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