A class of robust consensus algorithms with predefined-time convergence under switching topologies

12/18/2018 ∙ by R. Aldana-López, et al. ∙ iteso IEEE Université de Valenciennes et du Hainaut-Cambrésis (UVHC) 0

This paper addresses the robust consensus problem under switching topologies. Contrary to existing methods, the proposed approach provides decentralized protocols that achieve consensus for networked multi-agent systems in a predefined time. Namely, the protocol design provides a tuning parameter that allows setting the convergence time of the agents to a consensus state. An appropriate Lyapunov analysis exposes the capability of the current proposal to achieve predefined-time consensus over switching topologies despite the presence of bounded perturbations. Finally, the paper presents a comparison showing that the suggested approach subsumes existing fixed-time consensus algorithms and provides extra degrees of freedom to obtain predefined-time consensus protocols that are less over-engineered, i.e., the difference between the estimated convergence time and its actual value is lower in our approach. Numerical results are given to illustrate the effectiveness and advantages of the proposed approach.

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1 Introduction

The problem of cooperative control for multi-agent systems has received significant attention due to its wide range of applications in different fields such as biology, power grid, robotics, etc. In cooperative control, a fundamental challenge is the consensus problem, whose objective is to design decentralized protocols such that all the agents, in the network, interact with their neighbors to reach a common value Olfati-Saber et al. (2007), called the consensus equilibrium. Many decentralized consensus-based methods have been applied, for instance, to flocking Olfati-Saber (2006), formation control Oh et al. (2015); Ren (2007); Li and Wang (2013) and distributed resource allocation Xu et al. (2017a, b). Two consensus problems have been mainly addressed, the leader-follower consensus problem Defoort et al. (2015); Shi et al. (2018), where the agents along the network converge to the state of the leader (real or virtual) which communicates its state only to a subset of agents; and the leaderless consensus problem Zuo and Tie (2014); Ning et al. (2017); Zuo and Tie (2016), where the consensus equilibrium is a function of the initial conditions of the agents, for instance achieving consensus to the average value Olfati-Saber (2006), the average min-max value Cortés (2006); Li and Qu (2014), the median value Franceschelli et al. (2017), and the maximum or minimum value Liu et al. (2015) of the agents’ initial conditions.

Many existing consensus algorithms focus on asymptotic convergence (i.e. the settling time is infinite) Ren and Beard (2005) and finite-time convergence (i.e. the settling time estimate is finite but depends on the initial conditions) Wang and Xiao (2010); Zhao et al. (2018a). It is clear that a finite settling time is useful for several applications (a network of clusters, manufacturing systems, etc.). However, the initial conditions are usually unknown due to the decentralized architecture. Therefore, recently, there has been a great deal of attention in the research community on algorithms which solve the decentralized consensus problem in a finite-time with uniform convergence with respect to the initial value of the agents (see. e.g Gómez-Gutiérrez et al. (2018) and the references therein).

Two main approaches have been proposed to address this problem. The first one is based on the fixed-time stability theory of autonomous systems proposed in Polyakov (2012). Based on this class of systems, fixed-time consensus algorithms have been proposed either based on the homogeneity theory developed in Andrieu et al. (2008); Polyakov et al. (2016), as in Gómez-Gutiérrez et al. (2018), the settling-time estimation provided in Polyakov (2012), as in Zuo et al. (2014); Parsegov et al. (2013) or the subclass of fixed-time stable systems, with an upper bound of the settling time that is less conservative than the one in Polyakov (2012), presented in Parsegov et al. (2012), as in Parsegov et al. (2013); Zuo and Tie (2014); Defoort et al. (2015); Ning et al. (2017); Wang et al. (2017a). However, in spite of the advantages in terms of settling time estimation, this approach does not enable to easily arbitrarily preassigned the settling time. For this case, the consensus problem for agents with single integrator dynamics and affected by disturbances has been addressed in Zuo and Tie (2016); Ning et al. (2017).

The second approach is based on time-varying consensus protocols and provides consensus protocols with predefined-time convergence, where the convergence time is set a priori as a parameter of the protocol Yong et al. (2012); Liu et al. (2018); Wang et al. (2017b, 2018); Colunga et al. (2018); Zhao et al. (2018b). In these works, non-conservative upper-bound estimates of the settling time are provided. However, this approach presents some drawbacks. First, they are based on a common time-varying gain that is applied to each node on the network, which requires a common time-reference along the network. Second, this time-varying gain becomes singular at the predefined-time, such as methods based on the so-called time base generators Morasso et al. (1997) see for instance Yong et al. (2012). Third, methods, such as Liu et al. (2018); Zhao et al. (2018b), are based on a time-varying gain that is a piecewise-constant function that produces Zeno phenomenon Zhang et al. (2001), even in the unperturbed case. Additionally, the robustness, against external disturbances, of the methods proposed in Yong et al. (2012); Liu et al. (2018); Wang et al. (2017b, 2018); Zhao et al. (2018b) has not been analyzed.

In this paper, we address the robust predefined-time leaderless consensus problem for the cases where a common time reference is not available. Thus, we focus on developing autonomous protocols. Our approach is based on recent results on the upper bound estimation for the settling time of a class of fixed-time stable systems presented in Aldana-López et al. (2018) to propose a broader class of consensus protocols for perturbed first-order agents with a fixed-time convergence whose convergence upper bound is set a priori as a parameter of the system. In our opinion, Ning et al. (2017) is the closest approach in the literature, as it presents robust fixed-time consensus protocols that can be designed to satisfy time constraints. We will show that the results in Ning et al. (2017) and other fixed-time consensus protocols, such as Zuo and Tie (2014); Zuo et al. (2014); Wang et al. (2017a), are subsumed by our approach since our results allow more flexibility in the parameter selection to obtain, for instance, predefined-time consensus protocols where the slack between the real-convergence time and the estimated upper bound of the convergence time is lower. Contrary to previous fixed-time consensus protocols Parsegov et al. (2013); Zuo and Tie (2014); Zuo et al. (2014); Wang et al. (2017a); Zuo and Tie (2016); Ning et al. (2017) our approach allows to straightforwardly specify the upper bound of the convergence time as a parameter of the consensus protocol. We presents two robust consensus protocols. The first one is the one that is computationally simpler; we show that it presents predefined-time convergence under static networks and fixed-time convergence under switched dynamic networks. The second one, in the absence of disturbances, converges to a consensus state that is the average of the agents’ initial conditions. This algorithm is shown to be a robust predefined-time consensus algorithm for static and dynamic networks arbitrarily switching among connected graphs.

The rest of the manuscript is organized as follows. Section 2 introduces the preliminaries on graph theory, finite-time stability and predefined-time stability. Section 3 presents two approaches to address the robust consensus problem for dynamic networks. Section 4 presents a comparison with Ning et al. (2017) showing that having a broader class of predefined-time consensus algorithms provides greater flexibility to improve the performance of the consensus algorithms. This section also presents a qualitative comparison of both protocols with respect to other fixed-time consensus protocols previously proposed in the literature highlighting the contribution of our approach. Finally, Section 5 presents some concluding remarks.

2 Preliminaries

2.1 Graph Theory

The following preliminaries on graph theory are from Godsil and Royle (2001). In this paper, we will focus only on undirected graphs.

Definition 1.

A graph consists of a set of vertices and a set of edges where an edge is an unordered pair of distinct vertices of .

Writing denotes an edge and denotes that the vertex and vertex are adjacent or neighbors, i.e., there exists an edge . The set of neighbors of in the graph is expressed by .

Definition 2.

A path from to in a graph is a sequence of distinct vertices starting with and ending with such that consecutive vertices are adjacent. If there is a path between any two vertices of the graph then is said to be connected. Otherwise, it is said to be disconnected.

Definition 3.

A weighted graph is a graph together with a weight function .

Definition 4.

Let be a weighted graph such that has weight and let . Then, the adjacency matrix is an matrix where .

Definition 5.

Let be a graph, the Laplacian of is denoted by (or simply when the graph is clear from the context) and is defined as where with .

Remark 1.

For the Laplacian , there exists a factorization ( is known as the incidence matrix of  Godsil and Royle (2001)) where is an matrix, such that if is an edge with weight then the column of corresponding to the edge has only two nonzero elements: the th element is equal to and the th element is equal to . Clearly, the incidence matrix , satisfies . The Laplacian matrix

is a positive semidefinite and symmetric matrix. Thus, its eigenvalues are all real and non-negative.

When the graph is clear from the context we omit as an argument. For instance we write , , etc to represent the Laplacian, the incidence matrix, etc.

Lemma 1.

Godsil and Royle (2001) Let be a connected graph and its Laplacian. The eigenvalue

has algebraic multiplicity one with eigenvector

. The smallest nonzero eigenvalue of , denoted by satisfies .

It follows from Lemma 1 that for every , . is known as the algebraic connectivity of the graph . The distance between two distinct nodes and is the shortest path length between them. The diameter of is the longest distance between two distinct vertices.

For a path graph and cycle graph of nodes, and , can be computed as and , respectively. For a star graph , . Graphs that are more connected have a larger . A path graph is the “most nearly disconnected” connected graph of nodes.

Definition 6.

A switched dynamic network

is described by the ordered pair

where is a collection of graphs having the same vertex set and is a switching signal determining the topology of the dynamic network at each instant of time.

In this paper, we assume that is generated exogenously and that there is a minimum dwell time between consecutive switchings in such a way that Zeno behavior in network’s dynamic is excluded, i.e., there is a finite number of switchings in any finite interval.

2.2 On finite-time, fixed-time and predefined-time stability

Consider the system

(1)

where

is the system state, the vector

stands for the parameters of system (1) which are assumed to be constant, i.e., . Furthermore, there is no limit for the number of parameters, so can take any value in the natural number set . The function is nonlinear, and the origin is assumed to be an equilibrium point of system (1), so . The initial condition of this system is .

Definition 7.

Bhat and Bernstein (2000) The origin of (1) is globally finite-time stable if it is globally asymptotically stable and any solution of (1

) reaches the equilibrium point at some finite time moment, i.e.,

, where is called the settling-time function.

Definition 8.

Polyakov (2012) The origin of (1) is fixed-time stable if it is globally finite-time stable and the settling-time function is bounded, i.e. .

To address the fixed-time stability analysis, one approach is based on the homogeneity theory.

Definition 9.

Bhat and Bernstein (2005) A function is homogeneous of degree with respect to the standard dilation if and only if

A vector field , with , is homogeneous of degree with respect to the the standard dilation if

Definition 10.

Andrieu et al. (2008); Polyakov et al. (2016) A function , such that , is homogeneous in the limit with degree with respect to the standard dilation if , defined as

is homogeneous of degree with respect to the standard dilation.
A vector field is said to be homogeneous in the limit with degree with respect to the standard dilation if the vector field , defined as

(2)

is homogeneous of degree with respect to the standard dilation.

A characterization of fixed-time stability, based on the homogeneity theory, is given in the following theorem.

Theorem 1.

Andrieu et al. (2008); Polyakov et al. (2016) Let the vector field be homogeneous in the limit with degree and homogeneous in the -limit with degree (if both conditions are satisfied it is said that the vector field f(x) is homogeneous in the bi-limit). If for the dynamic systems , and the origin is globally asymptotically stable (where and are obtained from (2) with and , respectively), then the origin of is a globally fixed-time stable equilibrium.

However, despite the advantages of having a finite settling time which allows uniform convergence with respect to the initial conditions, the homogeneity based approach does not enable to easily arbitrarily preassigned the settling time. For some applications such as state estimation, dynamic optimization, consensus of cluster networks, among others, it would be convenient that the trajectories of system (1) reach the origin within a time , which can be defined in advance as a function of the system parameters, i.e., . This motivates the following definitions.

Definition 11.

Sánchez-Torres et al. (2014) Let the origin be fixed-time stable for system (1). The set of all the bounds of the settling-time function is defined as

Definition 12.

Sánchez-Torres et al. (2015, 2018) For the parameter vector of system (1) and a constant , the origin of (1) is said to be predefined-time stable if it is fixed-time stable and the settling-time function is such that

is called a predefined time.

Remark 2.

It would be desirable to choose not only as a bound of the settling-time function , but as the least upper bound, i.e., . However, this selection requires complete knowledge about the system, compromising its application to decentralized systems.

Let us recall some important results concerning predefined-time stability.

Theorem 2.

(Aldana-López et al., 2018, Theorem 1) Consider system

(3)

where , . The parameters of the system are the real numbers which satisfy the constraints and . Let be the parameter vector of (3). Then, the origin of system (3) is fixed-time stable and the settling time function satisfies , where

(4)

and is the Gamma function defined as  (Bateman et al., 1953, Chapter 1).

Using Theorem 2, one can obtain the following Lyapunov based characterization of predefined-time stable systems.

Theorem 3.

(Aldana-López et al., 2018, Theorem 3) Consider the nonlinear system

(5)

where is the system state, the vector stands for the system parameters which are assumed to be constant. The function is such that . Assume there exists a continuous radially unbounded function such that:

and the derivative of along the trajectories of (5) satisfies

(6)

where , , and is given in (4) and is the upper right-hand Dini derivative of Kannan and Krueger (2012).

Then, the origin of (5) is predefined-time-stable with predefined time .

3 Main results

3.1 Problem statement

Consider a multi-agent system consisting of agents with first-order dynamics given by

(7)

where and are the system state and control input respectively. Term represents external perturbations and is assumed to be bounded by a known positive constant , i.e.

(8)

The communication topology between agents is represented by the graph (which could be dynamic) where is a collection of graphs having the same vertex set and is the switching signal determining the topology of the dynamic network at each instant of time.

The control objective is to design decentralized consensus protocols (), based on available local information, such that the state of all the agents converges to an equilibrium, known as a consensus state , in a predefined-time , regardless of the initial conditions of the agents and the disturbance signals affecting each agent, i.e.

(9)
Remark 3.

If , the control objective is called as the predefined-time average consensus problem.

Remark 4.

There exist methods for predefined-time consensus in the literature. However, they are based on time-varying gains Yong et al. (2012); Liu et al. (2018); Wang et al. (2017b, 2018); Colunga et al. (2018); Zhao et al. (2018b), since the same gain must be applied to all agents, a common time-reference is needed along the network. Moreover, this time-varying gain becomes singular at the predefined-time, see for instance Yong et al. (2012) or as in Liu et al. (2018); Zhao et al. (2018b) where it may produce a Zeno behavior (infinite switching in a finite interval). These drawbacks are not present in our approach.

3.2 Robust predefined-time consensus algorithms

In this section, we propose and analyze two fixed-time consensus protocols, which have the following form

(10)

and

(11)

where , and . The constant and will be designed later to guarantee the convergence in a predefined-time , even in the presence of disturbances.

Remark 5.

Let . On the one hand, notice that the consensus protocol (10) is computationally simpler than consensus protocol (11) since at each time instance each agent only requires a single computation of the nonlinear function whereas for protocol  (11) an agent requires to perform one computation of for each neighbor. On the other hand, we will show that (10) is a predefined-time consensus protocol for static networks and a fixed-time consensus protocol for dynamic networks whereas control (11) is a predefined-time consensus protocol for static and dynamic networks

3.2.1 Convergence analysis under consensus protocol (10)

Let , where is given in (10), then can be written as with . Notice that, the dynamic of the network under the consensus algorithm (10) is given by

(12)

where, for , the function is defined as

(13)

and with .

Let us now study the stability of the closed-loop system (12). First, using the homogeneity theory, let us derive sufficient conditions for the design of control (10) such that the consensus is achieved in a fixed-time under switching topologies.

Lemma 2.

The vector field of (12) is homogeneous in the limit with degree and homogeneous in the -limit with degree with respect to the standard dilation.

Proof.

It follows straightforwardly from the definition of homogeneity in the bi-limit Andrieu et al. (2008). ∎

Theorem 4.

Let be a collection of connected graphs and let be a non-Zeno switching signal.
If

(14)

then, protocol (10) guarantees the fixed-time consensus on switched dynamic network under arbitrary switching signals .

Proof.

Since, according to Lemma 2 the vector field in (12) is homogeneous in the bi-limit then, according to Theorem 1, to show that fixed-time consensus is achieved it only remains to prove that (12) as well as

(15)

and

(16)

are asymptotically stable where, with and ,

and

To this aim, consider the (Lipschitz continuous) non-smooth Lyapunov function candidate

(17)

which is differentiable almost everywhere and positive definite. Let and for a nonzero interval then the time derivative of (17) along the trajectory of (12) is given by

Notice that, since and then and , and unless consensus is achieved, they cannot be both zero for a non zero interval, because since the graph is connected there is a path from agent to agent such that there is a node satisfying but . Thus if then . A similar argument follows to show that there is a node such that but and thus if then . Thus almost everywhere and consensus is achieved.

Fixed-time convergence for dynamic networks, under arbitrary switching, follows from the stability theory of switched systems (Liberzon, 2003, Theorem 2.1) by noticing that (17) is a common Lyapunov function for each subsystem evolving with connected. ∎

Remark 6.

Notice that Theorem 4 extends the results in Gómez-Gutiérrez et al. (2018), in which the fixed-time consensus problem was addressed using homogeneity for the case where and no disturbances are present.

Using an appropriate Lyapunov function, let us derive sufficient conditions for the design of control (10) such that the consensus is achieved in a predefined-time under a fixed topology.

Theorem 5.

Let be a connected graph and let

(18)

where

and is defined in Eq. (4), then, protocol (10) guarantees that consensus is achieved before a predefined-time .

Proof.

Let be a disagreement variable such that where is a consensus value, which is unknown but constant. Consider the Lyapunov function candidate

(19)

By noticing that , then it follows that

Let , therefore:

(20)

Using Lemma 7, the first term can be rewritten as:

where and Moreover, using Lemma 8, one can obtain

Expressing the disagreement variable as a linear combination of the eigenvectors of , the term can be bounded as

Therefore, by Lemma 5:

(21)

Furthermore, from the last two terms of (20) the following is obtained:

(22)

Therefore, the following inequality is obtained from (20), by combining (21) and (22):

(23)

Then, according to Theorem 3, protocol (10) guarantees that consensus is achieved before a predefined-time . ∎

Example 1.

Consider the multi-agent system (7) composed of agents with external perturbation . The communication topology, given in Figure 1 is undirected and static. The corresponding algebraic connectivity is . The initial conditions of the agents are randomly generated and are as follows:

According to Theorem 5, protocol (10) with , guarantees that the consensus is achieved before a predefined-time under the graph topology . Figure 2 shows the corresponding result. For this experiment the settling time is of 0.095s.

Figure 1: Static network used for Example 1.
time (s)
Figure 2: Convergence of the consensus algorithm for Example 1.
Example 2.

Consider the multi-agent system (7) composed of agents with external perturbation . The collection of communication topologies , given in Figure 2(a)-2(d) are undirected. The switched dynamic network evolves according to the switching signal given in Figure 4 which satisfies the minimum dwell time condition. The corresponding algebraic connectivity is

The initial conditions of the agents are randomly generated and are as follows:

According to Theorem 4, protocol (10) with , , , , ,

and guarantees that the consensus is achieved in a fixed-time under the switched dynamic network . Figure 4 shows the corresponding result.

(a)
(b)
(c)
(d)
Figure 3: Collection of communication topologies used for the switched dynamic network in Example 2.
Figure 4: Convergence of the consensus algorithm for Example 2 and the corresponding switching signal.

3.2.2 Convergence analysis under consensus protocol (11)

Similarly to the previous subsection, let us define . Notice that the dynamic of the network under the consensus algorithm (11) is given by

(24)

where, for , the function is defined as (13) and with .

Let us now study the stability of the closed-loop system (24). First, let us derive a useful lemma concerning average consensus in the absence of disturbance. Then, using an appropriate Lyapunov function, let us derive sufficient conditions for the design of control (11) such that the consensus is achieved in a predefined-time under switching topologies

Lemma 3.

Let be a switching dynamic network composed of connected graphs and assume that under the protocol (11) and in the absence of disturbance consensus is achieved at a time . Then, the consensus state is , i.e. consensus to the average is achieved.

Proof.

Let be the sum of the agent states. In the absence of disturbances, i.e. if , since , then . Thus, is constant during the evolution of the system, i.e. , .

Therefore, Thus, if , . Thus, and , i.e. consensus to the average is achieved. ∎

Theorem 6.

Let be a collection of connected graphs and let be a non-Zeno switching signal. If

(25)

with

and is defined in Eq. (4), then, protocol (11) guarantees that consensus is achieved before a predefined-time on switched dynamic networks under arbitrary switching signals . Moreover, in the absence of disturbances consensus to the average is obtained.

Proof.

Let be a disagreement variable such that where is a consensus value, which is unknown but constant. Consider the Lyapunov function candidate

(26)

To show that consensus is achieved on dynamic networks under arbitrary switchings, we will prove that (26) is a common Lyapunov function for each subsystem of the switched nonlinear system (24(Liberzon, 2003, Theorem 2.1). To this aim, assume that for . By noticing that , then it follows that

Let , therefore:

(27)

Using Lemma 7, the first term can be rewritten as:

where and . Moreover, it follows from Lemma 8 that

Therefore, by Lemma 5:

(28)

Furthermore, from the last two terms of (27), the following is obtained:

(29)

Therefore, the following inequality is obtained from (27), by combining (28) and (29):