A Nonlinear Consensus Protocol of Multiagent Systems Considering Measuring Errors

Abstract

In order to avoid a potential waste of energy during consensus controls in the case where there exist measurement uncertainties, a nonlinear protocol is proposed for multiagent systems under a fixed connected undirected communication topology and extended to both the cases with full and partial access a reference. Distributed estimators are utilized to help all agents agree on the understandings of the reference, even though there may be some agents which cannot access to the reference directly. An additional condition is also considered, where self-known configuration offsets are desired. Theoretical analyses of stability are given. Finally, simulations are performed, and results show that the proposed protocols can lead agents to achieve loose consensus and work effectively with less energy cost to keep the formation, which have illustrated the theoretical results.

1. Introduction

One of the most attractive benefits of multiagent systems is that many fractional, inexpensive, and simple agents working together can achieve the same objective as a monolithic, expensive, and complicated agent. Since certain formation configurations are often indispensable for such systems, the coordination problem in the area of formation control has attracted compelling interest in recent years. One critical issue arising from the coordination problem is to develop distributed control laws based on local information that enable all agents to reach an agreement on certain quantities of interest, which is known as the consensus problem.

In the context of multiagent systems, consensus problems have been studied widely in recent years. Reference [1] provided a theoretical explanation for the coordination behavior of a group of autonomous agents, and results implied that proper control models based only on neighbor information could motivate systems to achieve consensus. Reference [2] provided an overview of information consensus in multivehicle cooperative control, with several consensus control laws reviewed by describing some specific coordination applications. More challenging results were obtained by extending the simple first-order dynamics to second-order ones to better solve practical problems with some actual conditions being considered, such as actuator saturations and lacks of relative velocity measurements (see, e.g., [3]). And, more applications of consensus can be also found in varieties of research paper, such as flocking, attitude alignment, swarm moving, and containment control (see, e.g., [4–7]).

Most results of the aforementioned research are obtained in case where dynamics are linear. And in case where the dynamic agents are physical models, the design and analysis of nonlinear consensus protocols have also been studied, with input constraints of such systems being taken into account for more adaptive designs. Reference [8] introduced both linear and nonlinear protocols based on consensus for a group of dynamic agents with undirected information flow, and it implied from the definition of nonlinear functions that linear protocols are actually the special cases of the nonlinear ones. Reference [9] studied the performance of a class of special nonlinear protocols, which could be designed using any mean of order of the agents’ states instead of the average mean to form the feedback term, to achieve consensus. Many other research papers have also been done to explore nonlinear protocols, such as [10, 11], where nonlinear protocols were utilized to achieve faster convergence or apply with second-order dynamic agents, respectively.

However, sometimes, energy means a lot for some particular applications, such as certain applications for space, for example, formation keeping and attitude alignment. It is worthwhile to note that, although the aforementioned protocols are able to deal with the measurement uncertainties, they may not perform well in saving energy (which will be shown through comparisons in the later simulations). It can be known that, in the typical tight spacecraft formation keeping missions, the relative position measurements are of precision of centimeter number grade [12]. So, due to the existence of measurement uncertainties, it is unnecessary that the aforementioned protocols should act even when the measured position configuration errors are of precision of centimeter number grade. These overactive actions would cost energy continuously, but in the meanwhile, energy quantity influences the lifetimes of spacecraft for normal performances. Therefore, it is worthwhile to present a particular nonlinear protocol to address such uncertain problems for better saving of energy.

The outline of this paper is as follows. Some backgrounds on algebraic graph theory and linear protocol are given in Section 2. A nonlinear protocol without a reference is presented to achieve a loose consensus with its stability proof given together in Section 3.1. The protocol is extended to the case where all agents can communicate with a group references in Section 3.2. By introducing a set of sliding mode estimators to estimate a group reference for each agent, the protocol is easily extended to the case where only a portion of agents can communicate with a group reference in Section 3.3. Simulation results are presented in Section 4. Finally, concluding remarks are made in Section 5.

2. Backgrounds and Preliminaries

The m-dimensional open-loop dynamics of 𝑛 agents with fixed undirected communication links can be given by 𝐫̇_𝑖(𝑡)=𝐯_𝑖(𝑡)𝐯̇_𝑖(𝑡)=𝐮_𝑖(𝑡),𝑖∈{1,…,𝑛}. (1)

A weighted undirected graph 𝒢=(𝒱,ℰ) can be borrowed to model the interaction among these agents, where 𝒱={1,…,𝑛} is the node set and ℰ⊆𝒱×𝒱 is the edge set. If (𝑗,𝑖)∈ℰ, agents 𝑖 and 𝑗 can be called neighbors, which means that they can exchange information from each other. The weighted adjacency matrix 𝒜=[𝑎_𝑖𝑗]∈𝑅^𝑛×𝑛 associated with 𝒢 is defined as 𝑎_𝑖𝑗>0 if (𝑗,𝑖)∈ℰ, while 𝑎_𝑖𝑗=0 otherwise. Since (𝑗,𝑖)∈ℰ implies (𝑖,𝑗)∈ℰ, 𝑎_𝑖𝑗=𝑎_𝑗𝑖 holds. The Laplacian matrix ℒ=[ℓ_𝑖𝑗]∈𝑅^𝑛×𝑛 associated with 𝒢 is defined as ℓ_𝑖𝑖=∑^𝑛_{𝑗=1,𝑗≠𝑖}𝑎_𝑖𝑗, while ℓ_𝑖𝑗=−𝑎_𝑖𝑗 for any 𝑖≠𝑗. It is clear that 𝒜 is symmetric and ℒ is symmetric positive semidefinite.

The linear consensus protocol can be given by (see, e.g., [2, 13]) 𝐮_𝑖(𝑡)=−^𝑛_𝑗=1𝑎_𝑖𝑗𝐫_𝑖(𝑡)−𝐫_𝑗(𝑡−_𝑛𝑗=1)𝑎_𝑖𝑗𝛾_𝑐𝐯_𝑖(𝑡)−𝐯_𝑗(𝑡), (2) where the gain 𝛾_𝑐 is a positive scalar. It follows from [13] that the protocol (2) can guarantee the consensus of the multiagent system (1) under any undirected connected interaction graph and the system is asymptotically stable with 𝐫_𝑖(𝑡)→𝐫_𝑗(𝑡) and 𝐯_𝑖(𝑡)→0 asymptotically, as 𝑡→∞.

It is clear to note that protocol (2) would act even when the configuration errors are of the same precision as those of the measurements with costing energy unnecessarily. Assuming that measurement uncertainties are bounded, the mathematical forms can be written as follows: 𝐫_𝑖(𝑡)−𝐫_𝑖(𝑡)_∞≤𝜀_𝑟,𝐯_𝑖(𝑡)−𝐯_𝑖(𝑡)_∞≤𝜀_𝑣, (3) where 𝐫_𝑖(𝑡) and 𝐯_𝑖(𝑡) denote the true values, 𝜀_𝑟 and 𝜀_𝑣 denote the boundaries of errors, and ‖⋅‖_∞ denotes the ∞-norm operator (‖𝐩‖_∞ equals the maximum absolute value of the components of vector 𝐩).

3. Consensus Analysis

3.1. Nonlinear Protocol for Average

Intuitively thinking, when the configuration errors are of the same precision as those of the measurements, it is not convincing to control based on the measurements, and, instead, a conservative control strategy is considered. Introduce two nonlinear functions 𝜙(𝑧) and 𝜑(𝑧), which are defined componentwise as:𝜙(𝑧)=𝑧+𝜀_𝑟,𝑧∈−∞,−𝜀_𝑟,0,𝑧∈−𝜀_𝑟,𝜀_𝑟,𝑧−𝜀_𝑟,𝑧∈𝜀_𝑟,𝜑(𝑧)=,+∞𝑧+𝜀_𝑣,𝑧∈−∞,−𝜀_𝑣,0,𝑧∈−𝜀_𝑣,𝜀_𝑣,𝑧−𝜀_𝑣,𝑧∈𝜀_𝑣,+∞. (4)

Then the nonlinear consensus protocol is given by 𝐮_𝑖(𝑡)=−^𝑛_𝑗=1𝑎_𝑖𝑗𝜙𝐫_𝑖(𝑡)−𝐫_𝑗(𝑡−_𝑛𝑗=1)𝑎_𝑖𝑗𝛾_𝑐𝜑𝐯_𝑖(𝑡)−𝐯_𝑗(𝑡),∀𝑖∈{1,…,𝑛}. (5)

Before going on, a lemma and its proof are given as follows.

Lemma 1. Suppose that 𝜍∈𝑅^𝑚, 𝜑∈𝑅^𝑚, and 𝐂=[𝑐_𝑖𝑗]∈𝑅^𝑛×𝑛. If 𝐂 is symmetric and 𝜙(𝑧) is an odd function, then it follows that 12^𝑛𝑛_𝑖=1𝑗=1𝑐_𝑖𝑗𝜍_𝑖−𝜍_𝑗^𝑇𝜙𝜑_𝑖−𝜑_𝑗=^𝑛𝑛_𝑖=1𝑗=1𝑐_𝑖𝑗𝜍^𝑇_𝑖𝜙𝜑_𝑖−𝜑_𝑗. (6)

Proof. Considered 12^𝑛𝑛_𝑖=1𝑗=1𝑐_𝑖𝑗𝜍_𝑖−𝜍_𝑗^𝑇𝜙𝜑_𝑖−𝜑_𝑗=12^𝑛𝑛_𝑖=1𝑗=1𝑐_𝑖𝑗𝜍^𝑇_𝑖𝜙𝜑_𝑖−𝜑_𝑗−12^𝑛𝑛_𝑖=1𝑗=1𝑐_𝑖𝑗𝜍^𝑇_𝑗𝜙𝜑_𝑖−𝜑_𝑗=12^𝑛𝑛_𝑖=1𝑗=1𝑐_𝑖𝑗𝜍^𝑇_𝑖𝜙𝜑_𝑖−𝜑_𝑗+12^𝑛𝑛_𝑗=1𝑖=1𝑐_𝑗𝑖𝜍^𝑇_𝑗𝜙𝜑_𝑗−𝜑_𝑖=^𝑛𝑛_𝑖=1𝑗=1𝑐_𝑖𝑗𝜍^𝑇_𝑖𝜙𝜑_𝑖−𝜑_𝑗, (7) where, in order to get (7), we have used the facts that 𝑐_𝑖𝑗=𝑐_𝑗𝑖 and 𝜙(𝜑_𝑖−𝜑_𝑗)=−𝜙(𝜑_𝑗−𝜑_𝑖), and we also have switched the dummy variables 𝑖 and 𝑗 in the summation function.

Theorem 2. Using protocol (5) for multiagent system (1), then if the communication topology graph of (1) is undirected and connected, loose consensus can be achieved so that ‖𝐯_𝑖(𝑡)−𝐯_𝑗(𝑡)‖_∞≤𝜀_𝑣 and ‖𝐫_𝑖(𝑡)−𝐫_𝑗(𝑡)‖_∞≤𝜌 as 𝑡→∞, for all 𝑖,𝑗, where 𝜌 is a certain bounded scalar.

Proof. Denote 𝜓(𝑧) as the potential function of 𝜙(𝑧), which is also defined component-wise. One obtains that 𝜓(𝑧)=^𝑧₀𝜙(𝑠)𝑑𝑠. (8)
Introduce a Lyapunov function as 1𝑉=2^𝑛𝑛_𝑖=1𝑗=1𝑎_𝑖𝑗𝟏^𝑇₃𝜓𝐫_𝑖−𝐫_𝑗+12^𝑛_𝑖=1𝐯^𝑇_𝑖𝐯_𝑖. (9)
Then the time derivative of 𝑉 along the trajectories of (5) is given by 𝑉̇=12^𝑛𝑛_𝑖=1𝑗=1𝑎_𝑖𝑗𝐯_𝑖−𝐯_𝑗^𝑇𝜙𝐫_𝑖−𝐫_𝑗−^𝑛_𝑖=1𝐯^𝑇_𝑖^𝑛_𝑗=1𝑎_𝑖𝑗𝜙𝐫_𝑖−𝐫_𝑗+𝜑𝐯_𝑖−𝐯_𝑗=^𝑛𝑛_𝑖=1𝑗=1𝑎_𝑖𝑗𝐯^𝑇_𝑖𝜙𝐫_𝑖−𝐫_𝑗−^𝑛𝑛_𝑖=1𝑗=1𝑎_𝑖𝑗𝐯^𝑇_𝑖𝜙𝐫_𝑖−𝐫_𝑗+𝜑𝐯_𝑖−𝐯_𝑗=−^𝑛𝑛_𝑖=1𝑗=1𝑎_𝑖𝑗𝐯^𝑇_𝑖𝜑𝐯_𝑖−𝐯_𝑗1=−2^𝑛𝑛_𝑖=1𝑗=1𝑎_𝑖𝑗𝐯_𝑖−𝐯_𝑗^𝑇𝜑𝐯_𝑖−𝐯_𝑗≤0, (10) where we have used derivative rules to obtain the first equality, have used Lemma 1 to obtain the second and fourth equality, and have used the fact that (𝐯_𝑖−𝐯_𝑗)^𝑇𝜑(𝐯_𝑖−𝐯_𝑗) is nonnegative to get the final inequality.
Let 𝑆={(𝐫_𝑖−𝐫_𝑗,𝐯_𝑖−𝐯_𝑗)∣𝑉̇=0}. Since 𝑉̇≤0 along the trajectories of system, there exists 𝑇>0 such that 𝑉̇≡0 holds for all 𝑡>𝑇. Note that 𝑉̇≡0 means 𝜑(𝐯_𝑖−𝐯_𝑗)=0, for all 𝑖,𝑗. So, when the undirected graph is connected, 𝑉̇≡0 implies ‖𝐯_𝑖(𝑡)−𝐯_𝑗(𝑡)‖_∞≤𝜀_𝑣, for all 𝑖,𝑗. Therefore, it follows from (5) that 𝐯̇_𝑖(𝑡)=−^𝑛_𝑗=1𝑎_𝑖𝑗𝜙𝐫_𝑖(𝑡)−𝐫_𝑗(𝑡), (11) for all 𝑡>𝑇.
It can be seen from (11) that agent 𝑖 is forced towards the weighted average position of its neighbors and it is the same case with the other agents. This means that all agents tend to gather. Combining with ‖𝐯_𝑖(𝑡)−𝐯_𝑗(𝑡)‖_∞≤𝜀_𝑣, for all 𝑖,𝑗, one can obtain the conclusion that 𝐯̇_𝑖(𝑡) is bounded, for all 𝑖. It then follows from (11) that there must exists a bounded scalar 𝜌>0 such that ‖𝐫_𝑖(𝑡)−𝐫_𝑗(𝑡)‖_∞≤𝜌 holds for all 𝑡>𝑇. This completes the proof.

3.2. Consensus with All Accesses to a Group Reference

For the sake of being referred conveniently, if there exists a group reference which is available to all agents, the group reference can be labeled as a virtual agent 0, with 𝑎_𝑖0>0, for all 𝑖∈{1,…,𝑛} defined additionally as the weighted communication links. And it is worthwhile to note that the reference is not treated as a member of the n-agent system in this work actually, although it is labeled as number 0. Thus the loose consensus protocol for the real n-agent system is given by𝐮_𝑖(𝑡)=𝐯̇₀(𝑡)−^𝑛_𝑗=0𝑎_𝑖𝑗𝜙𝐫_𝑖(𝑡)−𝐫_𝑗(𝑡−_𝑛𝑗=0)𝑎_𝑖𝑗𝛾_𝑐𝜑𝐯_𝑖(𝑡)−𝐯_𝑗(𝑡),∀𝑖∈{1,…,𝑛}, (12) where 𝐫₀(𝑡) and 𝐯₀ are denoted as the position and velocity of the reference, respectively.

Theorem 3. Suppose that a group reference labeled as agent 0 is available to all agents of multiagent system (1). Using protocol (12) for multiagent system (1), then if the communication topology graph of (1) is undirected and connected, all agents can track the group reference and loose consensus can be achieved such that ‖𝐯_𝑖(𝑡)−𝐯₀(𝑡)‖_∞≤𝜀_𝑣 and ‖𝐫_𝑖(𝑡)−𝐫₀(𝑡)‖_∞≤𝜌 as 𝑡→∞, for all 𝑖,𝑗, where 𝜌 is a certain bounded scalar.

Proof. Denote 𝐫̃_𝑖=𝐫_𝑖−𝐫₀ and 𝐯_𝑖=𝐯_𝑖−𝐯₀, for all 𝑖∈{1,…,𝑛}. Thus, protocol (12) can be written as ̇𝐯_𝑖(𝑡)=−^𝑛_𝑗=0𝑎_𝑖𝑗𝜙𝐫̃_𝑖(𝑡)−𝐫̃_𝑗(𝑡−_𝑛𝑗=0)𝑎_𝑖𝑗𝛾_𝑐𝜑𝐯_𝑖(𝑡)−𝐯_𝑗(𝑡),∀𝑖∈{1,…,𝑛}. (13)
Note that (13) has the same expression form with (5), except for that the dummy variable 𝑗 in the summation function starts from 0 to 𝑛. Introduce a Lyapunov function as follows: 1𝑉=2^𝑛𝑛_𝑖=1𝑗=1𝑎_𝑖𝑗𝟏^𝑇₃𝜓𝐫̃_𝑖−𝐫̃_𝑗+^𝑛_𝑖=1𝑎_𝑖0𝟏^𝑇₃𝜓𝐫̃_𝑖+12^𝑛_𝑖=1𝐯_𝑖^𝑇𝐯_𝑖. (14)
Then the time derivative of 𝑉 along the trajectories of (13) is given by 𝑉̇=12^𝑛𝑛_𝑖=1𝑗=1𝑎_𝑖𝑗𝐯_𝑖−𝐯_𝑗^𝑇𝜙𝐫̃_𝑖−𝐫̃_𝑗+^𝑛_𝑖=1𝑎_𝑖0𝐯^𝑇_𝑖𝜙𝐫̃_𝑖−^𝑛_𝑖=1𝐯^𝑇_𝑖^𝑛_𝑗=1𝑎_𝑖𝑗𝜙𝐫̃_𝑖−𝐫̃_𝑗+𝜑𝐯_𝑖−𝐯_𝑗+𝑎_𝑖0𝜙𝐫̃_𝑖+𝜑𝐯_𝑖=^𝑛𝑛_𝑖=1𝑗=1𝑎_𝑖𝑗𝐯^𝑇_𝑖𝜙𝐫̃_𝑖−𝐫̃_𝑗+^𝑛_𝑖=1𝑎_𝑖0𝐯^𝑇_𝑖𝜙𝐫̃_𝑖−^𝑛𝑛_𝑖=1𝑗=1𝑎_𝑖𝑗𝐯^𝑇_𝑖𝜙𝐫̃_𝑖−𝐫̃_𝑗+𝜑𝐯_𝑖−𝐯_𝑗−^𝑛_𝑖=1𝑎_𝑖0𝐯^𝑇_𝑖𝜙𝐫̃_𝑖+𝜑𝐯_𝑖=−^𝑛𝑛_𝑖=1𝑗=1𝑎_𝑖𝑗𝐯^𝑇_𝑖𝜑𝐯_𝑖−𝐯_𝑗−^𝑛_𝑖=1𝑎_𝑖0𝐯^𝑇_𝑖𝜑𝐯_𝑖1=−2^𝑛𝑛_𝑖=1𝑗=1𝑎_𝑖𝑗𝐯_𝑖−𝐯_𝑗^𝑇𝜑𝐯_𝑖−𝐯_𝑗−^𝑛_𝑖=1𝑎_𝑖0𝐯^𝑇_𝑖𝜑𝐯_𝑖≤0, (15) where we have used derivative rules to obtain the first equality, have used Lemma 1 to obtain the third and last equality, and have used the fact that (𝐯_𝑖−𝐯_𝑗)^𝑇𝜑(𝐯_𝑖−𝐯_𝑗) and 𝐯_𝑖^𝑇𝜑(𝐯_𝑖) are both non-negative for all 𝑖 and 𝑗 to get the final inequality.
Note that 𝑉̇≡0 means that ‖𝐯_𝑖(𝑡)‖_∞≤𝜀_𝑣 and ‖𝐯_𝑖(𝑡)−𝐯_𝑗(𝑡)‖_∞≤𝜀_𝑣,for all 𝑖,𝑗, and, by employing the similar analysis in Section 3.1, one can obtain that ‖𝐫̃_𝑖(𝑡)‖_∞≤𝜌. Since 𝐫̃_𝑖=𝐫_𝑖−𝐫₀ and 𝐯_𝑖=𝐯_𝑖−𝐯₀, it is clear that the proof is completed.

3.3. Offset Consensus with Partial Access to a Reference

In practice, it is more common that only a portion of agents can access to the reference, while others cannot. Generally in this case, the linear consensus problem often used the same form of protocol as what is used in the case where the group reference is available to all agents (see, e.g., [3, 5, 6]). However, in this work, protocol (12) cannot be direct borrowed to apply for loose consensus, and the reasons are as follows: partial access to the reference means that there exist some 𝑖 such that 𝑎_𝑖0=0, and hence ∑^𝑛_𝑖=1𝑎_𝑖0𝐯^𝑇_𝑖𝜑(𝐯_𝑖)=0 does not guarantee 𝐯^𝑇_𝑖𝜑(𝐯_𝑖)=0 for all 𝑖 any longer, and thus 𝑉̇≡0 will not lead to the results that ‖𝐯_𝑖(𝑡)‖_∞≤𝜀_𝑣 for all 𝑖 (see the last equality of (15)). A worse outcome may occur, in which case, even though ‖𝐯_𝑖(𝑡)−𝐯_𝑗(𝑡)‖_∞≤𝜀_𝑣 holds for all 𝑖, there are still some 𝑖 for ‖𝐯_𝑖(𝑡)‖_∞>𝜀_𝑣 so that loose consensus is not achieved.

In order to better borrow the form of protocol (12), in this subsection, a set of 𝑛 distributed estimators are additionally designed as follows:𝐱̇_𝑖=−𝛽_𝑥sgn^𝑛_𝑖=1𝑎_𝑖𝑗𝐱_𝑖−𝐱_𝑗+𝑎_𝑖0𝐱_𝑖−𝐫₀,𝐲̇_𝑖=−𝛽_𝑦sgn^𝑛_𝑖=1𝑎_𝑖𝑗𝐲_𝑖−𝐲_𝑗+𝑎_𝑖0𝐲_𝑖−𝐯₀,𝐳̇̂_𝑖=−𝛽_𝑧sgn^𝑛_𝑖=1𝑎_𝑖𝑗𝐳̂_𝑖−𝐳̂_𝑗+𝑎_𝑖0𝐳̂_𝑖−𝐯̇₀,∀𝑖∈{1,…,𝑛}, (16a)(16b)(16c)where sgn(⋅) denotes the signum function, defined component-wise. 𝛽_𝑥, 𝛽_𝑦, and 𝛽_𝑧 are proper positive constants. 𝐱_𝑖, 𝐲_𝑖, and 𝐳̂_𝑖 are the 𝑖th estimation calculated by agent 𝑖, and denote the estimation of position, velocity, and acceleration of the group reference, respectively.

It is worthwhile to note that estimators (16a), (16b), and (16c) are actually three independent linear first-order information consensus protocols. Also, note that what they deal with are not the dynamics or control forces but the information (the understandings of the reference state for each agent), so uncertainties do not exist and thus linear form is suitable for these estimators.

Theorem 4. Suppose that 𝐯₀, 𝐯̇₀, and 𝐯̈₀ are bounded as ‖𝐯₀‖_∞≤𝜌_𝑥, ‖𝐯̇₀‖_∞≤𝜌_𝑦, and ‖𝐯̈₀‖_∞≤𝜌_𝑧, respectively. Using estimate algorithms (16a), (16b), and (16c) for multiagent system (1), consensus will be achieved in finite time such that 𝐱_𝑖→𝐱_𝑗→𝐫₀, 𝐲_𝑖→𝐲_𝑗→𝐯₀, and 𝐳̂_𝑖→𝐳̂_𝑗→𝐯̇₀, for all 𝑖,𝑗, if and only if both the following conditions hold:(i)the communication topology graph of (1) is undirected and connected, and there exists at least one 𝑎_𝑖0>0;(ii)arguments satisfy that 𝛽_𝑥>𝜌_𝑥, 𝛽_𝑦>𝜌_𝑦 and 𝛽_𝑧>𝜌_𝑧.

Proof (see [14] for more proving details). With 𝐫̃_𝑖=𝐱_𝑖−𝐫₀, we can rewrite (16a) as ̇̃𝐫_𝑖=−𝛽_𝑥sgn^𝑛_𝑖=1𝑎_𝑖𝑗𝐫̃_𝑖−𝐫̃_𝑗+𝑎_𝑖0𝐫̃_𝑖−𝐫̇₀(𝑡). (17)
Equation (17) can be written in matrix form as ̇̃𝐫=−𝛽_𝑥sgn𝐌⊗𝐈₃𝐫̃−𝟏_𝑛⊗𝐫̇₀(𝑡), (18) where 𝐫̃ is the column stack vector of 𝐫̃_𝑖, 𝑖=1,…,𝑛, 𝐌=ℒ+diag(𝑎₁₀,…,𝑎_𝑛0), and “⊗” denotes the Kronecker product operator.
Note that the fixed undirected graph 𝒢 is connected and at least one 𝑎_𝑖0 is positive; it follows that 𝐌 is symmetric positive definite.
Consider the Lyapunov function candidate 𝑉=(1/2)𝐫̃^𝑇(𝐌⊗𝐈₃)𝐫̃. The derivative of 𝑉 is 𝑉̇=𝐫̃^𝑇𝐌⊗𝐈₃−𝛽_𝑥sgn𝐌⊗𝐈₃𝐫̃−𝟏_𝑛⊗𝐫̇₀(𝑡)≤−𝛽_𝑥𝐌⊗𝐈₃𝐫̃₁+𝐫̇₀(𝑡)⋅𝐌⊗𝐈₃𝐫̃₁≤−𝛽_𝑥−𝜌₂𝐌⊗𝐈₃𝐫̃₁<0. (19)
Here, we use the fact that 𝐌 is symmetric positive definite, Holder’s inequality, and 𝛽_𝑥>𝜌₂. It then follows from Lyapunov stability theory that ‖𝐫̃‖→𝟎 as 𝑡→∞.
Note that 𝑉=(1/2)𝐫̃^𝑇(𝐌⊗𝐈₃)𝐫̃≤(1/2)𝜆_max(𝐌)‖𝐫̃‖²₂; it then follows that the derivative of 𝑉 satisfies 𝑉̇≤−𝛽_𝑥−𝜌₂𝐌⊗𝐈₃𝐫̃₁≤−𝛽_𝑥−𝜌₂𝐌⊗𝐈₃𝐫̃₂≤−𝛽_𝑥−𝜌₂𝜆_min(𝐌)‖𝐫̃‖₂≤−𝛽_𝑥−𝜌₂√2𝜆_min(𝐌)𝜆_max(𝐌)√𝑉. (20)
After some manipulation, we get that 2𝑉(𝑡)≤2𝑉(0)−𝛽_𝑥−𝜌₂√2𝜆_min(𝐌)𝜆_max(𝐌)𝑡. (21)
Therefore, there exists a finite time, which equals 𝐫̃^𝑇(0)(𝐌⊗𝐈₃)𝐫̃(0)𝜆_max(𝐌)/(𝛽_𝑥−𝜌₂)𝜆_min(𝐌) actually, such that 𝑉(𝑡)=0 holds for all 𝑡 after the finite time. Equations (16b) and (16c) can also be proved with no differences, and hence the proof is completed.

After the design of 𝑛 distributed estimators, all 𝑛 agents can have a consensus understanding of the state of the group reference after some finite time. In other words, after some finite time for such 𝑛 estimators to work, the group reference can be regarded as being available to all the 𝑛 agents by using their each estimation instead of the partial available group reference. And thus the remaining problem has been completely transformed to the case which is discussed in Section 3.2.

In this subsection, the configuration offset of each agent, which is denoted as 𝛿_𝑖 for all 𝑖 (it can be time varying, and it should be self-known by agent 𝑖. Sometimes heterogeneous agents may be desired to form a given relative configuration. e.g., a sighted agent should be aware of its role of sensing, and hence an offset in front of the formation would be set as default when such an agent gets ready for formation motion), is also additionally considered. The loose consensus protocol for the 𝑛 agents, which have a partial access to a reference and have self-known offsets, is given by 𝐮_𝑖(𝑡)=𝐳̂_𝑖+𝛿̈_𝑖−𝜙𝐫_𝑖(𝑡)−𝛿_𝑖−𝐱_𝑖−𝜑𝐯_𝑖(𝑡)−𝛿̇_𝑖−𝐲_𝑖−^𝑛_𝑗=1𝑎_𝑖𝑗𝜙𝐫_𝑖(𝑡)−𝛿_𝑖−𝐫_𝑗(𝑡)+𝛿_𝑗−^𝑛_𝑗=1𝑎_𝑖𝑗𝛾_𝑐𝜑𝐯_𝑖(𝑡)−𝛿̇_𝑖−𝐯_𝑗(𝑡)+𝛿̇_𝑗,∀𝑖∈{1,…,𝑛}. (22)

Theorem 5. After using (16a), (16b), and (16c) for multiagent system (1) for an enough time 𝑡<𝑇₀, employing protocol (22), if both the conditions in Theorem 4 hold, then system (1) will achieve loose consensus such that ‖𝐯_𝑖(𝑡)−𝛿̇_𝑖−𝐯₀(𝑡)‖_∞≤𝜀_𝑣 and ‖𝐫_𝑖(𝑡)−𝛿_𝑖−𝐫₀(𝑡)‖_∞≤𝜌 as 𝑡→∞, for all 𝑖,𝑗, where 𝜌 is a certain bounded scalar.

Proof. If both the conditions in Theorem 4 hold, then it follows from Theorem 4 that there exists 𝑇₀>0 such that 𝐱_𝑖→𝐱_𝑗→𝐫₀, 𝐲_𝑖→𝐲_𝑗→𝐯₀, and 𝐳̂_𝑖→𝐳̂_𝑗→𝐯̇₀, for all 𝑡>𝑇₀.
Thus, for all 𝑡>𝑇₀, we can use 𝐫₀, 𝐯₀, and 𝐯̇₀ to take the place of 𝐱_𝑖, 𝐲_𝑖, and 𝐳̂_𝑖, respectively, in (22). And by denoting 𝐫̃_𝑖=𝐫_𝑖−𝛿_𝑖 and 𝐯_𝑖=𝐯_𝑖−𝛿̇_𝑖, (22) can be written as follows: ̇𝐯_𝑖(𝑡)=𝐯̇₀−𝜙𝐫̃_𝑖(𝑡)−𝐫₀−𝜑𝐯_𝑖(𝑡)−𝐯₀−^𝑛_𝑗=1𝑎_𝑖𝑗𝜙𝐫̃_𝑖(𝑡)−𝐫̃_𝑗(𝑡−_𝑛𝑗=1)𝑎_𝑖𝑗𝛾_𝑐𝜑𝐯_𝑖(𝑡)−𝐯_𝑗(𝑡=̇𝐯₀)(𝑡)−^𝑛_𝑗=0𝑎_𝑖𝑗𝜙𝐫̃_𝑖(𝑡)−𝐫̃_𝑗(𝑡−_𝑛𝑗=0)𝑎_𝑖𝑗𝛾_𝑐𝜑𝐯_𝑖(𝑡)−𝐯_𝑗(𝑡). (23)
Note that, if one let all 𝑎_𝑖0 in (12) equal 1, then (12) can be totally equivalent to (23). Therefore, the same process of analysis in Section 3.2 can be borrowed to complete the proof, and hence detailed process can be omitted here. It can be followed from the conclusion of Theorem 3 that ‖𝐯_𝑖(𝑡)−𝐯₀(𝑡)‖_∞≤𝜀_𝑣 and ‖𝐫̃_𝑖(𝑡)−𝐫₀(𝑡)‖_∞≤𝜌. Note that 𝐫̃_𝑖=𝐫_𝑖−𝛿_𝑖 and 𝐯_𝑖=𝐯_𝑖−𝛿̇_𝑖, and it is clear that ‖𝐯_𝑖(𝑡)−𝛿̇_𝑖−𝐯₀(𝑡)‖_∞≤𝜀_𝑣 and ‖𝐫_𝑖(𝑡)−𝛿_𝑖−𝐫₀(𝑡)‖_∞≤𝜌, which means that the offset of agents will achieve loose consensus as 𝑡→∞.

4. Numerical Simulations

In the simulations, we will show that the proposed protocol (5) and (22) can lead to consensus firstly. Then by comparing protocols (5) with (2), we will show that, in the existence of measurement uncertainties, the proposed protocols in this paper are of benefit in saving energy.

A multiagent system which contains 6 agents is taken as an example, and they are considered in 1 dimension for simplicity, although the analyses throughout the paper are discussed in any dimensions of order 𝑚. Figure 1(a) shows the internal communication links in the system, while Figure 1(b) additionally contains the communication links of the external reference. The weights of each edge are all set to 0.8. Initial conditions of the 6 agents are all set randomly, which are 𝐫(𝑡₀)=[−4,5,−7,4,−3,3]^𝑇 m and 𝐯(𝑡₀)=[1.6,2.2,−1.5,3,−2.8,−0.8]^𝑇 m/s as an instantiation in particular. The arguments in function (4) are set as 𝜀_𝑟=0.05 and 𝜀_𝑣=0.02, which means that measurement uncertainties of positions are within 0.05 m and those of velocities are within 0.02 m/s. The damping scalar 𝛾_𝑐 is set to 2.5. The group reference is set as 𝑟₀(𝑡)=2sin(0.2𝑡), and the offsets of the 𝑖th agent from the reference are 𝑖 m, respectively, which are constant.

(a)

(b)

Figure 1 

Communication links of a multiagent system.

When a group reference does not exist, employing protocol (5) for the system, we get simulation results shown in Figures 2 and 3. It is clear that the 6 agents gather and then keep moving together with similar velocities around their average velocity. Since their positions and velocities are very similar but not exact coincident, we know that loose consensus is achieved. The trajectories results by applying linear protocol (2) are similar to Figures 2 and 3 with little difference actually, so they are omitted here.

Figure 2 

Position trajectories of all agents with (5).

Figure 3 

Velocity trajectories of all agents with (5).

When the reference exists, after using (16a), (16b), and (16c) to estimate the reference for a period of time, we employ protocol (22) for the system, then we get simulation results shown in Figures 4 and 5. It can be seen that, in the existence of the reference, the trajectories of agents are different from what have been shown in Figures 2 and 3. The position trajectories change like sinusoidal functions with amplitudes of about 4 m, and velocity trajectories also change like sinusoidal functions with amplitudes of about 0.8 m/s. Both the position trajectories and velocity trajectories are in accordance with the given reference. It can be known that agents track the reference successfully. And the difference among each position trajectory is their different offsets; the phenomenon shows that the self-known configuration offsets do have an effect. Loose consensus is achieved.

Figure 4 

Position trajectories of all agents with (22).

Figure 5 

Velocity trajectories of all agents with (22).

Then we compare linear protocol (2) and non-linear protocol (5) in the area of energy saving. We introduce a quantity that indicates the sum of the energy (e.g., fuels) of all the agents consuming to generate forces over time, defined as 𝐸(𝑡)=∫^𝑡₀∑^𝑛_𝑖=1|𝑢_𝑖|𝑑𝑥. Comparison results are shown in Figure 6. For clear statement, the simulation time (0~100 s) can be distinguished into two periods: in the first period, which can be called formation forming in this paper, agents try to achieve consensus from distinct initial states till the consensus achieved; and in the second one, which can be called formation keeping in this paper, agents maintain their similar states. It can be seen that, in the period of formation forming, there is nearly no difference between the two protocols. And as soon as the multiagent system gets into the period of formation keeping, protocol (5) performs better to save energy, which costs much less energy than protocol (2) does. The reason is obvious in that, since the measuring errors exist, linear protocol (2) finds it hard to make agents agree on the very exact average and thus cost energy continuously. As a comparison, nonlinear protocol (5) allows agents to achieve a loose agreement, and hence it avoids the unnecessary cost of energy. It is also worthwhile to note that protocol (5) is more effective in the keeping period but less effective in the forming period. This is because, in the forming period, compared to measurement uncertainties, the state differences among agents are large enough to mask the effect of the loose control. In this sense, the protocol proposed in this work will have no advantage compared to the linear one in the case where relative configuration of formation is desired to be time varying.

Figure 6 

Comparison of energy costs.

5. Conclusions

In this paper, in order to avoid a potential waste of energy in the case where there exist measurement uncertainties, a nonlinear protocol is proposed with a connected undirected communication topology. The protocol is then extended to both the cases where agents can have full or partial access to a group reference. Especially in the case of partial access, a set of distributed estimators is utilized to help all agents agree on the understandings of the reference, even though some agents cannot access to the reference directly. An additional condition where self-known configuration offsets are desired is also considered. Theoretical analyses of stability are given for the proposed protocols. Finally, simulations are performed by MATLAB, and results show that these protocols can lead agents to loose consensus and work effectively with less energy cost to keep formation, while there is little enhancement by using such protocols when agents are trying to form the formation.