19/04/2023titre présentation 1

Low Order Models

by the Modal Identification Method

(MIM)

Application to thermal control

Manuel Girault, Etienne Videcoq, Daniel Petit

Institut P’ • UPR CNRS 3346SP2MI • Téléport 2Boulevard Marie et Pierre Curie • BP 30179F86962 FUTUROSCOPE CHASSENEUIL Cedex

2nd International Forum on Flow ControlWorkshop - December 8-10, 2010

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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)

Introduction - Low Order Models (LOMs)

Conclusions & prospects

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Introduction - Low Order Models (LOMs): for what ?

Replace a large-sized model by a low-sized one

for specific tasks requiring very fast computations

Build a low-sized model ad-hoc from in-situ measurements

when classical modelling is difficult

Model type Large-sized Model Low Order Model

Features- N dof/eqs. (space

discretization)

- large computing time (3D)

- accuracy

- n dof/eqs. ( n << N )

- computing time

- memory size

Use Simulations

Understanding phenomena

- inverse problems

- control (open/closed loop)

CONTEXT

SISO process

- Tuning the parameters with the Ziegler-Nichols method, the Nyquist diagram

- Robust and easily understood algorithm

PI / PID controller

MIMO process

SISO PID tuning techniques

- Decoupling control strategies (static/dynamic)

- Iterative tuning methods

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Low Order Model

- State feedback control

SystemYm (noisy

measurements)

State estimatorEstimated state

Regulator

Actuators U

perturbation

Z ≈ Zd (desired)

Introduction - Low Order Models in control frame

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Part I - Modal Identification Method (MIM) for building LOMs

Optimization methods in MIM

MIM for linear systems

Thermal diffusion with constant properties

Forced heat convection

MIM & POD-G: common features & differences

Main features

LOMs for state feedback control

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Modal Identification Method: main features1Modal Identification Method: main features

Identify the Low Order Model parameters

Through an optimization procedure

1

3

Generate some numerical or experimental data for a set of known inputs

(Boundary Conditions, sources …)

2

From these numerical or experimental data

For a given physical problem

Define an adequate structure of equations for the Low Order Model

from local conservation equations

General methodology

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

Modal Identification Method: main features1Modal Identification Method: main features

From local conservation equations (PDEs)1

Low sized state vector1 ≤ n ≤ o(10)

Define an adequate structure of equations for the Low Order Model

Output vector

Input vector (BC, sources, …)In a control framework:

actuators & perturbations

State space representation

Local variables(velocity, temperature,concentration, etc.)

Space-differential operator (linear and/or nonlinear)

Boundary conditionsSources, …

Vector θ & matrix H are LOM parameters

(to be identified)

In a more general way

Vector withlinear and/or nonlinear contributions

n ODE

Structure of the Low Order Model

Modal Identification Method: main features1Modal Identification Method: main features

Identify the Low Order Model parameters: vector θ and matrix H3

Generate some numerical or experimental data for a known input vector 2

Physical System Output vector Y*(t)

Low Order Model

input vector U*(t) known

Optimization algorithms :• Ordinary Linear Least Squares (H)• Swarm Particle Optimization or

Quasi-Newton type method (θ)

Sum of squared residuals to be minimized

Output vector Y(t,θ,H)

Iterative procedure

Observables: simulated or measured data

Boundary Conditions,

Sources

Model: N ODE or

Real device

n ODE, 1≤ n ≤o(10) << N

Optimal parameters θ & H

These should be « rich » enough to contain the targeted dynamics of the system

Low Order Model building

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 algorithmCurrent θ known, U*(t) known ⇒ X(t) can be computed for all times

Modal Identification Method: main features2Optimization methods in MIM

Low Order Model(θ, H )opt = Argmin

𝕏 = [ X(t1) … X(tNt) ]𝕐* = [ Y*(t1) … Y*(tNt) ]Y(t) = H X(t) ∀t =1, … ,Nt ⟹ HT ≈ [𝕏𝕏T]-1 𝕏 𝕐*T

Modal Identification Method: main features3MIM for linear systems

Elementary Reduced Model (ERM) relative to each component Uk of input vector ULinearity

Global state vector X(t) of size Global output matrix H of size (q,n)Global diagonal state matrix F of size (n,n) Global input matrix G of size (n,p)

Superposition principle:

Known input U*k(t):

0

Other components of U : 0 ∀t ≥ 0t=0 t

Detailed Model ( order N )

or

Real System t

Y*(k)(t)

Elementary Reduced Model

( order 1≤ n(k) ≤o(10) << N )

X(k)(t) = F(k) X(k)(t) + 1(k)Uk(t)

Y(k)(t) = H(k) X(k)(t)

ERMs are assembled to form a global LOM for U(t) = [U1(t) … Up(t)]TSuperposition principle

X(t) = FX(t) + GU(t)

Y(t) = HX(t)

ERM for Uk

Global LOM∑

All components equal to 1

Low Order Model :

BC:

+ convective BC

+ convecto-radiative BC

Low Order Model:

Vector of nonlinearities:

Modal Identification Method: main features4Thermal diffusion with constant properties

X = Low sized state vectorSize n

1 ≤ n ≤ o(10)

→ Mass, momentum and energy conservation equations :

« thermal » LOM: state vector X(t), depending on « fluid » reduced state vector Z(t)

« fluid » LOM: state vector Z(t)

+ flow BC

+ thermal BC

Modal Identification Method: main features3Forced convection: Navier-Stokes + EnergyModal Identification Method: main features5Forced heat convection

6 – LOMs for control purposesModal Identification Method: main features6MIM & POD-G: common features & differences

Main differences

Common features LOMs are built from numerical or experimental data dynamical state equations show similar terms space-time decomposition of variables

Computation of an « empirical » basis, reduction by truncation in modes spectrum then insertion in local equations (Galerkin projection)⟹ weak coupling between 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 whole space domain 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 squared norm of the residuals between data and LOM outputs⟹ strong coupling between the building of the

projection space basis (H) and the building of the dynamical state equation (parameters θ)

Optimality for the chosen outputs and the inputs applied for the identification procedure

The model outputs may be 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): may be long (3D fields + high frequency sampling) or very short (a few outputs)

POD-Galerkin MIM

POD

MIM

Eigenvalue problem

Optimization procedure

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6 – LOMs for control purposesModal Identification Method: main features7LOMs for state feedback control

Linearization of a nonlinear LOM valid in a wider range of operating conditions

Build a linear LOM for small deviations from a specific working point

Classical state feedback control theory (LQR, LQE, LQG) relies on linear(ized) state space models

Assumption of small variations around specific working conditions

First order Taylor expansion of

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Modelling issues → « experimental modelling »

Experimental Low Order Model

State feedback thermal control

Control test case

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Experimental thermal control demonstrator1

Part II - THERMAL CONTROL OF A VENTILATED PLATE HEATED BY A MOBILE SOURCE VIA STATE FEEDBACK USING A LOM

9 thermocouples T1 … T9 on the rear side of the slab

Objective :

Control-command in real time of multi-input multi-output thermal systems

(regulation of temperature around nominal working conditions)

Tools :- Low Order Model built by MIM

from experiment

- Linear Quadratic Gaussian (LQG) Compensator

1Experimental thermal control demonstratorRack of fans

(perturbation)

Temperature measurements

Aluminum slab

Mobile radiative heat source3 actuators:

coordinates xs and ys

heat power P

Inaccurate knowledge of thermal conductivity k, emissivity ε, … Estimation of heat exchange coefficient h in both forced/natural convection Source modelling Non-linearities

Experimental building of a Low Order Model

2Modelling issues → « experimental modelling »

Rack of fans (perturbation)

Temperature measurements

The source covers an area 𝚪s(t)

whose center

may move with time

Aluminum slab

Mobile radiative heat source3 actuators:

coordinates xs and ys

heat power P

4 independent inputs: 3 actuators and 1 perturbation

4 Elementary Reduced Models, each one of order 2, identified by MIM

Global LOM of order n = 8

Temperatures in nominal working conditions P0, xs0, ys0, V0Temperature deviations

Low order state vector

Heat source Fan voltage disturbance

Deviations of temperatures to be controlledTemperatures for state estimation

3Experimental Low Order Model

diagonal

Closed-loop controller (LQG)

Temperatures in nominal working conditions P0, xs0, ys0, V0

Temperature deviations

Deviations of temperatures to be controlled

Temperatures for state estimation

4State feedback thermal control

4 independent inputs: 3 actuators and 1 perturbation

Heat source Fan voltage disturbance

Global LOM of order n = 8

Low order state vector

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Linear Quadratic Gaussian compensator

Computation of an estimate of the state vector :

Computation of the command correction vector :

Implicit time discretization

with

(Computed once and for all)

k=

k+

1

Easy computation of

at each time step thanks to

the LOM

4State feedback thermal control

Linear Quadratic Estimator (Kálmán filter)

Linear Quadratic Regulator

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Linear Quadratic Regulator (LQR)

Gain matrices Kr and Ke

ℓ = parameter to limit the command magnitude

P = solution of the algebraic Riccati matrix equation:

Low Order Model matrices

Kr and Ke easy to compute thanks to low-sized matrices of the LOM (n = 8)

Computed off-line, once and for all

Linear Quadratic Estimator (LQE)

= ratio between standard deviations of

measurement noise and fan voltage perturbation

S = solution of the algebraic Riccati matrix equation:

Resolution of nonlinear Riccati matrix equations OK up to model size

n about a few tenths

4State feedback thermal control

Low Order Model matrices

Objective and perturbation

Controlled temperatures

Regulation of temperature at three chosen points T4, T5 and T7

when the nominal ventilation level is perturbed

Perturbed fan voltage V: successive steps

of random magnitude

Controlled phase Uncontrolled phase

Nominal level

V0 = 8.5 V

5Control test case

10800 12000 13200 14400 15600 16800 180006

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8

9

10

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time (s)

Fa

n v

olt

ag

e (

V)

Measurement of to+ update of

Control algorithm

Transient regime

Controlled phase : 3600 s

Uncontrolled phase : 3600 s

t = 3 s

Implementation using Labview(measurements, Kálmán filter, regulator, actuators)

5Control test case

Fan voltage perturbation

Source motionPower change

Update of

computation of

Controller parameters:

• for the LQR: ℓ = 5x10-3

• for the LQE: =

2x10-2

Actuators

Heat power actuator

Nominal powerP0 = 100 W

Nominal fan voltageV0 = 8.5 V

Heat source abscissa

Nominal abcissax0 = 0

Nominal fan voltage V0 = 8.5 V

Heat source ordinate

5Control test case

10800 11400 12000 12600 13200 13800 14400-4

-3

-2

-1

0

1

2

3

4

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7

8

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10

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12

time (s)

So

urc

e a

bcis

sa

(m

m)

Fa

n v

olt

ag

e (

V)

10800 11400 12000 12600 13200 13800 1440040

60

80

100

120

140

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7

8

9

10

11

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time (s)

He

at

po

we

r (W

)

Fa

n v

olt

ag

e (

V)

Nominal ordinatey0 = 0

10800114001200012600132001380014400-17

-13

-9

-5

-1

3

7

11

6

7

8

9

10

11

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time (s)

So

urc

e o

rdin

ate

(m

m)

Fa

n v

olt

ag

e (

V)

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Temperatures

Controlled phase Uncontrolled phase

5Control test case

Temperatures T4, T5 and T7

Mean quadratic discrepancies (K)

10800 12000 13200 14400 15600 16800 1800024

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28

30

32

34

36

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T4T5T7 T4 nomT5 nomT7 nom

time (s)

Te

mp

era

ture

(°C

)

𝜎T 𝜎T4 𝜎T5 𝜎T7

controlled 0.17 0.13 0.11 0.24

uncontrolled 0.96 1.11 0.99 0.74

Conclusions

Recent PhD thesis about MIM (numerical works):⇒ Aerothermal transient reduced models (AIRBUS)

Application to 2D circular cylinder wake (Jérôme Ventura)

LOM (6 modes) able to reproduce vortex street in the range Re=[2000, 4000]

Experimental thermal control demonstrator⇒ 3 inputs – 3 outputs temperature regulation problem⇒ Low Order Model built by MIM from experimental data (n = 8 dof)⇒ Real-time control achievable thanks to the low-sized model (t = 2s up to now)

⇒ Model reduction in forced convection (steady velocity, unsteady heat transfer)

Application to thermal control downstream a backward-facing step (Yassine Rouizi)

LOM

time

Vy (m/s)

Steady LOM (7 modes) able to reproduce velocity field in the range Re=[100, 800]

LOM

CFD

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Closed loop thermal control in forced convection

Objective :

Control of a temperature profile around 320 K in a cross section downstream a backward facing-step

Tools :- Low Order Model built

from numerical data

- Linear Quadratic Gaussian (LQG) Compensator

Tin = 300 K + 𝛿Tin

𝜑2𝜑1

Actuators: heat fluxes Target: temperature profile

Perturbed inlet temperature

Conclusions

T(K)

time (s)

Numerical works by Yassine Rouizi

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Prospects

Control in mixed convection (with Laurent Cordier)⇒ Model reduction in mixed convection (MIM, POD)⇒ Control in the wake of a heated circular cylinder⇒ 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

Developments on the experimental demonstrator⇒ Control with 9 actuators (9 fans independently commandable)

and 3 perturbations (heat source power and displacements)⇒ Tracking control problems

Thermal control of a high precision geometrical measurement machine⇒ temperature regulation is needed in order to prevent dilatations in the machine⇒ project granted by Euramet (gathering of European National Measurement Labs)