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Low Order Models by the Modal Identification Method (MIM) Application to thermal control

2 nd International Forum on Flow Control Workshop - December 8-10, 2010. Low Order Models by the Modal Identification Method (MIM) Application to thermal control. Manuel Girault, Etienne Videcoq , Daniel Petit. Institut P’ • UPR CNRS 3346 SP2MI • Téléport 2

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Low Order Models by the Modal Identification Method (MIM) Application to thermal control

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  1. 2nd International Forum on Flow Control Workshop - December8-10, 2010 Low Order Modelsby the Modal Identification Method (MIM)Application to thermal control Manuel Girault, Etienne Videcoq, Daniel Petit Institut P’ • UPR CNRS 3346 SP2MI • Téléport 2 Boulevard Marie et Pierre Curie • BP 30179 F86962 FUTUROSCOPE CHASSENEUIL Cedex titre présentation

  2. Introduction - Low Order Models (LOMs) Part I – Modal Identification Method (MIM) for building LOMs Part II - Thermal control of a ventilated plate heated by a mobile source via state feedback using a LOM (experimental) Conclusions & prospects

  3. Introduction - Low Order Models (LOMs): for what ? Replace a large-sized model by a low-sized one for specifictasksrequiringveryfast computations Build a low-sized model ad-hoc from in-situ measurements whenclassicalmodellingisdifficult

  4. 1 CONTEXT Introduction - Low Order Models in control frame SISO process MIMO process SISO PID tuning techniques PI / PID controller - Decoupling control strategies (static/dynamic) - Iterative tuning methods - State feedback control Low Order Model - Tuning the parameterswith the Ziegler- Nicholsmethod, the Nyquistdiagram - Robust and easily understood algorithm perturbation ActuatorsU Z ≈ Zd(desired) System Ym(noisymeasurements) Estimated state Regulator State estimator

  5. Part I - Modal Identification Method (MIM) for building LOMs 1 Main features 2 Optimizationmethods in MIM 3 MIM for linear systems 4 Thermal diffusion with constant properties 5 Forced heat convection 6 MIM & POD-G: commonfeatures & differences LOMs for state feedback control 7

  6. 1 Modal Identification Method: main features Modal Identification Method: main features General methodology For a givenphysicalproblem Define an adequate structure of equations for the LowOrder Model from local conservation equations 1 2 Generatesomenumerical or experimental data for a set of known inputs (Boundary Conditions, sources …) 3 Identify the LowOrder Model parameters Fromthesenumerical or experimental data Through an optimizationprocedure • Objective functional 𝓙(1) is first minimized for order n =1 • ⟹ identification of a LOM of order 1 (single scalar equation) • n is then increased and the minimization of 𝓙(n), involving more • unknown parameters, leads to LOMs of higher order

  7. 1 Modal Identification Method: main features Modal Identification Method: main features Structure of the LowOrder Model 1 From local conservation equations (PDEs) Local variables (velocity, temperature, concentration, etc.) Boundary conditions Sources, … Space-differentialoperator (linear and/or nonlinear) Define an adequate structure of equations for the LowOrder Model n ODE Vectorθ & matrixH are LOM parameters (to beidentified) Vectorwith linear and/or nonlinear contributions State spacerepresentation Input vector (BC, sources, …) In a control framework: actuators & perturbations Lowsized state vector 1 ≤ n ≤ o(10) In a more generalway Output vector

  8. 1 Modal Identification Method: main features Modal Identification Method: main features LowOrder Model building 2 Generatesomenumerical or experimental data for a known input vector Theseshouldbe « rich » enough to contain the targeteddynamics of the system 3 Identify the LowOrder Model parameters: vectorθand matrixH Observables: simulated or measured data Output vector Physical System Y*(t) Model: N ODE or Real device Sum of squaredresiduals to beminimized input vectorU*(t)known LowOrder Model Boundary Conditions, Sources Output vector Y(t,θ,H) • Optimizationalgorithms : • OrdinaryLinear Least Squares (H) • SwarmParticleOptimization or Quasi-Newton type method (θ) n ODE, 1≤ n ≤o(10) << N Iterativeprocedure Optimal parametersθ & H

  9. 2 Optimizationmethods in MIM Modal Identification Method: main features (θ, H )opt = Argmin LowOrder Model • Y(t) is nonlinear with respect to θ • Y(t) is linear with respect to H 2 types of optimization methods are used for the minimization of • Nonlinear iterative method for the estimation of θ • deterministic method such as a Conjugate Gradient or Quasi-Newton method for instance • or stochastic method such as Particle Swarm Optimization, Genetic Algorithm, etc • an initial guess for θ is required • Ordinary (Linear) Least Squares (OLS) are used for the estimation of H • at each iteration of the above mentioned nonlinear iterative algorithm Currentθknown, U*(t) known ⇒ X(t)canbecomputed for all times Y(t) = H X(t)∀t =1, … ,Nt ⟹ HT≈ [𝕏𝕏T]-1 𝕏 𝕐*T 𝕏 = [ X(t1) … X(tNt) ] 𝕐* = [ Y*(t1) …Y*(tNt) ]

  10. 3 MIM for linearsystems Modal Identification Method: main features ElementaryReducedModel (ERM) relative to eachcomponent Uk of input vectorU Linearity Y*(k)(t) DetailedModel ( orderN ) or Real System Known input U*k(t): ERM for Uk t 0 t=0 Other components of U: 0 ∀t ≥ 0 t ERMs are assembled to form a global LOM for U(t) = [U1(t) … Up(t)]T Superposition principle ∑ X(k)(t) = F(k) X(k)(t) + 1(k)Uk(t) Y(k)(t) = H(k)X(k)(t) X(t) = FX(t) + GU(t) Y(t) = HX(t) ElementaryReduced Model ( order1≤n(k)≤o(10) << N) Global LOM All components equal to 1 Global state vector X(t) of size Superposition principle: Global diagonal state matrix F of size (n,n) Global input matrix G of size (n,p) Global output matrix H of size (q,n)

  11. 4 Thermal diffusion with constant properties Modal Identification Method: main features X = Lowsized state vector Size n 1 ≤ n ≤ o(10) BC: + convective BC LowOrder Model: + convecto-radiative BC LowOrder Model : Vector of nonlinearities:

  12. 3 5 Forced convection: Navier-Stokes + Energy Forcedheat convection Modal Identification Method: main features Modal Identification Method: main features →Mass, momentum and energy conservation equations : + flow BC + thermal BC « fluid » LOM: state vectorZ(t) • « thermal » LOM: state vectorX(t), dependingon « fluid » reducedstate vectorZ(t)

  13. 6 MIM & POD-G: commonfeatures & differences Modal Identification Method: main features 6 – LOMs for control purposes Eigenvalueproblem Common features • LOMs are builtfromnumerical or experimental data • dynamical state equations show similarterms • space-time decomposition of variables POD MIM Main differences Optimizationprocedure POD-Galerkin MIM • Computation of an « empirical » basis, reduction by truncation in modes spectrumthen insertion in local equations (Galerkin projection) • ⟹ weakcouplingbetween the building of the projection space basis (POD) and the building of the dynamical state equation (Galerkin projection) • POD is optimal in the sense of data compression (signal energy) • Data have to cover the wholespacedomain or at least a large part • Computing time for the LOM building function of the amount of data (Min(space, time)) • Identification of a LOM in state space by minimization of a squarednorm of the residualsbetween data and LOM outputs • ⟹ strongcouplingbetween the building of the projection space basis (H) and the building of the dynamical state equation (parametersθ) • Optimality for the chosen outputs and the inputs applied for the identification procedure • The model outputs maybe a selection of a few observables of interest (a single one is possible) • Computing time for the LOM building function of the amount of data (space x time): maybe long (3D fields + highfrequencysampling) or very short (a few outputs)

  14. 7 LOMs for state feedback control Modal Identification Method: main features 6 – LOMs for control purposes Classical state feedback control theory (LQR, LQE, LQG) relies on linear(ized) state spacemodels Assumption of small variations aroundspecificworking conditions Build a linear LOM for smalldeviationsfrom a specificworking point Linearization of a nonlinear LOM valid in a wider range of operating conditions First order Taylor expansion of

  15. Part II - THERMAL CONTROL OF A VENTILATED PLATE HEATED BY A MOBILE SOURCE VIA STATE FEEDBACK USING A LOM Experimental thermal control demonstrator 1 Modelling issues → « experimentalmodelling » 2 ExperimentalLowOrder Model 3 State feedback thermal control 4 Control test case 5

  16. 1 Experimental thermal control demonstrator Rack of fans (perturbation) Aluminumslab Objective : Control-command in real time of multi-input multi-output thermal systems (regulation of temperaturearound nominal working conditions) • Tools : • LowOrder Model built by MIM fromexperiment • LinearQuadraticGaussian (LQG) Compensator Mobile radiative heat source 3 actuators: coordinatesxs and ys heat power P Temperaturemeasurements 9 thermocouples T1 … T9 on the rearside of the slab

  17. Modelling issues → « experimentalmodelling » 2 Aluminumslab Rack of fans (perturbation) Mobile radiative heat source 3 actuators: coordinatesxs and ys heat power P Temperaturemeasurements The source covers an area 𝚪s(t) whose center may move with time • Inaccurateknowledge of thermal conductivityk, emissivityε, … • Estimation of heat exchange coefficient h in bothforced/natural convection • Source modelling • Non-linearities Experimental building of a LowOrder Model

  18. 3 ExperimentalLowOrder Model 4 independent inputs: 3 actuators and 1 perturbation Heat source Fan voltage disturbance 4 ElementaryReducedModels, each one of order 2, identified by MIM Global LOM of ordern = 8 diagonal Loworder state vector Temperatures for state estimation Deviations of temperatures to becontrolled Temperaturedeviations Temperatures in nominal working conditions P0, xs0, ys0, V0

  19. 4 State feedback thermal control 4 independent inputs: 3 actuators and 1 perturbation Heat source Fan voltage disturbance Global LOM of ordern = 8 Loworder state vector Temperatures for state estimation Temperaturedeviations Temperatures in nominal working conditions P0, xs0, ys0, V0 Deviations of temperatures to becontrolled Closed-loopcontroller (LQG)

  20. 4 State feedback thermal control LinearQuadraticGaussiancompensator Implicit time discretization Computation of an estimate of the state vector : k=k+1 LinearQuadraticEstimator (Kálmánfilter) with (Computed once and for all) Computation of the command correction vector : LinearQuadraticRegulator Easy computation of ateach time stepthanks to the LOM

  21. 4 State feedback thermal control Gain matrices KrandKe LinearQuadraticRegulator (LQR) LinearQuadraticEstimator(LQE) ℓ = parameter to limit the command magnitude P = solution of the algebraicRiccatimatrixequation: = ratio between standard deviations of measurement noise and fan voltage perturbation S = solution of the algebraicRiccatimatrixequation: LowOrder Model matrices LowOrder Model matrices Resolution of nonlinearRiccatimatrixequations OK up to model size n about a few tenths Kr and Keeasy to computethanks to low-sized matrices of the LOM (n = 8) Computedoff-line, once and for all

  22. 5 Control test case Objective and perturbation Regulation of temperature at three chosen points T4, T5 and T7 when the nominal ventilation level is perturbed Controlled phase Uncontrolled phase Nominal level V0 = 8.5 V Controlledtemperatures Perturbed fan voltage V: successive steps of random magnitude

  23. 5 Control test case Control algorithm ImplementationusingLabview (measurements, Kálmánfilter, regulator, actuators) Transient regime Controlled phase : 3600 s Uncontrolled phase : 3600 s t = 3 s Update of computation of Fan voltage perturbation Measurement of to + update of • Controller parameters: • for the LQR: ℓ = 5x10-3 • for the LQE: = 2x10-2 Source motion Power change

  24. 5 Control test case Actuators Nominal power P0 = 100 W Nominal fan voltage V0 = 8.5 V Heat power actuator Nominal abcissa x0 = 0 Nominal ordinate y0 = 0 Heat source abscissa Heat source ordinate Nominal fan voltage V0 = 8.5 V

  25. 5 Control test case Temperatures Controlled phase Uncontrolled phase Meanquadraticdiscrepancies (K) TemperaturesT4, T5 and T7

  26. Conclusions • Experimental thermal control demonstrator • 3 inputs – 3 outputs temperature regulation problem • LowOrder Model built by MIM from experimental data (n = 8 dof) • Real-time control achievablethanks to the low-sized model (t = 2s up to now) Vy (m/s) LOM • RecentPhDthesis about MIM (numericalworks): • Aerothermaltransientreducedmodels (AIRBUS) • Application to 2D circularcylinderwake (Jérôme Ventura) • LOM (6 modes) able to reproduce vortex street in the range Re=[2000, 4000] time • Model reduction in forced convection (steadyvelocity, unsteadyheattransfer) • Application to thermal control downstream a backward-facingstep (Yassine Rouizi) CFD Steady LOM (7 modes) able to reproducevelocityfield in the range Re=[100, 800] LOM

  27. Conclusions Closedloop thermal control in forced convection Numericalworks by Yassine Rouizi Target: temperature profile Actuators: heat fluxes 𝜑2 Tin = 300 K + 𝛿Tin Perturbedinlettemperature 𝜑1 T(K) Objective : Control of a temperature profile around 320 K in a cross section downstream a backwardfacing-step • Tools : • LowOrder Model builtfromnumerical data • LinearQuadraticGaussian (LQG) Compensator time (s)

  28. Prospects • Developments on the experimentaldemonstrator • Control with 9 actuators (9 fans independentlycommandable) • and 3 perturbations (heat source power and displacements) • Tracking control problems • Thermal control of a highprecisiongeometricalmeasurement machine • temperatureregulationisneeded in order to prevent dilatations in the machine • projectgranted by Euramet (gathering of European National MeasurementLabs) • Control in mixed convection (with Laurent Cordier) • Model reduction in mixed convection (MIM, POD) • Control in the wake of a heatedcircularcylinder • Experimental validation in the frame of the COMIFO project • (mixed & forced convection around bluff bodies) • Granted by the National Foundation for Research in Space and Aeronautics

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