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In this paper, we study the problem of synchronization over a network, with nonlinear components dynamics modeled in Lure form, and linear stochastic interaction among the components. To study this problem we utilize the stochastic version of Positive Real Lemma (PRL), which is used to provide a sufficient condition for synchronization of stochastic network systems. This sufficiency condition is a function of nominal (mean coupling) Laplacian eigenvalues, and the statistics of link uncertainty in the form of coefficient of dispersion (CoD). Robust control-based small-gain interpretation is provided for the derived sufficiency condition which allows 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 network synchronization. Our results indicate that significant role played by both the largest and the second smallest eigenvalue of the nominal Laplacian in synchronization of stochastic networks. Furthermore, 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.

In this paper, we propose a novel approach for the data-driven characterization of power system dynamics. The developed method of Extended Subspace Identification (ESI) is suitable for systems with output measurements when all the dynamics states are not observable. It is particularly applicable for power systems dynamic identification using Phasor Measurement Units (PMUs) measurements. As in the case of power systems, it is often expensive or impossible to measure all the internal dynamic states of system components such as generators, controllers and loads. PMU measurements capture voltages, currents, power injection and frequencies, which can be considered as the outputs of system dynamics. The ESI method is suitable for system identification, capturing nonlinear modes, computing participation factor of output measurements in system modes and identifying system parameters such as system inertia. The proposed method is suitable for measurements with a noise similar to realistic system measurements. The developed method addresses some of the known deficiencies of existing data-driven dynamic system characterization methods. The approach is validated for multiple network models and dynamic event scenarios with synthetic PMU measurements., Comment: 11 Pages, 10 figures, under review in IEEE transactions on Power Systems

In this paper, we propose a data-driven approach for uncertainty propagation and reachability analysis in a dynamical system. The proposed approach relies on the linear lifting of a nonlinear system using linear Perron-Frobenius (P-F) and Koopman operators. The uncertainty can be characterized in terms of the moments of a probability density function. We demonstrate how the P-F and Koopman operators are used for propagating the moments. Time-series data is used for the finite-dimensional approximation of the linear operators, thereby enabling data-driven approach for moment propagation. Simulation results are presented to demonstrate the effectiveness of the proposed method., Accepted in the 2020 American Control Conference, to be held in Denver, CO, USA on July 1-3, 2020

We propose the application of Koopman operator theory for the design of stabilizing feedback controller for a nonlinear control system. The proposed approach is data-driven and relies on the use of time-series data generated from the control dynamical system for the lifting of a nonlinear system in the Koopman eigenfunction coordinates. In particular, a finite-dimensional bilinear representation of a control-affine nonlinear dynamical system is constructed in the Koopman eigenfunction coordinates using time-series data. Sample complexity results are used to determine the data required to achieve the desired level of accuracy for the approximate bilinear representation of the nonlinear system in Koopman eigenfunction coordinates. A control Lyapunov function-based approach is proposed for the design of stabilizing feedback controller. A systematic convex optimization-based formulation is proposed for the search of control Lyapunov function. Several numerical examples are presented to demonstrate the application of the proposed data-driven stabilization approach.

In this paper, we study the optimal quadratic regulation problem for nonlinear systems. The linear operator theoretic framework involving the Koopman operator is used to lift the dynamics of nonlinear control system to an infinite dimensional bilinear system. Optimal quadratic regulation problem for nonlinear system is formulated in terms of the finite dimensional approximation of the bilinear system. A convex optimization-based approach is proposed for solving the quadratic regulator problem for bilinear system. Simulation results are presented to demonstrate the application of the developed framework.

In this paper, we propose a novel approach to compute the Koopman operator from sparse time series data. In recent years there have been considerable interests in operator theoretic methods for data-driven analysis of dynamical systems. Existing techniques for the approximation of the Koopman operator require sufficiently large data sets, but in many applications, the data set may not be large enough to approximate the operators to acceptable limits. In this paper, using ideas from robust optimization, we propose an algorithm to compute the Koopman operator from sparse data. We enrich the sparse data set with artificial data points, generated by adding bounded artificial noise and formulate the noisy robust learning problem as a robust optimization problem and show that the optimal solution is the Koopman operator with the smallest error. We illustrate the efficiency of our proposed approach in three different dynamical systems, namely, a linear system, a nonlinear system and a dynamical system governed by a partial differential equation.

We consider identification problems for stochastic nonlinear dynamical systems. An explicit sample complexity bound in terms of the number of data points required to recover the models to some certain precision is derived. Our results extend recent sample complexity results for linear stochastic dynamics. Our approach for obtaining sample complexity bounds for nonlinear dynamics relies on a linear, albeit infinite dimensional, representation of nonlinear dynamics provided by Koopman and Perron-Frobenius operators. We exploit the linear property of these operators to derive the sample complexity bounds. Such complexity bounds may play a significant role in data-driven learning and control of nonlinear dynamics. Several numerical examples are provided to highlight our theory.

In this paper, we propose linear operator theoretic framework involving Koopman operator for the data-driven identification of power system dynamics. We explicitly account for noise in the time series measurement data and propose robust approach for data-driven approximation of Koopman operator for the identification of nonlinear power system dynamics. The identified model is used for the prediction of state trajectories in the power system. The application of the framework is illustrated using an IEEE nine bus test system., Accepted for publication in IEEE Power and Energy System General Meeting 2019

In this paper, we propose a data-driven approach for control of nonlinear dynamical systems. The proposed data-driven approach relies on transfer Koopman and Perron-Frobenius (P-F) operators for linear representation and control of such systems. Systematic model-based frameworks involving linear transfer P-F operator were proposed for almost everywhere stability analysis and control design of a nonlinear dynamical system in previous works [1]–[3]. Lyapunov measure can be used as a tool to provide linear programming-based computational framework for stability analysis and optimal control design of a nonlinear system. In this paper, we show that the Lyapunov measure-based framework can extended to a data-driven setting, where the finite dimensional approximation of linear transfer P-F operator and optimal control can be obtained from time-series data. We exploit the positivity and Markov property of P-F operator to provide linear programming based approach for designing an optimally stabilizing feedback controller.

In this paper, we provide a systematic approach for the design of stabilizing feedback controllers for nonlinear control systems using the Koopman operator framework. The Koopman operator approach provides a linear representation for a nonlinear dynamical system and a bilinear representation for a nonlinear control system. The problem of feedback stabilization of a nonlinear control system is then transformed to the stabilization of a bilinear control system. We propose a control Lyapunov function (CLF)-based approach for the design of stabilizing feedback controllers for the bilinear system. The search for finding a CLF for the bilinear control system is formulated as a convex optimization problem. This leads to a schematic procedure for designing CLF-based stabilizing feedback controllers for the bilinear system and hence the original nonlinear system. Another advantage of the proposed controller design approach outlined in this paper is that it does not require explicit knowledge of system dynamics. In particular, the bilinear representation of a nonlinear control system in the Koopman eigenfunction space can be obtained from time-series data. Simulation results are presented to verify the main results on the design of stabilizing feedback controllers and the data-driven aspect of the proposed approach., Accepted for IEEE Conference on Decision and Control(CDC) 2018

In this paper, we present a novel operator theoretic framework for optimal placement of actuators and sensors in nonlinear systems. The problem is motivated by its application to control of nonequilibrium dynamics in the form of temperature in building systems and control of oil spill in oceanographic flow. The controlled evolution of a passive scalar field, modeling the temperature distribution or density of oil dispersant, is governed by a linear advection partial differential equation (PDE) with spatially located actuators and sensors. Spatial locations of actuators and sensors are optimized to maximize the controllability and observability regions of the linear advection PDE. Linear transfer Perron–Frobenius and Koopman operators, associated with the advective velocity field, are used to provide an analytical characterization for the controllable and observable spaces of the advection PDE. Set-oriented numerical methods are proposed for the finite dimensional approximation of the linear transfer operators. The finite dimensional approximation is shown to introduce weaker notion of controllability and observability, referred to as coarse controllability and observability. The finite dimensional approximation is used to formulate the optimization problem for the optimal placement of sensors and actuators. The optimal placement problem is a combinatorial optimization problem. However, the positivity property of the linear transfer operator is exploited to provide an exact solution to the optimal placement problem using greedy algorithm. Application of the framework is demonstrated for the placement of sensors in a building system for the detection of contaminants and for optimal release of dispersant location for control of contaminant in a Double Gyre velocity field. Simulation results reveal interesting connections between the optimal location of actuators and sensors, maximizing the controllability and observability regions respectively, and the coherent structures in the fluid flow.

In this paper, we provide a new algorithm for the finite dimensional approximation of the linear transfer Koopman and Perron-Frobenius operator from time series data. We argue that existing approach for the finite dimensional approximation of these transfer operators such as Dynamic Mode Decomposition (DMD) and Extended Dynamic Mode Decomposition (EDMD) do not capture two important properties of these operators, namely positivity and Markov property. The algorithm we propose in this paper preserve these two properties. We call the proposed algorithm as naturally structured DMD since it retains the inherent properties of these operators. Naturally structured DMD algorithm leads to a better approximation of the steady-state dynamics of the system regarding computing Koopman and Perron- Frobenius operator eigenfunctions and eigenvalues. However preserving positivity properties is critical for capturing the real transient dynamics of the system. This positivity of the transfer operators and it's finite dimensional approximation also has an important implication on the application of the transfer operator methods for controller and estimator design for nonlinear systems from time series data.

In this paper, we develop linear transfer Perron-Frobenius operator-based approach for optimal stabilization of stochastic nonlinear systems. One of the main highlights of the proposed transfer operator based approach is that both the theory and computational framework developed for the optimal stabilization of deterministic dynamical systems in [1] carries over to the stochastic case with little change. The optimal stabilization problem is formulated as an infinite-dimensional linear program. Set oriented numerical methods are proposed for the finite-dimensional approximation of the transfer operator and the controller. Simulation results are presented to verify the developed framework.

Summary 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 the 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. Copyright © 2014 John Wiley & Sons, Ltd.

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.

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.

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.

In this paper, we study the stabilization problem of a scalar nonlinear system over a packet-drop channel. We consider moment stability notions adopted from the ergodic theory of dynamical systems. The main result of this paper proves that q -moment exponential stability requires a minimal quality of service from the communications link. The necessary conditions presented in the paper relate the positive Lyapunov exponent of the open loop system and the largest probability of erasure. For the first time, the dependence on the Lyapunov exponent highlights the important role played by the global non-equilibrium dynamics in nonlinear stabilization over networks.

In our recent work [1], we introduced Lyapunov measure as a new tool to verify weaker set-theoretic notion of almost everywhere stability of stochastic nonlinear systems. A Linear transfer Perron-Frobenius operator for stochastic systems was used to provide an explicit formula for the Lyapunov measure, verifying almost everywhere almost sure stability of stochastic systems. The focus of this paper is on the computational aspect of the Lyapunov measure for stochastic systems. We used set-oriented numerical methods for the finite dimensional approximation of the linear operator and the Lyapunov measure. Stability results in the finite dimensional approximation space are also presented. In particular, we show the finite dimensional approximation leads to a further weaker notion of stability referred to as coarse stability.

In this paper, we prove the stochastic version of the Positive Real (PR) Lemma, to study the stability problem of nonlinear systems in Lure form with stochastic uncertainty. We study the mean square stability problem of systems in Lure form with stochastic parametric uncertainty affecting the linear part of the system dynamics. The stochastic PR Lemma result is then used to study the problem of synchronization of coupled Lure systems, with stochastic interaction over the network. We provide sufficiency condition for the synchronization of such 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). Under the assumption that the individual subsystems have identical dynamics, we show that the sufficiency condition is only a function of a single subsystem dynamics and mean network characteristics. This makes the sufficiency condition attractive from the point of view of computation for large size network systems. Interestingly, our results indicate that both the largest and the second smallest eigenvalue of the mean Laplacian play an important role in synchronization of complex dynamics, characteristic to nonlinear systems. Simulation results for network of coupled oscillators with stochastic link uncertainty are presented to verify the developed theoretical framework.

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.

Active controllers that allow the compressor to operate safely at its peak pressure rise point by preventing surge and controlling entry into rotating stall need to be globally stabilizing. Bifurcation methods have emerged as a useful tool for the analysis of global stability and control of nonlinear dynamical systems such as the axial compressor. In this paper we show how bifurcation analysis can be used effectively to obtain answers to questions of global stability of the compressor dynamics. Since the bifurcation method does not directly provide information on transient behaviour, we demonstrate how bifurcation results need to be carefully interpreted to be of use when dealing with practical systems. We then present a novel nonlinear bifurcation-based stall/surge controller that is globally stabilizing. With such a controller, it becomes possible to avoid surge entirely and to prevent abrupt entry into rotating stall. The controller also eliminates the hysteresis between entry into and recovery from a rotating stall, and maintains system stability under all perturbations, small and large.

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.

We state and prove a robustness result for the concept of almost everywhere uniform stability of an invariant attractor for a nonlinear autonomous differential equation. We also show that the concept of almost everywhere uniform stability rectifies the drawback of vanishing density on stable manifolds of hyperbolic equilibrium points which exists for a weaker concept of almost everywhere stability.

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.

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.

In this paper, we study the problem of control of nonlinear systems over an erasure channel. The stability and performance metric are adopted from the ergodic theory of random dynamical systems to study the almost sure and second moment stabilization problem. The main result of this paper proves that, while there are no limitations for the almost sure stabilization, fundamental limitations arise for the second moment stabilization. In particular, we provide a necessary condition for the second moment stabilization of multi-state single input nonlinear systems expressed in terms of the probability of erasure and positive Lyapunov exponents of the open loop unstable system. The dependence of the limitation result on the Lyapunov exponents highlights, for the first time, the important role played by the global non-equilibrium dynamics of the nonlinear systems in obtaining the performance limitation. This result generalizes the existing results for the stabilization of linear time invariant systems over erasure channels and differs from the existing Bode-like fundamental limitation results for nonlinear systems, which are expressed in terms of the eigenvalues of the linearization.

In this paper, we propose a novel approach based on the analysis of linear derivative map for the model reduction of nonlinear system with output measurement. With every nonlinear system one can associate a linear derivative map that evolves vectors on the tangent space. We employ the linear derivative map for the construction of observability gramian on the tangent space and then use it for the purpose of reduced order modeling of nonlinear system. Computation methods based on time domain simulation of system dynamics is proposed for the construction of the empirical gramian. Finally, examples demonstrating the application of the proposed method for the model reduction of dynamical systems are provided.

In this paper, we present a systematic approach inspired from the ergodic theory of dynamical system for the optimal control of complex fluid flows. The infinite dimensional Navier Stokes equation describing the complex fluid flow is first reduced to a finite set of coupled ordinary differential equations. We utilize Proper Orthogonal Decomposition techniques to obtain a reduced order model. The linear transfer, Perron Frobenius (P-F), operator-based Lyapunov measure framework is used for the nonlinear stability analysis and optimal control design of the reduced order system. Efficient numerical schemes that leverage the linearity of the framework have been developed for analysis and control design. The framework is utilized to analyze a benchmark problem in fluid dynamics: the control of recirculation in a two-dimensional channel flow with a backward facing step.

We provide a numerical approach to estimating nonlinear stochastic dynamic models from time-series data. After possible dimensional reduction, time-series data can be used to construct an empirical Markov model. Spectral analysis of the Markov model is then carried out to detect the presence of complex limit cycling, almost invariant, and bistable behavior in the model. Model parameters are expressed as a linear combination of basis functions over the phase space. A least squares minimization is used to fit the basis function coefficients in order to match the spectral properties of the respective Markov operators. The approach is demonstrated on the estimation of a nonlinear stochastic model describing combustion oscillation data.

In this paper, we propose a novel methodology for establishing fundamental limitations in nonlinear stabilization. To aid the analysis, we express the stabilization problem as control of Markov chains. Using Markov chains, we derive the limitations as certain maximum probability bounds or as positive conditional entropy of the certain signals in the feedback loop. The former is related to the infeasibility of the asymptotic stabilization in the presence of quantization and the latter to the Bode integral formula. In either cases, it is shown that uncertainty - associated here with the unstable eigenvalues of the linearization - leads to fundamental limitations.