Usman A. Khan

Distributing the Kalman Filter for Large-Scale Systems

This paper presents a distributed Kalman filter to estimate the state of a sparsely connected, large-scale, n -dimensional, dynamical system monitored by a network of N sensors. Local Kalman filters are implemented on nl-dimensional subsystems, nl Lt n, obtained by spatially decomposing the large-scale system. The distributed Kalman filter is optimal under an Lth order Gauss-Markov approximation to the centralized filter. We quantify the information loss due to this Lth-order approximation by the divergence, which decreases as L increases. The order of the approximation L leads to a bound on the dimension of the subsystems, hence, providing a criterion for subsystem selection. The (approximated) centralized Riccati and Lyapunov equations are computed iteratively with only local communication and low-order computation by a distributed iterate collapse inversion (DICI) algorithm. We fuse the observations that are common among the local Kalman filters using bipartite fusion graphs and consensus averaging algorithms. The proposed algorithm achieves full distribution of the Kalman filter. Nowhere in the network, storage, communication, or computation of n-dimensional vectors and matrices is required; only nl Lt n dimensional vectors and matrices are communicated or used in the local computations at the sensors. In other words, knowledge of the state is itself distributed.

Modeling of Future Cyber–Physical Energy Systems for Distributed Sensing and Control

This paper proposes modeling the rapidly evolving energy systems as cyber-based physical systems. It introduces a novel cyber-based dynamical model whose mathematical description depends on the cyber technologies supporting the physical system. This paper discusses how such a model can be used to ensure full observability through a cooperative information exchange among its components; this is achieved without requiring local observability of the system components. This paper also shows how this cyber-physical model is used to develop interactive protocols between the controllers embedded within the system layers and the network operator. Our approach leads to a synergistic framework for model-based sensing and control of future energy systems. The newly introduced cyber-physical model has network structure-preserving properties that are key to effective distributed decision making. The aggregate load modeling that we develop using data mining techniques and novel sensing technologies facilitates operations of complex electric power systems.

DILAND: An Algorithm for Distributed Sensor Localization With Noisy Distance Measurements

We present an algorithm for distributed sensor localization with noisy distance measurements (DILAND) that extends and makes the DLRE more robust. DLRE is a distributed sensor localization algorithm in Rm (m ¿ 1) introduced in our previous work (IEEE Trans. Signal Process., vol. 57, no. 5, pp. 2000-2016, May 2009). DILAND operates when: 1) the communication among the sensors is noisy; 2) the communication links in the network may fail with a nonzero probability; and 3) the measurements performed to compute distances among the sensors are corrupted with noise. The sensors (which do not know their locations) lie in the convex hull of at least m + 1 anchors (nodes that know their own locations). Under minimal assumptions on the connectivity and triangulation of each sensor in the network, we show that, under the broad random phenomena described above, DILAND converges almost surely (a.s.) to the exact sensor locations.

Modeling future cyber-physical energy systems

In this paper a model of a future combined cyber-physical energy system is introduced. We view such systems as the intertwined physical-cyber network interconnections of many non-uniform components, such as diverse energy sources and different classes of energy users, equipped with their own local cyber. This modeling approach is qualitatively different from the currently used models that do not explicitly account for the effects of sensing and communications. The proposed approach is based, instead, on representing all physical components as modules interconnected by means of an electric network. However, not all physical components can be modeled from first principles because of the extreme non-uniformity and the complexity of various classes of components. Instead, many components and/or groups of components have to be monitored and their models have to be identified using extensive signal processing, sensing, and model identification. We illustrate such combined cyber-physical models of key components, and use these to introduce a structure preserving model of a cyber-physical infrastructure of the interconnected system. Such a model becomes a basis for deciding what to sense and at which rate, what level of data mining is needed for which (groups of) physical modules to achieve predictable performance for cyber-physical future energy systems. This model rests on the premise that the performance of future energy systems can be shaped in major ways by means of broadly available cyber technologies. In order to make the most out of the available cyber technologies, the first step is to establish models which capture these interdependencies. This paper is a step in such direction.

On connectivity, observability, and stability in distributed estimation

We introduce a new model of social learning and distributed estimation in which the state to be estimated is governed by a potentially unstable linear model driven by noise. The state is observed by a network of agents, each with its own linear noisy observation models. We assume the state to be globally observable, but no agent is able to estimate the state with its own observations alone. We propose a single consensus-step estimator that consists of an innovation step and a consensus step, both performed at the same time-step. We show that if the instability of the dynamics is strictly less than the Network Tracking Capacity (NTC), a function of network connectivity and the observation matrices, the single consensus-step estimator results in a bounded estimation error. We further quantify the trade-off between: (i) (in)stability of the parameter dynamics, (ii) connectivity of the underlying network, and (iii) the observation structure, in the context of single timescale algorithms. This contrasts with prior work on distributed estimation that either assumes scalar dynamics (which removes local observability issues) or assumes that enough iterates can be carried out for the consensus to converge between each innovation (observation) update.

Nano Ferrites Microwave Complex Permeability and Permittivity Measurements by T/R Technique in Waveguide

There is a huge demand to accurately determine the magnetoelectrical properties of particles in the nano-sized regime due to the modern IC technology revolution and biomedical applications. In this paper, we present a microwave waveguide measurement technique for measuring complex permeability and permittivity of expensive nano-sized magnetic powder materials. We used a vector network analyzer to provide a standard TRL calibration for free space inside waveguide measurements. In order to maintain the recommended insertion phase range, a very thin prepared sample was loaded inside the calibrated waveguide. The loaded materials magnetic and dielectric effects were also considered in the cutoff wavelength calculation of the propagation constant of the TE10 wave from the geometrical dimensions of the waveguides. These provisions make the permeability and permittivity measurements more reliable than those found by commonly used techniques. We used six different compounds of nano-sized ferrite powders (Fe3O4, CuFe2O4, CuFe2O4Zn, F12NiO3Zn, BaFe12O19, and SrFe12O19), in which the average diameter of nano particles is less than 40 nm, for measurement purposes. We measured the complex permeability and permittivity from 3.95 to 5.85 GHz. The results show that the dielectric permittivity of these materials is quite different from that of solid-state materials.

Higher Dimensional Consensus: Learning in Large-Scale Networks

The paper considers higher dimensional consensus (HDC). HDC is a general class of linear distributed algorithms for large-scale networks that generalizes average-consensus and includes other interesting distributed algorithms, like sensor localization, leader-follower algorithms in multiagent systems, or distributed Jacobi algorithm. In HDC, the network nodes are partitioned into ¿anchors¿, nodes whose states are fixed over the HDC iterations, and ¿sensors¿, nodes whose states are updated by the algorithm. The paper starts by briefly considering what we call the forward problem by presenting the conditions for HDC to converge, the limiting state to which it converges, and what is its convergence rate. The main focus of the paper is the inverse or design problem, i.e., learning the weights or parameters of the HDC so that the algorithm converges to a desired prespecified state. This generalizes the well-known problem of designing the weights in average-consensus. We pose learning as a constrained nonconvex optimization problem that we cast in the framework of multiobjective optimization (MOP) and to which we apply Pareto optimality. We derive the solution to the learning problem by proving relevant properties satisfied by the MOP solutions and by the Pareto front. Finally, the paper shows how the MOP approach leads to interesting tradeoffs (speed of convergence versus performance) arising in resource constrained networks. Simulation studies illustrate our approach for a leader-follower architecture in multiagent systems.

Distributed Kalman Filters in Sensor Networks: Bipartite Fusion Graphs

We study the distributed Kalman filter in sensor networks where multiple sensors collaborate to achieve a common objective. Our motivation is to distribute the global model that comes from the state-space representation of a sparse and localized large-scale system into reduced coupled sensor-based models. We implement local Kalman filters on these reduced models, by approximating the Gaussian error process of the Kalman filter to be Gauss-Markov, ensuring that each sensor is involved only in reduced-order computations and local communication. We propose a generalized distributed Jacobi algorithm to compute global matrix inversion, locally, in an iterative fashion. We employ bipartite fusion graphs in order to fuse the shared observations and shared estimates across the local models.

Coordinated networked estimation strategies using structured systems theory

In this paper, we consider linear networked estimation strategies using the results from structured systems theory. We are interested in estimating a linear dynamical system where the observations are distributed over a network of agents. In this context, we devise both state fusion and observation fusion strategies that guarantee a stable estimator. We assume global observability, i.e., given all of the observations, the dynamical system is observable. To derive our results, we employ the genericity properties of dynamical systems that are studied in the structured systems theory. The genericity properties rely on the graphical properties of the dynamical systems and their outputs, and thus, depend on the zero and non-zero pattern of the system and output (observation) matrices. In particular, we study the generic observability of networked estimators and derive results on the topology of the agent communication graph to ensure a stable estimator. We then focus on the design of local estimator gains that results into iterative procedures to solve a Linear Matrix Inequality (LMI) with structural constraints.

On the Genericity Properties in Distributed Estimation: Topology Design and Sensor Placement

In this paper, we consider distributed estimation of linear, discrete-time dynamical systems monitored by a network of agents. We require the agents to exchange information with their neighbors only once per dynamical system time-scale and study the network topology sufficient for distributed observability. To this aim, we provide a novel measurement-based agent classification: Type- α,β, and γ, which leads to the construction of specific graph topologies: <i>G</i>_α and <i>G</i>_β. In particular, in <i>G</i>_α, every Type-α agent has a direct connection to every other agent, whereas, in <i>G</i>_β, every agent has a directed path to every Type-β agent. With the help of these constructs, we formulate an estimator where measurement and predictor-fusion are implemented over <i>G</i>_α and <i>G</i>_β, respectively, and show that the proposed scheme leads to distributed observability, i.e., observability of the distributed estimator. In order to characterize the estimator further, we show that Type-α agents only exist in systems with <i>S</i>-rank (maximal rank of zero/non-zero pattern) deficient system matrices. In other words, systems with full <i>S</i>-rank matrices only have Type-β agents, and thus, a strongly-connected (agent) network is sufficient for full <i>S</i>-rank systems-by the definition of <i>G</i>_β above; however strong-connectivity is not necessary, i.e., there exist weakly-connected networks that result in distributed observability. Furthermore, we show that for <i>S</i> -rank deficient systems, measurement-fusion over <i>G</i>_α is required, and predictor-fusion alone is insufficient. The approach taken in this paper is structural, i.e., we use the concept of structured systems theory and generic observability to derive the results. Finally, we provide an iterative method to compute the local estimator gain at each agent once the observability is ensured using the aforementioned construction.