Amit Diwadkar

Limitations for Nonlinear Observation Over Erasure Channel

In this technical note, 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 technical note 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 nonequilibrium dynamics on the attractor set 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.

Robust synchronization in nonlinear network with link failure uncertainty

The problem of synchronization of systems over a network, is a widely studied problem given the importance of synchronization phenomena, in various natural science and engineering applications. In this paper, we study one of the important aspect of this problem that is, robustness of synchronization to random link failure uncertainty. The link failure uncertainty is modeled as an on-off Bernoulli switch. The main results of this paper provide, for the first time, analytical conditions for the maximum tolerable link failure uncertainty to maintain mean square synchronization among the network components. The analytical conditions are expressed in terms of individual component dynamics, network properties, and link uncertainty. The main results of this paper can be used to determine, the weakest/strongest link in the network. Simulation results are provided to verify the main results of this paper.

Synchronization in complex network system with uncertainty

In this paper, we study the problem of synchronization in a network of nonlinear systems with scalar dynamics. The nonlinear systems are connected over linear network with stochastic uncertainty in their interactions. We provide a sufficient condition for the synchronization of such network system 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 provided 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 role played by the internal dynamics of the nonlinear systems, network topology, and uncertainty statistics for the network synchronization. We show that there exist important trade-offs between these various network parameters necessary to achieve synchronization. We provide simulation results in network system with internal dynamics modeling of agents moving in a double well potential function. The synchronization of network happens whereby the dynamics of the network system flip from one potential well to another at the backdrop of stochastic interaction uncertainty.

Stabilization of system in Lure form over uncertain channels

In this paper, we study the problem of stabilization of nonlinear system in Lure form with uncertainty at the input and output channels. The channel uncertainty is modeled using Bernoulli random variable. Generalization of Positive Real Lemma for stochastic systems are derived to prove the main result of this paper providing sufficient condition for the mean square exponential stability of the closed loop system with erasure channels at the input and output. We generalize this result to provide sufficient condition for stabilization over general uncertain channel at the input and perfect measurement channel at the output. The results in this paper provide synthesis method for the design of controller and observer that are robust to channel uncertainty. Due to nonlinear plant dynamics, the controller and observer design problem are coupled, however we provide explicit relation between the erasure probability of the input and output channels to maintain stability of the feedback control system.

Nonlinear observation over erasure channel

In this paper, we study the problem of state observation of nonlinear systems over an erasure channel. Stochastic notions of stability are adopted from ergodic theory of random dynamical systems for the analysis. We use stability with probability one and mean square exponential stability to analyze the stability property of observer error dynamics. The main results of this paper prove that there is no limitation for stabilization with probability one, however fundamental limitation arises for mean square exponential stabilization of the error dynamics. We provide necessary condition for the mean square exponential stability of the error dynamics, expressed in terms of the probability of channel erasure and the positive Lyapunov exponents of the system dynamics. Simulation results are presented to verify the main results of the paper.

Stability analysis and controller synthesis for continuous-time linear stochastic systems

In this paper, we derive results for the stochastic stability analysis and controller synthesis for continuous-time stochastic system. The important feature considered here is the multiplicative nature of the stochastic uncertainty in system dynamics. We generalize the existing small-gain type results for stability of discrete-time system with stochastic uncertainty in feedback loop to continuous-time dynamics. Further, LMI-based computable necessary and sufficient conditions are provided for the mean square stability of feedback system. The proposed stability analysis results are used for the synthesis of dynamic robust stabilizing feedback controller with stochastic uncertainty in the feedback loop between the plant and the controller. Fundamental limitation result for the mean square stabilization of system over stochastic channels is presented. Finally, we demonstrate the proposed framework on pendulum on a cart system.

Modeling and analysis of rotational freeplay nonlinearity of a 2D airfoil

Aeroelastic flutter is a dynamic instability of fluid-structural system in which the structure exhibits a sustained, often diverging oscillation. Flutter behavior is self-feeding and destructive. Nonlinearities such as freeplay in rigid-body rotational stiffness of the structural system can have an effect on the onset of flutter and its amplitude. In particular there is experimental evidence that as the amount of freeplay increases, the freestream velocity at which the flutter instability occurs decreases. In this paper, we develop a modeling framework that allows us to predict this dependence of flutter velocity on the freeplay parameter. We model the airfoil system with freeplay nonlinearity as a feedback interconnection of linear system and sector bounded nonlinearity. Freeplay in stiffness is practically approximated as a hyperbola nonlinearity. Eigenvalue analysis at equilibrium points is used to predict onset of flutter and characterize a Hopf bifurcation of the system from stable to limit cycle behavior. Spectral analysis us used to characterize the limit cycle behavior. This analysis indicates the flutter onset velocity to be a function of freeplay region length. Follow-on research correlating recently obtained wind tunnel results to a three-dimensional extension of the model is outlined.

Performance limitation in multi-agent estimation with limited sensor measurements

This paper studies the problem of performance limitations in state estimation of multi-agent systems with limited sensor measurement. We model the agents as LTV systems with erasure in output. We first look to design a state observer for a single agent with output measurement erasure, which guarantees a required performance criterion on the error dynamics. The main result shows that fundamental performance limitation arises in state observation of the agent dynamics when we use exponential mean square stability of the estimation error as the performance metric. This limitation is expressed in terms of the characteristic Lyapunov exponents of the agent dynamics and the uncertainty in the measurement. We then apply this result to the multi-agent estimation problem with limited sensor measurements. Separate cases for cooperative and noncooperative behavior among sensors is studied. Simulation results comparing these cases are given.

Vulnerability analysis of large-scale dynamical networks to coordinated attacks

We study the vulnerability of large-scale linear dynamical networks to coordinated attacks. We consider scenarios in which an attacker can tamper with the links connecting the network components and can also manipulate input injections at the nodes. When these two types of attacks take place simultaneously, the attack is referred to as a coordinated attack. We assume that network links are attacked with a certain probability and that malicious data is injected at the input ports. We employ Markov jump linear systems to model link-based attacks and the system input matrix to model data injection attacks. System theoretic vulnerability metrics developed in earlier work are used to analyze network vulnerability to coordinated attacks. These measures of vulnerability allow us to characterize the impact of coordinated attacks and the difficulty associated with detecting them. Finally, we analyze the vulnerability of coordinated attacks on the New England 39 bus power network.

Synchronization of nonlinear systems with dissipative nonlinearity over large-scale stochastic networks

In this paper, we study the synchronization of identical nonlinear systems over large-scale network with uncertainty in the interconnections. We consider a special class of nonlinear systems which have a dissipative nonlinearity and the stability of such systems can be analyzed using absolute stability theory tools like the Positive Real Lemma, Bounded Real Lemma and dissipativity theory. We extend this analysis to the stochastic setting over a network where the interconnection weights are drive by Wiener process with given mean and variance. To capture the stability of the synchronized state, we study the notion of mean square stability from stochastic stability theory and formulate a network size-independent sufficient condition based on the theory of stochastic dissipative systems. We also compute a heuristic margin of synchronization for the networked systems to indicate the tolerance to stochastic uncertainty in interconnection links.