low order models by the modal identification method (mim) application to thermal control
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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
2
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
3
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
1
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
5
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
2
3
4
5
6
1
7
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
14
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
15
Modelling issues → « experimental modelling »
Experimental Low Order Model
State feedback thermal control
Control test case
2
3
4
5
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
20
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
21
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
7
8
9
10
11
12
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
6
7
8
9
10
11
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
6
7
8
9
10
11
12
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
12
time (s)
So
urc
e o
rdin
ate
(m
m)
Fa
n v
olt
ag
e (
V)
25
Temperatures
Controlled phase Uncontrolled phase
5Control test case
Temperatures T4, T5 and T7
Mean quadratic discrepancies (K)
10800 12000 13200 14400 15600 16800 1800024
26
28
30
32
34
36
38
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
27
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
28
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)