1,191 research outputs found

    Contraction analysis of nonlinear systems and its application

    Get PDF
    The thesis addresses various issues concerning the convergence properties of switched systems and differential algebraic equation (DAE) systems. Specifically, we focus on contraction analysis problem, as well as tackling problems related to stabilization and synchronization. We consider the contraction analysis of switched systems and DAE systems. To address this, a transformation is employed to convert the contraction analysis problem into a stabilization analysis problem. This transformation involves the introduction of virtual systems, which exhibit a strong connection with the Jacobian matrix of the vector field. Analyzing these systems poses a significant challenge due to the distinctive structure of their Jacobian matrices. Regarding the switched systems, a time-dependent switching law is established to guarantee uniform global exponential stability (UGES). As for the DAE system, we begin by embedding it into an ODE system. Subsequently, the UGES property is ensured by analyzing its matrix measure. As our first application, we utilize our approach to stabilize time-invariant switched systems and time-invariant DAE systems, respectively. This involves designing control laws to achieve system contractivity, thereby ensuring that the trajectory set encompasses the equilibrium point. In oursecond application, we propose the design of a time-varying observer by treating the system’s output as an algebraic equation of the DAE system. In our study on synchronization problems, we investigate two types of synchronization issues: the trajectory tracking of switched oscillators and the pinning state synchronization. In the case of switched oscillators, we devise a time-dependent switching law to ensure that these oscillators effectively follow the trajectory of a time-varying system. As for the pinning synchronization problem, we define solvable conditions and, building upon these conditions, we utilize contraction theory to design dynamic controllers that guarantee synchronization is achieved among the agents

    Contraction analysis of nonlinear systems and its application

    Get PDF
    The thesis addresses various issues concerning the convergence properties of switched systems and differential algebraic equation (DAE) systems. Specifically, we focus on contraction analysis problem, as well as tackling problems related to stabilization and synchronization. We consider the contraction analysis of switched systems and DAE systems. To address this, a transformation is employed to convert the contraction analysis problem into a stabilization analysis problem. This transformation involves the introduction of virtual systems, which exhibit a strong connection with the Jacobian matrix of the vector field. Analyzing these systems poses a significant challenge due to the distinctive structure of their Jacobian matrices. Regarding the switched systems, a time-dependent switching law is established to guarantee uniform global exponential stability (UGES). As for the DAE system, we begin by embedding it into an ODE system. Subsequently, the UGES property is ensured by analyzing its matrix measure. As our first application, we utilize our approach to stabilize time-invariant switched systems and time-invariant DAE systems, respectively. This involves designing control laws to achieve system contractivity, thereby ensuring that the trajectory set encompasses the equilibrium point. In oursecond application, we propose the design of a time-varying observer by treating the system’s output as an algebraic equation of the DAE system. In our study on synchronization problems, we investigate two types of synchronization issues: the trajectory tracking of switched oscillators and the pinning state synchronization. In the case of switched oscillators, we devise a time-dependent switching law to ensure that these oscillators effectively follow the trajectory of a time-varying system. As for the pinning synchronization problem, we define solvable conditions and, building upon these conditions, we utilize contraction theory to design dynamic controllers that guarantee synchronization is achieved among the agents

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

    Get PDF
    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    Learning and Control of Dynamical Systems

    Get PDF
    Despite the remarkable success of machine learning in various domains in recent years, our understanding of its fundamental limitations remains incomplete. This knowledge gap poses a grand challenge when deploying machine learning methods in critical decision-making tasks, where incorrect decisions can have catastrophic consequences. To effectively utilize these learning-based methods in such contexts, it is crucial to explicitly characterize their performance. Over the years, significant research efforts have been dedicated to learning and control of dynamical systems where the underlying dynamics are unknown or only partially known a priori, and must be inferred from collected data. However, much of these classical results have focused on asymptotic guarantees, providing limited insights into the amount of data required to achieve desired control performance while satisfying operational constraints such as safety and stability, especially in the presence of statistical noise. In this thesis, we study the statistical complexity of learning and control of unknown dynamical systems. By utilizing recent advances in statistical learning theory, high-dimensional statistics, and control theoretic tools, we aim to establish a fundamental understanding of the number of samples required to achieve desired (i) accuracy in learning the unknown dynamics, (ii) performance in the control of the underlying system, and (iii) satisfaction of the operational constraints such as safety and stability. We provide finite-sample guarantees for these objectives and propose efficient learning and control algorithms that achieve the desired performance at these statistical limits in various dynamical systems. Our investigation covers a broad range of dynamical systems, starting from fully observable linear dynamical systems to partially observable linear dynamical systems, and ultimately, nonlinear systems. We deploy our learning and control algorithms in various adaptive control tasks in real-world control systems and demonstrate their strong empirical performance along with their learning, robustness, and stability guarantees. In particular, we implement one of our proposed methods, Fourier Adaptive Learning and Control (FALCON), on an experimental aerodynamic testbed under extreme turbulent flow dynamics in a wind tunnel. The results show that FALCON achieves state-of-the-art stabilization performance and consistently outperforms conventional and other learning-based methods by at least 37%, despite using 8 times less data. The superior performance of FALCON arises from its physically and theoretically accurate modeling of the underlying nonlinear turbulent dynamics, which yields rigorous finite-sample learning and performance guarantees. These findings underscore the importance of characterizing the statistical complexity of learning and control of unknown dynamical systems.</p

    Essays on noncausal and noninvertible time series

    Get PDF
    Over the last two decades, there has been growing interest among economists in nonfundamental univariate processes, generally represented by noncausal and non-invertible time series. These processes have become increasingly popular due to their ability to capture nonlinear dynamics such as volatility clustering, asymmetric cycles, and local explosiveness - all of which are commonly observed in Macroeconomics and Finance. In particular, the incorporation of both past and future components into noncausal and noninvertible processes makes them attractive options for modeling forward-looking behavior in economic activities. However, the classical techniques used for analyzing time series models are largely limited to causal and invertible counterparts. This dissertation seeks to contribute to the field by providing theoretical tools robust to noncausal and noninvertible time series in testing and estimation. In the first chapter, "Quantile Autoregression-Based Non-causality Testing", we investigate the statistical properties of empirical conditional quantiles of non-causal processes. Specifically, we show that the quantile autoregression (QAR) estimates for non-causal processes do not remain constant across different quantiles in contrast to their causal counterparts. Furthermore, we demonstrate that non-causal autoregressive processes admit nonlinear representations for conditional quantiles given past observations. Exploiting these properties, we propose three novel testing strategies of non-causality for non-Gaussian processes within the QAR framework. The tests are constructed either by verifying the constancy of the slope coefficients or by applying a misspecification test of the linear QAR model over different quantiles of the process. Some numerical experiments are included to examine the finite sample performance of the testing strategies, where we compare different specification tests for dynamic quantiles with the Kolmogorov-Smirnov constancy test. The new methodology is applied to some time series from financial markets to investigate the presence of speculative bubbles. The extension of the approach based on the specification tests to AR processes driven by innovations with heteroskedasticity is studied through simulations. The performance of QAR estimates of non-causal processes at extreme quantiles is also explored. In the second chapter, "Estimation of Time Series Models Using the Empirical Distribution of Residuals", we introduce a novel estimation technique for general linear time series models, potentially noninvertible and noncausal, by utilizing the empirical cumulative distribution function of residuals. The proposed method relies on the generalized spectral cumulative function to characterize the pairwise dependence of residuals at all lags. Model identification can be achieved by exploiting the information in the joint distribution of residuals under the iid assumption. This method yields consistent estimates of the model parameters without imposing stringent conditions on the higher-order moments or any distributional assumptions on the innovations beyond non-Gaussianity. We investigate the asymptotic distribution of the estimates by employing a smoothed cumulative distribution function to approximate the indicator function, considering the non-differentiability of the original loss function. Efficiency improvements can be achieved by properly choosing the scaling parameter for residuals. Finite sample properties are explored through Monte Carlo simulations. An empirical application to illustrate this methodology is provided by fitting the daily trading volume of Microsoft stock by autoregressive models with noncausal representation. The flexibility of the cumulative distribution function permits the proposed method to be extended to more general dependence structures where innovations are only conditional mean or quantile independent. In the third chapter, "Directional Predictability Tests", joint with Carlos Velasco, we propose new tests of predictability for non-Gaussian sequences that may display general nonlinear dependence in higher-order properties. We test the null of martingale difference against parametric alternatives which can introduce linear or nonlinear dependence as generated by ARMA and all-pass restricted ARMA models, respectively. We also develop tests to check for linear predictability under the white noise null hypothesis parameterized by an all-pass model driven by martingale difference innovations and tests of non-linear predictability on ARMA residuals. Our Lagrange Multiplier tests are developed from a loss function based on pairwise dependence measures that identify the predictability of levels. We provide asymptotic and finite sample analysis of the properties of the new tests and investigate the predictability of different series of financial returns.This thesis has been possible thanks to the financial support from the grant BES-2017-082695 from the Ministerio de Economía Industria y Competitividad.Programa de Doctorado en Economía por la Universidad Carlos III de MadridPresidente: Miguel ángel Delgado González.- Secretario: Manuel Domínguez Toribio.- Vocal: Majid M. Al Sadoo

    Beam scanning by liquid-crystal biasing in a modified SIW structure

    Get PDF
    A fixed-frequency beam-scanning 1D antenna based on Liquid Crystals (LCs) is designed for application in 2D scanning with lateral alignment. The 2D array environment imposes full decoupling of adjacent 1D antennas, which often conflicts with the LC requirement of DC biasing: the proposed design accommodates both. The LC medium is placed inside a Substrate Integrated Waveguide (SIW) modified to work as a Groove Gap Waveguide, with radiating slots etched on the upper broad wall, that radiates as a Leaky-Wave Antenna (LWA). This allows effective application of the DC bias voltage needed for tuning the LCs. At the same time, the RF field remains laterally confined, enabling the possibility to lay several antennas in parallel and achieve 2D beam scanning. The design is validated by simulation employing the actual properties of a commercial LC medium

    Nonlinear Dynamics Analysis and Control of Space Vehicles with Flexible Structures

    Get PDF
    Space vehicles that implement hardware such as antennas, solar panels, and other extended appendages necessary for their respective missions must consider the nonlinear rotational and vibrational dynamics of these flexible structures. Formulation and analysis of these flexible structures must account for the rigid-flexible coupling present in the system dynamics for stability analysis and control design. The system model is represented by a flexible appendage attached to a central rigid body, where the flexible appendage is modeled as a cantilevered Euler-Bernoulli beam. Discretization techniques, such as the assumed modes method and the finite element method, are used to model the coupled dynamics by transforming the partial differential equations of motion into a finite set of differential equations. State feedback control laws are designed to achieve stability and desired motion in the presence of rigid-flexible coupling. An optimal control law in the form of a linear quadratic regulator is presented and compared with a Lyapunov-based control law that guarantees asymptotic stability. Conventional and adaptive sliding mode control laws are also presented to account for any uncertainties in the linearized system model. Full-order and reduced-order observers are included in the control system to account for lack of velocity state measurements that are generally unavailable in real world applications

    Robust Out-of-Distribution Detection in Deep Classifiers

    Get PDF
    Over the past decade, deep learning has gone from a fringe discipline of computer science to a major driver of innovation across a large number of industries. The deployment of such rapidly developing technology in safety-critical applications necessitates the careful study and mitigation of potential failure modes. Indeed, many deep learning models are overconfident in their predictions, are unable to flag out-of-distribution examples that are clearly unrelated to the task they were trained on and are vulnerable to adversarial vulnerabilities, where a small change in the input leads to a large change in the model’s prediction. In this dissertation, we study the relation between these issues in deep learning based vision classifiers. First, we benchmark various methods that have been proposed to enable deep learning meth- ods to detect out-of-distribution examples and we show that a classifier’s predictive confidence is well-suited for this task, if the classifier has had access to a large and diverse out-distribution at train time. We theoretically investigate how different out-of-distribution detection methods are related and show that several seemingly different approaches are actually modeling the same core quantities. In the second part we study the adversarial robustness of a classifier’s confidence on out- of-distribution data. Concretely, we show that several previous techniques for adversarial robustness can be combined to create a model that inherits each method’s strength while sig- nificantly reducing their respective drawbacks. In addition, we demonstrate that the enforce- ment of adversarially robust low confidence on out-of-distribution data enhances the inherent interpretability of the model by imbuing the classifier with certain generative properties that can be used to query the model for counterfactual explanations for its decisions. In the third part of this dissertation we will study the problem of issuing mathematically provable certificates for the adversarial robustness of a model’s confidence on out-of-distribution data. We develop two different approaches to this problem and show that they have comple- mentary strength and weaknesses. The first method is easy to train, puts no restrictions on the architecture that our classifier can use and provably ensures that the classifier will have low confidence on data very far away. However, it only provides guarantees for very specific types of adversarial perturbations and only for data that is very easy to distinguish from the in-distribution. The second approach works for more commonly studied sets of adversarial perturbations and on much more challenging out-distribution data, but puts heavy restrictions on the architecture that can be used and thus the achievable accuracy. It also does not guar- antee low confidence on asymptotically far away data. In the final chapter of this dissertation we show how ideas from both of these techniques can be combined in a way that preserves all of their strengths while inheriting none of their weaknesses. Thus, this thesis outlines how to develop high-performing classifiers that provably know when they do not know

    Learning Stable Koopman Models for Identification and Control of Dynamical Systems

    Get PDF
    Learning models of dynamical systems from data is a widely-studied problem in control theory and machine learning. One recent approach for modelling nonlinear systems considers the class of Koopman models, which embeds the nonlinear dynamics in a higher-dimensional linear subspace. Learning a Koopman embedding would allow for the analysis and control of nonlinear systems using tools from linear systems theory. Many recent methods have been proposed for data-driven learning of such Koopman embeddings, but most of these methods do not consider the stability of the Koopman model. Stability is an important and desirable property for models of dynamical systems. Unstable models tend to be non-robust to input perturbations and can produce unbounded outputs, which are both undesirable when the model is used for prediction and control. In addition, recent work has shown that stability guarantees may act as a regularizer for model fitting. As such, a natural direction would be to construct Koopman models with inherent stability guarantees. Two new classes of Koopman models are proposed that bridge the gap between Koopman-based methods and learning stable nonlinear models. The first model class is guaranteed to be stable, while the second is guaranteed to be stabilizable with an explicit stabilizing controller that renders the model stable in closed-loop. Furthermore, these models are unconstrained in their parameter sets, thereby enabling efficient optimization via gradient-based methods. Theoretical connections between the stability of Koopman models and forms of nonlinear stability such as contraction are established. To demonstrate the effect of the stability guarantees, the stable Koopman model is applied to a system identification problem, while the stabilizable model is applied to an imitation learning problem. Experimental results show empirically that the proposed models achieve better performance over prior methods without stability guarantees

    Set-based state estimation and fault diagnosis using constrained zonotopes and applications

    Full text link
    This doctoral thesis develops new methods for set-based state estimation and active fault diagnosis (AFD) of (i) nonlinear discrete-time systems, (ii) discrete-time nonlinear systems whose trajectories satisfy nonlinear equality constraints (called invariants), (iii) linear descriptor systems, and (iv) joint state and parameter estimation of nonlinear descriptor systems. Set-based estimation aims to compute tight enclosures of the possible system states in each time step subject to unknown-but-bounded uncertainties. To address this issue, the present doctoral thesis proposes new methods for efficiently propagating constrained zonotopes (CZs) through nonlinear mappings. Besides, this thesis improves the standard prediction-update framework for systems with invariants using new algorithms for refining CZs based on nonlinear constraints. In addition, this thesis introduces a new approach for set-based AFD of a class of nonlinear discrete-time systems. An affine parametrization of the reachable sets is obtained for the design of an optimal input for set-based AFD. In addition, this thesis presents new methods based on CZs for set-valued state estimation and AFD of linear descriptor systems. Linear static constraints on the state variables can be directly incorporated into CZs. Moreover, this thesis proposes a new representation for unbounded sets based on zonotopes, which allows to develop methods for state estimation and AFD also of unstable linear descriptor systems, without the knowledge of an enclosure of all the trajectories of the system. This thesis also develops a new method for set-based joint state and parameter estimation of nonlinear descriptor systems using CZs in a unified framework. Lastly, this manuscript applies the proposed set-based state estimation and AFD methods using CZs to unmanned aerial vehicles, water distribution networks, and a lithium-ion cell.Comment: My PhD Thesis from Federal University of Minas Gerais, Brazil. Most of the research work has already been published in DOIs 10.1109/CDC.2018.8618678, 10.23919/ECC.2018.8550353, 10.1016/j.automatica.2019.108614, 10.1016/j.ifacol.2020.12.2484, 10.1016/j.ifacol.2021.08.308, 10.1016/j.automatica.2021.109638, 10.1109/TCST.2021.3130534, 10.1016/j.automatica.2022.11042
    • …
    corecore