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Passivity as a Design Tool for Group Coordination
Abstract
We pursue a group coordination problem where the objective is to steer the differences between output variables of the group members to a prescribed compact set. To stabilize this set we study a class of feedback rules that are implementable with local information available to each member. When the information flow between neighboring members is bidirectional, we show that the closed-loop system exhibits a special interconnection structure which inherits the passivity properties of its components. By exploiting this structure we develop a passivity-based design framework, which results in a broad class of feedback rules that encompass as special cases some of the existing formation stabilization and group agreement designs in the literature. The passivity approach offers additional design flexibility compared to these special cases, and systematically constructs a Lurie-type Lyapunov function for the closed-loop system. We further study the robustness of these feedback laws in the presence of a time-varying communication topology, and present a persistency of excitation condition which allows the interconnection graph to lose connectivity pointwise in time as long as it is established in an integral sense.
... Since the environment interaction has a scalar nature, the interconnection of dissipative subsystems yields a dissipative system under mild assumptions. This allows the stability analysis of complex systems by considering the properties of their subsystems and interconnections in a simplified way as shown in multi-agent [6], network [7] or cooperative [8] problems. ...
... For α-lose-less systems, taking any trajectory from x to x * (arriving at time t), we have from inequality (8) ...
... Proof of Proposition 3. From inequality (8) and passivity, we have 0 ≤ V a (0; α) = sup t,u −I α y T u ≤ V (0) = 0, which implies that (∀t > 0), (∀u ∈ U ), I α y T u(t) ≥ 0. The converse follows using the already noted fact that if a system Σ is α-positive real and β > α then Σ is β -positive real, together with Example 2. ...
... The recent decades have witnessed a substantial progress in analysis of synchrony mechanisms; the main results in this direction can be found in monographs [39,62,226,269,295] and numerous surveys, e.g. [57,215,223,251,261]; the most developed lines of research are concerned with the coupling matrix decomposition [88,235], master stability functions [229], contraction theory [73] and passivity [16,226,234]. Most of the available results either assume that the network topology is constant or impose restrictive assumptions on the dynamics of individual nodes (assuming them e.g. to be passive systems or linear systems with identical dynamics). ...
... The uniform cut-balance extends this "mutuality" principle to complementary groups of agents. 16 In the work [135], this property is simply called cut-balance. The term "uniform" emphasizes that the constant in (2.44) does not depend on time. ...
... for some , > 0, which condition (similar in spirit to "persistent excitation" conditions [16,20]) ...
This is my habilitation ("Doctor of Science") thesis, summarizing my most cited works on first-order, (or averaging) algorithms of multi-agent consensus, convergence of associated averaging inequalities, and applications to models of opinion dynamics. I started this journey in 2010 and it took me almost 12 years, but I promised to finish it to myself and my supervisor. I am proud of some results obtained in this period (only some of them are included into this text) and hope that this thesis will be later reworked into a monograph. The thesis was submitted in late 2021 and defended successfully in June 2022. This version includes some corrections, suggested by Prof. N.E. Barabanov.
Instead of the abstract, I prefer to write an acknowledgement, which is not appreciated in Russian theses but must be written somewhere.
This thesis is devoted to the memory of my parents, Liubov and Victor, of my scientific father - Vladimir Yakubovich, and my friend and coauthor Roberto Tempo, who was very enthusiastic about opinion formation modeling and whose influence on some my works was enormous.
This thesis would never have been completed without the help of Alexey Matveev whom I consider the most ingenious mathematician working in the field of systems and control. I am also very thankful to my uncle, aunts and cousins, to my friends and colleagues from Navis company and all universities where I was working or staying in the previous 12 years: Yuri, Yulia, Elena, Igor, Sergey, Fan, Ming, Manuel, Peyman, Fabrizio, Chiara, Francesco, Oleg and many others. Writing this thesis, I was thinking about you.
... In [14], the notion of passivity is used for group coordination. The control objective is to steer the differences of output variables of neighboring agents to a prescribed compact set. ...
... Since the dynamics of the external feedback are associated Preprint submitted to Automatica with the edges, the convergence analysis is related to the incidence matrix of the graph. For group coordination in [14], a necessary condition to achieve convergence of system states to a target set is the linear independence of the columns of the incidence matrix. However, in the context of formation control, this condition is often not satisfied, for example in the displacement-based formation with cyclic graphs. ...
... In addition, it also allows for modeling complex agent dynamics and is scalable to a large number of subsystems. The passivity framework for group coordination was introduced in [14]. ...
This paper proposes a passivity-based port-Hamiltonian (pH) framework for multi-agent displacement-based and rigid formation control and velocity tracking. The control law consists of two parts, where the internal feedback is to track the velocity and the external feedback is to achieve formation stabilization by steering variables of neighboring agents that prescribe the desired geometric shape. Regarding the external feedback, a general framework is proposed for all kinds of formations by means of the advantage that the pH model is energy-based and coordinate-free. To solve the issue that the incidence matrix is not of full column rank over cyclic graphs, the matrix property is used to prove the convergence to the target sets for the displacement-based formation, while for rigid formations, the algebraic conditions of infinitesimal rigidity are investigated to achieve asymptotic local stability. Furthermore, the rigid formation with heterogeneous constraints is further investigated under this framework and the asymptotic local stability is proved under a mild assumption. Simulations are performed to illustrate the effectiveness of the framework.
... Passivity-based approaches are widely used in the control of dynamic multi-agent networks as they enable the design of decentralised controllers which have desired plug-and-play characteristics, i.e. stability is ensured when subsystems are added/removed from the network. [1], [2]. Such approaches have been applied to networks involving congestion control [3] and chemical and biological systems [1]. ...
... Next, a method to find such controllers for linear systems is presented. Here, the model of Section II-B is linearised so that it can be written in the form (8). Similar relationships hold for (2). We let A V = i∈V A i , with similar definitions holding for B V , B uV , C V , A E , B E and C E . ...
Passivity-based approaches have been suggested as a solution to the problem of decentralised control design in many multi-agent network control problems due to the plug- and-play functionality they provide. However, it is not clear if these controllers are optimal at a network level due to their inherently local formulation, with designers often relying on heuristics to achieve desired global performance. On the other hand, solving for an optimal controller is not guaranteed to produce a passive system. In this paper, we address these dual problems by using inverse optimal control theory to formulate a set of sufficient local conditions, which when satisfied ensure that the resulting decentralised control policies are the solution to a network optimal control problem, while at the same time satisfying appropriate passivity properties. These conditions are then reformulated into a set of linear matrix inequalities (LMIs) which can be used to yield such controllers for linear systems. The proposed approach is demonstrated through a DC microgrid case study. The results substantiate the feasibility and efficacy of the presented method.
... Various physical systems including the aforementioned power networks possess passivity properties. Passivity and its variant concepts have already been witnessed as useful tools for agreement; see, e.g., [1][2][3]11,16,19,23]. In particular, the works [2,3,16] have studied output consensus problems based on shifted passivity. ...
... For the DC network (1), the passive output for both Krasovskii and shifted passivity is y in (4). (1), the results in [2,3,16] for output consensus analysis are not applicable. These papers impose special edge dynamics such that the interconnected system naturally possesses a consensus property. ...
Motivated by current sharing in power networks, we consider a class of output consensus (also called agreement) problems for nonlinear systems, where the consensus value is determined by external disturbances, e.g., power demand. This output consensus problem is solved by a simple distributed output feedback controller if a system is either Krasovskii or shifted passive, which is the only essential requirement. The effectiveness of the proposed controller is shown in simulation on an islanded DC power network.
... The central concept involves incorporating the intended trajectory into the Hamiltonian framework of the original form, and this allows the transformed system to remain in the same form as the original system. By employing passivitybased methods to stabilize the error system, the objective of tracking the desired trajectory is accomplished [51,52]. The noteworthy findings from [53] are outlined below. ...
Small bodies have the characteristics of noncooperative, irregular gravity, and complex terrain on the surface, which cause difficulties in successful landing for conventional landers. In this paper, a multinode flexible lander is put forward to address the problem. The dynamics of this new lander are constructed based on the port-Hamilton framework. The trajectory-tracking formation controller for the lander is designed in a passive way. The proposed dynamics and controller are further validated through numerical simulations. This research presents a fresh concept that holds inspiration for future design involving small-body landers.
... It is well known that there is an equivalence between the classical small gain and passivity theorems, (Anderson 1972). Passivity in turn has been useful in proving stability of adaptive, , Dasgupta et al. 1986, and multiagent systems (Arcak 2007). A future direction could also be to formulate passivity type theorems, perhaps by using a variation of dissipative type Lyapunov functions, e.g. by having the inner product of input and output in the stead of their norms. ...
This paper considers small gain theorems for the global asymptotic and exponential input-to-state stability for discrete time time-delay systems using Razumikhin-type Lyapunov function. Among other things, unlike the existing literature, it provides both necessary and sufficient conditions for exponential input-to-state stability in terms of the Razumikhin-type Lyapunov function and the small gain theorem. Previous necessary ad sufficient conditions were with the more computationally onerous, Krasovskii-type Lyapunov functions. The result finds application in the robust stability analysis of a graph-based distributed algorithm, namely, the biased min-consensus protocol, which can be used to compute the length of the shortest path from each node to its nearest source in a graph. We consider the biased min-consensus protocol under perturbations that are common in communication networks, including noise, delay and asynchronous communication. By converting such a perturbed protocol into a discrete time time-delay nonlinear system, we prove its exponential input-to-state stability under perturbations using our Razumikhin-type Lyapunov-based small gain theorem. Simulations are provided to verify the theoretical results.
... In particular, since feedback and parallel interconnections of passive systems are passive (and stable), complex nonlinear systems may be stabilized by ensuring passivity for individual systems and their controllers. In this context, it was shown in [8] that under a particular interaction topology (connected and undirected graph), control based on a well-designed potential function results in convergence to a desired set of equilibria for the group of agents, given that the individual agents are passive. For mechanical systems, control via passivity interacts with both the velocity and force of end-effectors. ...
In this work, we propose two cooperative passivity-based control methods for networks of mechanical systems. By cooperatively synchronizing the end-effector coordinates of the individual agents, we achieve cooperation between systems of different types. The underlying passivity property of our control approaches ensures that cooperation is stable and robust. Neither of the two approaches rely on the modeling information of neighbors, locally, which simplifies the interconnection of applicable systems and makes the approaches modular in their use. Our first approach is a generalized cooperative Interconnection-and-Damping Assignment passivity-based control (IDA-PBC) scheme for networks of fully actuated and underactuated systems. Our approach leverages the definition of end-effector coordinates in existing single-agent IDA-PBC solutions for underactuated systems to satisfy the matching conditions, independently of the cooperative control input. Accordingly, our approach integrates a large set of existing single-agent solutions and facilitates cooperative control between these and fully actuated systems. Our second approach proposes agent outputs composed of their end-effector coordinates and velocities to guarantee cooperative stability for networks of fully actuated systems in the presence of communication delays. We validate both approaches in simulation and experiments.
... In the last decades, agreement problems have been examined in different settings, including consensus [1], distributed optimization [2], and synchronization [3]. Among various solutions in the literature to agreement problems, those based on passivity are of particular interest, e.g., [4], [5], [6], [7], [8], [9], [10] because passivity is a property possessed by various physical systems. ...
In this letter, we consider nonlinear network systems under unknown disturbances and address the problem of mixed input/output consensus, i.e., consensus among disjoint sets of input and output nodes. We develop two control schemes based on different notions of passivity: 1) Krasovskii passivity and 2) shifted passivity. Furthermore, we propose an input consensus controller which is applicable to either Krasovskii or shifted passive systems. Finally, we validate the proposed controllers in simulation by achieving current sharing in a heterogeneous DC microgrid and power sharing in an AC power system, which are Krasovskii and shifted passive, respectively.
... Passivity theory, in particular, has been used extensively in the literature for understanding various fundamental physical properties of power system networks as well as the methods for controlling them by analyzing the dynamical models of synchronous electrical generators. For example, papers such as [3], [4] have derived results on modeling and stability properties of power grids using notion of passivity and differential passivity. Papers such as [5]- [8], on the other hand, have presented various centralized and distributed control algorithms for automatic generation control (AGC) of synchronous generators, based on the passivity of the input-output map from their turbine mechanical power input to their frequency T. Ishizaki deviations. ...
We study equilibrium-independent passivity properties of nonlinear dynamical models of electric power systems. The model of our interest comprises of a feedback interconnection of two subsystems, one being a first-order linear ordinary differential equation and the other being a set of nonlinear differential algebraic equations (DAE). We prove the following three facts by analyzing the nonlinear DAE subsystem. First, a lossless transmission network is necessary for guaranteeing equilibrium-independent passivity of the DAE. Second, the convexity of a strain energy function characterizes the largest set of equilibria over which this DAE subsystem is equilibrium-independent passive. Finally, we prove that the strain energy function of a power system with two-axis generator models is convex if and only if its flux linkage dynamics are stable, and the strain energy function of a classical generator model derived by a singular perturbation approximation of the flux linkage dynamics is convex. These novel findings are derived by elaborating on linearization and Kron reduction properties of the power system model. Numerical simulation of the IEEE 9-bus power system model demonstrates the practical implications of the various mathematical results.
... Generally, the UAV-payload connection types include active connections and passive connections [6]. Active connection involves equipping the vehicle with a gripper to grasp and hold the payload rigidly [7], while passive connection refers to suspending the payload through cables [8] or via a universal joint [9]. The gripper attachment increases the mass and inertia of the system considerably and thereby makes the system respond slowly. ...
This article studies the collaborative transportation of a cable-suspended pipe by two quadrotors. A force-coordination control scheme is proposed, where a force-consensus term is introduced to average the load distribution between the quadrotors. Since thrust uncertainty and cable force are coupled together in the acceleration channel, disturbance observer can only obtain the lumped disturbance estimate. Under the quasi-static condition, a disturbance separation strategy is developed to remove the thrust uncertainty estimate for precise cable force estimation. The stability of the overall system is analyzed using Lyapunov theory. Both numerical simulations and indoor experiments using heterogeneous quadrotors validate the effectiveness of thrust uncertainty separation and force-consensus algorithm.
... In (Proskurnikov, 2014), the problem of consensus of nonlinear agents of the 2nd order (double integrators) is investigated, in addition, nonstationary communication functions are considered. A number of papers consider the problems of adaptive synchronization of interconnected dynamic subsystems (Chopra & Spong, 2006;Arcak, 2007) and adaptive management of agents with significantly different dynamic properties. The influence of the bandwidth of the information exchange channel between agents on the stability of the agent system and its dynamic properties as a whole is studied (Andrievsky et al., 2010;Amelin, et al., 2019). ...
The use of a formation consisting of adaptive autonomous mobile agents allows solving a wide range of tasks that are often beyond the capabilities of individual agents. A multi-agent formation is a complex high-order dynamic system, so analyzing the stability of such a system is a complex task. At present, the problem of estimating the stability of a formation formed by substantially nonlinear high-order agents with variable dynamic parameters is not sufficiently considered. This task is particularly important for a formation that is affected by complex unstable environmental conditions, in particular for the formation of unmanned aerial vehicles (UAVs). A method for analyzing the stability of formations of nonlinear agents with different types and orders of transfer function has been developed for studying information exchange in UAV networks. The new approach is based on the use of the Popov frequency criterion and the piecewise linear approximation of the hodograph. A computational experiment was performed to analyze the stability of a formation with transfer functions of various types and orders from the 1st to the 10th. The conducted studies revealed a significant difference in the calculated boundary coefficients of formation stability in the linear and nonlinear modes, which confirms the need to analyze the nonlinear stability under the influence of strong destabilizing influences on the formation.
... One may check the synchronization results of Lur'e systems for recent relevant advances. The output synchronization [58] of Lur'e systems composed of a passive linear system and a static feedback nonlinearity usually requires passivity, which results in the necessary application of passivity-based approaches [133,9,111]. To overcome the limitations of the passivity-based methods, the algebraic connectivity of the considered network of systems is required to exceed a threshold. ...
This work presents new results on input-to-output stability conditions, robust synchronization, and state estimation for generalized Persidskii systems in the presence of external input/disturbance, as well as input-to-state stability analysis of those dynamics with time delays. The thesis starts from the problem formulation followed by a brief introduction and state-of-the-art in Chapter 1. Preliminary definitions and auxiliary results are summarized in Chapter 2. Chapter 3 focuses on input-to-output stability conditions and their application to robust synchronization of generalized Persidskii models. The synchronization conditions are illustrated by the example of the neural Hindmarsh-Rose model. Chapter 4 considers a state observer designed for generalized Persidskii systems with nonlinear measurements, state disturbances, and output noise. The theory of input-to-output stability is applied to obtain robust stability and convergence conditions for the estimation error. Two applications to a perturbed two-mass spring-damper system and a multi-group susceptible-infected-susceptible model are provided to demonstrate the efficacy and performances of the proposed observer. In Chapter 5, the delay-dependent input-to-state stability and stabilization conditions for time-delay generalized Persidskii systems are studied and formulated in termsof state-dependent matrix inequalities. Numerical examples of opinion dynamics and a modified Lotka-Volterra model illustrate the proposed results.
... The majority of results that have been obtained for homogeneous networks, approach the problem using control tools specifically adapted to deal with a distributed framework. Among them, we recall for instance a passivity approach in Arcak (2007), a dissipativity one in Stan and Sepulchre (2007) and an ISS one in Casadei et al. (2019a,b). High-gain techniques, inherited from high-gain observers theory or high-gain domination approaches (see, e.g., Chopra and Spong (2008) (2017)), form another notable class of solutions. ...
This thesis deals with the notion of incremental stability and its application in the contextof control design for nonlinear systems. The manuscript is divided into four main chapters,each of them dealing with different topics but strictly related among them. In the firstchapter, we study the notion of incremental stability for nonlinear control systems. Inshort, a system is said to be incrementally stable if trajectories starting from differentinitial conditions asymptotically converge towards each other. Such a notion is of interestdue to several properties that incrementally stable systems share, such as periodicity oftrajectories, robustness with respect to external perturbations, and many others. Amongthe different tools to study such a notion, we focus on the so-called (Riemaniann) ‘metricbased’approach. Despite the theory of incrementally stable systems is receiving a lotof interest from the worldwide control community, several open questions need to beanswered yet, concerning the analysis of incremental properties and the feedback controldesign achieving incremental stability.In the second chapter, we focus on the output regulation problem. The goal is to designa (dynamic) control law such that the output of a nonlinear system can asymptoticallytrack a reference and, at the same time, reject perturbations. In particular, we aim toachieve “global” output regulation, meaning that the regulation task must be achievedindependently of the initial conditions and on the amplitude of the external signals. Thechallenge is to guarantee the existence of a steady-state solution on which the regulationerror is zero for every value of the external signals, and the convergence of trajectoriestowards such a solution for every initial condition. While tools achieving regulation forminimum phase systems in normal form are well developed, much less is known for moregeneral classes of systems, especially when global regulation is the goal. Therefore, newtools need to be developed. In our approach, in particular, we cast the regulation taskinto the incremental framework and we tackle the problem with tools derived from thefirst chapter of the manuscript.In the third chapter, we focus on the multiagent synchronization problem. Here, weconsider a group of single identical entities which communicate among them through acommunication protocol. The objective is the design of a distributed coupling control lawsuch that these entities reach an agreement on their state evaluation. While the theory forlinear systems is well developed, many questions remain open for nonlinear ones. In ourapproach, we cast the synchronization problem into the incremental framework. Such a choice is motivated by the fact that, if agents are described by the same dynamical model,then the synchronization problem corresponds to the design of a distributed control lawsuch that different trajectories of the same differential equation asymptotically convergetowards each other.In the fourth and last chapter, we focus on two practical applications. In particular, weconsider two separate problems. The first problem is a robust output set-point trackingproblem for a power flow controller. A power flow controller is an electric circuit whoserole is to regulate the power of the lines to which it is attached, despite the uncertaintiesof the plant parameters and of the references to be tracked. The second problem is aperiodic trajectory tracking for a ventilation machine. A ventilation machine is a piece ofmedical equipment used to support patients breathing. The objective here is to design acontrol law such that the machine can track a periodic pressure signal representing thebreathing phase, despite the uncertainties in the plant.At the end of the last chapter, a summary of the thesis written in French is present.
... To achieve this type of operation, many theoretical studies have been conducted to provide plug and play property to the power distribution system. In order to have this property, it is useful to give passivity or dissipativity (Willems (1971), Willems (1972a), Willems (1972b)) to the distribution system (Arcak (2007), Qu and Simaan (2014)). ...
In this paper, we consider a power distribution system consisting of a straight feeder line. A nonlinear ordinary differential equation (ODE) model is used to describe the voltage distribution profile over the feeder line. At first, we show the dissipativity of the subsystems corresponding to active and reactive powers. We also show that the dissipation rates of these subsystem coincide with the distribution loss given by a square of current amplitudes. Moreover, the entire distribution system is decomposed into two subsystems corresponding to voltage amplitude and phase. As a main result, we prove the dissipativity of these subsystems based on the decomposition. As a physical interpretation of these results, we clarify that the phenomena related to the gradients of the voltage amplitude and phase are induced in a typical power distribution system from the dissipation equalities. Finally, we discuss a reduction of distribution losses by injecting a linear combination of the active and reactive powers as a control input based on the dissipation rate of the subsystem corresponding to voltage amplitude.
... Due to the widespread practical applications of multi-agent systems (MASs) in various fields such as material transportation, environmental detection and civilian tasks, they have attracted the attention of many scholars in the past few decades, and many achievements have been made in the research of MASs, including but not limited to [1][2][3][4][5][6]. The latest technological progress in the field of communication and computing has promoted the development of multi-agents, which can finish complex tasks that cannot usually be accomplished by a single agent. ...
In this paper, the obstacle avoidance problem-based leader–following formation tracking of nonholonomic wheeled mobile robots with unknown parameters of desired trajectory is investigated. First, the under-actuated system is transformed into a fully-actuated system by obtaining an auxiliary control variable using the transverse function. Second, by introducing a potential function for each obstacle, the influence of obstacles is considered in trajectory tracking, and the effect of the potential field on mobile robots is taken into account in the system tracking error. Third, the adaptive laws are designed to estimate the unknown parameters of the desired trajectory. Fourth, the results show that the formation error with respect to the actual position and orientation can be arbitrarily small by selecting appropriate design parameters. Finally, simulation examples are used to demonstrate that the proposed control scheme is effective.
... The passivity is a significant property of dynamical system, which gives promising application prospect in diverse fields, such as stability, synchronization and group coordination (Sheng and Zeng 2017;Arcak 2007;Wang et al. 2019a;Guo et al. 2014;Wang et al. , 2019bLi and Zheng 2020). Note that passivity is a useful tool for study of stability problems. ...
This paper investigates the fixed-time passivity of coupled quaternion-valued neural networks (CQVNNs) with multiple delayed couplings. Different from the finite-time passivity proposed in Wang et al. (IEEE Trans Cybern 50(5):2014–2025, 2020), a novel concept of fixed-time passivity is considered in this paper, where the settling time is independent on initial value. Compared with the existing literature,
in this work we extend the passivity problem of CNNs from real field to quaternion field. In particular, the multiple coupling delays and discontinuous activation functions are considered, which have not been investigated in previous papers yet. By designing appropriate controllers and applying inequality technique, sufficient conditions are obtained correspondingly for three different types of fixed-time passivity of CQVNNs. Finally, simulations are given to substantiate the correctness of the theoretical results.
... Since many actual systems contain complicated nonlinear dynamics, the researched system models have recently extended to nonlinear MASs [5][6][7][8]. According to the different communication relationships between agents, the consensus control can be divided into two categories: leaderless consensus control [9, 10] and leader-following consensus control [11][12][13][14]. It is shown in [1, 3] that the leader-following consensus control is more practical than the leaderless consensus control. ...
When the denial-of-service (DoS) attacks and the actuator faults exist simultaneously, this paper considers the problem of estimator-based adaptive consensus asymptotic tracking control for a class of constrained nonlinear multi-agent systems (MASs) with the state constraints. In light of the fact that the consensus tracking performance of the researched nonlinear MASs will be seriously impaired by the DoS attacks and actuator faults, a novel secure consensus asymptotic tracking control algorithm is developed. Due to only the output information of each agent can be obtained, the design difficulty of the security control algorithm is greatly increased. Following that, an estimator that approximates the unknown nonlinearities and observes the immeasurable system states is constructed for each agent. Furthermore, the barrier Lyapunov functions (BLFs) are introduced to solve the state constraint requirements of each agent, meanwhile the dynamic surface control (DSC) technology is used to overcome the difficulty of ``explosion of complexity'' in the backstepping process. The proposed event-triggered anti-attack controller guarantees that all closed-loop signals remain bounded, and the consensus tracking errors asymptotically converge to zero. At last, the proposed control algorithm is applied to a forced damped pendulums (FDPs) to verify the correctness of the theoretical result.
... More recent references [1] and [2] show some results on coordinated multiagent control. Consensus or consensus algorithm [3], [4], [5], formation control [6][7][8][9]. Traditionally, the system used a timed control method to adjust the actuator state at each sample time. Such time-controlled adjustment mechanisms can cause significant and frequent changes in the actuator state, leading to unnecessary energy consumption and actuator damage. ...
Gradual advancement of control technology gives rise to the studies of the stability of linear systems. The stability of the linear multiagent system is motivated by increasing utilization of agent dynamics together with the number of control protocols associated with each agent. To that respect, in this report, the idea of event-triggered control and the stability of the linear multiagent system under steady-state performance conditions will be presented. By applying a steady-state condition, the state of an agent in the closed-loop linear system will be discussed using fixed network topology both in discrete-time and continuous-time. Moreover, the system is also analyzed by employing the Lyapunov function methods, and then an average consensus of the system will also be realized. Finally, we will verify the system's average consensus and stability via a simulation example.
... In this paper, we examine the synchronization of discrete-time dynamic agents by utilizing the recently developed phase tool [18]- [21]. Within the literature, the works employing the positive realness and negative imaginariness provide the qualitative phase-type conditions for synchronization [13], [14], [22], [23]. We focus on the quantitative phase-type conditions. ...
... (3) Finally, let τ := − min i / ∈N (τ i (x i ) − τ i (x i )) ≥ 0, and for each s ∈ [1,2], define p(s) as ...
This paper examines how weak synaptic coupling can achieve rapid synchronization in heterogeneous networks. The assumptions aim at capturing the key mathematical properties that make this possible for biophysical networks. In particular, the combination of nodal excitability and synaptic coupling are shown to be essential to the phenomenon.
Passivity theory is one of the cornerstones of control theory providing a systematic way to study the stability of interconnected systems. It is well known that many systems are not passive, and must be passivized in order to be included in the framework of passivity theory. Input-output (loop) transformations are the most general tool for passivizing systems. In this paper, we propose a characterization of all possible input-output transformations that map a system with given shortage of passivity to a system with prescribed excess of passivity. We do so by using the connection between passivity theory and cones for SISO systems.
In this chapter, we study the problem of synchronization of homogeneous multi-agent systems in which the network is described by a connected graph. The dynamics of each single agent is described by an input-affine nonlinear system. We propose a feedback controller based on contraction analysis and Riemannian metrics. The starting point is the existence of a solution to a specific partial differential inequality which generalizes, in the nonlinear context, the well-known algebraic Riccati inequality. Further, we provide new results that allow to relax the Killing vector property so that to obtain a less stringent solution to be approximated with numerical methods. The proposed approach allows to design an infinite-gain margin feedback which is the fundamental ingredient to solve the problem of synchronization. We show that the synchronization problem is solved between two agents and we conjecture that the result holds for any connected graph. Simulations for a connected directed graph of Duffing oscillators corroborate the conjecture.
This research aims to develop a new learning-based flocking control framework that ensures inter-agent free collision. To achieve this goal, a leader-following flocking control based on a deep Q-network (DQN) is designed to comply with the three Reynolds’ flocking rules. However, due to the inherent conflict between the navigation attraction and inter-agent repulsion in the leader-following flocking scenario, there exists a potential risk of inter-agent collisions, particularly with limited training episodes. Failure to prevent such collision not only caused penalties in training but could lead to damage when the proposed control framework is executed on hardware. To address this issue, a control barrier function (CBF) is incorporated into the learning strategy to ensure collision-free flocking behavior. Moreover, the proposed learning framework with CBF enhances training efficiency and reduces the complexity of reward function design and tuning. Simulation results demonstrate the effectiveness and benefits of the proposed learning methodology and control framework.
This note proposes a passivity-based control method for trajectory tracking and formation control of nonholonomic wheeled robots without velocity measurements. Coordinate transformations are used to incorporate the nonholonomic constraints, which are then avoided by controlling the front end of the robot rather than the center of the wheel axle into the differential equations. Starting from the passivity-based coordination design, the control goals are achieved via an internal controller for velocity tracking and heading control and an external controller for formation in the port-Hamiltonian framework. This approach endows the resulting controller with a physical interpretation. To avoid unavailable velocity measurements or unreliable velocity estimations, we derive the distributed control law with only position measurements by introducing a dynamic extension. In addition, we prove that our approach is suitable not only for acyclic graphs but also for a class of non-acyclic graphs, namely, ring graphs. Simulations are provided to illustrate the effectiveness of the approach.
This chapter gives an overview of the synchronization of chaotic systems, generalized synchronization, multi-synchronization problem and their associated problems.
This paper considers the passivity-based boundary control for stochastic delay reaction–diffusion systems (SDRDSs) with boundary input–output. Delay-dependent sufficient conditions are obtained in the sense of expectation for SDRDSs by the use of Lyapunov functional method, for the input strict passivity and the output strict passivity, respectively. Then delay-dependent sufficient conditions are presented to ensure the uncertainty SDRDSs to achieve the robust passivity. Moreover, by utilising the techniques of Green formula and Poincare's inequality, the difficulty is handled caused by the boundary input and boundary controller. Besides, the troubles brought by the existence of the stochastic term and uncertain parameters are resolved by stochastic inequality techniques. Finally, numerical simulations are provided to show the validity of our theoretical results.
This paper proposes a novel adaptive spacing policy (ASP) of flocking control for multi-agent systems to accommodate complex and specific environments of engineering applications, such as connected and automated vehicles (CAVs). Compared with the existing rigid spacing policy (RSP) that uses a predefined fixed lattice, the ASP involves an ellipse interaction field around every agent, which is adaptive to the agent's speed and orientation. Leveraged by the proposed ASP, an augmented flocking control protocol can achieve heterogeneous lattice formations for a group of agents. The stability analysis proves that the flocking control protocol with the ASP can achieve speed consensus, cohesion, and separation asymptotically. Compared with the RSP, the simulation results demonstrate that the ASP-based flocking control can successfully achieve heterogeneous lattice formation, which displays the features of vehicle motions in CAV applications.
In this paper we consider stability of large scale interconnected nonlinear systems that satisfy a strict dissipativity property in terms of local storage and supply functions. Existing compositional stability criteria certify global stability by constructing a global Lyapunov function as the (weighted) sum of local storage functions. We generalize these results by unifying spatial composition, i.e., (weighted) sum of local supply functions is neutral, with temporal composition, i.e., (weighted) sum of supply functions over a time cycle is neutral. Two benchmark examples illustrate the benefits of the developed compositional stability criteria in terms of reducing conservatism and constrained distributed stabilization.
For state-dependent cooperation-competition networks, the output bipartite consensus problem of heterogeneous uncertain agents is studied in this paper. In our framework, all the agents are described by second-order continuous-time nonlinear systems with different intrinsic dynamics, and the agents' uncertainties are characterized by unknown parameters in the intrinsic term. Then, the edge evolution rules with hysteresis coefficients are proposed via the classification strategy. To solve this problem, the distributed Lyapunov-based redesign method with the potential function term is first applied to such second-order heterogeneous uncertain multi-agent system. Through the three-step correlation design, the explicit expressions of the distributed controllers and the unknown parameter estimators are obtained, and it is shown that the state-dependent cooperation-competition network is always connected and maintains structural balance for all time if an initial structurally balanced and connected network topology is provided. On this basis, it follows from the total Lyapunov function that output bipartite consensus can be achieved asymptotically for the heterogeneous uncertain multi-agent system. Finally, a numerical simulation is provided to validate the structural balance of state-dependent networks and output bipartite consensus of heterogeneous uncertain agents.
A sparsity pattern in
$\mathbb{R}^{n \times m}$
, for
$m\geq n$
, is a vector subspace of matrices admitting a basis consisting of canonical basis vectors in
$\mathbb{R}^{n \times m}$
. We represent a sparsity pattern by a matrix with
$0/\star$
-entries, where
$\star$
-entries are arbitrary real numbers and 0-entries are equal to 0. We say that a sparsity pattern has full structural rank if the maximal rank of matrices contained in it is
$n$
. In this paper, we investigate the degree of resilience of patterns with full structural rank: We address questions such as how many
$\star$
-entries can be removed without decreasing the structural rank and, reciprocally, how many
$\star$
-entries one needs to add so as to increase the said degree of resilience to reach a target. Our approach goes by translating these questions into max-flow problems on appropriately defined bipartite graphs. Based on these translations, we provide algorithms that solve the problems in polynomial time.
This paper studies the state agreement problem with the objective to ensure the asymptotic coincidence of all states of multiple nonlinear dynamical systems. The coupling structure of such systems is characterized in qualitative terms by means of a suitably deflned directed graph. Under a suitable sub- tangentiality assumption on the vector flelds of the systems, we obtain a necessary and su-cient graphical condition for their state agreement via nonsmooth analysis, with the invariance principle playing a central role. As applications, we study synchronization of coupled Kuramoto oscillators and synthesis of a rendezvous controller for a multi-agent system. Copyright c ∞2005 IFAC
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.
We provide a control-theoretic perspective on the design of distributed agreement protocols. First, we explore agreement-protocol analysis and design for a network of agents with single-integrator dynamics and arbitrary linear observations. One key contribution of our work is the analysis of protocols for networks with quite general observation topologies, including with multiple observations made by each agent. Another contribution is the development of techniques for agreement law design - i.e., for assignment of the dependence of the agreed-upon value on the initial states of the agents. Second, we explore agreement in a quasi-linear model with a stochastic protocol, which we call the controlled voter model. We motivate our study of this model, develop tests for whether agreement is achieved, and consider design of the agreement law. Finally, we provide some further thoughts regarding our control-theoretic perspective on agreement, including ideas for fault-tolerant protocol design using our approach.
The objective of this paper is to explore the feasibility of using multiple low-altitude, short endurance (LASE) unmanned air vehicles (UAVs) to cooperatively monitor and track the propagation of large forest fires. A real-time algorithm is described for tracking the perimeter of fires with an on-board infrared sensor. Using this algorithm, we develop a decentralized multiple-UAV approach to monitoring the perimeter of a fire. The UAVs are assumed to have limited communication and sensing range. The effectiveness of the approach is demonstrated in simulation using a six degree-of-freedom dynamic model for the UAV and a numerical propagation model for the forest fire. Salient features of the approach include the ability to monitor a changing fire perimeter, the ability to systematically add and remove UAVs from the team, and the ability to supply time-critical information to fire fighters.
In this article we specify an M-member "individual-based" continuous time swarm model with individuals that move in an n-dimensional space according to an attractant/repellent or a nutrient profile. The motion of each individual is determined by three factors: i) attraction to the other individuals on long distances; ii) repulsion from the other individuals on short distances; and iii) attraction to the more favorable regions (or repulsion from the unfavorable regions) of the attractant/repellent profile. The emergent behavior of the swarm motion is the result of a balance between inter-individual interactions and the simultaneous interactions of the swarm members with their environment. We study the stability properties of the collective behavior of the swarm for different profiles and provide conditions for collective convergence to more favorable regions of the profile.
The paper investigates the stability properties of mobile agent formations which are based on leader following. We derive nonlinear gain estimates that capture how leader behavior affects the interconnection errors observed in the formation. Leader-to-formation stability (LFS) gains quantify error amplification, relate interconnection topology to stability and performance, and offer safety bounds for different formation topologies. Analysis based on the LFS gains provides insight to error propagation and suggests ways to improve the safety, robustness, and performance characteristics of a formation.
A new infinitesimal sufficient condition is given for uniform global asymptotic stability (UGAS) for time-varying nonlinear systems. It is used to show that a certain relaxed persistency of excitation condition, called uniform δ-persistency of excitation (Uδ-PE), is sufficient for uniform global asymptotic stability in certain situations. Uδ-PE of the right-hand side of a time-varying differential equation is also shown to be necessary under a uniform Lipschitz condition. The infinitesimal sufficient condition for UGAS involves the inner products of the flow field with the gradients of a finite number of possibly sign-indefinite, locally Lipschitz Lyapunov-like functions. These inner products are supposed to be bounded by functions that have a certain nested, or triangular, negative semidefinite structure. This idea is reminiscent of a previous idea of Matrosov who supplemented a Lyapunov function having a negative semidefinite derivative with an additional function having a derivative that is "definitely nonzero" where the derivative of the Lyapunov function is zero. For this reason, we call the main result a nested Matrosov theorem. The utility of our results on stability analysis is illustrated through the well-known case-study of the nonholonomic integrator.
This paper deals with recent advances in developing direct methods for studying the transient stability problem of single-machine and multimachine power systems. The paper starts out with the construction of the mathematical model that is usually employed in the analyis of power system transient stability. Computer simulation methods are then briefly discussed, and it is indicated why accurate direct methods for transient stability investigations would be most welcome. It is shown that the classical direct methods, which are based on energy considerations, can be derived and generalized by means of Lyapunov's second method. The main purpose of the paper is to give an exposition of the interesting results that have been obtained by applying Lyapunov's second method to the transient stability problem of single-machine and multimachine power systems. In the final portion of the paper some areas for further research are discussed.
The aggregate motion of a flock of birds, a herd of land animals, or a school of fish is a beautiful and familiar part of the natural world. But this type of complex motion is rarely seen in computer animation. This paper explores an approach based on simulation as an alternative to scripting the paths of each bird individually. The simulated flock is an elaboration of a particle system, with the simulated birds being the particles. The aggregate motion of the simulated flock is created by a distributed behavioral model much like that at work in a natural flock; the birds choose their own course. Each simulated bird is implemented as an independent actor that navigates according to its local perception of the dynamic environment, the laws of simulated physics that rule its motion, and a set of behaviors programmed into it by the "animator." The aggregate motion of the simulated flock is the result of the dense interaction of the relatively simple behaviors of the individual simulated b...
. This paper presents a Converse Lyapunov Function Theorem motivated by robust control analysis and design. Our result is based upon, but generalizes, various aspects of well-known classical theorems. In a unified and natural manner, it (1) allows arbitrary bounded time-varying parameters in the system description, (2) deals with global asymptotic stability, (3) results in smooth (infinitely differentiable) Lyapunov functions, and (4) applies to stability with respect to not necessarily compact invariant sets. 1. Introduction. This work is motivated by problems of robust nonlinear stabilization. One of our main contributions is to provide a statement and proof of a Converse Lyapunov Function Theorem which is in a form particularly useful for the study of such feedback control analysis and design problems. We provide a single (and natural) unified result that: 1. applies to stability with respect to not necessarily compact invariant sets; 2. deals with global (as opposed to merely loca...
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 disagreement function that is a common Lyapunov function for the disagreement dynamics of a directed network with switching topology. A distinctive feature of this work is to address consensus problems for networks with directed information flow. We provide analytical tools that rely on algebraic graph theory, matrix theory, and control theory. Simulations are provided that demonstrate the effectiveness of our theoretical results.
The stability problem in power systems was mathematically formulated in Chapter I as one of ensuring that the state of the power system at the instant of clearing the fault is inside the region of stability (ROS) of the post-fault stable equilibrium point. Computation of ROS is perhaps the most difficult task in successfully using the Lyapunov/energy functions for stability analysis. In this chapter we explain the foundation of the theory underlying the characterization of the stability boundary of nonlinear autonomous dynamical systems and then indicate its application to power systems. We also provide a theoretical foundation to the potential energy boundary surface method. Both these theoretical results are due to Chiang et al. [1988]. A parallel development in characterizing ROS in the entire state space is due to Zaborsky et al. [1988]. In terms of application to realistic systems, three basic methods have proved to be successful in application.
This paper studies the state agreement problem with the objective to ensure the asymptotic coincidence of all states of multiple nonlinear dynamical systems. The coupling structure of such systems is characterized in qualitative terms by means of a suitably defined directed graph. Under a suitable subtangentiality assumption on the vector fields of the systems, we obtain a necessary and sufficient graphical condition for their state agreement via nonsmooth analysis, with the invariance principle playing a central role. As applications, we study synchronization of coupled Kuramoto oscillators and synthesis of a rendezvous controller for a multi-agent system.
1 Power System Stability in Single Machine System.- 1.1 Introduction.- 1.2 Statement of the Stability Problem.- 1.3 Mathematical Formulation of the Problem.- 1.4 Modeling Issues.- 1.5 Motivation Through Single Machine Infinite Bus System.- 1.6 Chapter Outline.- 2 Energy Functions for Classical Models.- 2.1 Introduction.- 2.2 Internal Node Representation.- 2.3 Energy Functions for Internal Node Models.- 2.4 Individual Machine and other Energy Functions.- 2.5 Structure Preserving Energy Functions.- 2.6 Alternative Form of the Structure Preserving Energy Function.- 2.7 Positive Definiteness of the Energy Integral.- 2.8 Tsolas-Araposthasis-Varaiya Model.- 3 Reduced Order Energy Functions.- 3.1 Introduction.- 3.2 Individual Machine and Group Energy Function.- 3.3 Simplified Form of the Individual Machine Energy Function.- 3.4 Cutset Energy Function.- 3.5 Example of Cutset Energy Function.- 3.6 Extended Equal Area Criterion (EEAC).- 3.7 The Quasi Unstable Equilibrium Point (QUEP) Method.- 3.8 Decomposition-Aggregation Method.- 3.9 Time Scale Energies.- 4 Energy Functions with Detailed Models of Synchronous Machines and Its Control.- 4.1 Introduction.- 4.2 Single Machine System With Flux Decay Model.- 4.3 Multi-Machine Systems With Flux Decay Model (Method of Parameter Variations).- 4.4 Lyapunov Functions for Multi-Machine Systems With Flux Decay Model.- 4.5 Multi-Machine Systems With Flux Decay Models and AVR.- 4.6 Energy Functions With Detailed Models.- 4.7 Lyapunov Function for Multi-Machine Systems With Flux Decay and Nonlinear Voltage Dependent Loads.- 5 Region of Stability in Power Systems.- 5.1 Introduction.- 5.2 Characterization of the Stability Boundary.- 5.3 Region of Stability.- 5.4 Method of Hyperplanes and Hypersurfaces.- 5.5 Potential Energy Boundary Surface (PEBS) Method.- 5.6 Hybrid Method Using the Gradient System.- 6 Practical Applications of the Energy Function Method.- 6.1 Introduction.- 6.2 The Controlling u.e.p. Method.- 6.3 Modifications to the Controlling u.e.p. Method.- 6.4 Potential Energy Boundary Surface (PEBS) Method.- 6.5 Mode of Instability (MOI) Method.- 6.6 Dynamic Security Assessment.- 7 Future Research Issues.- Appendix A 10 Machine 39 Bus System Data.- References.
The aim of these lecture notes is to provide a synthesis between classical input-output and closed-loop stability theory, in particular the small-gain and passivity theorems, and recent work on nonlinear H( and passivity-based control. The treatment of the theory of dissipative systems is the main aspect of these lecture notes. Fundamentals of passivity techniques are summarised, and it is shown that the passivity properties of different classes of physical systems can be unified within a generalised Hamiltonian framework. Key developments in linear robust control theory are extended to the nonlinear context using L2-gain techniques. An extensive treatment of nonlinear H( control theory is presented, emphasising its main structural features. Since the application of L2-gain techniques relies on solving Hamilton-Jacobi inequalities the structure of their solution sets and conditions for solvability are derived.
We consider differential inclusions where
a positive semidefinite function of the solutions satisfies a
class-${\mathcal{KL}}$ estimate
in terms of time and a second positive semidefinite function of the
initial condition.
We show that a smooth converse Lyapunov function, i.e., one whose
derivative along solutions can be
used to establish the class-${\mathcal{KL}}$ estimate, exists if and
only if the class-${\mathcal{KL}}$ estimate
is robust, i.e., it holds for a larger, perturbed differential
inclusion.
It remains an open question whether all class-${\mathcal{KL}}$
estimates are robust.
One sufficient condition for robustness is that the original
differential inclusion is locally Lipschitz.
Another sufficient condition is that the two positive semidefinite
functions agree and
a backward completability condition holds. These special cases unify
and generalize many results
on converse Lyapunov theorems for differential equations and
differential inclusions that have appeared in the literature.
Let G = (V, E) be a simple graph on vertex set V = {nu(1), nu(2),...., nu(n)}. Further let d(i) be the degree of nu(i) and N-i be the set of neighbors of nu(i). It is shown that max {d(i) + d(j) - \N-i boolean AND N-j\ : 1 less than or equal to i < i less than or equal to n, nu(i)nu(j) is an element of E} is an upper bound for the largest eigenvalue of the Laplacian matrix of G, where \N-i boolean AND N-j\ denotes the number of common neighbors between nu(i) and nu(j). For any G, this bound does not exceed the order of G. Further using the concept of common neighbors another upper bound for the largest eigenvalue of the Laplacian matrix of a graph has been obtained as max {root2(d(i)(2) + d(i)m(i)') : 1 less than or equal to i less than or equal to n}, where m(i)' = Sigma(j) {d(j) - \N-i boolean AND N-j\ : nu(i) nu(j) E/d(i)}.
Synchronization, collective behavior, and group cooperation have been the object of extensive recent research. A fundamental understanding of aggregate motions in the natural world, such as bird flocks, fish schools, animal herds, or bee swarms, for instance, would greatly help in achieving desired collective behaviors of artificial multi-agent systems, such as vehicles with distributed cooperative control rules. In [38], Reynolds published his well-known computer model of boids, successfully forming an animation flock using three local rules: collision avoidance, velocity matching, and flock centering. Motivated by the growth of colonies of bacteria, Viscek et al.[55] proposed a similar discrete-time model which realizes heading matching using information only from neighbors. Visceks model was later analyzed analytically [16, 52, 53]. Models in continuous-time [1, 22, 32, 33, 62] and combinations of Reynolds three rules [21, 34, 35, 49, 50] were also studied. Related questions can also be found e.g. in [3, 18, 20, 42], in oscillator synchronization [48], as well as in physics in the study of lasers [39] or of Bose-Einstein condensation [17].
We present a framework for coordinated and distributed control of
multiple autonomous vehicles using artificial potentials and virtual
leaders. Artificial potentials define interaction control forces between
neighboring vehicles and are designed to enforce a desired inter-vehicle
spacing. A virtual leader is a moving reference point that influences
vehicles in its neighborhood by means of additional artificial
potentials. Virtual leaders can be used to manipulate group geometry and
direct the motion of the group. The approach provides a construction for
a Lyapunov function to prove closed-loop stability using the system
kinetic energy and the artificial potential energy. Dissipative control
terms are included to achieve asymptotic stability. The framework allows
for a homogeneous group with no ordering of vehicles; this adds
robustness of the group to a single vehicle failure
As a distributed solution to multi-agent coordination, consensus or agreement problems have been studied extensively in the literature. This paper provides a survey of consensus problems in multi-agent cooperative control with the goal of promoting research in this area. Theoretical results regarding consensus seeking under both time-invariant and dynamically changing information exchange topologies are summarized. Applications of consensus protocols to multiagent coordination are investigated. Future research directions and open problems are also proposed.
We consider the problem of finding a linear iteration that yields distributed averaging consensus over a network, i.e., that asymptotically computes the average of some initial values given at the nodes. When the iteration is assumed symmetric, the problem of finding the fastest converging linear iteration can be cast as a semidefinite program, and therefore efficiently and globally solved. These optimal linear iterations are often substantially faster than several common heuristics that are based on the Laplacian of the associated graph.We show how problem structure can be exploited to speed up interior-point methods for solving the fastest distributed linear iteration problem, for networks with up to a thousand or so edges. We also describe a simple subgradient method that handles far larger problems, with up to 100 000 edges. We give several extensions and variations on the basic problem.
In the early days of nonlinear control theory most of the stability, optimality and uncertainty concepts were descriptive rather than constructive. This survey describes their ‘activation’ into design tools and constructive procedures. Structural properties of nonlinear systems, such as relative degree and zero dynamics, are connected to passivity, while dissipativity, as a finite -gain property, also appears in the disturbance attenuation problem, a nonlinear counterpart of robust linear control. Passivation-based designs exploit the connections between passivity and inverse optimality, and between Lyapunov functions and optimal value functions. Recursive design procedures, such as backstepping and forwarding, achieve certain optimal properties for important classes of nonlinear systems. The survey concludes with four representative applications. The selection of the topics and their interpretations are greatly influenced by the experience and personal views of the senior author.
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
This is a substantial revision of a much-quoted monograph, first published in 1974. The structure is unchanged, but the text has been clarified and the notation brought into line with current practice. A large number of 'Additional Results' are included at the end of each chapter, thereby covering most of the major advances in the last twenty years. Professor Biggs' basic aim remains to express properties of graphs in algebraic terms, then to deduce theorems about them. In the first part, he tackles the applications of linear algebra and matrix theory to the study of graphs; algebraic constructions such as adjacency matrix and the incidence matrix and their applications are discussed in depth. There follows an extensive account of the theory of chromatic polynomials, a subject which has strong links with the 'interaction models' studied in theoretical physics, and the theory of knots. The last part deals with symmetry and regularity properties. Here there are important connections with other branches of algebraic combinatorics and group theory. This new and enlarged edition this will be essential reading for a wide range of mathematicians, computer scientists and theoretical physicists.
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In this paper, we introduce linear and nonlinear consensus protocols for networks of dynamic agents that allow the agents to agree in a distributed and cooperative fashion. We consider the cases of networks with communication time-delays and channels that have filtering effects. We find a tight upper bound on the maximum fixed time-delay that can be tolerated in the network. It turns out that the connectivity of the network is the key in reaching a consensus. The case of agreement with bounded inputs is considered by analyzing the convergence of a class of nonlinear protocols. A Lyapunov function is introduced that quantifies the total disagreement among the nodes of a network. Simulation results are provided for agreement in networks with communication time-delays and constrained inputs.
Vicsek et al. proposed (1995) 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.
It has been shown that synchronization between two nonlinear
systems can be studied as a control-theory problem. We show that this
relationship can he extended to synchronization in arbitrary coupled
arrays of nonlinear systems. In particular, we use several well-known
stability conditions to obtain synchronization criteria in arbitrarily
coupled arrays: the passivity criterion, the circle criterion and a
result on observer design of Lipschitz nonlinear systems. We also study
how these synchronization criteria depend on the topology of the coupled
networks. In particular, we show that synchronization is improved by
using nonlocal connections or introducing random connections
We study a simple but compelling model of network of agents interacting via time-dependent communication links. The model finds application in a variety of fields including synchronization, swarming and distributed decision making. In the model, each agent updates his current state based upon the current information received from neighboring agents. Necessary and/or sufficient conditions for the convergence of the individual agents' states to a common value are presented, thereby extending recent results reported in the literature. The stability analysis is based upon a blend of graph-theoretic and system-theoretic tools with the notion of convexity playing a central role. The analysis is integrated within a formal framework of set-valued Lyapunov theory, which may be of independent interest. Among others, it is observed that more communication does not necessarily lead to faster convergence and may eventually even lead to a loss of convergence, even for the simple models discussed in the present paper.
Inspired by the so-called "bugs" problem from mathematics, we study the geometric formations of multivehicle systems under cyclic pursuit. First, we introduce the notion of cyclic pursuit by examining a system of identical linear agents in the plane. This idea is then extended to a system of wheeled vehicles, each subject to a single nonholonomic constraint (i.e., unicycles), which is the principal focus of this paper. The pursuit framework is particularly simple in that the n identical vehicles are ordered such that vehicle i pursues vehicle i+1 modulo n. In this paper, we assume each vehicle has the same constant forward speed. We show that the system's equilibrium formations are generalized regular polygons and it is exposed how the multivehicle system's global behavior can be shaped through appropriate controller gain assignments. We then study the local stability of these equilibrium polygons, revealing which formations are stable and which are not.
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.
We present a stable control strategy for groups of vehicles to move and reconfigure cooperatively in response to a sensed, distributed environment. Each vehicle in the group serves as a mobile sensor and the vehicle network as a mobile and reconfigurable sensor array. Our control strategy decouples, in part, the cooperative management of the network formation from the network maneuvers. The underlying coordination framework uses virtual bodies and artificial potentials. We focus on gradient climbing missions in which the mobile sensor network seeks out local maxima or minima in the environmental field. The network can adapt its configuration in response to the sensed environment in order to optimize its gradient climb.
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 general form of area inequality is used to develop a frequency domain criterion for absolute stability of single-nonlinear sampled-data (SNSD) systems. By reductions of the general form to specific area inequalities, it is suggested that all known and some new frequency domain criteria applicable to different subclasses of SNSD systems may be derived as special cases. One of the latter involving q < 0 is investigated to determine its significance and general applicability to SNSD systems. An example of a SNSD system with dead zone is used to illustrate some of the results presented.
A sufficient condition for stability of a class of sampled-data feedback systems containing a memory-less, nonlinear gain element is obtained. The new stability theorem for the class of systems discussed requires that the following relationship be satisfied on the unit circle: Re G^{ast}(z)[1 + q(z - 1)] + frac{1}{K} - frac{K'|q|}{2} | (z - 1)G^{ast}(z)|^{2} leq 0 . In this papers the stability criterion embodied in this theorem can be readily obtained from the frequency response of the linear plant. This method is essentially similar to Popov's method applied to the study of nonlinear continuous systems. Furthermore, Tsypkin's resuits for the discrete case are obtained as a special case when q=0 . Several examples are discussed, and the results are compared with Lyapunov's quadratic and quadratic plus integral forms as well as with other methods. For these examples, the results obtained from the new theorem yield less conservative values of gain than Lyapunov's method. Furthermore, for certain linear plants the new theorem also yields the necessary and sufficient conditions.
Consensus problems in networks of agents with switching topology and time-delaysStability of multiagent systems with time-dependent communication links
- R Olfati
- R Saber
- L Murray
- Moreau
R. Olfati-Saber and R. Murray, "Consensus problems in networks of agents with switching topology and time-delays," IEEE Trans. Autom. Control, vol. 49, no. 9, pp. 1520–1533, 2004. [6] L. Moreau, "Stability of multiagent systems with time-dependent communication links," IEEE Trans. Autom. Control, vol. 50, pp. 169–182, 2005.
Formation of vehicles in cyclic pursuit
- J Marshall
- M Broucke
- B Francis
J. Marshall, M. Broucke, and B. Francis, "Formation of vehicles
in cyclic pursuit," IEEE Trans. Autom. Control, vol. 49, no. 11, pp.
1963-1974, 2004.