A General Coupling Cost Function

5.4 Extensions of the Theory

5.4.1 A General Coupling Cost Function

In the theory of the previous sections, neighboring agents are coupled by terms in the integrated cost function. While the generalization of the form of the integrated cost

in Section 4.3 is restricted to quadratic, more general forms of the cost function are possible. For example, a non-quadratic coupling cost that is relevant for multi-vehicle formation stabilization and for which the decomposition is straightforward is given at the end of Chapter 6. In this section, we explore a generalization of the coupling integrated cost.

We begin with a single optimal control problem, for which there is defined an integrated cost functionL:R^nNa×R^mNa →Rthat is twice continuously differentiable and satisfies

L(z, u)≥0, ∀(z, u)∈R^nNa×R^mNa, and L(z, u) = 0 if and only if (z, u) = (z^c,0).

As before, we are interested in stabilizing the dynamics to the equilibrium point z^c with 0 =f(z^c,0).

By definition, a function l(x, y) is additively separable (or just separable) in x and y if there exists functionslx and ly such that l(x, y) =lx(x) +ly(y). A necessary condition for separability of a cost that is twice continuously differentiable is that

∂²l(x,y)

∂x∂y = 0 for all x and y. For simplicity, we assume that the single cost on the control is additively separable, using the notation

L(z, u) =L^z(z) +

Na

X

i=1

L^u_i(ui).

Consequently, we are interested in generic conditions for decomposing a general state dependent costL^z. First, we can define what it means for any two agents to be coupled in the state dependent cost. To simplify things, we make the following assumption.

Assumption 5.2 The cost L^z(z) is the sum of separable and nonseparable terms, each of which is nonnegative.

Definition 5.2 Agents i₁, ..., i_M ∈ {1, ..., N_a}, 2 ≤ M ≤ N_a, are coupled if there exists a nonnegative term in the cost L^z(z) that depends on at least one compo- nent of every vector z_i1, ..., z_iM and the term is not additively separable in any such components.

The definition rules out looking at the zero function as coupling any subset of agents, since zero is additively separable. Consistent with previous notation, let Ni be the set of neighbors of agent i, where each neighbor is coupled to i in the cost function L^z(z) in the sense of Definition 5.2. From the definition, it is clear thati∈ Nj if and only if j ∈ Nⁱ, for any i, j ∈ {1, ..., Na}. As before, let z−i = (zj1, ..., zj_|N

i|) be the concatenated vector of states of the neighbors of i, and denote Ni := |Ni|+ 1. We make another assumption without loss of generality.

Assumption 5.3 Every agent is coupled to at least one other agent, i.e., for every agent i∈ {1, ..., Na}, Ni ≥2.

By definition, every agent i ∈ {1, ..., N_a} is coupled to agents j₁, ..., j|Ni| ∈ Ni by nonseparable terms in L^z(z) and is coupled to no agents in the set {1, ..., Na} \ Ni. Proposition 5.1 There exists a nontrivial cost function L^z_i : R^nNi → R for every agent i∈ {1, ..., Na} such that

L^z(z) =

Na

X

i=1

L^z_i(z_i, z−i) and L^z_i(z_i, z_−i^c ) = 0 ⇒ z_i =z_i^c.

Proof: For any agent i coupled to agents j1, ..., jMi ∈ Ni, for some Mi with 1 ≤ M_i ≤ |Ni|, denote the nonseparable coupling term asg_i which exists and is nontrivial by Definition 5.2 and the assumption above. Further, there may be terms in L^z that are additively separable and depend only on components of z_i for any agenti. Now, define each L^z_i as the sum of every separable term in L^z that depends solely on zi (if such terms exist) and every nonseparable coupling term in L^z that depends onz_i and neighbors of i, normalizing each such term by the total number of agents coupled in the term (e.g., g_i/(M_i + 1) for the general coupling term g_i). By construction, the sum of the L^z_i functions recovers the cost L^z.

Now, L^z_i(z_i, z_−i^c ) = 0 always has at least the solution z_i = z^c_i. This follows since the L^z_i functions sum to give L^z and L^z(z^c) = 0. It remains to show that this is the unique solution. Let ¯z_i be the vector of states of all agents that are not i and not in Ni, i.e., not neighbors ofi. Next, define the function ¯L^z_i(z−i,z¯i) := L^z(z)−L^z_i(zi, z−i),

i.e., ¯L^z_i is all terms in L^z excluding those inL^z_i. Clearly, ¯L^z_i does not depend onzi. By assumption, we haveL^z(zi, z^c_−i) = 0, which holdsfor any z¯i. Setting the state of every agent in ¯zi to its equilibrium value, and denoting the resultant vector ¯z_i^c, we know that L^z(z^c) =L^z_i(z_i^c, z_−i^c ) + ¯L^z_i(z^c_−i,z¯^c_i) = 0 and therefore ¯L^z_i(z_−i^c ,z¯_i^c) = 0. So, setting

¯

zi = ¯z_i^c andz−i =z_−i^c , we have by assumption thatL^z_i(zi, z_−i^c ) + ¯L^z_i(z_−i^c ,z¯_i^c) = 0, which implies that

L(z^c₁, ..., z_i−1^c , zi, z^c_i+1, ..., z^c_Na) = 0.

Since L(z) = 0 if and only ifz =z^c, we have that L^z_i(zi, z^c_−i) = 0 implieszi =z_i^c. Proposition 5.1 is an existence statement, and the proof constructs a specific function L^z_i for each i. By construction, each such function is unique. A different choice for the weighting of each nonseparable coupling term, i.e., one other than normalizing by the number of agents coupled in the term, can be chosen, provided the weighting preserves the summation in the proposition. It is also noted that the second part of Proposition 5.1 can be considered a detectability condition.

The decomposition can now be performed in terms of the unique cost functions L^z_i constructed in the proof of Proposition 5.1. The decomposition is stated as a definition, similar to Definition 4.1.

Definition 5.3 Thedistributed integrated cost function for each agenti∈ {1, ..., Na} is defined as

L(zi, z−i, ui) =γ[L^z_i(zi, z−i) +L^u_i(ui)], where γ ∈(1,∞).

From Proposition 5.1 and the definition above,L(zi, z^c_−i, ui) = 0 if and only if zi =z_i^c and u_i = 0.

Remark 5.5 It may be that an agentiis coupled to all other agents{1, ..., Na}\{i}. In that case, the communication cost may be prohibitive and multi-hop information exchanges may be required. When accounting for delays and information loss due to communication, however, the multi-hop situation will not in general perform as well as when every agent has a direct link to every neighbor. The effects of real-time

communication delays and loss, in single and multi-hop situations, will be part of the ongoing research in this area.

In many multiagent systems, coupling between agents also occurs in constraints. In the next section, we explore the implications of coupling state constraints on the distributed receding horizon control implementation.