Flow Control: TD#3

Flow Control: TD#3

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

• examples

• Smith’s predictor

• special features

• a bit of mathematics

A brief introduction to time-delay systems

A telling example…

Natural loop of audio-phonatory control

TO BE OR NOT TO BE, THAT IS THE QUESTION: WHETHER ‘TIS

NOBLER IN THE MIND TO SUFFER THE SLINGS AND ARROWS OF OUTRAGEOUS FORTUNE. OR TO TAKE ARMS AGAINST A SEA OF TROUBLES, AND BY OPPOSING, END

THEM? TO DIE: TO SLEEP; NO MORE; AND BY A SLEEP

TO SAY WE END

Brief introduction to time-delay systems

A telling example…

Networked loop of audio-phonatory ctrl

G…

Brief introduction to time-delay systems

… and a palpable one.

Current interactive systems often take between 50 to 200ms to update the display in response to touch input.

Brief introduction to time-delay systems

as well as…

Brief introduction to time-delay systems

Strejc-Broïda models for process control

 frequently used in process engineering

 simple and generic approximation

 PID control? ... OK if t > 5T , poor effectiveness at t <

 Smith predictor or « Generalized PID »

Exemple du GV LAGIS

T

heating +

thermic transfer

Victor Broïda (Fr) 1969 The determination of large time-constants by step-response extrapolation. Automatica 5(5): 677-683 [IFAC President 1969-1972]

Vladimír Strejc (Cz) 1965 The physical realizability of an optimum Ν-parameter, discrete, linear control system determined in Wiener's sense. Kybernetika 1(5): 399-409

Brief introduction to time-delay systems

Some classical example

... of control engineering classes (1900’s)

PID ok

T

Brief introduction to time-delay systems

Another classics

T

Tele surgery: the Lindbergh operation,

07/09/2001

Constant RTT

< 200 msec Distance:

17 000 km

Com. cost ≈ 160k$/month

« The only restriction to the development of long-distance tele-surgery has to do, still today, with its cost. For tele-surgery, you must use a transcontinental ATM line, that you have to book during 6 monthes, at the price of about 1 million dollars. » Prof. J. Marescaux, Le Monde, January 6, 2010

Brief introduction to time-delay systems

More spectacular…

Brief introduction to time-delay systems

Networked control and communication delays

RTT (40km) Mean = 82 ms Maxi = 857 ms Mini = 1 ms

France - France

one week of RTT…

Brief introduction to time-delay systems

Networked control and communication delays

RTT (1640km) Maxi = 415 ms Mini = 70 ms RTT (1640km) Maxi = 415 ms Mini = 70 ms

France – North-Africa

one week of RTT…

Brief introduction to time-delay systems

transmission time + access time + packet loss + sampling…

Measurement channel

1 - Controller

2 - Plant

channel

Network

variable delay h

variable delay h

= 2 variable delays

can be estimated by Plant (time stamps + packet nb)

can be estimated

by Controller (time stamps + packet nb)

known / unknown ?

Hyp: Clock Synchro NTP, GPS



Our approach (2008-2012)

Brief introduction to time-delay systems



Our approach (2008-2012)

Brief introduction to time-delay systems

Networked control and communication delays

unshared CAN 2m: 200 µsec

bluetooth 2m: 40 msec

Internet: 100-400 msec

orbital stations: 0.4-7 sec underwater 1.7km: >2 sec

Other RTT approximated values:

Brief introduction to time-delay systems

Alternative to PID: the Smith Predictor

Example of a delayed measurement (sensor)

How could you implement a PID?

Use a simulation model?

Could you

make it robust?

Brief introduction to time-delay systems

Alternative to PID: the Smith Predictor

Equivalent scheme:

Controller structure:

?

Brief introduction to time-delay systems

Alternative to PID: the Smith Predictor

Equivalent scheme:

Controller structure:

Pros:

 Very simple

o H

= PID tuning

o case of a PI controller H

« PIR »

Cons:

 Restricted use!

o needs Open-Loop stability

o constant delay, known

predict = « advance » time using the model

Brief introduction to time-delay systems

Interpretation of « Predictor »

 Various techniques are stemming from this idea, including Smith’s

 such as the « Artstein’s transformation » Brief introduction to time-delay systems

Interpretation of « Predictor »

equivalence of controllability

Much a do about delay ?

delay

t x

h

t x

Brief introduction to time-delay systems

Much a do about delay?

drive

voltage u measured angle x

+ -

gap e ^{= 0 –} x

speed target angle

x

= 0

Brief introduction to time-delay systems

A crude example

drive u (t )

measured angle x

+ -

e

t

speed target angle

x

= 0

Brief introduction to time-delay systems

A crude example

ctrl.channel.



delay ~ h/2

meas.channel



delay~ h/2

received angle

x(t-h/2)

received control

e (t-h/2)

t x

x

0

?

?

t

t

t

t

t

t

t

t

t

t

t

Exercise…

Exercise… for my students, don’t worry ;-)

Brief introduction to time-delay systems

A crude example

?

w.r.t.

Brief introduction to time-delay systems

A crude example

(parenthèse...)



Brief introduction to time-delay systems

… Last, note that delay may also stabilize

 notion of « state » ?

initial variable

X(t)

t

Brief introduction to time-delay systems

Back to the crude example

Brief introduction to time-delay systems

Back to the crude example

(Shimanov’s notation, 1960)

function x

= state at time t vector x ( t ) = x

(0) solution at t

t

72

infinite dim. syst.

 notion of « state » ?

initial variable

X(t)

t

Functional state x

Brief introduction to time-delay systems

Back to the crude example

Re(s) Im(s)

poles?

infinite dim. syst.

infinite number of poles

© J.P. RICHARD 2011 74

frequency behaviour?

infinite dim.

phasis - 

phasis ^ 

+-

(BO)

+-

h

t x

h

t

(Bode, open loop)

log w gain

log w phasis

 = -p/2-jhw

Brief introduction to time-delay systems

Back to the crude example

Let’s sum up...

delay  strong influence on stability functional state

infinite number of eigenvalues (Hurwitz OK, no Routh)

important dephasing (

- )

… and, until now, it was the most simple:

constant delay

scalar, linear system 1

order derivative

What about variable delays h ( t ) ?

a counter-example...

Brief introduction to time-delay systems

« Crude », but not that simple?

1

p/

constant :  h  [0,1] iff  grey zone

variable : asymptot. stable iff  yellow zone :

a b

(T=1)

2

2

0

-2

-2

2

1 2

stable h ( t )

- unstable h =cte

unstable h ( t )

- stable h =cte

=1

for

if if

1 2 3 4 5 6 7 8 91

Brief introduction to time-delay systems

… and mind the variable delays!

Note that such a delay is very – very – classic, guess what it represents?

Brief introduction to time-delay systems

Yes! It corresponds to any sampling effect, even in non-periodic situation

Idea and stability analysis initiated in

[Fridman-Seuret-Richard Automatica 2004]

Then, improved in:

[Fridman Automatica 2010]

[Seuret Automatica 2012]

[Karafyllis, Krstić IEEE TAC 2012]

[Mazenc, Malisoff, Dinh Automatica 2013]

…

See

https://scholar.google.fr/scholar?hl=fr&as_sdt=0%2C5

&q=input+delay+approach&btnG=

x(t

) = x(t -[t - t

] ) = x(t - h(t) )

Thus… another statement of the packet loss problem

• maximum nb of successively lost packets  h

• piecewise-continuous delay with

delay

1 lost packet_

2 lost packets _{ }

0 ≤ h(t) ≤ h _max

h ( t ) ≤ 1

dt d

Brief introduction to time-delay systems

Yes! It corresponds to any sampling effect, even in non-periodic situation

Time-varying sampling: ^ any consequence?

[Zhang, Branicky, Phillips. - IEEE Ctrl.Syst.Mag. 2001]

Brief introduction to time-delay systems

Brief introduction to time-delay systems

[Gu, Kharitonov, Chen - Birkhauser 2003]

Time-varying sampling: ^ any consequence?

Stability of TDS in the linear time-invariant case

Instable (et « dégénéré »)

Exemple 2:

Re(s) Im(s)

Brief introduction to time-delay systems

Instable (et « dégénéré »)

Exemple 2:

Re(s) Im(s)

Stability of TDS in the linear time-invariant case

Brief introduction to time-delay systems

Instable (and « degenerate »)

+ Exemple 1:

Exemple 2:

+

Stability of TDS in the linear time-invariant case

Brief introduction to time-delay systems

Méthode de Walton et Marshall (1987)

Extrait de Borne, Dauphin, Richard, Rotella, Zambettakis

Analyse et régulation des processus industriels - Régulation continue. 495 pages, Edt. Technip 1993

tcroissant stabilise 𝑞(𝜔²)

𝜔²

𝜔_𝑗²

est instable ;

Stability : 1rst Lyapunov’s method

« small movements approximation »

Stability: case of small delays

« small delays approximation »