Gils, van, R.W.

Nijmeijer, H.

Speetjens, M.F.M.

English

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 Observer design for a nonlinear two-dimensional pool boilingsystemCitation for published version (APA):Gils, van, R. W., Speetjens, M. F. M., & Nijmeijer, H. (2011). Observer design for a nonlinear two-dimensionalpool boiling system. In C. Ionescu, R. De Keyser, & P. Guillaume (Eds.), Proceedings of the 30th BeneluxMeeting on Systems and Control, 15 - 17 March 2011, Lommel, Belgium (pp. 166-). Universiteit Gent.Document status and date:Published: 01/01/2011Document Version:Accepted manuscript including changes made at the peer-review stagePlease check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publicationGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverneTake down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.Download date: 08. Jul. 2024Observer design for a nonlinear two-dimensional pool boiling systemR.W. van Gils∗,1, M.F.M. Speetjens2 and H. Nijmeijer1Eindhoven University of Technology, PO Box 513, 5600 MB EindhovenDepartment of Mechanical Engineering,1Dynamics and Control,2Energy TechnologyEmail: ∗r.w.v.gils@tue.nl1 IntroductionIn pool-boiling systems heat is extracted from a heater bya pool of boiling liquid. The heat flux between heater andfluid is highly nonlinear with respect to heater temperature,described byqF(TF) (Figure 1) and results in a highly unsta-ble regime which must be stabilised to allow for high heatremoval rates. This can be accomplished by a control lawbased on the (nonmeasurable) spectral modes of the heaterstemperature field [1]. Application of this control law re-quires an observer for the temperature profile of the heater.qF (TF )TFFigure 1 Nonlinear heat fluxqF as function of interface temperatureTF .2 Pool boiling model descriptionThe heat transfer in the 2D rectangular nondimensionalheaterH := {(x,y) ∈ [0,1]× [0,D]}, see Figure 2, is con-sidered. Its temperature fieldT (x,y,t) is described by∂T∂ t(x,y,t) = κ∇2T (x,y,t). (1)The boundary conditions comprise adiabatic, i.e. perfectlyisolated, sidewalls onΓA := {(x,y)|x = 0,1}, a controlledheat supply onΓH := {(x,y)|y = 0} and the nonlinear heatextraction onΓF := {(x,y)|y = D}, i.e.∂T∂x∣∣∣∣ΓA= 0,∂T∂y∣∣∣∣ΓH= −1+ u(t)Λ,∂T∂y∣∣∣∣ΓF= −Π2qF (TF)Λ.(2)HereTF(x) := T (x,D) is the fluid-heater interface tempera-ture,Λ, D, Π2 andκ are positive parameters,qF(TF) is theboiling curve andu(t) is the input, i.e. an additional heatsupply at the bottom of the heater, see [1].yDx10∂T∂t= κ∇2Tboiling liquid: Λ∂T∂y= −Π2qF (TF )heat supply: −Λ∂T∂y= 1 + u(t)∂T∂x= 0∂T∂x= 0Figure 2 Two-dimensional rectangular heater.3 Observer designThe temperature can be measured atR + 1 points on theheater surface, given bỹxr = r/R, r = 0, . . . ,R, i.e. the sys-tem output equalsyr = TF(x̃r) for r = 0, . . . ,R. The observeris designed to be a copy of the system with output injectiononly on the boundary condition where the measurements areavailable. The obtained observer is given by (1) and (2) withthe stateZ(x,y,t), i.e. the estimate ofT (x,y,t), instead ofT (x,y,t) and with the boundary condition onΓF as∂Z∂y∣∣∣∣ΓF= −Π2qF(ZF )Λ+R∑r=0pr(x)(TF(x̃r)−ZF(x̃r)) , (3)with pr(x) the observer gain functions. The observer er-ror dynamics ofE = T − Z are analysed by linearisationof system and observer, with which local stability of thenonlinear error dynamics can be obtained. Figure 3 showsthe evolution of the output, forR = 2, meaning the outputequalsT (0,D,t), T (0.5,D,t) andT (1,D,t). The initial sys-tem state equals a uniform field ofT ≈ 4, whereas the initialobserver state equals zero. Although the initial differencesbetween system state, observer state and non-uniform equi-librium are large, the observer states converge to the systemstates and the equilibrium is stabilised.Figure 3 Evolution of the system output and the observer output.4 ConclusionAn observer for a nonlinear PDE system is designed. Theclosed-loop system is stabilised by the control law discussedin [1] and this observer, even for large initial perturbations.References[1] R. van Gils, et al. Feedback stabilisation of a two-dimensional pool-boiling system by modal control.Automatica,submitted for publication, 2010

Observer design for a nonlinear two-dimensional pool boiling system

 Observer design for a nonlinear two-dimensional pool boilingsystemCitation for published version (APA):Gils, van, R. W., Speetjens, M. F. M., & Nijmeijer, H. (2011). Observer design for a nonlinear two-dimensionalpool boiling system. In C. Ionescu, R. De Keyser, & P. Guillaume (Eds.), Proceedings of the 30th BeneluxMeeting on Systems and Control, 15 - 17 March 2011, Lommel, Belgium (pp. 166-). Universiteit Gent.Document status and date:Published: 01/01/2011Document Version:Accepted manuscript including changes made at the peer-review stagePlease check the document version of this publication:• A submitted manuscript is the version of the article upon submission and before peer-review. There can beimportant differences between the submitted version and the official published version of record. Peopleinterested in the research are advised to contact the author for the final version of the publication, or visit theDOI to the publisher's website.• The final author version and the galley proof are versions of the publication after peer review.• The final published version features the final layout of the paper including the volume, issue and pagenumbers.Link to publicationGeneral rightsCopyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright ownersand it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.            • Users may download and print one copy of any publication from the public portal for the purpose of private study or research.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying the publication in the public portal.If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, pleasefollow below link for the End User Agreement:www.tue.nl/taverneTake down policyIf you believe that this document breaches copyright please contact us at:openaccess@tue.nlproviding details and we will investigate your claim.Download date: 16. Nov. 2023Observer design for a nonlinear two-dimensional pool boiling systemR.W. van Gils∗,1, M.F.M. Speetjens2 and H. Nijmeijer1Eindhoven University of Technology, PO Box 513, 5600 MB EindhovenDepartment of Mechanical Engineering,1Dynamics and Control,2Energy TechnologyEmail: ∗r.w.v.gils@tue.nl1 IntroductionIn pool-boiling systems heat is extracted from a heater bya pool of boiling liquid. The heat flux between heater andfluid is highly nonlinear with respect to heater temperature,described byqF(TF) (Figure 1) and results in a highly unsta-ble regime which must be stabilised to allow for high heatremoval rates. This can be accomplished by a control lawbased on the (nonmeasurable) spectral modes of the heaterstemperature field [1]. Application of this control law re-quires an observer for the temperature profile of the heater.qF (TF )TFFigure 1 Nonlinear heat fluxqF as function of interface temperatureTF .2 Pool boiling model descriptionThe heat transfer in the 2D rectangular nondimensionalheaterH := {(x,y) ∈ [0,1]× [0,D]}, see Figure 2, is con-sidered. Its temperature fieldT (x,y,t) is described by∂T∂ t(x,y,t) = κ∇2T (x,y,t). (1)The boundary conditions comprise adiabatic, i.e. perfectlyisolated, sidewalls onΓA := {(x,y)|x = 0,1}, a controlledheat supply onΓH := {(x,y)|y = 0} and the nonlinear heatextraction onΓF := {(x,y)|y = D}, i.e.∂T∂x∣∣∣∣ΓA= 0,∂T∂y∣∣∣∣ΓH= −1+ u(t)Λ,∂T∂y∣∣∣∣ΓF= −Π2qF (TF)Λ.(2)HereTF(x) := T (x,D) is the fluid-heater interface tempera-ture,Λ, D, Π2 andκ are positive parameters,qF(TF) is theboiling curve andu(t) is the input, i.e. an additional heatsupply at the bottom of the heater, see [1].yDx10∂T∂t= κ∇2Tboiling liquid: Λ∂T∂y= −Π2qF (TF )heat supply: −Λ∂T∂y= 1 + u(t)∂T∂x= 0∂T∂x= 0Figure 2 Two-dimensional rectangular heater.3 Observer designThe temperature can be measured atR + 1 points on theheater surface, given bỹxr = r/R, r = 0, . . . ,R, i.e. the sys-tem output equalsyr = TF(x̃r) for r = 0, . . . ,R. The observeris designed to be a copy of the system with output injectiononly on the boundary condition where the measurements areavailable. The obtained observer is given by (1) and (2) withthe stateZ(x,y,t), i.e. the estimate ofT (x,y,t), instead ofT (x,y,t) and with the boundary condition onΓF as∂Z∂y∣∣∣∣ΓF= −Π2qF(ZF )Λ+R∑r=0pr(x)(TF(x̃r)−ZF(x̃r)) , (3)with pr(x) the observer gain functions. The observer er-ror dynamics ofE = T − Z are analysed by linearisationof system and observer, with which local stability of thenonlinear error dynamics can be obtained. Figure 3 showsthe evolution of the output, forR = 2, meaning the outputequalsT (0,D,t), T (0.5,D,t) andT (1,D,t). The initial sys-tem state equals a uniform field ofT ≈ 4, whereas the initialobserver state equals zero. Although the initial differencesbetween system state, observer state and non-uniform equi-librium are large, the observer states converge to the systemstates and the equilibrium is stabilised.Figure 3 Evolution of the system output and the observer output.4 ConclusionAn observer for a nonlinear PDE system is designed. Theclosed-loop system is stabilised by the control law discussedin [1] and this observer, even for large initial perturbations.References[1] R. van Gils, et al. Feedback stabilisation of a two-dimensional pool-boiling system by modal control.Automatica,submitted for publication, 2010

Gils, RW Rob  van

Speetjens, MFM Michel

Nijmeijer, H Henk

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Observer design for a nonlinear two-dimensional pool boiling system

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