Second-order Networks Research Articles

Explicit Characterization of Performance of a Class of Networked Linear Control Systems

We quantify performance of a class of interconnected linear dynamical networks with arbitrary nodal dynamics. It is shown that the steady-state variance of the output, which is equivalent to the squared $\mathcal {H}_2$ norm of the network, as a performance measure, can be expressed explicitly as a summation of a rational function over the Laplacian eigenvalues of the underlying graph of the network. Our results generalize the existing works on $\mathcal {H}_2$ -norm characterization for the first- and second-order consensus networks to general consensus-type linear dynamical networks with arbitrary nodal dynamics. We characterize several fundamental limits and scaling laws for the performance of this class of networks. To show the practical usefulness of our results, we discuss how to design decentralized observer-based output feedback mechanisms and how to analyze the performance of composite networks. Numerous examples support our theoretical contributions.

Adaptive Tracking Control of Cooperative Robot Manipulators With Markovian Switched Couplings

Many cooperative robotic systems have not only modeling heterogeneity and uncertainty but also switched couplings, causing control difficulties. Here, we develop a neural network adaptive control framework for cooperative robot manipulators with unknown Euler–Lagrange dynamics and Markovian switched couplings. Second-order Markovian switching networks are used for modeling such cooperative robotic systems, which admit a hybrid neural network control with a desired tracking performance. The hybrid neural network control scheme contains a distributed adaptive controller and a hybrid adaptation law, enabling learning in the closed-loop system. The position and velocity tracking errors are shown to be practically uniformly exponentially stable in the mean-square sense, respectively, guaranteeing the second-order practical tracking. The results also suggest that the neural weight evolves with practical convergence to the ideal, showing the effect of network structure on the adaptation capacity.

Fast Approximation of Coherence for Second-Order Noisy Consensus Networks.

It has been recently established that for second-order consensus dynamics with additive noise, the performance measures, including the vertex coherence and network coherence defined, respectively, as the steady-state variance of the deviation of each vertex state from the average and the average steady-state variance of the system, are closely related to the biharmonic distances. However, direct computation of biharmonic distances is computationally infeasible for huge networks with millions of vertices. In this article, leveraging the implicit fact that both vertex and network coherence can be expressed in terms of the diagonal entries of pseudoinverse L2† of the square of graph Laplacian, we develop a nearly linear-time algorithm to approximate all diagonal entries of L2† , which has a theoretically guaranteed error for each diagonal entry. The key ingredient of our approximation algorithm is an integration of the Johnson-Lindenstrauss lemma and Laplacian solvers. Extensive numerical experiments on real-life and model networks are presented, which indicate that our approximation algorithm is both efficient and accurate and is scalable to large-scale networks with millions of vertices.

Multi-Agent Trajectory-Tracking Flexible Formation via Generalized Flocking and Leader-Average Sliding Mode Control

This paper reports a flexible (or time-varying) multi-agent formation approach with average trajectory tracking for second-order integral multi-agent networks with single virtual leaders. The approach is developed by means of time-varying Olfati-Saber flocking algorithms, and sliding mode control (SMC) in terms of the leader-average dynamics. More precisely, SMC-specifying average trajectory tracking is combined with flexible multi-agent flocking driven by the Olfati-Saber flocking algorithms with time-varying weighting norm. Existence conditions and properties of the suggested multi-agent formation are examined rigorously, together with implementation formulas. It is shown that by designing the sliding surface and the time-varying weighting matrix appropriately, flexible formation with finite-time trajectory tracking can be achieved, free of control action chattering; moreover, the sliding mode control and formation control can be designed separately. Numerical examples are given to illustrate the main results.

Asynchronous impulsive control for consensus of second-order multi-agent networks

This paper addresses the asynchronous consensus problems of second-order multi-agent networks (MANs) with sampled-data communication, where each agent has its own time clock. An impulsive control is designed to achieve second-order consensus of the considered MANs under such asynchronous setting. By analyzing the presented impulsive control strategy with the discretization method, nonnegative matrix theory and graph theory, some sufficient consensus criteria, on the range of the sampling periods and the control gains of the presented control strategy, are established for multi-agent systems under the asynchronous setting. Finally, a simulation example is presented to show the effectiveness of the proposed control and the correctness of the theoretical analysis.

Coordinated Network Changes across the Catalytic Cycle of Alpha Tryptophan Synthase

Networks of noncovalent interactions are important for protein structural dynamics. We used nuclear magnetic resonance chemical shift covariance analyses on an inactive variant of the alpha subunit of tryptophan synthase to map amino acid interaction networks across its catalytic cycle. Although some network connections were common to every enzyme state, many of the network connections strengthened or weakened over the catalytic cycle; these changes were highly coordinated. These results suggest a higher level of network organization. Our analyses identified periodic, second-order networks that show highly coordinated interaction changes across the catalytic cycle. These periodic networks may help synchronize the sequence of structural transitions necessary for enzyme function. Molecular dynamics simulations identified interaction changes across the catalytic cycle, including those involving the catalytic residue Glu49, which may help drive other interaction changes throughout the enzyme structure. Similar periodic networks may direct structural transitions and allosteric interactions in other proteins.

Hybrid Functional Brain Network With First-Order and Second-Order Information for Computer-Aided Diagnosis of Schizophrenia

Brain functional connectivity network (BFCN) analysis has been widely used in the diagnosis of mental disorders, such as schizophrenia. In BFCN methods, brain network construction is one of the core tasks due to its great influence on the diagnosis result. Most of the existing BFCN construction methods only consider the first-order relationship existing in each pair of brain regions and ignore the useful high-order information, including multi-region correlation in the whole brain. Some early schizophrenia patients have subtle changes in brain function networks, which cannot be detected in conventional BFCN construction methods. It is well-known that the high-order method is usually more sensitive to the subtle changes in signal than the low-order method. To exploit high-order information among brain regions, we define the triplet correlation among three brain regions, and derive the second-order brain network based on the connectivity difference and ordinal information in each triplet. For making full use of the complementary information in different brain networks, we proposed a hybrid approach to fuse the first- and second-order brain networks. The proposed method is applied to identify the biomarkers of schizophrenia. The experimental results on six schizophrenia datasets (totally including 439 patients and 426 controls) show that the proposed method outperforms the existing brain network methods in the diagnosis of schizophrenia.

Cluster consensus in first and second-order continuous-time networks with input and communication delays

In this paper, we investigate the cluster consensus problem for first and second-order multi-agent systems with or without delay in continuous-time. Contrary to previous studies on cluster consensus in the literature, the clusters are assumed not to be pre-determined. One of the main contributions of the paper is to compute the number and members of the clusters for a first-order multi-agent network evolving over an arbitrary directed graph. Another novel result is its extension to systems with second-order dynamics. The effect of input and communication delay on cluster consensus is also studied in detail and it is shown that there is an upper bound of input delay whereas communication delay does not adversely alter the cluster consensus properties. Theoretical results are illustrated via several simulations.

Stabilization of Second-Order Memristive Neural Networks With Mixed Time Delays via Nonreduced Order.

In this brief, we investigate a class of second-order memristive neural networks (SMNNs) with mixed time-varying delays. Based on nonsmooth analysis, the Lyapunov stability theory, and adaptive control theory, several new results ensuring global stabilization of the SMNNs are obtained. In addition, compared with the reduced-order method used in the existing research studies, we consider the global stabilization directly from the SMNNs themselves without the reduced-order method. Finally, we give some numerical simulations to show the effectiveness of the results.

Guaranteed cost synchronization for second-order wireless sensor networks with given cost budgets

The current paper addresses leader-following guaranteed cost synchronization with the cost budget given previously for the second-order wireless sensor networks. The published researches on guaranteed cost synchronization design criteria usually are based on the linear matrix inequality (LMI) techniques and cannot take the cost budget given previously into consideration. Firstly, the current paper proposes a guaranteed cost synchronization protocol, which can realize the tradeoff design between the battery power consumption and the synchronization regulation performance. Secondly, for the case without the given cost budget, sufficient conditions for leader-following guaranteed cost synchronization are presented and an upper bound of the cost function is shown. Thirdly, for the case that the cost budget is given previously, the criterion for leader-following guaranteed cost synchronization is proposed. Especially, the value ranges of control gains in these criteria are determined, which means that the existence of control gains in synchronization criteria can be guaranteed, but the LMI techniques can only determine the gain matrix and cannot give the value ranges of control gains. Moreover, these criteria are only associated with the minimum nonzero eigenvalue and the maximum eigenvalue, which can ensure the scalability of the wireless sensor networks. Finally, numerical simulations are given to illustrate theoretical results.

High‐Q second‐order all‐pass delay network in CMOS

Analogue signal processing (ASP) is a promising alternative to digital signal processing techniques in future telecommunication and data-processing solutions. Second-order all-pass delay networks – the building blocks of ASPs – are currently primarily implemented in off-chip planar media, which is unsuited for volume production. In this study, a novel on-chip complementary metal–oxide–semiconductor (CMOS) second-order all-pass network is proposed that includes a post-production tuning mechanism. It is shown that automated tuning with a genetic local optimiser can compensate for CMOS process variation and parasitics, which make physical realisation otherwise infeasible. Measurements indicate a post-tuning bandwidth of 280 MHz, peak-to-nominal delay variation of 10 ns and magnitude variation of 3.1 dB. This is the first time that measurement results have been reported for an active inductorless on-chip second-order all-pass network with a delay Q-value larger than 1.

A Millimeter-Wave Second-Order All-Pass Delay Network in BiCMOS

Analog signal processing (ASP) is a promising alternative to DSP techniques in millimeter-wave (mm-wave) technologies such as 5G, with the second-order all-pass networks a key building block in ASPs. We present an active on-chip mm-wave second-order all-pass network in a 130-nm, 280-GHz $f_{\mathrm {max}}$ SiGe BiCMOS process with an effective bandwidth of 40 GHz, peak delay of 62 ps at 36 GHz, delay $Q_{D}$ value of 3.6, and a magnitude ripple of 1.4 dB. A layout-focused design methodology incorporating layout parasitics and process tolerances is followed. This is the first reported mm-wave bandwidth second-order all-pass network and the first monolithic microwave integrated all-pass network with a $Q_{D}$ value greater than 1.

Self-consistent method and steady states of second-order oscillators

The self-consistent method, first introduced by Kuramoto, is a powerful tool for the analysis of the steady states of coupled oscillator networks. For second-order oscillator networks complications to the application of the self-consistent method arise because of the bistable behavior due to the co-existence of a stable fixed point and a stable limit cycle, and the resulting complicated boundary between the corresponding basins of attraction. In this paper, we report on a self-consistent analysis of second-order oscillators which is simpler compared to previous approaches while giving more accurate results in the small inertia regime and close to incoherence. We apply the method to analyze the steady states of coupled second-order oscillators and we introduce the concepts of margin region and scaled inertia. The improved accuracy of the self-consistent method close to incoherence leads to an accurate estimate of the critical coupling corresponding to transitions from incoherence.

Some Necessary and Sufficient Conditions for Synchronization of Second-Order Interconnected Networks.

This paper presents some necessary and sufficient conditions for the synchronization of second-order interconnected networks, where fixed inner-linked connections with information communication exist. First, a novel derivation of the conditions for a second-order polynomial with complex coefficients to be Hurwitz is provided. Based on this, a sufficient and necessary condition is proposed for the synchronization of the coupled complex network. Next, the design method of the synchronization protocol is constructively given, where the upper and lower bounds of the control gains are obtained through the properties of the polynomial. In addition, we consider some special cases where the second-order model is a double integrator, or general multiagent model without fixed interactions. Finally, the simulation result is given to verify the theoretical analysis.

Exponential stability criteria for delayed second-order memristive neural networks

Abstract This brief presents a class of delayed second-order memristive neural networks (SMNNs). By building a new Lyapunov functional and employing some inequalities technique, we derive some new criteria ensuring global exponential stability of the delayed SMNNs. Compared with other researches on dynamics of neural networks (NNs), our system is described by second-order differential equations and memristor. In addition, we discuss the SMNNs directly and the main results given here without changing the SMNNs into the first-order differential systems. Finally, simulations are given to elaborate the validity of the obtained criteria.

Efficient belief propagation in second-order Bayesian networks for singly-connected graphs

Second-order Bayesian networks extend Bayesian networks by incorporating uncertainty in the conditional probabilities. This paper develops a method for inference in a binary second-order Bayesian network with a singly-connected graph that builds upon the message-passing algorithm for regular belief propagation by leveraging recent developments in subjective logic. The method applies the moment-matching approach to the Beta representation of the uncertain probabilities. We provide experimental analysis which shows that the introduced method effectively captures the bounds for the actual error in a consistent manner and, at the same time, does not decrease the efficiency of the performance compared to the other similar approaches.

Existence, properties and trajectory specification of generalised multi-agent flocking

ABSTRACTA class of flocking algorithms for second-order multi-agent networks with single virtual leaders are developed by including more weighting and navigating matrices in this paper. Flocking existence and properties under the generalised algorithms are investigated around control action boundedness, average trajectory, stability, controllability/observability and ultimate collision. Control significance and design principles of the algorithm matrices are summarised by means of Gronwall-type upper/lower bound inequalities. It is shown that multi-agent systems can form what we call the modified lattices such that flocking features in terms of convergence ratio, orientation, deformation, lattice scaling and subspace assignment can be manipulated with the algorithm matrices. Overall/local ultimate collisions due to dense initial aggregation and inadequate inter-agent distance evaluation are scrutinised. Moreover, by modelling the leader-average dynamics, the average transient/steady-state features of the multi-agent networks are expressed analytically and can be specified through the linear time-invariant (LTI) theory, say pole assignment, least quadratic regulation (LQR) and observer design. Numerical examples are included to verify the main results.

H∞ Performance of Mechanical and Power Networks

This paper investigates the robustness of two well-known applications of second-order consensus dynamics, namely mechanical and power networks. For uniform subsystem parameters, we derive expressions for the H∞ norms of mechanical and power networks, from external disturbances to body displacements and to generator phase angles, respectively. The closed-form expressions are in terms of the physical parameters (damping coefficients and inertias) of the dynamics and in terms of the spectrum of the grounded Laplacian matrix associated with the network. We then analyze the dependence of the H∞ norm of each network on both the network structure and the physical parameters. For a fixed network topology, we find that each system norm can be minimized by choosing the damping coefficient within a specified range. Theoretical contributions are verified via two illustrative examples for mechanical and power networks, in which we show that the network structure, number of the reference nodes and their location in the network can have considerable effects on the system H∞ norm.

Resilient consensus of second-order agent networks: Asynchronous update rules with delays

We study the problem of resilient consensus of sampled-data multi-agent networks with double-integrator dynamics. The term resilient points to algorithms considering the presence of attacks by faulty/malicious agents in the network. Each normal agent updates its state based on a predetermined control law using its neighbors’ information which may be delayed while misbehaving agents make updates arbitrarily and might threaten the consensus within the network. Assuming that the maximum number of malicious agents in the system is known, we focus on algorithms where each normal agent ignores large and small state values among its neighbors to avoid being influenced by malicious agents. The malicious agents are assumed to be omniscient in that they know the updating times and delays and can collude with each other. We deal with both synchronous and partially asynchronous cases with delayed information and derive topological conditions in terms of graph robustness.

Super-twisting-based continuous neural networks modelling of second-order interconnected systems

ABSTRACTThe aim of this work was to design a non-parametric model of interconnected systems represented by uncertain second-order systems with incomplete information (only the generalized position vector is measurable). Artificial neural networks appeared to be a plausible alternative to get a non-parametric representation of the aforementioned interconnected systems. The modelling strategy used a set of spatial distributed second-order continuous neural networks (CNN). Each node in the interconnected system was represented as a second-order continuous neural network added by the super-twisting discontinuous sliding mode algorithm. The non-parametric modelling problem was reduced to design a feasible expression for the CNN weights in order to reproduce the states (including the generalized derivative of position vector) of all the nodes dynamics together and simultaneously. The adaptive laws for the CNN weights ensured the convergence of the CNN trajectories to the states of the uncertain interconnected system. To investigate the qualitative behaviour of the suggested methodology, two numerical examples were proposed. The first one represents the interconnection of three mass–spring–damper mechanical systems. The second example considers the problem of the non-parametric modelling problem for a wave partial differential equation. A set of three-dimensional graphic representations were used to demonstrate the identification abilities achieved by the CNN designed in this study for the second case.