Linear Time-invariant Networks Research Articles

Reciprocity and Inter-Reciprocity: A Tutorial— Part I: Linear Time-Invariant Networks

We present a systematic treatment of reciprocity and inter-reciprocity in linear electrical networks, without resorting to the use of Tellegen’s theorem. In Part I, we discuss linear time-invariant networks. In Part II, we expand the applications of our approach to linear periodically time-varying (LPTV) networks.

Space–Time Sampling for Network Observability

Designing sparse sampling strategies are crucial in having resilient estimation and control in networked systems as they make sampling less expensive and networks more resilient w.r.t where and when samples are collected. It is shown under what conditions taking coarse samples from a network will contain the same amount of information as a finer set of samples. Our goal is to estimate the initial condition of linear time-invariant networks using a set of noisy measurements. The observability condition is reformulated as the frame condition, where one can easily trace location and time stamps of each sample. We compare estimation quality of various sampling strategies using estimation measures, which depend on spectrum of the corresponding frame operators. Using properties of the minimal polynomial of the state matrix, deterministic and randomized methods are suggested to construct observability frames. Intrinsic tradeoffs assert that collecting samples from fewer subsystems dictates taking more samples (in average) per subsystem. Three scalable algorithms are developed to generate sparse space-time sampling strategies with explicit error bounds.

Spatiotemporally Modulated Gyrators with Extended Bandwidth through Dual-Sideband Operation

One of the major challenges in the design of nonreciprocal devices is the application of magnetic biasing. This problem has motivated a flurry of work for the development of magnetless nonreciprocal devices, with spatiotemporal modulation being one of the most promising approaches to this end. The most fundamental nonreciprocal component is the gyrator, which can be the building block of other nonreciprocal components, like ciculators and isolators. Spatiotemporally modulated (STM) gyrators are typically based on a pair of frequency converters with different modulation phases, connected through a linear time-invariant network. Such gyrators are usually designed to operate with a single modulation sideband. In this paper we argue that this approach is suboptimal and show how adding just one more sideband can increase the bandwidth by $40\mathrm{%}$ with a lower modulation frequency and the same modulation amplitude. We present an implementation of the proposed approach based on a double-balanced circuit with Wheatstone bridges of varactors, which provides large isolation between the input signal and the sidebands. We provide a theoretical and an experimental analysis of the circuit with excellent agreement to each other. In addition to introducing a topology of STM gyrators with improved bandwidth compared with existing designs, our results highlight a potential connection between the bandwidth of STM devices and their harmonic order, which might also find applications in other STM systems, like isolators, circulators, and photonic topological insulators.

Structural Differences Between Healthy Subjects and Patients With Schizophrenia or Schizoaffective Disorder: A Graph and Control Theoretical Perspective.

The coordinated dynamic interactions of large-scale brain circuits and networks have been associated with cognitive functions and behavior. Recent advances in network neuroscience have suggested that the anatomical organization of such networks puts fundamental constraints on the dynamical landscape of brain activity, i.e., the different states, or patterns of regional activation, and transition between states the brain can display. Specifically, it has been shown that densely connected, central regions control the transition between states that are “easily” reachable (in terms of expended energy), whereas weakly connected areas control transitions to states that are hard-to-reach. Changes in large-scale brain activity have been hypothesized to underlie many neurological and psychiatric disorders. Evidence has emerged that large-scale dysconnectivity might play a crucial role in the pathophysiology of schizophrenia, especially regarding cognitive symptoms. Therefore, an analysis of graph and control theoretic measures of large-scale brain connectivity in patients offers to give insight into the emergence of cognitive disturbances in the disorder. To investigate these potential differences between patients with schizophrenia (SCZ), patients with schizoaffective disorder (SCZaff) and matched healthy controls (HC), we used structural MRI data to assess the microstructural organization of white matter. We first calculate seven graph measures of integration, segregation, centrality and resilience and test for group differences. Second, we extend our analysis beyond these traditional measures and employ a simplified noise-free linear discrete-time and time-invariant network model to calculate two complementary measures of controllability. Average controllability, which identifies brain areas that can guide brain activity into different, easily reachable states with little input energy and modal controllability, which characterizes regions that can push the brain into difficult-to-reach states, i.e., states that require substantial input energy. We identified differences in standard network and controllability measures for both patient groups compared to HCs. We found a strong reduction of betweenness centrality for both patient groups and a strong reduction in average controllability for the SCZ group again in comparison to the HC group. Our findings of network level deficits might help to explain the many cognitive deficits associated with these disorders.

Strong structural input and state observability of linear time-invariant systems: Graphical conditions and algorithms

The paper studies input and state observability (ISO) of discrete-time linear time-invariant network systems whose dynamics are affected by unknown inputs. More precisely, we aim at reconstructing the initial state and the sequence of unknown inputs from the system outputs, and we will use the term ISO when the input reconstruction is possible with delay one, namely the inputs up to time k−1 and the states up to time k can be obtained from the outputs up to time k, while the term unconstrained ISO will refer to the case where there is some arbitrary delay in the input reconstruction. We focus on the problem of s-structural ISO (resp. s-structural unconstrained ISO) wherein the objective is to find conditions such that for all system matrices that carry the same network structure, the resulting system is ISO (resp. unconstrained ISO). We provide first a graphical characterization for s-structural unconstrained ISO, and subsequently, sufficient conditions and necessary conditions for s-structural ISO. For the latter, under the assumption of zero feedthrough, these conditions coincide and characterise ISO. The conditions presented are in terms of existence of suitable uniquely restricted matchings in bipartite graphs associated with the structured system. In order to test these conditions, we present polynomial-time algorithms. Finally, we discuss an equivalent reformulation of the main conditions in terms of coloring algorithms as in the literature of zero forcing sets.

Fast Online Reinforcement Learning Control Using State-Space Dimensionality Reduction

In this article, we propose a fast reinforcement learning (RL) control algorithm that enables online control of large-scale networked dynamic systems. RL is an effective way of designing model-free linear quadratic regulator (LQR) controllers for linear time-invariant (LTI) networks with unknown state-space models. However, when the network size is large, conventional RL can result in unacceptably long learning times. The proposed approach is to construct a compressed state vector by projecting the measured state through a projective matrix. This matrix is constructed from online measurements of the states in a way that it captures the dominant controllable subspace of the open-loop network model. Next, an RL controller is learned using the reduced-dimensional state instead of the original state such that the resulting cost is close to the optimal LQR cost. Numerical benefits as well as the cyber-physical implementation benefits of the approach are verified using illustrative examples including an example of wide-area control of the IEEE 68-bus benchmark power system.

A Generalized Model for Linear-Periodically-Time-Variant Circulators

Magnetic-free non-reciprocity based on linear-periodically-time-variant (LPTV) circuits has received significant research and commercial attention since it could revolutionize wireless communications. LPTV circuits are formed by two main components: linear-time-invariant (LTI) networks and periodically-modulated switches. The modulated switches are the core elements to break the reciprocity of LTI networks. To understand and design LPTV circulators, a universal and intuitive analytical model is required. However, such model does not exist as it is extremely challenging to accurately model and fully understand the LPTV behaviour of energy storage networks. To address this limitation, this work introduces a novel analysis method, which is broadly applicable to any LPTV networks, and validates it experimentally. The novelty of this methodology comes from two main contributions: (1) modelling of the switch as a resistor in parallel with a current-controlled current source; (2) the decomposition of the LPTV network into the linear superposition of two LTI networks. We apply this technique to model the exact behaviour of an LPTV circulator in the frequency domain.

A General n-Port Network’s Equivalent Current Sources Theorem

In this paper a general n-port network’s equivalent current theorem has been derived out, for n = 1, 2…. the traditional Norton’s Theorem is only a special case of it for n=1. When an n-port passive linear time-invariant network is connected to another n-port linear time-invariant network which contained sinusoidal sources with same frequency, this theorem provides a new way to calculate the port-current of the n-port passive network. But the short-port currents of the n-port network contained sinusoidal sources must be known at first. In sinusoidal networks, currents are vector quantity or complex quantity, including magnitude and phase angle. Ammeter can only be used to measure the magnitude of the current, not including its phase angle. So it is impossible to get the short-port currents by the short-port experiment. Moreover the short-port experiment may cause some dangerous events. So a special method to get the short-port currents is introduced in this paper, First to find out the open-port voltage vector ( including magnitude and phase angle), by measuring the voltages magnitude between some two points of the open-port with a voltmeter and by drawing a series of voltage vector triangles that one side vector is the sum of other two side vectors , if the phase angle of one side vector in a triangle is known, the phase angles of the other side vectors in the same triangle can be decided. In the first triangle, the first open-port voltage vector is contained, its phase angle can be assigned to be zero, then the phase angles of the other two voltage vectors in the first triangle can be decided. In the second triangle, one of the two above voltage vectors is contained, then the phase angles of the other two voltage vectors in the second triangle can be decided. Thus go on step by step, all the open-port voltage vectors can be obtained. And the open-port voltage complex matrix has been obtained. The equation related the short-port current complex matrix and the open-port voltage complex matrix has been derived out in this paper. So the short-port current complex matrix can be obtained.

Network Identification With Latent Nodes via Autoregressive Models

We consider linear time-invariant networks with unknown topology where only a manifest subset of the nodes can be directly actuated and measured while the state of the remaining latent nodes and their number are unknown. Our goal is to identify the transfer function of the manifest subnetwork and determine whether interactions between manifest nodes are direct or mediated by latent nodes. We show that if there are no inputs to the latent nodes, the manifest transfer function can be approximated arbitrarily well in the $H_{\infty }$ -norm sense by the transfer function of an autoregressive model and present a least-squares estimation method to construct the autoregressive model from measured data. We show that the least-squares autoregressive method guarantees an arbitrarily small $H_{\infty}$ -norm error in the approximation of the manifest transfer function, exponentially decaying once the model order exceeds a certain threshold. Finally, we show that when the latent subnetwork is acyclic, the proposed method achieves perfect identification of the manifest transfer function above a specific model order as the length of the data increases. Various examples illustrate our results.

Robust Synchronization in Multi-Agent Networks With Unstable Dynamics

The problem of synchronization in heterogenous linear time-invariant networks is investigated. An approach blending the theory of integral quadratic constraints (IQCs) with the gap metric is proposed to study the problem within a unifying framework, where both the agents and communication channels can be dynamic, infinite-dimensional, unstable, and uncertain. Structural properties of the uncertainty are described by IQCs and exploited in the analysis to reduce conservatism. The homotopy employed in IQC analysis is defined with respect to the graph topology induced by the gap metric, whereby open-loop unstable dynamics are accommodated. Sufficient conditions for synchronism are provided, extending recent developments which have been shown to unify several existing synchronization analysis results in the literature.

Detecting severity of delamination in a lap joint using S-parameters

The scattering parameters (S-parameters) represent the frequency response of a two-port linear time-invariant network. Treating a lap joint structure instrumented with two piezoelectric wafer active transducers (PWaTs) as such a network, this paper investigates the application of the S-parameters for detecting the severity of delamination in the lap joint. The pulse-echo signal calculated from the reflection coefficients, namely the S11 and S22-parameters, can be divided into three signals, i.e. the excitation, resonant, and echo signals, based on their respective time spans. Analyzing the effects of the delamination on the resonant signal enables us to identify the resonance at which the resonant characteristics of the PWaTs are least sensitive to the delamination. Only at this resonance, we found that the reflection coefficients and the amplitude of the first arrival echo signal changed monotonously with the increase of the delamination length. This discovery is further validated by the time-domain pitch-catch signal calculated from the transmission coefficient (i.e. the S21-parameter). In addition, comparing the pulse-echo signals obtained from both PWaTs enables us to determine the side of the lap joint that the delamination is located at. This work establishes the S-parameters as an effective tool to evaluate the effects of damage on the PWaT resonant characteristics, based on which the PWaT resonance can be selected judiciously for damage severity detection. Correlating the reflection and transmission coefficients also provide addition validations that increase the detection confidence.

Identification of a Class of Nonlinear Dynamical Networks

Identification of dynamic networks has attracted considerable interest recently. So far the main focus has been on linear time-invariant networks. Meanwhile, most real-life systems exhibit nonlinear behaviors; consider, for example, two stochastic linear time-invariant systems connected in series, each of which has a nonlinearity at its output. The estimation problem in this case is recognized to be challenging, due to the analytical intractability of both the likelihood function and the optimal one-step ahead predictors of the measured nodes. In this contribution, we introduce a relatively simple prediction error method that may be used for the estimation of nonlinear dynamical networks. The estimator is defined using a deterministic predictor that is nonlinear in the known signals. The estimation problem can be defined using closed-form analytical expressions in several non-trivial cases, and Monte Carlo approximations are not necessarily required. We show, that this is the case for some block-oriented networks with no feedback loops and where all the nonlinear modules are polynomials. Consequently, the proposed method can be applied in situations considered challenging by current approaches. The performance of the estimation method is illustrated on a numerical simulation example.

SNEM: Full S-Parameter Synthesis From Near-End Measurement

A novel method called SNEM (full S-parameter synthesis from near-end measurement) is proposed to obtain the full S-parameter matrix by probing only a subset of all the ports of a passive linear time-invariant network. Other ports that are not probed need to be terminated by various impedance values. The traditional open-short-load method is a special case in the SNEM framework. A termination error correction algorithm and a guideline for choosing termination impedance values have also been developed to make the method robust, accurate, and efficient. Examples are included in this paper to demonstrate the applications of the SNEM method in integrated circuit package, on-die structure, and socket/connector characterizations. Simulation results also show that the SNEM method can be applied to power line communication system characterization. The SNEM method has been validated by both simulations and lab measurements.

A Fast Algorithm for Testability Analysis of Large Linear Time-Invariant Networks

Testability analysis is aimed at evaluating the degree of solvability of both the fault diagnosis and the parameter identification problems. As far as analog linear time-invariant circuits are concerned, various approaches, predicated on either fully numerical or symbolic techniques, have been proposed in the past for such purpose. However, the former are severely plagued by the effects of roundoff errors, whereas the latter soon become unacceptably time-consuming as the circuit size grows large. In the present paper, a novel algorithm is described, which completely avoids resorting to symbolic techniques, thereby overcoming the mentioned related drawbacks and, at the same time, it drastically reduces the computational issues posed by formerly proposed fully numerical methods. Such an algorithm provides information as to how many and which parameters can be diagnosed or identified and has been straightforwardly translated into a software for fully automated testability analysis, also briefly described herein. The aforementioned noteworthy features of the algorithm on which it is based render the software in question a successful tool for testability analysis of very large networks, as opposed to formerly proposed computer programs, which generally fail such a task. Examples of application of both the algorithm and its software counterpart are given.

普遍n口网络的等效电压源定理

本文研究在网络正弦稳态状况下将传统的单口网络的等效电压源定理(戴维南定理)推广为普遍n口网络的等效电压源定理。前者仅是后者n = 1的特例。 This paper presents a general theorem for an n-port linear time-invariant network contained several sources can be expressed by equivalent voltage sources which equal to the open port vol-tages and a passive network which is the original network when its contained sources don’t work. Obviously, the conventional Thevenin’s theorem is only a special case of it for n = 1 .

Realizing actual feedback control of complex network

Abstract In this paper, we present the concept of feedbackability and how to identify the Minimum Feedbackability Set of an arbitrary complex directed network. Furthermore, we design an estimator and a feedback controller accessing one MFS to realize actual feedback control, i.e. control the system to our desired state according to the estimated system internal state from the output of estimator. Last but not least, we perform numerical simulations of a small linear time-invariant dynamics network and a real simple food network to verify the theoretical results. The framework presented here could make an arbitrary complex directed network realize actual feedback control and deepen our understanding of complex systems.

Synchronization of Nonlinear Oscillators in an LTI Electrical Power Network

Sufficient conditions are derived for the global asymptotic synchronization of a class of identical nonlinear oscillators coupled through a linear time-invariant network. In particular, we focus on systems where oscillators are connected to a common node through identical branch impedances. For such networks, it is shown that the synchronization condition is independent of the number of oscillators and the value of the load impedance connected to the common node. Theoretical findings are then leveraged to control a system of parallel single-phase voltage source inverters serving an impedance load in an islanded microgrid application. The ensuing paradigm: i) does not necessitate communication between inverters, ii) is independent of system load, and iii) facilitates a modular design approach because the synchronization condition is independent of the number of oscillators. We present both simulation and experimental case studies to validate the analytical results and demonstrate the proposed application.

A synthetic network using current conveyors of the open-circuit impedance matrix of a linear time-invariant multiport

Previous research appearing in the literature required that all the realizable z parameters of the open-circuit impedance matrix of a linear time-invariant multiport network must be real, rational, stable functions. Although different z parameters could be selected independently without necessarily to having common poles, this also required that all the z parameters be minimum phase. A synthetic network using second-generation current conveyors (CCIIs), third-generation current conveyors (CCIIIs) and differential difference current conveyors (DDCCs) is proposed here, with each transfer impedance parameter being either minimum or nonminimum phase. Additionally, it can be designed automatically by computers since the current conveyor circuits, the transfer functions and numerical methods are available.

State‐space equations of regular and strictly topologically degenerate linear lumped time‐invariant networks: the multiport method

AbstractA new complete approach to the multiport formulation of the state‐space equations of uniquely solvable regular or strictly topologically degenerate linear lumped time‐invariant networks is presented. It is based on a Gedankenexperiment during which the topological structure of the original network is manipulated in various ways. The final method requires one to calculate the describing matrices of three homogeneous multiports (i.e. a capacitive, an inductive and a resistive one), which are obtained from the network of interest in a very simple manner. As a by‐product, the equivalent partitioned network is also derived. As an example of application, the state‐space equations of a fourth‐order strictly topologically degenerate network are provided. Copyright © 2001 John Wiley & Sons. Ltd.

Analysis of switching loads in networks using operational matrices

This letter presents an approach for rapid calculation of the steady state of time-varying electric networks based on the use of orthogonal series expansions. The method takes advantage of the operational properties available to most orthogonal series expansions, i.e., integration, differentiation, product, and coefficients matrices. An important feature of this approach is that an analytical expression for the steady state is provided. The approach may be seen as an extension of phasor analysis commonly used for the analysis of linear time invariant electric networks to periodic networks.