Robust Output Tracking for a Room Temperature Model with Distributed Control and Observation

We consider robust output regulation of a partial differential equation model describing temperature evolution in a room. More precisely, we examine a two-dimensional room model with the velocity field and temperature evolution governed by the incompressible steady state Navier-Stokes and advection-diffusion equations, respectively, which coupled together form a simplification of the Boussinesq equations. We assume that the control and observation operators of our system are distributed, whereas the disturbance acts on a part of the boundary of the system. We solve the robust output regulation problem using a finite-dimensional low-order controller, which is constructed using model reduction on a finite element approximation of the model. Through numerical simulations, we compare performance of the reduced-order controller to that of the controller without model reduction as well as to performance of a low-gain robust controller.Comment: 12 pages, 5 figures. Accepted for publication in the Proceedings of the 24th International Symposium on Mathematical Theory of Networks and Systems, 23-27 August, 202

Robust Output Regulation of the Linearized Boussinesq Equations with Boundary Control and Observation

We study temperature and velocity output tracking problem for a two-dimensional room model with the fluid dynamics governed by the linearized translated Boussinesq equations. Additionally, the room model includes finite-dimensional models for actuation and sensing dynamics, thus the complete model dynamics are governed by an ODE-PDE-ODE system. As the main result, we design a low-dimensional internal model based controller for robust output racking of the room model. Efficiency of the controller is demonstrated through a numerical example of velocity and temperature tracking.Comment: 26 pages, 9 figures, submitte

Boundary Stabilization of the Navier--Stokes Equations in the Case of Mixed Boundary Conditions

International audienc

Application of non-linear system identification approaches to modelling, analysis, and control of fluid flows.

Flow control has become a topic of great importance for several applications, ranging from commercial aircraft, to intercontinental pipes and skyscrapers. In these applications, and many more, the interaction with a fluid flow can have a significant influence on the performance of the system. In many cases the fluids encountered are turbulent and detrimental to the latter. Several attempts have been made to solve this problem. However, due to the non-linearity and infinite dimensionality of fluid flows and their governing equations, a complete understanding of turbulent behaviour and a feasible control approach has not been obtained. In this thesis, model reduction approaches that exploit non-linear system identification are applied using data obtained from numerical simulations of turbulent three-dimensional channel flow, and two-dimensional flow over the backward facing step. A multiple-input multiple-output model, consisting of 27 sub-structures, is obtained for the fluctuations of the velocity components of the channel flow. A single-input single-output model for fluctuations of the pressure coefficient, and two multiple-input single-output models for fluctuations of the velocity magnitude are obtained in flow over the BFS. A non-linear model predictive control strategy is designed using identified one- and multi-step ahead predictors, with the inclusion of integral action for robustness. The proposed control approach incorporates a non-linear model without the need for expensive non-linear optimizations. Finally, a frequency domain analysis of unmanipulated turbulent flow is perfumed using five systems. Higher order generalized frequency response functions (GFRF) are computed to study the non-linear energy transfer phenomena. A more detailed investigation is performed using the output FRF (OFRF), which can elucidate the contribution of the n-th order frequency response to the output frequency response

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