Christoforos Somarakis

Christoforos Somarakis, Yaser Ghaedsharaf, Nader Motee

We quantify the value-at-risk of inter-vehicle collision and detachment for a class of platoons, which are governed by second-order dynamics in presence of communication time-delay and exogenous stochastic noise. Closed-form expressions for the risk measures are obtained as functions of Laplacian eigen-spectrum as well as their fine explicit approximations using rational polynomial functions. We quantify several hard limits and fundamental tradeoffs among the risk measures, network connectivity, communication time-delay, and statistics of exogenous stochastic noise. Simultaneous presence of stochastic noise and time delay in a platoon imposes some idiosyncratic limitations on the behavior of collision and detachment risks, for instance, weakening (improving) network connectivity may result in lower (higher) levels of risk. Furthermore, a thorough risk analysis and comparison have been conducted for networks with specific graph topology. We support our theoretical findings via extensive simulations.

Yaser Ghaedsharaf, Milad Siami, Christoforos Somarakis et al.

We investigate notions of network centrality in terms of the underlying coupling graph of the network, structure of exogenous uncertainties, and communication time-delay. Our focus is on time-delay linear consensus networks, where uncertainty is modeled by structured additive noise on the dynamics of agents. The centrality measures are defined using the $\mathcal H_2$-norm of the network. We quantify the centrality measures as functions of time-delay, the graph Laplacian, and the covariance matrix of the input noise. Several practically relevant uncertainty structures are considered, where we discuss two notions of centrality: one w.r.t intensity of the noise and the other one w.r.t coupling strength between the agents. Furthermore, explicit formulas for the centrality measures are obtained for all types of uncertainty structures. Lastly, we rank agents and communication links based on their centrality indices and highlight the role of time-delay and uncertainty structure in each scenario. Our counter intuitive grasp is that some of centrality measures are highly volatile with respect to time-delay.

Christoforos Somarakis, Nader Motee

The focus of this paper is to quantify measures of aggregate fluctuations for a class of consensus-seeking multiagent networks subject to exogenous noise with alpha-stable distributions. This type of noise is generated by a class of random measures with heavy-tailed probability distributions. We define a cumulative scale parameter using scale parameters of probability distributions of the output variables, as a measure of aggregate fluctuation. Although this class of measures can be characterized implicitly in closed-form in steady-state, finding their explicit forms in terms of network parameters is, in general, almost impossible. We obtain several tractable upper bounds in terms of Laplacian spectrum and statistics of the input noise. Our results suggest that relying on Gaussian-based optimal design algorithms will result in non-optimal solutions for networks that are driven by non-Gaussian noise inputs with alpha-stable distributions. The manuscript has been submitted for publication to IEEE Transactions on Control of Network Systems. It is the extended version of preliminary paper included in the proceedings of the 2018 American Control Conference.

Guangyi Liu, Vivek Pandey, Christoforos Somarakis et al.

We develop a framework for studying and quantifying the risk of cascading failures in time-delay consensus networks, motivated by a team of agents attempting temporal rendezvous under stochastic disturbances and communication delays. To assess how failures at one or multiple agents amplify the risk of deviation across the network, we employ the Average Value-at-Risk as a systemic measure of cascading uncertainty. Closed-form expressions reveal explicit dependencies of the risk of cascading failure on the Laplacian spectrum, communication delay, and noise statistics. We further establish fundamental lower bounds that characterize the best-achievable network performance under time-delay constraints. These bounds serve as feasibility certificates for assessing whether a desired safety or performance goal can be achieved without exhaustive search across all possible topologies. In addition, we develop an efficient single-step update law that enables scalable propagation of conditional risk as new failures are detected. Analytical and numerical studies demonstrate significant computational savings and confirm the tightness of the theoretical limits across diverse network configurations.

Ion Matei, Johan de Kleer, Christoforos Somarakis et al.

To understand changes in physical systems and facilitate decisions, explaining how model predictions are made is crucial. We use model-based interpretability, where models of physical systems are constructed by composing basic constructs that explain locally how energy is exchanged and transformed. We use the port Hamiltonian (p-H) formalism to describe the basic constructs that contain physically interpretable processes commonly found in the behavior of physical systems. We describe how we can build models out of the p-H constructs and how we can train them. In addition we show how we can impose physical properties such as dissipativity that ensure numerical stability of the training process. We give examples on how to build and train models for describing the behavior of two physical systems: the inverted pendulum and swarm dynamics.

Hossein K. Mousavi, Christoforos Somarakis, Qiyu Sun et al.

Spectral decomposition of dynamical systems is a popular methodology to investigate the fundamental qualitative and quantitative properties of these systems and their solutions. In this chapter, we consider a class of nonlinear cooperative protocols, which consist of multiple agents that are coupled together via an undirected state-dependent graph. We develop a representation of the system solution by decomposing the nonlinear system utilizing ideas from the Koopman operator theory and its spectral analysis. We use recent results on the extensions of the well-known Hartman theorem for hyperbolic systems to establish a connection between the original nonlinear dynamics and the linearized dynamics in terms of Koopman spectral properties. The expected value of the output energy of the nonlinear protocol, which is related to the notions of coherence and robustness in dynamical networks, is evaluated and characterized in terms of Koopman eigenvalues, eigenfunctions, and modes. Spectral representation of the performance measure enables us to develop algorithmic methods to assess the performance of this class of nonlinear dynamical networks as a function of their graph topology. Finally, we propose a scalable computational method for approximation of the components of the Koopman mode decomposition, which is necessary to evaluate the systemic performance measure of the nonlinear dynamic network.

Yaser Ghaedsharaf, Milad Siami, Christoforos Somarakis et al.

We analyze performance of a class of time-delay first-order consensus networks from a graph topological perspective and present methods to improve it. The performance is measured by network's square of H-2 norm and it is shown that it is a convex function of Laplacian eigenvalues and the coupling weights of the underlying graph of the network. First, we propose a tight convex, but simple, approximation of the performance measure in order to achieve lower complexity in our design problems by eliminating the need for eigen-decomposition. The effect of time-delay reincarnates itself in the form of non-monotonicity, which results in nonintuitive behaviors of the performance as a function of graph topology. Next, we present three methods to improve the performance by growing, re-weighting, or sparsifying the underlying graph of the network. It is shown that our suggested algorithms provide near-optimal solutions with lower complexity with respect to existing methods in literature.

Christoforos Somarakis, Yaser Ghaedsharaf, Nader Motee

For the class of noisy time-delay linear consensus networks, we obtain explicit formulas for risk of large fluctuations of a scalar observable as a function of Laplacian spectrum and its eigenvectors. It is shown that there is an intrinsic tradeoff between risk and effective resistance of the underlying coupling graph of the network. The main implication is that increasing network connectivity, increases the risk of large fluctuations. For vector-valued observables, we obtain computationally tractable lower and upper bounds for joint risk measures. Then, we study behavior of risk measures for networks with specific graph topologies and show how risk scales with network size.