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Stability Analysis of Linear Systems with Generalized Frequency Variables and Its Applications to Formation Control
Abstract
A linear system with a generalized frequency variable denoted by G(s) is a system which is given by replacing transfer function's 's' variable in the original system G0(s) with a rational function 'Phi(s)', i.e., G(s) is defined by Go(Phi(s)). A class of large-scale systems with decentralized information structures such as multi-agent systems can be represented by this form. In this paper, we investigate fundamental properties of such a system in terms of controllability, observability, and stability. Specifically, we first derive necessary and sufficient conditions that guarantee controllability and observability of the system Q(s) based on those of subsystems Go(s) and 1/Phi(s). Then we present Nyquist-type stability criterion which can be reduced to a linear matrix inequality (LMI) feasibility problem. Finally, we apply the results to stability analysis of a class of formation control and confirm the effectiveness of the approach as a general framework which can unify variety of results in the field.
... In particular, ignoring the agent's dynamics may cause a potential problem where each agent cannot precisely track its designed trajectory or the global convergence of multiple agents, meaning that the designated formation may not be achieved. To overcome the abovementioned difficulties, Kwak et al. [21] and Kim et al. [22] developed simple diagrammatic Lyapunov and asymptotic formation stability criteria, which are the significant extension of a stability analysis method for linear systems with a generalized frequency variable proposed in [23,24] and investigated in [25]. The formation control scheme combined with a cyclic-pursuit-based distributed online path generator satisfying those stability criteria guarantees the required global convergence properties with theoretical rigor. ...
... Note that ϕ θ (0) = 0 from (9). Hence, the transformed transfer function H θ (s) = L θ (ϕ θ (s)) of L θ (s) is said to have a generalized frequency variable ϕ θ (s) (see [23,24] for details). Similarly, the d-and α-directional transfer functions H d (s) and H α (s) can be derived as follows: ...
... The situation for the dand α-directional stability is slightly different. Because A d in (11) and A α in (12) have n multiple eigenvalues at (−k d + j0) and (−k α + j0), respectively, the asymptotic stability can be analyzed by using the scheme given in [23], i.e., the necessary and sufficient asymptotic stability condition for H d (s) (H α (s)) is that all eigenvalues of ...
In this study, the cyclic pursuit formation stabilization problem in target-enclosing operations by multiple homogeneous dynamic agents is investigated. To this end, a Lyapunov D-stability problem is first formulated to cover the transient performance requirements for multi-agent systems. Then, a simple diagrammatic Lyapunov D-stability criterion for cyclic pursuit formation is derived. The formation control scheme combined with a cyclic-pursuit-based distributed online path generator satisfying this condition guarantees both the required transient performance and global convergence properties with theoretical rigor. It is shown that the maximization of the connectivity gain in a cyclic-pursuit-based online path generator can be reduced to an optimization problem subject to linear matrix inequality constraints derived using the generalized Kalman-Yakubovich–Popov lemma. This approach provides a permissible range of connectivity gain, which not only guarantees global formation stability/convergence properties but also satisfies the required performance specification. Several numerical examples are provided to confirm the effectiveness of the proposed method.
... The MIMO case is fully addressed in [7], [8] using integral quadratic constraints and in [9] with frequency-domain approaches. Also an approach based on the generalised frequency variable, proposed in [10], was used to obtain necessary and sufficient stability conditions for networks of homogeneous MIMO LTI systems in [11], where the robustness analysis encompasses heterogeneous systems of almost equal agents. ...
... Using the properties of the 2-norm and of the Kronecker product we have that (10). Recall from equation (6) ...
... ρ is an upper bound for the maximum eigenvalue of the Laplacian matrix L = BB , ζ (F, G) is defined as in (9) and K( jω) is defined as in (10). Proof: To show that inequality (15) implies inequality (8) for all k ∈ {1, . . . ...
... From the theoretical point of view, this paper is inspired by the glocal control concept [15]. A remarkable innovation of the glocal theory is the "generalized frequency variable" method (GFV) proposed by Hara group [16]- [19]. This method provides a systematic way to analyze the complex multi-agent systems, such as gene protein regulatory [20] and in-wheel-motor vehicle [21]. ...
... where the vector u and v are defined as in (10). Applying the matrix determinant lemma to (16), the condition (15) becomes ...
... Remark 4: Following the Lemma 3.1 and 3.2, and the Proposition 3.1 and 3.2 in [16], the nominal interconnected system is stable if -λK,1 is located in the stable domain given by ϕ1(s), and -λT,i is located in the stable domain given by ϕ2(s) for all i from 2 to N. In other words, the stability of the nominal system can be verified easily through two sets of inequalities established from the GFV ϕ1(s) and ϕ2(s). ...
Due to the increasing in size and complexity of the power networks, how to assure the stability of the overall system while maintaining high frequency control performance are nontrivial issues. To this end, this paper introduces a hierarchical decentralized control configuration for load frequency control (LFC) with the help of additional battery stations (BSs) which will be popular in future society. A command to each BS is distributed from a global controller that manages the average frequency deviation. On the other hand, each local controller is designed to suppress the local frequency. Utilizing the generalized frequency variable and the idea of shared model set, a method is proposed for stabilizing the LFC system modelled by the simple swing dynamics. Then, the proposed method is extended to the more practical model of LFC system considering the dynamics of turbines, governors and the management of area control error (ACE). We finally demonstrated the effectiveness of the proposal by numerical simulations.
... Each multi-agent system can be modeled as G(s) :=G(φ(s)) in which G(s) represents the overall dynamics and φ(s) is the GFV. Thanks to the GFV theory presented in [18] and [19], this system description can considerably reduce the burden of stability analysis. Our proposal, therefore, is shown tobe systematic and useful for practical application. ...
... Therefore, we can say that the system shown in Fig. 3(b) has the generalized frequency variable φ x (s). Hence, we can apply the results in [18] by defining the following domains in the complex plane: Thanks to the Propositions 1 and 2 in [19], the characterization of the stability region c x+ Ω can be provided by a purely algebraic condition or a numeric formula with a set of LMI conditions, which are established based on the coprime polynomials of H x (s). This point is an advance of our stability analysis in comparison with graphical condition based on the Nyquist stability criterion by Fax and Murray proposed in [20] as seen in the illustration section. ...
... The system shown in Fig. 4(a) has the generalized frequency variable φ y (s):=1/H y (s). Define the domains: According to the GFV theory in [18] and [19], we obtain the following proposition: ...
Tire force distribution is an important topic in motion control of electric vehicles. For years, many researchers have focused on optimal distribution by minimizing a certain cost function. However, no effort has been paid to the stability analysis of tire force distribution theoretically. Let the actuators (in-wheel-motors and steering motors) be the local agents, the electric vehicle can be seen as a special type of multi-agent system. The agents are not decoupled but they physically interact with each other via the vehicle body. As the increasing of the actuator number, stability analysis becomes more and more complex. By modeling the EV system with generalized frequency variable, we propose a systematic scheme that can considerably reduce the burden of stability analysis for tire force distribution.
... Regarding formation control with dynamic agents, Hara et al. [2007a] proposed a novel technique to analyze the characteristics of large-scale linear systems. To this end, they first introduced the notion of a linear system with a generalized frequency variable; this system denoted as G(s) is developed by just replacing transfer function's 's' variable in the original system L(s) with a rational function 'φ(s)', i.e., G(s) := L(φ(s)). ...
... In this paper, we first present an on-line path generator design method based on a cyclic pursuit scheme, which was proposed by Kim and Sugie [2007]. Then, based on the results of Hara et al. [2007a], we derive a stability condition which the above cyclic pursuit based on-line path generator should satisfy to guarantee the formation stability. This is described in relation to the pole locations of the developed path generator in the complex plane and the region which φ(s) (:= s/H(s)) maps the right-half complex plane to. ...
... (12) Note that the variable 's' in (11) characterizes the frequency properties of the transfer function L θ (s) and that G θ (s) is generated by just replacing 's' by 'φ(s)' in L θ (s). Hence, we say that the transformed transfer function G θ (s) = L θ (φ(s)) of L θ (s) has a generalized frequency 17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008 variable φ(s) (see Hara et al. [2007a] for details). The transfer functions G d (s), L d (s), G α (s) and L α (s) can be derived in a similar manner. ...
This paper studies a design methodology of a distributed cooperative controller for target-enclosing operations by multiple dynamic agents. To this end, we first present an on-line path generator design method based on a cyclic pursuit scheme. Then, we provide the stability condition which the developed path generator should satisfy. This condition is derived based on a simple stability analysis method for large-scale linear systems with generalized frequency variable. The formation control scheme combined with a cyclic pursuit based distributed on-line path generator satisfying the derived stability condition guarantees the required global convergence property with theoretical rigor. Simulation examples illustrate its distinctive features and the achievement of a desired pursuit pattern.
... Regarding formation control with dynamic agents, Hara et al. [2007a] proposed a novel technique to analyze the characteristics of large-scale linear systems. To this end, they first introduced the notion of a linear system with a generalized frequency variable; this system denoted as G(s) is developed by just replacing transfer function's 's' variable in the original system L(s) with a rational function 'φ(s)', i.e., G(s) := L(φ(s)). ...
... In this paper, we first present an on-line path generator design method based on a cyclic pursuit scheme, which was proposed by Kim and Sugie [2007]. Then, based on the results of Hara et al. [2007a], we derive a stability condition which the above cyclic pursuit based on-line path generator should satisfy to guarantee the formation stability. This is described in relation to the pole locations of the developed path generator in the complex plane and the region which φ(s) (:= s/H(s)) maps the right-half complex plane to. ...
... (12) Note that the variable 's' in (11) characterizes the frequency properties of the transfer function L θ (s) and that G θ (s) is generated by just replacing 's' by 'φ(s)' in L θ (s). Hence, we say that the transformed transfer function G θ (s) = L θ (φ(s)) of L θ (s) has a generalized frequency 17th IFAC World Congress (IFAC'08) Seoul, Korea, July 6-11, 2008 variable φ(s) (see Hara et al. [2007a] for details). The transfer functions G d (s), L d (s), G α (s) and L α (s) can be derived in a similar manner. ...
This paper studies a design methodology of a distributed cooperative controller for target-enclosing operations by multiple dynamic agents. To this end, we first present an on-line path generator design method based on a cyclic pursuit scheme. Then, we provide the stability condition which the developed path generator should satisfy. This condition is derived based on a simple stability analysis method for large-scale linear systems with generalized frequency variable. The formation control scheme combined with a cyclic pursuit based distributed on-line path generator satisfying the derived stability condition guarantees the required global convergence property with theoretical rigor. Simulation examples illustrate its distinctive features and the achievement of a desired pursuit pattern.
... Furthermore, we analyze the observability of the state-space model under applying the hierarchical decentralized observer. It turns out that our result is a more general version of the result in [15], although we adopt a different approach to derive a necessary and sufficient condition for the observability. ...
... where Ψ ∈ R P ×N , c G ∈ R pG×n and c L ∈ R pL×n . The system (A, C) is a generalization of the system dealt with in [15], [16], where the identical SISO subsystem is described by a generalized frequency variable. In order to explain (4), let us introduce two outputs of the i-th subsystem denoted by y G i = c G x i and y L i = c L x i . ...
... Corollary 1, which is a generalized result derived in [15] where the case of c G = c I is considered, gives a simple condition when the subsystem is SISO and c G ∈ R 1×n . Furthermore, from the point of view of determining the output matrices such as Ψ and c G , these statements are useful. ...
This paper proposes a novel type of decentralized observers for a network system, where identical linear subsystems are interconnected. For this system, we derive a state-space model with block-triangular structure, in which the dynamics of the interaction among the subsystems and the dynamics of each subsystem are decoupled. Based on the decoupled model, we design a hierarchical decentralized observer, where a kind of centralized observer estimates coarse information on interaction among the subsystems and a decentralized observer estimates the state of each subsystem. Furthermore, we derive a necessary and sufficient condition of the observability for the decentralized estimation under applying the hierarchical decentralized observer.
... Under these assumptions, Hara and co-authors have been able to describe the overall homogeneous multi-agent system dynamics as a linear system with generalized frequency variable [3], [4], [5], [6], [12], and to derive powerful results regarding controllability, H 2 -and H ∞ -norm computation, stability and stabilizability of the overall system. ...
... In this paper we consider the same control configuration as in [3], [4], but we drop the homogeneity assumption on the agents, thus considering the more realistic scenario when each agent is characterized by a distinct strictly proper transfer function. Consequently, the overall system is an interconnection of a diagonal transfer matrix and of a supervisory controller (also referred to in the literature as cooperative output feedback). ...
... In this paper we provide a complete characterization of the controllability (and, by duality, of the observability) property of the overall dynamic system, as well as some preliminary results about stability and stabilizability. Comparisons with the results derived in [3], [4], [6], [12] are performed, and results are illustrated with several examples. In section II we introduce the system model: the agents dynamics will be the plant, while the information exchange among the agents will be described by the supervisory controller. ...
In this paper we consider a supervisory control scheme for non-homogenous multi-agent systems. Each agent is modeled through an independent strictly proper SISO state space model, and the supervisory controller, representing the information exchange among the agents, is implemented in turn via a linear state-space model. Controllability of the overall system is characterized, and some preliminary results about stability and stabilizability are provided. The paper extends to non-homogenous multi-agent systems some of the results obtained in [2], [3], [4] for the homogenous case.
... and φ(s) is a scalar function about variable s. φ(s) is called the GFV, since φ(s) replaces the traditional frequency variable s in the transfer function G(s). The concept of LTI systems with GFVs was firstly came up by Hara's research group for providing a unifying theoretical framework of analysis and synthesis of multi-agent systems [6,7]. In recent years, LTI systems with GFVs have appeared in a variety of areas including gene regulatory networks [17,20], biomolecular communication networks [11], multi-robot formation control problems [6] as well as the control torque distribution of electric vehicles [35]. ...
... When the fractional-order multi-agent system is concerned , the GFVs φ(s) can be generalized to be fractional degree rational function as (2.2) [6,7]. For this case, it is difficult to draw the boundary of curve determined by φ(s). ...
A novel linear time-invariant (LTI) system model with fractional degree generalized frequency variables (FDGFVs) is proposed in this paper. This model can provide a unified form for many complex systems, including fractional-order systems, distributed-order systems, multi-agent systems and so on. This study mainly investigates the stability and robust stability problems of LTI systems with FDGFVs. By characterizing the relationship between generalized frequency variable and system matrix, a necessary and sufficient stability condition is firstly presented for such systems. Then for LTI systems with uncertain FDGFVs, we present a robust stability method in virtue of zero exclusion principle. Finally, the effectiveness of the method proposed in this paper is demonstrated by analyzing the robust stability of gene regulatory networks.
... The framework presented below allows to reduce the problem to a couple of simpler ones using the notion of consensus region. The criterion for scalar agents was first proposed by Polyak and Tsypkin in [26]; similar results were obtained later in [28] and [27]. Some other extensions may be found in [3] and [4]. ...
... Note that Ω is precisely the region of parameters λ such that the matrix A − λBK is Hurwitz stable. The function φ(s) is sometimes referred to as the generalized frequency variable [27], [28]. The details of determining the consensus region may be found in [26]; in the case of φ(s) = s 2 + γs, γ > 0, this region has form of the interior of a parabola in the complex plane: φ(jω) = −ω 2 + jγω, −∞ < ω < ∞. ...
The paper addresses the problem of consensus seeking among second-order linear agents interconnected in a specific ring topology. Unlike the existing results in the field dealing with one-directional digraphs arising in various cyclic pursuit algorithms or two-directional graphs, we focus on the case where some arcs in a two-directional ring graph are dropped in a regular fashion. The derived condition for achieving consensus turns out to be independent of the number of agents in a network.
... In recent years, consensus and synchronization problems among multi-agent systems have received an intensive interest in the literature, due to the variety of applications in many different areas including cooperative control of unmanned areal vehicles, formation control of mobile robots and communication among sensor networks. See, e.g., [11], [13], [16], [17], [24], [29]. Specifically, consensus refers to agents coming to a global agreement on a state value, by the exchange of information modeled by some communication graph. ...
... Assume that one of matrices A k in (5) is not Schur-Cohn, and assume without loss of generality that it is A N −1 . Consider the coordinate system in (11) with (13). Then, from the block diagonal structure ofĀ, since A N −1 is not Schur-Cohn, there exists a vector ω * ∈ R n (an eigenvector of one of the nonconverging natural modes) such that the solution to (11) from ...
This paper addresses the problem of consensus of multi-agent systems, consisting of a set of identical continuous-or discrete-time systems, connected through a network with fixed topology. The information exchanged over the communication network is the output of each system. First, necessary and sufficient conditions are given for the uniform global exponential stability of the consensus set. Then, the quality-fair delivery of media contents is dealt with as an application of the presented equivalent conditions, which lead to a control design paradigm consisting in a suitable linear static output feedback design problem. This problem is solved proposing an iterative design technique based on Linear Matrix Inequalities (LMIs), and its effectiveness is demonstrated via simulation results, where we compare our results with pre-existing approaches.
... It is difficult to design A directly such that the eigenvalue distribution of A meets the two requirements, especially when N is very large. Fortunately, the feedback system belongs to a class of LTI systems with generalized frequency variables [2,3], and hence the assignment of eigenvalues of A can be reduced to doing that of A (not A) in the associated stability region determined by h(s). For example, when h(s) is given by h 1 (s) = (2s + 1)/(s 2 + s + 1) and h 2 (s) = 2/(s 2 + 3s), the corresponding stability regions are the hatched one in left and right of Fig. 2, respectively. ...
... A two-layer hierarchical structure is realized in the sense that the agents in each subsystem exchange information using the local structure in the lower layer and the subsystems do aggregated information in the upper layer. 2 Consider M subsystems. The kth subsystem contains n k agents, where n k (k = 1, . . . ...
In this paper, we study consensus problem for hierarchical multi-agent dynamical systems with low rank interconnection by eigenvector-based connection. The system considered is more general than the existing one in a sense that the number of agents in each subsystem and the connection structure of each subsystem are allowed to be different from each other. We provide analytical expressions of the eigenvalue sets of the hierarchical interconnection matrices and systematic synthesis procedures for achieving the output consensus with Lyapunov stability. The results are applicable to more general class of hierarchical interconnection structures than previous work, and numerical examples with simulations confirm the effectiveness of the proposed design method.
... The problem of time-delay in multiagent systems with information influences has been studied by a number of authors in papers dedicated to consensus (see [14][15][16][17][18][19][20]). The second part of the book [14] provides an overview of some papers dedicated to delay in discrete-time systems, synchronization in networks with delayed connections, approximate consensus in networks with measurement delays, etc. ...
Coordination in multiagent systems with information influences is studied. In particular, a model of multiagent system in which information is transmitted with a constant delay for all agents is studied. Using the Nyquist criterion applied by Tsypkin for systems with delayed feedback, a formula is obtained for the boundary value of time-delay which is included as a parameter in the system of differential equations with an asymmetric constant Laplacian matrix. The condition of independence of stability from time-delay is founded. The results obtained generalize some previously results and can be applied in coordination analysis in a multiagent systems with complex protocol.
... In [73], the authors mention three contents of the stability for formation control: string stability, mesh stability, and leader-to-formation stability. In [74], the authors used the Nyquist-type criterion and solved linear matrix inequalities (LMIs) for multi-agent systems (MAS). In [75], the authors research the communication graph method. ...
In recent years, formation control for multiple unmanned vehicles becomes an active research topic that has received a lot of attention from scientists due to its superior advantages compared with other conventional systems. Algebraic graph and graph rigidity theories are the two main mathematical backgrounds of the formation control theory. The graph theory is used to describe the interconnections among vehicles in formation while rigid graph theory - an important subset of graph theory - ensured that the inter-vehicle distance constraints of the desired formation are enforced via the graph rigidity. This paper provides a comprehensive review of graph theory supporting formation control for groups of unmanned aerial vehicles (UAV) or swarm UAVs. The background of the theory and the recent developments of graph-theory-based formation control are reviewed. We provide a cohesive overview of the formation control and coordination of multiple vehicles. Finally, some challenges and future potential directions in formation control are discussed.
... The function φ(s) is sometimes referred to as the generalized frequency variable [49], [36]. The details of determining the consensus region may be found in [35]. ...
The paper addresses the problem of multi-agent communication in networks with regular directed ring structure. These can be viewed as hierarchical extensions of the classical cyclic pursuit topology. We show that the spectra of the corresponding Laplacian matrices allow exact localization on the complex plane. Furthermore, we derive a general form of the characteristic polynomial of such matrices, analyze the algebraic curves its roots belong to, and propose a way to obtain their closed-form equations. In combination with frequency domain consensus criteria for high-order SISO linear agents, these curves enable one to analyze the feasibility of consensus in networks with varying number of agents.
... For interconnections of SISO (single-input and single-output) LTI (linear time invariant) systems, robust stability conditions were provided by Kao et al. (2009) and Jönsson and Kao (2010) for particular topologies; by Lestas and Vinnicombe (2006) using the multivariable Nyquist criterion proposed by Desoer and Wang (1980) and the concept of S-hull; by Hara et al. (2007Hara et al. ( , 2009Hara et al. ( , 2014 and Hori et al. (2015) in the generalised frequency variable framework. Extensions of the above stability conditions to the MIMO (multiple-input and multiple-output) case were provided by Pates and Vinnicombe (2012) with a Nyquist-like approach and by Andersen et al. (2014) and Khong and Rantzer (2014) based on Integral Quadratic Constraints, also used by Pates and Vinnicombe (2017) for control design: these conditions are scalable because, to ensure stability of the whole network, it is enough to satisfy a local condition at each node. ...
We study dynamic networks described by a directed graph where the nodes are associated with MIMO systems with transfer-function matrix F(s), representing individual dynamic units, and the arcs are associated with MIMO systems with transfer-function matrix G(s), accounting for the dynamic interactions among the units. In the nominal case, we provide a topology-independent condition for the stability of all possible dynamic networks with a maximum connectivity degree, regardless of their size and interconnection structure. When node and arc transfer-function matrices are affected by norm-bounded homogeneous uncertainties, the robust condition for size- and topology-independent stability depends on the uncertainty magnitude. Both conditions, expressed as constraints for the Nyquist diagram of the poles of the transfer-function matrix H(s) = F(s)G(s), are scalable and can be checked locally to guarantee stability-preserving “plug-and-play” addition of new nodes and arcs.
... The passivity approach can be further applied to hierarchically decentralized frequency control systems with communication delays. The generalized frequency variable (GFV) was proposed in the last decade as one solution to analyze multi-agent systems [32], [33]. Its effectiveness was clarified by the applications of gene regulatory networks [34], platoon cars [35], and motion control of electric vehicles [36]. ...
How to practically maintain the frequency stability of large-scale power systems by a decentralized way is a simple but non-trivial question. In other words, is it possible to design any local controller without understanding the other controlled areas and with less understanding of network structure? With respect to the special properties of physical interaction between the local areas, this paper suggests two existing theories for tackling this issue. Firstly, passivity theory is shown to be a candidate for frequency control problem using swing equation. Based on the passivity of swing dynamics, it is possible to guarantee the system stability by designing for each local area a passive controller. We further extend the passivity approach to the hierarchically decentralized control system with unknown communication delay. Secondly, we discuss the application of generalized frequency variable (GFV) to the frequency control problem using area-control-error. Each local controller is designed such that each local subsystem follows a nominal model set. Utilizing GFV theory, we present a triad of conditions that sufficiently guarantee the system stability. The conditions can be tested conveniently by a limited set of inequalities established from the GFV and the eigenvalues of the physical interaction matrix. The effectiveness, limitation, and challenge of two theories are discussed by design examples with numerical simulations.
... Several different approaches have been taken, including evaluating the impact on individual control algorithms and determining maximum acceptable latency. Other methodologies for developing metrics have characterized latency as part of the overall dynamics of a multi-agent hierarchy, representing communications links as transfer functions between individual agents that are stabilized by a given control algorithm [19]. ...
Today's complex grid involves many interdependent systems. Various layers of hierarchical control and communication systems are coordinated, both spatially and temporally to achieve gird reliability. As new communication network based control system technologies are being deployed, the interconnected nature of these systems is becoming more complex. Deployment of smart grid concepts promises effective integration of renewable resources, especially if combined with energy storage. However, without a philosophical focus on resilience, a smart grid will potentially lead to higher magnitude and/or duration of disruptive events. The effectiveness of a resilient infrastructure depends upon its ability to anticipate, absorb, adapt to, and/or rapidly recover from a potentially catastrophic event. Future system operations can be enhanced with a resilient philosophy through architecting the complexity with state awareness metrics that recognize changing system conditions and provide for an agile and adaptive response. The starting point for metrics lies in first understanding the attributes of performance that will be qualified. In this paper, we will overview those attributes and describe how they will be characterized by designing a distributed agent that can be applied to the power grid.
... Several different approaches have been taken, including evaluating the impact on individual control algorithms and determining maximum acceptable latency. Other methodologies have characterized latency as part of the overall dynamics of a multi-agent hierarchy, representing communications links as transfer functions between individual agents that are stabilized by a given control algorithm [12]. ...
During a real-time power system event, a system operator needs to conservatively reduce operating limits while the changing system conditions are analyzed. The time it takes to develop new operating limits could affect millions of transmission system users, especially if this event is classified by NERC as a Category D type event (extreme events resulting in the loss of two or more bulk electric system elements). Controls for the future grid must be able to perform real-time analysis, identify new reliability risks, and set new SOLs (System Operating Limit) for real-time operations. In this paper we are developing “Resilience Metrics” requirements that describe how systems operate at an acceptable level of normalcy despite disturbances or threats. We
consider the interdependencies inherent in critical infrastructure systems and discuss some distributed resilience metrics that can be in current supervisory control and data acquisition (SCADA) to provide a level of state awareness. This level of awareness provides knowledge that can be used to characterize and reduce
the risk of cascading events. A “resilience power system agent” is proposed that provides attributes to measure and perform this metrics.
... To prove stability of the closed-loop systems, resulting from applying the protocols (3), (15), (17) to the agents (11), we employ the following stability criterion, elaborated by B.T. Polyak and Y.Z. Tsypkin [19] and reformulated later by S. Hara [9] using the concept of "generalized frequency variable". ...
In this paper we consider a special problem of formation control, that is, the problem of uniform spread (or deployment) of several identical agents on a line segment. This problem was exhaustively studied for first-order agents since the pioneering paper by I.A. Wagner and A.M. Bruckstein who considered it under the name of 'row straightening algorithm'. We extend it to the more realistic case of second-order agents.
... Several different approaches have been taken, including evaluating the impact on an individual control algorithm and determining maximum acceptable latency. Other methodologies have characterized latency as part of the overall dynamics of a multi-agent hierarchy, representing communications links as transfer functions between individual agents that are stabilized by a given control algorithm [22]. ...
“Resilience” describes how systems operate at an acceptable level of normalcy despite disturbances or threats. In this paper we first consider the interdependencies inherent in critical infrastructure systems and how resilience mitigates associated risks and then define “resilience” in distinction from convention control engineering. We then introduce the concepts “agent” and “multi-agent systems” (MAS) to consider the distributed nature of critical infrastructure control systems and illustrate the application of computational intelligence to MAS event-based dynamics (management, coordination) and time-based dynamics (execution) to manage policy and coordinate assets. In addition, we consider the optimal stabilization of the MAS and suggest the extension of graph theory to MAS execution layers. The closing discussion provides an overview of how to achieve critical infrastructure resilience through advanced control engineering.
... In applications such as sensor networks, coordination of mobile robots or UAVs, flocking and swarming in animal groups, dynamics of opinion forming, etc., the problem can be formulated as that of a group of agents exchanging information with the objective of reaching a common decision, a consensus, by resorting to distributed algorithms that make use of the information that each agent collects from neighboring agents (see, e.g. [2,11,12,15,19,20,21,24,25,27]. The interested reader is referred to [1] and [23] for a more complete list of references). ...
The aim of this paper is to address consensus and bipartite consensus for a group of homogeneous agents, under the assumption that their mutual interactions can be described by a weighted, signed, connected and structurally balanced communication graph. This amounts to assuming that the agents can be split into two antagonistic groups such that interactions between agents belonging to the same group are cooperative, and hence represented by nonnegative weights, while interactions between agents belonging to opposite groups are antagonistic, and hence represented by nonpositive weights. In this framework, bipartite consensus can always be reached under the stabilizability assumption on the state-space model describing the dynamics of each agent. On the other hand, (nontrivial) standard consensus may be achieved only under very demanding requirements, both on the Laplacian associated with the communication graph and on the agents’ description. In particular, consensus may be achieved only if there is a sort of “equilibrium” between the two groups, both in terms of cardinality and in terms of the weights of the “conflicting interactions” amongst agents.
Изучается задача согласования характеристик в многоагентной системе с информационными влияниями. В частности, изучена модель многоагентной системы, в которой информация между агентами передается с постоянным для всех агентов запаздыванием. С помощью критерия Найквиста, примененного Цыпкиным для систем с запаздыванием, получена формула для граничного значения запаздывания, входящего как параметр в систему дифференциальных уравнений с несимметричной постоянной лапласовской матрицей. Найдено условие, при котором устойчивость системы не зависит от запаздывания. Полученные результаты обобщают некоторые ранее полученные результаты и могут быть применены при анализе согласования характеристик в многоагентной системе со сложным протоколом.
The paper addresses the problem of multiagent communication in networks with regular directed ring structure. These can be viewed as hierarchical extensions of the classical cyclic pursuit topology. We show that the spectra of the corresponding Laplacian matrices allow exact localization on the complex plane. Furthermore, we derive a general form of the characteristic polynomial of such matrices, analyze the algebraic curves its roots belong to, and propose a way to obtain their closed-form equations. In combination with frequency domain consensus criteria for high-order SISO linear agents, these curves enable one to analyze the feasibility of consensus in networks with varying number of agents.
The use of consensus networks in networking has received great attention due to its wide array of applications in fields such as robotics, transportation, sensor networking, communication networking, biology, and physics. The focus of this paper is to study a generalization of consensus problems whereby the weights of network edges are no longer static gains, but instead are dynamic systems, leading to the notion of dynamic consensus networks. Specifically, we consider networks whose nodes are transfer functions (typically integrators) and whose edges are strictly positive real transfer functions representing dynamical systems that couple the nodes. We transform each concept of static graph theory into dynamic terms, out of which a generalized dynamic graph theory naturally emerges. We present a framework for dynamic graphs and dynamic consensus networks. This framework introduces the idea of dynamic degree, adjacency, incident, and Laplacian matrices in a way that naturally extends these concepts from the static case. We show that strictly positive realness of the edges is a sufficient condition for dynamic networks to be stable (i.e., to reach consensus). To study the spectral properties of dynamic networks, we introduce the Dynamic Grounded Laplacian matrix, which is used to estimate lower and upper bounds for the real parts of the smallest and largest non-zero eigenvalues of the dynamic Laplacian matrix. These bounds can be used to obtain stability conditions using the Nyquist graphical stability test for undirected dynamic networks controlled using distributed controllers. Numerical simulations are provided to verify the effectiveness of the results.
A class of networked dynamical systems with multiple homogeneous agents can be represented by a linear system with a generalized frequency variable. This paper is concerned with robust stability analysis of such class of systems under perturbations in the dynamics of agents. The perturbed agents no longer share the same dynamics and thus the analysis encompasses heterogeneous multi-agent systems consisting of slightly different agents. We first present nominal stability conditions for the system of interconnected MIMO agents. We then focus on robust stability, where we consider three types of perturbations, leading to homogeneous agents, heterogeneous agents with and without uncertain interconnections. For each case, we show that a necessary and sufficient condition for robust stability is given by an H∞-norm bound on a set of transfer functions parametrized by the eigenvalues of the interconnection matrix, provided the nominal interconnections have a certain structure. The robust stability condition is interpreted as the requirement that the eigenvalues of the interconnection matrix be located in a particular region on the complex plane. The usefulness of the results is demonstrated through an application to robust stability analysis for gene regulatory networks.
This paper is concerned with the analysis and synthesis of interconnected systems constructed from heterogeneous positive subsystems and a nonnegative interconnection matrix. We first show that admissibility, to be defined in this paper, is an essential requirement in constructing such interconnected systems. Then, we clarify that the interconnected system is admissible and stable if and only if a Metzler matrix, which is built from the coefficient matrices of positive subsystems and the nonnegative interconnection matrix, is Hurwitz stable. By means of this key result, we further provide several results that characterize the admissibility and stability of the interconnected system in terms of the Frobenius eigenvalue of the interconnection matrix and the weighted L1- induced norm of the positive subsystems again to be defined in this paper. Moreover, in the case where every subsystem is SISO, we provide explicit conditions under which the interconnected system has the property of persistence, i.e., its state converges to a unique strictly positive vector (that is known in advance up to a strictly positive constant multiplicative factor) for any nonnegative and nonzero initial state. As an important consequence of this property, we show that the output of the interconnected system converges to a scalar multiple of the right eigenvector of a nonnegative matrix associated with its Frobenius eigenvalue, where the nonnegative matrix is nothing but the interconnection matrix scaled by the steady-stage gains of the positive subsystems. This result is then naturally and effectively applied to formation control of multiagent systems with positive dynamics. This result can be seen as a generalization of a well-known consensus algorithm that has been basically applied to interconnected systems constructed from integrators.
In swarm robotics, a multitude of very simple robots often move to achieve pre-defined geometric patterns while communicating with each other. Robots are able to estimate their position and relocate themselves by obtaining other robots' position information. However, as is the often the case in wireless communication, robots cannot receive other robots' position information reliably. Thus, they need to deal with the packet loss problem and estimate the missing information in near-real time. Several nonlinear filters such as extended Kalman filters or particle filters have been widely used for many years. But these filters require enormous computational complexity, so it is difficult to be applied to low-cost mobile robots with limited computational and memory resources. To overcome this problem, we propose an extrapolating method with reduced computational cost yet high estimation accuracy. Specifically, we propose a novel exponential function concealment with linear blends, and validate its effectiveness and rate of convergence through extensive simulations.
This work presents a new necessary and sufficient condition for robust stability and robust performance of positive linear time invariant (LTI) systems with structured uncertainties. It is known that robustness analysis of LTI systems with structured uncertainties requires evaluation of structured singular value (SSV), which is known to be NP-hard. This paper shows that for positive systems, the structured singular value can be estimated efficiently using linear programming. Thus, the robustness analysis of positive systems is simplified to easily verifiable conditions that scale linearly with the dimensions of the system. This property finds great utility in analysis and synthesis of large scale positive systems with distributed control strategies.
Consideration was given to a special problem of controlling a formation of mobile agents, that of uniform deployment of several identical agents on a segment of the straight line. For the case of agents obeying the first-order dynamic model, this problem seems to be first formulated in 1997 by I.A. Wagner and A.M. Bruckstein as “row straightening.” In the present paper, the straightening algorithm was generalized to a more interesting case where the agent dynamics obeys second-order differential equations or, stated differently, it is the agent’s acceleration (or the force applied to it) that is the control.
This chapter is concerned with a problem to identify the structures of linear NW systems from input/output data, where the systems are supposed to possess a smaller number of actuators/sensors compared with the number of the nodes. In this chapter, two researches on such a NW structure identification are introduced with their comparison: One is a study proposing a characteristic-polynomial-based method using knock-out procedure, and the other is a study on the reconstruction of the dynamical structure function of the NW system.
A certain class of large-scale systems with distributed information structures such as multi-Agent dynamical systems can be represented by a class of linear time-invariant (LTI) systems with generalized frequency variables. Stability and robust stability conditions against uncertainties in the agent dynamics for such systems have been derived. However robust stability analysis of such systems with certain uncertainties in the interconnection has not been investigated yet. In this paper, we investigate the robust conditions against matrix polytope uncertainties. As the main results we derive exposed edge type robust stability conditions and examine how to check them.
In this paper, we study the consensus (and synchronization) problem for multi-agent linear dynamic systems. All the agents have identical linear dynamics which can be of any order, and only the output information of each agents is delivered throughout the communication network. In particular, it is shown that consensus is reached when the information processing filter k(s) is designed so that it stabilizes λig(s) where g(s) is the dynamics of the agents, and λi are the non-zero eigenvalues of the Laplacian representing the communication graph. We also compute the asymptotic trajectory of the agents, which is the outcome of the agreement among the agents, and depends on the initial conditions of the agents. As a showcase, some specific design of k(s) is given for the first-order and the second-order consensus problems, respectively.
In this paper, for an undirected network consisting of identical multi-dimensional linear subsystems in which the network structure is unknown and the number of input/output ports is less than that of subsystems, we propose a novel method to identify the strength of the connectivity between specified two subsystems. The proposed method is on the basis of the node knock-out procedure, and does not need any information on the other connectivities.
For a class of networked systems consisting of heterogeneous multi-dimensional subsystems, this paper proposes a novel knock-out-based method to identify the strength of the interaction between specified two nodes based on the input-output data of the networked systems, under the condition that the dynamics of the corresponding nodes is known. Our method does not require any information on dynamics of the other nodes as well as network structure among the other nodes. We also extend this method to the case in which the dynamics of the two nodes whose edge is identified is changed instead of the knock-out operation, i.e., to a knock-down-based identification method.
In this paper, controllability properties of networks of diffusively coupled linear systems are considered through the controllability Gramian. For a class of passive linear systems, it is shown that the controllability Gramian can be decomposed into two parts. The first part is related to the dynamics of the individual systems whereas the second part is dependent only on the interconnection topology, allowing for a clear interpretation and efficient computation of controllability properties for a class of networked systems. Moreover, a relation between symmetries in the interconnection topology and controllability is given. The results are illustrated by an example.
In this paper, based on the node knock-out procedure, for a networked system consisting of identical multi-dimensional subsystems in which the network structure is unknown and the number of input/output nodes is less than that of subsystems, we propose a novel method to identify the strength of the interaction between nodes even if information on the other nodes is not unknown.
This paper is concerned with the analysis of large-scale interconnected systems constructed from positive subsystems and a nonnegative interconnection matrix. We first show that the interconnected system is admissible and stable if and only if a Metzler matrix built from the coefficient matrices of the positive subsystems and the interconnection matrix is Hurwitz stable. By means of this key lemma, we further provide several results that characterize the admissibility and stability of interconnected systems in terms of the weighted L1-induced norm of each positive subsystem and the Frobenius eigenvalue of the interconnection matrix. Moreover, in the case where every subsystem is SISO, we provide explicit conditions under which the interconnected system has the property of persistence, i.e., the state of the interconnected system converges to a unique strictly positive vector (up to a strictly positive constant multiplicative factor) irrespective of nonnegative and nonzero initial states. We illustrate the effectiveness of the persistence results via formation control of multi-agent systems.
A class of large-scale, multi-agent systems with decentralized information structures can be represented by a linear system with a generalized frequency variable. In this paper, we investigate fundamental properties of such systems, stability, and -stability, exploiting the dynamical structure. Specifically, we first show that such system is stable if and only if the eigenvalues of the connectivity matrix lie in a region of the complex plane specified by the generalized frequency variable. The stability region is characterized in terms of polynomial inequalities, leading to an algebraic stability condition. We also show that the stability test can be reduced to a linear matrix inequality (LMI) feasibility problem involving generalized Lyapunov inequalities and that the LMI result can be extended for robust stability analysis of systems subject to uncertainties in the interconnection matrix. We then extend the result to -stability analysis to meet practical requirements, and provide a unified treatment of -stability conditions for ease of implementation. Finally, numerical examples illustrate utility of the stability conditions for the analysis of biological oscillators and for the design of cooperative stabilizers.
This paper deals with a hierarchical consensus problem in large scale
systems. We first define the hierarchical consensus and propose a fairly
general model of the system. We then define the incidence matrix which
expresses interconnection property between layers, where we focus on the
rank of the matrix. In order to examine the relationship between the
rank of inter-layer incidence matrix and consensus performance, we
analytically derive the eigenvalue distribution of the sysytem matrix in
the case of circulant incidence matrix. The resultant distribution shows
that the low-rank interconnection leads to the faster consensus in terms
of rate of convergence and damping, which is confirmed by numerical
examples with simulations.
Consideration was given to generalization of one of the formation control algorithms, that of equidistant arrangement of agents over a fixed interval. In distinction to the earlier approaches that are based on the equations of the first order, a second-order algorithm was proposed. It was proved to be stable and with proper selection of the adjusted parameter able to provide a higher rate of convergence in comparison with its first-order counterparts. Relation was demonstrated between the problem of arrangement over an interval and the classical problem of consensus. Examples of the results of modeling were presented.
This paper is concerned with symbolic computation in analysis and synthesis for multi-agent dynamical systems, where all the agents share a common linear time-invariant (LTI) system h(s). Such a system can be represented by the LTI system with generalized frequency variable φ(s) :=1/h(s). There are a very few results on the analysis and synthesis except stability test, since the system is large and more complex than the standard LTI system. Actually, most of the problems are nonconvex in general, and hence numerical computations do not work so well. Instead, appropriate use of symbolic computations may help to solve the problems. This paper treat two types of analysis problems on robustness of the system, namely H∞-norm computation and D-stability test, and show that symbolic computation including quantifier elimination (QE) method effectively works for solving the problems.
This is the first of a two-part paper that investigates the stability properties of a system of multiple mobile agents with double integrator dynamics. In this first part we generate stable flocking motion for the group using a coordination control scheme which gives rise to smooth control laws for the agents. These control laws are a combination of attractive/repulsive and alignment forces, ensuring collision avoidance and cohesion of the group and an aggregate motion along a common heading direction. In this control scheme the topology of the control interconnections is fixed and time invariant. The control policy ensures that all agents eventually align with each other and have a common heading direction while at the same time avoid collisions and group into a tight formation.
This paper proposes a new expression of frequency transformation for linear continuous-time state-space systems. The proposed frequency transformation preserves the controllability Gramian and the observability Gramian of prototype state-space systems, and thus allows us to easily realize various kinds of state-space systems keeping the same realization as that of prototype state-space systems. Our result is derived from a modified state-space formulation of frequency transformation and an appropriate state-space representation of reactance functions
This is the second of a two-part paper, investigating the stability properties of a system of multiple mobile agents with double integrator dynamics. In this second part, we allow the topology of the control interconnections between the agents in the group to vary with time. Specifically, the control law of an agent depends on the state of a set of agents that are within a certain neighborhood around it. As the agents move around this set changes, giving rise to a dynamic control interconnection topology and a switching control law. This control law consists of a a combination of attractive/repulsive and alignment forces. The former ensure collision avoidance and cohesion of the group and the latter result to all agents attaining a common heading angle, exhibiting flocking motion. Despite the use of only local information and the time varying nature of agent interaction which affects the local controllers, flocking motion is established, as long as connectivity in the neighboring graph is maintained.
This paper proves the invariance of the second-order modes of continuous-time systems, such as analog filters, under general frequency transformation, that is, under the reactance function. The invariance is proved by obtaining the state-space representation of the transformed system, and its controllability/observability grammians in terms of those of the original system and the reactance system. This invariance property is parallel to that of discrete-time systems such as digital filters proved by Mullis and Roberts.
We study the performance of spatially invariant plants interconnected through a static network. Such structures commonly arise in multi-agent systems and systems described by partial differential equations with constant coefficients. We introduce a new notion of Bode sensitivity integral for such interconnections of systems. We comment on a new waterbed effect in two dimensions, space and time and evaluate examples to show the role of Bode integral as a fundamental limitation in feedback interconnection of distributed systems
We introduce the concept of matrix-valued effective resistance for undirected matrix-weighted graphs. Effective resistances are defined to be the square blocks that appear in the diagonal of the inverse of the matrix-weighted Dirichlet graph Laplacian matrix. However, they can also be obtained from a "generalized" electrical network that is constructed from the graph, and for which currents, voltages and resistances take matrix values. Effective resistances play an important role in several problems related to distributed control and estimation. They appear in least-squares estimation problems in which one attempts to reconstruct global information from relative noisy measurements; as well as in motion control problems in which agents attempt to control their positions towards a desired formation, based on noisy local measurements. We show that in either of these problems, the effective resistances have a direct physical interpretation. We also show that effective resistances provide bounds on the spectrum of the graph Laplacian matrix and the Dirichlet graph Laplacian. These bounds can be used to characterize the stability and convergence rate of several distributed algorithms that appeared in the literature
In this paper, we discuss consensus problems for networks of dynamic agents with fixed and switching topologies. We analyze three cases: 1) directed networks with fixed topology; 2) directed networks with switching topology; and 3) undirected networks with communication time-delays and fixed topology. We introduce two consensus protocols for networks with and without time-delays and provide a convergence analysis in all three cases. We establish a direct connection between the algebraic connectivity (or Fiedler eigenvalue) of the network and the performance (or negotiation speed) of a linear consensus protocol. This required the generalization of the notion of algebraic connectivity of undirected graphs to digraphs. It turns out that balanced digraphs play a key role in addressing average-consensus problems. We introduce disagreement functions for convergence analysis of consensus protocols. A disagreement function is a Lyapunov function for the disagreement network dynamics. We proposed a simple disa
We consider the problem of cooperation among a collection of vehicles performing a shared
task using intervehicle communication to coordinate their actions. We apply tools from graph theory to relate the topology of the communication network to formation stability. We prove a Nyquist criterion that uses the eigenvalues of the graph Laplacian matrix to determine the effect
of the graph on formation stability. We also propose a method for decentralized information
exchange between vehicles. This approach realizes a dynamical system that supplies each vehicle
with a common reference to be used for cooperative motion. We prove a separation principle
that states that formation stability is achieved if the information flow is stable for the given
graph and if the local controller stabilizes the vehicle. The information flow can be rendered
highly robust to changes in the graph, thus enabling tight formation control despite limitations
in intervehicle communication capability.
In a recent Physical Review Letters article, Vicsek et al. propose a simple but compelling discrete-time model of n autonomous agents (i.e., points or particles) all moving in the plane with the same speed but with different headings. Each agent's heading is updated using a local rule based on the average of its own heading plus the headings of its "neighbors." In their paper, Vicsek et al. provide simulation results which demonstrate that the nearest neighbor rule they are studying can cause all agents to eventually move in the same direction despite the absence of centralized coordination and despite the fact that each agent's set of nearest neighbors change with time as the system evolves. This paper provides a theoretical explanation for this observed behavior. In addition, convergence results are derived for several other similarly inspired models. The Vicsek model proves to be a graphic example of a switched linear system which is stable, but for which there does not exist a common quadratic Lyapunov function.
A simple model with a novel type of dynamics is introduced in order to investigate the emergence of self-ordered motion in systems of particles with biologically motivated interaction. In our model particles are driven with a constant absolute velocity and at each time step assume the average direction of motion of the particles in their neighborhood with some random perturbation () added. We present numerical evidence that this model results in a kinetic phase transition from no transport (zero average velocity, ) to finite net transport through spontaneous symmetry breaking of the rotational symmetry. The transition is continuous since is found to scale as with .