Implementation of Delayed-Feedback Controllers on Continuous Systems and Analysis of their Response under Primary Resonance Excitations

During the last three decades, a considerable amount of research has been directed toward understanding the influence of time delays on the stability and stabilization of dynamical systems. From a control perspective, these delays can either have a compounding and destabilizing effect, or can actually improve controllers\u27 performance. In the latter case, additional time delay is carefully and deliberately introduced into the feedback loop so as to augment inherent system delays and produce larger damping for smaller control efforts. While delayed-feedback algorithms have been successfully implemented on discrete dynamical systems with limited degrees of freedom, a critical issue appears in their implementation on systems consisting of a large number of degrees of freedom or on infinite-dimensional structures. The reason being that the presence of delay in the control loop renders the characteristic polynomial of the transcendental type which produces infinite number of eigenvalues for every discrete controller\u27s gain and time delay. As a result, choosing a gain-delay combination that stabilizes the lower vibration modes can easily destabilize the higher modes. To address this problem, this dissertation introduces the concept of filter-augmented delayed-feedback control algorithms and applies it to mitigate vibrations of various structural systems both theoretically and experimentally. In specific, it explores the prospect of augmenting proper filters in the feedback loop to enhance the robustness of delayed-feedback controllers allowing them to simultaneously mitigate the response of different vibration modes using a single sensor and a single gain-delay actuator combination. The dissertation goes into delineating the influence of filter\u27s dynamics (order and cut-off frequency) on the stability maps and damping contours clearly demonstrating the possibility of effectively reducing multi-modal oscillations of infinite-dimensional structures when proper filters are augmented in the feedback loop. Additionally, this research illustrates that filters may actually enhance the robustness of the controller to parameter\u27s uncertainties at the expense of reducing the controller\u27s effective damping. To assess the performance of the proposed control algorithm, the dissertation presents three experimental case studies; two of which are on structures whose dynamics can be discretized into a system of linearly-uncoupled ordinary differential equations (ODEs); and the third on a structure whose dynamics can only be reduced into a set of linearly-coupled ODEs. The first case study utilizes a filter-augmented delayed-position feedback algorithm for flexural vibration mitigation and external disturbances rejection on a macro-cantilever Euler-Bernoulli beam. The second deals with implementing a filter-augmented delayed-velocity feedback algorithm for vibration mitigation and external disturbances rejection on a micro-cantilever sensor. The third implements a filter-augmented delayed-position feedback algorithm to suppress the coupled flexural-torsional oscillations of a cantilever beam with an asymmetric tip rigid body; a problem commonly seen in the vibrations of large wind turbine blades. This research also fills an important gap in the open literature presented in the lack of studies addressing the response of delay systems to external resonant excitations; a critical issue toward implementing delayed-feedback controllers to reduce oscillations resulting from persistent harmonic excitations. To that end, this dissertation presents a modified multiple scaling approach to investigate primary resonances of a weakly-nonlinear second-order delay system with cubic nonlinearities. In contrast to previous studies where the implementation is confined to the assumption of linear feedback with small control gains; this effort proposes an approach which alleviates that assumption and permits treating a problem with arbitrarily large gains. The modified procedure lumps the delay state into unknown linear damping and stiffness terms that are function of the gain and delay. These unknown functions are determined by enforcing the linear part of the steady-state solution acquired via the Method of Multiple Scales to match that obtained directly by solving the forced linear problem. Through several examples, this research examines the validity of the modified procedure by comparing its results to solutions obtained via a Harmonic Balance approach demonstrating the ability of the proposed methodology to predict the amplitude, softening-hardening characteristics, and stability of the resulting steady-state responses

Research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, numerical analysis and computer science

This report summarizes research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, numerical analysis, and computer science during the period April l, 1988 through September 30, 1988

Measurements, Models, Systems and Design

531 s.

Robust Output Regulation of Thermal Fluid Flows

Tässä väitöskirjassa tarkastellaan fluidien säätöä matemaattisen systeemiteorian näkökulmasta tutkimalla fluidien lämpötilan ja nopeuden kehitystä kuvaavia matemaattisia malleja. Tarkastellut mallit sisältävät ainakin yhden osittaisdifferentiaaliyhtälön ja saattavat lisäksi sisältää tavallisia differentiaaliyhtälöitä. Väitöskirjassa tarkasteltuja malleja voidaan käyttää esimerkiksi mallintamaan lämmitys- vesijohto- ja ilmastointilaitteiden (LVI) toimintaa. Väitöskirjan tavoitteena on suunnitella matemaattisiin malleihin perustuvia automaattisia säätäjiä, jotka takaavat mitatun fluidin lämpötilaan tai nopeuteen liittyvän ominaisuuden käyttäytyvän halutulla tavalla. Suunniteltujen säätäjien käytännöllisyyteen kiinnitetään huomiota läpi väitöskirjan, sillä erityisesti osittaisdifferentiaaliyhtälöitä sisältävät mallit saattavat johtaa vain teoriassa toimiviin säätöratkaisuihin. Väitöskirjassa suunnitellut säätäjät perustuvat sisäisen mallin periaatteeseen ja takaavat robustin säätötavoitteen toteutumisen regulointivirheen takaisinkytkentää hyödyntäen. Tarkastellut systeemit voivat olla joko lineaarisia tai epälineaarisia, ja kulloinkin käytetyn säätäjän suorituskykyä havainnollistetaan numeeristen simulaatioiden avulla. Regulointivirheen takaisinkytkentään perustuvat säätäjät muodostavat säätösignaalin fluidista suoritettavien lämpötila- tai nopeusmittausten perusteella ja takaavat mitatun suureen suppenevan halutulle sinimuotoiselle radalle asymptoottisesti. Robustisuuden ansiosta säätöratkaisu toimii pienistä systeemimallin virheistä tai sinimuotoisista häiriösignaaleista huolimatta. Väitöskirjan merkittävin kontribuutio on esitettyjen ulostulosäätöön käytettävien säätäjien perustuminen regulointivirheen takaisinkytkentään. Aiemmat väitöskirjassa tarkastelluille fluidimalleille esitetyt säätöratkaisut perustuvat joko tilatakaisinkytkentään tai tarkastelevat stabilointia. Väitöskirjan säätäjien edut näihin säätäjiin nähden ovat regulointivirheen takaisinkytkennän käyttö ja saavutetun ulostulosäädön robustisuus.In this thesis, we consider control of fluids from the perspective of mathematical systems theory by studying mathematical models which describe evolution of velocity and temperature of fluids. The models consist of at least one partial differential equation and in some cases also include ordinary differential equations and can be used to describe temperature and velocity properties related to for example heating, ventilation and air conditioning (HVAC). Our goal is to design automatic controllers that are based on properties of the fluid models and ensure that, given time, certain measured temperature or velocity quantities of the model behave as desired. Throughout this thesis we focus on practical implementability of the proposed controller designs, since that cannot be taken as granted especially for models including partial differential equations. We design error feedback controllers, based on the so-called internal model principle, for robust output regulation of linear and nonlinear thermal fluid flow models and illustrate the controllers’ performance using numerical simulations. The error feedback controllers operate based on measurements of fluid temperature or velocity at some parts of the spatial domain, and robustness means that the controllers reject disturbances and tolerate model uncertainties in addition to forcing the measured quantity to a desired sinusoidal trajectory given time. The main contribution of this thesis comes from our focus on robust output regulation using error feedback controllers. For the considered thermal fluid flow models, the existing control solutions focus on the problem of stabilization or use state feedback controllers. That is, the controllers of this thesis have the advantages of error feedback compared to state feedback and robustness of the achieved output regulation

Robust Output Regulation of Euler-Bernoulli Beam Models

In this thesis, we consider control and dynamical behaviour of flexible beam models which have potential applications in robotic arms, satellite panel arrays and wind turbine blades. We study mathematical models that include flexible beams described by Euler-Bernoulli beam equations. These models consist of partial differential equations or combination of partial and ordinary differential equations depending on the loads and supports in the model. Our goal is to influence the models by control inputs such as external applied forces so that measured deflection profiles of the beams in the models behave as desired. We propose dynamic controllers for the output regulation, where the measurements from the models track desired reference signals in the given time, of flexible beam models. The controller designs are based on the so-called internal model principle and they utilize difference between measurement and desired reference trajectory. Moreover, the controllers are robust in the sense that they can achieve output regulation despite external disturbances and model uncertainties. We also study the output regulation problem when there are certain limitations on the control input. In particular, we generalize the theory of output regulation for dynamical systems described by ordinary differential equations subject to input constraints to a particular class of systems described by partial differential equations. We present set of solvability conditions and a linear output feedback controller for the output regulation

Nonlinear Adaptive Control of Drilling Processes

This work deals with the modeling and control of automated drilling operations. Advances in drilling automation are of substantial importance because improvements in drilling control algorithms will result in more efficient drilling, which is beneficial from both economic and environmental points of view. While the primary application of the results is extraction of natural resources, potentially there exists a wide range of applications, including offshore exploration, archaeological research, and automated extraterrestrial mining, where implementation of new methods and control algorithms for drilling processes can bring substantial benefits. The main contribution of the thesis is development of new methods and algorithms for control of drilling processes in industrial drilling systems, ensuring stability and high performance characteristics. The problems of regulation of vertical penetration rate and drilling power in rotary drilling systems are solved; as a result, stability and vibration mitigation is ensured. A number of challenges is addressed, such as complexity and nonlinearity of the drilling model, lack of information about environment and parameters of the drilling system itself, and poor communication between downhole sensors and ground-level equipment. Several cases are considered, depending on the amount of information that is available in advance or in real time. Two mathematical models of the drilling system are investigated: one is finite-dimensional, and another is a distributed parameter model. Several solutions are proposed for both of them, using methods of adaptive, robust, and sliding mode control, and comparisons are made. Feasibility and efficiency of the proposed control algorithms are confirmed by simulations in MATLAB/Simulink

Proactive Quality Control based on Ensemble Forecast Sensitivity to Observations

Despite recent major improvements in numerical weather prediction (NWP) systems, operational NWP forecasts occasionally suffer from an abrupt drop in forecast skill, a phenomenon called "forecast skill dropout." Recent studies have shown that the "dropouts" occur not because of the model's deficiencies but by the use of flawed observations that the operational quality control (QC) system failed to filter out. Thus, to minimize the occurrences of forecast skill dropouts, we need to detect and remove such flawed observations. A diagnostic technique called Ensemble Forecast Sensitivity to Observations (EFSO) enables us to quantify how much each observation has improved or degraded the forecast. A recent study (Ota et al., 2013) has shown that it is possible to detect flawed observations that caused regional forecast skill dropouts by using EFSO with 24-hour lead-time and that the forecast can be improved by not assimilating the detected observations. Inspired by their success, in the first part of this study, we propose a new QC method, which we call Proactive QC (PQC), in which flawed observations are detected 6 hours after the analysis by EFSO and then the analysis and forecast are repeated without using the detected observations. This new QC technique is implemented and tested on a lower-resolution version of NCEP's operational global NWP system. The results we obtained are extremely promising; we have found that we can detect regional forecast skill dropouts and the flawed observations after only 6 hours from the analysis and that the rejection of the identified flawed observations indeed improves 24-hour forecasts. In the second part, we show that the same approximation used in the derivation of EFSO can be used to formulate the forecast sensitivity to observation error covariance matrix R, which we call EFSR. We implement the EFSR diagnostics in both an idealized system and the quasi-operational NWP system and show that it can be used to tune the R matrix so that the utility of observations is improved. We also point out that EFSO and EFSR can be used for the optimal assimilation of new observing systems

Towards a solution of the closure problem for convective atmospheric boundary-layer turbulence

We consider the closure problem for turbulence in the dry convective atmospheric boundary layer (CBL). Transport in the CBL is carried by small scale eddies near the surface and large plumes in the well mixed middle part up to the inversion that separates the CBL from the stably stratified air above. An analytically tractable model based on a multivariate Delta-PDF approach is developed. It is an extension of the model of Gryanik and Hartmann [1] (GH02) that additionally includes a term for background turbulence. Thus an exact solution is derived and all higher order moments (HOMs) are explained by second order moments, correlation coefficients and the skewness. The solution provides a proof of the extended universality hypothesis of GH02 which is the refinement of the Millionshchikov hypothesis (quasi- normality of FOM). This refined hypothesis states that CBL turbulence can be considered as result of a linear interpolation between the Gaussian and the very skewed turbulence regimes. Although the extended universality hypothesis was confirmed by results of field measurements, LES and DNS simulations (see e.g. [2-4]), several questions remained unexplained. These are now answered by the new model including the reasons of the universality of the functional form of the HOMs, the significant scatter of the values of the coefficients and the source of the magic of the linear interpolation. Finally, the closures 61 predicted by the model are tested against measurements and LES data. Some of the other issues of CBL turbulence, e.g. familiar kurtosis-skewness relationships and relation of area coverage parameters of plumes (so called filling factors) with HOM will be discussed also