We consider trajectories that satisfy the following coordinate-wise condition
for some quadratic energy function . Intuitively, this just requires that when a coordinate of changes, this change happens in a way that does not increase the energy function, but there is no requirement about the magnitude of the decrease, if any. We show that condition (1) guarantees convergence of
to a constant vector when the functionis positive definite, and partly extend this result to classes of positive semi-definite functions. We prove in particular that convergence is still guaranteed when the matrix defining is a graph Laplacian, so that is a measure of the “disagreement” between the .
Condition (1) appears naturally in certain multi-agent dynamics, including in the platooning problem we will analyze in Section III, which involves control laws with deadzones to remove potential instabilities resulting from incoherent measurements of the same distance by different agents. And indeed, we came across condition (1) when trying to establish the convergence of the system of Section III.
Classical approaches for establishing convergence based on energy functions rely on variation of the Lyapunov - Kraskowski - LaSalle theorems [15, 13]. These approaches apply to dynamical systems of the form , where the vector field often satisfies some (uniform) continuity condition [8, 4]. For example, LaSalle theorem guarantees (under certain conditions) the convergence of solutions of to an invariant set, but not necessarily to a point, provided that is nonincreasing everywhere . Convergence to a single point is only guaranteed under additional conditions, such as being sufficiently negative, by one of the original Lyapunov theorems. Another result for time-varying systems guarantees convergence to 0 if and is not identically 0 on any trajectory other than that staying at 0 . Crucially, in all these results, the relevant conditions on the decrease of and on the smoothness of and must be satisfied globally, and not just for the trajectory considered, otherwise the results do not hold.
Particularly in cyber-physical systems or systems involving discrete computations or events, there may not be a natural way of defining a global evolution of the form , for instance when external noise or control is present. Therefore may not contain all the information required to determine and would thus not qualify as “the state of the system” in a classical sense. The speed may indeed depend on various elements related to the history of , communications with other systems, random or arbitrary events, etc. Certain works on consensus overcome this difficulty by defining a trajectory-dependent equivalent vector field to hide the complexity of the process, i.e. a vector field for which holds for that specific trajectory only[9, 12, 14]. But it can be challenging to define equivalent fields satisfying the global conditions required to apply the classical convergence results, (continuity, decrease of , suitable invariant sets…), and we indeed did not succeed in applying this approach to general trajectories satisfying (1). Moreover, we would argue that this is a cumbersome and unnatural step. Extensions of Lyapunov results to differential inclusions could also not be directly applied to (1) as they require pre-defining the invariant sets to which would converge [1, 7]. Hence we think it is in many cases relevant to analyze the convergence of trajectories based purely on their properties, and not on those of a vector field or differential inclusion they follow.
Standard trajectory-focused techniques do not allow establishing convergence solely based on (1). Observe it implies that
so that is non-increasing and hence converging, but it is well known that does not imply the convergence of in general. There is here no guarantee on the decrease rate of , even relative to the magnitude of , as the gradient and can be orthogonal or arbitrarily close to being orthogonal. Hence the total decrease of cannot be directly bounded relative to the length of the trajectory, which would have guaranteed a finite length of the trajectory. We can thus a priori not exclude that would keep varying while approaching a level set . Moreover, these level sets are not compact when is only positive semi-definite.
In the following section we establish our new convergence result based on condition 1. Afterwards, we demonstrate its application for a platoon formation problem.
Ii Convergence result
For the simplicity of exposition, we state our main convergence result for quadratic functions of the form and particularize condition (1) to these functions, but extension to general quadratic functions is immediate by a applying a constant offset for some vector .
Let be a symmetric positive semi-definite matrix, and be an arbitrary absolutely continuous function, also implying that exists almost everywhere. Suppose that the following two conditions (particularizing (1) to ) holds for every :
If is positive definite, converges to a constant vector .
If is positive semi-definite and no nonzero vector of its kernel has a zero component (), then either converges to a constant vector or every accumulation point of lies in .
We note that condition (b) can only be satisfied if has dimension 1. Indeed, if
are linearly independent vectors in, one can always find a nontrivial linear combination for which for any given .
We first show that (b) implies (a): Indeed, with , it follows from (2) that so that always remains in the set . When is positive definite, this set is compact and has thus at least one accumulation point. Supposing that would not converge, (b) implies that every accumulation point of would be in the kernel of , i.e. would be equal to 0, which implies that would converge to 0 since it would be the only accumulation point. In , convergence of a continuous trajectory is indeed equivalent to the existence of one single accumulation point. We therefore only need to prove (b) in the sequel.
Consider the hyperplane
orthogonal to the th row of . If there is no accumulation point, or exactly one accumulation point, meaning that converges to a constant vector, the statement (b) holds trivially. We suppose to the contrary that there exist more than one accumulation point. We select an arbitrary accumulation point contained in the smallest possible number of hyperplanes and denote this smallest possible number by . We will show that . Without loss of generality, we can assume the indices are ordered in such a way that
We can choose such that two following two conditions hold: (i) and (ii) there is at least one other accumulation point outside of (otherwise would be the only accumulation point).
Due to the existence of this other accumulation point, must infinitely often leave while getting infinitely often into . More precisely, there exists a diverging sequence of disjoint time intervals such that , and for every . We define , so . Our proof relies on the following two lemmas, which will establish that .
for . As a consequence, for .
We first show that the distance between and converges to 0 for every . If it was not the case, there would be an infinite subsequence at a distance larger than from . Since the are by definition in the compact set , this sequence would admit an accumulation point that would have positive distance at least from . Moreover, this accumulation point could not belong to any with because these sets have no intersection with . Hence we would have an accumulation point that belongs to less than sets , which contradicts the selection of as an accumulation point of belonging to the smallest possible number of .
As a consequence the distance between and every , converges to 0, and a similar argument shows the same result for . This implies by definition of that of . The final implication of the Lemma follows from the boundedness of .
for . As a consequence, for .
For , we know is bounded away from by at least . Hence, if , then for some . So since , we have
Condition (2) implies that and have opposite signs whenever they are both nonzero (and also for all other indices ), hence we obtain from the previous inequality:
where we remind that . This last inequality holds for every , so that
as is non-increasing and the overall decrease of is finite. Therefore, there holds as which we wanted to show. The last implication of the Lemma follows from the boundedness of .
for some positive . If this was not the case, knowing that , an accumulation point of would reveal a vector with and which contradicts our condition on the kernel in part (b) of the theorem statement. In turn, knowing that is positive semi-definite and (5), we get
for some positive . This is in contradiction with (4). Hence we must have , meaning that belongs to all and thus to . Since was selected as belonging to the smallest number of all others accumulation points also belong to all and thus to , which establishes the claim (b). This also implies claim (a) as explained in the first part of the proof.
Observe that condition (b) of Theorem 1 does not guarantee the existence of an accumulation point. And in case there is a single accumulation point, it may not be in , as the trajectory could for example stop anywhere (and thus converge) without violating (2). However, in case the trajectory has multiple accumulation points, they all belong to . It remains open to determine if (i) the condition on vectors with 0 entries in the kernel of can be relaxed, and (ii) if trajectories satisfying condition (b) may indeed diverge or have multiple accumulation points.
The particular case of Laplacian matrices
It is possible to obtain stronger results for a specific class of positive semi-definite matrices: the (connected) graph Laplacians. A symmetric matrix is a Laplacian if all its off-diagonal entries are non-positive, and if each of its rows sums to 0: if , and . A Laplacian is positive semi-definite, and has rank if the corresponding graph is connected, that is, every node can be reached from any other one in the graph defined by associating a node to each and connecting two nodes if . Laplacians play a major role in various disciplines, including in algebraic graph theory , and are particularly important in consensus and synchronization applications, see e.g., [11, 20, 17] to name a few.
Laplacians have two properties of special interest in our context. First, observe that
that is, is a weighted sum of the differences between and the other coordinates. Second,
i.e., the associated quadratic function is a weighted sum of the square differences between the , and is thus a measure of the “disagreement” in . This also shows that the kernel of a Laplacian is spanned by the vector 1, since the quadratic form above is 0 iff all are equal (recalling that the corresponding graph is connected). We leverage these ideas to show that (2) implies convergence when the matrix is a Laplacian.
Let be a Laplacian whose corresponding graph is connected, and let be an arbitrary absolutely continuous trajectory. If
for every , then converges to a constant vector . Moreover, for all so that
We first show that and evolve monotonously. Note that the maximum of finitely many absolutely continuous functions is also absolutely continuous, and thus has a derivative almost everywhere. Let be an arbitrary time at which this derivative and that of all the exists, and let . Then we have from (6)
and (7) implies for all that .
By the continuity of all coordinates, there is a small enough such that for any we have , i.e., if , then for all in a small interval around . This means that for any we have
Consequently when taking the limit we get for one (or more) , and we have seen that for all so the same has to hold true for . Hence the absolutely continuous function has a nonpositive derivative almost everywhere, which implies it is non-increasing. An analogous reasoning can be applied for .
As a consequence always remains in the compact set and has thus at least one accumulation point .
Let us now assume, to argue by contradiction, that does not converge. Observe that the kernel of is the set , and it follows thus from Theorem 1 that the accumulation point lies in , with for some . Since is an accumulation point, for every there exist a time at which for every . In particular, and . The monotonicity of and implies then that for all . Since we can chose arbitrarily small, converges to , which contradicts our assumption. So must indeed converge to some , and the monotonicity of and implies (8).
Iii Application to Platoons with bounded disturbances
In this section, we study how to utilize condition (1) in designing a decentralized motion control scheme for the problem of keeping inter-agent distances in multi-vehicle-agent platoons at pre-defined desired values, using noisy inter-agent relative measurements, as considered in . The paper  has proposed a deadzone based switching control scheme to solve this problem, guaranteeing to have the agent positions kept bounded, robustly to distance measurement noises with a known upper bound. In , solution of the problem with the proposed control scheme is formally established only for two-agent platoons. Formal analysis for platoons with higher number of agents is left incomplete, ending with a conjecture on the agent positions being kept bounded and the inter-agent distances converging to certain intervals (balls) centered at the desired values, with radii proportional to the noise upper bound. The conjecture was supported by partial analysis for specific cases and simulation test results. The control scheme proposed in  is later adapted to the cooperative adaptive cruise control (CACC) problem of keeping a desired spacing between the consequent agents of a vehicle-platoon in , introducing a moving frame of reference and considering the vehicle dynamics of the agents. Next, we revisit the problem considered in  in a more general setting to be defined in the following subsection, and propose an approach based on generation of agent trajectories satisfying the condition (1).
We consider a set of agents each with a position A connected undirected graph represents the possible sensing capabilities: ( implies that can sense the relative position of with some noise, and vice-versa). A particular case of graph is the “chain graph”, with .
The measures are subject to disturbance, so that if there is an edge then agent can sense , where is an arbitrary disturbance satisfying , for some known . The are measurable, but not necessarily continuous. For each in we are given a desired distance , and the ideal objective would be that for each , . Those distances are supposed realizable, i.e., there exist such that for all . This implies in particular . This realizability constraint is automatically satisfied for the chain graph and for trees in general. For more general graphs, small mismatches of could also be modeled as being part of the disturbances.
In the absence of communication between agents, it has been observed that use of individual agent controllers in certain classical forms, such as proportional and proportional-integral, will lead to instabilities due to inconsistencies between the measurements of the inter-agent distance, as discussed in [3, 2]. Consider for example two agents 1, 2 with , and suppose while , i.e., agent overestimates its distance to . Then one can verify that if the agents use the same proportional controller based on the distance they sense, i.e., if each agent uses the control law , where is the index of the other agent, then we will have , and hence the average position will move to infinity. In the next subsection, we design a non-hierarchical control law for guaranteeing that all remain bounded, and that all constraints are (asymptotically) satisfied.
Iii-B Control Law
For robustness to effects of the disturbances , we use non-linear threshold functions. Such a simple threshold function is
But any nondecreasing function for which if and only if can be used. These imply in particular that for every if .
The aim in our control law design is to have the agent move only when there is no doubt that it moves in the right direction. For each agent, we propose the control law
where is the degree of in the graph . Since differs from by at most , this control law implies that will be negative (resp. positive) if and only if is positive (resp. negative) for sure. We will show that (9) guarantees convergence of to constant positions where the distance constraints are approximately satisfied, with errors that depend on and the properties of the graph.
A similar control law was introduced independently in the context of consensus with unknown bounded disturbance in . However, the final step of the convergence proof of , establishing convergence based on a condition akin to (1) is inaccurate111Specifically, equation (18) in , which the last arguments of the proof rely on, does not hold in general, indicating again the need for convergence results based on condition (1)..
We note that the issue of convergence is central here. For example, the similar looking control law
where the thresholds are applied to measurement as opposed to control actions, is observed to be inappropriate because agents would not necessarily converge to constant positions; for certain they can indeed oscillate for ever. An example of such oscillations is presented in Fig. 1 for a platoon with chain sensing graph with agents, where the initial positions are , , , , , , the desired distances (all in meter). For the sensor disturbances we take , let be a pulse signal with magnitude , bias , period sec, and pulse width sec, and let be a pulse signal with magnitude , bias , period sec, and pulse width sec. Finally, is sec phase delayed version of . The control law is that proposed in , i.e., (10), with
Consider agents , with positions at each time instant , and a connected undirected sensing graph as detailed in Section III-A. Under control law (9), for any class of nondecreasing functions for which if and only if ,
converges : exists, and satisfies
where is the degree of agent in and the bound on the disturbance.
For every agent and all time there holds
Let us perform a change of variable, defining . Noting that , we have
where satisfies, , and is the Laplacian matrix of the graph: , if and are connected, and 0 else. By definition of and in view of the bound , can be positive only if is negative, and vice versa, so that . Moreover, It is easy to confirm that is absolutely continuous as it is the integral of a measurable locally bounded function. Hence Theorem 1 shows that converges to some and remains at all time in , which implies the convergence of and the inclusion (13).
We now prove that , which implies (12), by contradiction. Suppose this condition does not hold, and without loss of generality, that . Since converges to , there is a time after which for some , and thus we have , since satisfies, . Since is non-decreasing, this means there is a time after which , where the last inequality follows from . As a result, would remain negative and bounded away from 0 for all time , in contradiction with its convergence to . Hence we must have and thus (12).
The particularization of equation (12) to the line graph implies when applied to node 1, and
when applied to node , so that . An induction argument shows then
where the second element in the min is obtained by starting the induction from the end of the platoon.
In this paper we have analyzed processes where the dynamics is not implicitly determined by (the gradient of) an energy function, but where that only serves as a barrier, leaving more freedom for the possible trajectory.
This framework beautifully matches the scenario of platoon formation, where the control of the dynamics has to be more conservative as it needs robustness as a priority over having an optimal configuration.
In both setups we have confirmed convergence of the processes when the energy function is quadratic described by a positive definite or Laplacian matrix. In a more general quadratic positive semi-definite case we have shown a partial concentration result, but we suspect much more is true.
These trajectory-based convergence results opens multiple perspectives: A straightforward challenge is to determine whether it is possible for a trajectory satisfying (2) to diverge or to have multiple accumulation points when has rank at most (and is not a Laplacian). Similarly, whether the absence of zero entries in the vectors of the kernel of , required in condition (b) of Theorem 1, can be relaxed. One obvious extension to broader context is to consider more general energy functions than quadratic ones.
Condition (2) can also be interpreted as requiring and to belong to a same cone among a finite set of cone. This insight could be use to derive more general conditions with more general and/or position dependent cones.
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