Showing papers in "IEEE Transactions on Control of Network Systems in 2014"

Convex Relaxation of Optimal Power Flow—Part I: Formulations and Equivalence

Abstract: This tutorial summarizes recent advances in the convex relaxation of the optimal power flow (OPF) problem, focusing on structural properties rather than algorithms. Part I presents two power flow models, formulates OPF and their relaxations in each model, and proves equivalence relationships among them. Part II presents sufficient conditions under which the convex relaxations are exact.

Detection of Faults and Attacks Including False Data Injection Attack in Smart Grid Using Kalman Filter

Abstract: By exploiting the communication infrastructure among the sensors, actuators, and control systems, attackers may compromise the security of smart-grid systems, with techniques such as denial-of-service (DoS) attack, random attack, and data-injection attack. In this paper, we present a mathematical model of the system to study these pitfalls and propose a robust security framework for the smart grid. Our framework adopts the Kalman filter to estimate the variables of a wide range of state processes in the model. The estimates from the Kalman filter and the system readings are then fed into the $\chi^{2}$ -detector or the proposed Euclidean detector. The $\chi^{2}$ -detector is a proven effective exploratory method used with the Kalman filter for the measurement of the relationship between dependent variables and a series of predictor variables. The $\chi^{2}$ -detector can detect system faults/attacks, such as DoS attack, short-term, and long-term random attacks. However, the studies show that the $\chi^{2}$ -detector is unable to detect the statistically derived false data-injection attack. To overcome this limitation, we prove that the Euclidean detector can effectively detect such a sophisticated injection attack.

Controllability Metrics, Limitations and Algorithms for Complex Networks

Abstract: This paper studies the problem of controlling complex networks, i.e., the joint problem of selecting a set of control nodes and of designing a control input to steer a network to a target state. For this problem, 1) we propose a metric to quantify the difficulty of the control problem as a function of the required control energy, 2) we derive bounds based on the system dynamics (network topology and weights) to characterize the tradeoff between the control energy and the number of control nodes, and 3) we propose an open-loop control strategy with performance guarantees. In our strategy, we select control nodes by relying on network partitioning, and we design the control input by leveraging optimal and distributed control techniques. Our findings show several control limitations and properties. For instance, for Schur stable and symmetric networks: 1) if the number of control nodes is constant, then the control energy increases exponentially with the number of network nodes; 2) if the number of control nodes is a fixed fraction of the network nodes, then certain networks can be controlled with constant energy independently of the network dimension; and 3) clustered networks may be easier to control because, for sufficiently many control nodes, the control energy depends only on the controllability properties of the clusters and on their coupling strength. We validate our results with examples from power networks, social networks and epidemics spreading.

Convex Relaxation of Optimal Power Flow—Part II: Exactness

Abstract: This tutorial summarizes recent advances in the convex relaxation of the optimal power flow (OPF) problem, focusing on structural properties rather than algorithms. Part I presents two power flow models, formulates OPF and their relaxations in each model, and proves equivalence relations among them. Part II presents sufficient conditions under which the convex relaxations are exact.

Minimal Controllability Problems

Abstract: Given a linear system, we consider the problem of finding a small set of variables to affect with an input so that the resulting system is controllable. We show that this problem is NP-hard; indeed, we show that even approximating the minimum number of variables that need to be affected within a multiplicative factor of clog n is NP-hard for some positive c. On the positive side, we show it is possible to find sets of variables matching this in approximability barrier in polynomial time. This can be done with a simple greedy heuristic which sequentially picks variables to maximize the rank increase of the controllability matrix. Experiments on Erdos-Renyi random graphs that demonstrate this heuristic almost always succeed at finding the minimum number of variables.

Optimal Resource Allocation for Network Protection Against Spreading Processes

Abstract: We study the problem of containing spreading processes in arbitrary directed networks by distributing protection resources throughout the nodes of the network. We consider that two types of protection resources are available: 1) preventive resources able to defend nodes against the spreading (such as vaccines in a viral infection process) and 2) corrective resources able to neutralize the spreading after it has reached a node (such as antidotes). We assume that both preventive and corrective resources have an associated cost and study the problem of finding the cost-optimal distribution of resources throughout the nodes of the network. We analyze these questions in the context of viral spreading processes in directed networks. We study the following two problems: 1) given a fixed budget, find the optimal allocation of preventive and corrective resources in the network to achieve the highest level of containment and 2) when a budget is not specified, find the minimum budget required to control the spreading process. We show that both the resource allocation problems can be solved in polynomial time using geometric programming (GP) for arbitrary directed graphs of nonidentical nodes and a wide class of cost functions. We illustrate our approach by designing optimal protection strategies to contain an epidemic outbreak that propagates through an air transportation network.

An $O(1/k)$ Gradient Method for Network Resource Allocation Problems

Abstract: We present a fast distributed gradient method for a convex optimization problem with linear inequalities, with a particular focus on the network utility maximization (NUM) problem. Most existing works in the literature use (sub)gradient methods for solving the dual of this problem which can be implemented in a distributed manner. However, these (sub)gradient methods suffer from an O(1/√k) rate of convergence (where k is the number of iterations). In this paper, we assume that the utility functions are strongly concave, an assumption satisfied by most standard utility functions considered in the literature. We develop a completely distributed fast gradient method for solving the dual of the NUM problem. We show that the generated primal sequences converge to the unique optimal solution of the NUM problem at rate O(1/k).

Formation Control for High-Order Linear Time-Invariant Multiagent Systems With Time Delays

Abstract: Formation control problems for high-order linear time-invariant multiagent systems with time delays are investigated. First, a general time-varying formation control protocol is proposed. Then, based on consensus approaches, necessary and sufficient conditions for multiagent systems to achieve a given time-varying formation are presented. An explicit expression of the time-varying formation reference function is also given. It is shown that the motion modes of the formation reference can be specified. Furthermore, necessary and sufficient conditions for formation feasibility are proposed. An approach to expand the feasible formation set and an algorithm to design the protocol for multiagent systems to achieve time-varying formations are provided, respectively. Finally, numerical simulations are presented to demonstrate theoretical results.

Quantized Subgradient Algorithm and Data-Rate Analysis for Distributed Optimization

Abstract: In this paper, we consider quantized distributed optimization problems with limited communication capacity and time-varying communication topology. A distributed quantized subgradient algorithm is presented with quantized information exchange between agents. Based on a proposed encoder-decoder scheme and a zooming-in technique, the optimal solution can be obtained without any quantization errors. Moreover, we explore how to minimize the quantization level number for quantized distributed optimization problems. In fact, the optimization problem can be solved with five-level quantizers in the switching topology case, while it can be solved with three-level quantizers in the fixed topology case.

Compositional Transient Stability Analysis of Multimachine Power Networks

Abstract: During the normal operation of a power system, all the voltages and currents are sinusoids with a frequency of 60 Hz in America and parts of Asia or of 50 Hz in the rest of the world. Forcing all the currents and voltages to be sinusoids with the right frequency is one of the most important problems in power systems. This problem is known as the transient stability problem in the power systems literature. The classical models used to study transient stability are based on several implicit assumptions that are violated when transients occur. One such assumption is the use of phasors to study transients. While phasors require sinusoidal waveforms to be well defined, there is no guarantee that waveforms will remain sinusoidal during transients. In this paper, we use energy-based models derived from first principles that are not subject to hard-to-justify classical assumptions. In addition to eliminate assumptions that are known not to hold during transient stages, we derive intuitive conditions ensuring the transient stability of power systems with lossy transmission lines. Furthermore, the conditions for transient stability are compositional in the sense that one infers transient stability of a large power system by checking simple conditions for individual generators.

Projection-Based Model Reduction of Multi-Agent Systems Using Graph Partitions

Abstract: In this paper, we establish a projection-based model reduction method for multiagent systems defined on a graph. Reduced order models are obtained by clustering the vertices (agents) of the underlying communication graph by means of suitable graph partitions. In the reduction process, the spatial structure of the network is preserved and the reduced order models can again be realized as multiagent systems defined on a graph. The agents are assumed to have single-integrator dynamics and the communication graph of the original system is weighted and undirected. The proposed model reduction technique reduces the number of vertices of the graph (which is equal to the dynamic order of the original multi-agent system) and yields a reduced order multiagent system defined on a new graph with a reduced number of vertices. This new graph is a weighted symmetric directed graph. It is shown that if the original multiagent system reaches consensus, then so does the reduced order model. For the case that the clusters are chosen using an almost equitable partition (AEP) of the graph, we obtain an explicit formula for the ${\mbi{\cal {H}}}_{\bf 2}$ -norm of the error system obtained by comparing the input–output behaviors of the original model and the reduced order model. We also prove that the error obtained by taking an arbitrary partition of the graph is bounded from below by the error obtained using the largest AEP finer than the given partition. The proposed results are illustrated by means of a running example.

On the Internal Model Principle in the Coordination of Nonlinear Systems

Abstract: The role of the internal model principle is investigated in this paper for the coordination of relative-degree-one and relative-degree-two nonlinear systems. For relative-degree-one systems that are incrementally (output-feedback) passive, we propose internal-model-based distributed control laws which guarantee output synchronization to an invariant manifold driven by autonomous synchronized internal models. For relative-degree-two systems, we consider a different internal-model-based distributed control framework for solving a formation control problem where the agents have to track a reference signal available only to the leader agent. In both cases, the local controller is also able to reject the disturbance signals generated by a local exosystem.

Collective Circular Motion of Unicycle Type Vehicles With Nonidentical Constant Velocities

Abstract: This paper addresses formation control problems for heterogeneous groups of unicycle type mobile agents with fixed cruising speed. The heterogeneity in the group is caused by the cruising speeds being nonidentical, which complicates the motion coordination problem but is of practical relevance, for example, in unmanned aerial vehicle applications. We show that two different types of collective circular motion are possible in such groups: 1) a circular motion with common angular frequency and different radius for each agent; or 2) a circular motion with common radius but different angular frequency for each agent. For the first motion type, the orientation of all vehicles can additionally be coordinated such that an agreement or a balanced configuration is achieved. We present suitable control laws for each of these motion coordination tasks. These control laws explicitly take into account the nonidentical velocities and guarantee convergence to the desired configurations. Numerical examples illustrate all results.

Consensus and Coherence in Fractal Networks

Abstract: We consider first- and second-order consensus algorithms in networks with stochastic disturbances. We quantify the deviation from consensus using the notion of network coherence, which can be expressed as an $H_{2}$ norm of the stochastic system. We use the setting of fractal networks to investigate the question of whether a purely topological measure, such as the fractal dimension, can capture the asymptotics of coherence in the large system size limit. Our analysis for first-order systems is facilitated by connections between first-order stochastic consensus and the global mean first passage time of random walks. We then show how to apply similar techniques to analyze second-order stochastic consensus systems. Our analysis reveals that two networks with the same fractal dimension can exhibit different asymptotic scalings for network coherence. Thus, this topological characterization of the network does not uniquely determine coherence behavior. The question of whether the performance of stochastic consensus algorithms in large networks can be captured by purely topological measures, such as the spatial dimension, remains open.

Distributed Weight Balancing Over Digraphs

Abstract: A weighted digraph is balanced if, for each node, the sum of the weights of the edges outgoing from that node is equal to the sum of the weights of the edges incoming to that node. Weight-balanced digraphs play a key role in a number of applications, including cooperative control, distributed optimization, and distributed averaging. We propose distributed algorithms that operate over static topologies, for solving the weight-balancing problem when the weights are either non-negative real numbers or when they are restricted to be non-negative integers. For the case of real weights, the proposed algorithm is shown to admit a geometric convergence rate. For the case of integer weights, the proposed algorithm is shown to converge after a finite number of iterations that we explicitly bound. We also provide examples to illustrate the operation, performance, and potential advantages of the proposed algorithms.

Stabilization of Uncertain Systems With Finite Data Rates and Markovian Packet Losses

Abstract: This paper studies stabilization of uncertain systems via data-rate-limited and lossy communication channels We consider linear systems where the parameters are given as intervals and the exact values are unavailable Communication between the plant and the controller is restricted in the sense that transmitted signals should be represented in finite bits and may be randomly lost While most of the existing works assume that the loss process is independent and identically distributed, we model it as a two-state Markov chain, which can deal with more practical situations including bursty dropouts The central question investigated is how large the data rate, the loss probabilities, and the uncertainty bounds should be to make the system stable We derive a necessary condition and a sufficient condition for stability The conditions provide limitations characterized by the product of the eigenvalues of the nominal plant In particular, for scalar plants, the conditions coincide with each other Furthermore, we introduce a nonuniform quantizer, whose quantization cells are designed to minimize the effect of the uncertainty on state estimation The quantizer can reduce the required data rate compared with the conventional uniform one

Multiagent Flocking Under General Communication Rule

Abstract: In this paper, we consider a multiagent system consisting of mobile agents with second-order dynamics. The communication network is determined by a general interaction rule based on the distance between agents. The goal of this paper is to determine practical conditions ensuring that the agents asymptotically agree on a common velocity, i.e., a flocking behavior is achieved. Unlike previous studies on the topic, our results simultaneously satisfy the three following features: 1) our conditions apply to a model, which does not require long distance communication; 2) they only depend on the initial positions and velocities of the agents; and 3) most importantly, our results allow for the disconnection of communication links which are not necessary for flocking. To circumvent the difficulty arising from the state dependent dynamics, a suitable bounding process is used. We apply our result to two cases, where communication takes place either within deterministic or stochastic distance radiuses. Our result is illustrated through simulations.

Collective Decision-Making in Ideal Networks: The Speed-Accuracy Tradeoff

Abstract: We study collective decision-making in a model of human groups, with network interactions, performing two alternative choice tasks. We focus on the speed-accuracy tradeoff, i.e., the tradeoff between a quick decision and a reliable decision, for individuals in the network. We model the evidence aggregation process across the network using a coupled drift-diffusion model (DDM) and consider the free response paradigm in which individuals take their time to make the decision. We develop a reduced DDM as a decoupled approximation to the coupled DDM and characterize its efficiency. We determine high probability bounds on the error rate and the expected decision time for the reduced DDM. We show the effect of the decision-maker's location in the network on their decision-making performance under several threshold selection criteria. Finally, we extend the coupled DDM to the coupled Ornstein-Uhlenbeck model for decision-making in two alternative choice tasks with recency effects, and to the coupled race model for decision-making in multiple alternative choice tasks.

To Go or Not to Go: On Energy-Aware and Communication-Aware Robotic Operation

Abstract: We consider the scenario where a mobile robot needs to visit a number of points of interest (POIs) in a workspace, gathers its generated bits of information, and successfully transmits them to a remote station, while operating in a realistic communication environment, minimizing its total energy consumption (including both motion and communication costs), and under time and reception quality constraints. We are interested in the co-optimization of the communication and motion strategies of the robot such that it finds the optimal trajectory (the order in which it visits all of the POIs) and optimally co-plans its communication and motion strategies, including motion speed, stop times, communication transmission rate, and power. By co-optimizing the usage of both communication and motion energy costs and using realistic probabilistic link metrics that go beyond the commonly used disk models, we show how the overall problem can be posed as a mixed integer linear program (MILP) and characterize several properties of the co-optimized solution. For instance, we derive conditions under which the optimal trajectory becomes the minimum-length trajectory as well as conditions under which the trajectory deviates from the minimum length one to visit areas with very high connectivity. We further characterize if/when it is beneficial for the robot to incur motion energy to find a better spot for communication. Moreover, we derive conditions that relate the co-optimized communication and motion strategies and clearly show the interplay between the two. Finally, our simulation results with real channel and motion parameters confirm the analysis and show considerable energy savings.

Message Passing Optimization of Harmonic Influence Centrality

Abstract: This paper proposes a new measure of node centrality in social networks, the Harmonic Influence Centrality (HIC), which emerges naturally in the study of social influence over networks. Using an intuitive analogy between social and electrical networks, we introduce a distributed message passing algorithm to compute the HIC of each node. Although its design is based on theoretical results which assume the network to have no cycle, the algorithm can also be successfully applied on general graphs.

Optimal Routing and Energy Allocation for Lifetime Maximization of Wireless Sensor Networks With Nonideal Batteries

Abstract: An optimal control approach is used to solve the problem of routing in sensor networks where the goal is to maximize the network's lifetime. In our analysis, the energy sources (batteries) at nodes are not assumed to be “ideal” but rather behaving according to a dynamic energy consumption model, which captures the nonlinear behavior of actual batteries. We show that in a fixed topology case there exists an optimal policy consisting of time-invariant routing probabilities, which may be obtained by solving a set of relatively simple nonlinear programming (NLP) problems. We also show that this optimal policy is, under very mild conditions, robust with respect to the battery model used. Further, we consider a joint routing and initial energy allocation problem over the network nodes with the same network lifetime maximization objective. We prove that the solution to this problem is given by a policy that depletes all node energies at the same time and that the corresponding energy allocation and routing probabilities are obtained by solving an NLP problem. Numerical examples are included to illustrate the optimality of the time-invariant policy and its robustness with respect to the battery model used.

An Asynchronous Consensus-Based Algorithm for Estimation From Noisy Relative Measurements

Abstract: In this paper, we address the problem of optimal estimating the position of each agent in a network from relative noisy vectorial distances with its neighbors by means of only local communication and bounded complexity, independent of network size and topology. We propose a consensus-based algorithm with the use of local memory variables which allows asynchronous implementation, has guaranteed exponential convergence to the optimal solution under simple deterministic and randomized communication protocols, and requires minimal packet transmission. In the randomized scenario, we then study the rate of convergence in expectation of the estimation error and we argue that it can be used to obtain upper and lower bound for the rate of converge in mean square. In particular, we show that for regular graphs, such as Cayley, Ramanujan, and complete graphs, the convergence rate in expectation has the same asymptotic degradation of memoryless asynchronous consensus algorithms in terms of network size. In addition, we show that the asynchronous implementation is also robust to delays and communication failures. We finally complement the analytical results with some numerical simulations, comparing the proposed strategy with other algorithms which have been recently proposed in the literature.

Optimal Control of Scalar Conservation Laws Using Linear/Quadratic Programming: Application to Transportation Networks

Abstract: This article presents a new optimal control framework for transportation networks in which the state is modeled by a first order scalar conservation law. Using an equivalent formulation based on a Hamilton-Jacobi (H-J) equation and the commonly used triangular fundamental diagram, we pose the problem of controlling the state of the system on a network link, in a finite horizon, as a Linear Program (LP). We then show that this framework can be extended to an arbitrary transportation network, resulting in an LP or a Quadratic Program. Unlike many previously investigated transportation network control schemes, this method yields a globally optimal solution and is capable of handling shocks (i.e., discontinuities in the state of the system). As it leverages the intrinsic properties of the H-J equation used to model the state of the system, it does not require any approximation, unlike classical methods that are based on discretizations of the model. The computational efficiency of the method is illustrated on a transportation network.

Pattern Synchronization of Nonlinear Heterogeneous Multiagent Networks With Jointly Connected Topologies

Abstract: In this paper, a decentralized control algorithm is proposed for a group of heterogeneous agents with nonlinear dynamics such that it achieves synchronization in diverse collective motion patterns. With the aid of a group of deliberately designed linear homogeneous reference models, the synchronization problem is converted into a class of regulation problem. The regulation problem is hence solved using the nonlinear output feedback control technique. The conversion and, hence, the decentralized synchronization algorithm are valid when the multiagent system is jointly connected in terms of its time-varying network topologies. The effectiveness of the algorithm is verified through theoretical analysis.

Distributed Control with Low-Rank Coordination

Abstract: A common approach to distributed control design is to impose sparsity constraints on the controller structure. Such constraints, however, may greatly complicate the control design procedure. This paper puts forward an alternative structure, which is not sparse yet might nevertheless be well suited for distributed control purposes. The structure appears as the optimal solution to a class of coordination problems arising in multiagent applications. The controller comprises a diagonal (decentralized) part, complemented by a rank-one coordination term. Although this term relies on information about all subsystems, its implementation only requires a simple averaging operation.

Remote Estimation With Noisy Measurements Subject to Packet Loss and Quantization Noise

Abstract: In this paper, we consider the problem of designing coding and decoding schemes to estimate the state of a scalar stable stochastic linear system subject to noisy measurements and in the presence of a wireless communication channel between the sensor and the estimator. In particular, we consider a communication channel which is prone to packet loss and includes quantization noise due to its limited capacity. We study two scenarios: the first with channel feedback and the second with no channel feedback. More specifically, in the first scenario the transmitter is aware of the quantization noise and the packet loss history of the channel, while in the second scenario the transmitter is aware of the quantization noise only. We show that in the first scenario, the optimal strategy among all possible linear encoders corresponds to the transmission of the Kalman filter innovation, similar to the differential pulse-code modulation (DPCM) technique used in digital communications. In the second scenario, we show that there is a critical packet loss probability above which it is better to transmit the state rather than the innovation. We also propose a heuristic strategy based on the transmission of a convex combination of the state and the Kalman filter innovation which is shown to provide a performance close to the one obtained with channel feedback.

An optimal transmission strategy for Kalman filtering over packet dropping links with imperfect acknowledgements

Abstract: This paper presents a novel design methodology for optimal transmission policies at a smart sensor to remotely estimate the state of a stable linear stochastic dynamical system. The sensor makes measurements of the process and forms estimates of the state using a local Kalman filter. The sensor transmits quantized information over a packet dropping link to the remote receiver. The receiver sends packet receipt acknowledgments back to the sensor via an erroneous feedback communication channel which is itself packet dropping. The key novelty of this formulation is that the smart sensor decides, at each discrete time instant, whether to transmit a quantized version of either its local state estimate or its local innovation. The objective is to design optimal transmission policies in order to minimize a long term average cost function as a convex combination of the receiver's expected estimation error covariance and the energy needed to transmit the packets. Under high resolution quantization assumptions, the optimal transmission policy is obtained by the use of dynamic programming techniques. Using the concept of submodularity, the optimality of a threshold policy in the case of scalar systems with perfect packet receipt acknowledgments is proved. Suboptimal solutions and their structural results are also discussed. Numerical results are presented illustrating the performance of the optimal and suboptimal transmission policies.

Observability in Connected Strongly Regular Graphs and Distance Regular Graphs

Abstract: This paper concerns the study of observability in consensus networks modeled with strongly regular graphs or distance regular graphs. We first give a Kalman-like simple algebraic criterion for observability in distance regular graphs. This criterion consists in evaluating the rank of a matrix built with the components of the Bose–Mesner algebra associated with the considered graph. Then, we define some bipartite graphs that capture the observability properties of the graph to be studied. In particular, we show that necessary and sufficient observability conditions are given by the nullity of the so-called local bipartite observability graph (respectively, the local unfolded bipartite observability graph) for strongly regular graphs (respectively, the distance regular graphs). When the nullity cannot be derived directly from the structure of these bipartite graphs, the rank of the associated bi-adjacency matrix enables evaluating observability. Eventually, as a byproduct of the main results, we show that nonobservability can be stated just by comparing the valency of the graph to be studied with a bound computed from the number of vertices of the graph and its diameter. Similarly, nonobservability can also be stated by evaluating the size of the maximum matching in the aforementioned bipartite graphs.

On Endogenous Random Consensus and Averaging Dynamics

Abstract: Motivated by various random variations of Hegselmann-Krause model for opinion dynamics and gossip algorithm in an endogenously changing environment, we propose a general framework for the study of endogenously varying random averaging dynamics, i.e.\ an averaging dynamics whose evolution suffers from history dependent sources of randomness. We show that under general assumptions on the averaging dynamics, such dynamics is convergent almost surely. We also determine the limiting behavior of such dynamics and show such dynamics admit infinitely many time-varying Lyapunov functions.

Distributed Coordination of Self-Organizing Mechanisms in Communication Networks

Abstract: The fast development of the self-organizing networks (SON) technology in mobile networks renders critical the problem of coordinating SON functionalities operating simultaneously. SON functionalities can be viewed as control loops that may need to be coordinated to guarantee conflict-free operation, to enforce stability of the network, and to achieve performance gain. This paper proposes a distributed solution for coordinating SON functionalities. It uses Rosen's concave games framework in conjunction with convex optimization. The SON functionalities are modeled as linear ordinary differential equations. The stability of the system is first evaluated using a basic control theory approach. The coordination solution consists in finding a linear map (called coordination matrix) that stabilizes the system of SON functionalities. It is proven that the solution remains valid in a noisy environment using stochastic approximation. A practical example involving three different SON functionalities deployed in base stations of a long-term evolution network demonstrates the usefulness of the proposed method.