Continuous-time quantized consensus: convergence of Krasovskii solutions
This note studies a network of agents having continuous-time dynamics with quantized interactions and time-varying directed topology. Due to the discontinuity of the dynamics, solutions of the resulting ODE system are intended in the sense of Krasovskii. A limit connectivity graph is defined, which encodes persistent interactions between nodes: if such graph has a globally reachable node, Krasovskii solutions reach consensus (up to the quantizer precision) after a finite time. Under the additional assumption of a time-invariant topology, the convergence time is upper bounded by a quantity which depends on the network size and the quantizer precision. It is observed that the convergence time can be very large for solutions which stay on a discontinuity surface.
Problems of consensus and coordination in networks have been widely studied during the last decade using a blend of tools from control theory and graph theory. While linear consensus systems based on time-invariant networks are easy to understand, things become harder when the network topology depends on time, or when communication between nodes is affected by limited precision due to bandwidth constraints. Consensus problems have been studied on time-dependent networks by a vast literature: we refer the reader to the early works [23, 22], as well as to the books [5, 21] for an introduction and to  for recent related results. On the other hand, coordination and consensus have also been studied in systems subject to limited-precision effects, i.e., to quantization. Most authors have focused on a variety of problems for discrete-time systems, including the analysis of convergence assuming static quantizers  and the design of effective dynamic quantization schemes . Quantized continuous-time systems, instead, have attracted attention more recently. Controllers based on quantizing the differences between the states of connected nodes are studied in  under the assumption that the network topology is a tree, and in  using binary quantizers in a leader-following framework. Quantized communication of states is instead considered in  for static topologies and in  for dissipative systems.
In the analysis of continuous-time quantized dynamics, the inherent discontinuity of the system right-hand side entails some mathematical difficulties, which are discussed in  and . The latter paper considers a simple continuous-time average consensus dynamics with time-invariant topology and uniform static quantizers, and demonstrates that choosing a suitable definition of solution is essential to ensure that solutions are defined for all times and thus to permit a meaningful convergence analysis. A natural and effective choice are Krasovskii solutions, which indeed are complete for every initial condition and converge to approximate consensus conditions under mild assumptions.
Statement of contributions
After this literature review, we are able to present the contribution of this paper. We study a coordination task for a network of agents having a scalar continuous-time dynamics, assuming that
the interaction between the agents is weighted by time-dependent coefficients which represent a dynamical communication network; and
connected agents can exchange information about their states only through a (static) quantizer.
Due to the quantization constraint, the goal of consensus between states can only be approximated up to the quantizer precision. Our main contribution consists in a sufficient condition for finite-time convergence of Krasovskii solutions to the best achievable approximation. This condition, presented in Theorem 1, is based on the connectivity of a suitable limit graph. Compared with the referenced literature, our convergence result holds (i) under milder assumptions on the network connectivity; and (ii) for a larger class of quantizers. Additionally, in Section 4 the convergence result is specialized to uniform quantizers and to average-preserving dynamics. With the further assumption of time-invariant topology, we also derive an upper bound on the convergence time, which is inversely proportional to the quantizer precision and is exponentially increasing with the network size. The tightness of this bound is discussed in view of ad hoc examples and of the evidences in the literature. We leave outside the scope of this paper the analysis of controllers based on quantization of differences, as well as the design of optimal controllers and quantizers, either dynamic or static.
2 Mathematical tools: Graphs and ODEs
In this section we provide some background in differential equations and graph theory. For our analysis it is necessary to recall from  a certain notion of solution to a –possibly discontinuous– differential equation, which is based on defining a suitable differential inclusion. Given111The symbols , , denote the sets of integer, real, nonnegative and positive numbers, respectively. denotes an -dimensional Euclidean space. Writing , where is a set of cardinality , we are indexing the components in the set . Given , the set of the (integer) multiples of is denoted by . and the differential equation , we say that solves this differential equation in the Krasovskii sense if is absolutely continuous and for almost every time in the interval satisfies the differential inclusion , where
with denoting the convex closure and the Euclidean ball of radius centered in . Here and elsewhere in the paper, “almost every” means “except in a set of zero Lebesque measure”. A solution is said to be complete if Note that we will also apply the Krasovskii operator to autonomous functions . An example of the convexification induced by is provided later in Figure 1. A similar notion is that of Filippov solution [2, Definition 6], which is quite common in the literature but will not be used here. Indeed, every Filippov solution is also a Krasovskii solution, so that the results in this paper apply a fortiori to Filippov solutions.
Our analysis also involves graphs and weighted graphs. We introduce here the main notions which we shall use later: the reader is referred to the literature, for instance to  or to the book , for a more complete introduction. Given a finite set of vertices (or nodes) , a (directed) graph is a pair where is the set of edges (or arcs). A weighted graph is triple which includes a weighted adjacency matrix with the consistency condition that if and only if We also assume that for all The Laplacian matrix associated to is a matrix such that if and . A sink is a node with no outgoing edge –that is, such that does not contain any edge of the form A path (of length ) from to in is an ordered list of edges in the form If such a path exists, we say that can be reached from . A cycle is a path from a node to itself. A graph is said to be connected if for every pair of nodes , either can be reached from or can be reached from . Instead, a graph is said to be strongly connected if every two nodes can be reached from each other. Given any directed graph we can consider its strongly connected components, namely maximal strongly connected subgraphs , with set of vertices and set of arcs such that the sets form a partition of . These components may have connections among each other: in order to encode these connections we define a directed graph with set of vertices such that there is an arc from to if there is an arc in from a vertex in to a vertex in . We observe that (i) has no cycle; (ii) is connected and has one sink if and only if there exists in a globally reachable node, i.e., a node which can be reached from every other node.
3 Problem statement and main result
In this section we introduce the dynamics of interest, and we state and prove our main convergence result. Let there be agents, indexed in a set , and for any pair , let be a measurable function, with . These interaction functions naturally lead to the following definitions. For every time , we consider a weighted interaction graph , such that the -th component of the matrix is the value and if and only if Given the function , we define –following – an unbounded interactions graph by
We observe that is the graph whose edges connect the nodes which are connected in for an infinite duration of time.
For , we let be a real variable and consider the dynamics
where is a quantizer mapping real numbers into a discrete222A subset is said to be discrete if all its points are isolated. Examples include the set of the integers and every finite subset of . Note that if has no limit point in , then is discrete. set. System (1) can also be rewritten in vector form as
where is the state vector, is the Laplacian matrix associated to the weighted adjacency matrix and by a slight notational abuse, is defined to operate componentwise on vectors. We consider for (1) solutions in the sense of Krasovskii, which we have defined in the previous section, and thanks to the linearity of the Krasovskii operator , we have that a Krasovskii solution to (1) is an absolutely continuous function of time which satisfies for almost every time the differential inclusion
By the current assumptions of boundedness on the functions , for any there exists a complete Krasovskii solution to (1), such that Note, however, that there can be more than one of such solutions. In the rest of this paper, whenever we refer to a solution, we mean a complete solution.
After these preliminary observations, we are ready to state and prove that system (1) reaches quantized consensus equilibria in finite time, provided the unbounded interactions graph has a globally reachable node.
Theorem 1 (Finite-time quantized consensus).
Let be a subset of with no limit point and be a non-decreasing function. Let be a Krasovskii solution to (1). If is connected and has only one sink, then there exist and such that, for every ,
Without loss of generality, we may think of the elements of as indexed in a set of consecutive integers, in such a way that and if and only if . Let As there is no limit point of , then
Given the solution and , we define the following time-dependent subset of indices
and we let and . By definition, and belong to and we denote and The dynamics (1) implies that, at almost every time and for all ,
In particular, if , then . Hence, for all ; similarly, we can deduce that Our proof aims at showing that actually increases until the system reaches an equilibrium: there exist and such that . The same conclusion can be reached by an analogous argument based on . Note that for all it holds , but sets of the form need not to be empty, in particular when some is at a discontinuity point of .
In view of the last remark, we denote for brevity and , and we start our argument by considering the set and claiming that
We show this fact by contradiction. Let be the discontinuity point of such that if and only if Assume, by contradiction, that there exists an agent such that and . Then there are three consequences: (i) by continuity, there exists such that ; (ii) consequently, ; (iii) for , and since , then necessarily . But (ii) implies that, for a set of times of positive measure, which is a contradiction. We conclude that (3) holds and that if an agent reaches (from the right), she necessarily has to stop at the border of the corresponding interval.
Next, we want to prove that there exist times when the inclusion (3) is strict. We define the set of the agents whose state is “strictly larger” than as
and If is finite, then for every . Then and we conclude that and , completing the proof. Otherwise, we proceed with our argument and assume333If , then there is nothing to prove: since in this case , we can start our argument from by contradiction that for all . We also temporarily assume that is strongly connected: the argument will be extended at the end of the proof. Then, thanks to the strong connectivity of , we can find an arc such that belongs to and does not. As a consequence of (3), for all and by contradiction we know that for all Notice that for almost every ,
where is the realization of the inclusion in (2). Define and Then
where in the last inequality we have used the fact that if , then
and is a measurable function taking values in Let denote the pre-image of under . If , then necessarily and the proof is completed since . Otherwise, we aim to show that the right-hand side of (4) is divergent as . If has infinite measure, divergence is clear from the assumption Otherwise, and instead has infinite measure: we want to use this fact, together with a lower bound on . To obtain such an estimate, we note that Equation (3) implies that for almost every such that . Then, for almost every it holds , and the equality
where is defined as follows. Let be such that and for all in a set of times of infinite measure either or . In the former case , in the latter By the connectivity assumption, there exists an infinite-measure set of times such that for there is a path in from to a node in , and by a recursive reasoning along this path, we conclude that for it holds . From (4) and the last inequality we can deduce
This inequality implies that diverges as , which contradicts the fact that for all We conclude that there exists such that for all it holds and Repeating this argument for every element of , we obtain that there exists such that . Afterwards, the same reasoning which has been applied to can be applied, with straightforward modifications, to , , …, showing that there exists a sequence of times such that Since , then the sequence of ’s must be finite. This implies that there exist and such that , under the assumption of strong connectivity of .
In order to complete the proof, we still have to relax the connectivity condition. If is not strongly connected, the above argument may fail, because at some time it may be impossible to find an arc coming out of the set of minima –say, the set . But in such a case, necessarily the sink component is a subset of . Then, since it is assumed that there is only one sink, it is still possible to conclude by applying the analogous argument based on the maximal value . ∎
Note that the assumptions of Theorem 1 about are satisfied, for instance, when is a finite set or when . The latter important special case is the topic of the next section.
4 Uniform quantizers
In this section, we assume that the states are communicated via a uniform quantizer, and we derive from Theorem 1 a more precise convergence result. After that, we study the case of average-preserving dynamics, and we estimate the convergence time
Let then be the uniform quantizer with precision , that is the map such that
The maps and are illustrated in Figure 1.
Corollary 2 (Uniform quantizers).
Since , Theorem 1 implies that there exist a nonnegative time and an integer such that for all it holds
This fact is equivalent to the statement of the corollary. ∎
4.1 Average consensus
In many applications one is concerned, rather than with mere convergence, with convergence to a certain target value, which is a function of the initial condition. For instance, the target can be the average of the initial states: this problem is referred to as the average consensus problem, and is studied in the next result.
Corollary 3 (Average-preserving dynamics).
By linearity, for almost every
By the assumption on the ’s, this implies that for almost every , so that the average is preserved. Corollary 2 then implies that . If in particular then it is clear that . Otherwise, being at the border of the interval, necessarily all must coincide. ∎
Note that Corollary 3 provides a formula for the limit (quantized) value, and also a sufficient condition to achieve exact consensus between the states. Corollary 3 improves on earlier convergence results available in the literature about average consensus of Krasovskii solutions (cf. [10, Proposition 4]), as it shows finite-time convergence for every initial condition and allows for time-dependent topologies.
4.2 Convergence time
In order to estimate the convergence time in Corollary 2, we restrict ourselves to consider time-invariant topologies, in the following sense. We assume that for every pair , either for all or for all , so that we may write and .
Proposition 4 (Estimate of ).
Then, considering the sequence of ’s, we argue that for every , as every quantization interval contains at most agents. On the other hand, needs not to be larger than . These remarks prove the statement. ∎
Next, we want to discuss the tightness of estimate (7), in terms of the dependence on and on . The parameter represents the quantizer precision and, in view of Corollary 2, also the accuracy which is achievable in approximating the consensus. The bound (7) allows for a convergence time which is polynomial in : the following example shows that there exist families of solutions which meet the bound, exhibiting a convergence time proportional to Indeed, for every we can find a weighted graph and an initial condition such that for a certain solution such that ,
Example 1 (Slow convergence: ).
We let , and we assume the topology to be a line graph, namely
Note that the resulting dynamics (1) preserves the average of the states. Regarding the initial condition, we assume for all . In the analysis of the resulting system, we think of the agents as arranged on a line and we only describe the evolution of the leftmost agents (1,2,…,), the evolution of the others being symmetrical. For early positive times, all agents are still except agent which moves to the right with constant speed . Then, at time we have that , that is agent reaches the border of the first quantization interval. Since , there is one Krasovskii solution such that for , is constant while agent 2 moves to the right until it reaches , so that and . Then, for the only agent on the move is again agent , until . At time , the two agents have the same state After this time, agents 3, 2 and 1 move to the right during three successive time intervals, so that at they are all collocated as By repeating this reasoning, we observe that the constructed solution reaches the limit configuration of Corollary 3 at time
Since then ∎
On the other hand, is the number of agents, and the bound (7) allows for a convergence time which is exponential in . The following example provides a family of solutions such that
for a positive constant . We observe that in order to have an exponential-in- convergence time, the solution must stay on a discontinuity of the right-hand side for a finite duration of time.
Example 2 (Slow convergence: ).
We let and we assume that, given
We also assume that the quantizer is uniform with and that the initial condition is
Note that is on a discontinuity point of for : then the Krasovskii convexification is nontrivial and we have , denoting the convexified values as . One can immediately verify that there exists a Krasovskii solution having the following properties:
for every , it holds that and if ;
for all and for ;
almost always for ;
at time the agents reach quantized consensus in the interval .
Then (8) follows choosing ∎
The qualitative behavior of the convergence time of Krasovskii solutions, outlined above, should be contrasted with that of nonquantized consensus dynamics. Let be the time for convergence within a precision in a suitable norm. Then, consensus dynamics without quantization typically yield a logarithmic dependence on ,
where is a constant which depends on the initial condition and on the topology of the interaction graph, and entails a dependence on which is at most polynomial.
We conclude that our theoretical results predict a qualitative degradation of convergence speed due to quantization. However, Proposition 4 is intrinsically a worst-case result, and not every solution needs to achieve the performance bound. Indeed, it is argued in [10, Remark 5] that, far from the equilibria, the quantized dynamics converges exponentially fast and has the same rate of convergence as the nonquantized linear consensus dynamics. This is confirmed by simulations reported in the same paper, which show logarithmic convergence times in both cases. These remarks entail no contradiction: far away from the equilibria the quantized dynamics is well approximated by the nominal linear dynamics, and the effect of quantization can be studied as a bounded disturbance (cf. [14, 3, 15]). On the other hand, in a neighborhood of the equilibria the approximation is no longer good and the consequences of quantization may fully come out, as we have shown above.
5 Summary and future work
This paper has demonstrated that a mathematical framework combining graph theory and Krasovskii differential inclusions can be useful to solve problems of distributed control with quantized communication. Complete Krasovskii solutions of quantized consensus dynamics exist for any initial condition, and it is possible to study their converge to equilibria of “practical consensus”. Under a mild connectivity assumption, which translates to the unbounded interactions graph the usual connectivity condition for consensus on static networks, solutions are shown to reach a neighborhood of consensus after a finite time. The size of such neighborhood only depends on the quantizer, and can thus be made arbitrarily small by design. On the other hand, the convergence time can be exponentially increasing in the number of nodes for some solutions which slide on a surface of discontinuity of the dynamics.
A few natural generalizations of the present work would be of interest: we briefly mention three of them.
In this paper, the states of the agents are communicated through a non-smooth map which is a quantizer, that is, whose range is a discrete space. However, our proof technique based on monotonicity properties seems to be promising for studying convergence of systems featuring more general non-smooth interaction maps.
Theorem 1 states sufficient conditions for consensus: is it then natural to ask whether these assumptions are necessary. While it is clear that the connectedness of is necessary for consensus, we believe that the argument of Theorem 1 can be extended in such a way to relax the non-degeneracy assumption . A sufficient connectivity condition would then be: there exist , and a graph which has a globally reachable node and is such that if , then for every . We leave the proof of this extension to future research. On the other hand, when is not connected but is cut-balanced in the sense of [17, Assumption 1], we expect results of partial consensus and clusterization [4, 9].
In this work, connectivity is a function of time determined by an exogenous signal. However, there are applications in which connectivity between agents is state-dependent. Which would be the convergence properties of quantized continuous-time dynamics on a state-dependent network described by interaction functions of type ? This investigation may have broad applications, including rendezvous and coordination problems in robotic networks where the ability to communicate depends on the robot locations [5, 25], and modeling opinion dynamics with limited verbalization capabilities  in social networks.
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