Chapter 7
Consensus in Heterogeneous Multi-Agent Systems
Gentlemen can live harmoniously together even though they have heterogeneous characters, but non-gentlemen of the same character live in discord.
Confucius (551–479 BC), Analects of Confucius
Multi-agent systems (MASs) may contain heterogeneous network channels and/or heterogeneous agent dynamics. Even for systems with identical plants in agents, diverse input delays generate heterogeneous agent dynamics, and diverse communication delays result in heterogeneous network channels. Consensus problems in heterogeneous MASs are studied in this chapter. High-order consensus is defined for a class of high-order heterogeneous MASs. A necessary and sufficient condition is given for the existence of high-order consensus solutions to the considered class of systems. The condition shows that for systems with diverse communication delays, high-order consensus does not require that the self-delay of each agent to be equal to the corresponding communication delay. The frequency-domain scalability analysis method developed in Chapter 3 is applied to first-order and second-order MASs with diverse input delays and communication delays.
7.1 Integrator Agent System with Diverse Input and Communication Delays
In this section, we first consider the consensus problem for the integrator agent system with diverse input delays based on undirected graphs. Due to the heterogeneousness caused by the diverse input delays, the consensus problem of such a system can not be converted into an ordinary stability problem by using the technique developed in Chapter 6. However, we will show that it can still be considered as a semi-stability problem if the interconnection topology graph is connected. Therefore, using the frequency-domain analysis theory developed in Chapter 3, we develop various scalable consensus conditions, which uses only local information of each agent. Finally, by considering the consensus problem for digraph-based systems with both diverse communication delays and diverse input delays, we will show that consensus condition is dependent on input delays but independent of communication delays when the digraph contains a globally reachable node.
7.1.1 Consensus in Discrete-Time Systems
Consider a discrete-time multi-agent system with topology graph G = (V, E, A) and integrator agents given by
where and denote the state and the control input of agent i at time instant k, respectively. The topology graph G = (V, E, A) of the system can be directed or undirected, depending on context. The consensus protocol for the system is given by
(7.2)
where Ni denotes the neighbors of agent i, and aij > 0 is the adjacency element of A in the graph G = (V, E, A).
When each agent is subject to an input delay Di, system (7.1) becomes
Under diverse communication delays, the consensus protocol becomes
where τij represents the communication delay from agent j to agent i.
Under protocol (7.4), multi-agent system (7.3) is said to achieve a consensus asymptotically if
where is a constant.
The closed-loop system of (7.3) and (7.4) is
Let x(k) = [x1(k), , xn(k)]T, and
Then, equation (7.5) can be rewritten as a time-delayed system in a vector form
where , and nd = n(n + 1). Obviously, , which is the Laplacian matrix of the topology graph.
The characteristic equation of system (7.6) is given by
The equilibrium set of system (7.6) is defined by
When L is singular, Xe is a continuum of equilibrium points. Assume that the interconnection topology of the system is described by a connected undirected graph or a digraph containing a globally reachable node. Then, by Theorem 1.9, the Laplacian matrix L has a simple eigenvalue 0, i.e., det (L) = 0 and rank(L) = n − 1. By the definition of L we also have L1n = 0. So, all the elements in Xe can be represented as c1n where c is any constant. Therefore, system (7.6) achieves a consensus asymptotically, if the solution of the system starting from any given initial states , asymptotically converges to an element in Xe. According to this analysis the following lemma can be easily proved.
Lemma 7.1 If the solutions of equation (7.7) have modulus less than unity except for a root at z = 1, then system (7.6) with a connected undirected graph or a digraph containing a globally reachable node achieves a consensus asymptotically.
This lemma implies that under the assumption that the graph is connected or the digraph contains a globally reachable node the first-order agent system with diverse input delays and communication delays achieves a consensus asymptotically if the closed-loop system is steady semi-stable with z = 1 as a simple pole.
7.1.2 Consensus under Diverse Input Delays
In this subsection we consider the consensus problem for multi-agent systems with input delays only. In this case, the closed-loop form (7.5) of the system reduces to
The following theorem gives a scalable consensus condition for multi-agent systems with diverse input delays
Theorem 7.2 Assume that system (7.8) of n agents is based on an undirected and connected graph G = (V, E, A) with symmetric weights. The system achieves a consensus asymptotically if
Proof. Taking the z-transformation of system (7.8) and writing it in vector form, we get
Note that L in (7.10) is a positively semi-definite matrix since an undirected graph is considered. The characteristic equation is
(7.11)
Define . Then, we will prove that all the zeros of p(z) have modulus less than unity except for a zero at z = 1.
Let z = 1, then p(1) = det (L). Since G = (V, E, A) is connected, by Theorem 1.9, zero is a simple eigenvalue of L, i.e., det (L) = 0 and rank(L) = n − 1. Thus, p(z) indeed has a zero at z = 1.
To prove that the system is steady semi-stable with z = 1 as a simple pole it suffices to prove that the zeros of have modulus less than unity. By the general Nyquist stability criterion for discrete-time systems (Theorem 2.22), this is the case if the eigenloci of
do not enclose the point (− 1, j0) for all ω [− π, π]. To show this we rewrite F(jω) as
where . Since
by Lemma 2.34 we have
Since the spectral radius of any matrix is bounded by its maximum absolute row sum according to Corollary 2.31 of Gershgorin's disc lemma, it follows from the condition (7.9) that
Now, from Theorem 3.16 we conclude that eigenloci of F(jω) do not enclose the point (− 1, j0) for all ω [− π, π], which implies that the zeros of p(z) have modulus less than unity except for a zero at z = 1. Theorem 7.2 is thus proved by Lemma 7.1.
Remark. By using Corollary 2.32 instead of Corollary 2.31 of Gershgorin's disc lemma in the proof, we know that system (7.8) achieves a consensus asymptotically if there exists such that
(7.12)
Of course, this condition can be less conservative than (7.9) and reduces to it if H = I.
Theorem 7.2 gives a scalable delay-dependent consensus condition. This condition suggests that under large input delays, small interconnection weights and small numbers of neighbors increase the possibility of achieving consensus if the interconnection topology graph is connected.
Now, we apply the result of Theorem 7.2 to study the effect of diverse input delays on some important systems that have been extensively investigated in literature.
First, let us consider Vicsek's model, which describes a group of agents moving in the plane with the same line velocity (Vicsek et al. 1995). When the headings of the agents are close to each other, the local updating rule for the headings may be approximated by the following linearized equation (Jadbabaie, Lin and Morse 2003)
(7.13)
with and where ni denotes the number of the neighbors of agent i. By applying Theorem 7.2 to this model, it is easy to get its consensus condition as follows:
Obviously, when n > 1, condition (7.14) holds only if Di = 0. This implies that Vicsek's model is very sensitive to input delays. To overcome this problem we proposed a modified linearized Vicsek's model with input delays as
where εi > 0 is an adjustable interconnection gain. From Theorem 7.2, we get the following corollary.
Corollary 7.3 Suppose that the interconnection topology graph of system (7.15) is connected. Then, the system achieves a consensus asymptotically if
Remark 1. When Di = 0, from (7.16) it follows that
which always holds if εi ≤ 1. This implies that the linearized Vicsek's model in its original form (εi = 1) can achieve a consensus asymptotically if and only if the interconnection topology graph of the system is connected. When Di ≥ 1, inequality (7.16) holds only if εi < 1. So, the introduction of small εi is necessary for enhancing the robustness of the linearized Vicsek's model against input delays.
Remark 2. Corollary 7.3 clearly shows the relationship of input delays, interconnection weights and number of neighbors: for large input delays one should use small interconnection weights or have small numbers of neighbors when the graph is kept connected.
Similarly, we can also apply Theorem 7.2 to Moreau's model (Moreau 2005) with input delays
where denotes the positive weight corresponding to the edge eij in the weighted graph G. The following result is a direct corollary of Theorem 7.2.
Corollary 7.4 Suppose that the interconnection topology graph of system (7.17) is connected and has symmetric weights. Then, the system achieves a consensus asymptotically if
Remark. Obviously, (7.18) always holds if Di = 0. This implies that a symmetric Moreau's model can achieve a consensus asymptotically if and only if the interconnection topology graph of the system is connected. For Di ≥ 1, from (7.18) we get a sufficient consensus condition of Moreau's model as
(7.19)
7.1.3 Consensus under Diverse Communication Delays and Input Delays
In this subsection, we consider multi-agent systems with both communication delays and input delays. Actually, if the communication delays τij are symmetric, i.e., they satisfy the requirement (2.26), then all the results given in Section 7.1.2 can be extended to systems with both communication delays and input delays without any difficulty.
In general, however, the diversity of communication delays destroys the symmetry of the system even if the graph is undirected with symmetric weights. This implies that the tools used for Theorem 7.2, which are mainly referred to Lemma 2.34 and Theorem 3.16, are no longer applicable. So, in the following analysis we will use Greshgorin's disk theorem (Lemma 2.30) to estimate matrix eigenvalues, which does not require the symmetry. Note that the interconnection topology studied in this subsection can be a digraph with asymmetric weights.
Let us first introduce the following lemma as a preliminary result.
Lemma 7.5 The following inequality
holds for all non-negative integers D and all ω [− π, π].
Proof. First of all, we claim that
holds for any non-negative integer D. Indeed, by denoting , we have x (0, 1] for any non-negative integer D. Thus, inequality (7.21) is equivalent to the well-known inequality , where x (0, 1].
Now, we note that We just need to prove (7.20) for all ω (0, π] because the left-hand side of (7.20) is an even function for ω [− π, π].
When , let . Calculating the derivative of h(ω) with respect to ω yields Obviously, we have , i.e., h(ω) is not increasing for all . Since h(0) = 0, we have h(ω) ≤ 0, i.e., for all . Since , we get .
When , we have for all non-negative integers D. So, from (7.21), we get
for all and all non-negative integers D. The lemma is proved.
Theorem 7.6 Consider multi-agent system (7.3) with protocol (7.4). Assume that the interconnection topology digraph G = (V, E, A) of the system has a globally reachable node. Then the system achieves a consensus asymptotically if
Proof. The closed-loop system of (7.3) with (7.4) is given by (7.5). Taking the z-transformation of the system (7.5), we get
where Xi(z) is the z-transformation of xi(k). Define an n × n matrix as follows:
Obviously, , which is the Laplacian matrix. Then, (7.23) can be written as where X(z) = [X1(z), , Xn(z)]T. Define
Then, we will prove that all the zeros of p(z) have modulus less than unity except for a zero at z = 1 in the following.
Let z = 1, . Since G = (V, E, A) has a globally reachable node, by Theorem 1.9, zero is a simple eigenvalue of L, i.e., det (L) = 0 and rank(L) = n − 1. Thus, p(z) indeed has a simple zero at z = 1.
Now, we prove that the zeros of have modulus less than unity. Based on the general Nyquist stability criterion (Corollary 2.20), the zeros of f(z) have modulus less than unity, if the eigenloci of , i.e., , do not enclose the point (− 1, j0) for ω [− π, π]. By Greshgorin's disk lemma (Lemma 2.30), we have for all ω [− π, π], where
Further, we can show that
where . Now, define
The Nyquist plot of Gi(ω) for ω [− π, π] is illustrated by Figure 7.1. Note that Gi(ω) is just the center of the disc . So, does not enclose the point (− 1, j0) for all ω [− π, π] as long as the point (− a, j0) with a ≥ 1 is not in the disc for all ω [− π, π], i.e., holds for all ω [− π, π] when a ≥ 1.
From (7.24), we have
Because holds for ω [− π, π] by Lemma 7.5, it follows from (7.22) that
Thus,
i.e.,
holds for all ω [− π, π] when a ≥ 1.
Now, we have proved that the zeros of p(z) have modulus less than unity except for a zero at z = 1. Therefore, Theorem 7.6 is proved by Lemma 7.1.
Remark 1. Noticing the inequality (7.21), we know that the condition (7.22) is more conservative than the condition (7.9) given by Theorem 7.2, which is even necessary and sufficient for the case of a single-link, two-node network with equal delay. However, it is still scalable, and moreover, it is applicable to the systems based on digraphs with asymmetric weights.
Remark 2. When there are no input delays, i.e., Di = 0, the consensus condition (7.22) reduces to
which implies that the system can achieve a consensus asymptotically if and only the interconnection topology digraph of the system has a globally reachable node and (7.25) holds regardless of the existence of diverse communication delays.
Now, let us apply Theorem 7.6 to study the linearized Vicsek model and Moreau's model.
With communication delays and input delays, the linearized Vicsek model given in Jadbabaie, Lin and Morse (2003) becomes
where ni denotes the number of the neighbors of agent i.
From Theorem 7.6 we get the following corollary for Vicsek's model (7.26).
Corollary 7.7 Assume that the interconnection topology of system (7.26) has a globally reachable node. System (7.26) achieves a consensus asymptotically if
Remark. On the one hand, when Di = 0, inequality (7.27) holds automatically. This implies that the convergence of the consensus protocol given by Vicsek's model is independent of communication delays provided the graph has a globally reachable node. This coincides with the result given in Wang and Slotine (2006). Cao et al. (2006) extended this result to the case when the graph is jointly rooted. On the other hand, inequality (7.27) holds only for Di = 0. This may suggest that Vicsek's model is very sensitive to input delays. To enhance its robustness against input delays, one should introduce small weights as shown in model (7.15).
Similarly, Moreau's model given in (Moreau 2005) with communication delays and input delays can be written as
The following result is a direct corollary of Theorem 7.6.
Corollary 7.8 If the interconnection topology digraph of system (7.28) has a globally reachable node, then the system achieves a consensus asymptotically if
Remark. Obviously, (7.29) can be rewritten as . So, it always holds if Di = 0. But for Di > 0 it holds only for some appropriately designed weights .
7.1.4 Continuous-Time System
Consider a continuous-time multi-agent system with diverse input delays and communication delays
It is easy to show that under the assumption that the topology graph is connected or the topology digraph contains a globally reachable node, the system achieves a consensus asymptotically if it is steady semi-stable with s = 0 as a simple pole.
Using the theory developed in Chapter 3 (Theorem 3.10), in a similar way to that shown in the proof of Theorem 7.2 one can get the consensus condition for continuous-time systems with input delays.
Theorem 7.9 Suppose that the topology graph of system (7.30) is connected with symmetric weights. Then, the system achieves a consensus asymptotically if
Remark. It is easy to see that in the case when all the delays Ti are the same for all , the condition (7.31) reduces to the result given by Olfati-Saber and Murray (2004).
The continuous-time system with diverse input and communication delays can be written as
Through a similar procedure to that used in the proof of Theorem 7.6, one can get a consensus condition for continuous-time systems with diverse communication and input delays.
Theorem 7.10 If the interconnection topology digraph of system (7.32) has a globally reachable node, then the system achieves a consensus asymptotically if
Remark. When there is no input delays, i.e., Ti = 0, the consensus condition (7.33) always holds, which implies that the system can achieve a consensus asymptotically if and only the interconnection topology digraph of the system has a globally reachable node regardless of the existence of diverse communication delays. This conclusion coincides with existing results in references such as Blondel et al. (2005) and Wang and Slotine (2006).
7.1.5 Simulation Study
Example 7.1 Symmetric system.
Consider a system of eighty agents described by the modified linearized Vicsek model (7.15). The interconnection topology for the agents is a closed ring (Figure 7.2). Note that under non-zero input delays, the linearized Vicsek's model with unity weights (εi = 1) has no consensus. In simulations, we choose the coupling weights as
By Corollary 7.3, the admissible values of the input delays are
Simulations validate that under the admissible input delays the system achieves consensus asymptotically. We depict the boundary values of the input delay versus the coupling weights in Figure 7.3.
Note that the bound of input delay determined by Corollary 7.3 may not apply to asymmetric systems. For example, with each agent's input delay at boundary, we introduce an identical communication delay, T ≥ 1, which destroys the conjugate symmetry of the system matrix in the frequency domain, between each pair of neighboring agents. Then, the simulation shows that the system has no asymptotic consensus.
Now, we can use Theorem 7.6 to get a consensus condition for the system with both input delays and communication delays. By Theorem 7.6, the admissible values of the input delays are
The boundary values of input delays estimated by Theorem 7.6 are also shown in Figure 7.3. The boundary determined by Theorem 7.6 is lower than the boundary given by Corollary 7.3, which implies that Theorem 7.6 is more conservative than Corollary 7.3 for symmetric systems. However, with input delays in the area determined by Theorem 7.6, the asymptotic consensus of the system is robust to arbitrary communication delays.
Example 7.2 Asymmetric system.
Consider the multi-agent system (7.5) with an interconnection digraph shown by Figure 7.4. The weights of the directed paths are: a12 = 0.1, a16 = 0.05, a23 = 0.15, a36 = 0.1, a43 = 0.05, a45 = 0.1, a56 = 0.15, a62 = 0.15, and the corresponding communication delays are: τ12 = 5, τ16 = 3, τ23 = 4, τ36 = 4, τ43 = 4, τ45 = 6, τ56 = 6, τ62 = 5. Simulation shows that the system is quite sensitive to input delays, and it cannot converge to any consensus when Ti > 3. This is an asymmetric system to which Theorem 7.2 does not apply. Using Theorem 7.6, we get that Ti ≤ 2 is a sufficient consensus condition which is independent of communication delays. We choose Ti = 2, i = 1, 2, 3, 4, 5, 6, in the simulation. The multi-agent system converges to a consensus as shown by Figure 7.5.
7.2 Double Integrator System with Diverse Input Delays and Interconnection Uncertainties
7.2.1 Leader-Following Consensus Algorithm
Consider a multi-agent system with interconnection topology graph G = (V, E, A). There are n agents with diverse input delays
where , , , and Ti > 0 are the position, velocity, acceleration and input delay, respectively, of agent i.
For system (7.34), the leader-following coordination control strategy is adopted in this section. Let the dynamics of the leader be determined by
where is the position of the leader, and is a constant which represents the desired velocity for all the agents.
Then, the consensus protocol for the first-order multi-agent system (Olfati-Saber and Murray (2004) can be easily extended to the leader-following system as follows
where ki > 0 and γ > 0, Ni denotes the neighbors of agent i, aij > 0 is the adjacency element of A in the digraph G = (V, E, A), and bi is the linking weight from agent i to the leader (7.35). Note that bi > 0 if there is a directed edge from agent i to the leader; otherwise, bi = 0. Let for notation convenience.
Remark. Protocol (7.36) can be used only for following a leader with a constant velocity, or a leader with a velocity which is time-varying but asymptotically approaching to a constant. If the leader's velocity is not converging to any constant, then each agent should estimate its neighbors' accelerations, and the stability analysis of that kind of consensus protocol will be much more complicated.
With consensus protocol (7.36), the closed-loop form of system (7.34) is given by
The following lemma gives some structural property of the leader-following system.
Lemma 7.13 Assume that the interconnection topology graph of n agents together with the leader in system (7.37) has the leader as a globally reachable node. Then, the matrix L + B has no zero eigenvalues, where L is the Laplacian matrix of the interconnection topology of n agents without the leader.
Proof. Consider the interconnection topology graph with n + 1 nodes corresponding to the n agents of system (7.37) and the leader. Obviously the Laplacian matrix of this topology is given by
where . Since the leader is a globally reachable node in the graph, we have , by Theorem 1.9. Taking elementary column transforms for by adding all the other columns to the first column as follows
we get rank(L + B) = n. The lemma is proved.
7.2.2 Consensus Condition under Symmetric Coupling Weights
Let
Then, from (7.37) it follows that
Taking the Laplace transform of (7.38), one gets
Denote
Then, using the framework of Lee and Spong (2006) one can get a sufficient condition of consensus as
However, it can be shown that this condition is so conservative that it gives an empty set of available control parameters for the second-order multi-agent system with input delays. Let us rewrite (7.40) as
where , and
Then, the condition (7.41) is equivalent to
(7.43)
Actually, such a condition never holds for any κi > 0 when γ > 0, Ti > 0.
Let us derive a less conservative consensus condition for system (7.37) with symmetric coupling weights.
For all , we denote
As shown in Chapter 3 (Proposition 3.26), ω0(i) is the critical point of the frequency response of Wi(s) from clockwise part to anti-clockwise part.
Let be the agent which has the maximal input delay constant , i.e.,
(7.44)
Now, we are in a position to present some sufficient consensus conditions for the second-order multi-agent system with input delays.
Theorem 7.14 Assume that system (7.37) is composed of n agents and a leader with a static interconnection topology that has the leader as a globally reachable node, and the topology graph has symmetric weights, i.e., aij = aji. For each agent the following preconditions are assumed:
Then, all the agents in the system asymptotically converge to the leader's state, if
where is the gain margin of the transfer function Wi(s) defined in (7.42).
Proof. Writing (7.39) in the vector form, we can get the characteristic equation of system (7.38) as
where L is the Laplacian matrix corresponding to the interconnection topology for all the agents without the leader.
Define . To prove Theorem 7.14 it suffices to prove that all the zeros of F(s) are in the open left half of the complex plane.
Let s = 0. Then F(0) = det (γdiag{ki}(L + B)). Because the interconnection topology composed of the n agents together with the leader has the leader as a globally reachable node, F(0) ≠ 0 by Lemma 7.13.
Now, define . We will prove that all the zeros of p(s) are inside the LHP. Based on the general Nyquist stability criterion (Corollary 2.20), all the zeros of p(s) lie inside the LHP, if the eigenloci
do not enclose the point (− 1, j0) for all .
For the symmetric weights (aij = aji), L + B = (L + B)T. Hence, based on Lemma 2.34, we have
Since the spectral radius of any matrix is bounded by its largest absolute row sum, it follows from the condition (7.47) that
Therefore, from Theorem 3.32 it follows that
i.e., the eigenloci of do not enclose the point (− 1, j0) for all , which implies that the zeros of F(s) are all inside the LHP. Theorem 7.14 is thus proved.
Remark. The result of Theorem 7.14 can be extended to systems with both communication delays and input delays without any difficulty if the diverse communication delays τij are symmetric, i.e., they satisfy the requirement (2.26).
7.2.3 Robust Consensus under Asymmetric Perturbations
The consensus condition given by Theorem 7.14 depends on the strict symmetry of the Laplacian matrix L. In practice, however, perturbations of coupling weights may occur and destroy the symmetry. In the following, we study the robustness of the consensus protocol against asymmetric perturbations.
Suppose that the symmetric coupling weights of system (7.37) are subject to some asymmetric perturbations, denoted by for each one. Then the system becomes
where aij = aji, and aij + δij > 0 hold for j Ni.
A robust consensus condition of the perturbed system is given by the following theorem.
Theorem 7.15 Assume that the nominal part of system (7.49), i.e., the system without asymmetric weight perturbations δij, converges to the leader's states asymptotically. Let
where and
Then, the agents in the perturbed system (7.49) converge to the leader's states asymptotically, if
where denotes the largest singular value of matrix, and Δ = {Δij} is the asymmetric perturbation matrix, which is defined as follows
Proof. Under the same variable transformation as used in the previous subsection
it is easy to get the characteristic equation of system (7.49) as
Since the system (7.49) without asymmetric weight perturbations δij converges to the leader's states asymptotically, the roots of the characteristic equation (7.48) all lie inside the LHP, i.e., the zeros of det (s2I + s2KD(s)(L + B)) lie inside the LHP, and det (L + B) ≠ 0.
In the following, we will prove that the roots of equation (7.52) are all inside the LHP.
First we show that equation (7.52) has no roots at s = 0. Indeed, by setting ω = 0 we get from (7.51) that . This implies that
So, it follows that
or equivalently,
This proves that equation (7.52) has no roots at s = 0. Therefore, the characteristic equation (7.52) can be equivalently rewritten as
The feedback diagram corresponding to the characteristic equation (7.53) is demonstrated in Figure 7.6.Using the linear fractional transformation, the diagram in Figure 7.6 can be equivalently transformed into the form shown by Figure 7.7, where M(s) is given by equation (7.50).
The characteristic equation of the closed-loop system in Figure 7.7 is
Obviously, D(s) has no poles in the open RHP. Thus, ΔM(s) has no poles in the open RHP. According to the general Nyquist stability criterion (Corollary 2.20), the roots of the characteristic equation (7.54) all lie inside the LHP, as long as the eigenloci of ΔM(s), i.e., λ(ΔM(jω)), do not enclose the point (− 1, j0) for .
From condition (7.51) it follows that
(7.55)
Hence, λ(ΔM(jω)) does not enclose the point (− 1, j0) for all , i.e., the roots of the characteristic equation (7.54) all lie inside the LHP. Therefore, the closed-loop system in Figure 7.7 is asymptotically stable, and the agents in (7.49) converge to the leader's states asymptotically. Theorem 7.15 is proved.
7.2.4 Simulation Study
Example 7.16 Design procedure based on Theorem 7.14
Consider a system (7.37) of five agents and one leader described by (7.35). The interconnection topology is described in Figure 7.8. Obviously, the leader is globally reachable. Assume that the input delays for the agents are: T1 = 0.5(s), T2 = 1.0(s), T3 = 0.7(s), T4 = 0.6(s) and T5 = 0.8(s). The weights of the edges are: a12 = a21 = 0.30, a25 = a52 = 0.70, a13 = a31 = 0.10, a34 = a43 = 1.10, a42 = a24 = 0.50, b5 = 1.50.
In the following, we design parameters γ and ki in the consensus protocol (7.36) so that the agents converge to the leader's state asymptotically.
Step 1: Choosing γ.
The condition (7.45) requires γ (0, 0.4495/Ti), , which implies that
Since T2 is the maximal input delay, we have . Denote , and the condition (7.46) can be represented as Ei > 0, i = 1, 2, 3, 4, 5. With the given input delays, Ei is a function of γ. The curves of Ei on γ are shown in Figure 7.9.
From Figure 7.9 it is clear that the condition Ei > 0, i = 1, 2, 3, 4, 5, holds if
According to (7.56) and (7.57), we can choose γ = 0.10 to guarantee the conditions (7.45) and (7.46).
Step 2: Choosing ki.
For the transfer functions
using the MATLAB® simulator we obtain the inverses of their gain margins as , , , , . From the condition (7.47), the constraints on ki can be calculated as k1 (0, 3.788), k2 (0, 0.498), k3 (0, 0.906), k4 (0, 0.801), k5 (0, 0.651), We choose k1 = 3.4, k2 = k3 = k4 = k5 = 0.4 for the simulation.
With the parameters chosen above and the initial states generated randomly, the agents in the system (7.37) asymptotically converge to the leader's state as shown in Figure 7.10.
Since there are no other theoretic results to compare with, we test the conservatism of our results by simulation. The procedure is as follows. Setting k2, k3, k4 and k5 at our theoretic boundary values as 0.498, 0.906, 0.801 and 0.651 respectively, we increase k1 from our boundary value 3.788 until the system has no consensus. Then we find the computational margin for k1 is k1m = 7.519. Using similar procedures we can obtain the other marginal gains as k2m = 0.797, k3m = 1.088, k4m = 0.954, k5m = 0.773. Note that unlike our theoretical results, these computational margins cannot be used in the consensus protocol simultaneously.
Example 7.17 System with interconnection uncertainties.
Consider the multi-agent systems (7.49) of five agents and one leader described by (7.35) with the same interconnection topology as Example 7.16 (see Figure 7.8). For simplicity, we choose the same aij, bi, input delays Ti and control parameters γ and as given in Example 7.16. From Theorem 7.14, the system (7.49) without asymmetric weight perturbations converges to the leader's states asymptotically, and the zeros of det (s2I + s2KD(s)(L + B)) lie inside the LHP. Using the MATLAB® simulator, we obtain that the largest value of on ω (− ∞, ∞) is . From Theorem 7.15, if the largest singular value of the asymmetric disturbance matrix Δ, i.e., , satisfies , the closed system in Figure 7.6 with Δ is asymptotically stable. For example, when
one can check that , and aij + δij > 0 for j Ni. Therefore, with the Laplacian matrix L + Δ and the initial states generated randomly, the agents in (7.49) converge to the leader's states asymptotically as shown in Figure 7.11.
7.3 High-Order Consensus in High-Order Systems
7.3.1 System Model
Suppose the interconnection topology of the system is described by a digraph G = (V, E, A) with |V| = n. We assume that the interconnection topology of the system is a connected undirected graph or a digraph containing a globally reachable node. Then, by Theorem 1.9, the Laplacian matrix L has a simple eigenvalue 0, i.e., det (L) = 0 and rank(L) = n − 1. Moreover, the definition of L implies that L · 1n = 0.
Let the model of the ith agent () be given by the following transfer function
where and denote the Laplace transformation of the output and input, respectively, of the ith agent; Ti is the input delay; ν and mi are positive integers satisfying mi ≥ ν; are system parameters. For any non-negative integer k, denotes by the kth-order derivative of the output yi(t) of the ith agent.
Definition 7.18 Multi-agent system (7.58) is said to reach the rth-order consensus asymptotically if
for system solutions from any admissible initial conditions, and
for system solutions from some admissible initial conditions.
Note that when r = 0, the above definition implies that
where is a constant. In this case we say the system achieves a constant consensus. So, constant consensus can be regarded as the zeroth-order consensus by Definition 7.18. We also note by this definition the constant c can not identically be zero for all initial conditions, i.e., it excludes the case of trivial consensus which actually implies that each agent is asymptotically stabilized.
Let τij be the communication delay from agent j to agent i. Then, at time instance t the information obtained by agent i from agent j is instead of . Let the consensus protocol be
where κi > 0, are some control parameters, and τ'ij denotes the estimation of communication delay τij used by agent i. Sometimes τ'ij, j Ni are also referred to as self-delays of agent i. We will show that the high-order consensus can be achieved even though the self-delays are not equal to the communication delays.
7.3.2 Consensus Condition
It is easy to get the closed-loop form of system (7.58) with protocol (7.62) as
(7.63)
Taking the Laplace transform under zero initial condition for the above equation yields
(7.64)
Let
(7.65)
(7.66)
(7.67)
(7.68)–(7.69)
and
(7.71)
Then, the closed-loop system can be shown by Figure 7.12.The frequency-domain model of the closed-loop system is given by
and the return difference equation of the system is given by
According to (2.42), direct use of the equation
(7.74)
may ignore cancelation between the poles of the closed-loop system and the poles of the open-loop system at s = 0. To avoid these cancelations we consider the following equation
as the characteristic equation of the system. Obviously,
Therefore, all the non-zero solutions of (7.75) are contained in the solutions of (7.76), and vice versa.
The following proposition gives an obvious condition for justifying if the open-loop poles at s = 0 enter into the set of the closed-loop poles.
Proposition 7.19 Suppose H(s)KL(s) is analytic at s = 0. If rank[H(s)KL(s)] = n − 1, for all at which H(s)KL(s) is analytic, then
where g(s) has neither poles nor zeros at s = 0.
Proof. The proposition can be easily proved by converting H(s)KL(s) into a diagonal matrix through a similar transformation.
Proposition 7.19 shows that s = 0 is a repeated pole with multiplicity r + 1 of the closed-loop system. This also implies that the solution of the closed-loop system can be expressed as
where are constant vectors, are vectors whose elements are polynomials of t, λk are zeros of g(s), and M is a finite or infinite positive integer. Furthermore, if Reλi < 0, we have as t→ ∞. We call the steady state of the system.
Lemma 7.20 Assume that det (sr+1I + H(s)KL(s)) = sr+1g(s), where g(s) has neither poles nor zeros at s = 0. Then, for any s ≠ 0, g(s) = 0 if and only if
Proof. The lemma is obvious due to the equation (7.76).
By Definition 2.7, a linear time-invariant system is said to be steady semi-stable if all its characteristic roots are inside the LHP or at the origin of the complex plane. By Lemma 7.20, to verify the steady semi-stability of the system, we just need to check if all the zeros of (7.78) have negative real parts. Note that the generalized Nyquist stability criterion (Theorem 2.19 and Theorem 2.20) can be used for this purpose.
Denote for convenience, and denote by the kth-order derivative of .
Theorem 7.21 Assume that the multi-agent system (7.58) with consensus protocol (7.62) is steady semi-stable, and the transfer function is analytic in a neighborhood of s = 0. Then, the rth-order consensus is reached at the steady state if , for all , and for all at which is analytic.
Proof. By Proposition 7.19 we know that the system has zeros at s = 0 with multiplicity r + 1, and the system solution can be expressed by (7.77). With the assumption of the steady semi-stability, from (7.77) it is easy to see that
and
when t→ ∞. So, conditions (7.60) and (7.61) in Definition 7.18 are already satisfied. We just need to check (7.59).
Let s be in the neighborhood D of s = 0 in which is analytic. By the Taylor formula for functions of a complex variable we have
where
(7.82)
(7.83)
Substituting (7.81) into (7.73) yields
Note that the above equation holds in the neighborhood D of s = 0. When the system is steady semi-stable, Y(s) is analytic for Res > 0. Then, by taking the inverse Laplace transformation of the equation, we have
(7.84)
for t→ ∞. Applying (7.79) to the above equation we get
when t→ ∞.
Differentiating (7.80)r times yields
Substituting (7.80) and (7.86) into (7.85) yields
where
(7.88)
Equation (7.87) holds if and only if .
From Er = 0 we get Cr span(1n) because and . Using the result Cr span(1n) and the assumption , from Er−1 = 0 it follows that which implies Cr−1 span(1n). Conducting this procedure to the end, i.e., E0 = 0, we get . Thus, we have y(t) span(1n) when t→ ∞, which, by (7.80), also implies that .
7.3.3 Existence of High-Order Consensus Solutions
Theorem 7.21 gives some sufficient consensus conditions for the system. Here we try to find a necessary and sufficient condition of the existence of high-order consensus solutions. First, we show that the condition given by Proposition 7.19 can be further weakened as follows.
Proposition 7.22 Suppose is analytic in a neighborhood of s = 0. Then,
with g(s) having neither poles nor zeros at s = 0, if and only if , and there exists a non-zero constant vector α span(1n) such that , , and .
Proof. Let s be in the neighborhood of s = 0, denoted by D, in which is analytic. By the Taylor formula for functions of a complex variable, we have
(7.89)
where
(7.90)
(7.91)
It is clear that
if and only if there exists a normalized non-singular matrix (i.e., det (T) = 1) such that one column (say without loss of generality, the first one) of T−1L1(s)T is zero, i.e., [T−1L1(s)T](:, 1) = 0. Denote by α the first column of T. It is easy to get
Obviously, [T−1L1(s)T](:, 1) = 0, ∀ s D if and only if , .
Denoting by pij the (i, j)th elements of T−1L1(s)T for , we have
And denoting by gij the (i, j)th elements of T−1L2(s)T, we have
Since H(s)KL(s) is analytic at s = 0, all the elements pij () are analytic at s = 0. Hence, g(s) is also analytic at s = 0.
Now, we prove g(s) has no zeros at s = 0, i.e., g(0) ≠ 0 if and only if and . Denote . From (7.93) we have
because det (T−1) = 1. Obviously, g(0) ≠ 0 if and only if
and
From (7.92) we know that
and hence if and only if . Simple calculation shows that . So g(0) ≠ 0 if and only if , and . Finally, we note that the non-zero vector α span(1n). This is indeed the case. Since we have proved , the null space of is one-dimensional. From the property of Laplacian matrix, L(0) · 1n = 0, we get . So, α span(1n). The proposition is proved.
Remark. When , a sufficient condition for is , where c is a non-zero constant. And a more sufficient condition for is .
Exercises 7.23 Suppose and . Show α : = c1n with c ≠ 0 is not in .
When s = 0 is a repeated pole with multiplicity r + 1 of the closed-loop system, the solution of the closed-loop system can be expressed by equation (7.77)
Now, we are ready to present the following theorem.
Theorem 7.24 Assume that the multi-agent system (7.58) with consensus protocol (7.62) is steady semi-stable and the transfer function is analytic in a neighborhood of s = 0. Then, the rth-order consensus is reached at the steady state if and only if , , , and .
Proof. (Sufficiency) Suppose , , , and . Then, by Proposition 7.22 we know that the system has zeros at s = 0 with multiplicity r + 1, and the system solution can be expressed by (7.77). The rest of the sufficiency part of the this proof is as the same as given in the proof of Theorem 7.21.
(Necessity) Suppose that the system reaches the consensus defined by (7.59), (7.60) and (7.61). Note that (7.60), (7.61) and the steady semi-stability assumption imply that the system has zeros at s = 0 of multiplicity r + 1. So, by Proposition 7.22, the necessity is obvious.
7.3.4 Constant Consensus
For the case of constant consensus, i.e., the case when r = 0, without loss of generality, we let bi0 = 1 for all . Then, the protocol (7.62) reduces to the following form
the closed-loop system equation becomes
and H(s) defined in (7.70) takes the form
where
(7.95)
Let us apply Theorem 7.24 to the case of constant consensus. For this case, Theo-rem 7.24 requires: (1) the marginal stability of the closed-loop system, (2) and . Since κi ≠ 0, requirement (2) implies that rank[L(0)] = n − 1 and L(0) · 1 = 0. It is well known that this is equivalent to the connectivity condition for the interconnection graph, i.e., the digraph has a globally reachable node.
To check the marginal stability of the closed-loop system, we may make some loop transformation on the system as shown by Figure 2.8. Denote by the open-loop transfer function matrix after the loop transformation. Then, based on the extended spectral radius theorem for steady semi-stability (Theorem 2.24), we get the following result immediately.
Theorem 7.26 Consider the multi-agent system (7.58) with the protocol (7.94). Assume the interconnection digraph has a globally reachable node, then, the system achieves a constant consensus, if
(7.96)
Since L(s) defined in (7.72) can be rewritten as
where
and
the system diagram shown in Figure 7.12 can be equivalently transformed as shown in Figure 7.13. In this case we have
(7.98)
where
(7.99)
Straightforward calculation shows that
So, the requirement (7.97) of Theorem 7.25 is satisfied. Therefore, under the assumption that the interconnection digraph has a globally reachable node, a sufficient condition for achieving a constant consensus is
(7.100)
By using Corollary 2.31 of Gershgorin's disc lemma, a more conservative but scalable condition is
(7.101)
This condition was first given by Lee and Spong (2006).
Similarly, from Theorem 2.27 and Theorem 2.29 we can also get the following two theorems for the constant consensus problem.
Theorem 7.26 Consider the multi-agent system (7.58) with the protocol (7.94). Assume the interconnection digraph has a globally reachable node, then, the system achieves a constant consensus, if
(7.102)
(7.103)
Theorem 7.27 Consider the multi-agent system (7.58) with the protocol (7.94). Assume the interconnection digraph has a globally reachable node, then, the system achieves a constant consensus, if
(7.104)
(7.105)
7.3.5 Consensus in Ideal Networks
Before applying Theorem 7.24 to ideal networks, i.e., networks with zero communication delays and constant channel dynamics, let us review the following fact from graph theory.
Let G be a digraph having at least one globally reachable node. If we choose only one globally reachable node, say , of G, and cut off all the edges from , then obviously is still a globally reachable node of G. But, if we do this operation for a node which is not globally reachable in G, or simultaneously do this operation for two or more globally reachable nodes, then G has no globally reachable node anymore. This is equivalent to saying, if is a Laplacian of a digraph having at least one globally reachable node, then rank[diag{b1, . . . , bn}L] = n − 1 if and only if , or and is a globally reachable node of G, where denotes the set of integers from 1 to n excluding j.
Now, let us consider a multi-agent system based on an ideal network. Suppose the topology digraph G contains at least one globally reachable node. Then, we have rank[L] = n − 1 and L · 1n = 0. In this case we have τij = τ'ij = 0 and L(s) = L. From L · 1n = 0 it is easy to get , and . Finally, by denoting , we know that if and only if gi(0) ≠ 0, or and is a globally reachable node of digraph G. Thus, we get the following result as a corollary of Theorem 7.24.
Theorem 7.28 In an ideal network with zero communication delays and constant channel dynamics, the rth-order consensus will be achieved for a steady semi-stable system at its steady state if and only if the topology graph contains at least one globally reachable node, and the agents's dynamics and the protocol satisfy gi(0) ≠ 0, or and is a globally reachable node of digraph G.
7.4 Integrator-Chain Systems with Diverse Communication Delays
7.4.1 Matching Condition for Self-Delay
Consider the multi-agent system, whose agents are chains of integrators, i.e., mi = ν, and hi(s) = 1. In this case gi(s) = bi(s), and thus we have
Since κi > 0 and bi0 ≠ 0, we have when the topology graph contains a globally reachable node. Now, we check the condition , . For the second-order consensus (ν = 2), this condition implies that
And generally, the consensus condition for the νth-order consensus can be obtained as
Equation (7.106) or (7.107) uncovers a very interesting fact that the second-order or high-order consensus does not necessarily require τ'ij = τij. We call condition (7.107) the matching condition for self-delays.
7.4.2 Adaptive Adjustment of Self-Delay
Since (7.106) does not require each self-delay to be equal to the communication delay, for the second-order consensus each agent can set an identical self-delay, i.e., τ'ij = τ'i, ∀ j Ni, and adjust τ'i so that (7.106) is satisfied. In the following, we propose a simple algorithm for the adaptive adjustment of the self-delay.
The performance index can be proposed as
where t0 is an initial time instant from which the adaptive adjustment begins, T > 0 is a large enough constant of time interval.
The algorithm of adaptive adjustment of τ'i is as follows:
(7.112)
(7.113)
The proposed algorithm is essentially a gradient algorithm which is usually just locally convergent. To get the equilibria of the algorithm, one can set . Then, it follows
Since yi(t) → C0 + C1t, from (7.114) it follows that
or
Obviously, (7.115) is exactly (7.106) corresponding to the desired equilibrium; (7.116) implies that the velocities of all the agents go to zero, which is the zeroth-order consensus (constant consensus) state for the second-order system.
7.4.3 Simulation Study
In this subsection several simulation experiments are conducted through two numerical examples to verify the obtained theoretical results.
Example 7.29 Consensus of a system of heterogeneous agents.
Consider a system of five agents described by
Obviously, agent 1 is absolutely different from the other four agents which are all double-integrators. The agents are interconnected by a digraph shown in Figure 7.14. The weighted adjacency matrix is given as follows
Example 7.30 Consensus with unknown communication delays.
Now, we study the consensus protocols for unknown communication delays. To show the correctness of the theoretical results, we consider the case of identical double-integrator agents, which are extensively studied in the literature. Let
The interconnection topology graph is assumed to be the same as Figure 7.14. The adjacency matrix is given by
The diverse communication delays are set as
It should be noted that the convergence of both the adaptive algorithm and the stability of the overall system heavily depend on the values of the adaptive gain β and the initial value of the delay estimation τ'1(0). Generally speaking, the larger β is, the smaller the upper bound of admissible initial value of τ'1 is; when the initial value of the delay estimation is sufficiently close to the value satisfying the matching condition, the adaptive gain can be arbitrarily large (see Table 7.1).
7.5 Notes and References
The study on consensus problems with communication delays can be traced in the research of distributed computation algorithms back to as early as the 1980s (Tsitsiklis, Bertsekas and Athans 1986). Similar problems are also studied in the context of synchronization of coupled oscillators (Yeung and Strogatz 1999). Recently, Olfati-Saber and Murray (2004) reviewed the consensus problem of the first-order multi-agent system and proposed a consensus protocol in which each agent delays its own measurement of state by the same value as the communication delay so that it could be matched by the delayed states of its neighbors. Olfati-Saber and Murray (2004) obtained a delay-dependent sufficient condition of constant consensus for the system with a uniform communication delay. Lin et al. (2007) extended the consensus protocol of Olfati-Saber and Murray (2004) to the second-order multi-agent system, and also obtained a delay-dependent sufficient condition of the second-order consensus. However, since each self-delay is equal to the corresponding communication delay in their protocol and only the unified delay bound is considered, the multi-agent systems studied in Olfati-Saber and Murray (2004) and Lin et al. (2007) are essentially homogeneous.
Lee and Spong (2006) considered a consensus protocol without self-delay and obtained a delay-independent sufficient condition of constant consensus for high-order heterogeneous systems with diverse communication delays by using Gershgorin's disc lemma and the small-gain theorem shown in Chapter 2. By using a similar technique, Wang and Elia (2008) also considered the consensus protocol without self-delay and gave a delay-independent sufficient condition of constant consensus for the first-order multi-agent systems with heterogeneous dynamic communication channels. Arcak (2007), Chopra and Spong (2006) studied heterogeneous MASs based on the passivity theory, and developed a general framework for the design of group coordination control of systems with nonlinear dynamical agents, which is applicable to the constant-consensus problem. Lestas and Vinnicombe (2006, 2007, 2010) also considered the constant-consensus problem for heterogeneous systems, and introduced the notion of S-hull, a relaxation of the convex hull of a set in the complex plane, to overcome the conservativeness of small-gain-like or passivity-like stability results. Using the frequency-domain analysis theory developed in Chapter 3, Section 7.1 gives scalable delay-dependent conditions of constant consensus for the first-order multi-agent system with diverse input delays and communication delays while Section 7.2 gives scalable delay-dependent conditions of the second-order consensus for the second-order multi-agent system with diverse input delays and asymmetric perturbations in communication channels. Sections 7.1 and 7.2 are mainly based on the results of Tian and Liu (2008, 2009).
Actually, the existence of constant consensus depends only on the connectivity of the interconnection topology of MASs (Ren and Beard 2005; Wang, Cheng and Hu 2008; Xiao and Wang 2007). The values of self-delays introduced by agents in consensus protocols may lead to instability of the consensus solution (Papachristodoulou, Jadbabaie and Mijünz 2010) but they do not influence the existence of the constant-consensus solution. Therefore, the main focus of the above-mentioned references on high-order heterogeneous MASs is on the stability instead of the existence of the set of consensus solutions.
It can be shown that an inappropriate value of self-delay may lead to the non-existence of high-order consensus solution. To guarantee the existence of high-order consensus solutions, currently existing consensus protocols introduce self-delays which are exactly equal to the corresponding communication delays (see, e.g., Hu and Hong 2007). In practice, however, communication delays can only be estimated approximately. Section 7.3 proposes a high-order consensus protocol based on the estimation of communication delays and investigates the existence of high-order consensus solution of high-order heterogeneous MASs under such a protocol. Section 7.4 presents a simple algorithm for on-line adjusting self-delays to guarantee the existence of high-order consensus solutions. These two sections are mainly taken from Tian and Zhang (2012) except for Section 7.3.4.
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