Title: Model reduction and thermal regulation by Model Predictive - - PowerPoint PPT Presentation

• 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: 𝑼 = 𝑩𝑼 𝒖 + 𝑪𝑽 𝒖 𝒁 𝒖 = 𝑫𝑼(𝒖) N ODE Thermal modeling : Energy balance:

𝛼. 𝑙 𝑁 𝛼𝑈 𝑁, 𝑢 + 𝑄

𝑘 𝑢

𝑊

𝑘

𝜓𝑘(𝑁)

𝑜𝑅 𝑘=1

= 𝜍 𝑁 𝐷𝑞 𝑁 𝜖𝑈 𝜖𝑢 𝑁, 𝑢 , ∀ 𝑁 ∈ 𝛻

Boundary conditions :

𝑙 𝑁 𝛼𝑈 𝑁, 𝑢 . 𝑜 = ℎ 𝑈

𝑏 𝑢 − 𝑈 𝑁, 𝑢

, ∀𝑁 ∈ Γ 𝑙 𝑁 𝛼𝑈 𝑁, 𝑢 . 𝑜 = 𝐵𝑗(𝑢)𝜊𝑗(𝑁) 𝑇𝑗

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 𝑌 𝑢 = 𝑮𝑌 𝑢 + 𝑯𝑉 𝑢 𝒁 𝒖 = 𝑰𝑌 𝑢 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
• ptimization algorithms:

 Ordinary Linear Least Squares (𝑰)  Particle Swarm Optimization (𝑮, 𝑯)

Model Identification Method

6 Output vector 𝒁𝑬𝑵(𝒖) Physical system COMSOL model ( N ODE ) Output vector 𝒁𝑺𝑵(𝑢, 𝑮, 𝑯, 𝑰) Reduced model 𝑌 𝑢 = 𝑮𝑌 𝑢 + 𝑯𝑉(𝑢) 𝒁𝑺𝑵 𝒖 = 𝑰𝑌(𝑢) 𝑜 ODE, 1 ≤ 𝑝 10 ≪ 𝑂 Input vector 𝑽𝑩 Deviation criterion to minimize 𝐾 = 𝒁𝑺𝑵 − 𝒁𝑬𝑵

𝑀2²

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 15 10

• 3

10

• 2

10

• 1

Model order

id

n , K

100000 200000 300000 400000 500000 600000 2 5 Time, s Heat power, W P1 P2 P3 P4 A1 A2 A3 A4

(a)

𝑉 𝑢 = 𝐵1(𝑢) ⋮ 𝐵4(𝑢) 𝑄

1(𝑢)

⋮ 𝑄

4(𝑢)

𝒁 𝑢 = 𝑈

1

⋮ 𝑈

18

𝝉𝒋𝒆𝒐 = (𝒁𝒔𝒏𝒋 𝒖𝒌 − 𝒁𝒋

𝑬𝑵 𝒖𝒌 )𝟑 𝑶𝒖 𝒌=𝟐 𝒓 𝒋=𝟐

𝒓 × 𝑶𝒖

Quadratic criterion: Where: 𝑟 is the number of observables 𝑂𝑢 the number of time steps

Identification of the RM

8

0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 100000 200000 300000 400000 500000 600000 0.2 0.4 0.6 0.8 Time, s Temperature, K T5(DM) T5(RM 5) T5(RM 10) T6(DM) T6(RM 5) T6(RM 10) T7 (DM) T7(RM 5) T7(RM 10) T8(DM) T8(RM 5) T8(RM 10) (a) (b) (c) (d)

State feedback control:

9

𝑌 𝑢 = 𝐺𝑌 𝑢 + 𝐻𝐵𝑉𝐵 𝑢 + 𝐻𝑄𝑉𝑄(𝑢) 𝑍𝑆𝑁 𝑢 = 𝐼𝑌(𝑢) 𝑎 𝑢 = 𝑈5, 𝑈6, 𝑈7, 𝑈8 𝑈 = 𝐼𝑨𝑌(𝑢)

system

Model Predictive Controller Linear Quadratic Estimator ( Kalman filter)

Disturbance 𝑉𝑄 Actuators 𝑉𝐵

𝑍𝑛

𝑌

Reduced Model

Estimated state

𝑎(𝑢)

Temperatures to control : 𝑉𝐵 𝑢 = 𝐵1 𝑢 , 𝐵2 𝑢 , 𝐵3 𝑢 , 𝐵4 𝑢

𝑈

𝑉𝑄 𝑢 = 𝑄

1 𝑢 , 𝑄2 𝑢 , 𝑄3 𝑢 , 𝑄 4 𝑢 𝑈

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 𝑂𝑞. 𝑙 𝑙 + 𝑂𝑞

Prediction horizon past future Predicted output Optimal input Reference trajectory

Dynamical system issued from time discretization of the state-space representation: Z 𝑙 = 𝜴𝑌(𝑙) + 𝜟𝑉𝐵(𝑙 − 1) + 𝜤UA(𝑙)

• 𝑙 is the current time index
• 𝑂𝑞 is the prediction horizon

Z(k)= 𝑎(𝑙 + 1) ⋮ 𝑎(𝑙 + 𝑂𝑞) UA(k)= ∆𝑉𝐵(𝑙) ⋮ ∆𝑉𝐵 (𝑙 + 𝑂𝑞 − 1)

Model Predictive Control

11

The control law issued from the minimization of a quadratic functional is : UA(𝑙) = (𝜤𝑈𝜤 + 𝜇𝐽)−1𝜤𝑈 Z𝑠𝑓𝑔 𝑙 − 𝜴𝑌 𝑙 − 𝜟𝑉𝐵 𝑙 − 1 𝐾 = Z𝑼Z + 𝜇UA

𝑼UA

The performance index to minimize : 𝜇 is a penalty parameter. 𝑌(𝑙) 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

𝜏𝑨 = 1 4 × 𝑂𝑢 (𝑎𝑗 𝑢𝑘 )2

𝑗∈{1,4} 𝑂𝑢 𝑘=1 1/2

Control parameters : Temperatures to control: 𝑈5, 𝑈6, 𝑈7, 𝑈8 Reduced model: RM10 Control time step: ∆𝑢 = 1𝑡 Prediction horizon: 𝑂𝑞 = 1 Standard deviation of the measurement noise: 𝜏𝑛 = 0.002 𝐿 The mean quadratic discrepancy between the desired (0K) and the

• btained temperatures deviations:

• 4

4 Perturbation, W 1000 2000 3000 4000 5000 6000 7000

• 2

2 Control, W P1 P2 P3 P4 A1 A2 A3 A4

• 3

3 Power, W A1 A2 A3 A4

(c) (d) (e)

Controlled phase Uncontrolled phase

Test case 1

13

𝜏𝑞 = 2.99 𝑋 Standard deviation of the perturbation:

Without control With control

𝝉𝒜, 𝑳 0.1051 0.0065 0.0024

• 0.2
• 0.1

0.1 0.2 Temperature, K 1000 2000 3000 4000 5000 6000 7000

• 0.2
• 0.1

0.1 0.2 Time(s) Temperature, K T5 T6 T7 T8 T5 T6 T7 T8

𝝁𝟐 = 𝟐𝟏−𝟒 𝝁𝟑 = 𝟐𝟏−𝟕

Test case 2

14

Standard deviation of the perturbation: 𝜏𝑞 = 6.42𝑋

• 10
• 5

5 10 Perturbation, W

• 5
• 2

2 5 Control, W 1000 2000 3000 4000 5000 6000 7000

• 5

2 2 5 Control, W P1 P2 P3 P4 A1 A2 A3 A4 A1 A2 A3 A4 Controlled phase Uncontrolled phase

• 0.4

0.4 0.8 Temperature, K 1000 2000 3000 4000 5000 6000 7000

• 0.4

0.4 0.8 Time, s Temperature, K T5 T6 T7 T8 T5 T6 T7 T8

Without control With control

𝝉𝒜, 𝑳 0.4572 0.0120 0.0041 𝝁𝟐 = 𝟐𝟏−𝟒 𝝁𝟑 = 𝟐𝟏−𝟕

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: 𝑌 𝑢 = 𝐺𝑌 𝑢 + 𝐻𝐵𝑉𝐵 𝑢 + 𝐿

𝑔(𝑍𝑆𝑁 𝑢 − 𝐼𝑌

) The correction is done through the Kalman gain given by : 𝐿

𝑔 = 1

𝛽2 𝑇𝐼𝑈 Where 𝑇 (∈ ℝ𝑜×𝑜) is the solution of Riccati equation: 𝑇𝐺𝑈 + 𝐺𝑇 − 1 𝛽2 𝑇𝐼𝑈𝐼𝑇 + 𝐻𝑄𝐻𝑄

𝑈 = 0

𝛽 =

𝜏𝑛 𝜏𝑞 is the ratio between the standard deviations of measurements and

heat source disturbances