REDCHO: Robust Exact Dynamic Consensus of High Order

This article addresses the problem of average consensus in a multi-agent system when the desired consensus quantity is a time varying signal. Recently, the EDCHO protocol leveraged high order sliding modes to achieve exact consensus under a constrained set of initial conditions, limiting its applicability to static networks. In this work, we propose REDCHO, an extension of the previous protocol which is robust to mismatch in the initial conditions, making it suitable to use cases in which connection and disconnection of agents is possible. The convergence properties of the protocol are formally explored. Finally, the effectiveness and advantages of our proposal are shown with concrete simulation examples showing the benefits of REDCHO against other methods in the literature.


Problem statement
Consider a multi-agent system distributed in a network G. In the following G is an undirected connected graph of n nodes characterized by its incidence matrix D or its adjacency matrix A [12,Chapter 8]. We consider that each agent i has access to a local time varying signal u i (t) ∈ R. Additionally, each agent is capable of communicating with their neighbors according to a communication topology defined by G. Moreover, each agent i runs a local observer with output y i (t) = [y i,0 (t), . . . , y i,m (t)] T ∈ R m+1 where m is a desired system order. In order to reduce communication burden, we require all agents to share only y i,0 instead of the whole y i (t). The goal of the system is to achieve the following property.
Remark 2 Assume that all agents are provisioned with an observer complying the properties described before. Moreover, assume all agents have a local physical system with dynamics of relative degree m. Then, the local observer's output y i (t) can be used locally at each agent to solve a DAT problem. In this case, a local feedback control can be designed using the ideas from [20,11,24].
It is easy to show that the EDCHO algorithm from [1] is a particular limiting case of REDCHO and can be recovered by choosing γ 0 = · · · = γ m = 0 and θ = 1. However, EDCHO assumes that n i=1 x i,µ (t 0 ) = 0 is satisfied. This condition breaks easily, specially when agents connect or disconnect from the network. The main result of this work, which is formally stated and shown in Section 7, is that using the gains k 0 , . . . , k m designed as in [1,Theorem 7] and under Assumption 3, then REDCHO algorithm works even when n i=1 x i,µ (t 0 ) = 0 and thus achieves robust EDC. In order to formally show these facts, we provide some auxiliary results.

Towards convergence of REDCHO
The REDCHO algorithm (2) can be written in partially vectorized form as: where we define X µ (t) := [x 1,µ (t), . . . , x n,µ (t)] T , Y µ (t) := [y 1,µ (t), . . . , y n,µ (t)] T , U (t) := [u 1 (t), . . . , u n (t)] T and D is the incidence matrix of G. Moreover, let X(t) : Using this notation we obtain the fully vectorized form of the algorithm: Both partial and fully vectorized versions of the algorithm will be used throughout this work. Moreover, note that G is the observability matrix of the pair (Γ, C) which is invertible. Then, the dynamics of Y(t) result iṅ We will show that Y(t) converges towards the average consensus vector ū(t)1 T n , . . . ,ū (m) (t)1 T n T asymptotically achieving EDC. This analysis is performed by decomposing Y(t) = (I m+1 ⊗1 n )ȳ(t)+Ỹ(t) ∈ R (m+1)n in the consensus componentȳ(t) = (I m+1 ⊗ 1 T n /n)Y(t) ∈ R m+1 and in the consensus errorỸ(t) = (I m+1 ⊗ P )Y(t) ∈ R (m+1)n with P = (I n − (1/n)1 n 1 T n ). Therefore, convergence of Y(t) can be established by means of showing thatȳ(t) converges exponentially to [ū(t), . . . ,ū (m) (t)] T andỸ(t) converges in finite time to the origin as we do in the following sections. First, we provide some results regarding structural properties of the matrices Γ, G and their relation to the signals u i (t) and w i (t) from Assumption 3. These notions will be useful in subsequent proofs.
since CB = 0 and continuing this procedure to obtain (8) concluding the proof. PROOF. Let the change of variables from Lemma 4 as U(t) = (G ⊗ I n )W(t). Then, Comparing with (7) completes the proof.

Convergence of the consensus components of REDCHO
In this section we show the behaviour ofȳ(t) = (I m+1 ⊗ 1 T n /n)Y(t) as given in the following result.

Convergence of the consensus error
In this section we show the behaviour ofỸ(t) = (I m+1 ⊗ P )Y(t) where P = (I n − (1/n)1 n 1 T n ). First, we obtain how the dynamics ofỸ(t) relate to EDCHO.
Now, if we show that Z(t) converge to the origin in finite time, the same conclusion will apply toỸ(t). Note that for given γ 0 , . . . , γ m , P W m+1 (t)/θ m+1 ∈ [−L, L] n /θ m+1 ⊆ [−L, L] n under Assumption 3 and θ ≥ 1. Therefore, comparing (11) with the EDCHO error system (A.1) in Appendix A, it would be the case that Z(t) reaches the origin if γ 0 = · · · = γ m = 0. In the following we will use homogeneity to show that even with those terms, stability of REDCHO will still be valid locally. To do so, we will decompose the right hand side of (11) in two parts, one similar to the right hand side of the EDCHO error system in (A.1) and the other with the remaining linear terms.
is complied for some c > 0 so thatV (Z) ≤ −θcV (Z) k−1 k for any Z ∈ R 0 . If β M ≤ 0, then (13) is complied for c = β m regardless of Z so that we can set R 0 = R n(m+1) . On the other hand, if β M > 0 then (13) for any Z ∈ R 0 regardless of β M . Note that due to Proposition 18, V (Z) is continuously differentiable and we can write 1 is a constant for fixed γ 0 , . . . , γ m . Thus, β m /(2β M ) = θ(β m /(2β M )) can be made arbitrarily big by increasing θ, so that R 0 can be made arbitrarily big as well. Finally, note that since (k−1)/k ∈ (0, 1), then V (Z) will reach the origin in finite time [4,Corolary 4.25] and so will Z(t) whenever Z(t 0 ) ∈ R 0 , completing the proof.
The previous result shows that trajectories ofŻ(t) ∈ θ(H(Z(t)) + Q(Z(t))) reach the origin if Z(t 0 ) ∈ R 0 which motivates to study if diverging trajectories can be obtained for some Z(t 0 ) / ∈ R 0 . In the following, we show that this is not possible and only a terminal bounded error is allowed.
Lemma 9 Let the conditions of Lemma 8 be satisfied. Thus, for any initial conditions Z(t 0 ) ∈ R n(m+1) , there exists T > 0 and a bounded neighborhood of the origin R ∞ such that solution ofŻ(t) ∈ θ(H(Z(t)) + Q(Z(t))) comply Z(t) ∈ R ∞ for t ≥ T + t 0 . Moreover, such neighborhood can be made arbitrarily big by increasing θ.

Convergence of REDCHO
In this section we formally state the main result of this work.
. Now, note that Lemma 8 implies the existence of a neighborhood R 0 such that if Z(t 0 ) ∈ R 0 with Z(t) := (ΘG −1 ⊗I n )Ỹ(t) then convergence of Z(t) is achieved towards the origin. Hence, consider the biggest ball of radius R(θ), B 0 (θ) = {Z ∈ R n(m+1) : Z T Z ≤ R(θ) 2 } such that B 0 (θ) ⊆ R 0 and R(θ) is an increasing function of θ due to the last part of Lemma 8. Now, This implies that convergence ofỸ(t) towards the origin happens for anyỸ where the previous region can be made arbitrarily big by increasing θ. This implies the existence of R ∈ R n(m+1) as required by the theorem. An identical argument can be made for R ′ ∈ R n(m+1) but using Lemma 9 instead to conclude uniformly bounded trajectories forỸ(t) implying uniformly bounded steady state error around dynamic consensus. Furthermore, Lemma 6 imply thatȳ(t) converge asymptotically towardsū(t) = (I m+1 ⊗1 T n /n)U(t) = ū(t)1 T n , . . . ,ū (m) (t)1 T n T . Hence, when Y(t 0 ) ∈ R, then Y(t) converge asymptotically towards (I m+1 ⊗1 n )ū(t). Equivalently, Y µ (t) →ū (µ) (t)1 n . Since no initialization condition is required, then (2) achieves robust EDC.

Remark 11
Note that since the γ 0 , . . . , γ m are fixed, the class of signals for which Assumption 3 is complied can be checked before-hand, so that the method remains fully distributed. On the other hand, showing the same stability properties in the case when all agents have different parameters γ i 0 , . . . , γ i m > 0, i ∈ {1, . . . , n} require more complicated computations, but is straightforward using similar arguments as in this work. Thus, only the case with a single set of parameters for all agents is provided here for simplicity.

Simulation examples
In the following we show some simulation scenarios designed to show the properties of the REDCHO protocol. The simulations were implemented using explicit Euler method with time step h = 10 −6 over (2).
Example 2 In order to show the robustness properties of the REDCHO protocol when the topology suffers from sudden changes, we simulated (2) for the network topology shown in Figure 4 which changes from G t<5 to G t≥5 at t = 5. Consider the same configuration and signals as in the previous example. Figure 5 shows how the outputs of the REDCHO algorithm converge to EDC approximately at t = 0.5. At t = 5 the topology changes, but the REDCHO protocol manages to make all agents, the first four and the new ones, converge to EDC again even when the states didn't comply neither 4 i=1 x i,µ (0) = 0 nor 8 i=1 x i,µ (5) = 0 as required in [1]. For comparison, consider the protocol in [1] obtained by setting γ 0 = γ 1 = γ 2 = 0 in REDCHO. Moreover, initial conditions are changed so that 4 i=1 x i,µ (0) = 0. Figure 6 shows the trajectories for the protocol in this case, where EDC is achieved before t = 5. However, when the new agents merge to the network, the agents output converge to consensus towards a signal that diverges.
In the previous examples we showed the effectiveness of REDCHO against its non-robust version in [1]. In the following we compare against other state of the art dynamic consensus methods, with particular focus on terminal precision for the EDC goal.

Example 3
In this example, we compare REDCHO with the Boundary-layer (B-layer) approach from [23] and the High-Order Linear protocol (HOL) from [20]. Both previous methods are able to achieve consensus towards the average signal and its derivatives, but are not robust and require to share a whole vector between agents. In addition, we compare with the First-Order Linear (FOL) protocol in [13] and the First-Order Sliding Mode (FOSM) protocol in [10]. Both protocols are robust, but cannot obtain the derivatives of the average signal by construction. Thus, a robust exact differentiator [15] is applied locally at each agent to obtain derivatives of the average signal.
In this setting, consider G constructed as a ring topology of n = 20 agents. Similarly as before, consider signals u i (t) = a i cos(ω i t) where the a i , ω i are not shown for brevity. Note that all approaches can handle these type of signals, with at least a bounded terminal consensus error regardless of the order of the algorithm. An order of  m = 2 was used for REDCHO, B-layer and HOL, and an exact differentiator of order m is applied for FOL and FOSM. Hence, all algorithms are able to obtain up to the second derivative of the average signal. All algorithms were implemented with parameters of similar magnitude, chosen such to roughly match same settling time for the sake of fairness. Moreover, we show the resulting consensus errors for all algorithms with h = 10 −6 as shown in Figure 7 and h = 10 −3 in Figure 8 to show how they degrade as the discretization becomes coarser. In addition, we simulated that agent 1 fails at t = 25 and resets its state, allowing us to evaluate robustness of the algorithms.
As it can be observed, HOL and FOL methods have similar low precision in all cases before t = 25. The reason is that neither of these methods are able to achieve exact convergence for sinusoidal signals. However, their performance does not degrade significantly when the time step is increased. On the other hand, it can be noted that the FOSM approach have better performance than the linear approaches when h = 10 −6 due to its theoretically exact convergence. However, it degrades significantly when h is increased as shown in Figure 8. The reason is that this method suffers from the chattering effect which is amplified for the higher order derivatives due to the exact differentiator. Note that the B-layer and REDCHO approaches have similar performance before t = 25 with both sampling step sizes and outperform the other methods with at least one order of magnitude of precision improvement when h = 10 −3 as shown in Figure 8. However, after t = 25 both B-layer and HOL converge to consensus only up to a constant error due to their lack of robustness as shown by the Y 0 (t) −ū(t)1 curves in both Figures 7 and 8. Although other methods manage to recover from the failure of agent 1, REDCHO is the one with the best performance in all cases.

Conclusions
In this work we proposed the REDCHO protocol. This new protocol achieves exact consensus towards the average of time varying signals and its derivatives distributed through a network. Proofs of convergence of the algorithm are given even when agents connect or disconnect from the network. Simulation scenarios were designed to confirm the advantages of the proposed protocol. Still, the proposed methodology works only when the changes in the network are isolated events. An analysis for general uncertain, time-varying networks with persistent fast changes will be explored in future work.

Remark 13
Note that the gains k 0 , . . . , k m from the previous result can be obtained using the parameters for the robust exact differentiator from [15] through a procedure described in [1, Section 6].

B Homogeneous differential inclusions
In this section, we consider dynamical systems characterized by set-valued maps F : R n ⇒ R n instead of typical vector fields. Moreover, we assume some regularity conditions on such maps called the basic conditions. We say that a set valued map F : G ⇒ R n satisfies the basic conditions if for all x ∈ G the set F (x) is non-empty, bounded, closed, convex and the map F is upper semi-continuous in x [2, Chapter 2.7]. The Filippov regularization for vector fields [8], commonly used to study discontinuous dynamical systems, satisfy the basic conditions by construction. In the following, let ∆ r (λ) = diag([λ r1 , . . . , λ rn ]) where r = [r 1 , . . . , r n ] are called the weights and λ > 0. For any x ∈ R n , the vector ∆ r (λ)x = [λ r1 x 1 , . . . , λ rn x n ] T is called its standard dilation (weighted by r). The following are some definitions and results of interest regarding the so called r-homogenety with respect to the standard dilation.
Definition 14 (Homogeneous scalar functions) [4, Definition 4.7] A scalar function V : R n → R is said to be r-homogeneous of degree d if V (∆ r (λ)x) = λ d V (x) for any x ∈ R n .
Definition 15 (Homogeneous set-valued fields) [4, Definition 4.20] A set-valued vector field F : R n ⇒ R n is said to be r-homogeneous of degree d if F (∆ r (λ)x) = λ d ∆ r (λ)F (x) for any x ∈ R n .