Modelling and control of coupled infinite-dimensional systems

First, we consider two classes of coupled systems consisting of an infinite-dimensional part [sigma]d and a finite-dimensional part [sigma]f connected in feedback. In the first class of coupled systems, we assume that the feedthrough matrix of [sigma]f is 0 and that [sigma]d is such that it becomes well-posed and strictly proper when connected in cascade with an integrator. Under several assumptions, we derive well-posedness, regularity and exact (or approximate) controllability results for such systems on a subspace of the natural product state space. In the second class of coupled systems, [sigma]f has an invertible first component in its feedthrough matrix while [sigma]d is well-posed and strictly proper. Under similar assumptions, we obtain well-posedness, regularity and exact (or approximate) controllability results as well as exact (or approximate) observability results for this class of coupled systems on the natural state space. Second, we investigate the exact controllability of the SCOLE (NASA Spacecraft Control Laboratory Experiment) model. Using our theory for the first class of coupled systems, we show that the uniform SCOLE model is well-posed, regular and exactly controllable in arbitrarily short time when using a certain smoother state space. Third, we investigate the suppression of the vibrations of a wind turbine tower using colocated feedback to achieve strong stability. We decompose the system into a non-uniform SCOLE model describing the vibrations in the plane of the turbine axis, and another model consisting of a non-uniform SCOLE system coupled with a two-mass drive-train model (with gearbox), in the plane of the turbine blades. We show the strong stabilizability of the first tower model by colocated static output feedback. We also prove the generic exact controllability of the second tower model on a smoother state space using our theory for the second class of coupled systems, and show its generic strong stabilizability on the energy state space by colocated feedback

Stability properties of coupled impedance passive LTI systems

We study the stability of the feedback interconnection of two impedance passive linear time-invariant systems, of which one is finite-dimensional. The closed-loop system is well known to be impedance passive, but no stability properties follow from this alone. We are interested in two main issues: (1) the strong stability of the operator semigroup associated with the closed-loop system, (2) the input-output stability (meaning transfer function in H∞) of the closed-loop system. Our results are illustrated with the system obtained from the non-uniform SCOLE (NASA Spacecraft Control Laboratory Experiment) model representing a vertical beam clamped at the bottom, with a rigid body having a large mass on top, connected with a trolley mounted on top of the rigid body, via a spring and a damper. Such an arrangement called a tuned mass damper (TMD), is used to stabilize tall buildings. We show that the SCOLE-TMD system is strongly stable on the energy state space and that the system is input-output stable from the horizontal force input to the horizontal velocity output

Null boundary controllability of a 1-dimensional heat equation with an internal point mass

We consider a linear hybrid system composed by two rods of equal length connected by a point mass. We show that the system is null controllable with Dirichlet and Neumann controls. The results are based on a careful spectral spectral analysis together with the moment method.Comment: 12 pages, typos corrected, added references, matches version to be submitted to Systems and Control Letter

Mini-Workshop: Recent Developments on Approximation Methods for Controlled Evolution Equations

This mini-workshop brought together mathematicians engaged in partial differential equations, functional analysis, numerical analysis and systems theory in order to address a number of current problems in the approximation of controlled evolution equations

Strong asymptotic stability of a compactly coupled system of wave equations

AbstractWe prove the well-posedness and study the strong asymptotic stability of a compactly coupled system of wave equations with a nonlinear feedback acting on one end only

Simultaneous exact controllability and some applications

Published versio

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

Abstract book

Welcome at the International Conference on Differential and Difference Equations & Applications 2015. The main aim of this conference is to promote, encourage, cooperate, and bring together researchers in the fields of differential and difference equations. All areas of differential & difference equations will be represented with special emphasis on applications. It will be mathematically enriching and socially exciting event. List of registered participants consists of 169 persons from 45 countries. The five-day scientific program runs from May 18 (Monday) till May 22, 2015 (Friday). It consists of invited lectures (plenary lectures and invited lectures in sections) and contributed talks in the following areas: Ordinary differential equations, Partial differential equations, Numerical methods and applications, other topics

Control of large offshore wind turbines.

Several control strategies are proposed to improve overall performances of conventional (geared equipped) and hydrostatic offshore wind turbines. Firstly, to maximise energy capture of a conventional turbine, an adaptive torque control technique is proposed through simplifying the conventional extremum seeking control algorithm. Simulations are conducted on the popular National Renewable Energy Laboratory (NREL) monopile 5-MW baseline turbine. The results demonstrate that the simplified ESC algorithms are quite effective in maximising power generation. Secondly, a TMD (tuned mass damper) system is configured to mitigate loads on a monopile turbine tower whose vibrations are typically dominated by its first mode. TMD parameters are obtained via H2 optimisation based on a spatially discretised tower-TMD model. The optimal TMDs are assessed through simulations using the NREL monopile 5-MW baseline model and achieve substantial tower load reductions. In some cases it is necessary to damp tower vibrations induced by multiple modes and it is well-known that a single TMD is lack of robustness. Thus a control strategy is developed to suppress wind turbine’s vibrations (due to multiple modes) using multiple groups of TMDs. The simulation studies demonstrate the superiority of the proposed methods over traditional ones. Thirdly, the NREL 5-MW baseline turbine model is transformed into a hydrostatic wind turbine (HWT). An H∞ loop-shaping torque controller and a light detection and ranging-based linear-parameter-varying anti-windup pitch controller are designed for the HWT. The tests on a monopile HWT model indicate good tracking behaviours of the torque controller and much improved performances of the linear-parameter-varying pitch controller over a gain-scheduled PI pitch controller. Finally, the hydraulic reservoir of a barge HWT is made into a bidirectional-tuned- liquid-column-damper (BTLCD) to suppress barge pitch and roll motions. The simulation results validate the effectiveness of the optimal BTLCD reservoir in reducing the tower loads and power fluctuations