Interconnected Dynamical Systems Research Articles

A tangential interpolation framework for the AAA algorithm

AbstractInterpolation‐based data‐driven methods, such as the Loewner framework or the Adaptive Antoulas‐Anderson (AAA) algorithm, are established and effective approaches to find a realization of a dynamical system from frequency response data (measurements of the system's transfer function). If a system‐theoretic representation of the original model is not available or unfeasible to evaluate efficiently, such reduced realizations enable effective analysis and simulation. This is especially relevant for models of interconnected dynamical systems, which typically have a high number of inputs and outputs to model their coupling conditions correctly.Tangential interpolation is an established strategy to construct accurate reduced‐order models while ensuring a reasonably small size even if many inputs and/or outputs have to be considered. In this contribution, we evaluate the applicability and effectiveness of data‐driven interpolation methods to compute reduced‐order models of dynamical systems with many inputs and outputs. Additionally, we extend AAA to a tangential interpolation setting and thus enable the use of AAA‐like methods in the context of transfer function interpolation for systems with many inputs and outputs.

Network Structure Identification Based on Measured Output Data Using Koopman Operators

This paper considers the identification problem of network structures of interconnected dynamical systems using measured output data. In particular, we propose an identification method based on the measured output data of each node in the network whose dynamic is unknown. The proposed identification method consists of three steps: we first consider the outputs of the nodes to be all the states of the dynamics of the nodes, and the unmeasurable hidden states to be dynamical inputs with unknown dynamics. In the second step, we define the dynamical inputs as new variables and identify the dynamics of the network system with measured output data using Koopman operators. Finally, we extract the network structure from the identified dynamics as the information transmitted via the network. We show that the identified coupling functions, which represent the network structures, are actually projections of the dynamical inputs onto the space spanned by some observable functions. Numerical examples illustrate the validity of the obtained results.

Distributed SDRE control design for a class of nonlinear interconnected dynamic systems

Distributed SDRE control design for a class of nonlinear interconnected dynamic systems

Distributed Kalman Filtering for Interconnected Dynamic Systems.

This article is concerned with the distributed Kalman filtering problem for interconnected dynamic systems, where the local estimator of each subsystem is designed only by its own information and neighboring information. A decoupling strategy is developed to minimize the impact of interconnected terms on the estimation performance, and then the recursive and distributed Kalman filter is derived in the minimum mean-squared error sense. Moreover, by using Lyapunov criterion for linear time-varying systems, stability conditions are presented such that the designed estimator is bounded. Finally, a heavy duty vehicle platoon system is employed to show the effectiveness and advantages of the proposed methods.

Route to synchronization in coupled phase oscillators with frequency-dependent coupling: Explosive or continuous?

Interconnected dynamical systems often transition between states of incoherence and synchronization due to changes in system parameters. These transitions could be continuous (gradual) or explosive (sudden) and may result in failures, which makes determining their nature important. In this study, we abstract dynamical networks as an ensemble of globally coupled Kuramoto-like phase oscillators with frequency-dependent coupling and investigate the mechanisms for transition between incoherent and synchronized dynamics. The characteristics that dictate a continuous or explosive route to synchronization are the distribution of the natural frequencies of the oscillators, quantified by a probability density function g(ω), and the relation between the coupling strength and natural frequency of an oscillator, defined by a frequency-coupling strength correlation function f(ω). Our main results are conditions on f(ω) and g(ω) that result in continuous or explosive routes to synchronization and explain the underlying physics. The analytical developments are validated through numerical examples.

Distributed Estimation for Interconnected Dynamic Systems Under Binary Sensors

This paper is concerned with the distributed state estimation problem for a class of interconnected dynamic systems, where several binary sensors are deployed to observe each subsystem. The judgement of switching instant for binary sensor is based on the certain estimates and sensed variables. In this case, two innovation sequences which combine thresholds and state estimates are given to extract valid information from binary measurement. Then, a distributed recursive estimator is given on the basis of weighting fusion criteria and bounded recursive optimization. The interconnected gains of state estimator are designed by minimizing the impacts of neighboring terms, while the optimal local gains and weighting fusion matrices are designed by constructing convex optimization problems. Moreover, the asymptotic stability of squared estimation error is proved by means of sequential analysis. Finally, two illustrative examples are employed to show the effectiveness of the proposed methods.

Distributed adaptive control for optimal tracking of uncertain interconnected dynamical systems

Distributed adaptive control for optimal tracking of uncertain interconnected dynamical systems

Discretization and Model Reduction Error Estimation of Interconnected Dynamical Systems

In this paper a novel approach for estimating both discretization and model order reduction error is presented. Discretization is carried out using finite element method. Finite element models are converted to linear time-invariant state space models. Using the generalized square root balanced truncation method a series of reduced models are obtained. It is shown that for special class of passive and dissipative dynamical systems upper a-posteriori reduction error bound can be modeled less conservatively. Estimation of both discretization and model reduction error is further refined using v-gap metric. Quality of the obtained reduced order model is verified in the frequency domain. The proposed error estimation approach is demonstrated on a numerical example.

Stackelberg Population Dynamics: A Predictive-Sensitivity Approach

Hierarchical decision-making processes traditionally modeled as bilevel optimization problems are widespread in modern engineering and social systems. In this work, we deal with a leader with a population of followers in a hierarchical order of play. In general, this problem can be modeled as a leader–follower Stackelberg equilibrium problem using a mathematical program with equilibrium constraints. We propose two interconnected dynamical systems to dynamically solve a bilevel optimization problem between a leader and follower population in a single time scale by a predictive-sensitivity conditioning interconnection. For the leader’s optimization problem, we developed a gradient descent algorithm based on the total derivative, and for the followers’ optimization problem, we used the population dynamics framework to model a population of interacting strategic agents. We extended the concept of the Stackelberg population equilibrium to the differential Stackelberg population equilibrium for population dynamics. Theoretical guarantees for the stability of the proposed Stackelberg population learning dynamics are presented. Finally, a distributed energy resource coordination problem is solved via pricing dynamics based on the proposed approach. Some simulation experiments are presented to illustrate the effectiveness of the framework.

On a security vs privacy trade-off in interconnected dynamical systems

We study a security problem for interconnected systems, where each subsystem aims to detect local attacks using local measurements and information exchanged with neighboring subsystems. The subsystems also wish to maintain the privacy of their states and, therefore, use privacy mechanisms that share limited or noisy information with other subsystems. We quantify the privacy level based on the estimation error of a subsystem’s state and propose a novel framework to compare different mechanisms based on their privacy guarantees. We develop a local attack detection scheme without assuming the knowledge of the global dynamics, which uses local and shared information to detect attacks with provable guarantees. Additionally, we quantify a trade-off between security and privacy of the local subsystems. Interestingly, we show that, for some instances of the attack, the subsystems can achieve a better detection performance by being more private. We provide an explanation for this counter-intuitive behavior and illustrate our results with examples.

The Use of Partial Stability in the Analysis of Interconnected Systems

Abstract In this paper, we develop sufficient conditions for uniform asymptotic stability of interconnected dynamical systems that are not in cascade form. We show that the stability analysis of a two-subsystem interconnection can be reduced to ensuring the stability of the first nonisolated subsystem with respect to its own state vector (partial stability) and the stability of the isolated second subsystem. In addition, based on the above results, we provide a control design framework for nonlinear systems where the control objective reduces to stabilization of only a part of the system state while guaranteeing the stability for the entire state of the system. We validate the efficacy of the proposed control framework via simulations and experiments using the wheeled mobile robot platform.

A Decomposition-Based Optimization Algorithm for Combined Plant and Control Design of Interconnected Dynamic Systems

AbstractStrong coupling of the physical and control parts within complex dynamic systems should be addressed by integrated design approaches that can manage such interactions. Otherwise, the final solution will be suboptimal or even infeasible. Combined design and control (co-design) methods can tackle this issue by managing the mentioned interactions and can result in superior optimal solutions. Current co-design methods are applicable to simplified non-interconnected systems; however, these methods might be impractical or even impossible to apply to real-world interconnected dynamic systems, hindering designers from obtaining the system-level optimal solutions. This work addresses this issue by developing an optimization algorithm which combines a decomposition-based optimization strategy known as analytical target cascading (ATC) with a co-design-centric formulation of multidisciplinary dynamic system design optimization (MDSDO). Considering the time-dependent linking variables among the dynamic systems’ components, a new consistency measure has also been proposed to manage such quantities in the optimization process. Finally, a plug-in hybrid electric vehicle powertrain, representative of an interconnected dynamic system, has been studied to validate the new algorithm’s results against the conventional all-at-once (AAO) MDSDO. Although the numerical results from the ATC-MDSDO slightly deviate from those in the AAO-MDSDO, this method can play a crucial role as a benchmark when the AAO solution is unattainable or a distributed design paradigm is required.

Basal ganglia oscillations as biomarkers for targeting circuit dysfunction in Parkinson's disease.

Oscillations are a naturally occurring phenomenon in highly interconnected dynamical systems. However, it is thought that excessive synchronized oscillations in brain circuits can be detrimental for many brain functions by disrupting neuronal information processing. Because synchronized basal ganglia oscillations are a hallmark of Parkinson's disease (PD), it has been suggested that aberrant rhythmic activity associated with symptoms of the disease could be used as a physiological biomarker to guide pharmacological and electrical neuromodulatory interventions. We here briefly review the various manifestations of basal ganglia oscillations observed in human subjects and in animal models of PD. In this context, we also review the evidence supporting a pathophysiological role of different oscillations for the suppression of voluntary movements as well as for the induction of excessive motor activity. In light of these findings, it is discussed how oscillations could be used to guide a more precise targeting of dysfunctional circuits to obtain improved symptomatic treatment of PD.

Decentralized finite-horizon suboptimal control for nonlinear interconnected dynamic systems using SDRE approach

This paper introduces a new approach to ensure the decentralized horizon suboptimal control of interconnected nonlinear systems based on the decentralized finite-state-dependent Riccati equation. T...

Improving Entropy Estimates of Complex Network Topology for the Characterization of Coupling in Dynamical Systems.

A new measure for the characterization of interconnected dynamical systems coupling is proposed. The method is based on the representation of time series as weighted cross-visibility networks. The weights are introduced as the metric distance between connected nodes. The structure of the networks, depending on the coupling strength, is quantified via the entropy of the weighted adjacency matrix. The method has been tested on several coupled model systems with different individual properties. The results show that the proposed measure is able to distinguish the degree of coupling of the studied dynamical systems. The original use of the geodesic distance on Gaussian manifolds as a metric distance, which is able to take into account the noise inherently superimposed on the experimental data, provides significantly better results in the calculation of the entropy, improving the reliability of the coupling estimates. The application to the interaction between the El Niño Southern Oscillation (ENSO) and the Indian Ocean Dipole and to the influence of ENSO on influenza pandemic occurrence illustrates the potential of the method for real-life problems.

Order reduction of hierarchical interconnected dynamical systems

Abstract. The simulation of large scale nonlinear dynamical interconnected systems, as they arise in all modern engineering disciplines, is a usual task. Due to the high complexity of the considered systems, the principle of thinking in hierarchical structures is essential and common among engineers. Therefore, this contribution proposes an approach for the numerical simulation of large systems, which keeps the hierarchical system structure alive during the entire simulation process while simultaneously exploiting it for order reduction purposes. This is accomplished by embedding the trajectory piecewise linear order reduction scheme in a modified variant of the component connection modeling for building interconnected system structures. The application of this concept is demonstrated by means of a widely used benchmark example and a modified variant of it.

Embedded order reduction of hierarchical systems within an mCCM framework

PurposeThe purpose of this paper is to present the embedding of linear and nonlinear order reduction methods in a theoretical framework for handling hierarchically interconnected dynamical systems.Design/methodology/approachBased on the component connection modeling (CCM), a modified framework called mCCM for describing interconnected dynamic systems especially with hierarchical structures is introduced and used for order reduction purposes. The balanced truncation method for linear systems and the trajectory piecewise linear reduction scheme are used for the order reduction of systems described within the mCCM framework.FindingsIt is shown that order reduction methods can be embedded into the context of interconnected dynamical systems with the benefit of having a further degree of freedom in form of the hierarchical level, on which the order reduction is performed.Originality/valueThe aspect of hierarchy within system descriptions is exploited for order reduction purposes. This distinguishes the presented approach from common methods, which already start with single large-scale systems without explicitly considering hierarchical structures.

Dissipativity reinforcement in interconnected systems

This paper focuses on the reinforcement of the quantitative performance in interconnected dynamical systems. The following problem is addressed that concerns dissipativity reinforcement via interconnection: Find a class of subsystems and their interconnection rule such that the L2 gain bound of the entire interconnected system is reduced compared with that of each individual subsystem. We assume that each subsystem has a special passivity property that is characterized by two parameters, and has a bounded L2 gain. Then, the feedback connection and the more general interconnection of the subsystems are expressed by the transition of the two parameters inheriting the same passivity property. In addition, the L2 gain bound of the entire interconnected system, estimated with the parameters, is strictly reduced and becomes less than that of each subsystem. Finally, special interconnection rules are considered to show that the scale-expansion of the interconnected system, i.e., increasing the number of subsystems, gradually reduces the L2 gain bound.

A Lyapunov approach to stability analysis of partial synchronization in delay-coupled networks

Abstract Networks of interconnected dynamical systems may exhibit a so-called partial synchronization phenomenon, which refers to synchronous behaviors of some but not all of the systems. The patterns of partial synchronization are often characterized by partial synchronization manifolds, which are linear invariant subspace of the state space of the network dynamics. Here, we propose a Lyapunov-Krasovskii approach to analyze the stability of partial synchronization manifolds in delay-coupled networks. First, the synchronization error dynamics are isolated from the network dynamics in a systematic way. Second, we use a parameter-dependent Lyapunov-Krasovskii functional to assess the local stability of the manifold, by employing techniques originally developed for linear parameter-varying (LPV) time-delay systems. The stability conditions are formulated in the form of linear matrix inequalities (LMIs) which can be solved by several available tools.

Synchronization and Dynamic Consensus of Heterogeneous Networked Systems

We present an analysis framework for the study of synchronization of heterogeneous nonlinear systems interconnected over networks described by directed graphs. Heterogeneous systems may have totally different dynamical models, albeit of the same dimension, or may possess equal models with different lumped parameters. We show that their behavior, when network-interconnected, is fully characterized in terms of two properties whose study may be recasted in terms of the stability analysis of two corresponding interconnected dynamical systems that evolve in orthogonal spaces: on one hand, we have the so-called emergent dynamics and, on the other, the synchronization error dynamics. Based on this premise, we introduce the concept of dynamic consensus and we present results on robust stability which assess the conditions for practical asymptotic synchronization of networked systems and characterize their collective behavior. To illustrate our main theoretical findings we broach a brief case-study on mutual synchronization of heterogeneous chaotic oscillators.