i q i v =z i z μ i 1R μ i 2 = zi z 7 where z i = z i μ i 1Rμ i 2, i =1,,N Then, Problem 2 becomes Problem 3 Problem 3: Design a distributed feedback F i T i T such that e i,i=1,,n [defined in 7], asymptotically tend to zero Then, our main objective becomes designing controller F i T i T such that the closed-loop system satisfying lim t e i t = Thus, the considered leader-following formation problem Problem 1 can be formulated as output regulation problem Problem 3

4 82 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART B: CYBERNETICS, VOL 42, NO 3, JUNE 212 Considering 2, it can be shown that z i = z i = i + B i F i T i T, i =1,,N 8 where νi ω i = i sinθ i d i ωi 2 cosθ i ν i ω i cosθ i d i ωi 2 sinθ, i 1 M B i = i cosθ i d i J i sinθ i 1 d M i sinθ i i J i cosθ i Obviously, matrices B i, i =1,,N, are always invertible since their determinants are given by d i /M i J i Therefore, assigning F i T i T such that F i T i T = B i i + u i, u i =u i1 u i2 T, i =1,,N 9 produces the input output linearization dynamics z i = u i, i =1,,N where u i is the control input of the transformed systems Take z i = ζ i =ζ i1 ζ i2 T, i =1,,N, and with the linearizing controller 9, the dynamics of the robot in the new coordinates are given by z i =ζ i ζ i =u i θ i = 1 d i ζ i1 sinθ i + 1 d i ζ i2 cosθ i i=1,,n 1 where θ i, i =1,,N, represents the unobservable states in the input output linearization internal dynamics [26] Note that the third equation of 1 can be obtained as follows From 2, we have νi cosθ ż i = i d i sinθ i ω i ζi1 = 11 ν i sinθ i +d i cosθ i ω i ζ i2 Then, we get ν i = ζ i1 cosθ i +ζ i2 sinθ i ω i = 1 d i ζ i1 sinθ i + 1 d i ζ i2 cosθ i Together with θ i = ω i, we can obtain the third equation of 1 The zero dynamics are stable because θ i =when ζ i = Since they are not asymptotically stable, during a transient the zero dynamics may grow large However, the state of the zero dynamics θ i the steering angle of the robot is 2π periodic Therefore, it does not make any real danger when the zero dynamics are not asymptotically stable Since the input output map of the system 1 is linear, it is possible to analytically solve for the equations of the manifold on which the tracking error for desired trajectory q i v is zero Set z 11 v = = z 1N v = z 1 v = Fv, z 21 v = = z 2N v = z 2 v = F Γv, u 1 v = = u N v =uv = F Γ 2 v Then, we have z 1i v = Fv z 2i v = z 1iv v Γv = F Γv u i v = z 2iv v Γv = F Γ 2 v i =1,,N Note that these are only a part of the manifold corresponding to the linear portion of the dynamics 1 For the part corresponding to the unobservable dynamics, we have the following assumption Assumption 1: There exist mappings ℶ i v with ℶ i =, i =1,,N, such that ℶ i v Γv = 1 z 2i,1 vsinℶ i v v d i + 1 d i z 2i,2 vcosℶ i v, i =1,,N 12 This assumption is needed for the solvability of the output regulation problem In other words, 12 is part of the regulator equation [Francis Byrnes Isidori FBI equations [8], [9], which provides necessary and sufficient condition for the solvability of the considered problem In fact, it is difficult to solve 12 Fortunately, one can always approximate the mapping ℶ i v in 12 arbitrarily closely [16] However, we do not need to solve 12 to be able to implement the controller It is sufficient for it to exist In the following, we will give a simple explanation to Assumption 1 more details can be seen in [8] and [9] Since the system 1 has an output zeroing manifold u i v = F Γ 2 v as defined by the solution of u i v = z 2i v/ vγv, there exists a stable feedback control u i such that the output zeroing manifold M = z i,ζ i,θ i z i,ζ i,θ i = z 1i v, z 2i v, ℶ i v} is also a stable center manifold of the closed-loop system, which is contained in the kernel of the mapping et = e 1 t T,,e N t T Here, we list other assumptions needed to solve the considered leader-following formation problem Assumption 2: Node the exosystem is always globally reachable in Ḡσt Assumption 3: The real parts of the eigenvalues of matrix Γ defined in 4 are nonnegative Denote A = V D ISTRIBUTED CONTROLLER DESIGN FOR SWITCHING CASES I2, B =, C =I I 2 2 Then, the linear model of the system 1 and the regulated error 7 can be written as ẑi = Aẑ i + Bu i ẑ i = z i ζ i e i = Cẑ i + Fv First, we will construct a steady-state generator and a canonical internal model candidate [8], [17] for the considered multirobot systems 1 with output u i

5 WANG et al: SWITCHING MULTIROBOT SYSTEMS VIA INTERNAL MODEL 821 Setting F Ẑ =, U = F Γ 2 13 F Γ it satisfies the regulator equation as follows: ẐΓ =A Ẑ + BU CẐ + F = 14 The following lemma shows the existence of an observable steady-state generator and a canonical internal model candidate for the considered multirobot systems 1 Lemma 4: Under Assumption 3, it is always possible to find an observable steady-state generator and a canonical internal model candidate with output u i = Uv,i =1,,N,forthe multirobot systems 1, independent of any switching σt Proof: Denote the minimal polynomial of Γ as and then polyγ = λ m + γ 1 λ m + + γ m λ + γ m Θ =T U UΓ UΓ m 15 where U is defined in 13, and T is a nonsingular matrix to be determined Take T 1 1 Φ = 1, Ψ =, γ m γ m γ 2 γ 1 Φ Ψ Φ=, Ψ= 16 Φ Ψ Therefore, the multirobot systems 1 has a linear observable steady-state generator Θ, Υ, Ξ} with output u i as follows: Θ= Θ Θ 17 Υ=TΦT Ξ=ΨT Then, we propose a special class of internal model candidate based on the constructed steady-state generator 17 Pick any matrices M R m m, Mg R m 1 with M, M g is controllable and M is Hurwitz with disjoint spectra with Φ Set M = Then, we claim that M M, M g = Mg Mg η i = Mη i + M g u i, η i R 2m 18 is an internal model candidate of the system with output u i, i =1,,N Since the spectra of the matrices Φ and M are disjoint and Ψ, Φ is observable, according to [8], there exists a unique and nonsingular matrix T such that T Φ M T = M g Ψ 19 T Set T =, and then T MΘ+M g U = MΘ+M g ΨT Θ = T ΦT Θ =ΥΘ Therefore, 18 is an internal model candidate of the multirobot systems 1 with output u i for i =1,,N From Lemma 4, system 1 has an observable steady-state generator 17 and an internal model candidate 18 with output u i, i =1,,N Then, we obtain an augmented system, ie, ẑ i = Aẑ i + Bu i η i = Mη i + M g u i i =1,,N 2 e i = Cẑ i + Fv To solve the leader-following formation problem of the switching multirobot systems 1, we construct the following observer-based feedback: u i =ΨT η i + K ξ ξ i η i = Mη i + M g ΨT η i + K ξ ξ i 21 ξ i =A ξ + B ξ K ξ ξ i + L ξ e iv ê iv, where A BΨT A ξ = M + M g ΨT B, B ξ = M g e iv = j N i a ij Cẑ i Cẑ j +a i Cẑ i + Fv, 22 ê iv = j N i a ij Cξ i Cξ j +a i Cξ i 23 with ξ i as the estimation of ẑ i and K ξ and L ξ to be determined later Letting x c = colx 1,η 1,x 2,η 2,,x N,η N,ξ 1,,ξ N, we have the closed-loop system under the feedback 21 as follows: ẋc = A σ c x c + Bc σ v 24 e = C c x c + D c v with A σ c = IN A ξ I N B ξ K ξ H σ L ξ C ξ I N A ξ +B ξ K ξ H σ L ξ C ξ where A ξ and B ξ are defined in 22, and 25 C ξ =C 26 Bc σ =,C c =I N C ξ,d c = 1 F H σ L ξ F

6 822 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART B: CYBERNETICS, VOL 42, NO 3, JUNE 212 For the closed-loop systems 24, the following result can be obtained noting that the result was obtained in [23] Lemma 5: In addition to Assumptions 2 and 3, we suppose that 1 A p c, p Pare Hurwitz, and there exists a unique matrix X c satisfying Xc Γ=A σ c X c + Bc σ 27 C c X c + D c =, with A σ c, Bc σ, C c, D c defined in 24 2 There is a common Lyapunov function V x c for x c = A σ c x c, x c = x c X c v 28 with V 28 negative definite Then, the leader-following formation problem of the considered systems 1 with switching topology can be solved under controller 21 Note that Lemma 5 is only a general result without any design procedure In the following, we will give our main result and show the solvability of K ξ, L ξ in the controller 21 Theorem 1: Let Assumptions 1, 2, and 3 hold Then, a group of robots 1, 2, 4, and 7 can achieve global leaderfollowing formation via distributed output feedback controller 9 with u i defined in 21 Proof: Step 1: There is a common regulation matrix X c satisfying the regulator equation 27 with switching topology under controller 21 Set Ẑ Θ Ẑ X c = 29 Θ where Ẑ is defined in 13, and Θ is defined in 17 From 14, Ẑ and U, depending only on A, B, C, and F, are independent of σt According to 15 and 17, Θ is also independent of σt Therefore, X c is independent of σt In the following, we will show that X c is the solution of the regulator equation in 27, with A σ c, Bc σ, C c, and D c defined in 24 Since Ẑ is the solution of the regulator equation 14, then ẐΓ =AẐ + BU From the proof of Lemma 4, we have Then U =ΨT Θ ẐΓ =AẐ + BΨT Θ 3 According to the proof of Lemma 4, we have M + M g ΨT Θ = T ΦT Θ 31 with Note that Thus Φ Θ=, Θ Θ Θ =T U UΓ UΓ m U UΓ UΓ m = UΓ UΓ 2 UΓ m ΘΓ = M + M g ΨT Θ 32 Recalling that Ẑ is the solution of the regulator equation in 14, and U =ΨT Θ, one has Then C ξ Ẑ Θ + F =, C ξ =C Ẑ Θ H σ L ξ C ξ Ẑ + H σ L ξ F = 33 Θ According to 3, 32, and 33, X c is the solution of the regulator equation in 27, with A σ c, Bc σ, C c, and D c defined in 24 Based on above analysis, there is a common regulation matrix X c Obviously, we can obtain 28 Then, according to Lemma 5, we only need to find a common Lyapunov function and prove the convergence Step 2: There is a common Lyapunov function for system 28 with A σ c and X c given in 25 and 29, respectively Set I ˆx c = x I I c Then, we have IN A ˆx c = ξ +B ξ K ξ I N B ξ K ξ I N A ξ H σ L ξ C ξ ˆx c 34 Let T σ be an orthogonal transformation such that U σ = T σ H σ Tσ is a diagonal matrix with the eigenvalues of H σ along the diagonal Clearly, U σ I N =T σ I N H σ I N Tσ I N Setting I x c = ˆx T σ I c N we obtain IN A x c = ξ +B ξ K ξ I N B ξ K ξ x I N A ξ U σ L ξ C ξ c 35

7 WANG et al: SWITCHING MULTIROBOT SYSTEMS VIA INTERNAL MODEL 823 In the following, we will construct the matrices K ξ, L ξ to guarantee the stability of the system 35 Set x c = x c1,, x cn T, and then 35 becomes x ci = Aξ + B ξ K ξ B ξ K ξ A ξ λ iσ L ξ C ξ x ci, i =1,,N where λ iσ is the ith eigenvalue of H σ Since A λi BΨT B M + M g ΨT λi M g A λi B = I I M λi M g ΨT I and M is Hurwitz together with A, B is stabilizable, we can get that A ξ,b ξ, defined in 22, is stabilizable Then, there exists K ξ such that is Hurwitz Considering A λi A ξ + B ξ K ξ 36 BΨT M + M g ΨT λi C 37 it has full rank for all λ σφ due to the detectability of C, A and the fact that M + M g ΨT = T ΦT Using the decomposition A λi BΨT M + M g ΨT λi C = A λi B M λi M g I I C ΨT A λi B rank =6, λ ΛΓ, C it easily follows that 37 has full rank for all λ σφ Thus is detectable, where A ξ, C ξ 38 C ξ =C 39 Since 38 is detectable, according to Lemma 2, A T ξ CT ξ C ξp ξ is stable, where P ξ is the unique solution of the following Riccati equation: Set where A ξ P ξ + P ξ A T ξ P ξ C T ξ C ξ P ξ + I = 4 L T ξ = max 1, 1 λ } C ξ P ξ 41 Under Assumption 2, according to Lemma 1, all eigenvalues of the matrices H p p P are positive Moreover, since the set P is finite, λ > is fixed Since 36 is Hurwitz, there exist positive definite matrices P and Q such that P A ξ + B ξ K ξ +A ξ + B ξ K ξ T P = Q From 38 and Lemma 2, all real parts of the eigenvalues of A ξ λh σ L ξ C ξ [L ξ defined in 41] are negative since those for A T ξ αct ξ C ξp ξ [P ξ defined in 4] are so for any α 1, where λh σ denotes any eigenvalue of matrix H σ Set Āξ = A ξ λh σ L ξ C ξ Then P ξ Ā T ξ + ĀξP ξ = I + P ξ Cξ T C ξ P ξ 2λH σ max 1, 1 λ } P ξ Cξ T C ξ P ξ Therefore which implies that P ξ Ā T ξ + ĀξP ξ I/2 P ξ C T ξ C ξ P ξ, P ξ Ā ξ +ĀT ξ P ξ Q ξ, Q ξ := P ξ where Q ξ is obviously positive definite Take a Lyapunov function for system 34, ie, where 2 /2 C T ξ C ξ 43 V ˆx c =ˆx T c I N P ˆx c 44 P /ϖ P = P ξ with ϖ> to be determined Clearly V =ˆx T c I N P ˆx c = x T c I N P x c = N x T cip x ci because Tσ T = Tσ Therefore, the form of V is a candidate of a common Lyapunov function independent of switching The interconnection graph associated with H p, p P is unchanged and connected on an interval [t i,t i+1 Therefore, A p c is constant in the interval Consider the derivative of V with t [t i,t i+1 where V 35 = i=1 N x T Q x ci ci 45 i=1 Q /ϖ Π Q = Π T, Π=P Q /ϖb ξ K ξ ξ From Lemma 3, the positive definite of Q can be guaranteed by the positive definite of the matrices λ=mineigenvalues of H p,p P, Assumption 2 holds} 42 Q /ϖ, Q ξ ϖπ T Q Π

8 824 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART B: CYBERNETICS, VOL 42, NO 3, JUNE 212 We proceed to show Q ξ ϖπ T Q Π is positive definite Since Q ξ ϖπ T Q Π=Q ξ 1 ϖ [P B ξ K ξ ] T Q [P B ξ K ξ ] it is positive definite when ϖ is sufficiently large Therefore, Q is positive definite Recalling the dwell-time assumption, it follows from 45 that V ˆλV/ P 2 t with ˆλ = mineigenvalues of Q}, which implies system 35 is asymptotically stable, thus system 34 is asymptotically stable According to the analysis of Lemma 5, the leader-following formation problem for system 1 is solved via distributed output feedback 21 with K ξ, L ξ defined in 36 and 41, respectively Therefore, controller 9 with u i defined in 21 solves the leader-following formation for the considered multirobot systems 1, 2, 4, and 7 Many existing results on robotic networks eg, [21] assume that all the robots can get access to the information of the exosystem and the interconnection topology between agents is fixed For this case, we have the following result, whose control design is the same as 1 proposed in [21] based on observer design Corollary 1: If all the robots can get the information from the leader and Assumptions 1 and 3 hold, then the corresponding controller for each robot becomes u i =ΨT η i + K ξ ξ i η i = Mη i + M g K ξ ξ i +ΨT η i A BΨT ξ i = M + M g ΨT ξ i B + K ξ ξ i + L ξ Cẑ i Cξ i M g for i =1,,N Here is an example for illustration Example 1: Consider a group of seven robots modeled in the form of 1 and an exosystem v1 = v 2 v 2 = which generates a reference trajectory The robots will follow the reference trajectory in a given formation The relative position for the robots is uniquely defined by the parameters μ 1 1 =, μ i 1 =2, i =2,,7, and μ 1 2 =, μ 2 2 = π/3, μ 3 2 =2π/3, μ 4 2 = π, μ 5 2 = 2π/3, μ 6 2 = π/3, μ 7 2 = Let h i = μ i 1Rμ i 2 Then, the relative position vectors for the robots with respect to the virtual leader are given as follows: h 1 =,, h 2 =1, 3, h 3 =, 3, h 4 = 2,, h 5 =, 3, h 6 =1, 3, and h 7 =2, To make the group of robots track the reference trajectory in a given formation, we take the regulated output as e i = z i v h i for agent i i =1,,7 In fact, z i v h i can be viewed as the planned path for robot i In this example 1 Γ= 1 1 Ẑ =, U = 1, Fw = I 2, Take 1 Φ=, Ψ= Then T =, T 2 = Then, the system 1 has a linearly observable steady-state generator Θ i, Υ, Ξ} based on 17 Also, we can take M =, M g = Then, the dynamic output feedback 21 solves the distributed robust distributed output regulation problem of the considered systems 1, where K ξ and L ξ satisfying 36 and 41, respectively, are given by K ξ =, L ξ = max 1, 1 λ } For simulation, the switchings between the two interconnection topologies are periodically carried out in the following order: G 1, G 2, G 1, G 2,} with switching period t =1 G 1 is described by a graph with adjacency weights as follows: a 23, a 32, a 14, a 41, a 34, a 43, a 45, a 54, a 46, a 64, a 57, a 75, a 1, a 2 are 1, while other weights are set as zero The eigenvalues of H 1 with G 1 are 136, 58, 1, 17, 244, 291, and 522 G 2 is described with a 23, a 32, a 14, a 41, a 31, a 13, a 52, a 25, a 46, a 64, a 57, a 75, and a 1 as 1 and other weights as zero The corresponding eigenvalues of H 2 with G 2 are 186, 268, 1, 2, 247, 373, and 434 Then, λ =136 Fig 1 demonstrates the effectiveness by showing positions of the points interested of the seven agents Fig 2 shows the relative distances of the six robots with respect to the central robot Note that it is equal to 2 in our example Fig 3 is the regulated error [described in 7] of the seven robots lim t e i t =, i =1,,7, is our control objective

9 WANG et al: SWITCHING MULTIROBOT SYSTEMS VIA INTERNAL MODEL 825 VI CONCLUSION In this paper, we have analyzed the leader-following formation problem of multirobot systems with switching interconnection topology We solved the problem from an output regulation viewpoint With the canonical internal-model-based dynamic feedback, the considered formation control problem with switching interconnection topology is solved Unfortunately, the presented method in its current form does not guarantee collision avoidance It needs careful further consideration and might be a fruitful topic for further research Fig 1 Fig 2 Fig 3 Positions of the seven robots Relative distances of the six robots with respect to the central robot Regulated error of the seven robots REFERENCES [1] D V Dimarogonas and K H Johansson, Further results on the stability of distance-based multi-robot formations, in Proc Amer Control Conf, Jun 29, pp [2] B A Francis and W M Wonham, The internal model principle of control theory, Automatica, vol 12, no 5, pp , Sep 1976 [3] J A Fax and R M Murray, Information flow and cooperative control of vehicle formation, IEEE Trans Autom Control,vol49,no9,pp , Sep 24 [4] C Godsil and G Royle, Algebraic Graph Theory New York: Springer- Verlag, 21 [5] R A Horn and C R Johnson, Matrix Theory Cambridge, UK: Cambridge Univ Press, 1986 [6] W Dong and J A Farrell, Cooperative control of multiple nonholonomic mobile agents, IEEE Trans Autom Control, vol 53, no 6, pp , Jul 28 [7] M Egerstedt and X Hu, Formation constrained multi-agent control, IEEE Trans Robot Autom, vol 17, no 6, pp , Dec 21 [8] J Huang, Nonlinear Output Regulation: Theory & Applications Phildelphia, PA: SIAM, 24 [9] A Isidori and C Byrnes, Output regulation of nonlinear systems, IEEE Trans Autom Control, vol 35, no 2, pp , Feb 199 [1] A Jadbabaie, J Lin, and A Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Trans Autom Control, vol 48, no 6, pp , Jun 23 [11] J Wang, D Cheng, and X Hu, Consensus of multi-agent linear dynamic systems, Asian J Control, vol 1, no 2, pp , 28 [12] Z Li, Z Duan, G Chen, and L Huang, Consensus of multiagent systems and synchronization of complex networks: A unified viewpoint, IEEE Trans Circuits Syst I, Reg Papers, vol 57, no 1, pp , Jan 21 [13] Z Meng, W Ren, Y Cao, and Z You, Leaderless and leader-following consensus with communication and input delays under a directed network topology, IEEE Trans Syst, Man, Cybern B, Cybern, vol 41, no 1, pp 75 88, Feb 211 [14] G Chen and F L Lewis, Distributed adaptive tracking control for synchronization of unknown networked Lagrangian systems, IEEE Trans Syst, Man, Cybern B, Cybern, vol 41, no 3, pp , Jun 211 [15] G Lafferriere, A Williams, J Caughman, and J J P Veerman, Decentralized control of vehicle formations, Syst Control Lett, vol 54, no 9, pp , Sep 25 [16] J Huang and W J Rugh, An approximate method for the nonlinear servomechanism, IEEE Trans Autom Control, vol 37, no 9, pp , Sep 1992 [17] V O Nikiforov, Adaptive non-linear tracking with complete compensation of unknown disturbances, Eur J Control, vol4, no2,pp , 1998 [18] R Olfati-Saber, Flocking for multi-agent dynamic systems: Algorithms and theory, IEEE Trans Autom Control, vol 51, no 3, pp 41 42, Mar 26 [19] W M Wonham, Linear Multivariable Control New York: Springer- Verlag, 1985 [2] Z Xi and Z Ding, Global decentralised output regulation for a class of large-scale nonlinear systems with nonlinear exosystem, IET Control Theory Appl, vol 1, no 5, pp , Sep 27 [21] V Gazi, Formation control of mobile robots using decentralized nonlinear servomechanism, in Proc 12th Mediterranean Conf Control Autom, Kusadasi, Turkey, Jun 24 [22] X Wang, Coverage boundary of unknown environment using mobile sensor networks, Int J Control, vol 84, no 4, pp , Apr 211 [23] X Wang, Y Hong, J Huang, and Z Jiang, A distributed control approach to a robust output regulation problem for multi-agent linear systems, IEEE Trans Autom Control, vol 55, no 12, pp , Dec 21

10 826 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART B: CYBERNETICS, VOL 42, NO 3, JUNE 212 [24] Z Lin, B Francis, and M Maggiore, Necessary and sufficient graphical conditions for formation control of unicycles, IEEE Trans Autom Control, vol 5, no 1, pp , Jan 25 [25] Y Hong, J Hu, and L Gao, Tracking control for multi-agent consensus with an active leader and variable topology, Automatica, vol 42, no 7, pp , Jul 26 [26] J E Slotine and W Li, Applied Nonlinear Control Englewood Cliffs, NJ: Prentice-Hall, 1991 [27] W Ni and D Cheng, Leader-following consensus of multi-agent systems under fixed and switching topologies, Syst Control Lett, vol59,no3/4, pp , Mar/Apr 21 Wei Ni received the PhD degree from the Chinese Academy of Sciences, Beijing, China, in 21 He is currently a Lecturer with the School of Science, Nanchang University, Nanchang, China His research interests include switched systems and complex systems control Xiaoli Wang received the PhD degree from the Chinese Academy of Sciences, Beijing, China, in 21 She is currently a Lecturer with School of Information and Electrical Engineering, Harbin Institute of Technology at Weihai, Weihai, China Her current research interests include system modeling and multiagent systems Xinsheng Wang received the PhD degree in control science and engineering from the Harbin Institute of Technology, Weihai, China, in 22 She is currently an Associate Professor with the Harbin Institute of Technology at Weihai Her current research interests include multiagent coordination and control system desig