On networked Euler–Lagrangian systems consensus under switching topologies

This paper studies the consensus of multiple Euler–Lagrangian systems with dynamic uncertainties and the challenges to be solved lie on weak interaction, time-varying interaction and parametric uncertainties. The overlapped problem makes the control protocol of the networked Euler–Lagrangian system hard to analyze and design and to resolve the issue, a novel reference, as well as an adaptive protocol, is proposed. In addition, the concept of integral-  p stability is employed. Under the control of the proposed protocol, networking coupled Euler–Lagrange systems achieve synchronization which means inter-subsystems state errors converge to zero with a mild assumption of the union of switching topologies containing a directed spanning tree. The numerical simulations verify the effectiveness of the proposed protocol.


INTRODUCTION
The past decade has witnessed the rapid development of multiagent systems (MAS) [1]. In particular, the consensus control for multi-agent networked systems had been brought under the prime focus of research in various fields, including multiagent formation [2], manipulation [3], and exploration [4,5]. Through information interaction among agents, the network is able to achieve an agreement in which the protocol serves as the role of a commander. Recently, the consensus protocol of MAS has been applied successfully in Euler-Lagrangian systems [6][7][8]. Reviewing the existing literature of Euler-Lagrangian MAS , some fundamental problems are most-discussed, for example interaction topology [9], communication delays [10], model uncertainties [3]. The control of Euler-Lagrangian MASs is the generalized control of a single Euler-Lagrangian system. For a single Euler-Lagrange system, it is usually intertwined with non-linearity and dynamic uncertainties, which has been widely investigated by many articles and some techniques are introduced to cope with dynamic uncertainties, for example variable structure [11], adaptive control [12], neural network [12] et al. For networked Euler-Lagrange MASs, many protocols has been proposed for achieving consensus [13], flocking [7], or robustness with respect to communication delays [14,15]. Based on the above theoretic techniques, many investigations have been done and have proposed various schemes to achieve a consensus of multiple Euler-Lagrangian systems. For instance, integral bounded input and bounded output (iBIBO) stability [16], small gain theory [17], passivity [18]. Technically speaking, the investigations of the consensus share a common goal to a certain extent, which is achieving the overall stability of the networks with the help of the protocols.
To achieve the goal and based on the above techniques, there are two widely used schemes: passivity-based scheme [17] and dynamic-compensator-based scheme [19]. The passivity-based adaptive scheme results in the consequence that the positions of the systems converge to the origin in the presence of gravitational torques. On the other hand, the dynamic-compensatorbased scheme in [19] designs defined interacting information which is decoupled to physical states, such as positions or velocities. Due to the exchanging information not coupled to the physical states of the followers, it results in that the overall networked system losses manipulability [20], which means the consensus is sensitive to external input. Obviously, in most practical application scenes, external inputs or perturbation may be inevitable and the method of dynamics compensation is limited. To resolve the manipulability problem, physical quantities are expected to be included in exchange information. In this sense, the consensus problem for Lagrangian systems using physically coupled information is still an open question.
The problem of various interaction topologies is also a hot topic in the field of multiple Euler-Lagrange MAS. The interaction graphs among the multiple Euler-Lagrange systems can be grouped into two categories. The first category of schemes achieves the consensus of robotic systems on undirected interaction graphs [21]. The second category of schemes achieves the consensus of the robotic systems on the more general directed graphs [8]. However, in many scenarios, the topologies of interactions are not invariant. Namely, the topologies of MAS may be changing with respect to time. Many articles investigate the case of switching topologies [19,[22][23][24][25]. Considering the problem of asymptotic tracking control for a class of uncertain switched non-linear systems, [26] proposes a fuzzy adaptive control strategy that can guarantee the local asymptotic tracking performance. Moreover, [27] considered the issues including arbitrary switchings, unmodelled dynamics, input saturation, unknown dead-zone output, dynamic disturbances, and unmeasurable states. In [28], the switched stochastic non-linear systems in a pure-feedback form are considered and an adaptive fuzzy output-feedback control scheme is proposed to achieve the tracking control of such a class of the above switched systems. The main challenge of the switching topologies, compared with its counterparts, is avoiding differentiating the discontinuous quantities resulted from the topology switching and the associated stability analysis which also is the motivation of this paper. Moreover, in the field of multiple Euler-Lagrange MAS, in many cases, the communication flows are inevitably delayed when communication links are unreliable or the bandwidth is limited. Therefore, it is essential and practical to investigate communicating delays in the multirobot system. Taking the delays into consideration, some studies have presented various solutions. Liu and Chopra [29] study the so-called controlled synchronization for networked multi-robot systems with uncertainties existing in dynamics and constant communicating delays. Wang [30] proposes a passivity-based controlled scheme to synchronize the multiagent robotic system with constant time delays, the dynamic uncertainties and the kinematic uncertainties. [31] investigates cooperative tracking control problems not using the task-space velocities and even with the consideration of communicating constant delays. By the above-mentioned methods, the delay problem in the interagent information exchanges is solved to a certain extent. But the problem of unknown time-varying delays in switching topologies is still an open issue to be solved.
To resolve the above-mentioned unresolved issues, we propose a novel adaptive protocol with integral- p stability. Specifically, we consider the case of switching topologies whose union contains a directed spanning tree and meanwhile there are unknown bounded time-varying delays existing in the network. The novelties of this research lie in the following three aspects.
(a) The exchanging information. In contrast with the common framework, one may notice that the exchanging information among agents, formed by the references (of velocity or of position), results in the lack of manipulability. Different from the many existing results, the consensus problem for multiple uncertain Lagrange systems relies on the coupled actions, guaranteeing the all agents responsive to external input. (b) The switching topologies. There are some results on consensus [32], synchronization [14], or flocking [9] which consider physically coupled information but as the author's best effort, there is no attempt on switching topologies pointing towards the similar research perspective. (c) Unknown time-varying delays. Time delay, a quite common phenomenon in the network field, challenges the stability of the overall networking system. Much of the literature is concerning coping with constant delays. The time-varying delays, however, are not well resolved in the Euler-Lagrange network as well as the associated stability.
To show the consensus under switching topologies with time-varying delays, we introduce several new input-output propositions concerning linear time-varying systems. These new input-output properties are referred to as uniform integral- p stability. Distinguishing from the standard  p stability [33], integral- p stability describes the relationship between the integral of the input and the output and also with the integral- p stability concerning marginally stable linear time-invariant systems. By the introduced new tools, the convergence of the consensus errors is rigorously shown under the condition that the union of the graphs contains a directed spanning tree. The proposed protocol only uses the physically coupled action as exchanging information, compared with the work in [19]. The proposed protocol ensures that the positions of the systems converge to a common value, compared with the passivity-based adaptive schemes whose the consensus equilibrium is the origin in the presence of gravitational torques, as is demonstrated in [4,14].

Graph theory
Graph theory is the fundamental of this paper which needs to be introduced. A graph can be conveniently used to represent the information flow [34] between agents. A brief introduction will be given in the context that n Euler-Lagrange systems. Let  = (, ,  ) to represent an undirected graph or directed graph (digraph) of order n with the set of nodes  () = {v 1 , v 2 , … , v n }, the set of edges  ⊂  × , and a weighted adjacency matrix  with non-negative adjacency elements and w i j = 0 otherwise. Furthermore, conventionally, assume there is no self-loop contained, that is i ∉  i , and hence for all i ∈ , w ii = 0. The Laplacian matrix L = [l i j ] is defined as In the case of switching topology, the interaction graph among the systems is time-varying. All candidate graphs can be formulated by the set where j ∈ [1, m] is the index of subgraphs, m denotes the number of possible graphs. The switching function is given by (t ) : [t 0 , +∞) → j , which is a piecewise constant switching signal. Then the switching communication graph is denoted by  (t ) . The dwell time t n+1 − t n is bounded by its lower bound .

Model of Euler-Lagrange systems
The dynamics of the ith Euler-Lagrange system can be written as [35] where q i ∈ ℝ n is a vector of generalized coordinates and the measuring output; n denotes the dimension of generalized coordinate; H i (q i ) denotes inertia matrix which is positive-define and symmetric; g i (q i ) is the gravitational force; is Coriolis and centrifugal matrix; i ∈ ℝ n×1 is the vector of torques produced by the actuators associated with the ith system. The centrifugal matrix C i (q i ,̇q i ) satisfies the following property [36]: The dynamics of an Euler-Lagrange rigid body system can be linearizing parameterized as follows.
Property 1. The dynamic equation of robot manipulator can be linearized as follows wherėi,̈i ∈ ℝ n×1 ; Y d,i (q i ,̇q i ,̇i,̈i ) ∈ ℝ n×p 3 is called dynamic regression matrix; the vector d ∈ ℝ p 3 ×1 includes all unknown dynamic parameters which are constant; p 3 is the dimension of vector d [37].

MAIN RESULTS
We develop new protocols to cope with the interaction graph of switching topologies, even with time-varying delays. These results present in the following two subsections.

The case of switching topologies
By define the following differentiable vector z i ∈ R m , one getṡ where is positive constant. Then construct the sliding vector as follows where s i ∈ R n×1 and z i is the reference velocity. Note that z i obtained from the integral of (4) is crucial for the case of switching topologies due to the fact that the reference velocity is differentiable when topology is switching. The result given here does not depend on the topology information with the relative velocities being incorporated for cope with the situation of the switching topologies, while in the case of a fixed topology it is possible to design a controller without involving relative velocity measurement (see, e.g. [38]). In addition, it might be noted that if the topology was fixed, the reference velocity z i defined by (4) would be quite similar to the one in [39] with the incorporation of the integral of the sliding vector except for a constant offset; but here the design perspective is different with defining the reference acceleration first and then deriving the reference velocity by an integral operation. This, on the one hand, facilitates the handling of switching topologies (e.g. avoiding differentiating discontinuous quantities), but on the other hand, introduces challenges for performing stability analysis.
Remark 1. By the definition of z i given by (4) using the pure integral, the reference velocity is differentiable even if the interaction topology is switching, compared with the existing results which usually define the reference velocity as with T i j denoting the delay which leads to the fact thaṫz i usually involves the derivative of the unknown delay. The scheme proposed herein is independent of the topology information with the relative velocities being incorporated for handling the case of switching topologies while in the case of a fixed topology it is possible to design a controller without involving relative velocity measurement.
Remark 2. Besides one may note that if the topology was fixed, the reference velocity z i defined by (4) would be quite similar to the design which is with the incorporation of the integral of the sliding vector except for a constant offset, that is [39]; but the proposed design perspective is different with defining the reference acceleration first and then deriving the reference velocity by integral operation. Such the design overcomes the difficulties in the design of switching topologies. But it introduces challenges for performing stability analysis as well.
The adaptive controller is given as where K i and Γ i are symmetric positive definite matrices nd̂i is the estimate of i . The adaptive controller given by (6) leads to the following dynamics that describes the behaviour of the i-th system [39] (7) defines a system which we refer to as integral-cascade system in that by integrating the first subsystem with respect to time, the overall system becomes a typical cascade system in terms of the cascade variable s i , that is Due to the integral operation in the first subsystem in Equation (8), the stability is difficult to analyze. Before analyzing the stability of the cascade system, the following proposition is necessary, which describes the integral-input-output properties of linear time varying systems.

Proposition 1. Consider a linear time-varying system with an external inpuṫy
where y ∈ R m 0 is the output, M (t ) ∈ R m 0 ×m 0 is the system coefficient matrix and is uniformly bounded, and u ∈ R m 0 serves as the external input. Under the assumption of the LTI system is uniformly exponentially stable, it holds that if ∫ t 0 u( )d ∈  ∞ then y ∈ ∞. Namely, the system (9) is iBIBO. It also holds that if ∫ Following typical practice of input-output properties of uniformly exponentially stable linear time varying systems reported by [40], one may derive thatȳ ∈ L ∞ whenū ∈ L ∞ . It immediately leads to y =ȳ + ∫ t 0 u( )d ∈ L ∞ . For p = [1, ∞), we consider c = 0 first. It is well known that the exponential stability of the systeṁy = M (t )ȳ implies that there exist the positive constants b 1 and b 2 such that [41,42] where Φ(t, t 0 ) denotes the transition matrix of the time-varying system. From existing literature [41,42], it can be easily verified that ifū ∈ L p thenȳ ∈ L p and y =ȳ + (10) and it shares same procedures then. □ Remark 3. The L p stability formulated in Proposition 1 as well as the proof extends the existing results for linear time-varying systems, for example [33]. The main difference is that the L p stability introduced is of the relation between the output and integral of the input (in contrast with [41,42] and also with the standard iISS in [43]), and we refer to these integral-inputoutput properties as uniform integral-L p stability. The advantage of Proposition 1 mainly lies in the new perspective, then the stability issues associated with the integral-cascade framework are fully addressed.
Now we are at the position to present the stability analysis.

Theorem 1.
For directed interaction network with switching topologies whose union of the graph containing a spanning tree and whose intervals are uniformly bounded, the adaptive protocol (6) guarantees the consensus of networked Euler-Lagrange systems. Namely, q i − q j → 0 anḋq i → 0 as t → ∞, ∀i, j = 1, … , n.
Proof. The Lyapunov-like candidate can be And differentiating V along the trajectories of the system yields V i = −s T i K i s i ≤ 0. Not hard to derive s i ∈ L 2 ∩ L ∞ and i ∈ L ∞ . Now, define a new sliding vector Then (4) can be rewritten aṡ For all agents in the network, the following holdṡ where is the Laplacian matrix at instant t . By define the following two error vectors̃= where Ψ(t ) is the associated matrix depending on L . Due to the boundedness of s i ,s ∈ L 2 ∩ L ∞ . Note thaṫi = −Ψ(t )̄i is uniformly exponentially stable. Based on Proposition 1, one may easily derivē∈ L 2 ∩ L ∞ . From the definition of̄, one Following similar processes, one hasq → 0 anḋq → 0. Therefore, from Barbarlet's lemma,q → 0. Now rewrite (4), one haṡ Easy to verify the boundedness of Z * . Then it yields that z i ,̇z i ∈ L 2 ∩ L ∞ . Bẏq i = z i + s i , easy to knoẇq i ∈ L 2 ∩ L ∞ . Recalling the second dynamic Equation (7),̇s i ∈ L ∞ holds and yields s i is uniformly continuous. Furthermore, s i → 0 as t → ∞, ∀i. Therefore, we can finally derivėq i → 0 as t → ∞, ∀i. □

The case of switching topologies with time varying delays
In this section, we consider the case that the interaction among the systems involves unknown time-varying communication delays in addition to switching topologies. The delays are assumed to be piece-wisely continuous and uniformly bounded, at the same time, the upper bounds of the delays are not required to be known. That is, the delays can be arbitrarily large and time-varying and the only requirement is the boundedness. We first introduce the following propositions and lemma, mainly for addressing the presence of time-varying communication delays.

Proposition 2. Consider a linear time-varying system with uniformly bounded time-varying delays and an external inpuṫ
where x is the state, y is the output, F D ( * ) is a linear mapping with F D (x) containing delayed version of x and the time-varying delays are uniformly bounded, C is the output matrix that is bounded, and u acts as the external input. Suppose that the linear time-varying system with u = 0 yields an output that uniformly exponentially converges to zero. Then

Proof. To facilitate analysis, one can define
The input-output property of (19) depends on the properties of F D (u + ) whose central element involves a variable u (19), by the standard input-output properties of linear systems, one may easily derive that y + ∈ L ∞ and thus y = y + + Cu + ∈ L ∞ . In the case that ∫ t 0 u( )d +c ∈ L p , p ∈ [1, ∞), similarly, we may also (19) with the above definition holds as well. In order to verify the above analysis, we may first consider the case of ∫ t 0 u( )d +cinL 2 . Definingc i the i-th elements ofc, one has with r = − T k ( ). Equation (20) guarantees u + i (r ) +c i ∈ L 2 . In the case of ∫ t 0 u( )d +c ∈ L p , p ∈ [1, ∞), p ≠ 2, similarly, it can be obtained that u + i (r ) +c i ∈ L p . Then in accordance with the standard input-output properties of linear systems, it is not hard to derive that y + ∈ L p since the input F D (u + ) ∈ L p and y ∈ L p . □

Proposition 3. Consider a uniformly marginally stable linear timevarying system of the first kind with uniformly bounded time-varying delays and an external inputẋ
where x is the state, F D ( * ) is a linear mapping with F D (x) containing delayed version of x and the time-varying delays are uniformly bounded and u acts as the external input. Then1

that is the system (17) is uniformly integral-bounded-input bounded stable.
For addressing the consensus with both the communication delays and switching topologies, we define the vector z i bẏ Theorem 2. If there exists an infinite number of uniformly bounded intervals [t ip , t ip+1 ), p = 1, 2, … with t i1 = t 0 satisfying the property that the union of the interaction graphs in each interval contains a directed spanning tree, then the adaptive controller given by (6) with z i being given by (22) ensures the consensus of the n Lagrangian systems with bounded unknown time-varying communication delays, that iṡq i −q j → 0 anḋ q i → 0 as t → ∞, ∀, j = 1, … , n.
Proof. The proof of Theorem 2 is similar with that of Theorem 1 but the main difference caused by the delay process iṡi ] +̇s i , i = 1, … , n. (23) and accordingly one may derive where F * D (⋅) is an associated delayed linear mapping and C is output matrix which is bounded. Follow the similar procedure of Theorem 1, we need to analyze the properties of (24). Likewise, it is easy to verify s ∈ L 2 ∩ L ∞ . And the output of (24) uniformly converges to zero wheṅs = 0, anḋis also uniformly convergent to zero. Besides, with the help of Proposition 2, (24) is integral-bounded-input bounded-output stable. Namely , s ∈ L ∞ yields E ∈ L ∞ . With the help of Proposition 3, (24) is integral-bounded-input bounded stable. Namely, s ∈ L ∞ yields F * D ( ) ∈ L ∞ . Following similar procedure in the proof of Theorem 1, one can finally obtain that q i − q j → 0 anḋq i = 0 as t → ∞. □ Remark 4. To provide more insights of this paper, we would like to discuss the result of the proposed work as follows.
Regarding the interaction between individual system dynamics and the network constraints, generally speaking, the simplified dynamics (at the aspects of uncertainty, non-linearity, and relative degree etc.) allows a milder requirement towards the network interconnection strength. High degree of topology switching, weak interconnection among the nodes, and timevarying communication delays typically weaken the network interconnection strength. In this work, we attempt to render the controlled system dynamics as simple as possible (yet still with practically acceptable control efforts) so as to tolerate the weak network interconnection (e.g. directed switching topologies and unknown time-varying communication delays). The limit is the possibilities of accommodating jointly-connected directed switching topologies and arbitrary unknown timevarying delays in (static) consensus of uncertain Lagrangian systems, and the costs required seem to be practically acceptable, namely, full state (position and velocity) measurement of the system and communication of the composite of position and velocity among the systems (with the dimension equal to the system DOFs).

SIMULATION RESULTS
To verify the effectiveness of the proposed Euler-Lagrange MAS protocol (4)-(6), the following simulations are conducted. The agent Euler-Lagrange system considered here is a robotic system, specifically, an Omni robot ( Figure 1). The D-H parameters of the Omni robots are given in Table 1. The dynamics of a Omni robot can be formulated in Equation (1) and where in which c i = cos(q i ), s i = sin(q i ), c 2,i = cos(2q i ), s 2,i = sin(2q i ), H 11 = a 1 + a 2 c 2,3 + a 3 s 2,3 + a 4 c 3 + a 5 s 3 . The sum matrix of Coriolis, centrifugal and gravitational vectors can be formulated by where In this simulation, that is for Omni manipulator nodes, the typical dynamics parameters values a 1 -a 8 are listed in Table 2. Based on the typical dynamics parameters, the parameters on different nodes, however, are slightly different for the purpose of better verification on adaptivity to uncertainties of the Euler-Lagrangian models.

FIGURE 2 Switching candidates of interaction graphs
To verify the synchronization under switching topologies (Theorem 1) and switching topologies with time-varying delays (Theorem 2), three candidate interaction graphs of the nodes are used and the interaction among the nodes switches in these graphs which are shown in Figure 2. In the figures, the different graphs are piled up and are demonstrated by the four layers z = 20, 15, 10. The layer z = 5 denotes the union of the interaction graphs.
In the first simulating test, six Omni robots under the above switching interaction graph are considered, whose union of the graph containing a spanning tree. And the switching intervals are set as 50 ms which is uniformly bounded. The set of tunable parameters in Equation (8) in this simulation is chosen as: = 3, Γ i = 4.5I 5 , K i = 10I 2 , i = 1, … , 6 and the coupling strength i j = 1, j ∈  i , i j = 0, otherwise. The estimation̂i of dynamic unknown vector is randomly given for each node. To verify the consensus of the robotic subsystems, nodes sinusoidal external perturbation is imposed on each Omni systems. The simulation results are presented in Figures 3-5. From the simulation results in Figure 3, when the switching interval is set as 30 ms, it can be easily seen that all joints in each robot converge to the same value and it means that the position consensus has been achieved. From Figure 4, the interval increases to a large one which is 500 ms and it shows that all agents can achieve consensus despite the fact that joint-1 of agent-1 moves toward opposite direction of consensus (around 6 s). To verify the effectiveness of the proposed protocol, under the same simulation parameters above, the distributed controllers which are based on invariant interaction topology are also employed and the results are demonstrated in Figure 5. It is not hard to see that the sub-robotic-systems fail to achieve consensus under the switching topologies case.
To show the effectiveness of the proposed scheme, namely, the results presented in Theorem 2, the second simulation is conducted. The interaction time varying delays d 12 (t ) = d 25 (t ) = 0.3 + 0.5 sin(t ∕2)s, d 65 (t ) = d 43 (t ) = 0.1 + 0.6 sin(t ∕2)s, d 51 (t ) = d 26 (t ) = 0.6 + 0.7 sin(t ∕2)s. After adding the delays on the interaction links, the switching topologies interval is set as 100 ms. Then we conduct the second simulation and the results are presented in Figure 6.

CONCLUSIONS
This paper examines the consensus problem of multiple Euler-Lagrangian systems under switching topologies. The new adaptively distributed controllers are presented to cope with the variant networking interaction including the two cases switching graphs and switching graphs with time-varying delays. Based on the proposed controllers, the main results of this paper can be summarized as (1) the networked Euler-Lagrangian systems with parametric uncertainties can achieve convergence under switching topologies; (2) the consensus of the above systems is generalized to the case of time-varying delays existing in interactions. It is noteworthy that a new theoretical analysis tool referred to as uniform integral- p stability is introduced to analyze the stability and convergence of the closed-loop system.