FORMATION INPUT-TO-STATE STABILITY. Herbert G. Tanner and George J. Pappas

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1 Copyright 2002 IFAC 5th Triennial Worl Congress, Barcelona, Spain FORMATION INPUT-TO-STATE STABILITY Herbert G. Tanner an George J. Pappas Department of Electrical Engineering University of Pennsylvania Philaelphia, PA 902 Abstract: This paper introuces the notion of formation input-to-state stability in orer to characterize the internal stability of leaer-follower formations, with respect to inputs receive by the formation leaer. Formation ISS is a weaker form of stability than string stability since it oes not require inter-agent communication. It relates group input to internal state of the group through the formation graph ajacency matrix. In this framework, ifferent formation structures can be analyze an compare in terms of their stability properties an their robustness. Keywors: Formations, graphs, interconnecte systems, input-to-state stability.. INTRODUCTION Formation control problems have attracte increase attention following the avances on communication an computation technologies that enable the evelopment of istribute, multi-agent systems. Direct fiels of application inclue automate highway systems (Varaiya, 993; Swaroop an Herick, 996; Yanakiev an Kanellakopoulos, 996, reconnaissance using wheele robots (Balch an Arkin, 998, formation flight control (Mesbahi an Haaegh, 200; Bear et al., 2000 an sattelite clustering (McInnes, 995. For coorinating the motion of a group of agents, three ifferent interconnection architectures have been consiere, namely behavior-base, virtual structure an leaer-follower. In behavior base approach (Balch an Arkin, 998; Lager et al., 994; Yun et al., 997, several motion premitives are efine for each agent an then the group behavior is generate as a weighte sum of these primary behaviors. Behavior base control schemes are usually har to analyze formally, although some attempts have been mae (Egerstet, In leaer-follower approaches (Bear et al., 2000; Desai an Kumar, 997; Tabuaa et al., 200; Fierro et al., 200, one agent is the leaer of the formation an all other agents are require to follow the leaer, irectly or inirectly. Virtual structure type formations (Tan an Lewis, 997; Egerstet an Hu, 200, on the other han, usually require a centralize control architecture. Balch an Arkin (998 implement behavior - base schemes on formations of unmanne groun vehicles an test ifferent formation types. Yun et al. (997 evelop elementary behavior strategies for maintaining a circular formation using potential fiel methos. Egerstet an Hu (200 aopt a virtual structure architecture in which the agents follow a virtual leaer using a centralize potential-fiel control scheme. Fierro et al. (200 evelop feeback linearizing controllers for the control of mobile robot formations in which each agent is require to follow one or two leaers. Tabuaa et al. (200 investigate the conitions uner which a set of formation constraints can be satisfie given the ynamics of the agents an consier the problem of obtaining a consistent group abstraction for the whole formation. This paper focuses on a ifferent problem: given a leaer-follower formation, investigate how the leaer input affects the internal stability of the overall for-

2 mation. Stability properties of interconnecte systems have been stuie within the framework of string stability (Swaroop an Herick, 996; Yanakiev an Kanellakopoulos, 996. String stability actually requires the attenuation of errors as they propagate in the formation. However, sting stability conitions are generally restrictive an generally require inter-agent communication. It is known, for instance (Yanakiev an Kanellakopoulos, 996 that string stability in autonomous operation of an AHS with constant intervehicle spacing, where each vehicle receives information only with respect to the preceing vehicle, is impossible. We therefore believe that a weaker notion of stability of interconnecte system that relates group objectives with internal stability woul be useful. Our approach is base on the notion of input-to-state stability (Sontag an Wang, 995 an exploits the fact that the cascae interconnection of two input-to-state stable systems is itself input-to-state stable (Khalil, 996; Krstić et al., 995. This property allows the propagation of input-to-state gains through the formation structure an facilitates the calculation of the total group gains that characterize the formation performance in terms of stability. Formation ISS is a weaker form of stability than string stability, in the sense that it oes not require inter-agent communication an relies entirely on position feeback only (as oppose to both position an velocity feeback from each leaer to its follower. We represent the formation by means of a formation graph (Tabuaa et al., 200. Graphs are especially suite to capture the interconnections (Tabuaa et al., 200; Fierro et al., 200 an information flow (Fax an Murray, 200 within a formation. The propose approach provies a means to link the formation leaer s motion or the external input to the internal state an the ajacency matrix of the formation. It establishes a metho for comparing stability properties of ifferent formation schemes. The rest of the paper is organize as follows: in section 2 the efinitions for formation graphs an formation input-to-state stability (ISS are given. Section 3 establishes the ISS properties of an leaer-follower interconnection an in section 4 it is shown how these properties can be propagate from one formation graph ege to another to cover the whole formation. Section 5 provies examples of two stucturally ifferent basic formation configurations an inicates how interconnection ifferences affect stability properties. In section 6 results are summarize an future research irections are highlighte. 2. FORMATION GRAPHS A formation is being moele by means of a formation graph. The graph representation of a formation allows a unifie way of capturing both the ynamics of each agent an the inter-agent formation specifications. All agent ynamics are suppose to be expresse by linear, time invariant controllable systems. Formation specifications take the form of reference relative positions between the agents, that escribe the shape of the formation an assign roles to each agent in terms of the responcibility to preserve the specifications. Such an assignment imposes a leaer-follower relationship that leas to a ecentralize control architecture. The assignment is expresse as a irecte ege on the formation graph (Figure Fig.. An example of a formation graph Definition 2.. (Formation Graph. A formation graph F =(V,E,D is a irecte graph that consists of: A finite set V = {v,...,v l } of l vertices an a mapping v i T R n that assignes to each vertice an LTI control system escribing the ynamics of a particular agent: 3 3 ẋ i = A i x i + B i u i where x i R n is the state of the agent accociate with vectice v i, u i R m is the agent control input an A i R n n,b i R m m is a controllable pair of matrices. A binary relation E V V representing a leaer-follower link between agents, with (v i,v j E whenever the agent associate with vectice v i is to follow the agent of v j. A finite set of formation specifications D inexe by the set E, D = { ij } (vi. For each ege,v j E (v i,v j, ij R n, enotes the esire relative istance that the agent associate with vectice v i has to maintain from the agent associate with agent v j. Our iscussion specializes in acyclic formation graphs. This implies that there can be at least one agent v L that can play the role of a leaer (i.e. a vectice with no outgoing arrow. The input of the leaer can be use to control the evolution of the whole formation. The

3 graph is orere starting from the leaer an following a breath-first numbering of its vertices. For every ege (v i,v j we associate an error vector that expresses the eviation from the specification prescribe for that ege: z ij x j x i ij R ni j The formation error z is efine as the augmente vector forme by concatenating the error vectors for all eges (v i,v j E: z [ z e ]e E A natural way to represent the connectivity of the graph is by means of the ajacency matrix, A. We will therefore consier the mapping E R l l that assigns to the set E of orere vertice pairs (v i,v j the ajacency matrix A E R l l. Our aim is to investigate the stability properties of the formation with respect to the input u L of the formation leaer. We thus nee to efine the kin of stability in terms of which the formation will be analyze: Definition 2.2. (Formation Input-to-State Stability. A formation is calle input-to-state stable iff there is a class KL function β an a class K function γ such that for any initial formation error z(0 an for any boune inputs of the formation leaer u L ( the evolution of the formation error satisfies: ( z(t β(z(0,t+γ supu L ( By investigating the formation input-to-state stability we establish a relationship between the amplitue of the input of the formation leaer an the evolution of the formation errors. This will provie upper bouns for the leaers input in orer for the formation shape to be maintaine insie some esire specifications. Further, it will allow to characterize an compare formations accoring to their stability properties. 3. EDGE INPUT-TO-STATE STABILITY In the leaer-follower configuration, one agent is require to follow another by maintaining a constant istance, x j x i = ij. If agent i is require to follow agent j, then this objective is naturally pursue by applying a follower feeback control law that epens on the relative istance between the agents. For x i = x j ij to be an equilibrium of the close loop control system: ẋ i = A i x i + B i u i it shoul hol that A i (x j ij R(B i ; otherwise the follower cannot be stabilize at that istance from its leaer. Suppose that there exists an e ij such that B i e ij = A i (x j ij. Then the following feeback law can be use for the follower: u i = (x j x i ij +e ij leaing to the close loop ynamics: ẋ i =(A i (x i x j + ij Then the error ynamics of the i- j pair of leaerfollower becomes: ż ij =(A i z ij + ẋ j which can be written, assuming that agent j follows agent k: ż ij =(A i z ij + g ij (2 where g ij (A j B j K j z jk. The stability of the follower is thus irectly epenent on the matrix (A i B i, the eigenvalues of which can be arbitrarily chosen, an the interconnection term g ij. The interconnection term can be boune as follows: g ij λ M (A j B j K j z jk where λ M ( is the maximum eigenvalue of a given matrix. If is chosen so that A i is Hurwitz, then the solution of the Lyapunov equation: P i (A i +(A i T P i = I provies a symmetric an positive efinite matrix P i an a natural Lyapunov function caniate V i = x T i P i x i for the interconnection ynamics (2 that satisfies: λ m x i V i λ M x i where λ m ( an λ M ( enote the minimum an maximum eigenvalue of a given matrix, respectively. For the erivative of V i : V i x i 2 + 2λ M λ M (A j B j K j x i z jk ( θx i 2 0 for all x i 2λ M λ M (A j B j K j θ z jk where θ (0,. Viewing (2 as a perturbe system: z ij (t ( λm 2 θ zij 2λ (0e M t λ m + 2(λ M 3 2 λ M (A j B j K j z (λ m jk 2 θ Equation (3 implies that (2 is input-to-state stable with respect to z jk as input an where θ β i (r,t=r β i e 2λ M t (3 γ i (r= γ i r (4 ( λm (P β i = i 2 λ m γ i = 2(λ M 3 2 λ M (A j B j K j (λ m 2 θ

4 4. FROM EDGE STABILITY TO FORMATION STABILITY An important property of input-to-state stability is that it is preserve in cascae connections. The property allows propagation of ISS properties from one agent to another, all the way up to the formation leaer. This proceure will yiel the global input gains of the leaer an give a measure of the sensitivity of the formation shape with respect to the input applie at the leaer. In the previous section it was shown that uner the assumption of pure state feeback, a formation graph ege is input-to-state stable. The gain functions for the cascae interconnection are given as: ẋ = f (t,x,x 2,u ẋ 2 = f 2 (t,x 2,u β (r,t=β (2β (r, t 2 +2γ (2β 2 (r,0, t 2 +γ (2β 2 (r, t 2 + β 2 (r,t, γ(r=β (2γ i (2γ 2 (r+2r,0 +γ (2γ 2 (r+2r+γ 2 (r. From the general expressions an (3-(4 the leaer j - follower i composite ISS gains become: ( β ij (r,t= 2 β 2 i e θ 2λ M t + 4 γ i βi β j e θ 4λ M t +2 γ i β j e θ 4λ M (P j t + β j e θ t 2λ M (P j r γ ij (r= (4 β i γ i γ j + 4 β i γ i + 2 γ i γ j + 2 γ i + γ j r Base on the above we efine: β ij β ij (,0=2 β 2 i + 4 γ i βi β j + 2 γ i β j + β j γ ij γ ij (=4 β i γ i γ j + 4 β i γ i + 2 γ i γ j + 2 γ i + γ j Although the β ij function for any connecte i, j vertices on the formation graph has to be calculate through successive compositions, the γ ij function can be algorithmically calculate: Let a i be the i row of the n n ajacency matrix A of the formation graph, F =(V, E, D. Define: β 0 [ ] T β βn, γ 0 [ ] T γ γ n an for the k + iteration let β k+ [ ] k+ k+ T β β n, γ k+ [ γ k+ γ n k+ ] T be given recursively as: where γ k+ = γ k + n n k c k γ, β k+ = β k + n n k c k β n n k =[0 0 0] T }{{}, n k c k γ = 2(2a n k β k a n k γ k γ k n k + 2a n k β k a n k γ k +a n k γ k γ n k k + a n k γk, c k β = 2(2a β n k k a n k γ k k β n k + 2a n k γ k k β n k +a n k β k The algorithm terminates in at most n steps, since this is maximum path length in a irecte acyclic graph with n vectices. It results in a γ vector containing the formation ISS input gains of each agent. These are the gains that express the sensitivity of the formation stability with respect to the ynamic interaction at each agent ue to interconnection. Thus if agent i follows agent j, then the steay state of the formation errors, (z k ss of all eges k between the escenants of agent i is boune: (z k ss γ i z ij The steay state error for the whole formation will then be boune by a linear function of the norm of the leaer s velocity: z ss γ ẋ 5. EXAMPLES Consier the case of three agents where one is assigne to be a leaer of the formation an the other two have to be followers. Denote the leaer by v an the followers by v i, i = 2,3. Suppose that the task is for v 2 to follow v keeping a constant istance an v 3 to follow v 2 keeping the same constant istance (Figure 2. Fig. 2. Serial interconnection 2 3 A simple one imensional moel of the formation coul be given as: ẋ = u ẋ 2 = k( + x ẋ 3 = k( + x 2 x 3 where x i, i =,2,3 are each agent absolute coorinates, u is the leaer s velocity, is the inter-agent esire istance, an k is the gain of the feeback law which is base on relative position measurements. By a change of coorinates,

5 z = x z 2 = x 2 x z 3 = x 3 2 the formation equations can be written as: 2 3 ż = u ż 2 = kz 2 u ż 3 = kz 3 + kz 2 For the 2 interconnection, a Lyapunov function caniate coul be: V 2 2 z2 2 Obviously, 2 z 2 2 V 2 2 z 2 2. Then, V 2 = kz z 2 u Select any θ (0, an write V 2 as V 2 = k( θz 2 2 kθz z 2 u k( θz 2 2, for z 2 > sup u kθ, θ (0,. Then it follows that, z 2 z 2 (0 e 2k( θt + sup u kθ = β 2 ( z 2 (0,t+γ 2 (sup u The ISS input-gain function for agent v 2 is γ 2 = sup u kθ Similarly, for agent v 3 a Lyapunov function caniate coul be: V 3 (z 3 2 z2 3 an its time erivative woul then be V 3 (z 3 = kz kz 3 z 2 For z 3 > sup z 2 (τ θ it follows that γ 3 = sup z 2 (τ θ Then the formation, as a cascae connection of the subsystems of agents v 2 an v 3, is input-to-state stable with u = 6 + 6kθ + θ γ(sup kθ 2 sup u In the secon case, both agents v 2 an v 3 are to follow the leaer at esignate istances that are for v 2 an 2 for v 3. The formation shape is the same as in the previous case, however the interconnections are ifferent as reflecte in the ifferent formation graphs (Figure 3. The formation equations in this case are: ẋ = u ẋ 2 = k( + x ẋ 3 = k(2 + x 2 x 3 which by a change of variables: Fig. 3. Parallel interconnection z = x z 2 = x 2 x z 3 = x 3 2 For the same Lyapunov caniate functions use in the previous case, it follows that: β 2 = z 2 (0 e 2( θt, γ 2 = kθ sup u β 3 = z 3 (0 e 2( θt, γ 3 = kθ sup u an the input gain for the formation is: γ(sup u = 2 kθ sup u It can be shown analytically that the secon formation can outperform the first in terms of the magnitue of relative errors with respect to the leaer s velocity. Specifically, if we enote by γ s the input-to-state gain of the first interconnection connection an by γ p the input-to-state gain of the secon interconnection, γ s sup u = 6 + 6kθ + θ kθ 2 6kθ + 7θ = kθ 2 = 6k kθ kθ 2 kθ = 6. CONCLUSIONS 6θ + 6kθ + θ kθ 2 γ p sup u In this paper, the notion of formation input-to-state stability has been introuce. This form of stability can be use to characterize the internal state of a formation that has a leaer-follower achitecture, an establishes a link between the motion of the leaer of the formation or its external input an the shape of the formation. Formation ISS is a weaker form of stability than string stability, in the sense that it oes not require inter-agent communication an relies entirely on position feeback only (as oppose to both position an velocity feeback from each leaer to its follower. Moreover, it establishes a link between the formation internal state an the outsie worl. In the propose framework, ifferent formation structures can be analyze an compare in terms of their stability properties. Future work is irecte towars investigating the effect of (limite inter-agent communication on formation stability an consistent ways of group abstractions that are base on the formation ISS properties.

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