- General issues, passive vs active - Control issues: optimality and learning vs robustness and rough model

Transcription

1 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

2 A brief introduction to time-delay systems examples Smith s predictor special features a bit of mathematics

3 A telling example 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 Natural loop of audio-phonatory control

4 A telling example G Networked loop of audio-phonatory ctrl

5 and a palpable one. Current interactive systems often take between 50 to 200ms to update the display in response to touch input.

6 as well as

7 Some classical example Strejc-Broïda models for process control frequently used in process engineering (inertial phenomena) simple and generic approximation (if there is no oscillation / instability) PID control?... OK if t > 5T, poor effectiveness at t < T Smith predictor or «Generalized PID» (only if open-loop stability) heating + thermic transfer T Victor Broïda (Fr) 1969 The determination of large time-constants by step-response extrapolation. Automatica 5(5): [IFAC President ] Vladimír Strejc (Cz) 1965 The physical realizability of an optimum Ν-parameter, discrete, linear control system determined in Wiener's sense. Kybernetika 1(5): Exemple du GV LAGIS

8 Another classics... of control engineering classes (1900 s) T T PID ok

9 More spectacular Tele surgery: the Lindbergh operation, 07/09/2001 Constant RTT < 200 msec Distance: 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

10 Networked control and communication delays one week of RTT RTT (40km) Mean = 82 ms Maxi = 857 ms Mini = 1 ms France - France

11 Networked control and communication delays one week of RTT RTT (1640km) Maxi = 415 ms Mini = 70 ms France North-Africa

12 Our approach ( ) transmission time + access time + packet loss + sampling = 2 variable delays known / unknown? can be estimated by Plant (time stamps + packet nb) 1 - Controller Actuating 2 - Plant variable delay h 1 channel Network Measurement variable delay h 2 channel can be estimated by Controller (time stamps + packet nb) Hyp: Clock Synchro NTP, GPS

13 Our approach ( )

14 Networked control and communication delays Other RTT approximated values: unshared CAN 2m: bluetooth 2m: Internet: orbital stations: underwater 1.7km: 200 µsec 40 msec msec sec >2 sec

15 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?

16 Alternative to PID: the Smith Predictor Controller structure:? Equivalent scheme:

17 Alternative to PID: the Smith Predictor Pros: Cons: Very simple Restricted use! o H 1 = PID tuning o needs Open-Loop stability o case of a PI controller H 1 «PIR» o constant delay, known Controller structure: Equivalent scheme:

18 Interpretation of «Predictor» predict = «advance» time using the model Various techniques are stemming from this idea, including Smith s

19 Interpretation of «Predictor» such as the «Artstein s transformation» equivalence of controllability

20 Much a do about delay? x t delay x h t

21 A crude example target angle x c = 0 gap e = 0 x voltage u drive measured angle x + - speed

22 A crude example target angle x c = e (t ) ctrl.channel. delay~h/2 received control e(t-h/2) u (t ) drive measured angle x speed received angle x(t-h/2) meas.channel delay~h/2

23 A crude example Exercise for my students, don t worry ;-)? t 0 x 0 x? t 1 t 2 t 3 t 4 t 6 0 t 5 t 7 t 8 t t 9 t 10

24 A crude example w.r.t.?

25 (parenthèse...) Last, note that delay may also stabilize

26 Back to the crude example notion of «state»? initial variable X(t) generating a unique solution from time t

27 Back to the crude example (Shimanov s notation, 1960) t notion of «state»? initial variable X(t) generating a unique solution from time t function x t = state at time t vector x(t) = x t (0) solution at t Functional state x t 72 infinite dim. syst.

28 Back to the crude example Im(s) poles? Re(s) infinite number of poles infinite dim. syst.

29 Back to the crude example + - (BO) + - gain x frequency behaviour? (Bode, open loop) phasis log w log w x h t phasis = -p/2-jhw h t phasis - J.P. RICHARD infinite dim.

30 «Crude», but not that simple? 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 rst order derivative What about variable delays h(t)? a counter-example...

31 and mind the variable delays! b for = p/ a variable : asymptot. stable iff yellow zone : (T=1) -2 if if 1 stable h(t)<1 - unstable h=cte<1 constant : h [0,1] iff grey zone 2 unstable h(t)<1 - stable h=cte<1 Note that such a delay is very very classic, guess what it represents?

32 Yes! It corresponds to any sampling effect, even in non-periodic situation x(t k ) = x(t -[t - t k ]) = x(t - h(t)) 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 &q=input+delay+approach&btng=

33 Yes! It corresponds to any sampling effect, even in non-periodic situation Thus another statement of the packet loss problem delay 1 lost packet 2 lost packets 0 h(t) h max maximum nb of successively lost packets h max piecewise-continuous delay with d dt h(t) 1

34 Time-varying sampling: any consequence? [Zhang, Branicky, Phillips. - IEEE Ctrl.Syst.Mag. 2001]

35 Time-varying sampling: any consequence? [Gu, Kharitonov, Chen - Birkhauser 2003]

36 Stability of TDS in the linear time-invariant case Exemple 1: Exemple 2: Im(s) Instable (et «dégénéré») Re(s)

37 Stability of TDS in the linear time-invariant case Exemple 1: Exemple 2: Im(s) Instable (et «dégénéré») Re(s)

38 Stability of TDS in the linear time-invariant case Exemple 1: Exemple 2: + + Instable (and «degenerate»)

39 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 (ω 2 ) t croissant stabilise ω j 2 ω 2

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41 est instable ;

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43 Stability : 1rst Lyapunov s method «small movements approximation»

44 Stability: case of small delays «small delays approximation»