H∞ and H2 norms of 2D mixed continuous-discrete-time systems via rationally-dependent complex Lyapunov functions

This paper addresses the problem of determining the H∞ and H2 norms of 2-D mixed continuous-discrete-time systems. The first contribution is to propose a novel approach based on the use of complex Lyapunov functions with even rational parametric dependence, which searches for upper bounds on the sought norms via linear matrix inequalities (LMIs). The second contribution is to show that the upper bounds provided are nonconservative by using Lyapunov functions in the chosen class with sufficiently large degree. The third contribution is to provide conditions for establishing the tightness of the upper bounds. The fourth contribution is to show how the numerical complexity of the proposed approach can be significantly reduced by proposing a new necessary and sufficient LMI condition for establishing positive semidefiniteness of even Hermitian matrix polynomials. This result is also exploited to derive an improved necessary and sufficient LMI condition for establishing exponential stability of 2-D mixed continuous-discrete-time systems. Some numerical examples illustrate the proposed approach. It is worth remarking that nonconservative LMI methods for determining the H∞and H2 norms of 2-D mixed continuous-discrete-time systems have not been proposed yet in the literature.postprin

Robust stability and performance analysis of 2D mixed continuous-discrete-time systems with uncertainty

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Rates of mixing in models of fluid devices with discontinuities

In the simplest sense, mixing acts on an initially heterogeneous system, transforming it to a homogeneous state through the actions of stirring and diffusion. The theory of dynamical systems has been successful in improving understanding of underlying features in fluid mixing, and how smooth stirring fields, coherent structures and boundaries affect mixing rates. The main stirring mechanism in fluids at low Reynolds number is the stretching and folding of fluid elements, although this is not the only mechanism to achieve complicated dynamics. Mixing by cutting and shuffling occurs in many situations, for example in micro–fluidic split and recombine flows, through the closing and re-orientation of values in sink–source flows, and within the bulk flow of granular material. The dynamics of this mixing mechanism are subtle and not well understood. Here, mixing rates arising from fundamental models capturing the essence of discontinuous, chaotic stirring with diffusion are investigated. In purely cutting and shuffling flows it is found that the number of cuts introduced iteratively is the most important mechanism driving the approach to uniformity. A balance between cutting, shuffling and diffusion achieves a long-time exponential mixing rate, but similar mechanisms dominate the finite time mixing observed through the interaction of many slowly decaying eigenfunctions. The time to achieve a mixed condition varies polynomially with diffusivity rate κ, obeying t ∝ κ^{−η} . For the transformations meeting good stirring criteria, η < 1. Considering the time to achieve a mixed condition to be governed by a balance between cutting, shuffling, and diffusion derives η ∼ 1/2, which shows good agreement with numerical results. In stirring fields which are predominantly chaotic and exponentially mixing, it is observed that the addition of discontinuous transformations contaminates mixing when the stretching rates are uniform, or close to uniform. The contamination comes from an increase in scales of the concentration field by the reassembly of striations when cut and shuffled. Mixing stemming from this process is unpredictable, and the discontinuities destroy the possibility to approximate early mixing rates from stretching histories. A speed up in mixing rate can be achieved if the discontinuity aids particle transport into islands of the original transformation, or chops and rearranges large striations generated from highly non-uniform stretching. The long-time mixing rates and time to achieve a mixed condition are shown to behave counter-intuitively when varying the diffusivity rate. A deceleration of mixing with increasing diffusion coefficient is observed, sometimes overshooting analytically derived bounds

Relaxing Fundamental Assumptions in Iterative Learning Control

Iterative learning control (ILC) is perhaps best decribed as an open loop feedforward control technique where the feedforward signal is learned through repetition of a single task. As the name suggests, given a dynamic system operating on a finite time horizon with the same desired trajectory, ILC aims to iteratively construct the inverse image (or its approximation) of the desired trajectory to improve transient tracking. In the literature, ILC is often interpreted as feedback control in the iteration domain due to the fact that learning controllers use information from past trials to drive the tracking error towards zero. However, despite the significant body of literature and powerful features, ILC is yet to reach widespread adoption by the control community, due to several assumptions that restrict its generality when compared to feedback control. In this dissertation, we relax some of these assumptions, mainly the fundamental invariance assumption, and move from the idea of learning through repetition to two dimensional systems, specifically repetitive processes, that appear in the modeling of engineering applications such as additive manufacturing, and sketch out future research directions for increased practicality: We develop an L1 adaptive feedback control based ILC architecture for increased robustness, fast convergence, and high performance under time varying uncertainties and disturbances. Simulation studies of the behavior of this combined L1-ILC scheme under iteration varying uncertainties lead us to the robust stability analysis of iteration varying systems, where we show that these systems are guaranteed to be stable when the ILC update laws are designed to be robust, which can be done using existing methods from the literature. As a next step to the signal space approach adopted in the analysis of iteration varying systems, we shift the focus of our work to repetitive processes, and show that the exponential stability of a nonlinear repetitive system is equivalent to that of its linearization, and consequently uniform stability of the corresponding state space matrix.PhDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/133232/1/altin_1.pd

Essays on the nonlinear and nonstochastic nature of stock market data

The nature and structure of stock-market price dynamics is an area of ongoing and rigourous scientific debate. For almost three decades, most emphasis has been given on upholding the concepts of Market Efficiency and rational investment behaviour. Such an approach has favoured the development of numerous linear and nonlinear models mainly of stochastic foundations. Advances in mathematics have shown that nonlinear deterministic processes i.e. "chaos" can produce sequences that appear random to linear statistical techniques. Till recently, investment finance has been a science based on linearity and stochasticity. Hence it is important that studies of Market Efficiency include investigations of chaotic determinism and power laws. As far as chaos is concerned, there are rather mixed or inconclusive research results, prone with controversy. This inconclusiveness is attributed to two things: the nature of stock market time series, which are highly volatile and contaminated with a substantial amount of noise of largely unknown structure, and the lack of appropriate robust statistical testing procedures. In order to overcome such difficulties, within this thesis it is shown empirically and for the first time how one can combine novel techniques from recent chaotic and signal analysis literature, under a univariate time series analysis framework. Three basic methodologies are investigated: Recurrence analysis, Surrogate Data and Wavelet transforms. Recurrence Analysis is used to reveal qualitative and quantitative evidence of nonlinearity and nonstochasticity for a number of stock markets. It is then demonstrated how Surrogate Data, under a statistical hypothesis testing framework, can be simulated to provide similar evidence. Finally, it is shown how wavelet transforms can be applied in order to reveal various salient features of the market data and provide a platform for nonparametric regression and denoising. The results indicate that without the invocation of any parametric model-based assumptions, one can easily deduce that there is more to linearity and stochastic randomness in the data. Moreover, substantial evidence of recurrent patterns and aperiodicities is discovered which can be attributed to chaotic dynamics. These results are therefore very consistent with existing research indicating some types of nonlinear dependence in financial data. Concluding, the value of this thesis lies in its contribution to the overall evidence on Market Efficiency and chaotic determinism in financial markets. The main implication here is that the theory of equilibrium pricing in financial markets may need reconsideration in order to accommodate for the structures revealed

The complex structured singular value

A tutorial introduction to the complex structured singular value (μ) is presented, with an emphasis on the mathematical aspects of μ. The μ-based methods discussed here have been useful for analysing the performance and robustness properties of linear feedback systems. Several tests for robust stability and performance with computable bounds for transfer functions and their state space realizations are compared, and a simple synthesis problem is studied. Uncertain systems are represented using Linear Fractional Transformations (LFTs) which naturally unify the frequency-domain and state space methods

Mathematical and Numerical Aspects of Dynamical System Analysis

From Preface: This is the fourteenth time when the conference “Dynamical Systems: Theory and Applications” gathers a numerous group of outstanding scientists and engineers, who deal with widely understood problems of theoretical and applied dynamics. Organization of the conference would not have been possible without a great effort of the staff of the Department of Automation, Biomechanics and Mechatronics. The patronage over the conference has been taken by the Committee of Mechanics of the Polish Academy of Sciences and Ministry of Science and Higher Education of Poland. It is a great pleasure that our invitation has been accepted by recording in the history of our conference number of people, including good colleagues and friends as well as a large group of researchers and scientists, who decided to participate in the conference for the first time. With proud and satisfaction we welcomed over 180 persons from 31 countries all over the world. They decided to share the results of their research and many years experiences in a discipline of dynamical systems by submitting many very interesting papers. This year, the DSTA Conference Proceedings were split into three volumes entitled “Dynamical Systems” with respective subtitles: Vibration, Control and Stability of Dynamical Systems; Mathematical and Numerical Aspects of Dynamical System Analysis and Engineering Dynamics and Life Sciences. Additionally, there will be also published two volumes of Springer Proceedings in Mathematics and Statistics entitled “Dynamical Systems in Theoretical Perspective” and “Dynamical Systems in Applications”

Reproducing kernel Hilbert space compactification of unitary evolution groups

A framework for coherent pattern extraction and prediction of observables of measure-preserving, ergodic dynamical systems with both atomic and continuous spectral components is developed. This framework is based on an approximation of the generator of the system by a compact operator W-tau on a reproducing kernel Hilbert space (RKHS). The operator W-tau is skew-adjoint, and thus can be represented by a projection-valued measure, discrete by compactness, with an associated orthonormal basis of eigenfunctions. These eigenfunctions are ordered in terms of a Dirichlet energy, and provide a notion of coherent observables under the dynamics akin to the Koopman eigenfunctions associated with the atomic part of the spectrum. In addition, W-tau generates a unitary evolution group {e(tW tau)}t epsilon R on the RKHS, which approximates the unitary Koopman group of the system. We establish convergence results for the spectrum and Borel functional calculus of W-tau as tau -> 0(+), as well as an associated data-driven formulation utilizing time series data. Numerical applications to ergodic systems with atomic and continuous spectra, namely a torus rotation, the Lorenz 63 system, and the Rossler system, are presented. (C) 2021 The Author(s). Published by Elsevier Inc.Peer reviewe