We introduce the Multi-Model Parameterized Koopman (MMPK) framework, a novel end-2-end data-driven modeling and control pipeline for enabling autonomous navigation in Uncrewed Ground Vehicles. MMPK builds upon the Koopman Extended Dynamic Mode Decomposition (KEDMD) algorithm, offering a flexible model- and control-adaptation in the presence of time-varying uncertainties with both ego-vehicle and operational-environment parameters. Unlike traditional methods, MMPK addresses challenges such as overfitting and reliance on a singular global model by adopting a set of pose-agnostic representations of positional data and curvature-parameterized Koopman models, thereby effectively mitigating data bias. The end-2-end unified pipeline encompasses an offline MMPK modeling and an online outer-loop control design consisting of model-based trajectory planning and linear Model Predictive Control adapted to switched Koopman dynamics. The performance of the proposed pipeline is verified via simulation and experimental testing using a 1/5th scale Ackermann-steered ground vehicle platform (AgileX Hunter SE) and benchmark driving profiles. Comparative evaluations demonstrate MMPK’s superior path-tracking capabilities and the effectiveness of its local planning strategy in bridging the Model-Sim-Real gap.
We introduce a novel approach for safe control design based on the density function. A control density function (CDF) is introduced to synthesize a safe controller for a nonlinear dynamic system. The CDF can be viewed as a dual to the control barrier function (CBF), a popular approach used for safe control design. While the safety certificate using the barrier function is based on the notion of invariance, the dual certificate involving the density function has a physical interpretation of occupancy. This occupancy-based physical interpretation is instrumental in providing an analytical construction of density function used for safe control synthesis. The safe control design problem is formulated using the density function as a quadratic programming (QP) problem. In contrast to the QP proposed for control synthesis using CBF, the proposed CDF-based QP can combine both the safety and convergence conditions to target state into single constraints. Further, we consider robustness against uncertainty in system dynamics and the initial condition and provide theoretical results for robust navigation using the CDF. Finally, we present simulation results for safe navigation with single integrator and double-gyre fluid flow-field examples, followed by robust navigation using the bicycle model and autonomous lane-keeping examples.
We introduce a novel approach for safe control design based on the density function. A control density function (CDF) is introduced to synthesize a safe controller for a nonlinear dynamic system. The CDF can be viewed as a dual to the control barrier function (CBF), a popular approach used for safe control design. While the safety certificate using the barrier function is based on the notion of invariance, the dual certificate involving the density function has a physical interpretation of occupancy. This occupancy-based physical interpretation is instrumental in providing an analytical construction of density function used for safe control synthesis. The safe control design problem is formulated using the density function as a quadratic programming (QP) problem. In contrast to the QP proposed for control synthesis using CBF, the proposed CDF-based QP can combine both the safety and convergence conditions to target state into single constraints. Further, we consider robustness against uncertainty in system dynamics and the initial condition and provide theoretical results for robust navigation using the CDF. Finally, we present simulation results for safe navigation with single integrator and double-gyre fluid flow-field examples, followed by robust navigation using the bicycle model and autonomous lane-keeping examples.
In this paper, we establish a connection between the spectral theory of the Koopman operator and the solution of the Hamilton Jacobi (HJ) equation. The HJ equation occupies a central place in systems theory, and its solution is of interest in various control problems, including optimal control, robust control, and input-output analysis. One of the main contributions of this paper is to show that the Lagrangian submanifolds, which are fundamental objects for solving the HJ equation, can be obtained using the spectral analysis of the Koopman operator. We present two different procedures for the approximation of the HJ solution. We utilize the spectral properties of the Koopman operator associated with the uncontrolled dynamical system and Hamiltonian systems that arise from the HJ equation to approximate the HJ solution. We present a convex optimization-based computational framework with convergence analysis for approximating the Koopman eigenfunctions and the Lagrangian submanifolds. Our solution approach to the HJ equation using Koopman theory provides for a natural extension of results from linear systems to nonlinear systems. We demonstrate the application of this work for solving the optimal control problem. Finally, we present simulation results to validate the paper’s main findings and compare them against linear quadratic regulator and Taylor series based approximation controllers.
In this paper, we study the stability problem of a stochastic, nonlinear, discrete-time system. We introduce a linear transfer operator-based Lyapunov measure as a new tool for stability verification of stochastic systems. Weaker set-theoretic notion of almost everywhere stochastic stability is introduced and verified, using Lyapunov measure-based stochastic stability theorems. Furthermore, connection between Lyapunov functions, a popular tool for stochastic stability verification, and Lyapunov measures is established. Using the duality property between the linear transfer Perron-Frobenius and Koopman operators, we show the Lyapunov measure and Lyapunov function used for the verification of stochastic stability are dual to each other. Set-oriented numerical methods are proposed for the finite dimensional approximation of the Perron-Frobenius operator; hence, Lyapunov measure is proposed. Stability results in finite dimensional approximation space are also presented. Finite dimensional approximation is shown to introduce further weaker notion of stability referred to as coarse stochastic stability. The results in this paper extend our earlier work on the use of Lyapunov measures for almost everywhere stability verification of deterministic dynamical systems (“Lyapunov Measure for Almost Everywhere Stability”, {\it IEEE Trans. on Automatic Control}, Vol. 53, No. 1, Feb. 2008).
Contrary to on-road autonomous navigation, off-road autonomy is complicated by various factors ranging from sensing challenges to terrain variability. In such a milieu, data-driven approaches have been commonly employed to capture intricate vehicle-environment interactions effectively. However, the success of data-driven methods depends crucially on the quality and quantity of data, which can be compromised by large variability in off-road environments. To address these concerns, we present a novel workflow to recreate the exact vehicle and its target operating conditions digitally for domain-specific data generation. This enables us to effectively model off-road vehicle dynamics from simulation data using the Koopman operator theory, and employ the obtained models for local motion planning and optimal vehicle control. The capabilities of the proposed methodology are demonstrated through an autonomous navigation problem of a 1:5 scale vehicle, where a terrain-informed planner is employed for global mission planning. Results indicate a substantial improvement in off-road navigation performance with the proposed algorithm (5.84x) and underscore the efficacy of digital twinning in terms of improving the sample efficiency (3.2x) and reducing the sim2real gap (5.2%).
The paper is about characterizing the stability boundary of an autonomous dynamical system using the Koopman spectrum. For a dynamical system with an asymptotically stable equilibrium point, the domain of attraction constitutes a region consisting of all initial conditions attracted to the equilibrium point. The stability boundary is a separatrix region that separates the domain of attraction from the rest of the state space. For a large class of dynamical systems, this stability boundary consists of the union of stable manifolds of all the unstable equilibrium points on the stability boundary. We characterize the stable manifold in terms of the zero-level curve of the Koopman eigenfunction. A path-integral formula is proposed to compute the Koopman eigenfunction for a saddle-type equilibrium point on the stability boundary. The algorithm for identifying stability boundary based on the Koopman eigenfunction is attractive as it does not involve explicit knowledge of system dynamics. We present simulation results to verify the main results of the paper.
Abstract
This paper introduces a novel approach based on density functions for safe navigation in dynamic environments with time-varying obstacles and targets. We analytically construct time-varying density functions to address the challenges of dynamic safe navigation. Using this approach, we develop a safe feedback controller that ensures safety and almost everywhere stability in dynamic environments. The proposed framework is demonstrated on complex robotic systems, including the safe navigation of a quadruped robot with time-varying obstacles and sensor uncertainty, as well as a robotic arm tracking a time-varying target with static obstacles.
This letter develops the machinery of Koopman-based Model Predictive Control (KMPC) design, where the Koopman derived model is unable to capture the real nonlinear system perfectly. We then propose to use an MPC-based reinforcement learning within the Koopman framework combining the strengths of MPC, Reinforcement Learning (RL), and the Koopman Operator (KO) theory for an efficient data-driven control and performance-oriented learning of complex nonlinear systems. We show that the closed-loop performance of the KMPC is improved by modifying the KMPC objective function. In practice, we design a fully parameterized KMPC and employ RL to adjust the corresponding parameters aiming at achieving the best achievable closed-loop performance.
We consider the problem of navigating a nonlinear dynamical system from some initial set to some target set while avoiding collision with an unsafe set. We extend the concept of density function to control density function (CDF) for solving navigation problems with safety constraints. The occupancy-based interpretation of the measure associated with the density function is instrumental in imposing the safety constraints. The navigation problem with safety constraints is formulated as a quadratic program (QP) using CDF. The existing approach using the control barrier function (CBF) also formulates the navigation problem with safety constraints as QP. One of the main advantages of the proposed QP using CDF compared to QP formulated using CBF is that both the convergence/stability and safety can be combined and imposed using the CDF. Simulation results involving the Duffing oscillator and safe navigation of Dubin car models are provided to verify the main findings of the paper.
We present data-driven methods for power system transient stability analysis using a unit eigenfunction of the Koopman operator. We show that the Koopman eigenfunction with unit eigenvalue can identify the region of attraction of the post-fault stable equilibrium. We then leverage this property to estimate the critical clearing time of a fault. We provide two data-driven methods to estimate said eigenfunction; the first method utilizes time averages over long trajectories, and the second method leverages nonparametric learning of system dynamics over reproducing kernel Hilbert spaces with short bursts of state propagation. Our methods do not require explicit knowledge of the power system model, but require a simulator that can propagate states through the power system dynamics. Numerical experiments on three power system examples demonstrate the efficacy of our method.
We consider the problem of navigating a nonlinear dynamical system from some initial set to some target set while avoiding collision with an unsafe set. We extend the concept of density function to control density function (CDF) for solving navigation problems with safety constraints. The occupancy-based interpretation of the measure associated with the density function is instrumental in imposing the safety constraints. The navigation problem with safety constraints is formulated as a quadratic program (QP) using CDF. The existing approach using the control barrier function (CBF) also formulates the navigation problem with safety constraints as QP. One of the main advantages of the proposed QP using CDF compared to QP formulated using CBF is that both the convergence/stability and safety can be combined and imposed using the CDF. Simulation results involving the Duffing oscillator and safe navigation of Dubin car models are provided to verify the main findings of the paper.
In neurological networks, the emergence of various causal interactions and information flows among nodes is governed by the structural connectivity in conjunction with the node dynamics. The information flow describes the direction and the magnitude of an excitatory neuron’s influence to the neighbouring neurons. However, the intricate relationship between network dynamics and information flows is not well understood. Here, we address this challenge by first identifying a generic mechanism that defines the evolution of various information routing patterns in response to modifications in the underlying network dynamics. Moreover, with emerging techniques in brain stimulation, designing optimal stimulation directed towards a target region with an acceptable magnitude remains an ongoing and significant challenge. In this work, we also introduce techniques for computing optimal inputs that follow a desired stimulation routing path towards the target brain region. This optimization problem can be efficiently resolved using non-linear programming tools and permits the simultaneous assignment of multiple desired patterns at different instances. We establish the algebraic and graph-theoretic conditions necessary to ensure the feasibility and stability of information routing patterns (IRPs). We illustrate the routing mechanisms and control methods for attaining desired patterns in biological oscillatory dynamics.
The paper is about the computation of the principal spectrum of the Koopman operator (i.e., eigenvalues and eigenfunctions). The principal eigenfunctions of the Koopman operator are the ones with the corresponding eigenvalues equal to the eigenvalues of the linearization of the nonlinear system at an equilibrium point. The main contribution of this paper is to provide a novel approach for computing the principal eigenfunctions using a path-integral formula. Furthermore, we provide conditions based on the stability property of the dynamical system and the eigenvalues of the linearization towards computing the principal eigenfunction using the path-integral formula. Further, we provide a Deep Neural Network framework that utilizes our proposed path-integral approach for eigenfunction computation in high-dimension systems. Finally, we present simulation results for the computation of principal eigenfunction and demonstrate their application for determining the stable and unstable manifolds and constructing the Lyapunov function.
We present a data-driven approach to propagate uncertainty in initial conditions through the dynamics of an unknown system in a reproducing kernel Hilbert space (RKHS). The uncertainty in initial conditions is represented through its kernel mean embedding (KME) in the RKHS. For a discrete-time Markovian dynamical system, we utilize the conditional mean embedding (CME) operator to encode the underlying dynamics. Learning in RKHS often incurs prohibitive data storage requirements. To circumvent said limitation, we propose an algorithm to propagate uncertainty via a learned sparse CME operator, and provide theoretical guarantees on the approximation error for the embedded distribution with time. We empirically study our algorithm over illustrative dynamical systems and power systems.
We provide a data-driven framework for optimal control of a continuous-time stochastic dynamical system. The proposed framework relies on the linear operator theory involving linear Perron-Frobenius (P-F) and Koopman operators. Our first results involving the P-F operator provide a convex formulation to the optimal control problem in the dual space of densities. This convex formulation of the stochastic optimal control problem leads to an infinite-dimensional convex program. The finite-dimensional approximation of the convex program is obtained using a data-driven approximation of the P-F operator. Our second results demonstrate the use of the Koopman operator, which is dual to the P-F operator, for the stochastic optimal control design. We show that the Hamilton Jacobi Bellman (HJB) equation can be expressed using the Koopman operator. We provide an iterative procedure along the lines of a popular policy iteration algorithm based on the data-driven approximation of the Koopman operator for solving the HJB equation. The two formulations, namely the convex formulation involving P-F operator and Koopman based formulation using HJB equation, can be viewed as dual to each other where the duality follows due to the dual nature of P-F and Koopman operators. Finally, we present several numerical examples to demonstrate the efficacy of the developed framework.
Optimization problems emerging in most of the real-world applications are dynamic, where either the objective function or the constraints change continuously over time. This article proposes projected primal–dual dynamical system approaches to track the primal and dual optimizer trajectories of an inequality constrained time-varying (TV) convex optimization problem with a strongly convex objective function. First, we present a dynamical system that asymptotically tracks the optimizer trajectory of an inequality constrained TV optimization problem. Later, we modify the proposed dynamics to achieve the convergence to the optimizer trajectory within a fixed time. The asymptotic and fixed-time convergence of the proposed dynamical systems to the optimizer trajectory is shown via the Lyapunov-based analysis. Finally, we consider the TV extended Fermat–Torricelli problem of minimizing the sum-of-squared distances to a finite number of nonempty, closed, and convex TV sets, to illustrate the applicability of the projected dynamical systems proposed in this article.
This paper presents a novel approach for safe control synthesis using the dual formulation of the navigation problem. The main contribution of this paper is in the analytical construction of density functions for almost everywhere navigation with safety constraints. In contrast to the existing approaches, where density functions are used for the analysis of navigation problems, we use density functions for the synthesis of safe controllers. We provide convergence proof using the proposed density functions for navigation with safety. Further, we use these density functions to design feedback controllers capable of navigating in cluttered environments and high-dimensional configuration spaces. The proposed analytical construction of density functions overcomes the problem associated with navigation functions, which are known to exist but challenging to construct, and potential functions, which suffer from local minima. Application of the developed framework is demonstrated on simple integrator dynamics and fully actuated robotic systems. Our project page with implementation is available at https://github.com/ clemson-dira/density_feedback_control
Ground vehicles operate under different driving conditions, which require the analysis of varying parameter values. It is essential to ensure the vehicle’s safe operation under all these conditions of the parameter variation. In this paper, we investigate the safe operating limits of a ground vehicle by performing the reachability analysis for varying parameters using the Koopman spectrum approach. The reachable set is computed using the Koopman principal eigenfunctions obtained from a convex optimization formulation for different values of the parameter. We consider the two degrees-of-freedom nonlinear quarter-car model to simulate the vehicle’s dynamics. Based on the obtained reachable sets, we provide the mean and variance computation framework with parametric uncertainty. The results show that the reachable set for each value provides valuable information regarding the safe operating limits of the vehicle and can assist in developing safe driving strategies.
The paper is about L2-gain computation and the small-gain theorem for nonlinear input-output systems. We show that the Koopman operator’s spectrum can provide conditions for L2-gain guarantees and small-gain theorem-based stability of interconnection over a large region of the state space. The large region in the state space can be characterized in terms of the region where Koopman eigenfunctions and the solution of the Hamilton Jacobi equation are well defined. The connection of system L2-gain to the spectral properties of the Koopman operator has led to a novel approach, based on the approximation of the Koopman spectrum, for the computation of the L2-gain and stability verification of the interconnected system. We present simulation results including application of the developed framework to a power system example.
The paper is about the computation of the principal spectrum of the Koopman operator (i.e., eigenvalues and eigenfunctions). The principal eigenfunctions of the Koopman operator are the ones with the corresponding eigenvalues equal to the eigenvalues of the linearization of the nonlinear system at an equilibrium point. The main contribution of this paper is to provide a novel approach for computing the principal eigenfunctions using a path-integral formula. Furthermore, we provide conditions based on the stability property of the dynamical system and the eigenvalues of the linearization towards computing the principal eigenfunction using the path-integral formula. Further, we provide a Deep Neural Network framework that utilizes our proposed path-integral approach for eigenfunction computation in high-dimension systems. Finally, we present simulation results for the computation of principal eigenfunction and demonstrate their application for determining the stable and unstable manifolds and constructing the Lyapunov function.
This paper presents a data-driven framework to discover underlying dynamics on a scaled F1TENTH vehicle using the Koopman operator linear predictor. Traditionally, a range of white, gray, or black-box models are used to develop controllers for vehicle path tracking. However, these models are constrained to either linearized operational domains, unable to handle significant variability or lose explainability through end-2-end operational settings. The Koopman Extended Dynamic Mode Decomposition (EDMD) linear predictor seeks to utilize data-driven model learning whilst providing benefits like explainability, model analysis and the ability to utilize linear model-based control techniques. Consider a trajectory-tracking problem for our scaled vehicle platform. We collect pose measurements of our F1TENTH car undergoing standard vehicle dynamics benchmark maneuvers with an OptiTrack indoor localization system. Utilizing these uniformly spaced temporal snapshots of the states and control inputs, a data-driven Koopman EDMD model is identified. This model serves as a linear predictor for state propagation, upon which an MPC feedback law is designed to enable trajectory tracking. The prediction and control capabilities of our framework are highlighted through real-time deployment on our scaled vehicle.
This paper presents the implementation of off-road navigation on legged robots using convex optimization through linear transfer operators. Given a traversability measure that captures the off-road environment, we lift the navigation problem into the density space using the Perron-Frobenius (P-F) operator. This allows the problem formulation to be represented as a convex optimization. Due to the operator acting on an infinite-dimensional density space, we use data collected from the terrain to get a finite-dimension approximation of the convex optimization. Results of the optimal trajectory for off-road navigation are compared with a standard iterative planner, where we show how our convex optimization generates a more traversable path for the legged robot compared to the suboptimal iterative planner.
The path-tracking control performance of an autonomous vehicle (AV) is crucially dependent upon modeling choices and subsequent system-identification updates. Traditionally, automotive engineering has built upon increasing fidelity of white- and gray-box models coupled with system identification. While these models offer explainability, they suffer from modeling inaccuracies, non-linearities, and parameter variation. On the other end, end-to-end black-box methods like behavior cloning and reinforcement learning provide increased adaptability but at the expense of explainability, generalizability, and the sim2real gap. In this regard, hybrid data-driven techniques like Koopman Extended Dynamic Mode Decomposition (KEDMD) can achieve linear embedding of non-linear dynamics through a selection of “lifting functions”. However, the success of this method is primarily predicated on the choice of lifting function(s) and optimization parameters. In this study, we present an analytical approach to construct these lifting functions using the iterative Lie bracket vector fields considering holonomic and non-holonomic constraints on the configuration manifold of our Ackermann-steered autonomous mobile robot. The prediction and control capabilities of the obtained linear KEDMD model are showcased using trajectory tracking of standard vehicle dynamics maneuvers and along a closed-loop racetrack.
Accurate terrain mapping is of paramount importance for motion planning and safe navigation in unstructured terrain. LIDAR sensors provide a modality, in the form of a 3D point cloud, that can be used to estimate the elevation map of the surrounding environment. But, working with the 3D point cloud data turns out to be challenging. This is primarily due to the unstructured nature of the point clouds, relative sparsity of the data points, occlusions due to negative slopes and obstacles, and the high computational burden of traditional point cloud algorithms. We tackle these problems with the help of a learning-based, efficient data processing approach for vehicle-centric terrain reconstruction using a 3D LIDAR. The 3D LIDAR point cloud is projected on the ground plane, which is processed by a generative adversarial network (GAN) architecture in the form of an image to fill in the missing parts of the terrain heightmap. We train the GAN model on artificially generated datasets and show the method’s effectiveness by means of the reconstructed terrains.
In this paper, we propose a novel data-driven approach for learning and control of quadrotor UAVs based on the Koopman operator and extended dynamic mode decomposition (EDMD). Building observables for EDMD based on conventional methods like Euler angles (to represent orientation) is known to involve singularities. To address this issue, we employ a set of physics-informed observables based on the underlying topology of the nonlinear system. We use rotation matrices to directly represent the orientation dynamics and obtain a lifted linear representation of the nonlinear quadrotor dynamics in the SE(3) manifold. This EDMD model leads to accurate prediction and can be generalized to several validation sets. Further, we design a linear model predictive controller (MPC) based on the proposed EDMD model to track agile reference trajectories. Simulation results show that the proposed MPC controller can run as fast as 100 Hz and is able to track arbitrary reference trajectories with good accuracy. Implementation details can be found in \url{this https URL}.
Transfer operators provide a rich framework for representing the dynamics of very general, nonlinear dynamical systems. When interacting with reproducing kernel Hilbert spaces (RKHS), descriptions of dynamics often incur prohibitive data storage requirements, motivating dataset sparsification as a precursory step to computation. Further, in practice, data is available in the form of trajectories, introducing correlation between samples. In this work, we present a method for sparse learning of transfer operators from ββ-mixing stochastic processes, in both discrete and continuous time, and provide sample complexity analysis extending existing theoretical guarantees for learning from non-sparse, i.i.d. data. In addressing continuous-time settings, we develop precise descriptions using covariance-type operators for the infinitesimal generator that aids in the sample complexity analysis. We empirically illustrate the efficacy of our sparse embedding approach through deterministic and stochastic nonlinear system examples.
We consider the problem of optimal navigation control design for navigation on off-road terrain. We use traversability measure to characterize the degree of difficulty of navigation on the off-road terrain. The traversability measure captures the property of terrain essential for navigation, such as elevation map, terrain roughness, slope, and terrain texture. The terrain with the presence or absence of obstacles becomes a particular case of the proposed traversability measure. We provide a convex formulation to the off-road navigation problem by lifting the problem to the density space using the linear Perron-Frobenius (P-F) operator. The convex formulation leads to an infinite-dimensional optimal navigation problem for control synthesis. The finite-dimensional approximation of the infinite-dimensional convex problem is constructed using data. We use a computational framework involving the Koopman operator and the duality between the Koopman and P-F operator for the data-driven approximation. This makes our proposed approach data-driven and can be applied in cases where an explicit system model is unavailable. Finally, we demonstrate the application of the developed framework for the navigation of vehicle dynamics with Dubin’s car model.
This paper considers the problem of optimizing robot navigation with respect to a time-varying objective encoded into a navigation density function. We are interested in designing state feedback control laws that lead to an almost everywhere stabilization of the closed-loop system to an equilibrium point while navigating a region optimally and safely (that is, the transient leading to the final equilibrium point is optimal and satisfies safety constraints). Though this problem has been studied in literature within many different communities, it still remains a challenging non-convex control problem. In our approach, under certain assumptions on the time-varying navigation density, we use Koopman and Perron-Frobenius Operator theoretic tools to transform the problem into a convex one in infinite dimensional decision variables. In particular, the cost function and the safety constraints in the transformed formulation become linear in these functional variables. Finally, we present some numerical examples to illustrate our approach, as well as discuss the current limitations and future extensions of our framework to accommodate a wider range of robotics applications.
This paper is concerned with data-driven optimal control of nonlinear systems. We present a convex formulation to the optimal control problem (OCP) with a discounted cost function. We consider OCP with both positive and negative discount factor. The convex approach relies on lifting nonlinear system dynamics in the space of densities using the linear Perron-Frobenius (P-F) operator. This lifting leads to an infinite-dimensional convex optimization formulation of the optimal control problem. The data-driven approximation of the optimization problem relies on the approximation of the Koopman operator using the polynomial basis function. We write the approximate finite-dimensional optimization problem as a polynomial optimization which is then solved efficiently using a sum-of-squares-based optimization framework. Simulation results are presented to demonstrate the efficacy of the developed data-driven optimal control framework.
Information flow among nodes in a complex network describes the overall cause-effect relationships among the nodes and provides a better understanding of the contributions of these nodes individually or collectively towards the underlying network dynamics. Variations in network topologies result in varying information flows among nodes. We integrate theories from information science with control network theory into a framework that enables us to quantify and control the information flows among the nodes in a complex network. The framework explicates the relationships between the network topology and the functional patterns, such as the information transfers in biological networks, information rerouting in sensor nodes, and influence patterns in social networks. We show that by designing or re-configuring the network topology, we can optimize the information transfer function between two chosen nodes. As a proof of concept, we apply our proposed methods in the context of brain networks, where we reconfigure neural circuits to optimize excitation levels among the excitatory neurons.
In this work we approach the dual optimal reach-safe control problem using sparse approximations of Koopman operator. Matrix approximation of Koopman operator needs to solve a least-squares (LS) problem in the lifted function space, which is computationally intractable for fine discretizations and high dimensions. The state transitional physical meaning of the Koopman operator leads to a sparse LS problem in this space. Leveraging this sparsity, we propose an efficient method to solve the sparse LS problem where we reduce the problem dimension dramatically by formulating the problem using only the non-zero elements in the approximation matrix with known sparsity pattern. The obtained matrix approximation of the operators is then used in a dual optimal reach-safe problem formulation where a linear program with sparse linear constraints naturally appears. We validate our proposed method on various dynamical systems and show that the computation time for operator approximation is greatly reduced with high precision in the solutions.
This letter proposes using the Koopman operator for reachability analysis of an autonomous dynamical system. In particular, we demonstrate the application of spectral analysis of the Koopman operator involving eigenfunctions and eigenvalues in the approximate computation of forward and backward reachable sets for an autonomous dynamical system. The formal guarantees for the approximate reachable sets are provided using the Hausdorff distance between sets that measure how far the approximate reachable set is from the true reachable set. A computational framework based on convex optimization is provided to compute the Koopman spectrum and the approximate reachable set. Finally, we present simulation results to demonstrate the application of the developed framework.
In this paper, we present an approach based on the spectral analysis of the Koopman operator for the approximate solution of the Hamilton Jacobi equation that arises while solving the optimal control problem. It is well-known that one can associate a Hamiltonian dynamical system with the Hamilton Jacobi equation. Furthermore, the Lagrangian submanifold of the Hamiltonian dynamical system play a fundamental role in solving the Hamilton Jacobi equation. We show that the principal eigenfunctions of the Koopman operator associated with the Hamiltonian dynamical system can be used in constructing the Lagrangian submanifold, thereby approximating the solution of the Hamilton Jacobi equation. We present simulation results to verify the main findings of the paper.
We present an analytical and computational framework using the theory of Koopman operator to design dual-mode model predictive control (MPC) for nonlinear control systems. Dual-mode MPC incorporates stability and performance constraints in the controller and remains a challenging problem to solve for systems involving nonlinear dynamics. We exploit the spectral properties of the Koopman operator to provide a systematic approach for the design of optimal control, guaranteeing the stability of the feedback control system. The optimal cost function is also used for characterizing the maximal invariant set, which serves as the terminal set in the design of finite-horizon MPC. The finite horizon MPC relies on lifting nonlinear system dynamics using Koopman eigenfunctions to provide a linear approach for the design of MPC with state and input constraints.
In this paper, we present results for the data-driven control oriented modeling of complex dynamics that arise in phase separation of multi-phase immiscible system. This phase separation phenomena is modeled using a complicated nonlinear partial differential equations which are not suitable for control. We use simulation data generated from a detailed physics-based model for the data-driven modeling. We employ Koopman-based lifting for the identification of linear models from the data both under controlled and uncontrolled setting. Spectral analysis of Koopman and its adjoint Perron-Frobenius operator helps us identify invariant structure and dominant modes for the reduced-order representation from the data. Simulation results are presented for the reconstruction and validation of the controlled dynamical model.
Quadruped locomotion over soft deformable terrain is a challenging problem. It is difficult to accurately model the nonlinear dynamics associated with the terrain-leg interaction during the stance phase of the gait motion. In order to overcome this challenge, we utilize Koopman spectral theory to obtain a linear dynamical system in the space of observable functions. We propose an experimental framework to obtain a data-driven Koopman model of the quadruped's leg dynamics over deformable terrains as a switched dynamical system. Experimental results show that the learned switched system model can be used to predict gait trajectories on unknown terrain. Furthermore, we exploit the linearity of the Koopman operator to extract the complex leg-terrain interaction dynamics. Finally, we show that the Koopman generator has a unique spectrum associated with each terrain, making terrain classification possible without the need for foot force sensors.
This letter considers the optimal control problem of nonlinear systems under safety constraints with unknown dynamics. Departing from the standard optimal control framework based on dynamic programming, we study its dual formulation over the space of occupancy measures. For control-affine dynamics, with proper reparametrization, the problem can be formulated as an infinite-dimensional convex optimization over occupancy measures. Moreover, the safety constraints can be naturally captured by linear constraints in this formulation. Furthermore, this dual formulation can still be approximately obtained by utilizing the Koopman theory when the underlying dynamics are unknown. Finally, to develop a practical method to solve the resulting convex optimization, we choose a polynomial basis and then relax the problem into a semi-definite program (SDP) using sum-of-square (SOS) techniques. Simulation results are presented to demonstrate the efficacy of the developed framework.
Forced oscillations is one of the critical challenges faced by the evolving power systems. In this paper, we present a system theoretic approach based on the concept of controllability Gramian to identify the source of forced oscillation. The Gramians are constructed empirically using time-series simulation data in the training phase. These Gramian matrices are combined with the empirically constructed Gramian obtained from the real-time data with forced oscillation in the formulation of optimization problem to identify the source location. The proposed optimization problem is convex and can be solved efficiently for large scale system. We present the problem formulation, implementation strategies, and case studies for the proposed approach. The proposed data-driven methodology addresses the limitations of existing energy flow method and other data-driven methodologies.
Testing any new safety technology of Autonomous Vehicles (AV) and Advanced Driver Assistance Systems (ADAS) requires simulation-based validation and verification. The specific scenarios used for testing, outline incidences of accidents or near-miss events. In order to simulate these scenarios, specific values for all the above parameters are required including the ego vehicle model. The ‘criticality’ of a scenario is defined in terms of the difficulty level of the safety maneuver. A scenario could be over-critical, critical, or under-critical. In over-critical scenarios, it is impossible to avoid a crash whereas, for under-critical scenarios, no action may be required to avoid a crash. The criticality of the scenario depends on various parameters e.g. speeds, distances, road/tire parameters, etc. In this paper, we propose a definition of criticality metric and identify the parameters such that a scenario becomes critical. The criticality of a scenario should be independent of the controller or the driver model. Hence, we use an optimal control as the ‘best’ candidate. The proposed approach has three key steps - 1) obtain optimal control for given dynamic and static constraints, 2) compute the probability of a crash assuming small variations in model parameters and control action, and 3) compute occupancy metric over the criticality parameters design space. The occupancy metric, which is related to the value function of the optimal control, defines the criticality of the scenario. The key benefit of this approach is a clear definition of criticality metric which reflects the probability of collision. The proposed approach is demonstrated using an example of an obstacle avoidance maneuver.
Transfer operators such as the Koopman and the Perron-Frobenius operators provide valuable insights into the properties of nonlinear dynamical systems. Recent work has shown that non-parametric approximations of these operators can be constructed over reproducing kernel Hilbert space (RKHS) with data. These kernel transfer operators can then be written as functions of covariance and cross-covariance operators associated with the data generated by the dynamical system. In this paper, we study sparse kernel learning methods for kernel transfer operators. Specifically, we study sample complexity guarantees for coherency-based sparsification and demonstrate its efficacy over an example dynamical system.
A time-varying (TV) optimization problem arises in many real-time applications, where the objective function or constraints change continuously with time. Consequently, the optimal points of the problem at each time instant form an optimal trajectory and hence tracking the optimal trajectory calls for the need to solve the TV optimization problem. A second-order continuous-time gradient-flow approach is proposed in this paper to track the optimal trajectory of TV convex optimization problems in fixed-time irrespective of the initial conditions. Later on we present a second-order nonsmooth dynamical system to solve the TV convex optimization problem in fixed time that does not require the exact information about the time rate of change of the cost function gradient. It makes the non-smooth dynamical system robust to the temporal variation in the gradient of the cost function. Two numerical examples are considered here for the simulation-based validation of the proposed approaches.
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.
We consider the problem of navigation with safety constraints. The safety constraints are probabilistic, where a given set is assigned a degree of safety, a number between zero and one, with zero being safe and one being unsafe. The deterministic unsafe set will arise as a particular case of the proposed probabilistic description of safety. We provide a convex formulation to the navigation problem with probabilistic safety constraints. The convex formulation is made possible by lifting the navigation problem in the dual space of density using linear transfer operator theory methods involving Perron-Frobenius and Koopman operators. The convex formulation leads to an infinite-dimensional feasibility problem for probabilistic safety verification and control design. The finite-dimensional approximation of the optimization problem relies on the data-driven approximation of the linear transfer operator.
In this paper, a distributed resource allocation problem is considered where multiple agents want to allocate network resources among themselves while optimizing certain performance index. The first continuous-time distributed second-order gradient algorithm is proposed for resource allocation over static (non-switching) graphs under synchronous protocol. The algorithm is able to converge to the optimal solution of the problem with exponential convergence rate under suitable assumptions. Finally, a numerical example of a distributed estimation in wireless sensor networks is given by using the algorithm in order to demonstrate the results.
Classically, the optimal control problem in the presence of an adversary is formulated as a two-player zero-sum differential game or an H∞ control problem. The solution to these problems can be obtained by solving the Hamilton-Jacobi-Issac equation (HJIE). We provide a novel Koopman-based expression of the HJIE, where the solutions can be obtained through the approximation of the Koopman operator itself. In particular, we developed a data-driven and model based policy iteration algorithm for approximating the optimal value function using a finite-dimensional approximation of the Koopman operator and generator.
This article is about the data-driven computation of optimal control for a class of control affine deterministic nonlinear systems. We assume that the control dynamical system model is not available, and the only information about the system dynamics is available in the form of time-series data. We provide a convex formulation for the optimal control problem (OCP) of the nonlinear system. The convex formulation relies on the duality result in the dynamical system’s stability theory involving density function and Perron–Frobenius operator. We formulate the OCP as an infinite-dimensional convex optimization program. The finite-dimensional approximation of the optimization problem relies on the recent advances made in the Koopman operator’s data-driven computation, which is dual to the Perron–Frobenius operator. Simulation results are presented to demonstrate the application of the developed framework.
We consider a class of nonlinear control synthesis problems where the underlying mathematical models are not explicitly known. We propose a data-driven approach to stabilize the systems when only sample trajectories of the dynamics are accessible. Our method is founded on the density function based almost everywhere stability certificate that is dual to the Lyapunov function for dynamic systems. Unlike Lyapunov based methods, density functions lead to a convex formulation for a joint search of the control strategy and the stability certificate. This type of convex problem can be solved efficiently by invoking the machinery of the sum of squares (SOS). For the data-driven part, we exploit the fact that the duality results in the stability theory of the dynamical system can be understood using linear Perron-Frobenius and Koopman operators. This connection allows us to use data-driven methods developed to approximate these operators combined with the SOS techniques for the convex formulation of control synthesis. The efficacy of the proposed approach is demonstrated through several examples.
A model-based, data-driven control framework is introduced within the context of autonomous driving in this study. We propose a data-driven control algorithm that combines autonomous system identification using model-free learning and robust control using a model-based controller design. We present a full solution framework that is capable to automatically generate tire-friction limit path while performing system identification of a vehicle with unknown dynamics. We then design modelbased control which is actively learned from a data-driven approach. Based on our new system identification algorithm, we can approximate an accurate, explainable, and linearized system representation in a high-dimensional latent space, without any prior knowledge of the system. To validate the algorithm, we conduct the model predictive control of an autonomous vehicle based on the augmented system identification on a scaled racing vehicle. The result indicates that we are able to design control in the lifted space to achieve tasks in path control and obstacle avoidance. The automatic path generation combined with the data driven control requires no a-priori knowledge of the vehicle and also proved to be effective that only requires less than 5 laps to design an low lap-time trajectory while identified a system that is able to achieve minimum lap time without extra learning episodes.
The paper is about the optimal control of a stochastic dynamical system. We provide a convex formulation to the optimal control problem involving a stochastic dynamical system. The convex formulation is made possible by writing the stochastic optimal control problem in the dual space of densities involving the Fokker-Planck or Perron-Frobenius generator for a stochastic system. The convex formulation leads to an infinite-dimensional convex optimization problem for optimal control. We exploit Koopman and Perron-Frobenius generators' duality for the stochastic system to construct the finite-dimensional approximation of the infinite-dimensional convex problem. We present simulation results to demonstrate the application of the developed framework.
We consider an optimal control synthesis problem for a class of control-affine nonlinear systems. We propose Sum-of-Square based computational framework for optimal control synthesis. The proposed computation framework relies on the convex formulation of the optimal control problem in the dual space of densities. The convex formulation to the optimal control problem is based on the duality results in dynamical systems’ stability theory. We used the Sum-of-Square based computational framework for the finite-dimensional approximation of the convex optimization problem. The efficacy of the developed framework is demonstrated using simulation results.
In this paper, relations between distributed consensus-based optimization and a network resource allocation problem are considered. It is shown that first-order gradient algorithm for distributed consensus-based optimization can be used for finding an optimal solution of distributed resource allocation with synchronous protocol under weaker assumptions than those given in the literature. Moreover, second-order gradient algorithm for distributed consensus-based optimization is presented that can be employed for solving distributed resource allocation problems. As a result, several algorithms used for distributed consensus-based optimization can now be applied to derive distributed algorithms for resource allocation. It is shown that first and second order gradient algorithms for distributed resource allocation can be utilized for finding an optimal solution of distributed consensus-based optimization as well. The results presented in this paper can be applied to time-varying or random directed networks with or without synchronous protocols with arbitrary initialization. Finally, a numerical example of a distributed estimation in wireless sensor networks is given to demonstrate the results.
This paper presents an adaptive damping controller for wind power plants in which the turbines are equipped with doubly-fed induction generators. The controller is designed to respond to an input control signal that is triggered according to the system operating conditions. A processing unit continuously estimates the electromechanical modes of oscillation based on real-time streaming data acquired from a phasor measurement unit that is strategically positioned on the grid. The decision to trigger (or not trigger) the control signal is automatic, based on the relative damping of the dominant mode. The modes are estimated using the dynamic mode decomposition algorithm with time-delay embedding. Numerical simulations performed on the two-area system demonstrate that the proposed controller enhances the rotor angle stability for both small-signal and large disturbances, and is adaptive to changing grid conditions.
This paper presents an information-theoretic approach for model reduction for finite time simulation. Although system models are typically used for simulation over a finite time, most of the metrics (and pseudo-metrics) used for model accuracy assessment consider asymptotic behavior e.g., Hankel singular values and Kullback-Leibler(KL) rate metric. These metrics could further be used for model order reduction. Hence, in this paper, we propose a generalization of KL divergence-based metric called n-step KL rate metric, which could be used to compare models over a finite time horizon. We then demonstrate that the asymptotic metrics for comparing dynamical systems may not accurately assess the model prediction uncertainties over a finite time horizon. Motivated by this finite time analysis, we propose a new pragmatic approach to compute the influence of a subset of states on a combination of states called information transfer (IT). Model reduction typically involves the removal or truncation of states. IT combines the concepts from the n-step KL rate metric and model reduction. Finally, we demonstrate the application of information transfer for model reduction. Although the analysis and definitions presented in this paper assume linear systems, they can be extended for nonlinear systems.
Recent studies have demonstrated the potential of flexible loads in providing frequency response services. However, uncertainty and variability in various weather-related and end-use behavioral factors often affect the demand-side control performance. This work addresses this problem with the design of a demand-side control to achieve frequency response under load uncertainties. Our approach involves modeling the load uncertainties via stochastic processes that appear as both multiplicative and additive to the system states in closed-loop power system dynamics. Extending the recently developed mean square exponential stability (MSES) results for stochastic systems, we formulate multi-objective linear matrix inequality (LMI)-based optimal control synthesis problems to not only guarantee stochastic stability, but also promote sparsity, enhance closed-loop transient performance, and maximize allowable uncertainties. The fundamental trade-off between the maximum allowable (critical) uncertainty levels and the optimal stochastic stabilizing control efforts is established. Moreover, the sparse control synthesis problem is generalized to the realistic power systems scenario in which only partial-state measurements are available. Detailed numerical studies are carried out on IEEE 39-bus system to demonstrate the closed-loop stochastic stabilizing performance of the sparse controllers in enhancing frequency response under load uncertainties; as well as illustrate the fundamental trade-off between the allowable uncertainties and optimal control efforts.
We consider optimal control synthesis problems for control-affine systems under safety constraints. We are interested in the task of optimally driving a system from an initial set to a desired state and avoiding some predefined unsafe sets at the same time. Building on a weaker notion of stability of dynamic systems relying on the Lyapunov density, we formulate the optimal control synthesis problem into a convex optimization problem in the space of densities. In this formulation, the safety constraints are naturally mapped to convex constraints on the densities. To improve scalability, the densities are parameterized by polynomials or rational functions. The Sum-of-Squares technique is then employed to efficiently solve the optimal control problems and verify the safety constraints. Numerical examples are presented to demonstrate the efficacy of the proposed framework.
The problem of finding a point in Rn, from which the sum-of-distances to a finite number of nonempty, closed and convex sets is minimum is called generalized Fermat-Torricelli Problem (FTP). In applications, along with the point that minimizes sum-of-distances, it is important to know the points in the convex sets at which the minimum sum-of-distances is achieved. Various formulations existing in literature do not involve finding the optimal points in the convex sets. In this letter, we formulate a non-smooth convex optimization problem, with both the point/set of points which yields the minimum sum-of-distances as well as the corresponding points in the convex sets as primal variables. We term this problem as extended FTP (eFTP). We adopt non-smooth projected primal-dual dynamical approach to solve this problem. The proposed dynamical system can exhibit a continuum of equilibria. Hence we show semistability of the set of optimal points, which is the pertinent notion of stability for such systems. A distributed implementation of the primal-dual dynamical system is also presented in this letter. Four illustrative examples are considered for the simulation based validation of the solution proposed for eFTP.
In this paper, we provide a novel approach to capture causal interaction in a dynamical system from time series data. In Sinha and Vaidya (in: IEEE conference on decision and control, pp 7329–7334, 2016), we have shown that the existing measures of information transfer, namely directed information, Granger causality and transfer entropy, fail to capture the causal interaction in a dynamical system and proposed a new definition of information transfer that captures direct causal interactions. The novelty of the information transfer definition used in this paper is the fact that it can differentiate between direct and indirect influences Sinha and Vaidya (2016). The main contribution of this paper is to show that the proposed definition of information transfers in Sinha and Vaidya (2016) and Sinha and Vaidya (in: Indian control conference, pp 303–308, 2017) can be computed from time series data, and thus, the direct influences in a dynamical system can be identified from time series data. We use transfer operator theoretic framework, involving Perron–Frobenius and Koopman operators for the data-driven approximation of the system dynamics and computation of information transfer. Several examples, involving linear and nonlinear system dynamics, are presented to verify the efficiency of the developed algorithm.
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, and the principle of inverse optimality is used to comment on the optimality of the designed stabilizing feedback controller for the bilinear system. 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.
We develop a data-driven, model-free approach for the optimal control of the dynamical system. The proposed approach relies on the Deep Neural Network (DNN) based learning of Koopman operator for the purpose of control. In particular, DNN is employed for the data-driven identification of basis function used in the linear lifting of nonlinear control system dynamics. The controller synthesis is purely data-driven and does not rely on a priori domain knowledge. The OpenAI Gym environment, employed for Reinforcement Learning-based control design, is used for data generation and learning of Koopman operator in control setting. The method is applied to two classic dynamical systems on OpenAI Gym environment to demonstrate the capability.
Recent focus on robustness to adversarial attacks for deep neural networks produced a large variety of algorithms for training robust models. Most of the effective algorithms involve solving the min-max optimization problem for training robust models (min step) under worst-case attacks (max step). However, they often suffer from high computational cost from running several inner maximization iterations (to find an optimal attack) inside every outer minimization iteration. Therefore, it becomes difficult to readily apply such algorithms for moderate to large size real world data sets. To alleviate this, we explore the effectiveness of iterative descent-ascent algorithms where the maximization and minimization steps are executed in an alternate fashion to simultaneously obtain the worst-case attack and the corresponding robust model. Specifically, we propose a novel discrete-time dynamical system-based algorithm that aims to find the saddle point of a min-max optimization problem in the presence of uncertainties. Under the assumptions that the cost function is convex and uncertainties enter concavely in the robust learning problem, we analytically show that our algorithm converges asymptotically to the robust optimal solution under a general adversarial budget constraints as induced by ℓp norm, for 1≤p≤∞. Based on our proposed analysis, we devise a fast robust training algorithm for deep neural networks. Although such training involves highly non-convex robust optimization problems, empirical results show that the algorithm can achieve significant robustness compared to other state-of-the-art robust models on benchmark data sets.
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.
Sensors in buildings are used for a wide variety of applications such as monitoring air quality, contaminants, indoor temperature, and relative humidity. These are used for accessing and ensuring indoor air quality, and also for ensuring safety in the event of chemical and biological attacks. It follows that optimal placement of sensors become important to accurately monitor contaminant levels in the indoor environment. However, contaminant transport inside the indoor environment is governed by the indoor flow conditions which are affected by various uncertainties associated with the building systems including occupancy and boundary fluxes. Therefore, it is important to account for all associated uncertainties while designing the sensor layout. The transfer operator based framework provides an effective way to identify optimal placement of sensors. Previous work has been limited to sensor placements under deterministic scenarios. In this work we extend the transfer operator based approach for optimal sensor placement while accounting for building systems uncertainties. The methodology provides a probabilistic metric to gauge coverage under uncertain conditions. We illustrate the capabilities of the framework with examples exhibiting boundary flux uncertainty.
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.
This paper investigates utilizing a heterogeneous group of thermostatically controlled loads (TCLs) for long-term demand response applications. The steady-state services are achieved through manipulating the stored thermal-energy with minimal impact on devices' switching rates and the operating duty-cycles. The Markov chain abstraction method has been developed in literature for aggregating the TCLs at fixed temperature set-point. In this paper, an extended Markov model (EMM) is proposed to account for the dynamics involved in modifying various set-point magnitudes in both directions. The EMM is formulated online based on linear mapping and fast restructuring to Markov chains developed offline at fixed set-points, where a training process is used to construct each Markov chain. Set-point adjustments force devices to operate in a synchronized pattern, causing the aggregated power to oscillate or traverse extreme conditions. Therefore, model predictive control with direct ON/OFF switching capability is proposed to apply the set-point change sequentially and control devices' movement toward the new operating set-point. The performance of the proposed modeling and control techniques are compared against existing methods which rely on the direct ON/OFF control solely rather than adjusting the thermal-energy level.
In this paper, we present a novel approach to identify the generators and states responsible for the small-signal stability of power networks. To this end, the newly developed notion of information transfer between the states of a dynamical system is used. In particular, using the concept of information transfer, which characterizes influence between the various states of a dynamical system, we identify the generators and states which are responsible for causing instability of the power network. While characterizing influence from state to state, information transfer can also describe influence from state to modes thereby generalizing the well known notion of participation factor while at the same time overcoming some of the limitations of the participation factor. The developed framework is applied to reproduce known results for the three bus system, identifying the various causes of instabilities, and is extended to IEEE 39 bus system.
In the paper, we consider the problem of robust approximation of transfer Koopman and Perron–Frobenius (P–F) operators from noisy time-series data. In most applications, the time-series data obtained from simulation or experiment are corrupted with either measurement or process noise or both. The existing results show the applicability of algorithms developed for the finite-dimensional approximation of the deterministic system to a random uncertain case. However, these results hold only in asymptotic and under the assumption of infinite data set. In practice, the data set is finite, and hence it is important to develop algorithms that explicitly account for the presence of uncertainty in data set. We propose a robust optimization-based framework for the robust approximation of the transfer operators, where the uncertainty in data set is treated as deterministic norm bounded uncertainty. The robust optimization leads to a min–max type optimization problem for the approximation of transfer operators. This robust optimization problem is shown to be equivalent to regularized least-square problem. This equivalence between robust optimization problem and regularized least-square problem allows us to comment on various interesting properties of the obtained solution using robust optimization. In particular, the robust optimization formulation captures inherent trade-offs between the quality of approximation and complexity of approximation. These trade-offs are necessary to balance for the proposed application of transfer operators, for the design of optimal predictor. Simulation results demonstrate that our proposed robust approximation algorithm performs better than some of the existing algorithms like extended dynamic mode decomposition (EDMD), subspace DMD, noise-corrected DMD, and total DMD for systems with process and measurement noise.
Accurate and rapid monitoring of indoor air quality is critical to ensure occupant safety in the built environment. This is especially important in events where hazardous substances are released, and prompt estimation of contaminant distribution will facilitate quick evacuation and control response. The built environment is usually equipped with a finite set of sensors that measure local concentration of contaminants. The goal is to use this streaming dataset to estimate the contaminant concentration distribution in the complete domain. We accomplish this by integrating two powerful concepts. We utilize an operator theoretic approach – specifically the Perron-Frobenius (P-F) operator – to model the contaminant transport. Previous work has shown that the PF approach is a fast, effective, and accurate paradigm for sensor placement and contaminant transport prediction. The PF approach is integrated with an Ensemble Kalman Filter to rapidly estimate contaminant distribution under unknown release scenarios, given minimal sensor data. The framework is illustrated for two scenarios: a 2D problem involving an office space, and a 3D problem involving a furnished hotel room. Both examples show that the contaminant distribution is accurately predicted within a few sensor measurement cycles. The general applicability of the framework is illustrated by testing the framework for multiple, unknown release locations. This approach provides a unified, extendable framework for rapid contaminant estimation.
We discover discrete-time counterpart of the continuous-time saddle point algorithm developed in [1] for solving robust optimization problems. Under the assumption that the cost function is convex in the decision variable and uncertainties enter concavely in the robust optimization problem, we prove global asymptotic convergence of the saddle-point algorithm to the robust optimal solution. The sub-gradient nature of the proposed discrete-time algorithm allows us to handle robust optimization problems with discontinuous cost function and constraint.
We propose a dynamical system-based approach for solving robust optimization problems. The well-known continuous-time dynamical system for solving deterministic optimization problems arises in the form of primal-dual gradient dynamics where the vector field is derived as the gradient of the Lagrangian. The new continuous-time dynamical system we introduce for solving robust optimization problems differs from the primal-dual dynamics in the sense that the vector field is not derived as the gradient of the Lagrangian function. We call this new dynamical system as saddle point dynamics. In the saddle point dynamics, the uncertain variable arises as a dynamical state. For a general class of robust optimization problem, where the cost function is convex in decision variable and concave in uncertain variable, we show that the robust optimal solution can be recovered as a globally asymptotically stable equilibrium point of the saddle point dynamical system. Simulation results are presented to demonstrate the capability of this new dynamical system to solve various robust optimization problems. We also compare our proposed approach with existing methods based on robust counterpart and scenario-based random sampling.
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.
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.
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.
Maintaining indoor environment quality requires availability of effective estimation and control measures. In hazardous substances release scenario, a prompt estimation of contaminant distribution in the space is essential to enable quick control action. In this paper, we discuss the use of Perron-Frobenius (P-F) operator based approach for the design of an estimator to efficiently track the contaminant in an indoor environment. While the contaminants are evolving under the influence of nonlinear fluid flow field, the linear nature of the P-F operator is exploited for the design of estimator dynamics. In particular, the linear nature of the P-F framework is used for the design of a Kalman filtering algorithm to track the contaminants under an impulsive release scenario. Simulation results involving International Energy Agency (IEA) two dimensional building system model are presented to verify the main findings of the paper.
We propose a robust optimization-based approach for the optimal voltage regulation of a distribution system in the presence of renewable solar uncertainty. The variability in renewable solar is modeled as deterministic but norm-bounded uncertainty. The structure of the uncertainty entering in the optimization problem is exploited to propose a primal-dual gradient dynamics to solve the robust optimization problem. Simulation results are presented involving a realistic three-phase unbalanced IEEE 13-bus distribution test system to demonstrate the applications of the developed framework.
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.
We introduce navigation measure as a new tool to solve the motion planning problem in the presence of static obstacles. Existence of navigation measure guarantees collision-free convergence at the final destination set beginning with almost every initial condition with respect to the Lebesgue measure. Navigation measure can be viewed as a dual to the navigation function. While the navigation function has its minimum at the final destination set and peaks at the obstacle set, navigation measure takes the maximum value at the destination set and is zero at the obstacle set. A linear programming formalism is proposed for the construction of navigation measure. Set-oriented numerical methods are utilised to obtain finite dimensional approximation of this navigation measure. Application of the proposed navigation measure-based theoretical and computational framework is demonstrated for a motion planning problem in a complex fluid flow.
The recent large-scale penetration of renewable energy in power networks has also introduced with it a risk of random variability. This new source of power uncertainty can fluctuate so substantially that the traditional base-point forecast and control scheme may fail to work. To address this challenge, we study the so-called robust AC optimal power flow (AC-OPF) so as to provide robust control solutions that can immunize the power system against the intermittent renewables. In this paper we generalize the continuous-time primal-dual gradient dynamics approach to solve the robust AC-OPF. One advantage of the proposed approach is that it does not require any convexity assumptions for the decision variables during the dynamical evolution. This paper first derives a stability analysis for the primal-dual dynamics associated with a generic robust optimization, and then applies the primal-dual dynamics to the robust AC-OPF problem. Simulation results are also provided to demonstrate the effectiveness of the proposed approach.
In the paper, we consider the problem of robust approximation of transfer Koopman and Perron-Frobenius (P-F) operators from noisy time series data. In most applications, the time-series data obtained from simulation or experiment is corrupted with either measurement or process noise or both. The existing results show the applicability of algorithms developed for the finite dimensional approximation of deterministic system to a random uncertain case. However, these results hold true only in asymptotic and under the assumption of infinite data set. In practice the data set is finite, and hence it is important to develop algorithms that explicitly account for the presence of uncertainty in data-set. We propose a robust optimization-based framework for the robust approximation of the transfer operators, where the uncertainty in data-set is treated as deterministic norm bounded uncertainty. The robust optimization leads to a min-max type optimization problem for the approximation of transfer operators. This robust optimization problem is shown to be equivalent to regularized least squares problem. This equivalence between robust optimization problem and regularized least squares problem allows us to comment on various interesting properties of the obtained solution using robust optimization. In particular, the robust optimization formulation captures inherent tradeoffs between the quality of approximation and complexity of approximation. These tradeoffs are necessary to balance for the proposed application of transfer operators, for the design of optimal predictor. Simulation results demonstrate that our proposed robust approximation algorithm performs better than the Extended Dynamic Mode Decomposition (EDMD) and DMD algorithms for a system with process and measurement noise.
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.
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 novel approach to capture causal interaction in a linear dynamical system from time-series data. In [1], we have shown that the existing measures of information transfer, namely directed information, Granger causality and transfer entropy fail to capture true causal interaction in a dynamical system and proposed a new definition of information transfer that captures true causal interaction. The main contribution of this paper is to show that the proposed definition of information transfer in [1] [2] can be computed from time-series data and the computed information measure allows the identification of causal interaction and network topology in a dynamical system. The data-driven algorithm for computation of information transfer for linear systems relies on a robust optimization formulation of transfer operator theoretic framework. The proposed technique is applied to a number of different examples to establish its efficiency.
Dynamical system-based linear transfer Perron-Frobenius (P-F) operator framework is developed to address analysis and design problems in the building system. In particular, the problems of fast contaminant propagation and optimal placement of sensors in uncertain operating conditions of indoor building environment are addressed. The linear nature of transfer P-F operator is exploited to develop a computationally efficient numerical scheme based on the finite dimensional approximation of P-F operator for fast propagation of contaminants. The proposed scheme is an order of magnitude faster than existing methods that rely on simulation of an advection-diffusion partial differential equation for contaminant transport. Furthermore, the system-theoretic notion of observability gramian is generalized to nonlinear flow fields using the transfer P-F operator. This developed notion of observability gramian for nonlinear flow field combined with the finite dimensional approximation of P-F operator is used to provide a systematic procedure for optimal placement of sensors under uncertain operating conditions. Simulation results are presented to demonstrate the applicability of the developed framework on the IEA-annex 2D benchmark problem.
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 analyze the performance of the primal-dual gradient dynamics algorithm in the presence of stochastic communication channel uncertainty. In contrast to the existing results on the analysis of discretized primal-dual gradient dynamics with communication channel uncertainty, the main contribution of this work is in analyzing the stochastic continuous-time primal-dual gradient dynamics. Primal-dual gradient dynamics for distributed optimization are naturally modeled as a continuous-time dynamical system and analysis of this dynamics help us understand fundamental limitations and trade-offs between the cost function, network topology, and channel uncertainty for distributed optimization. We analyze the mean square stochastic stability of primal-dual gradient dynamics with communication channel uncertainty. The network topology is said to be more robust for distributed optimization if it can tolerate maximum variance of communication uncertainty. One of the important results of this paper is to show the existence of an optimal number of neighbors individual agent should have for robust distributed optimization. The optimal number is a function of network topology and cost function. If the network has more or less number of neighbors than the optimal number, then the network performance degrades. Simulation results involving nearest network topology are presented to verify the main conclusion of this paper.
In this paper, we present a novel approach to identify the generators and states responsible for the small-signal stability of power networks. To this end, the newly developed notion of information transfer between the states of a dynamical system is used. In particular, using the concept of information transfer, which characterizes influence between the various states and a linear combination of states of a dynamical system, we identify the generators and states which are responsible for causing instability of the power network. While characterizing influence from state to state, information transfer can also describe influence from state to modes thereby generalizing the well-known notion of participation factor while at the same time overcoming some of the limitations of the participation factor. The developed framework is applied to study the three bus system identifying various cause of instabilities in the system. The simulation study is extended to IEEE 39 bus system.
This paper investigates utilizing a heterogeneous group of thermostatically controlled loads (TCLs) for long-term demand response applications. The steady-state services are achieved through manipulating the stored thermal-energy with minimal impact on devices' switching rates and the operating duty-cycles. The Markov chain abstraction method has been developed in literature for aggregating the TCLs at fixed temperature set-point. In this paper, an extended Markov model (EMM) is proposed to account for the dynamics involved in modifying various set-point magnitudes in both directions. The EMM is formulated online based on linear mapping and fast restructuring to Markov chains developed offline at fixed set-points, where a training process is used to construct each Markov chain. Set-point adjustments force devices to operate in a synchronized pattern, causing the aggregated power to oscillate or traverse extreme conditions. Therefore, model predictive control with direct ON/OFF switching capability is proposed to apply the set-point change sequentially and control devices' movement toward the new operating set-point. The performance of the proposed modeling and control techniques are compared against existing methods which rely on the direct ON/OFF control solely rather than adjusting the thermal-energy level.
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 analyze the performance of the primal-dual gradient dynamics algorithm in the presence of stochastic communication channel uncertainty. In contrast to the existing results on the analysis of discretized primal-dual gradient dynamics with communication channel uncertainty, the main contribution of this work is in analyzing the stochastic continuous-time primal-dual gradient dynamics. Primal-dual gradient dynamics for distributed optimization are naturally modeled as a continuous-time dynamical system and analysis of this dynamics help us understand fundamental limitations and trade-offs between the cost function, network topology, and channel uncertainty for distributed optimization. We analyze the mean square stochastic stability of primal-dual gradient dynamics with communication channel uncertainty. The network topology is said to be more robust for distributed optimization if it can tolerate maximum variance of communication uncertainty. One of the important results of this paper is to show the existence of an optimal number of neighbors individual agent should have for robust distributed optimization. The optimal number is a function of network topology and cost function. If the network has more or less number of neighbors than the optimal number, then the network performance degrades. Simulation results involving nearest network topology are presented to verify the main conclusion of this paper.
Volatile organic compounds, particulate matter, airborne infectious disease, and harmful chemical or biological agents are examples of gaseous and particulate contaminants affecting human health in indoor environments. Fast and accurate methods are needed for detection, predictive transport, and contaminant source identification. Markov matrices have shown promise for these applications. However, current (Lagrangian and flux based) Markov methods are limited to small time steps and steady-flow fields. We extend the application of Markov matrices by developing a methodology based on Eulerian approaches. This allows construction of Markov matrices with time steps corresponding to very large Courant numbers. We generalize this framework for steady and transient flow fields with constant and time varying contaminant sources. We illustrate this methodology using three published flow fields. The Markov methods show excellent agreement with conventional PDE methods and are up to 100 times faster than the PDE methods. These methods show promise for developing real-time evacuation and containment strategies, demand response control and estimation of contaminant fields of potential harmful particulate or gaseous contaminants in the indoor environment.
We introduce navigation measure as a new tool to solve the motion planning problem in the presence of static obstacles. Existence of navigation measure guarantees collision-free convergence at the final destination set beginning with almost every initial condition with respect to the Lebesgue measure. Navigation measure can be viewed as a dual to the navigation function. While the navigation function has its minimum at the final destination set and peaks at the obstacle set, navigation measure takes the maximum value at the destination set and is zero at the obstacle set. A linear programming formalism is proposed for the construction of navigation measure. Set-oriented numerical methods are utilised to obtain finite dimensional approximation of this navigation measure. Application of the proposed navigation measure-based theoretical and computational framework is demonstrated for a motion planning problem in a complex fluid flow.
Sensors are an integral part of buildings to sense pollutants, identifying extreme events and for maintaining comfort. Traditionally, identification of the sensor locations is via solution of inverse problem (computationally intensive) or using some standard thumb rules which might not be suitable for every building. In recent years,the concepts of non-linear control theory is integrated with fluid dynamics to develop Perron-Frobenious(PF) operator based approaches to design sensor placement strategies. This approach alleviates the shortcomings of the previous approaches. The current paper extends the PF framework to account for uncertainty of airflow in the building to compute more robust sensor location maps. The developed approach is demonstrated for a IEA-2D benchmark problem. The algorithm is easily extensible to the complex buildings and can account for various uncertainties.
Stability analysis of power system is a problem of immense importance in power community. Identification of the cause for instability is a relevant problem and has been studied widely. In this work we provide a novel approach, using the concept of information transfer in a dynamical system, to identify the states and the generators which are most responsible for instability in a given power network. Our developed notion of information transfer is physically motivated and has been previously shown to capture the true notion of causality and influence. In this paper, we use information transfer measure to characterize causal interactions and influence in a power network. In particular, we identify the dynamic states of the generators (and the generator) which are most responsible for the instability. We further determine the states which contribute most to system oscillations and these findings reflect the physical intuitions of power network. Moreover, the analysis of information transfer from generators to load identifies which generators are most responsible for load fluctuations.
In this paper, we demonstrate the fragility of decentralized load-side frequency algorithm proposed in [1] against stochastic parametric uncertainty in power network model. The stochastic parametric uncertainty is motivated through the presence of renewable energy resources in power system model. We show that relatively small variance value of the parametric uncertainty affecting the system bus voltages cause the decentralized load-side frequency regulation algorithm to become stochastically unstable. The critical variance value of the stochastic bus voltages above which the decentralized control algorithm become mean square unstable is computed using an analytical framework developed in [2], [3]. Furthermore, the critical variance value is shown to decrease with the increase in the cost of the controllable loads and with the increase in penetration of renewable energy resources. Finally, simulation results on IEEE 68 bus system are presented to verify the main findings of the paper.
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.
Classification is at the very center of the supervised learning. In this work, we propose a novel algorithm to classify the test data set with the aid of a vector field, emanating from the training data set. In particular, the vector field is constructed such that the location of each training data point becomes a local minimum of the potential. The test data points are allowed to evolve under the influence of the velocity field, generated by the training data set, and thereby would be converging to the domain of attractions of different classes. The proposed approach avoids explicit computation of the separating hyper-plane like Support Vector Machine, which becomes difficult, if the structure of the separating hyper-plane is nonlinear. The proposed method is specially suited for online learning problems, as the model training does not involve any additional time. Comparative simulation studies are presented over data sets coming from three practical Machine Learning benchmark problems.
In this paper, we show through examples, how the existing definitions of information transfer, namely directed information and transfer entropy fail to capture true causal interaction between states in control dynamical system. Furthermore, existing definitions are shown to be too weak to have any implication on two of the most fundamental concepts in system theory, namely controllability and observability. We propose a new definition of information transfer, based on the ideas from dynamical system theory, and show that this new definition can not only capture true causal interaction between states, but also have implication on system controllability and observability properties. In particular, we show that non-zero transfer of information from input-to-state and state-to-output implies structural controllability and observability properties of the control dynamical system respectively. Analytical expression for information transfer between state-to-state, input-to-state, state-to-output, and input-to-output are provided for linear system. There is a natural extension of our proposed definition to define information transfer over n time steps and average information transfer over infinite time step. We show that the average information transfer in feedback control system between plant output and input is equal to the entropy of the open loop dynamics thereby re-deriving the Bode fundamental limitation results using the proposed definition of transfer.
This paper, for the first time, explores the use of data-driven method to model the traffic dynamics of an interstate highway system with high resolution probe data. The dynamic mode decomposition and spatio-temporal pattern network were used to analyze traffic patterns on a 290 mile interstate highway across Iowa. The results show the data-driven methods can effectively detect the changes in traffic system dynamics during different time of day. Traffic dynamics during morning peak hours, evening peak hours and off-peak were very different on the studied road. In contrast, the trends over multiple months were similar during the same time periods. The study also found that inclement weather had a significant impact on the system dynamics. In future, the proposed methodology can be used to gain insights in the system dynamics of a traffic network. These models will be instrumental in optimal traffic control, traffic sensor placement and other policy decisions affecting the capacity of the network.
Most people spend approximately 90% of their lives indoors. Thus, designing effective ventilation systems is essential to mitigating problems with indoor air quality. The measures of mechanical ventilation design performance considered in this study are age of air, air residual life time, air residence time, and ventilation effectiveness. This paper presents two different methods to help quantify these measures. The first method is based on transport equations, where a continuous representation of these quantities are calculated. The second method is based on Markov matrices, where a discrete representation of these quantities are calculated. We show 1) how both the continuous and discrete methods are related and 2) that the age of air and residual life time are adjoints. A new transport equation for residual life time along with methods for these quantities using Markov matrices are established. The two approaches are validated and compared using previously established experimental data. The results show that both approaches provide similar results. Using these techniques allows for the quantities of residual life time and residence time to be integrated into the design processes. This paper provides a simple framework that enables designers to get a comprehensive picture of the ventilation systems they design.
In this paper, we investigate the problem of conformation change in a network of coupled oscillator system with double well internal potential. Conformation change refers to the phenomena where all the oscillators in the network make a transition from one potential well to another potential well under the influence of external perturbation. We propose a novel approach based on information transfer in network dynamical systems to understand this phenomenon. We consider a heterogeneous network system where the internal dynamics of the oscillators are assumed to be nonidentical and the interconnecting Laplacian can also be asymmetric. The objective is to determine which of the network oscillator is most influential in driving the conformation dynamics. We show that the net information transfer of individual oscillators can be used to determine the most influential oscillator which can drive the entire network from one well to another well of the potential. We use three different network topologies to verify our proposed framework.
The synchronization of nonlinear systems connected over large-scale networks has gained popularity in a variety of applications, such as power grids, sensor networks and biology. Stochastic uncertainty in the interconnections is a ubiquitous phenomenon observed in these physical and biological networks. We provide a size-independent network sufficient condition for the synchronization of scalar nonlinear systems with stochastic linear interactions over large-scale networks. This sufficient condition, expressed in terms of nonlinear dynamics, the Laplacian eigenvalues of the nominal interconnections and the variance and location of the stochastic uncertainty, allows us to define a synchronization margin. We provide an analytical characterization of important trade-offs between the internal nonlinear dynamics, network topology and uncertainty in synchronization. For nearest neighbour networks, the existence of an optimal number of neighbours with a maximum synchronization margin is demonstrated. An analytical formula for the optimal gain that produces the maximum synchronization margin allows us to compare the synchronization properties of various complex network topologies.
Air quality has been an important issue in public health for many years. Sensing the level and distributions of impurities help in the control of building systems and mitigate long term health risks. Rapid detection of infectious diseases in large public areas like airports and train stations may help limit exposure and aid in reducing the spread of the disease. Complete coverage by sensors to account for any release scenario of chemical or biological warfare agents may provide the opportunity to develop isolation and evacuation plans that mitigate the impact of the attack. All these scenarios involve strategic placement of sensors to promptly detect and rapidly respond. This paper presents a data driven sensor placement algorithm based on a dynamical systems approach. The approach utilizes the finite dimensional Perron-Frobenius (PF) concept. The PF operator (or the Markov matrix) is used to construct an observability gramian that naturally incorporates sensor accuracy, location constraints, and sensing constraints. The algorithm determines the response times, sensor coverage maps, and the number of sensors needed. The utility of the procedure is illustrated using four examples: a literature example of the flow field inside an aircraft cabin and three air flow fields in different geometries. The effect of the constraints on the response times for different sensor placement scenarios is investigated. Knowledge of the response time and coverage of the multiple sensors aides in the design of mechanical systems and response mechanisms. The methodology provides a simple process for place sensors in a building, analyze the sensor coverage maps and response time necessary during extreme events, as well as evaluate indoor air quality. The theory established in this paper also allows for future work in topics related to construction of classical estimator problems for the sensors, real-time contaminant transport, and development of agent dispersion, contaminant isolation/removal, and evacuation strategies.
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 work, we develop a data-driven method for the diagnosis of damage in mesoscale mechanical structures using an array of distributed sensor networks. The proposed approach relies on comparing intrinsic geometries of data sets corresponding to the undamaged and damaged states of the system. We use a spectral diffusion map approach to identify the intrinsic geometry of the data set. In particular, time series data from distributed sensors is used for the construction of diffusion maps. The low dimensional embedding of the data set corresponding to different damage levels is obtained using a singular value decomposition of the diffusion map. We construct appropriate metrics in the diffusion space to compare the different data sets corresponding to different damage cases. The developed algorithm is applied for damage diagnosis of wind turbine blades. To achieve this goal, we developed a detailed finite element-based model of CX-100 blade in ANSYS using shell elements. Typical damage, such as crack or delamination, will lead to a loss of stiffness, is modeled by altering the stiffness of the laminate layer. One of the main challenges in the development of health monitoring algorithms is the ability to use sensor data with a relatively small signal-to-noise ratio. Our developed diffusion map-based algorithm is shown to be robust to the presence of sensor noise. The proposed diffusion map-based algorithm is advantageous by enabling the comparison of data from numerous sensors of similar or different types of data through data fusion, hereby making it attractive to exploit the distributed nature of sensor arrays. This distributed nature is further exploited for the purpose of damage localization. We perform extensive numerical simulations to demonstrate that the proposed method can successfully determine the extent of damage on the wind turbine blade and also localize the damage. We also present preliminary results for the application of the developed algorithm on the experimental data. These preliminary results obtained using experimental data are promising and is a topic of our ongoing investigation.
There is increased research trend towards the use of Phasor Measurement Units (PMUs) for real-time stability monitoring and active feedback control of power system. In this paper, we address the problem of control of inter-area oscillations when the measurements from Phasor Measurement Units (PMUs) are corrupted with noise. Unlike existing results, we assume that the noise enters multiplicatively in system measurements. We provide systematic procedure based on the solution of Linear Matrix Inequalities (LMI) for the mean square stochastic stability verification of power system with measurement uncertainties. Furthermore, an optimization-based procedure is proposed for the synthesis of measurement-based feedback controller robust to wide area measurement noise. Finally, simulation results are provided to demonstrate the application of the developed framework on WSCC 9 bus system.
In this paper, we introduce novel measure based on information to define influence in a dynamical system. The objective is to determine how a particular state (or a linear combination of states) in dynamical system influence or participate in the dynamics of another state (or a linear combination of states). An important parameter in determining the influence is the definition of the influence. We propose information or entropy-based measure for influence characterization. In particular, state x is said to influence or participate in the dynamics of state y, if the evolution of state x results in change in entropy or information content of state y. This work builds on our prior work on formalism for information transfer in dynamical network [1]. We discuss the applications of the developed framework for influence characterization in power system and social network. For power system, the proposed influence measure is used for the computation of participation factor of individual generator to the inter-area oscillation mode of the power system. For social network application, we use a Twitter network for influence characterization. The influence measure is used to understand the distribution of influential nodes and for influence-based clustering of the Twitter network.
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.
Predicting the movement of contaminants in the indoor environment has applications in tracking airborne infectious disease, ventilation of gaseous contaminants, and the isolation of spaces during biological attacks. Markov matrices provide a convenient way to perform contaminant transport analysis. However, no standardized method exists for calculating these matrices. A methodology based on set theory is developed for calculating contaminant transport in real-time utilizing Markov matrices from CFD flow data (or discrete flow field data). The methodology provides a rigorous yet simple strategy for determining the number and size of the Markov states, the time step associated with the Markov matrix, and calculation of individual entries of the Markov matrix. The procedure is benchmarked against scalar transport of validated airflow fields in enclosed and ventilated spaces. The approach can be applied to any general airflow field, and is shown to calculate contaminant transport over 3000 times faster than solving the corresponding scalar transport partial differential equation. This near real-time methodology allows for the development of more robust sensing and control procedures of critical care environments (clean rooms and hospital wards), small enclosed spaces (like airplane cabins) and high traffic public areas (train stations and airports).
In this paper, we discover a novel approach for defining information transfer in a linear network dynamical system. We provide entropy based characterization of the information transfer where the information transfer from state x to state y is measured by the amount of entropy/uncertainty that is transferred from state x to y over one time step. Our proposed definition of information transfer is based on three axioms. The first axiom has to do with zero information transfer, which says that if state x is not connected (or appears) in the dynamics of y then information transfer from x → y is zero. The second axiom captures the asymmetric nature of information transfer i.e., x is not connected to the dynamics of y but y is connected to the dynamics of x then information transfer from x → y is zero but the transfer from y → x is not zero. The third axiom is on information conservation. Information conservation axiom says that if y space can be split into two subspace, y1 and y2, then the information transfer from x → y will be equal to the sum of the information transfers from x → y1 and x → y2 provided y1 and y2 are “dynamical independent”. Similar conservation property also applies for the case where x is split into two parts x1 and x2 with y intact. We provide an analytical expression for information transfer satisfying these three axioms. Preliminary results are provided for identifying information-based most influential nodes and clusters in network system with small world network topology.
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 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.
In this paper, we present results for the vulnerability analysis of a power network to stochastic link failure attacks. We assume a network links are subjected to attacks where the link-based attack is modeled as stochastic perturbation to link weight. The objective is to determine which links in the network can tolerate the least amount of stochastic perturbation to maintain stochastic stability of a power network. We develop a system theoretical-based, analytical, and computation framework that allows us to rank links in the order of their critical importance to maintain stochastic stability of the network. The computational approach relies on solving a Linear Matrix Inequality (LMI). The developed framework is applied to a structure preserving model of the power network. The structure preserving model allows for the representation of original network topology, thereby identifying critical links connecting generators and load buses. Simulations are performed on IEEE 14 bus system to demonstrate the application of the developed framework.
In this paper, we study the stability problem of a stochastic, nonlinear, discrete-time system. We introduce a linear transfer operator-based Lyapunov measure as a new tool for stability verification of stochastic systems. Weaker set-theoretic notion of almost everywhere stochastic stability is introduced and verified, using Lyapunov measure-based stochastic stability theorems. Furthermore, connection between Lyapunov functions, a popular tool for stochastic stability verification, and Lyapunov measures is established. Using the duality property between the linear transfer Perron-Frobenius and Koopman operators, we show the Lyapunov measure and Lyapunov function used for the verification of stochastic stability are dual to each other. The results in this paper extend our earlier work on the use of Lyapunov measures for almost everywhere stability verification of deterministic dynamical systems [1].
In this paper, we develop data-driven method for the diagnosis of damage in mechanical structures using an array of distributed sensors. The proposed approach relies on comparing intrinsic geometry of data sets corresponding to the undamage and damage state of the system. We use spectral diffusion map approach for identifying the intrinsic geometry of the data set. In particular, time series data from distributed sensors is used for the construction of diffusion map. The low dimensional embedding of the data set corresponding to different damage level is done using singular value decomposition of the diffusion map to identify the intrinsic geometry. We construct appropriate metric in diffusion space to compare the different data set corresponding to different damage cases. The application of this approach is demonstrated for damage diagnosis of wind turbine blades. Our simulation results show that the proposed diffusion map-based metric is not only able to distinguish the damage from undamage system state, but can also determine the extent and the location of the damage.
In this letter, we present an algorithm for rotor angle stability monitoring of power system in real-time. The proposed algorithm is model-free and can make use of high resolution phasor measurement units (PMUs) data to provide reliable, timely information about the system's stability. The theoretical basis behind the proposed algorithm is adopted from dynamical systems theory. In particular, the algorithm approximately computes the system's Lyapunov exponent (LE), thereby measuring the exponential convergence or divergence rate of the rotor angle trajectories. The LE serves as a certificate of stability where the positive (negative) value of the LE implies exponential divergence (convergence) of nearby system trajectories, hence, unstable (stable) rotor angle dynamics. We also show the proposed model-free algorithm can be used for the identification of the coherent sets of generators. The simulation results are presented to verify the developed results in the paper on the modified IEEE 162-bus system.
In this paper, we introduce a novel entropy-based metric to characterize the fault-induced delayed voltage recovery (FIDVR) phenomenon. In particular, we make use of Kullback-Leibler (KL) divergence to determine both the rate and the level of voltage recovery following a fault or disturbance. The computation of the entropy-based measure relies on voltage time-series data and is independent of the underlying system model used to generate the voltage time-series. The proposed measure provides quantitative information about the degree of WECC voltage performance violation for FIDVR phenomenon. The quantitative measure for violation allows one to compare the voltage responses of different buses to various contingencies and to rank order them, based on the degree of violation.
Contingency analysis has been an integral part of power system planning and operations. Dynamic contingency analysis is often performed with offline simulation studies, due to its intense computational effort. Due to a large number of possible system variations, covering all combinations in planning studies is very challenging. Contingencies must be chosen carefully to cover a wider group of possibilities, while ensuring system security. This paper proposes a method to classify dynamic contingencies into different clusters, according to their behavioral patterns, in particular, with respect to voltage recovery patterns. The most severe contingency from each cluster becomes the representative of other contingencies in the corresponding cluster. Using the information of contingency clusters, a new concept called dynamic voltage control area (DVCA) is derived. The concept of DVCA will address the importance of the location of dynamic reactive reserves. Simulations have been completed on the modified IEEE 162-bus system to test and validate the proposed method.
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.
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.
Voltage stability plays major role in many blackout incidents. It has been recommended to develop online tools that can assess stability of the system and help in reducing the occurrence of large scale blackouts. This paper proposes a model free algorithm that can assess the short term voltage stability of the system in real time based on voltage measurements from PMU's. This approach uses simple calculations to determine the Lyapunov Exponent whose sign indicates the stability (negative) or instability (positive) of the system. This method is implemented in a real time cyber-physical test bed that is simulating the WECC 9 Bus system in RTDS that is interfaced with an SEL-421 PMU transmitting the voltage samples data over the Ethernet to OpenPDC which is interfaced to Matlab via a database. The setup is demonstrated and the algorithm is successful in detecting the stability of the system in real time.
We consider the problem of controllability degradation in dynamical networks subjected to malicious attacks. Attacks on the networks are assumed to be in the form of the removal of interconnection links. We formulate an optimization problem that seeks sets of links whose removal causes maximal degradation in the rank of the controllability gramian. We apply the alternating direction method of multipliers and sequential convex programming to find local solutions to this combinatorial optimization problem. We provide illustrative examples to demonstrate the utility of our results.
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.
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.
Numerical solutions for the optimal feedback stabilization of discrete time dynamical systems is the focus of this technical note. Set-theoretic notion of almost everywhere stability introduced by the Lyapunov measure, weaker than conventional Lyapunov function-based stabilization methods, is used for optimal stabilization. The linear Perron-Frobenius transfer operator is used to pose the optimal stabilization problem as an infinite dimensional linear program. Set-oriented numerical methods are used to obtain the finite dimensional approximation of the linear program. We provide conditions for the existence of stabilizing feedback controls and show the optimal stabilizing feedback control can be obtained as a solution of a finite dimensional linear program. The approach is demonstrated on stabilization of period two orbit in a controlled standard map.
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. Copyright © 2012 John Wiley & Sons, Ltd.
We prove a necessary and sufficient condition for the existence of Lyapunov density for a system of coupled autonomous ordinary differential equations. In particular, we characterize the kinds of couplings that preserve almost everywhere uniform stability of the origin provided the isolated systems have an almost everywhere uniformly stable equilibrium point at the origin.
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.
Heating ventilation and air conditioning (HVAC) systems in residential and commercial buildings make up 16% of the United States energy consumption. Utilizing natural ventilation strategies is a low energy solution to reduce the energy used by building environmental control systems. Design of effective natural ventilation strategies is challenging because of inherent stochasticity in interior (machine loads, number of people) and exterior conditions (wind load, outside temperature). However, by exploiting the natural dynamics of building systems, efficient design and control seems possible. We explore this idea by introducing a stochastic approach to analyze the natural dynamics of building systems (under natural ventilation) by explicitly incorporating the effects of stochastic wind speeds and stochastic internal loads. We show that complex dynamics in the form of bi-stable behavior emerges when considering a single zone building with stochastic inputs. We show that neglecting these complex stochastic dynamics leads to inaccurate predictions in the thermal response, especially for natural ventilation. We compute the sensitivity of the system with respect to various system parameters which provide insight into developing robust design guidelines. The techniques presented aid in the design process, and are a step toward adaptive, efficient, robust control of natural ventilation systems.
Cyber threats are serious concerns for power systems. This paper proposes a risk assessment framework to enhance the resilience of power systems against cyber attacks. The Duality Element Relative Fuzzy Evaluation Method (DERFEM) is used to evaluate quantitatively security vulnerabilities within cyber systems of power systems; the Attack Graph is employed to identify intrusion scenarios that exploit multiple vulnerabilities; an System Stability Monitoring and Response System (SSMARS) is developed to monitor the impact of intrusion scenarios on power system dynamics in real time. SSMARS is a novel PMU application for the Smart Grid. It is designed be implemented in a RTO/ISO control center to assess and enhance power system security on line. SSMARS calculates the Conditional Lyapunov Exponents (CLEs) in real time based on the Phasor Measurement Unit (PMU) data. Power system stability is predicted through the values of CLEs. Control actions based on CLEs are suggested if power system instability is likely to happen. The effectiveness of SSMARS is illustrated with the IEEE 39 bus system model.
In this paper, we provide a systematic convex programming-based approach for the optimal locations of static actuators and sensors for the control of nonequilibrium dynamics. The problem is motivated with regard to its application for 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 the density of oil dispersant, is governed by the 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 of the linear advection PDE. Linear transfer Perron-Frobenius and Koopman operators, associated with the advective velocity field, are used to provide analytical characterization for the controllable and observable spaces of the advection PDE. Set-oriented numerical methods are used for the finite dimensional approximation of the transfer operators and in the formulation of the optimization problem. Application of the framework is demonstrated for the optimal placement of actuators for the release of dispersant for oil spill control.
We propose a system theoretic approach to the identification and mitigation of vulnerabilities to cyber attacks, in networks of dynamical systems. Using the controllability and observability gramians, we define a network's vulnerability in terms of the impact of an attack input and the degree of difficulty with which this impact can be detected. In this framework, a network is deemed as vulnerable if it is easy for an attacker to steer it to a certain state and yet such a state is hard to observe through the network's sensing mechanisms. We propose strategies for finding the optimal location of a small number of sensors that minimize the network's vulnerability. Such strategies are obtained as the solution of convex optimization problems, formulated so as to strike a balance between maximal reduction of the system's vulnerability and employing a minimal number of sensors. The utility of the developed framework is demonstrated on a standard IEEE nine bus power system network model.
We develop a model-free approach for the short-term voltage stability monitoring of a power system. Finite time Lyapunov exponents are used as the certificate of stability. The time-series voltage data from phasor measurement units (PMU) are used to compute the Lyapunov exponent to predict voltage stability in real time. Issues related to practical implementation of the proposed method, such as phasor measurement noise, communication delay, and the finite window size for prediction, are also discussed. Furthermore, the stability certificate in the form of Lyapunov exponents is also used to determine the stability/instability contributions of the individual buses to the overall system stability and for computation of critical clearing time. Simulation results are provided for the IEEE 162-bus system to demonstrate the application of the developed method.
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 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.
In this paper, we propose a novel approach based on tools from system theory and ergodic theory of dynamical systems for the identification of critical interactions responsible for the emergence of complex dynamics in network systems. We consider a network system with multiple uncertain random parameters and operating in non-equilibrium. The objective is to determine which of these multiple parameters are critical for maintaining the stability of non-equilibrium dynamics. We provide necessary condition, expressed in terms of variance of uncertainty and nominal system dynamics, to maintain the stability of network system. The condition is used for rank ordering uncertain parameters and to provide margin of stability for network system. The proposed method is applied for the identification of parameters responsible for limit cycle oscillations in biochemical network involved in yeast cell glycolysis and for robust synchronization in network of Kuramoto oscillators with uncertainty in coupling parameters.
We study the problem of actuator and sensor placement in a linear advection partial differential equation (PDE). The problem is motivated by its application to actuator and sensor placement in building systems for the control and detection of a scalar quantity such as temperature and contaminants. We propose a gramian based approach to the problem of actuator and sensor placement. The special structure of the advection PDE is exploited to provide an explicit formula for the controllability and observability gramian in the form of a multiplication operator. The explicit formula for the gramian, as a function of actuator and sensor location, is used to provide test criteria for the suitability of a given sensor and actuator location. Furthermore, the solution obtained using gramian based criteria is interpreted in terms of the flow of the advective vector field. In particular, the almost everywhere stability property of the advective vector field is shown to play a crucial role in deciding the location of actuators and sensors. Simulation results are performed to support the main results of this paper.
We prove a KAM-type result for the persistence of two-dimensional invariant tori in perturbations of integrable action–angle–angle maps with degeneracy, satisfying the intersection property. Such degenerate action–angle–angle maps arise upon generic perturbation of three-dimensional volume-preserving vector fields, which are invariant under volume-preserving action of when there is no motion in the group action direction for the unperturbed map. This situation is analogous to degeneracy in Hamiltonian systems. The degenerate nature of the map and the unequal number of action and angle variables make the persistence proof non-standard. The persistence of the invariant tori as predicted by our result has implications for the existence of barriers to transport in three-dimensional incompressible fluid flows. Simulation results indicating existence of two-dimensional tori in a perturbation of swirling Hill’s spherical vortex flow are presented.
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 -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 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.
Dynamics of flutter is an important consideration in the design of aircraft structures. Flutter is an unstable self-excitation of the structure due to an undesirable coupling of structural elasticity and aerodynamics. Flutter is very difficult to predict and its occurrence can lead to catastrophic structural failure. The dynamics of flutter are affected by several factors including nonlinearities in structural stiffness, damping, and free-play in control surfaces. The free-play nonlinearity in control surfaces mechanisms is similar to the backlash in gears. Such nonlinearity introduces persistent limit cycle oscillations (LCO's) and significantly affects the onset of flutter. The impact of free-play on the flutter speed and frequency is not fully understood and is an active area of research. Historically, very conservative estimates have been used for the allowable free-play. The current military specification limit for free-play is based on the wind tunnel tests performed in 1950's at the Wright Air Development Center (WADC). The key contribution of this paper lies in gaining deeper understanding of free-play dynamics to enable more accurate modeling of free-play and predict its impact on flutter speed and frequency. The proposed modeling methodology is validated via close agreement of the simulation with WADC test data. Energy-based novel approach is presented for life cycle assessment and to predict flutter instability.
Rotor angle instability, if not corrected in time, may cause cascaded load and generator tripping events which will lead to a major blackout. In recent years, the Phasor Measurement Units (PMUs) have been undergoing fast deployment toward a hierarchical Wide-Area Measurement System (WAMS). Wide-area early warning systems for rotor angle instability are being developed. This paper proposes an online monitoring scheme for rotor angle stability based on PMU data. A loss of synchronism is predicted by Lyapunov Exponents. The relationship between rotor angle stability and the Maximal Lyapunov Exponent (MLE) is established. A computationally efficient algorithm for MLE calculation is developed for on-line applications. Simulation results on a 200-bus model are used to validate the effectiveness of the proposed method.
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 that exists for a weaker concept of almost everywhere stability.
In this paper, we propose a model-free approach for short term voltage stability monitoring of power system. Our data-driven approach makes use of Phasor Measurement Units (PMU) data to generate the stability certificate for verifying the voltage stability. We employ Lyapunov exponent, a stability tool adapted from ergodic theory of dynamical system, to generate the stability certificate. The time-series voltage data from PMU is used for the online computation of Lyapunov exponent. The proposed method can not only be used to determine the voltage stability of the entire system but can also be used to determine stability/instability contribution of individual buses to the overall system stability. Simulation results are presented on WSCC nine bus system using different load models.
Online monitoring of rotor angle stability in wide area power systems is an important task to avoid out-of-step instability conditions. In recent years, the installation of phasor measurement units (PMUs) on the power grids has increased significantly and, therefore, a large amount of real-time data is available for online monitoring of system dynamics. This paper proposes a PMU-based application for online monitoring of rotor angle stability. A technique based on Lyapunov exponents is used to determine if a power swing leads to system instability. The relationship between rotor angle stability and maximal Lyapunov exponent (MLE) is established. A computational algorithm is developed for the calculation of MLE in an operational environment. The effectiveness of the monitoring scheme is illustrated with a three-machine system and a 200-bus system model.
We study the problem of actuator and sensor placement in a linear advection partial differential equation (PDE). The problem is motivated by its application to actuator and sensor placement in building systems for the control and detection of a scalar quantity such as temperature and contaminants. We propose a gramian based approach to the problem of actuator and sensor placement. The special structure of the advection PDE is exploited to provide an explicit formula for the controllability and observability gramian in the form of a multiplication operator. The explicit formula for the gramian, as a function of actuator and sensor location, is used to provide test criteria for the suitability of a given sensor and actuator location. Furthermore, the solution obtained using gramian based criteria is interpreted in terms of the flow of the advective vector field. In particular, the almost everywhere uniform stability property and ergodic properties of the advective vector field are shown to play a crucial role in deciding the location of actuators and sensors. Simulation results are performed to support the main results of this paper.
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.
The problem of almost everywhere stability of a nonlinear autonomous ordinary differential equation is studied using a linear transfer operator framework. The infinitesimal generator of a linear transfer operator (Perron–Frobenius) is used to provide stability conditions of an autonomous ordinary differential equation. It is shown that almost everywhere uniform stability of a nonlinear differential equation, is equivalent to the existence of a non-negative solution for a steady state advection type linear partial differential equation. We refer to this non-negative solution, verifying almost everywhere global stability, as Lyapunov density. A numerical method using finite element techniques is used for the computation of Lyapunov density.
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.
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.
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.
This paper is concerned with computational methods for Lyapunov-based stabilization of an attractor set of a nonlinear dynamical system. Based upon a stochastic representation of deterministic dynamics, a Lyapunov measure is used for these purposes. The paper poses and solves the co-design problem of jointly obtaining a control Lyapunov measure and a state feedback controller. The computational framework employs set-oriented numerical techniques. Using these techniques, the resulting co-design problem is shown to lead to a finite number of linear inequalities. These inequalities determine the feasible set of the solutions to the co-design problem. A particular solution can be efficiently obtained using methods of linear programming.
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.