Decentralized Formation Control with A Quadratic Lyapunov Function

# Decentralized Formation Control with A Quadratic Lyapunov Function

Xudong Chen This work has been partially supported by grant NSF ECCS 13-51586.Xudong Chen is with Department of Electrical and Computer Engineering and Coordinated Science Laboratory, University of Illinois at Urbana-Champaign, email: xdchen@illinois.edu
###### Abstract

In this paper, we investigate a decentralized formation control algorithm for an undirected formation control model. Unlike other formation control problems where only the shape of a configuration counts, we emphasize here also its Euclidean embedding. By following this decentralized formation control law, the agents will converge to certain equilibrium of the control system. In particular, we show that there is a quadratic Lyapunov function associated with the formation control system whose unique local (global) minimum point is the target configuration. In view of the fact that there exist multiple equilibria (in fact, a continuum of equilibria) of the formation control system, and hence there are solutions of the system which converge to some equilibria other than the target configuration, we apply simulated annealing, as a heuristic method, to the formation control law to fix this problem. Simulation results show that sample paths of the modified stochastic system approach the target configuration.

## I Introduction

Undirected formation control has been one of the most studied subjects in multi-agent systems. The formation control model is described by two characteristics: one is an undirected graph describing the pattern of interaction, and the other is a set of scalar functions each of which describes the interaction law between a pair of adjacent agents. A detailed description is given below

Let be an undirected connected graph of vertices . Let be the set of vertices adjacent to vertex . We then consider the motion of agents in by

 ddt→ai=∑j∈Viuij⋅(→ai−→aj),∀i=1,⋯,N (1)

Each with is a scalar function describing how adjacent agents and interact with each other. We require that be identical with for all , in other words, interactions among agents are reciprocal. In its most general form, each could be a function of , and possibly the time variable as well. In this case, we have proved in [11] that if is connected, then system (1), treated as a centralized control system, is approximately path-controllable over an open dense subset of the configuration space. Yet, if we regard system (1) as a decentralized control system, i.e, each agent only accesses part of the information, then there is a restriction on what variables each can depend on. For example, it is often assumed that each agent knows agent if and only if is its neighbor, i.e, . Then in this case each is at most a function of and , and possibly the time .

Over the last decade, there have been many solid works about using system (1) to achieve decentralized formation control. Questions about the level of interaction laws that are necessary for organizing such systems, questions about system convergence, questions about counting and locating stable equilibria, and questions about the issues of robustness and etc. have all been treated to some degree (see, for example, [2, 3, 4, 5, 6, 7, 8, 9, 1, 10, 11, 12]). For example, a popular decentralized algorithm, known as the Krick’s law [3, 4, 5, 6, 7, 8, 9], is that we assume each agent measures the mutual distance between himself and its neighbors . The control law is then given by for all where is the prescribed distance between and in the target formation. By following this decentralized algorithm, system (1) will then be a gradient system with respect to the potential function . In fact, it has been shown that if each is a continuous function depending only on the distance , then the resulting system is always a gradient system [2, 10]. However, the associated potential function often has multiple local minima, and in some cases, the number of local minima has an exponential growth with respect to the number of agents (see, for example, [6, 12]).

Also, we note that if each depends only on the mutual distance as is the case if we adopt the Krick’s law for , then how a configuration is embedded into the Euclidean space is not relevant, only the shape of the formation matters. In other words, if a configuration is an equilibrium associated with system (1), then any rotation or translation of the configuration will also be an equilibrium. In any of such case, the group of rigid motions is introduced to describe this phenomena. Two configurations will be recognized as the same target formation if they are in the same orbit with respect to the group action.

In this paper, as in many earlier work on this problem, we will investigate system (1), treated as a decentralized formation control system, by equipping it with a set of new control laws. What distinguishes this paper from others is that in addition to the shape of the target configuration, we also emphasize its Euclidean embedding. To be more precise, we let and be two configurations with the same centroid, i.e, , and we distinguish and in the sense that these two configurations are recognized as the same target formation if and only if . We impose the condition that and have the same centroid because of the fact that the centroid of a configuration in an undirected formation control system is invariant along the evolution regardless of the choice of the control laws.

The decentralized control law is then designed for each agent so that the solution of the control system may converge to the target configuration. In particular, we show that there is a quadratic Lyapunov function associated with system (1) whose unique local (global) minimum point is the target configuration. But we also note (and we will see later in the paper) that there may exist a continuum of equilibria for system (1), thus a solution of system (1) may fail to converge to the global minimum point. To fix this problem, we then modify the formation control laws by adding noise terms. This is an application of simulated annealing to formation control systems. Simulation results then show that sample paths of the modified stochastic system approach the global minimum point.

The rest of this paper is organized as follows. In section 2, we will first specify what information each agent knows, or in other words what variables can depend on. Then we will introduce the decentralized formation control law, and establish the convergence of system (1). In section 3, we will explore one of the limitations of this formation control model by showing that there may exist a continuum of equilibria of system (1). For simplicity, we will only focus on trees as a special type of network topologies. In this special case, we show that there is a simple condition for determining whether a configuration is an equilibrium or not, and thus there is a geometric characterization of the set of equilibria of system (1). The existence of continuum of equilibria poses a problem about the convergence of system (1) to the target configuration. In section 4, we will focus on fixing this problem by applying the technique of simulated annealing to the algorithm. Simulation results then show that a typical sample path will converge to the target configuration.

## Ii A Lyapunov Approach for Decentralized Formation Control

In this section, we will introduce the decentralized formation control law and show that system (1), when equipped with this control law, converges to the set of equilibria. But before that, we need to be clear about what we mean by a decentralized formation control system. So in the first part of this section, we will specify what information each agent knows, i.e, what variables each can depend on. Also we will specify how the information of the target configuration is distributed among agents.

### Ii-a Information distribution among the agents

We first introduce the underlying space of system (1). As interactions among agents are reciprocal, the centroid of a configuration is always invariant along the evolution in an undirected formation control system, so we may as well assume that the centroid of a configuration is located at the origin. The configuration space , as the underlying space of system (1), is then defined by

 P:={p=(→a1,⋯,→aN)∈Rn×N∣∣N∑i=1→ai=0} (2)

It is clear that is a Euclidean space of dimension . In this paper, we assume that

 q=(→b1,⋯,→bN)∈P (3)

is the target configuration, i.e, each is the target position for agent . We will now specify what information each agent can access. In this paper, if is an edge of , we then assume that

• agent knows ;

• agent is able to measure at any time ;

Consequently, if is an edge of , then we require that each scalar function depend only on , and possibly the time variable .

### Ii-B The decentralized formation control law

Suppose is an edge of , we then let the control law be defined as

 uij:={⟨→bj−→bi,→aj−→ai⟩/|→aj−→ai|2−1→aj≠→ai0→aj=→ai (4)

where is the standard inner-product of two vectors, and is the standard Euclidean norm of a vector. We note that in this definition, if we exchange roles of vertex and vertex , then will be identical with . This is consistent with our assumption that the formation control system is undirected. The main result of this section we will prove is about the convergence of system (1) as stated below.

###### Theorem 1

Let be the decentralized control law defined by expression (4). Then there is a quadratic Lyapunov function associated with system (1) defined as

 Φ(→a1,⋯,→aN):=N∑i=1|→ai−→bi|2 (5)

Let be a solution of system (1), then

 ddtΦ(p(t))=−∑(i,j)∈Eu2ij⋅|→ai(t)−→aj(t)|2≤0 (6)

The derivative is zero if and only if is an equilibrium.

{proof}

This proof is done by explicit computation. We check that

 (7)

It is then clear that the time derivative is zero if and only if each is zero which implies that is an equilibrium. Remark I. There may exist multiple equilibria of system (1). In fact, as we will see in the next section that in the case is a tree graph there exists a continuum of equilibria. Nevertheless, there is only one local (and also global) minimum point of the potential function which is . It thus suggests that we apply simulated annealing to this formation control law as we will discuss in the last section of the paper.
Remark II. Notice that the potential function approaches infinity as goes to infinity. On the other hand, we have

 Φ(p(t))=|p(t)−q|≤|p(0)−q|=Φ(p(0)) (8)

So each solution of system (1) has to remain in a bounded set, and thus converges to the set of equilibria. In other words, no agent escapes to infinity along the evolution.

## Iii Existence of Continuum of Equilibria

In this section, we will explore the set of equilibria of system (1). The main purpose of doing this is to illustrate one of the limitations of this formation control law. It is well-known that if is a Lyapunov function for a dynamical system and is a unique equilibrium, then is stable and all solutions of the system will converge to . However, this is not the case here, i.e, the target configuration will not be the unique equilibrium of system (1). As we will see in this section there may exist a continuum of equilibria of system (1). For simplicity, we will only focus on the case where the interaction pattern is a tree graph. We focus on this special class of interaction patterns because in this case there is a simple condition telling us whether a configuration is an equilibrium or not. In particular, we will use this condition to characterize the set of equilibria of system (1) in a geometric way.

A path in a graph is a finite sequence of edges which connects a sequence of vertices. A simple path then refers to a path which does not have repeated vertices, and a circle refers to a path without repeated vertices or edges, other than the repetition of the starting and ending vertices. An undirected graph is a tree if any two vertices of are connected by a unique simple path, i.e, there is no circle in . Each tree graph can be inductively built up starting with one vertex, and then at each step, we join a new vertex via one new edge to an existing vertex. This, in particular, implies that each tree graph has a leaf, i.e, a vertex of degree one. An example of a tree graph is given in Figure 1.

### Iii-a Equilibrium condition

In this part, we show that if the graph is a tree graph, then the set of equilibria associated with system (1) can be characterized by a simple condition stated below.

###### Lemma 2

Let be a tree graph, then is an equilibrium associated with system (1) if and only if for all .

{proof}

It is clear that if each is zero, then is an equilibrium. We now prove the other direction, and the proof is done by induction on the number of agents.
Base case. Suppose , then . So if is an equilibrium, then either or , but they both imply .
Inductive step. Assume the lemma holds for with , and we prove for the case . Since each tree graph has at least one leaf, we may assume that vertex is a leaf of and it joins the graph via edge . Suppose is an equilibrium, then we must have

 ˙→ak=u1k⋅(→ak−→a1)=0 (9)

Then by the same arguments we used for proving in the base case, we conclude that . Now let

 V′:={1,⋯,k−1} (10)

and let be the subgraph induced by , i.e, for any two vertices and in , the pair is an edge of if and only if it is an edge of . It is clear that is also a tree graph. Let be a sub-configuration of consisting of agents , then is also an equilibrium under with . This holds because is an equilibrium and meanwhile , so the agent doesn’t attract or repel any agent in . By induction, we have for all . This then, combined with the condition , establishes the proof.

### Iii-B Geometry of the set of equilibria

In this part, we will use the equilibrium condition to characterize the set of equilibria of system (1).

###### Theorem 3

Let be a tree graph. Let be the set of equilibria of system (1) with defined by expression (4). Then there is a diffeomorphism of given by

 K≈ΠN−1i=1Sn−1 (11)

as a product of copies of the unit sphere in .

{proof}

The proof of the theorem will again carried out by induction on the number of agents. So we first prove for the case , and the inductive step will be given after that.
Base case. We show that Theorem 3 holds in the case . Suppose is an equilibrium, then by Lemma 2, we have , this then implies

 ⟨→b1−→b2,→a1−→a2⟩=|→a1−→a2|2 (12)

The set of equilibria associated with system (1) is characterized by equation (12), together with the condition that . Let be a subset of defined by

 W:={→v∈Rn∣∣⟨→b1−→b2,→v⟩=|→v|2} (13)

Then it is clear that is diffeomorphic to . To see this, we define a map by

 φ:→v↦12(→v,−→v) (14)

It is clear that the map is a diffeomorphism. We will now show that the set is itself a sphere in . Let

 ⎧⎨⎩→b:=12(→b1−→b2)r:=12|→b1−→b2| (15)

and let be the sphere of radius centered at in , i.e,

 (16)

It is clear by computation that

 Sr(→b)=W (17)

In fact, if lies inside , then

 ⟨→b1−→b2,→v⟩>|→v|2 (18)

and if lies outside , then

 ⟨→b1−→b2,→v⟩<|→v|2 (19)

This then completes the proof of the base case.
Inductive step. We will now use induction to prove Theorem 3. We assume that the theorem holds for with , and we prove for the case . We again assume that vertex is a leaf of and it joins the graph via edge . Let be the subgraph of induced by vertices , then is a tree graph. Let be a sub-configuration of consisting of agents . Let be a subset of defined by

 K′:={p′=(→a1,⋯,→ak−1)∣∣p′ is an equilibrium} (20)

The equilibria set is then characterized by the condition that is an equilibrium, together with the condition that . Since these two conditions are independent of each other, there is a diffeomorphism of given by

 K≈K′×Sn−1 (21)

We may translate each in so that the centroid of is zero after translation. Since is a tree graph, by induction the set is diffeomorphic to , and hence is diffeomorphic to .

One may ponder at this point whether the existence of continuum of equilibria is a consequence of the fact that a tree graph is not a rigid graph. However, it is not the case. For example, if we consider three agents evolving on a plane with being the complete graph, one can then show that the set of equilibria is diffeomorphic to a disjoint union of two circles. Though at this moment we do not have a statement about the set of equilibria in the most general case, the tree-graph cases, as well as the three-agents example suggest that it may be inevitable for system (1) to possess a continuum of equilibria.

## Iv Simulated Annealing on Formation Control

In the previous section, we have showed that there may exist a continuum of equilibria of system (1) under the proposed formation control law. This certainly affects the efficacy of the algorithm because there may exist a solution of system (1) which converges to an equilibrium other than the target configuration. In this section, we will focus on fixing the problem. In view of the fact that there is only one local (global) minimum of the quadratic Lyapunov function which is the target configuration, we attempt to apply simulated annealing, as a heuristic method, to the decentralized formation control algorithm. In particular, if we add an appropriate noise term to each , the resulting stochastic system is then described by

 d→ai=∑j∈Vi(uijdt+λij(t)dwij(t))⋅(→ai−→aj),∀i=1,⋯,N (22)

where the are independent standard Wiener processes, and is a scalar function of time and defined by

 λij(t):={c1exp(−c2t)/|→ai−→aj|if →ai≠→aj0otherwise (23)

with and positive constants. As decays along time, the impact of the noise tends to zero as goes to infinity. With these noise terms, we expect that the centroid of the stochastic formation control system is still invariant along the evolution because otherwise the entire configuration may drift to some place which is neither predictable nor controllable. Fortunately this is the case here as stated in the next theorem.

###### Theorem 4

The centroid of the configuration is invariant along the evolution of the stochastic formation control system described by expression (22).

{proof}

Let be a function defined by

 ϕ(z1,⋯,zN):=N∑i=1zi (24)

Let be the -th coordinate of agent . We need to show that

 dϕ(xk1,⋯,xkN)=0 (25)

for all . Let be a vector collecting the -th coordinates of all the agents, i.e,

 →xk:=(xk1,⋯,xkN) (26)

By defining vector , we can rewrite the system equation (22) in a matrix form as

 d→xk=U→xkdt+∑(i,j)∈EΛij(t)→xkdwij (27)

where is a symmetric matrix of zero-column/row-sum with the -th, , entry defined by

 Uij:={−uijif (i,j)∈E0otherwise (28)

and each is also a symmetric matrix of zero-column/row-sum defined by

 Λij(t)=λij(t)⋅(→ei→eTi+→ei→eTi−→ei→eTj−→ej→eTi) (29)

where is the standard basis of .

We now apply the Itō rule, and get the stochastic differential equation for as follows

 dϕ(→xk)=⟨∂ϕ(→z)∂→z∣∣z=→xk,U→xk⟩dt+12∑(i,j)∈E⟨Λij→xk,∂2ϕ(z)∂→z2∣∣→z=→xkΛij→xk⟩dt+∑(i,j)∈E⟨∂ϕ(z)∂→z,Λij→xk⟩dwij (30)

Notice that for any , we have

 ⎧⎪ ⎪⎨⎪ ⎪⎩∂ϕ(→z)∂→z=→e∂2ϕ(z)∂→z2=0 (31)

where is a vector of all ones, and

 {U→e=0Λij→e=0∀(i,j)∈E (32)

So then all inner-products in equation (30) vanish. This completes the proof.

We now give some examples of this stochastic formation control system, and illustrate how sample paths of this stochastic system evolve over time .
Examples. Consider five agents , , , and evolving in . Let be the target configuration given by

 →b1=(0,0)→b2=(−1,1)→b3=(1,1)→b4=(1,−1)→b5=(−1,−1) (33)

We will work with two network topologies, one is a star graph which is a special type of tree graph and the other is a circle. Details are described below
1. Star as the network topology. We assume that is a star graph with defined by

 E={(1,2),(1,3),(1,4),(1,5)} (34)

We then pick an initial condition given by

 →a1(0)=(1,2)→a2(0)=(−2,−1)→a3(0)=(1,−1)→a4(0)=(3,−2)→a5(0)=(−3,2) (35)

In Figure 2, we show how the value of the quadratic Lyapunov function evolves over time. The smooth curve (the green one) refers to the solution of system (1) where there is no noise term added into the control law. As this solution converges to an equilibrium, so we see from the figure that the green curve converges to a constant line along the evolution. Also it is clear that the solution, with the initial condition given by equation (35), does not converge to the target configuration. On the other hand, the ragged curve refers to the solution of system (22) where we have added noise terms into it. In the simulation, we have chosen and . We see from the figure that approaches zero, in a stochastic way, along time which implies that approaches .

2. Circle as the network topology. We assume that is now a circle with defined by

 E={(1,2),(2,3),(3,4),(4,5),(1,5)} (36)

We adopt the same initial condition given by expression (35). Figure 3 shows how evolves over time . Similarly, we see that if there is no noise term, then the solution of system (1) does not converge to the target configuration. On the other hand, the sample path approaches along the evolution. The two parameters and are again chosen to be and respectively.

These two examples have demonstrated that simulated annealing can be used to modify the formation control law in order to achieve global convergence to the target configuration. More provable facts are needed at this moment for this heuristic algorithm.

## V Conclusion

In this paper, we have proposed a decentralized formation control law for agents to converge to a target configuration in the physical space. In particular, there is a quadratic Lyapunov function associated with the formation control system whose unique local (also global) minimum point is the target configuration. We then focussed on one of the limitations of this formation control model, i.e, there may exist a continuum of equilibria of the system, and thus there are solutions of system (1) which do not converge to the target configuration. To fix this problem, we then applied the technique of simulated annealing to the formation control law, and showed that the modified stochastic system preserves one of the basic properties of the undirected formation control system, i.e, the centroid of the configuration is invariant along the evolution over time. We then worked on two simple examples of the stochastic system. Simulations results showed that sample paths approach to the target configuration.

## Vi Acknowledgements

The author here thanks Prof. Ali Belabbas, Prof. Tamer Başar, as well as the reviewers of the earlier draft of this work for their comments on this paper.

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