K. Bouderbala
M. Girault
E. Videcoq
H. Nouira
D. Petit
J. Salgado
Title: Model reduction and thermal
regulation by Model Predictive Control of
a new cylindricity measurement
apparatus
1 Laboratoire Commun de Métrologie (LNE-CNAM), Laboratoire National de
Métrologie et d'Essais (LNE), 1 Rue Gaston Boissier, 75015 Paris, France,
2 Institut P’, Téléport 2, 1 avenue Clément Ader, BP 40109, F86961 Futuroscope
Chasseneuil cedex
Symposium on Temperature and Thermal Measurements in Industry and science, 2013
1. Experimental device
2. Model reduction
Modal Identification Method
Identification of the parameters of the reduced model
3. State feedback control
Model Predictive Control (MPC)
4. Control test cases
5. Conclusion and prospects
Contents
Context
3
Issues :
• Heat dissipation
• Thermal dilatation
Objective :
Real time control to reduce the effects of temperature
variation
Tools:
Reduced model built by
Modal Identification Method
Model Predictive Control
Model reduction
4
Resolution of the heat transfer equation
Spatial discretization: - Finite elements
- Finite volumes
- Finite differences
State space representation:
𝑻 = 𝑨𝑻 𝒕 + 𝑩𝑼 𝒕
𝒀 𝒕 = 𝑪𝑻(𝒕)
N ODE
Thermal modeling :
Energy balance:
𝛻. 𝑘 𝑀 𝛻𝑇 𝑀, 𝑡 + 𝑃𝑗 𝑡
𝑉𝑗𝜒𝑗(𝑀)
𝑛𝑄
𝑗=1
= 𝜌 𝑀 𝐶𝑝 𝑀𝜕𝑇
𝜕𝑡𝑀, 𝑡 ,
∀ 𝑀 ∈ 𝛺
Boundary conditions :
𝑘 𝑀 𝛻𝑇 𝑀, 𝑡 . 𝑛 = ℎ 𝑇𝑎 𝑡 − 𝑇 𝑀, 𝑡 , ∀𝑀 ∈ Γ
𝑘 𝑀 𝛻𝑇 𝑀, 𝑡 . 𝑛 =𝐴𝑖(𝑡)𝜉𝑖(𝑀)
𝑆𝑖
Issues:
• Memory
• Large computation time
• Unsuitable for real-time control
Solution
• Find a model reproducing the behaviour of the system with
a small number of differential equations (model reduction)
Model Identification Method (MIM)
5
1. Definition of a suitable structure of the reduced model
Modal state-space representation
𝑋 𝑡 = 𝑭𝑋 𝑡 + 𝑮𝑈 𝑡 𝒀 𝒕 = 𝑯𝑋 𝑡 n ODE, n << N
2. Generation of numerical output data for a set of known input signals
3. Identification of the parameters of the reduced model ( F,G,H ) through
optimization algorithms:
Ordinary Linear Least Squares (𝑯)
Particle Swarm Optimization (𝑭, 𝑮)
Model Identification Method
6
Output vector
𝒀𝑫𝑴(𝒕)
Physical
system
COMSOL
model
( N ODE )
Output vector
𝒀𝑹𝑴(𝑡, 𝑭, 𝑮,𝑯)
Reduced model
𝑋 𝑡 = 𝑭𝑋 𝑡 + 𝑮𝑈(𝑡) 𝒀𝑹𝑴 𝒕 = 𝑯𝑋(𝑡)
𝑛 ODE, 1 ≤ 𝑜 10 ≪ 𝑁
Input vector
𝑼𝑨 Deviation criterion to minimize
𝐽 = 𝒀𝑹𝑴 − 𝒀𝑫𝑴𝐿2
²
Optimization
algorithms
Boundary conditions,
sources
Observables:
simulated
Iterative procedure
Known vector
Model Identification Method scheme
Identification of the RM
1 2 3 4 5 6 7 8 9 10 11 12 13 14 1510
-3
10-2
10-1
Model order
idn
, K
0 100000 200000 300000 400000 500000 6000000
2
5
Time, s
Heat
po
wer, W
0
0.2
0.4
0.6
0.8
0
0.2
0.4
0.6
0.8
0
0.2
0.4
0.6
0.8
0 100000 200000 300000 400000 500000 6000000
0.2
0.4
0.6
0.8
Time, s
Tem
peratu
re, K
P1
P2
P3
P4
A1
A2
A3
A4
T5(DM) T
5(RM 5) T
5(RM 10)
T6(DM) T
6(RM 5) T
6(RM 10)
T7 (DM) T
7(RM 5) T
7(RM 10)
T8(DM) T
8(RM 5) T
8(RM 10)
(a)
(e)
(b)(d) (c)
𝑈 𝑡 =
𝐴1(𝑡)⋮
𝐴4(𝑡)𝑃1(𝑡)
⋮𝑃4(𝑡)
𝒀 𝑡 =
𝑇1⋮
𝑇18
𝝈𝒊𝒅𝒏 =
(𝒀𝒓𝒎𝒊 𝒕𝒋 − 𝒀𝒊𝑫𝑴 𝒕𝒋 )𝟐
𝑵𝒕𝒋=𝟏
𝒒𝒊=𝟏
𝒒 × 𝑵𝒕
Quadratic criterion:
Where:
𝑞 is the number of observables
𝑁𝑡 the number of time steps
Identification of the RM
8
0
0.2
0.4
0.6
0.8
0
0.2
0.4
0.6
0.8
0
0.2
0.4
0.6
0.8
0 100000 200000 300000 400000 500000 6000000
0.2
0.4
0.6
0.8
Time, s
Tem
pera
ture
, K
T5(DM) T
5(RM 5) T
5(RM 10)
T6(DM) T
6(RM 5) T
6(RM 10)
T7 (DM) T
7(RM 5) T
7(RM 10)
T8(DM) T
8(RM 5) T
8(RM 10)
(a)
(b)
(c)
(d)
State feedback control:
9
𝑋 𝑡 = 𝐹𝑋 𝑡 + 𝐺𝐴𝑈𝐴 𝑡 + 𝐺𝑃𝑈𝑃(𝑡)
𝑌𝑅𝑀 𝑡 = 𝐻𝑋(𝑡)
𝑍 𝑡 = 𝑇5, 𝑇6, 𝑇7, 𝑇8𝑇 = 𝐻𝑧𝑋(𝑡)
system
Model
Predictive
Controller
Linear Quadratic
Estimator ( Kalman
filter)
Disturbance 𝑈𝑃
Actuators 𝑈𝐴
𝑌𝑚
𝑋
Reduced Model
Estimated state
𝑍(𝑡)
Temperatures to control :
𝑈𝐴 𝑡 = 𝐴1 𝑡 , 𝐴2 𝑡 , 𝐴3 𝑡 , 𝐴4 𝑡 𝑇
𝑈𝑃 𝑡 = 𝑃1 𝑡 , 𝑃2 𝑡 , 𝑃3 𝑡 , 𝑃4 𝑡 𝑇
Where :
= input vector of actuators (surface heaters)
= input vector of perturbations
(laser interferometers)
Model predictive control
10
Principle :
A model of the system is used to predict plants
behavior and choose the best control in the sense of
some cost function within constraints.
The future response of the plant is predicted over a
Prediction horizon 𝑁𝑝.
𝑘 𝑘 + 𝑁𝑝
Prediction horizon
past future
Predicted output
Optimal input
Reference trajectory
Dynamical system issued from time discretization of the state-space representation:
Z 𝑘 = 𝜳𝑋(𝑘) + 𝜞𝑈𝐴(𝑘 − 1) + 𝜣UA(𝑘)
• 𝑘 is the current time index
• 𝑁𝑝 is the prediction horizon
Z(k)=
𝑍(𝑘 + 1)⋮
𝑍(𝑘 + 𝑁𝑝) UA(k)=
∆𝑈𝐴(𝑘)⋮
∆𝑈𝐴 (𝑘 + 𝑁𝑝 − 1)
Model Predictive Control
11
The control law issued from the minimization of a quadratic functional is :
UA(𝑘) = (𝜣𝑇𝜣 + 𝜆𝐼)−1𝜣𝑇 Z𝑟𝑒𝑓 𝑘 − 𝜳𝑋 𝑘 − 𝜞𝑈𝐴 𝑘 − 1
𝐽 = Z𝑻Z+ 𝜆UA
𝑻UA
The performance index to minimize :
𝜆 is a penalty parameter.
𝑋(𝑘) is obtained by using a linear quadratic estimator (Kalman filter)
The matrices Ψ, Γ, Θ depend on the matrices F,G,H of the reduced model.
Control test case :
12
𝜎𝑧 =1
4 × 𝑁𝑡 (𝑍𝑖 𝑡𝑗 )2
𝑖∈{1,4}
𝑁𝑡
𝑗=1
1/2
Control parameters :
Temperatures to control:
𝑇5, 𝑇6, 𝑇7, 𝑇8
Reduced model:
RM10
Control time step:
∆𝑡 = 1𝑠
Prediction horizon:
𝑁𝑝 = 1
Standard deviation of the
measurement noise:
𝜎𝑚 = 0.002 𝐾
The mean quadratic discrepancy between the desired (0K) and the
obtained temperatures deviations:
-4
0
4
Pert
urb
ati
on
, W
0 1000 2000 3000 4000 5000 6000 7000
-2
0
2
Co
ntr
ol, W
P1
P2
P3
P4
A1
A2
A3
A4
-3
0
3
Po
wer,
W
A1
A2
A3
A4(c)
(d)
(e)
Controlled phase Uncontrolled phase
Test case 1
13
𝜎𝑝 = 2.99 𝑊
Standard deviation of the perturbation:
Without
control With control
𝝈𝒛, 𝑲 0.1051 0.0065 0.0024
-0.2
-0.1
0
0.1
0.2
Tem
pera
ture
, K
0 1000 2000 3000 4000 5000 6000 7000-0.2
-0.1
0
0.1
0.2
Time(s)T
em
pera
ture
, K
T5
T6
T7
T8
T5
T6
T7
T8
𝝀𝟏 = 𝟏𝟎−𝟑
𝝀𝟐 = 𝟏𝟎−𝟔
Test case 2
14
Standard deviation of the perturbation:
𝜎𝑝 = 6.42𝑊 -10
-5
0
5
10
Pert
urb
ati
on
, W
-5
-202
5
Co
ntr
ol, W
0 1000 2000 3000 4000 5000 6000 7000
-5
202
5
Co
ntr
ol, W
P1
P2
P3
P4
A1
A2
A3
A4
A1
A2
A3
A4
Controlled phase Uncontrolled phase
-0.4
0
0.4
0.8
Tem
pera
ture
, K
0 1000 2000 3000 4000 5000 6000 7000-0.4
0
0.4
0.8
Time, s
Tem
pera
ture
, K
T5
T6
T7
T8
T5
T6
T7
T8
Without
control With control
𝝈𝒛, 𝑲 0.4572 0.0120 0.0041
𝝀𝟏 = 𝟏𝟎−𝟑
𝝀𝟐 = 𝟏𝟎−𝟔
Conclusions and prospects
15
• Numerical generation of input-output data
• A reduced model built with the MIM from numerical generation
• Thermal regulation of temperature in 4 points of the structure
• Study the effects of the MPC parameters
• Identify a reduced model from experimental data
• Increase the number of controlled points
Thank you for your
attention
Linear quadratic estimator
17
State estimation:
𝑋 𝑡 = 𝐹𝑋 𝑡 + 𝐺𝐴𝑈𝐴 𝑡 + 𝐾𝑓(𝑌𝑅𝑀 𝑡 − 𝐻𝑋 )
The correction is done through the Kalman gain given by :
𝐾𝑓 =1
𝛼2 𝑆𝐻𝑇
Where 𝑆 (∈ ℝ𝑛×𝑛) is the solution of Riccati equation:
𝑆𝐹𝑇 + 𝐹𝑆 −1
𝛼2 𝑆𝐻𝑇𝐻𝑆 + 𝐺𝑃𝐺𝑃𝑇 = 0
𝛼 =𝜎𝑚
𝜎𝑝 is the ratio between the standard deviations of measurements and
heat source disturbances