Flow Control: TD4-2

- Control issues: optimality and learning vs robustness and rough model - Model-based control: linear model, nonlinear control

- Linear model, identification - Sliding Mode Control

- Delay effect

- Time-delay systems - Introduction to delay systems

- Examples

- Much a do about delay? Some special features + a bit of maths - Time-varying delay

- Model-based control: nonlinear model, nonlinear control - Overview of MF’s PhD: Sliding Mode Control - Application to the airfoil

- Application to the Ahmed body (MF and CC’PhDs)

…

- Machine Learning and model-free control: + 4h with Thomas Gomez

Baptiste Plumejeau Marc Lippert ¹

Laurent Keirsbulck ¹ Franck Kerhervé ³ Andrey Polyakov ² Jean-Pierre Richard ² Jean-Marc Foucaut

¹ LAMIH, ² CRIStAL, INRIA, ³ pPRIME, ⁴ IPSA, ⁵ LMFL.

Rear face

PROBLEM

Flow configuration

𝐹_𝐷 =

𝑆_𝑤

𝑃𝑡_∞ − 𝑃𝑡_𝑠𝑤 𝑑𝜎+1 2𝜌𝑉_∞²

𝑆_𝑤

𝑉_𝑧² 𝑉_∞²+𝑉_𝑧²

𝑉_∞² 𝑑𝜎 −1 2𝜌𝑉_∞²

𝑆_𝑤

1 − 𝑉_𝑥 𝑉_∞

2

𝑑𝜎

3

PROBLEM

Active control

Methods of flow control

Passive control

(Small variation in the geometric configuration) Limitations of design requirements

C_AIR LOUNGE

4

PROBLEM

Physical system: Case I – Ahmed body

Physical system: Case II – Vehicle

Characteristics:

Closed-loop wind tunnel

Max. velocity 60m/s 200km/h)

Optimal test section:

2m x 2m, length 10m

5

𝐻 = 0.135𝑚 𝑊 = 0.170𝑚 𝐿 = 0.370𝑚

𝑟 = 0.05𝑚 𝒈 = 𝟎. 𝟎𝟑𝟓𝒎

𝑈

= 10𝑚/𝑠

𝑅𝑒

= 9𝑥10

W

H

L

7

W

H

L

2000 double frame pictures 7Hz repetition rate Interrogation window 16x16𝑝𝑖𝑥𝑒𝑙𝑠² 50% overlapping

PIV domain

3.7H x 1.8H

6

Jet velocity 𝑉_𝑗

Steady jet velocity 𝑉_𝑗0 = 16𝑚/𝑠

8

18

𝑠 𝑡 = 𝑓 𝑠, 𝑡 + 𝑔 𝑠, 𝑡 . 𝑏(𝑡)

Actuation 𝑏(𝑡)

Experimental plant

Controller

Sensor 𝑠 𝑡 = 𝐹_𝑑(𝑡)

ref 𝑠^∗

19

Feingesicht et al. (2017) Int. J. Robust Nonlinear Control

Experimental plant Control law

𝑃(𝑡)

𝐹_𝑑(𝑡)

Sensors

𝑈_∞

𝑏 𝑡 = 1 𝑖𝑓 𝜎 𝑡 − 𝜎^∗ < 0 0 𝑖𝑓 𝜎 𝑡 − 𝜎^∗ > 0

𝑆^∗= 𝐹_𝑑

20

• Wait for s(t) to arrive to s* • Keep s(t) at s*

21

Sensors

22

𝑃(𝑡)

𝐹_𝑑(𝑡)

𝑈_∞

𝑏 𝑡 = 1 𝑖𝑓 𝜎 𝑡 − 𝜎^∗ < 0 0 𝑖𝑓 𝜎 𝑡 − 𝜎^∗ > 0

𝑆^∗= 𝐹_𝑑

Reaching phase Sliding phase

𝑠 𝑡 = 𝑓 𝑠, 𝑡 + 𝑔 𝑠, 𝑡 . 𝑏(𝑡)

23

• Wait for s(t) to arrive to s* • Keep s(t) at s*

Feingesicht et al. (2017) Int. J. Robust Nonlinear Control

t

𝑠 𝑡 = 𝛼𝑠(𝑡) + 𝛽𝑏(𝑡) 𝜎

= 𝑠(𝑡)

𝑏 𝑡 =

1 𝑖𝑓 𝜎 𝑡 − 𝜎

< 0 0 𝑖𝑓 𝜎 𝑡 − 𝜎

> 0

𝑠 𝑡 = 𝑓 𝑠, 𝑡 + 𝑔 𝑠, 𝑡 . 𝑏(𝑡) 𝑠 𝑡 = 𝛼𝑠(𝑡) + 𝛽𝑏(𝑡)

t

𝑏 𝑡 =

0 𝑖𝑓 𝜎 𝑡 − 𝜎

> 0 𝜎

= 𝑠(𝑡)

25

𝑠 𝑡 = 𝑓 𝑠, 𝑡 + 𝑔 𝑠, 𝑡 . 𝑏(𝑡) 𝑠 𝑡 = 𝛼𝑠(𝑡) + 𝛽𝑏(𝑡)

t

𝑏 𝑡 =

1 𝑖𝑓 𝜎 𝑡 − 𝜎

< 0 0 𝑖𝑓 𝜎 𝑡 − 𝜎

> 0 𝑠 𝑡 = 𝛼𝑠(𝑡) + 𝛽𝑏(𝑡 − ℎ)

𝜎

= 𝑠 𝑡 + 𝛽

𝑡−ℎ 𝑡

𝑏 𝑝 𝑑𝑝

26

𝑠 𝑡 = 𝑓 𝑠, 𝑡 + 𝑔 𝑠, 𝑡 . 𝑏(𝑡) 𝑠 𝑡 = 𝛼𝑠(𝑡) + 𝛽𝑏(𝑡)

t

Sensor with delay

𝑠 𝑡 = 𝛼

𝑠 𝑡 − ℎ − 𝛼

𝑠 𝑡 + (𝛽 − 𝛾𝑠 𝑡 − ℎ + 𝛾 𝑡 − 𝜏 )𝑏(𝑡 − ℎ)

Sensor without delay

𝐹𝐼𝑇 % = 1 −

𝑆

− 𝑆

2

𝑆

− 𝑆

2

𝑥 100% = 53%

𝛼₁ = 27.37 𝛼₂ = 32.70 𝛽 = 1.97 𝛾 = 1.92 𝜏 = 0.18 ℎ = 0.01

Feingesicht et al. (2017) Int. J. Robust Nonlinear Control

𝑠 𝑡 = 𝛼𝑠(𝑡) + 𝛽𝑏(𝑡 − ℎ)

𝜎^∗ = 𝑠 𝑡 + 𝛾

𝑡−𝜏+ℎ 𝑡

𝑠 𝑝 𝑑𝑝 +

𝑡−ℎ 𝑡

𝛼₁𝑠 𝑝 + (𝛽 − 𝛾𝑠 𝑝 + 𝛾𝑠 𝑝 − 𝜏 + ℎ 𝑏(𝑝))𝑑𝑝

𝑏 𝑡 =

1 𝑖𝑓 𝜎 𝑡 − 𝜎

< 0 0 𝑖𝑓 𝜎 𝑡 − 𝜎

> 0

28 𝑡⁺ = 𝑡 𝑈_∞/𝐻

PROBLEM

Physical system: Case I – Ahmed body

Physical system: Case II – Vehicle

Race track located at Clastres (North of France)

𝐹_𝐷 =

𝑆_𝑤

𝑃_𝑡^∞ − 𝑃_𝑡^𝑆𝑤 𝑑𝜎 − 𝜌𝑈_∞² 2 𝑆_𝑤

𝑈_𝑦² 𝑈 _∞

2

+ 𝑈_𝑧² 𝑈 _∞

2

𝑑𝜎 + 𝜌𝑈_∞² 2 𝑆_𝑤

1 − 𝑈_𝑥² 𝑈 _∞

2

𝑑𝜎 Onorato’s relation

Onorato, M. et al (1984) SAE, SP-569,International congress and Exposition, Detroit: pp. 85-93

𝐹_𝐷 =

𝑆_𝑤

𝑃_𝑡^∞ − 𝑃_𝑡^𝑆𝑤 𝑑𝜎 Onorato’s relation

Direction I: departure to arrival Direction II: arrival to departure

Overview

- General issues, passive vs active…

- Control issues: optimality and learning vs robustness and rough model - Model-based control: linear model, nonlinear control

- Linear model, identification - Sliding Mode Control

- Delay effect

- Time-delay systems - Introduction to delay systems

- Examples

- Much a do about delay? Some special features + a bit of maths - Time-varying delay

- Model-based control: nonlinear model, nonlinear control - Overview of MF’s PhD: Sliding Mode Control - Application to the airfoil

- Application to the Ahmed body (MF and CC’PhDs)

…

- Machine Learning and model-free control: + 4h with Thomas Gomez

Conferences:

Nonlinear Control for Turbulent Flows, IFAC 2017 World Congress, Toulouse, July 2017 M. Feingesicht, A. Polyakov, F. Kerherve and J. P. Richard

Model-Based Feedforward Optimal Control Applied to a Turbulent Separated Flow, IFAC 2017 M. Feingesicht, A. Polyakov, F. Kerherve and J. P. Richard

A bilinear input-output model with state-dependent delay for separated flow ctrl, ECC 2016 Aalborg M. Feingesicht, C. Raibaudo, A. Polyakov, F. Kerherve and J. P. Richard

Reducing Car Consumption by Means of a Closed-loop Drag Control C. Chovet, M. Feingesicht et al., V^EHICULAR2018, Venice

Patent:

Dispositif de contrôle actif du recollement d’un écoulement sur un profil.

M. Feingesicht, A. Polyakov, F. Kerherve and J. P. Richard

Book:

Mathématiques pour l’ingénieur (chap.6: Systèmes à retard) https://hal.archives-ouvertes.fr/hal-00519555

J. P. Richard et al., 2009 (available online)

Spotter Twingo GT

https://youtu.be/V1BQaCjO7to

Dreams…