arXiv subjects
Publications and source records attributed to Xudong Chen.
A weighted digraph is balanced if the sums of the weights of the incoming and of the outgoing edges are equal at each vertex. We show that if these sums are integers, then the edge weights can be integers as well.
We derive conditions on the products of stochastic matrices guaranteeing the existence of a unique limit invariant distribution. Belying our approach is the hereby defined notion of restricted triangulated Laman graphs. The main idea is the following: to each triangle in the graph, we assign a stochastic matrix. Two matrices can be adjacent in a product only if their corresponding triangles share an edge in the graph. We provide an explicit formula for the limit invariant distribution of the product in terms of the individual stochastic matrices.
In this paper, we study the problem of robust global synchronization of resetting clocks in multi-agent networked systems, where by robust global synchronization we mean synchronization that is insensitive to arbitrarily small disturbances, and which is achieved from all initial conditions. In particular, we aim to address the following question: Given a set of homogeneous agents with periodic clocks sharing the same parameters, what kind of information flow topologies will guarantee that the resulting networked systems can achieve robust global synchronization? To address this question, we rely on the framework of robust hybrid dynamical systems and a class of distributed hybrid resetting algorithms. Using the hybrid-system approach, we provide a partial solution to the question: Specifically, we show that one can achieve robust global synchronization with no purely discrete-time solutions in any networked system whose underlying information flow topology is a rooted acyclic digraph. Such a result is complementary to the existing result [1] in which strongly connected digraphs are considered as the underlying information flow topologies of the networked systems. We have further computed in the paper the convergence time for a networked system to reach global synchronization. In particular, the computation reveals the relationship between convergence time and the structure of the underlying digraph. We illustrate our theoretical findings via numerical simulations towards the end of the paper.
We study the problem of robust global synchronization of pulse-coupled oscillators (PCOs) over directed graphs. It is known that when the digraphs are strongly connected, global synchronization can be achieved by using a class of deterministic set-valued reset controllers. However, for large-scale networks, these algorithms are not scalable because some of their tuning parameters have upper bounds of the order of O(1/N), where N is the number of agents. This paper resolves this scalability issue by presenting several new results in the context of global synchronization of PCOs with more general network topologies using deterministic and stochastic hybrid dynamical systems. First, we establish that similar deterministic resetting algorithms can achieve robust, global, and fixed-time synchronization in any rooted acyclic digraph. Moreover, in this case we show that the synchronization dynamics are now scalable as the tuning parameters of the algorithm are network independent, i.e., of order O(1). However, the algorithms cannot be further extended to all rooted digraphs. We establish this new impossibility result by introducing a counterexample with a particular rooted digraph for which global synchronization cannot be achieved, irrespective of the tuning of the reset rule. Nevertheless, we show that if the resetting algorithms are modified by accommodating an Erdos-Renyi type random graph model, then the resulting stochastic resetting dynamics will guarantee global synchronization almost surely for all rooted digraphs and, moreover, the tunable parameters of the dynamics are network independent. Stability and robustness properties of the resetting algorithms are studied using the tools from set-valued hybrid dynamical systems. Numerical simulations are provided at the end of the paper for demonstration of the main results.
We introduce in the paper a novel observability problem for a continuum ensemble of nonholonomic control systems with unknown population density. We address the problem by focussing on a prototype of such ensemble system, namely, the ensemble of Bloch equations. The dynamics of the equations are structurally identical, but show variations in Larmor dispersion and radio frequency (rf) inhomogeneity. We assume that the initial state of every individual system is unknown and, moreover, the population density of these individual systems is also unknown. Furthermore, we assume that at any time, there is only one scalar measurement output at our disposal. The measurement output integrates a certain observation function, common to all individual systems, over the continuum ensemble. The observability problem we pose in the paper is thus the following: Whether one is able to use the common control input (i.e., the rf field) and the single measurement output to estimate the initial states of the individual systems and, moreover, to identify the population density? Amongst other things, we establish a sufficient condition for the ensemble system to be observable: We show that if the common observation function is any harmonic homogeneous polynomial of positive degree, then the ensemble system is observable. The main focus of the paper is to demonstrate how to leverage tools from representation theory of Lie algebras to address the observability problem. Although the results we establish in the paper are for the specific ensemble of Bloch equations, the approach we develop along the analysis can be generalized to investigate observability of other general ensembles of nonholonomic control systems with a single, integrated measurement output.
We call a finite, spanning set of a semi-simple real Lie algebra a distinguished set if it satisfies the following property: The Lie bracket of any two elements out of the set is, up to some constant, another element in the set; conversely, for any element in the set, there are two elements out of the set whose Lie bracket is, up to some constant, the given element. We show in the paper that every semi-simple real Lie algebra has a distinguished set.
By utilizing tools from game theory, we develop a novel multi-period-multi-company demand response framework considering the interactions between companies (sellers of energy) and their consumers (buyers of energy). We model the interactions in terms of a Stackelberg game, where companies set their prices and consumers respond by choosing their demands. We show that the underlying game has a unique equilibrium at which the companies maximize their revenues while the consumers maximize their utilities subject to their local constraints. Closed-form expressions are provided for the optimal strategies of all players. Based on these solutions, a power allocation game has been formulated, which is shown to admit a unique pure-strategy Nash equilibrium, for which closed-form expressions are also provided. This equilibrium is found under the assumption that companies can freely allocate their power across the time horizon, but we also demonstrate that it is possible to relax this assumption. We further provide a fast distributed algorithm for the computation of all optimal strategies using only local information. We also study the effect of variations in the number of periods (subdivisions of the time horizon) and the number of consumers. As a consequence, we are able to find an appropriate company-to-consumer ratio when the number of consumers participating in demand response exceeds some threshold. Furthermore, we show, both analytically and numerically, that the multi-period scheme provides incentives for energy consumers to participate in demand response, compared to the single-period framework studied in the literature. In our framework, we provide a condition for the minimum budgets consumers need, and carry out case studies using real life data to demonstrate the benefits of the approach, which show potential savings of up to $30\%$ and equilibrium prices that have low volatility.
Today the gold standard for in vivo imaging through scattering tissue is the point-scanning two-photon microscope (PSTPM). Especially in neuroscience, PSTPM is widely used for deep-tissue imaging in the brain. However, due to sequential scanning, PSTPM is slow. Temporal focusing microscopy (TFM), on the other hand, focuses femtosecond pulsed laser light temporally, while keeping wide-field illumination, and is consequently much faster. However, due to the use of a camera detector, TFM suffers from the scattering of emission photons. As a result, TFM produces images of poor spatial resolution and signal-to-noise ratio (SNR), burying fluorescent signals from small structures such as dendritic spines. In this work, we present a data-driven deep learning approach to improve resolution and SNR of TFM images. Using a 3D convolutional neural network (CNN) we build a map from TFM to PSTPM modalities, to enable fast TFM imaging while maintaining high-resolution through scattering media. We demonstrate this approach for in vivo imaging of dendritic spines on pyramidal neurons in the mouse visual cortex. We show that our trained network rapidly outputs high-resolution images that recover biologically relevant features previously buried in the scattered fluorescence in the TFM images. In vivo imaging that combines TFM and the proposed 3D convolution neural network is one to two orders of magnitude faster than PSTPM but retains the high resolution and SNR necessary to analyze small fluorescent structures. The proposed 3D convolution deep network could also be potentially beneficial for improving the performance of many speed-demanding deep-tissue imaging applications such as in vivo voltage imaging.
A conjunctive Boolean network (CBN) is a finite state dynamical system, whose variables take values from a binary set, and the value update rule for each variable is a Boolean function consisting only of logic AND operations. We investigate the asymptotic behavior of CBNs by computing their periodic orbits. When the underlying digraph is strongly connected, the periodic orbits of the associated CBN has been completely understood, one-to-one corresponding to binary necklaces of a certain length given by the loop number of the graph. We characterize in the paper the periodic orbits of CBNs over an arbitrary weakly connected digraphs. We establish, among other things, a new method to investigate their asymptotic behavior. Specifically, we introduce a graphical-approach, termed system reduction, which turns the underlying digraph into a special weakly connected digraph whose strongly connected components are all cycles. We show that the reduced system uniquely determines the asymptotic behavior of the original system. Moreover, we provide a constructive method for computing the periodic orbit of the reduced system, which the system will enter for a given but arbitrary initial condition.
We propose in the paper a novel framework for using a common control input to simultaneously steer an infinite ensemble of networked control systems. We address the problem of co-designing information flow topology and network dynamics of every individual networked system so that a continuum ensemble of such systems is controllable. To keep the analysis tractable, we focus in the paper on a special class of ensembles systems, namely ensembles of multi-agent formation systems. Specifically, we consider an ensemble of formation systems indexed by a parameter in a compact, real analytic manifold. Every individual formation system in the ensemble is composed of $N$ agents. These agents evolve in $\mathbb{R}^n$ and can access relative positions of their neighbors. The information flow topology within every individual formation system is, by convention, described by a directed graph where the vertices correspond to the $N$ agents and the directed edges indicate the information flow. For simplicity, we assume in the paper that all the individual formation systems share the same information flow topology described by a common digraph $G$. Amongst other things, we establish a sufficient condition for approximate path-controllability of the continuum ensemble of formation systems. We show that if the digraph $G$ is strongly connected and the number $N$ of agents in each individual system is great than $(n + 1)$, then every such system in the ensemble is simultaneously approximately path-controllable over a path-connected, open dense subset.
Ensemble control deals with the problem of using a finite number of control inputs to simultaneously steer a large population (in the limit, a continuum) of control systems. Dual to the ensemble control problem, ensemble estimation deals with the problem of using a finite number of measurement outputs to estimate the initial condition of every individual system in the ensemble. We introduce in the paper an innovative class of ensemble systems, termed distinguished ensemble systems, and establish sufficient conditions for controllability and observability of such systems. Every distinguished ensemble system has two key components, namely distinguished control vector fields and codistinguished observation functions. Roughly speaking, a set of vector fields is distinguished if it is closed (up to scaling) under Lie bracket, and moreover, every vector field in the set can be obtained by a Lie bracket of two vector fields in the same set. Similarly, a set of functions is codistinguished to a set of vector fields if the Lie derivatives of the functions along the given vector fields yield (up to scaling) the same set of functions. We demonstrate in the paper that the structure of a distinguished ensemble system can significantly simplify the analysis of ensemble controllability and observability, and can be used as a principle for ensemble system design. We further address in the paper the existence of a distinguished ensemble system for a given manifold. Specifically, we show that every connected, semi-simple Lie group admits a set of distinguished vector fields, together with a set of codistinguished functions. The proof is constructive, leveraging structure theory of semi-simple real Lie algebras and representation theory. Examples will be provided along the presentation illustrating key definitions and main results.
Transmission control protocol (TCP) congestion control is one of the key techniques to improve network performance. TCP congestion control algorithm identification (TCP identification) can be used to significantly improve network efficiency. Existing TCP identification methods can only be applied to limited number of TCP congestion control algorithms and focus on wired networks. In this paper, we proposed a machine learning based passive TCP identification method for wired and wireless networks. After comparing among three typical machine learning models, we concluded that the 4-layers Long Short Term Memory (LSTM) model achieves the best identification accuracy. Our approach achieves better than 98% accuracy in wired and wireless networks and works for newly proposed TCP congestion control algorithms.
General detectors follow the pipeline that feature maps extracted from ConvNets are shared between classification and regression tasks. However, there exists obvious conflicting requirements in multi-orientation object detection that classification is insensitive to orientations, while regression is quite sensitive. To address this issue, we provide an Encoder-Decoder architecture, called Rotated Feature Network (RFN), which produces rotation-sensitive feature maps (RS) for regression and rotation-invariant feature maps (RI) for classification. Specifically, the Encoder unit assigns weights for rotated feature maps. The Decoder unit extracts RS and RI by performing resuming operator on rotated and reweighed feature maps, respectively. To make the rotation-invariant characteristics more reliable, we adopt a metric to quantitatively evaluate the rotation-invariance by adding a constrain item in the loss, yielding a promising detection performance. Compared with the state-of-the-art methods, our method can achieve significant improvement on NWPU VHR-10 and RSOD datasets. We further evaluate the RFN on the scene classification in remote sensing images and object detection in natural images, demonstrating its good generalization ability. The proposed RFN can be integrated into an existing framework, leading to great performance with only a slight increase in model complexity.
We investigate the joint actuator-sensor design problem for stochastic linear control systems. Specifically, we address the problem of identifying a pair of sensor and actuator which gives rise to the minimum expected value of a quadratic cost. It is well known that for the linear-quadratic-Gaussian (LQG) control problem, the optimal feedback control law can be obtained via the celebrated separation principle. Moreover, if the system is stabilizable and detectable, then the infinite-horizon time-averaged cost exists. But such a cost depends on the placements of the sensor and the actuator. We formulate in the paper the optimization problem about minimizing the time-averaged cost over admissible pairs of actuator and sensor under the constraint that their Euclidean norms are fixed. The problem is non-convex and is in general difficult to solve. We obtain in the paper a gradient descent algorithm (over the set of admissible pairs) which minimizes the time-averaged cost. Moreover, we show that the algorithm can lead to a unique local (and hence global) minimum point under certain special conditions.
In this paper, we investigate the geometry of a general class of gradient flows with multiple local maxima. we decompose the underlying space into disjoint regions of attraction and establish the adjacency criterion. The criterion states a necessary and sufficient condition for two regions of attraction of stable equilibria to be adjacent. We then apply this criterion on a specific type of gradient flow which has as many as n! local maxima. In particular, we characterize the set of equilibria, compute the index of each critical manifold and moreover, find all pairs of adjacent neighbors. As an application of the adjacency criterion, we introduce a stochastic version of the double bracket flow and set up a Markov model to approximate the sample path behavior. The study of this specific prototype with its special structure provides insight into many other difficult problems involving simulated annealing.
A Boolean network is a finite dynamical system, whose variables take values from a binary set. The value update rule for each variable is a Boolean function, depending on a selected subset of variables. Boolean networks have been widely used in modeling gene regulatory networks. We focus in this paper on a special class of Boolean networks, termed as conjunctive Boolean networks. A Boolean network is conjunctive if the associated value update rule is comprised of only AND operations. It is known that any trajectory of a finite dynamical system will enter a periodic orbit. We characterize in this paper all periodic orbits of a conjunctive Boolean network whose underlying graph is strongly connected. In particular, we establish a bijection between the set of periodic orbits and the set of binary necklaces of a certain length. We further investigate the stability of a periodic orbit. Specifically, we perturb a state in the periodic orbit by changing the value of a single entry of the state. The trajectory, with the perturbed state being the initial condition, will enter another (possibly the same) periodic orbit in finite time steps. We then provide a complete characterization of all such transitions from one periodic orbit to another. In particular, we construct a digraph, with the vertices being the periodic orbits, and the (directed) edges representing the transitions among the orbits. We call such a digraph the stability structure of the conjunctive Boolean network.
A Boolean network is a finite state discrete time dynamical system. At each step, each variable takes a value from a binary set. The value update rule for each variable is a local function which depends only on a selected subset of variables. Boolean networks have been used in modeling gene regulatory networks. We focus in this paper on a special class of Boolean networks, namely the conjunctive Boolean networks (CBNs), whose value update rule is comprised of only logic AND operations. It is known that any trajectory of a Boolean network will enter a periodic orbit. Periodic orbits of a CBN have been completely understood. In this paper, we investigate the orbit-controllability and state-controllability of a CBN: We ask the question of how one can steer a CBN to enter any periodic orbit or to reach any final state, from any initial state. We establish necessary and sufficient conditions for a CBN to be orbit-controllable and state-controllable. Furthermore, explicit control laws are presented along the analysis.
We consider the actuator placement problem for linear systems. Specifically, we aim to identify an actuator which requires the least amount of control energy to drive the system from an arbitrary initial condition to the origin in the worst case. Said otherwise, we investigate the minimax problem of minimizing the control energy over the worst possible initial conditions. Recall that the least amount of control energy needed to drive a linear controllable system from any initial condition on the unit sphere to the origin is upper-bounded by the inverse of the smallest eigenvalue of the associated controllability Gramian, and moreover, the upper-bound is sharp. The minimax problem can be thus viewed as the optimization problem of minimizing the upper-bound via the placement of an actuator. In spite of its simple and natural formulation, this problem is difficult to solve. In fact, properties such as the stability of the system matrix, which are not related to controllability, now play important roles. We focus in this paper on the special case where the system matrix is positive definite. Under this assumption, we are able to provide a complete solution to the optimal actuator placement problem and highlight the difficulty in solving the general problem.