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

Title: Model reduction and thermal regulation by Model Predictive Control of a new cylindricity measurement apparatus K. Bouderbala M. Girault E. Videcoq H. Nouira D. Petit J. Salgado 1 Laboratoire Commun de M é trologie (LNE-CNAM), Laboratoire National de M é trologie 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

Contents 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

Context Issues : • Heat dissipation • Thermal dilatation Tools:  Reduced model built by Modal Identification Method  Model Predictive Control Objective : Real time control to reduce the effects of temperature variation 3

Model reduction Thermal modeling : Resolution of the heat transfer equation Energy balance: 𝑜 𝑅 + 𝑄 𝑘 𝑢 = 𝜍 𝑁 𝐷 𝑞 𝑁 𝜖𝑈 Spatial discretization: - Finite elements 𝛼. 𝑙 𝑁 𝛼𝑈 𝑁, 𝑢 𝜓 𝑘 (𝑁) 𝜖𝑢 𝑁, 𝑢 , 𝑊 𝑘 - Finite volumes 𝑘=1 ∀ 𝑁 ∈ 𝛻 - Finite differences Boundary conditions : State space representation: 𝑙 𝑁 𝛼𝑈 𝑁, 𝑢 . 𝑜 = ℎ 𝑈 𝑏 𝑢 − 𝑈 𝑁, 𝑢 , ∀𝑁 ∈ Γ 𝑼 = 𝑩𝑼 𝒖 + 𝑪𝑽 𝒖 𝑙 𝑁 𝛼𝑈 𝑁, 𝑢 . 𝑜 = 𝐵 𝑗 (𝑢)𝜊 𝑗 (𝑁) 𝒁 𝒖 = 𝑫𝑼(𝒖) 𝑇 𝑗 N ODE 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) 4

Model Identification Method (MIM) 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 optimization algorithms:  Ordinary Linear Least Squares ( 𝑰 )  Particle Swarm Optimization ( 𝑮 , 𝑯 ) 5

Model Identification Method Physical system Observables: COMSOL Output vector simulated model 𝒁 𝑬𝑵 (𝒖) Known vector ( N ODE ) Input vector 𝑽 𝑩 Deviation criterion to minimize 𝒁 𝑺𝑵 − 𝒁 𝑬𝑵 𝐾 = 𝑀 2 ² Reduced model Boundary conditions, 𝑌 𝑢 = 𝑮𝑌 𝑢 + 𝑯𝑉(𝑢) sources 𝒁 𝑺𝑵 𝒖 = 𝑰𝑌(𝑢) Output vector 𝑜 ODE, 1 ≤ 𝑝 10 ≪ 𝑂 𝒁 𝑺𝑵 (𝑢, 𝑮, 𝑯, 𝑰) Optimization algorithms Iterative procedure Model Identification Method scheme 6

Identification of the RM 𝐵 1 (𝑢) (a) P 1 P 2 P 3 P 4 A 1 A 2 A 3 A 4 5 ⋮ 𝑈 1 Heat power, W 𝐵 4 (𝑢) ⋮ 𝒁 𝑢 = 𝑉 𝑢 = 𝑄 1 (𝑢) 𝑈 18 ⋮ 2 𝑄 4 (𝑢) 0 0 100000 200000 300000 400000 500000 600000 -1 Time, s 10 Quadratic criterion: 𝑬𝑵 𝒖 𝒌 ) 𝟑 𝒓 𝑶 𝒖 (𝒁𝒔𝒏 𝒋 𝒖 𝒌 − 𝒁 𝒋 𝝉 𝒋𝒆𝒐 = 𝒋=𝟐 𝒌=𝟐 𝒓 × 𝑶 𝒖 n , K -2 10  id Where: 𝑟 is the number of observables 𝑂 𝑢 the number of time steps -3 10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Model order

Identification of the RM 0.8 T 5 (DM) T 5 (RM 5) T 5 (RM 10) (a) 0.6 0.4 0.2 0 0.8 T 6 (DM) T 6 (RM 5) T 6 (RM 10) Temperature, K (b) 0.6 0.4 0.2 0 0.8 T 7 (DM) T 7 (RM 5) T 7 (RM 10) (c) 0.6 0.4 0.2 0 0.8 T 8 (DM) T 8 (RM 5) T 8 (RM 10) (d) 0.6 0.4 0.2 0 0 100000 200000 300000 400000 500000 600000 Time, s 8

State feedback control: 𝑌 𝑢 = 𝐺𝑌 𝑢 + 𝐻 𝐵 𝑉 𝐵 𝑢 + 𝐻 𝑄 𝑉 𝑄 (𝑢) Temperatures to control : 𝑎 𝑢 = 𝑈 5 , 𝑈 6 , 𝑈 7 , 𝑈 8 𝑈 = 𝐼 𝑨 𝑌(𝑢) 𝑍 𝑆𝑁 𝑢 = 𝐼𝑌(𝑢) Where : 𝑈 𝑉 𝐵 𝑢 = 𝐵 1 𝑢 , 𝐵 2 𝑢 , 𝐵 3 𝑢 , 𝐵 4 𝑢 = input vector of actuators (surface heaters) 𝑈 𝑉 𝑄 𝑢 = 𝑄 1 𝑢 , 𝑄 2 𝑢 , 𝑄 3 𝑢 , 𝑄 4 𝑢 = input vector of perturbations (laser interferometers) Disturbance 𝑉 𝑄 𝑎(𝑢) Actuators 𝑉 𝐵 system 𝑍 𝑛 Model Linear Quadratic Estimated state Predictive Estimator ( Kalman 𝑌 Controller filter) Reduced Model 9

Model predictive control Prediction horizon Principle : past future A model of the system is used to predict plants Reference trajectory behavior and choose the best control in the sense of some cost function within constraints. Predicted output The future response of the plant is predicted over a Optimal input Prediction horizon 𝑂 𝑞 . 𝑙 𝑙 + 𝑂 𝑞 Dynamical system issued from time discretization of the state-space representation: Z 𝑙 = 𝜴𝑌(𝑙) + 𝜟𝑉 𝐵 (𝑙 − 1) + 𝜤 UA (𝑙) 𝑙 is the current time index • 𝑂 𝑞 is the prediction horizon • 𝑎(𝑙 + 1) ∆𝑉 𝐵 (𝑙) ⋮ ⋮ Z (k)= UA (k)= 𝑎(𝑙 + 𝑂 𝑞 ) ∆𝑉 𝐵 (𝑙 + 𝑂 𝑞 − 1) 10

Model Predictive Control The matrices Ψ, Γ, Θ depend on the matrices F,G,H of the reduced model. The performance index to minimize : 𝐾 =  Z 𝑼  Z + 𝜇 UA 𝑼 UA The control law issued from the minimization of a quadratic functional is : UA (𝑙) = (𝜤 𝑈 𝜤 + 𝜇𝐽) −1 𝜤 𝑈 Z 𝑠𝑓𝑔 𝑙 − 𝜴𝑌 𝑙 − 𝜟𝑉 𝐵 𝑙 − 1 𝜇 is a penalty parameter. 𝑌(𝑙) is obtained by using a linear quadratic estimator (Kalman filter) 11

Control test case : Control parameters : Temperatures to control: 𝑈 5 , 𝑈 6 , 𝑈 7 , 𝑈 8 Reduced model: RM10 Control time step: ∆𝑢 = 1𝑡 Prediction horizon: 𝑂 𝑞 = 1 Standard deviation of the measurement noise: 𝜏 𝑛 = 0.002 𝐿 The mean quadratic discrepancy between the desired (0K) and the obtained temperatures deviations: 1/2 𝑂 𝑢 1 (𝑎 𝑗 𝑢 𝑘 ) 2 𝜏 𝑨 = 4 × 𝑂 𝑢 𝑘=1 𝑗∈{1,4} 12

Test case 1 Standard deviation of the perturbation: P 1 P 2 P 3 P 4 Perturbation, W 4 𝜏 𝑞 = 2.99 𝑋 0 T 5 T 6 T 7 T 8 -4 0.2 Temperature, K A 1 A 2 A 3 A 4 3 (c) 0.1 Power, W 0 0 𝝁 𝟐 = 𝟐𝟏 −𝟒 -0.1 (d) -3 -0.2 T 5 T 6 T 7 T 8 A 1 A 2 A 3 A 4 0.2 2 Temperature, K Control, W (e) 0.1 0 0 𝝁 𝟑 = 𝟐𝟏 −𝟕 -0.1 -2 -0.2 0 1000 2000 3000 4000 5000 6000 7000 0 1000 2000 3000 4000 5000 6000 7000 Time(s) Uncontrolled phase Controlled phase Without With control control 𝝉 𝒜 , 𝑳 0.1051 0.0065 0.0024 13

Test case 2 P 1 P 2 P 3 P 4 Perturbation, W Standard deviation of the perturbation: 10 5 𝜏 𝑞 = 6.42𝑋 0 -5 -10 T 5 T 6 T 7 T 8 0.8 A 1 A 2 A 3 A 4 Temperature, K 5 Control, W 0.4 2 0 -2 𝝁 𝟐 = 𝟐𝟏 −𝟒 0 -5 -0.4 A 1 A 2 A 3 A 4 T 5 T 6 T 7 T 8 0.8 5 Control, W Temperature, K 2 0.4 0 2 𝝁 𝟑 = 𝟐𝟏 −𝟕 -5 0 0 1000 2000 3000 4000 5000 6000 7000 -0.4 Uncontrolled phase 0 1000 2000 3000 4000 5000 6000 7000 Controlled phase Time, s Without With control control 𝝉 𝒜 , 𝑳 0.4572 0.0041 0.0120 14

Conclusions and prospects • 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 15

Thank you for your attention

Linear quadratic estimator State estimation: 𝑔 (𝑍 𝑆𝑁 𝑢 − 𝐼𝑌 𝑢 = 𝐺𝑌 𝑢 + 𝐻 𝐵 𝑉 𝐵 𝑢 + 𝐿 ) 𝑌 The correction is done through the Kalman gain given by : 𝑔 = 1 𝛽 2 𝑇𝐼 𝑈 𝐿 Where 𝑇 ( ∈ ℝ 𝑜×𝑜 ) is the solution of Riccati equation: 𝑇𝐺 𝑈 + 𝐺𝑇 − 1 𝑈 = 0 𝛽 2 𝑇𝐼 𝑈 𝐼𝑇 + 𝐻 𝑄 𝐻 𝑄 𝜏 𝑛 𝛽 = 𝜏 𝑞 is the ratio between the standard deviations of measurements and heat source disturbances 17