Event-triggered robust cooperative stabilization in nonlinearly interconnected multiagent systems

European Journal of Control 48 (2019) 9–20

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European Journal of Control journal homepage: www.elsevier.com/locate/ejcon

Event-triggered robust cooperative stabilization in nonlinearly interconnected multiagent systems Vahid Rezaei∗, Margareta Stefanovic Department of Electrical and Computer Engineering, University of Denver, 2155 E Wesley Ave, Denver, CO 80208, USA

a r t i c l e

i n f o

Article history: Received 2 July 2018 Revised 12 December 2018 Accepted 19 January 2019 Available online 25 January 2019 Recommended by Prof. T Parisini Keywords: Event-triggered control Cooperative control Robust control Large-scale systems Multiagent systems Cyber-physical systems

a b s t r a c t By way of the graph theoretic tools, we model closed-loop large-scale systems as two-layer multiagent systems with the agent layer describing nonlinearly interconnected dynamics and the control layer designed to stabilize the system by communications in agents’ neighborhoods. We consider a scenario in which the effect of dynamical interactions over the agent layer appears through some nonlinear functions which are in the range space of the control input matrix, yet the underlying agent layer interconnection topology and the associated nonlinearities are not known to the control layer designer. We develop a static linear cooperative algorithm and, having endowed each local controller with several buffers, we schedule the control layer information exchange events in a non-periodic non-synchronous manner. We combine graph theoretic and event triggering ideas with an optimal control formulation, and propose a design procedure to obtain the required robust control gain. We prove that all trajectories of the closedloop multiagent system will exponentially converge to the origin in the presence of time-varying nonlinearly interconnected modeling uncertainties over agent layer. For the same cooperative stabilization algorithm, trading off with the exponential convergence to a neighborhood around the origin, we further show that a practical event triggering mechanism can be developed to better deal with the effect of modeling mismatch on the number of triggering events. We also prove the proposed event triggering mechanisms do not exhibit Zeno behavior to ensure the feasibility of the proposed ideas in the sense of reduced communication load and, thereby, energy consumption. © 2019 European Control Association. Published by Elsevier Ltd. All rights reserved.

1. Introduction With advances in networked systems science, networked control and cyber-physical systems have emerged as new directions in studying large-scale infrastructures where the functionality of networked control systems depends on their limited energy resources, and cyber physical systems are notable for being subject to uncertainties. The centralized and decentralized strategies have been widely used for the stabilization purpose in large-scale systems which can be extended to both networked control and cyber-physical systems, and the existing distributed algorithms in the literature of networked control systems are based on the systems theoretic tools. Recently, with the improvements in embedded sensing, computation, and communication technologies, graph theoretic tools have been successfully used to ensure agreement of multiagent systems (MASs) in a distributed fashion; however, the application of graph theoretic tools for the stabilization in largescale network of cyber-physical systems is not well-studied yet. In ∗

Corresponding author. E-mail addresses: [email protected] (V. Rezaei), [email protected] (M. Stefanovic).

the rest of this section, we briefly overview these topics and lay a foundation for energy efficient graph theoretic cooperative stabilization in large-scale infrastructures with communication capabilities. Control of large-scale systems received considerable attention within the control systems society. Thinking about the physical interaction of subsystems’ dynamics as a key feature of large-scale systems, two major configurations have been studied: centralized and decentralized control schemes. Although decentralized techniques may result in lower performance compared to the centralized approaches, low computation and implementation costs have made them popular to use in large-scale systems (see [31] as a comprehensive reference for control of large-scale systems). However, large-scale systems may be subject to undesirable “decentralized fixed modes” and “quotient fixed modes” where the former cannot be manipulated by any linear time-invariant (LTI) decentralized controllers and the latter cannot be changed by any decentralized controllers (time-invariant, time-varying, linear, and nonlinear) [49]. It is known that these types of large-scale systems can be completely controlled whenever the information flow is complete [19,55]. Several ideas have been proposed to handle these unwanted modes; however, designing such non-decentralized

https://doi.org/10.1016/j.ejcon.2019.01.004 0947-3580/© 2019 European Control Association. Published by Elsevier Ltd. All rights reserved.

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V. Rezaei and M. Stefanovic / European Journal of Control 48 (2019) 9–20

controllers can be a tedious task. Recently, the structurally constrained controllers have been proposed by Sojoudi et al. [49] in which the authors use the exact mathematical model of large-scale system (including the interconnection topology) to find the fixed modes, use them to identify the required inter-controller information flow structure, develop a transformation to reformulate the problem as a decentralized control design task for the transformed system, design decentralized “sub”-controllers, and find the structured control system by transferring back to the original coordinate. The aforementioned structurally constrained technique requires some carefully selected sub-controllers to “communicate” with each other. With this in mind, we turn our attention toward the distributed consensus of MASs in which each agent is equipped with appropriate sensing, computation, and communication devices so that a common decision can be made by all agents through communication within each agent’s neighborhood. Unlike large-scale systems, MASs are in general considered to be a group of individual dynamical systems that operate toward a common goal (consensus or agreement value). The initial research was mainly focused on understanding the relationship between graph and (linear) systems theory in MASs of single and double integrators [39,42]. More complicated scenarios have been recently proposed to consider the uncertainty in the mathematical model of agents as a practical issue. Of those, we may point out the Refs. [53] and [1] where each agents’ modeling uncertainty is a function of the agent’s own variables. The consensus problem has also been investigated for MASs with interconnected dynamics in [38] with completely known dynamics, and in [9] where the model of MAS is subject to structured interconnected linear uncertainties. However, these results are limited to linear multiagent systems while a more realistic scenario may result in nonlinear modeling uncertainties. For example, Li et al. [28] proposed an adaptive nonlinear approach to ensure consensus in MASs of linear agents with (non-interconnected) nonlinear modeling uncertainties. Several other nonlinear control techniques have been proposed for linear and nonlinear MASs (see [30] and [4], respectively). Nevertheless, the theoretically strong nonlinear techniques are still unpopular in industry due to, e.g., additional design and implementation complexities compared to linear techniques [46]. Recently, along with the advances in embedded sensing, computation, and communication devices [8,12]; cyber-physical systems have emerged as way of studying infrastructures with the aforementioned sensing, computation, and communication cyber aspects in addition to the physical interaction of large-scale systems. This is a multidisciplinary topic; however, within the scope of our paper, there is an interest in discussing the behavior of cyber-physical systems using systems- and distributed controltheoretic tools. For example, Egerstedt [18] has studied the relationship between the architecture and global control objectives using graph theoretic tools for cyber-physical systems with completely known mathematical models. However, assuming completely known cyber-physical system might be a physically restrictive assumption [3]. For MASs with interconnected (non-) linear modeling uncertainties, this issue has been addressed by Rezaei and Stefanovic [43,44] as distributed decoupling problems to cancel the effect of physical couplings on the behavior of dynamically stable local agents. Nevertheless, these references rely on the existence of limitless energy resources in their control design approaches in which agents might continuously exchange information with their neighbors over wireless communication network. The sampled-data control approaches consume less energy compared to the aforementioned continuous measurement-based techniques. In particular, unlike the periodic sampling-based ideas which continuously take and transmit the measured samples based on a fixed (small) sampling interval, the aperiodic sampling

allows the wireless communication network to stay in sleep mode until the time that an updated measurement is requested by the controller. (See [22,23,40] and the references therein.) The event triggering-based control strategies have been discussed within the context of large-scale systems [31]. Without any discussion on the Zeno behavior, i.e., exclusion of infinite triggering events over a finite time, Stocker and Lunze [50] relied on the measured physical interconnection signals and proposed an event-triggered control approach for completely known linear model of large-scale systems. Moreover, finding the approximated model at each subsystem’s level could be a tedious task. From a viewpoint similar to that of the previous reference, Demir and Lunze [15] designed a non-event-triggered control schemes for both completely known MASs and large-scale systems through a transformationbased decomposition approach; however, the event-triggered result was limited to the physically decoupled MASs. With an emphasis on the limited energy of wireless networks, this topic is closely related to the networked control systems that has been widely discussed in the literature. For example, the pioneering work [56] proved that an interconnected LTI system could be stabilized based on a state-dependent event-triggered broadcast of information using detailed knowledge about the underlying interconnection topology and the associated interconnection sub-matrices. This methodology necessarily ends with the same communication and interconnection topologies satisfying a special condition to ensure perfect decoupling of subsystems (using an appropriately designed distributed controller). Based on a mixed constant and time-dependent threshold function in the triggering mechanism and for a readily designed control gain (e.g., using the ideas of [56]), a level of tolerable imperfect decoupling was obtained in [21] based on the concept of diagonal dominance in the literature of multi-input multi-output multivariable systems. Ref. [57] proposed an analysis strategy for event triggered control of interconnected nonlinear systems whose completely known LTI counterpart, without perfect decoupling assumption, resulted in a local synthesis scenario with high-dimension linear matrix inequality dependent on detailed knowledge about the underlying interconnection topology as well as its size. Additionally, as pointed out in [57], Zeno-freeness could be lost in a vicinity of the origin. In a different study, De Persis et al. [16] designed a Zeno-free event triggering mechanism and analyzed the resulting event-triggered closed-loop interconnected nonlinear systems where the impact of each subsystem’s state variable on its neighbors was characterized in terms of input-to-state stability (e.g., see [14]). Similarly, Liu and Jiang [29] used the concept of input-to-state stability for designing an event triggering mechanism and analyzing the closed-loop system stability. However, as pointed in [16] (Remark 1), generalizing these analyses ideas to the controller design procedures is not a straightforward task. Focusing back on graph theoretic ideas, part of the eventtriggered approaches (for single dynamical systems) has already been generalized to ensure consensus in MASs with limited energy resources. In this regard, MASs of single or double integrators are studied in order to understand the effect of event-driven data broadcast of agents’ information in their neighborhoods on the consensus as well as Zeno-free behavior of the closed-loop MAS (e.g., see [17,26,48,60,61]). Recently, event-triggered consensus in MASs with LTI dynamics [2,20] and LTI MASs with structured interconnected linear modeling uncertainties [10] as well as physically decoupled LTI agents subject to nonlinear modeling uncertainties [54] have also received attention. While the former linear scenarios were addressed using linear distributed algorithms, the effect of nonlinearities in the latter reference was handled by an adaptive nonlinear distributed protocol. In any case, a particular attention must be paid to the Zeno-freeness of the resulting closed-loop MAS which in some instances, e.g., using pure

V. Rezaei and M. Stefanovic / European Journal of Control 48 (2019) 9–20

state-dependent triggering mechanisms, might be problematic for any graph-theoretic methods. For further details on this issue, we refer to [20] and [51] where this problem has been respectively addressed by adding a constant bias or time-dependent term to the triggering threshold function. In this paper, we consider a class of Lur’e nonlinear large-scale systems with local LTI subsystems subject to partially-known timevarying interconnected nonlinearities which model the physical couplings among subsystems. We focus on a matched scenario in which the nonlinearities are in the range space of the control input matrix. We assume each subsystem is equipped with local sensing, computation, and communication devices and propose the cooperative stabilization problem for interconnected MASs (i.e., largescale systems) in the presence of unknown interconnection. From a cyber-physical system viewpoint, we use graph theoretic tools and propose a two-layer structure with agent and control layers. In this setup, agents are physically coupled to each other over an agent layer with unknown interconnection topology whose effect appears through some unknown nonlinearities in the state space models and, over a separate control layer communication graph, controllers exchange information in their neighborhoods to ensure robust global exponential convergence of all trajectories to the origin. In this two-layer approach, the control layer topology is chosen independent from the agent layer such that we are able to systematically develop an appropriate control gain for all cooperative, decentralized, and centralized configurations using a single design procedure. In particular, we propose a fictitious robust control problem for the completely known linear part of the interconnected dynamics, and develop a fixed- and low-dimension formulation to design the required static feedback gain to be used in a linear control protocol. We further add an event triggering mechanism to each controller in order to non-periodically take samples of each agent’s state variable and non-synchronously transmit agents’ information in their neighborhoods over the control layer and save their limited (energy) resources. We use fundamental concepts of optimal control theory to prove the robustness of the two-layer MAS with respect to unknown time-varying nonlinear interconnections and analyze the Zeno-free behavior of the proposed event-triggered mechanism. Adding a bias term to the triggering mechanism, we prove a similar result is achievable by making a trade-off between the number of triggering events and practical stability, i.e., exponential convergence to a vicinity of origin. We further discuss robust performance by finding a worstcase bound on the exponential convergence rate and guaranteed quadratic upper-bound based on a linear quadratic regulatory optimal control formulation. The rest of this paper is organized as follows. We first introduce the mathematical notation and preliminary concepts in Section 2. Then, in Section 3, we formulate the problem, propose the main contributions of this paper on event-triggered robust stabilization of two-layer MASs, and discuss the resulting guaranteed performance. In Section 4, we validate the proposed ideas through simulation studies. Finally, we summarize this paper in Section 5.

2. Notation and preliminaries We use standard notation in this paper. In particular, R denotes the set of real-valued scalars, R+ non-negative real-valued scalars, Rm×n real-valued m × n matrices, Z+ the set of positive integers, 0 a matrix of all zeros with appropriate dimension, 1N an N × 1 vector of all ones, diag{.} a (block) diagonal matrix of the elements in {.}, col{xi } an aggregated vector of xi for all i, and A  0 () a positive (semi-) definite matrix A ∈ Rn×n . The symbols λmin (A ) ∈ R and λmax (A ) ∈ R stand for the minimum and maximum eigenvalues of A. The symbol |N | denotes the cardinality of the set N , X the

2-norm of vector X ∈ Rn or induced 2-norm X ∈ Rn×n .

11



λmax (X T X ) of matrix

Fact 1. For any vectors x, y ∈ Rn and scalar a ∈ R+ , the inequality −2xT y ≤ axT x + 1a yT y holds. We use a graph G to visualize agent layer physical interconnection or control layer communication topologies. G is made by a node set V and an edge set E , and can be represented by an adjacency matrix A = [ai j ] ∈ RN×N where aij = 0 indicates the existence of an edge from node j to node i with aij as the weight, i.e., (ν j , νi ) ∈ E for i, j ∈ V = {1, 2, . . . , N} where N denotes the number of agents or control nodes. We use sub- or super-scripts a for agent layer and c for control layer. We point out that, while the non-zero edge weights over the control layer must be nonnegative scalars, i.e., aci j ≥ 0 in Ac = [aci j ], we do not impose such a restriction on the agent layer and accept both positive and negative scalars, i.e., aai j ∈ R in Aa = [aai j ]. For the agent layer graph, unlike the conventional definition of adjacency matrix [33], self loops are lumped into Aa by admitting non-zero aii for i ∈ {1, 2, . . . , N}. These two modifications provide a sufficient flexibility for the modeling of large-scale systems as agent layer dynamics [31]. For the control layer, we introduce a conventional graph Laplacian ma trix Lc = [licj ] ∈ RN×N by licj = −aci j and liic = Nj=1 aci j , and a diagonal matrix Bc = diag{bc } where bc = col {bci }, bci > 0 if there is a self loop around the ith node, bci = 0 otherwise. Now, the effect of self loops are lumped in to a modified Laplacian matrix Hc = Lc + Bc ∈ RN×N . Both agent and control layer graphs can be disconnected. However, we assume that at least one node in each connected component of Gc has a self loop such that, considering the symmetry of Lc for undirected control layer graphs, all eigenvalues of Hc are real-valued positive scalars. This assumption is required inspired by the discussion on the completeness of information flow in [19,55] (see Section 1), and the positiveness of all eigenvalues can be proved by following the steps of Lemma 4 in [36]. The strategy proposed in Section 3 treats the control layer communication graph Gc as a design degree of freedom, thereby, it is a completely known graph. However, the induced 2-norm Aa  is the only required information about the physical interconnection topology. As a result, the agent layer owner can prevent sharing detailed knowledge about the underlying system topology to the control layer designer.

3. Event-triggered robust cooperative stabilization In this section, we explain the main contributions of this paper. In Section 3.1, we formulate the graph theoretic cooperative stabilization problem for MASs with unknown time-varying nonlinearly interconnected agent layer dynamics. In Section 3.2, we first prove that an appropriately designed static linear cooperative algorithm with aperiodic and non-synchronous information broadcasting events will exponentially steer all trajectories of the two-layer MAS to the origin. We then propose a practical approach to improve the robustness of event triggering mechanism with respect to modeling mismatch such as additive measurement noise while guaranteeing boundedness of all trajectories. We also prove these event triggering mechanisms are Zeno free to show the feasibility of the proposed strategies by excluding he possibility of continuous broadcast of information for all agents. Additionally, in Section 3.3, we analyze the performance of two-layer (closed-loop) multiagent system in terms of exponential convergence rate and guaranteed quadratic cost.

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3.1. Problem statement We consider an MAS of N Lur’e agents:

x˙ i (t ) = Axi (t ) + Bui (t ) + B fi (zi (t ), t )  zi (t ) = Czi aai j x j

(1)

j∈Ni a

with matched nonlinearities which are in the range space of the control input matrix, i ∈ {1, 2, . . . , N} denotes the agent number, xi ∈ Rnx the ith agent’s state variable, ui ∈ Rnu control input, and zi ∈ Rnz coupling variable. Also, A ∈ Rnx ×nx and B ∈ Rnx ×nu are the known system matrices such that the pair (A, B) realizes a stabilizable state-space model. The unknown nonlinear functions fi : Rnz × R+ → Rnu are piecewise continuous in time and Lipschitz on Rnz , and satisfy fi (0, t ) = 0. These conditions ensure the existence and uniqueness of solutions to the above nonlinear differential equations for i ∈ {1, 2, . . . , N} ([24], Section 3.1); however, we consider a scenario under which the Lipschitz constant is unknown to the control layer designer. Instead of this global information that is satisfied for any two points in the state space, we only require the knowledge about γ fi ≥ 0 which satisfies a (weaker) point-wise condition fi (zi (t), t)2 ≤ γ fi zi 2 for each agent. Note that, while the Lipschitz constant for fi satisfies this point-wise norm condition, γ fi does not necessarily satisfy the Lipschitz inequality for all pairs of points in the state space. Therefore, developing a method that uses point-wise norm condition can be beneficial in the sense that it can be less conservative than a Lipschitz-based approach. We also need to mention that if these conditions are satisfied over a local domain that includes the origin, the results of this paper will be valid only over the domain of attraction. The unknown coupling matrices Czi ∈ Rnz ×nx satisfy Czi 2 ≤ γ ci with known real-valued scalars γ ci ≥ 0. We define γ f max i {γ fi } and γ c max i {γ ci } which are two known real-valued scalars. An index set Ni a determines the ith agent’s physical neighbors over Ga , i.e., those agents whose dynamics have direct effect on the ith agent’s dynamics. We consider a situation in which the matrix A can be nonHurwitz and the agent layer interconnection is described by an unknown graph topology Ga with self loops and both positive and negative interconnection edge weights. The control objective is ensuring global exponential stabilization of the interconnected MAS of agents (1):

lim xi (t ) = 0

t→∞

∀i ∈ {1 , 2 , . . . , N }

(2)

using the non-periodic and non-synchronous broadcast of information in agents’ neighborhoods over the control layer communication topology Gc . This control layer topology might be disconnected; however, each of its connected components has at least one self loop to ensure all real-valued eigenvalues μci of the modified Laplacian matrix Hc are positive scalars to be sorted as 0 < μc1 ≤ μc2 ≤ · · · ≤ μcN . 3.2. Main results In this subsection, we design a Zeno-free event-triggered cooperative controller that ensures the objective (2) is exponentially achieved for a class of MASs with unstable interconnected timevarying nonlinear agent layer dynamics (1). We further improve the robustness of the triggering mechanism with respect to the additive measurement noise by proposing a Zeno-free practical (modified) event triggering strategy. We are interested in designing two-layer MASs with nonperiodically and non-synchronously scheduled communication in agents’ neighborhoods over the control layer graph Gc . As will be seen, this two-layer structure provides flexibility to determine the

control configuration while relaxing the need for detailed knowledge about the agent layer interconnection topology for the control design purpose. To this end, we equip each agent with |Ni c | + 1 buffers (memory spaces) in which Ni c denotes the ith agent’s neighboring set over Gc (different from the physical neighboring set Ni a in (1)). In particular, the first memory space can be explained as follows:

xˆi (t ) = xi (tki ),

t ∈ [tki , tki +1 )

(3)

{tki }

t0i

where = 0 and denotes the ith agent’s triggering time sequence for k ∈ Z+ , and the other |Ni c | buffers are described by

xˆij (t ) = x j (tkj ),

t ∈ [tkj , tkj+1 )

j ∈ Ni c

and

(4)

in which t0 = 0 and {tk } represents the jth agent’s triggering time sequence. We implement the following linear cooperative algorithm: j



ui (t ) = K

j





(xˆi (t ) − (t )) + xˆij

bci xˆi

(t )

(5)

j∈Ni c

where K ∈ Rnu ×nx denotes the control gain matrix to be designed. While we focus on a cooperative stabilization scenario where the control layer operator directly uses the absolute values of only a few agents’ sampled measurements for the control purpose in (5), we notice that the proposed formulation further covers decentralized and centralized control configurations as two special cases: when Ni c = ∅ and bci > 0 for all i ∈ {1, 2, . . . , N}, (5) models a decentralized control scenario and, when Ni c = {1, 2, . . . , N}\{i} for all i and all bci > 0, (5) models a centralized controller with allto-all communication. In any case, although the proposed implementation scenario makes the agent-level (local) processing more complicated compared to the equivalent cooperative algorithm with continuous monitoring, the overall energy consumption might be significantly decreased since it enables the agents to sample and broadcast information in a non-periodic and non-synchronous manner at the required time for the stability. For example, Ploennigs et al. [40] showed the non-periodic sampling consumes up to 80% less energy compared to the periodic approaches for a building automation case study, and Willig [58] discussed that processing systems consume less energy than the data transmission network. We focus on a delay-free communication network as is the case in the literature of MASs [2], e.g., assuming the agent layer dynamics are slow compared to the data transmission rate over Gc , and rewrite the cooperative algorithm (5) as follows to be used for the analysis purpose in the rest of this paper:



ui (t ) = K





(xˆi (t ) − xˆ j (t )) + bci xˆi (t )

(6)

j∈Ni c

We define the local state update errors i  xˆi − xi where the time variable t is dropped for the sake of readability and space issues, and rewrite (6) as follows:



ui = K



( xi − x j ) +

j∈Ni c

bci xi

+





(i −  j ) + 

bci i

j∈Ni c

which results in the following closed-loop agent dynamics:





x˙ i = Axi + BK

 + BK



(xi − x j ) + bci xi

j∈Ni c

 j∈Ni c



(i −  j ) + 

bci i

+ B f i ( zi , t )

(7)

V. Rezaei and M. Stefanovic / European Journal of Control 48 (2019) 9–20

13

Now, the aggregated closed-loop interconnected MAS is modeled by

gain that solves the fictitious robust control problem (9) for the nominal dynamics in (8).

x˙ = A¯ x + (Hc  BK )x + (Hc  BK ) + (IN  B ) f (z )

min







Nominal dynamics





Triggering effect





v i ∈Uvi

Physical coupling

subject to

z = Cz A¯a x

(8)

where A¯ = IN  A, Cz = diag{Czi }, A¯a = Aa  Inx , x = col {xi },  = col {i }, f (z ) = col { fi (zi )}, z = col {zi }, and i ∈ {1, 2, . . . , N}. Based on the nominal dynamics in (8) for a trigger-free scenario, a coupled control signal ut f = (Hc  K )x must be designed to stabilize the interconnected MAS dynamics using (Hc  K ) ∈ RNnu ×Nnx (note that Hc is also a design degree of freedom). Although this is solvable for an MAS with a few agents, the computational complexity may significantly increase in MASs with a high number of agents. Based on the literature of distributed control in physically decoupled MASs (i.e., fi = 0 in (1)), this dimension issue could be addressed through some two-step design approaches. In the first step, a control gain K could be designed to stabilize the local agent dynamics (i.e., to have a Hurwitz matrix A + BK in (1) with fi = 0). Then, in the second step, a correction scalar c could be obtained based on the knowledge about the control layer communication topology Gc in order to either (a) re-scale the control gain of the first step via c > 0 and implement the rescaled control gain cK for all agents [59], or (b) correct the relative measurements to be fed back into the distributed control unit [27]. We use Hc to determine the control configuration as discussed after (5), decompose the coupled triggering-free control signal as ut f = (Hc  Inu )v and v = K¯ x = (IN  K )x, and rewrite the nominal dynamics of (8) as follows:

x˙ = A¯ x + B¯ v







+

Networked nominal model

B¯ H¯ c v

(9)



Fictitious uncertainty

such that the stabilizing candidate gain K of (8) can be designed in one step by finding a completely decoupled (diagonal) control signal v for the fictitious robust control problem in (9) (see [43]). c Here, B¯ = IN  μc1 B, H¯ c = ( H μc1 − IN )  Inu  0, and μc1 > 0 is the smallest eigenvalue of Hc . The word “fictitious” refers to the fact that the control layer communication topology Gc is completely known, but we will not use it in Design procedure 1 to be able to design a single gain K for cooperative as well as decentralized and centralized configurations in a unified manner. Also, the word “networked” is used to distinguish the networked nominal model (9) (under the effect of Gc or μc1 in B¯ ) from the aggregated nominal dynamics x˙ = A¯ x + (IN  B )u that could be found based on (1). Now, we use the networked nominal model in (9) and develop a systematic procedure to design the aggregated decoupled virtual control signal v = K¯ x ensuring exponential stability of (9) subject to the fictitious modeling uncertainty B¯ H¯ c v. Let Q ∈ Rnx ×nx and R ∈ Rnu ×nu be two positive definite design matrices. We introduce R¯ f = IN  R f , and R f = γ f γc λmax (R )Aa 2 Inx for which the following inequality holds:

f T (z )R¯ f (z ) ≤ xT R¯ f x where R¯ = IN  R, Q f = Q + R f μ  0 where R f μ =

1

μ2c1

R f , and P ∈

Rnx ×nx be the solution of the following algebraic Riccati equation (ARE):

AT P + PA + Q f − μ2c1 P BR−1 BT P = 0

(10)

and, furthermore, define Uvi as the set of all admissible (static state feedback) stabilizing controllers for the ith networked nominal dynamics in (9). Design procedure 1. Let vi = Kxi minimize the modified LQR problem (11) for i ∈ {1, 2, . . . , N}. Then, K = −μc1 R−1 BT P is a candidate

J (xi (0 )) =



∞

t=0

(xTi (t )Q f xi (t ) + vTi (t )Rvi (t ))dt

x˙ i = Axi + μc1 Bvi

(11)

The proof of last statement in this design procedure can be induced from the proof of Theorem 1, and is omitted for brevity. We, however, need to mention that the “modified” LQR refers to two facts: (a) we necessarily should use a modified state weighting matrix Qf in the minimization problem, although we still can arbitrarily choose the weighting matrices Q and R; and (b) the optimal control problem is solved subject to the (local) networked nominal dynamics which should be regarded as the modification of agents’ nominal dynamics in (1). Note that the pairs (A, μc1 B) and (Qf , R) are the same for all agents i ∈ {1, 2, . . . , N} such that the resulting stabilization gain K is also the same for all agents; thus, we need to solve the Design procedure 1 only for once. Also, since the state space realization (A, B) is stabilizable and μc1 is a real-valued positive scalar, the ARE (10) with the pair (A, μc1 B) has a unique positive definite solution P if the pair (Q 1f /2 , A ) is observable (which can be ensured by design). Such an optimal control theoretic formulation can be further modified to design the control layer topology Gc , i.e., the neighboring sets Ni c and communication weights aci j , in the cooperative controller (5). In that case, the control layer topology can be appropriately designed to not only determine the control configuration, but also address stability and performance issues. An extension of [45] for an MAS with highdimension agents (1) is left as future work. Note that the solution of ARE (10) can be found by reformulating that ARE as a single matrix inequality whose dimension is fixed and depends only on the order of local agents (1) (e.g., see [5]). Then, in this sense, the Design procedure 1 computationally outperforms the proposed approach by Cheng and Ugrinovskii [10] where N linear matrix inequalities should be simultaneously solved in which the dimension of each matrix depends on the number of agents as well as the order of local agent’s dynamics (see Theorem 1 in [10] for MASs with structured linear interconnection). Furthermore, we point out that the Design procedure 1 results in an MAS of networked nominal dynamics which satisfies the following fundamental property of optimal systems. Property 1. Based on the fundamental properties of optimal control systems, the following equalities hold for the closed-loop networked nominal model in (9) with v = K¯ x found in Design procedure 1:

xT Q¯ f x + vT R¯ v + J¯xT (A¯ x + B¯ v ) = 0 and 2vT R¯ + J¯xT B¯ = 0 ¯ where K¯ = IN  K, Q¯ f = IN  Q f , R¯ = IN  R, J¯x = ∂∂xJ , and J¯(x ) = N i=1 J (xi ).

The above MAS-level property is obtained by an aggregation of local optimality conditions that could be immediately found based on the modified LQR problem in Design procedure 1 for i from 1 to N ([25], Section 3.11), and noting the fact that the networked nominal model in (9) is completely diagonal (decoupled) and the optimal solution in Design procedure 1 can be written as J (xi ) = xTi P xi . Now that a candidate control gain K is designed for the cooperative algorithm (5), we propose an event triggering mechanism to determine the information broadcast time sequence t0i = 0 and {tki } of the ith agent for k ∈ Z+ :

tki +1 = inf{t > tki

| i (t )2 ≥ κ (t )}

(12)

which is a non-periodic agent-level and non-synchronous MASlevel scheduler. Also, the threshold function κ (t ) = κ1 e−σ t is specified by two real-valued design scalars κ 1 > 0 and σ ∈ (0, ρ V ) where

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ρV =

μ2cN 2 1 T λmax (P ) (λmin (Q ) − a1 μ2 λmax (K RK )), c1

a1 ∈ ( 0,

μ2c1 λmin (Q ) ) μ2cN λ2max (K T RK )

is a real-valued design scalar to ensure ρ V > 0, and μcN > 0 is the largest eigenvalue of the modified Laplacian matrix Hc associated to the control layer communication graph Gc . The Zeno phenomenon is ruled out from this event triggering mechanism if an infinite number of triggering events does not occur in a finite time, i.e., tk → ∞ as k → ∞ [51]. It means that none of the agents will continuously update and broadcast their information. In the next theorem, we prove that through the use of buffers (3) and (4), the cooperative stabilizing algorithm (5) over the control layer communication graph Gc together with the event triggering mechanism (12) stabilize the origin of MASs with interconnected time-varying nonlinear agent layer dynamics (1). Theorem 1. Let the control gain be found using Design procedure 1. In the partially-unknown interconnected nonlinear MAS of agents (1), the cooperative linear algorithm (5) equipped with buffers (3) and (4) over the control layer communication topology Gc ensures robust exponential convergence of all trajectories to the origin with no Zeno behavior if the agents update and broadcast their measurements using the event triggering mechanism (12). Proof. This proof is derived in two steps for the stability of origin and Zeno-free behavior of the closed-loop interconnected MAS with stabilization algorithm (5) and event triggering mechanism (12). In the first step, to prove robust exponential convergence of all trajectories to the origin, we introduce the following candidate Lyapunov function:

V¯ (x ) = xT P¯ x  0 where P¯ = IN  P . We know the optimal cost of the modified LQR problem (11) can be written as J (xi (0 )) = xTi (0 )P xi (0 ). Since V¯ (x(t )) = J¯(x(t )) for J¯(x(t )) of Property 1, we also know the pair (x, v = K¯ x ) satisfies the equalities of Property 1 by replacing J¯ by V¯ . Now, the time derivative of V¯ along the unknown trajectories of (8) can be upper-bounded as follows:



1 V¯˙ = V¯xT x˙ = V¯xT A¯ x + B¯ v + B¯ H¯ c v + B¯ (Hc  K ) + B¯ fμ

μc1

= −xT Q¯ f x − vT R¯ v − 2vT R¯ H¯ c v −

2

2

μc1

c

μc1

vT R¯ (Hc  K ) − 2vT R¯ fμ

Based on Fact 1, we have:



Hc2

μ2c1

Since ρ V > σ > 0 is satisfied by design, this latter inequality on state variables allows us to conclude exponential convergence of all (unknown) trajectories of the closed-loop interconnected MAS (1), (3)–(5), and (12) to the origin as t → ∞. In the second step, we prove the Zeno-free behavior of the twolayer MAS by finding a lower bound on the time-interval between any two successive triggering events. Based on (7) and the definition i = xˆi − xi , the triggering errors evolve according to:





˙ i = −Axi − BK



( xi − x j ) +

j∈Ni c





−BK

bci xi



(i −  j ) + 

bci i

− B fi

(16)

j∈Ni c

with the initial conditions i (tki ) = 0 on each interval t ∈ [tki , tki +1 ) for i ∈ {1, 2, . . . , N}. Taking the norm of both sides, we find:

˙ i  ≤ Axi  + Hc BK x + Hc BK  + B f  (17) Based on (15), we already know the norm of state trajectories is bounded as follows:



x(t )2 ≤

 κ1 N λmax (P ) x ( 0 )2 + e−σ t =: b1 e−σ t λmin (P ) a1 λmin (P )(ρV − σ ) (18)

due to 0 < σ < ρ V that is satisfied by design. We also know 1 √ i (t ) ≤ κ1 e− 2 σ t always holds using the event triggering mech-



anism (12), and conclude  (t ) ≤ κ1 Ne− 2 σ t . Thus, the norm inequality (17) is rewritten as follows: 1







 κ1 N )Hc BK  + b1 γ f γc Aa  ˙i  B is a strictly positive scalar. Additionally, since ˙ i  = i  i  ≥  T 2i ˙ i d = dt ( iT i ) = dtd (i  ), we find: 2  where

b2 =

b1 A + (

b1 +

d 1 (i (t ) ) ≤ b2 e− 2 σ t dt



 (K T RK )2 x +

1 T   a1

(13)

2

which, since ≤ Nκ (t) as a consequence of event triggered mechanism (12), is further rewritten as follows:

N κ (t ) V¯˙ ≤ −ρV V + a1

(14)

Relying on the comparison Lemma [24], the time response V¯ (t ) satisfies the following inequality:

V¯ (t ) ≤ e−ρV t V¯ (0 ) +

(15)

i

 K T RK 

where fμ = μ1 f is a scaled aggregated coupling function. Note c1 that the inequality is obtained using −xT P¯ B¯ (Hc  R−1 )B¯ T P¯ x¯  0 T ¯ T ¯ c because H¯ c = Hc  Inu = ( H μc1 − IN )  Inu  0, and x R f μ x ≥ f μ R f μ where R¯ f μ = IN  R f μ and Rfμ is defined in Design procedure 1.

V¯˙ ≤ −xT Q¯ x + a1 xT

λmax (P ) x(0 )2 e−ρV t λmin (P ) κ1 N + (e−σ t − e−ρV t ) a1 λmin (P )(ρV − σ )

1

vT R¯ (Hc  K ) H

≤ −xT Q¯ x − 2xT

x(t )2 ≤

˙ i (t ) ≤ b2 e− 2 σ t

μc1 ≤ −xT Q¯ x − (v + fμ )T R¯ (v + fμ ) − (xT R f μ x − fμ R¯ fμ ) −

which, using Rayleigh–Ritz inequality for V¯ (t ) = xT P¯ x, results in

κ1 N (e−σ t − e−ρV t ) a1 (ρV − σ )

whose time response over any interval [tki , tki +1 ) is bounded as follows (based on the comparison Lemma [24]):

i (t ) ≤

2b2

σ

( e− 2 σ tk − e− 2 σ t ) 1

1

i

Based on the event triggering mechanism (12), i (tki +1 ) ≥ 1 √ κ1 e− 2 σ tk+1 must be satisfied to trigger the (next) event at time tki +1 . Therefore, we find:

√

σ κ1 2b2

e− 2 σ tk+1 ≤ e− 2 σ tk − e− 2 σ tk+1 1

i

1

i

1

i

which results in the following inequality on any two successive event triggering times:

tki +1 − tki ≥

2

σ

ln(1 +

√

σ κ1 2b2

)>0

(19)

This strictly positive lower-bound guarantees the Zeno-free behavior of the closed-loop interconnected MAS with event-triggering mechanism (12). 

V. Rezaei and M. Stefanovic / European Journal of Control 48 (2019) 9–20

In Theorem 1, we proved the exponential stability of the origin for MASs of agents (1) with partially-known interconnected nonlinearities. In the literature of MASs, it is shown that the time length between the successive triggering events can be better controlled or the Zeno behavior can be excluded by adding a constant bias term κ 0 > 0 to the threshold function of the event triggering mechanism:

tki +1 = inf{t > tki : i 2 ≥ κ (t ) + κ0 },

t0i = 0.

(20)

For example, see [2,21,48] and, in particular, [20] where the authors used such a constant term to ensure Zeno freeness. Regarding our objective in (2), instead of exponential stability, this modification results in the convergence of all trajectories to a neighborhood around the origin:

lim xi (t ) ≤ b p

∀i ∈ {1 , 2 , . . . , N }

t→∞

(21)

which is conceptually close to the practical stability or boundedness of all trajectories. This is well-known in the literature of adaptive control in order to, e.g., better deal with the chattering that may arise due to unknown model of dynamical systems [6]. Chattering in adaptive control, in its spirit, is similar to Zeno behavior which has already been excluded from the two-layer MAS of this paper in the presence of partially-known nonlinear interconnection. However, inspired by this concept, the practical stability enables us to benefit from a triggering mechanism with fewer events in the presence of additive measurement noise mismatch in addition to the interconnected time-varying nonlinear modeling uncertainties. In particular, such a modification might be interesting for interconnected MASs where each agent adds a bit of noise to the measurement in order to preserve its privacy (e.g., see the discussion on differential privacy in [11]). In the next corollary, we prove boundedness of all trajectories and Zeno-free behavior for a closed-loop MAS with agent layer dynamics (1), buffers (3) and (4), cooperative stabilizing algorithm (5), and practical event triggering mechanism (20). (Variables and parameters are defined similar to those of Theorem 1.) Corollary 1. (Practical stability) Based on the same setup as in Theorem 1, Zeno-free robust cooperative practical stabilization (21) is guaranteed for the closed-loop interconnected MAS if all agents update and broadcast their measurements according to the practical event triggering mechanism (20). Proof. To prove practical stability, we start from (13) and find:

N N κ1 e−σ t + κ0 V¯˙ ≤ −ρV V¯ + a1 a1





(22)



The same as (14)

Consequently,

V¯ (t ) ≤ e−ρV t V¯ (0 ) +

κ1 N κN (e−σ t − e−ρV t ) + 0 (1 − e−ρV t ) a1 (ρV − σ ) a1 ρV

which, based on the Rayleigh–Ritz inequality, results in:

x(t )2 ≤

λmax (P ) x(0 )2 e−ρV t λmin (P ) κ1 N + (e−σ t − e−ρV t ) a1 λmin (P )(ρV − σ ) κ0 N + (1 − e−ρV t ) a1 ρV λmin (P )

(23)

or

x(t )2 ≤ b1 e−σ t +

κ0 N κ0 N ≤ b1 + =: b3 (24) a1 ρV λmin (P ) a1 ρV λmin (P )

where b1 is defined in the proof of Theorem 1 (see (18)). Now, based on (17), we find:

d (i  ) ≤ ˙ i  ≤ b4 dt

b4 =

where





15

b3 (A + Hc BK  +



γ f γc Aa B ) +

(κ0 + κ1 )NHc BK  is a strictly positive scalar. Hence, over each interval [tki , t ), we find i  ≤ b4 (t − tki ) that results in √

tki +1 − tki ≥

κ0

b4

>0

(25)

√ because i (tki +1 ) ≥ κ0 holds at the (next) triggering time tki +1 based on the practical event triggering mechanism (20). Additionally, for thisZeno-free two-layer MAS, the left inequality in (24) gives b p = (21).



κ0 N a1 ρV λmin (P )

as a conservative practical bound in

We need to point out that we have used an identical triggering mechanism (12) or (20) for all agents in order to simplify the derivations of this paper by avoiding an excessive number of new parameters and variables. However, note that all cooperative stability developments will remain valid for the case of non-identical triggering mechanisms where each agent has its own κ 0i , κ 1i , σ i > 0 for i ∈ {1, 2, . . . , N}. In that scenario, the stability statements in the proof of Theorem 1 and Corollary 1 must be written based on κ (t) ← κ max (t) to be obtained using κ0 ← κ0max = maxi {κ0i }, κ1 ← κ1max = maxi {κ1i }, and σ ← σimin = mini {σi }. While the strictly positive lower-bound (19) in the proof of Theorem 1 is limited to identical σi = σ ∈ (0, ρV ), the Zeno-freeness in Corollary 1 will remain valid for non-identical σ i if we use κ0 ← κ0min = mini {κ0i } in (25). 3.3. Discussion In the previous subsection, we proposed a graph theoretic formulation to design a cooperative stabilization algorithm which could further cover both centralized and decentralized configurations as special cases. We proved that either exponential or practical stability is achievable when the control gain is designed using the proposed formulation. In this subsection, we also establish a relationship between the guaranteed performance of the closedloop interconnected system and the proposed event-triggered information broadcast idea. Two different criteria have been widely discussed for graph theoretic distributed methods in the literature of MASs: maximum convergence rate and minimum quadratic cost. Regarding the former criterion, Refs. [41] and [13] numerically optimized the graph topology to achieve the fastest convergence to the agreement value in MASs of single integrators, and Ogiwara et al. [37] addressed the same issue by proposing various scenarios for the number of nodes and edges in MASs of single integrators. Regarding the latter criterion, we first need to note that the local optimality refers to the case with a block diagonal state weighting matrix and the global optimality points to the state weighting matrices with nonzero off-diagonal terms. From this viewpoint, for a special selection of weighting matrices in the linear quadratic regulatory cost formulation, [7] optimized the global consensus performance using an all-to-all communication in MASs of single or double integrators. Assuming a direct leader-to-follower communication for all agents, [32,52] optimized the sum of decoupled local (agent-level) cost functions in MASs of single and double integrators. Moreover, based on the inverse-optimal control formulation, Movric and Lewis [34] designed a cooperative tracking algorithm which could optimize the global linear quadratic regulatory cost function. Nevertheless, all of these results are limited to MASs with completely known physically decoupled dynamics (i.e., a group of individual agents). We cannot prove the “optimality” of ideas in Section 3.2 due to the presence of modeling uncertainty in (1). Instead, in the rest of this subsection, we analyze the robust performance of the proposed closed-loop interconnected MAS in terms

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V. Rezaei and M. Stefanovic / European Journal of Control 48 (2019) 9–20

of guaranteed exponential convergence rate and upper-bound on the given global quadratic cost function. For the convergence rate analysis, according to the upper-bound (18), we need to note  thatσ the event triggering mechanism (12) results in x(t ) ≤ b1 e− 2 t as the worst-case upper-bound on all trajectories or, equivalently, the robust exponential convergence rate 12 σ for all agents in the two-layer MAS. Based on the first inequality in (24), a similar discussion holds for the convergence of all trajectories to the bounded neighborhood of origin whenever the practical rule (20) is used as the event triggering mechanism. This essentially means that, using event triggering mechanisms (12) and (20), the designer is able to manipulate the worst-case bound on the magnitude of all state trajectories and their convergence rate toward the origin or a neighborhood around it. For the guaranteed-cost analysis of the results in Section 3.2, let the cost function be specified by

J¯(x(0 )) =



∞

0

(xT Q¯ g x + uT R¯ g u )dt ≤



∞

0

(xT Q¯ x + uT R¯ u )dt

(26)

where Q¯ g ∈ RNnx ×Nnx and R¯ g ∈ RNnu ×Nnu are (symmetric) positive definite matrices with nonzero off diagonal terms to define a global cost function J¯, and Q¯ and R¯ are two block diagonal matrices satisfying Q¯ = IN  Q  Q¯ g and R¯ = IN  R  R¯ g . Also, u = (Hc  K )xˆ is defined using the aggregated MAS model (8), and note that the weighting matrices Q and R satisfy the conditions of Design procedure 1. Based on (13) in the proof of Theorem 1, we know the following inequality holds for the two-layer MAS with event triggering mechanism (12):





V¯˙ ≤ −xT Q − a1

Hc2

μ2c1



 (K T RK )2

x+

κ1 N a1

e −σ t

tinuous measurement-based approach corresponding to the nominal guaranteed cost. In this sense, the second term in the above guaranteed cost correspond to the 2-norm of deviations from the nominal trajectories. A similar discussion can be made for the practical event triggering mechanism (20) regarding the cost of convergence to a practical neighborhood around the origin, i.e., since limt→∞ x(t ) = 0 is not satisfied, we must consider a cost function in which, e.g., xT Q¯ x = 0 whenever the practical convergence is achieved in limit. Then, regarding the additional parameter κ 0 , we may follow the same intuition and notice that an increase in this parameter will increase the guaranteed cost. We also point out that increasing κ 0 and κ 1 and decreasing σ will, in general, end in reduced communication load which might be beneficial for saving energy. Therefore, while the above analyses can be used for the tuning of parameters and matrices in the modified LQR formulation (11) of Design procedure 1 and event triggering mechanism (12) or (20) in the presence of time-varying nonlinearly interconnected modeling uncertainties, we leave the (guaranteed) performance-oriented design problem as future work. In particular, we emphasize that a co-design of controller gain and triggering mechanism might be an interesting alternative solution in this regard (e.g., see [47]). 4. Simulation verification In this section, we validate the theoretical results of Section 3 through various simulation studies. We consider an MAS of seven agents (1) with oscillatory LTI dynamics:



0 A= −1

1 0



 

B=

and the following (unknown) time-varying nonlinearities

which, adding to both sides of this inequality, results 2 d in xT Q¯ x + uT R¯ u ≤ − dt (xT P¯x ) + a1 xT ( H2c  (K T RK )2 )x + uT R¯ u +

f1 (z1 , t ) = 0.7 cos(t )z1

κ1 N −σ t because V¯˙ = V¯xT x˙ and V¯x = 2xT P¯ . We now take an integral a1 e over t ∈ [0, ∞), and find

f5 (z5 ) = −0.5 sin(z5 )

uT R¯ u

μc1

J (x(0 )) ≤ xT (0 )P¯ x(0 ) + +

κ1 N a1





∞

0

 



xT Hc2 

a1

μ

2 c1



(K T RK )2 + K T RK

e−σ t dt

x (27)

noticing the fact that limt→∞ x(t ) = 0. Let us define b5 = μ2cN λmax ( a21 (K T RK )2 + K T RK ), Now, using the exponential converμc1

gence of state trajectories in (18), the above inequality is further rewritten as follows:



J (x(0 )) ≤ xT (0 )P¯ x(0 ) + b1 b5 + ≤ xT (0 )P¯ x(0 ) +

κ1 N a1

a1 b1 b5 + κ1 N a1 σ



∞

e−σ t dt

1 min

f2 (z2 ) = −0.4 sin(z2 )

f3 (z3 , t ) = 0.5 sin(t ) tanh(z3 )

f4 (z4 ) = −0.4 tanh(z4 )

f6 (z6 , t ) = 0.4 sin(t ) sin(z6 )

f 7 ( z7 ) = 0.6z7 in which the (unknown) coupling matrices are given by C1 = [0, 1.5], C2 = [0, 1], C3 = [0, 0.5], C4 = [0, 0.5], C5 = [0, 0.5], C6 = [0, 1], C7 = [0, 1.5]. The agent layer coupling graph’s modified adjacency matrix is given by:

⎡

2 ⎢−2 ⎢0 ⎢ Aa = ⎢ ⎢0 ⎢0 ⎣0 0

−1 0 −1 0 0 0 −1

0 0 0 0 −1 0 0

0 0 0 0 1 0 0

0 0 0 0 0 −1 0

0 0 0 0 0 0 0

⎤

−1 0⎥ 0⎥ ⎥ −1⎥ ⎥ 0⎥ ⎦ 0 2

0

where the first term on the right-hand inequality is a trigger-free guaranteed cost that can be obtained using continuous monitoring in agents’ neighborhoods, and the second term characterizes the effect of event-triggered broadcast of information. This indicates that the guaranteed cost of an event-triggered strategy is always higher than that of continuous monitoring. We now conclude that, for a fixed controller and number of agents, increasing κ 1 or decreasing σ will increase the guaranteed cost. This κ N is because b1 = λλmax((PP)) x(0 )2 + a λ (P1)(ρ −σ ) and the inequality min

0 1

V

0 < σ < ρ V holds by design. Intuitively, based on the event triggering mechanism (12), this finding is well-aligned with the fact that increasing κ 1 and decreasing σ will, in general, increase the interevent time length and also increase the deviation from the con-

which results in an interconnected agent layer dynamics with unstable behavior, unbounded trajectories, as depicted in Fig. 1. (The physical interconnection of agent layer dynamics is shown by black items in Fig. 2.) We consider two isolated areas for the control communication purpose due to, e.g., the geographical distance of agents in area 1 from those of area 2. In this configuration, agents i ∈ {1, 2, 3, 4} belong to the first group with Laplacian matrix Lc1 and agents i ∈ {5, 6, 7} are associated to the second group with Lc2 :

⎡

Lc1

1 ⎢−1 =⎣ 0 0

−1 2 −1 0

0 −1 2 −1

⎤

0 0⎥ −1⎦ 1

 Lc2 =

1 −1 0

−1 2 −1



0 −1 . 1

We further define Bc = diag{0, 1, 1, 0, 0, 1, 0} to represent self loops over the control layer graph, and find the modified Laplacian

V. Rezaei and M. Stefanovic / European Journal of Control 48 (2019) 9–20

17

Table 1 The max t xi (t) for t ∈ [20, 25] to approximately measure the effect of noise mismatch and event triggering mechanism on the closeness of trajectories to the origin.

Agent Agent Agent Agent Agent Agent Agent

1 2 3 4 5 6 7

Mechanism (12)

Mechanism (20)

Noise-free (10−8 )

Noisy (10−6 )

Noise-free (10−5 )

Noisy (10−5 )

2.5210 3.1744 3.3983 1.9486 1.9650 3.2299 1.5233

1.7783 2.0415 2.1268 1.7511 1.4338 1.6516 1.3660

3.7612 1.5533 1.0558 0.3668 0.7175 1.1586 1.3838

9.2514 1.9173 1.4253 9.8980 8.1596 1.3873 1.0026

Table 2 Total number of triggering events under different noise scenarios in addition to the agent layer modeling uncertainties.

Fig. 1. Diverging trajectories of the open-loop (control and communication free) interconnected MAS corresponding to the agent layer in Fig. 2.

Agent Agent Agent Agent Agent Agent Agent

1 2 3 4 5 6 7

Mechanism (12)

Mechanism (20)

Noise-free

Noisy

Noise-free

Noisy

87 116 124 55 62 114 87

296 413 446 298 258 360 234

58 95 72 35 49 92 82

70 124 100 60 56 139 87

Fig. 2. Two-layer MAS: (black) agent-layer dynamics with unknown time-varying interconnected nonlinearities, and (blue) control layer communication topology. Solid lines represent permanent (physical) connections and dashed lines stand for the non-periodic discontinuous communications which are scheduled by the eventtriggered mechanisms (12) or (20). Letters a and c, respectively, denote agent and controller which are connected by solid red lines to show they are co-located. Over agent layer, the two-way coupling between agents 1 and 2 indicates the potentially different coupling weights in each direction and, over the control layer, the twoway communications emphasize on the non-synchronous information broadcast by agents. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

matrix Hc = diag{Lc1 , Lc2 } + Bc . The control layer communication topology is shown by blue items in Fig. 2. We design the stabilization gain K of the cooperative algorithm (5) based on the Design procedure 1, and the discussion in Section 3.3, and investigate four simulation scenarios to validate the results of Section 3. Two simulations scenarios are proposed for each of the event triggering mechanisms (12) and (20) without and with additive measurement noise xnoise = xi + wi to be fed i into the control block (5) in which, except the white noise wi with power 10−8 and the bias term κ0 = 10−5 in (20), all other parameters are the same. In Table 1, we numerically compare the maximum of the 2norm of state variables over the time interval [20,25] to approximately measure the closeness of all trajectories to the origin as t → ∞ in (2) or (21). In Table 2, we report the number of trig-

Fig. 3. Two-layer MAS with event triggering mechanism (12), noise-free: Evolution of xi (t) in the first 15 s (we ignore the next 10 s for the sake of visibility). Top to bottom are associated with agents 1–7.

gering events under all simulation scenarios to understand the effectiveness of proposed ideas in the absence and presence of measurement noise. Using the original mechanism (12) or the practical one (20), the two-layer MAS is robustly stable with respect to the interconnected time-varying nonlinear modeling uncertainties in agent layer dynamics, However, as can be interpreted based on these tables, the practical event triggering mechanism (20) provides a degree of freedom so that, compared to the nonpractical method (12), the information broadcast scheduling is lesssensitive to the additive measurement noise mismatch.

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V. Rezaei and M. Stefanovic / European Journal of Control 48 (2019) 9–20

Fig. 4. Two-layer MAS with event triggering mechanism (12), noise-free: triggering time sequence for all agents i ∈ {1, 2, . . . , 7}. The Zeno-freeness can be induced from Table 2 too.

Fig. 6. Two-layer MAS with event triggering mechanism (12), noisy measurements: Triggering time sequence for all agents i ∈ {1, 2, . . . , 7}. The major effect of noise is on the number of triggering events in the last piece of simulation when the effect of noise is comparable to xi (t). A zoomed-in view is shown in Fig. 7.

Fig. 7. Zoomed-in view of the result in Fig. 6 to show the Zeno-free behavior of the non-practical event triggering mechanism (12) in the presence of measurement noise mismatch in the second simulation study (see Table 2): i ∈ {1, 2, . . . , 7} denotes the number of agents. Fig. 5. Two-layer MAS with event triggering mechanism (12), noisy measurements: Evolution of xi (t) in the first 15 s (we ignore the last 10 s for the sake of visibility). Top to bottom are associated with agents 1–7. As seen, the effect of noise is negligible in the transient part when xi  is relatively bigger than the magnitude of noise.

In the rest of this section, we provide a few pictorial descriptions for the aforementioned simulation scenarios. For the first numerical study, based on the event triggering mechanism (12) in the absence of noise, Fig. 3 shows the evolution xi  → 0 (as t → ∞) for all trajectories in the first 15 s and Fig. 4 depicts the triggering time sequence of all agents during the entire 25 s of simulation (see Table 2 for the total number of triggering events). For the second simulation, in the presence of additive white noise, Figs. 5 and 6 present the result of using event triggering mechanism (12) as in the previous numerical study. Based on these figures, we note that the major effect of measurement noise appears in the last 10 s of simulation where the norm of all trajectories are close to the origin

within a comparable range to the magnitude of noise. Fig. 7 provides a zoomed-in view of the triggering event distribution to validate the Zeno-free behavior of the closed-loop system (as was reported in Table 2). In the third simulation, we incorporate the practical event triggering mechanism (20) with a very small bias κ0 = 10−5 in the absence of noise. For the sake of comparison, we show the evolution of norms xi  in Fig. 8 and the triggering time sequences of agents in Fig. 9. Finally, in the fourth simulation, we investigate the benefits of using practical event triggering mechanism (20) in the presence of additive measurements noise. Fig. 10 shows the evolution of xi  and, in particular, Fig. 11 shows the number of triggering events which has been significantly decreased compared to that of Fig. 6 due to the bias term κ 0 in practical mechanism (20) (also see Table 2).

V. Rezaei and M. Stefanovic / European Journal of Control 48 (2019) 9–20

Fig. 8. Two-layer MAS with practical event triggering mechanism (20), noise-free: evolution of xi (t) in the first 15 s (we ignore the last 10 s for the sake of visibility). Top to bottom are associated with agents 1–7. As seen, the transient behavior is similar to the non-practical case in Fig. 3 because κ 0 is too small. We refer to Table 1 in order to see the effect of κ 0 on the norm of state variables at time t ≥ 20 s.

Fig. 9. Two-layer MAS with practical event triggering mechanism (20), noise-free: triggering time sequence for all agents i ∈ {1, 2, . . . , 7}. With a negligible increase in the worst-case value of xi  (reported in Table 1), the practical mechanism (20) decreases the number of triggering events in the last piece of simulation where xi  is sufficiently small (compared to Fig. 4).

5. Summary We consider the cooperative stabilization problem in multiagent systems with interconnected time-varying nonlinear modeling uncertainties. From a cyber-physical systems’ viewpoint, we propose a two-layer structure in which agents are physically coupled to each other over the agent layer and controllers cooperatively exchange information over the control layer. The proposed

19

Fig. 10. Two-layer MAS with practical event triggering mechanism (20), noisy measurements: evolution of xi (t) in the first 15 s (we ignore the last 15 s for the sake of visibility). Top to bottom are associated with agents 1–7.

Fig. 11. Two-layer MAS with practical event triggering mechanism (20), noisy measurements: triggering time sequence for all agents i ∈ {1, 2, . . . , 7}. With a negligible increase in the worst-case value of xi  (see Table 1), the practical mechanism (20) dramatically decreases the number of triggering events compared to the result of Fig. 6.

graph theoretic formulation enables us to choose the control layer topology independently from the unknown agent layer topology and consider the well known decentralized and centralized control configurations as special cases. Combining this graph theoretic formulation with optimal control ideas, we propose a single low-dimension optimal control formulation in order to systematically design the robust gain for all cooperative, decentralized, and centralized configurations. The fixed dimension of this design formulation is independent of the size of multiagent system and the control configuration. We equip the control layer with a triggering mechanism which limits each agent’s information broadcast to its

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V. Rezaei and M. Stefanovic / European Journal of Control 48 (2019) 9–20

own (local) buffer-based estimation error. We prove all trajectories of this two-layer (closed-loop) multiagent system are steered to the origin in spite of the non-periodic and non-synchronous communication over the control layer and in the presence of modeling uncertainties over the agent layer. We further propose a practical event triggering mechanism which handles the effect of unmodeled measurement noise on the total number of triggering events, and prove robust global exponential convergence of all trajectories to a neighborhood around the origin. We validate these theoretical results through various simulation studies. Power systems [35] can be viewed as potential applications; further application-oriented studies are left as future work.

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