arXiv subjects

Amit Diwadkar

Publications and source records attributed to Amit Diwadkar.

6 recordsLinked to original sources

arXiv2016

Mean Square Stability Analysis of Stochastic Continuous-time Linear Networked Systems

In this technical note, we study the mean square stability-based analysis of stochastic continuous-time linear networked systems. The stochastic uncertainty is assumed to enter multiplicatively in system dynamics through input and output channels of the plant. Necessary and sufficient conditions for mean square exponential stability are expressed in terms of the input-output property of deterministic or nominal system dynamics captured by the {\it mean square} system norm and variance of channel uncertainty. The stability results can also be interpreted as a small gain theorem for continuous-time stochastic systems. Linear Matrix Inequalities (LMI)-based optimization formulation is provided for the computation of mean square system norm for stability analysis. For a special case of single input channel uncertainty, we also prove a fundamental limitation result that arises in the mean square exponential stabilization of the continuous-time linear system. Overall, the contributions in this work generalize the existing results on stability analysis from discrete-time linear systems to continuous-time linear systems with multiplicative uncertainty. Simulation results are presented for WSCC $9$ bus power system to demonstrate the application of the developed framework.

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arXiv2015

Mean sqaure synchronization in large scale nonlinear networks with uncertain links

In this paper, we study the problem of synchronization with stochastic interaction among network components. The network components dynamics is nonlinear and modeled in Lure form with linear stochastic interaction among network components. To study this problem we first prove the stochastic version of Positive Real Lemma (PRL). The stochastic PRL result is then used to provide sufficient condition for synchronization of stochastic network system. The sufficiency condition for synchronization, is a function of nominal (mean) coupling Laplacian eigenvalues and the statistics of link uncertainty in the form of coefficient of dispersion (CoD). Contrary to the existing literature on network synchronization, our results indicate that both the largest and the second smallest eigenvalue of the mean Laplacian play an important role in synchronization of stochastic networks. Robust control-based small-gain interpretation is provided for the derived sufficiency condition which allow us to define the margin of synchronization. The margin of synchronization is used to understand the important tradeoff between the component dynamics, network topology, and uncertainty characteristics. For a special class of network system connected over torus topology we provide an analytical expression for the tradeoff between the number of neighbors and the dimension of the torus. Similarly, by exploiting the identical nature of component dynamics computationally efficient sufficient condition independent of network size is provided for general class of network system. Simulation results for network of coupled oscillators with stochastic link uncertainty are presented to verify the developed theoretical framework.

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arXiv2014

Limitations and tradeoffs in synchronization of large-scale networks with uncertain links

We study synchronization in scalar nonlinear systems connected over a linear network with stochastic uncertainty in their interactions. We provide a sufficient condition for the synchronization of such network systems expressed in terms of the parameters of the nonlinear scalar dynamics, the second and largest eigenvalues of the mean interconnection Laplacian, and the variance of the stochastic uncertainty. The sufficient condition is independent of network size thereby making it attractive for verification of synchronization in a large size network. The main contribution of this paper is to provide analytical characterization for the interplay of roles played by the internal dynamics of the nonlinear system, network topology, and uncertainty statistics in network synchronization. We show there exist important tradeoffs between these various network parameters necessary to achieve synchronization. We show for nearest neighbor networks with stochastic uncertainty in interactions there exists an optimal number of neighbors with maximum margin for synchronization. This proves in the presence of interaction uncertainty, too many connections among network components is just as harmful for synchronization as the lack of connection. We provide an analytical formula for the optimal gain required to achieve maximum synchronization margin thereby allowing us to compare various complex network topology for their synchronization property.

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arXiv2014

Limitations for nonlinear observation over erasure channel

In this paper, we study the problem of state observation of nonlinear systems over an erasure channel. The notion of mean square exponential stability is used to analyze the stability property of observer error dynamics. The main results of this paper prove, fundamental limitation arises for mean square exponential stabilization of the observer error dynamics, expressed in terms of probability of erasure, and positive Lyapunov exponents of the system. Positive Lyapunov exponents are a measure of average expansion of nearby trajectories on an attractor set for nonlinear systems. Hence, the dependence of limitation results on the Lyapunov exponents highlights the important role played by non-equilibrium dynamics in observation over an erasure channel. The limitation on observation is also related to measure-theoretic entropy of the system, which is another measure of dynamical complexity. The limitation result for the observation of linear systems is obtained as a special case, where Lyapunov exponents are shown to emerge as the natural generalization of eigenvalues from linear systems to nonlinear systems.

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arXiv2014

Stabilization of linear time varying system over uncertain channels

In this paper, we study the problem of control of discrete-time linear time varying systems over uncertain channels. The uncertainty in the channels is modeled as a stochastic random variable. We use exponential mean square stability of the closed-loop system as a stability criterion. We show that fundamental limitations arise for the mean square exponential stabilization for the closed-loop system expressed in terms of statistics of channel uncertainty and the positive Lyapunov exponent of the open-loop uncontrolled system. Our results generalize the existing results known in the case of linear time invariant systems, where Lyapunov exponents are shown to emerge as the generalization of eigenvalues from linear time invariant systems to linear time varying systems. Simulation results are presented to verify the main results of this paper.

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arXiv2014

Control of systems in Lure form over erasure channel

In this paper, we study the problem of control of discrete-time nonlinear systems in Lure form over erasure channels at the input and output. The input and output channel uncertainties are modeled as Bernoulli random variables. The main results of this paper provide sufficient condition for the mean square exponential stability of the closed loop system expressed in terms of statistics of channel uncertainty and plant characteristics. We also provide synthesis method for the design of observer-based controller that is robust to channel uncertainty. To prove the main results of this paper, we discover a stochastic variant of the well known Positive Real Lemma and principle of separation for stochastic nonlinear system. Application of the results for the stabilization of system in Lure form over packet-drop network is discussed. Finally a result for state feedback control of a Lure system with a general multiplicative uncertainty at actuation is discussed.

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