Plant tuning: a robust Lyapunov approach Franco Blanchini 1 , - - PowerPoint PPT Presentation

▶
plant tuning a robust lyapunov approach
SMART_READER_LITE
LIVE PREVIEW

Plant tuning: a robust Lyapunov approach Franco Blanchini 1 , - - PowerPoint PPT Presentation

Jan 15, 2023 249 likes •1.25k views

Plant tuning: a robust Lyapunov approach Franco Blanchini 1 , Gianfranco Fenu 2 , Giulia Giordano 3 , Felice Andrea Pellegrino 2 1 Dipartimento di Scienze Matematiche, Informatiche e Fisiche University of Udine, Italy; 2 Dipartimento di Ingegneria

slide-1
SLIDE 1

pollo

Plant tuning: a robust Lyapunov approach

Franco Blanchini 1, Gianfranco Fenu 2, Giulia Giordano 3, Felice Andrea Pellegrino 2

1Dipartimento di Scienze Matematiche, Informatiche e Fisiche

University of Udine, Italy;

2Dipartimento di Ingegneria e Architettura

University of Trieste, Italy

3Linnaeus Center and Department of Automatic Control

University of Lund, Sweden

January 10, 2017

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-2
SLIDE 2

pollo

Uumidity-temperature regulation

ω I T h

T = φ(I,ω), h = ψ(I,ω),

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-3
SLIDE 3

pollo

Uumidity-temperature regulation

We know that

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-4
SLIDE 4

pollo

Uumidity-temperature regulation

We know that the temperature T is a decreasing function of the fan speed ω and an increasing function of the current I;

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-5
SLIDE 5

pollo

Uumidity-temperature regulation

We know that the temperature T is a decreasing function of the fan speed ω and an increasing function of the current I; the relative humidity h is a decreasing function of both the fan speed ω and the current I.

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-6
SLIDE 6

pollo

Uumidity-temperature regulation

We know that the temperature T is a decreasing function of the fan speed ω and an increasing function of the current I; the relative humidity h is a decreasing function of both the fan speed ω and the current I. Available information: ∂T

∂I ∂T ∂ω ∂h ∂I ∂h ∂ω

  • ∈

+m1 −m2 −m3 −m4

  • ,

where m1,m2,m3,m4 ∈ [ε,µ].

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-7
SLIDE 7

pollo

Plant tuning without a model

TUNER y u

g

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-8
SLIDE 8

pollo

Plant tuning without a model

TUNER y u

g

∂g ∂u ∈ M =

  • M =

r

∑

i=1

αiMi,

r

∑

i=1

αi = 1, αi ≥ 0

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-9
SLIDE 9

pollo

Problem formulation

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-10
SLIDE 10

pollo

Problem formulation

Problem Given the static plant y = g(u), where g : Rq → Rp, p ≤ q, assume that, for some (unknown) ¯ u, 0 = g(¯ u). and that the following inclusion holds: Gu . = ∂g ∂u

  • ∈ M .

where M is a polytopic set. Find a dynamic algorithm such that y(t) → 0 and u(t) → ¯ u

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-11
SLIDE 11

pollo

Main result

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-12
SLIDE 12

pollo

Main result

Assumption Robust non–singularity. Any matrix in the polytope M is right invertible.

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-13
SLIDE 13

pollo

Main result

Assumption Robust non–singularity. Any matrix in the polytope M is right invertible. Theorem Under Assumption 2, Problem 1 can be solved with a control scheme of the form ˙ u(t) = v(t), v(t) = Φ(y(t)), with added input variable v(t).

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-14
SLIDE 14

pollo

A no–solution

Observation The dynamics of the output y can be described by ˙ y = Gu ˙ u = Guv

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-15
SLIDE 15

pollo

A no–solution

Observation The dynamics of the output y can be described by ˙ y = Gu ˙ u = Guv This would be a driftless system if Gu were known.

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-16
SLIDE 16

pollo

A no–solution

Observation The dynamics of the output y can be described by ˙ y = Gu ˙ u = Guv This would be a driftless system if Gu were known. Take v = −G −1

u y, then

˙ y = −y and y(t) → 0 exponentially (Newton’s method).

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-17
SLIDE 17

pollo

A no–solution

Observation The dynamics of the output y can be described by ˙ y = Gu ˙ u = Guv This would be a driftless system if Gu were known. Take v = −G −1

u y, then

˙ y = −y and y(t) → 0 exponentially (Newton’s method). u(t) converges to ¯ u such that G(¯ u) = 0.

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-18
SLIDE 18

pollo

Sketch of proof p = q

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-19
SLIDE 19

pollo

Sketch of proof p = q

Cheating and regretting ....

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-20
SLIDE 20

pollo

Sketch of proof p = q

Cheating and regretting .... Consider the Lyapunov like function V (y) = 1 2y⊤y, ˙ V = y⊤ ˙ y = y⊤Gu ˙ u = y⊤Guv.

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-21
SLIDE 21

pollo

Sketch of proof p = q

Cheating and regretting .... Consider the Lyapunov like function V (y) = 1 2y⊤y, ˙ V = y⊤ ˙ y = y⊤Gu ˙ u = y⊤Guv. Take the “fake” control v = −γ(y)G −1

u y

˙ V = −γ(y)y⊤y < 0, for y = 0 (∗)

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-22
SLIDE 22

pollo

Sketch of proof p = q

Cheating and regretting .... Consider the Lyapunov like function V (y) = 1 2y⊤y, ˙ V = y⊤ ˙ y = y⊤Gu ˙ u = y⊤Guv. Take the “fake” control v = −γ(y)G −1

u y

˙ V = −γ(y)y⊤y < 0, for y = 0 (∗) Claim There exists v(y,Gu) = γ(y)G −1

u y ≤ ξ such that (*) holds.

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-23
SLIDE 23

pollo

Sketch of proof p = q

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-24
SLIDE 24

pollo

Sketch of proof p = q

Game theoretic–interpretation: (S. Gutman, G. Leitmann 1976)

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-25
SLIDE 25

pollo

Sketch of proof p = q

Game theoretic–interpretation: (S. Gutman, G. Leitmann 1976) Good player (control): chooses v ≤ ξ to minimize ˙ V ;

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-26
SLIDE 26

pollo

Sketch of proof p = q

Game theoretic–interpretation: (S. Gutman, G. Leitmann 1976) Good player (control): chooses v ≤ ξ to minimize ˙ V ; Bad player (misfortune): chooses Gu ∈ M to maximize ˙ V .

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-27
SLIDE 27

pollo

Sketch of proof p = q

Game theoretic–interpretation: (S. Gutman, G. Leitmann 1976) Good player (control): chooses v ≤ ξ to minimize ˙ V ; Bad player (misfortune): chooses Gu ∈ M to maximize ˙ V . µ+ = min

v≤ξ max Gu∈M y⊤Guv

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-28
SLIDE 28

pollo

Sketch of proof p = q

Game theoretic–interpretation: (S. Gutman, G. Leitmann 1976) Good player (control): chooses v ≤ ξ to minimize ˙ V ; Bad player (misfortune): chooses Gu ∈ M to maximize ˙ V . µ+ = min

v≤ξ max Gu∈M y⊤Guv

µ− = max

Gu∈M min v≤ξ y⊤Guv

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-29
SLIDE 29

pollo

Sketch of proof p = q

Game theoretic–interpretation: (S. Gutman, G. Leitmann 1976) Good player (control): chooses v ≤ ξ to minimize ˙ V ; Bad player (misfortune): chooses Gu ∈ M to maximize ˙ V . µ+ = min

v≤ξ max Gu∈M y⊤Guv

µ− = max

Gu∈M min v≤ξ y⊤Guv

This game has a saddle–point µ− = µ+ = y⊤M∗(y)v∗(y)

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-30
SLIDE 30

pollo

Sketch of proof p = q

Game theoretic–interpretation: (S. Gutman, G. Leitmann 1976) Good player (control): chooses v ≤ ξ to minimize ˙ V ; Bad player (misfortune): chooses Gu ∈ M to maximize ˙ V . µ+ = min

v≤ξ max Gu∈M y⊤Guv

µ− = max

Gu∈M min v≤ξ y⊤Guv

This game has a saddle–point µ− = µ+ = y⊤M∗(y)v∗(y) Then v = Φ(y) = v∗(y)

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-31
SLIDE 31

pollo

Facts

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-32
SLIDE 32

pollo

Facts

The computation of v∗ requires solving a convex optimization problem on–line with domain M .

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-33
SLIDE 33

pollo

Facts

The computation of v∗ requires solving a convex optimization problem on–line with domain M . If we take the Euclidean norm, then it is quadratic programming min

M∈M y⊤M2

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-34
SLIDE 34

pollo

Facts

The computation of v∗ requires solving a convex optimization problem on–line with domain M . If we take the Euclidean norm, then it is quadratic programming min

M∈M y⊤M2

If we take the ∞ norm, then it is linear programming min

M∈M y⊤M1

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-35
SLIDE 35

pollo

Facts

The computation of v∗ requires solving a convex optimization problem on–line with domain M . If we take the Euclidean norm, then it is quadratic programming min

M∈M y⊤M2

If we take the ∞ norm, then it is linear programming min

M∈M y⊤M1

v is not continuous, however its integral u is continuous.

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-36
SLIDE 36

pollo

The humidity–temperature regulation continued

J = + − − −

  • ,

ε ≤ Jik ≤ µ

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-37
SLIDE 37

pollo

The humidity–temperature regulation continued

J = + − − −

  • ,

ε ≤ Jik ≤ µ ∆T = gT(I,ω) = (0.1I 2 −0.5ω)−20, ∆h = gω(I,ω) = (−2 10−4I 2 −2.5 10−2ω +0.9)−0.5,

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-38
SLIDE 38

pollo

The umidity–temperature regulation

5 10 15 20 25 30 5 10 15 20 25 t [min] T [oC] 5 10 15 20 25 30 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t [min] h

Temperature T [◦C] (top) and relative humidity h (bottom)

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-39
SLIDE 39

pollo

The umidity–temperature regulation

5 10 15 20 25 30 2 4 6 8 10 12 14 16 18 20 t [min] I, ω

Inputs: u1 = I current (blue), u2 = ω, fan speed (green)

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-40
SLIDE 40

pollo

Electric network

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-41
SLIDE 41

pollo

Electric network

VA VB

  • =
  • 1+ R1

RA + R1 R3

− R1

R3

−R2

R3

1+ R2

RB + R2 R3

−1 V1 V2

  • .
  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-42
SLIDE 42

pollo

Electric network

VA VB

  • =
  • 1+ R1

RA + R1 R3

− R1

R3

−R2

R3

1+ R2

RB + R2 R3

−1 V1 V2

  • .

Let: y1 = VA − ¯ VA, y2 = VB − ¯ VB, y1 y2

  • =

a+c +d b c e +b +d u1 u2

  • −

¯ VA ¯ VB

  • ,
  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-43
SLIDE 43

pollo

Electric network

VA VB

  • =
  • 1+ R1

RA + R1 R3

− R1

R3

−R2

R3

1+ R2

RB + R2 R3

−1 V1 V2

  • .

Let: y1 = VA − ¯ VA, y2 = VB − ¯ VB, y1 y2

  • =

a+c +d b c e +b +d u1 u2

  • −

¯ VA ¯ VB

  • ,

Assume 0 < ε ≤ a,b,c,d,e ≤ µ, the Jacobian is robustly non–singular.

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-44
SLIDE 44

pollo

Electric network

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 1 2 3 4 5 6 7 8 t [s] u [V]

Input variables: u1 [V] (blue), u2 [V] (green).

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-45
SLIDE 45

pollo

Electric network

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.5 1 1.5 2 2.5 t [s] y [V]

Output variables; y1 [V] (blue) and y2 [V] (green) and desired levels (red).

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-46
SLIDE 46

pollo

Flow networks

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-47
SLIDE 47

pollo

Flow networks

    y1 y2 y3 y4     =     1 −1 −1 −1 1 −1 1 −1 1 1 1             φ1(u1) φ2(u2) φ3(u3) φ4(u4) φ5(u5) φ6(u6)         −     r1 r2 r3 r4    

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-48
SLIDE 48

pollo

Flow networks

φk(uk) = αkuk +βk arctan(uk),

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-49
SLIDE 49

pollo

Flow networks

φk(uk) = αkuk +βk arctan(uk), Outgoing flow y (top) and control flow u (bottom).

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-50
SLIDE 50

pollo

Extensions

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-51
SLIDE 51

pollo

Extensions

Constraints on y and u can be considered.

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-52
SLIDE 52

pollo

Extensions

Constraints on y and u can be considered. Implicit representation: u = h(y)

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-53
SLIDE 53

pollo

Implicit representation

Flow tuning with pressure drop

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-54
SLIDE 54

pollo

Implicit representation

Flow tuning with pressure drop ak = ψk

  • p0 −Ψ
  • 3

∑

i=1

qi

  • ,qk
  • ,
  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-55
SLIDE 55

pollo

Implicit representation

Formula: if ∂h(y) ∂y ∈ M then h(y)−h(0) = Myy, for some My ∈ M

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-56
SLIDE 56

pollo

Implicit representation

Formula: if ∂h(y) ∂y ∈ M then h(y)−h(0) = Myy, for some My ∈ M u = h(y) ⇒ v . = ˙ u = ∂h(y) ∂y ˙ y

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-57
SLIDE 57

pollo

Implicit representation

Formula: if ∂h(y) ∂y ∈ M then h(y)−h(0) = Myy, for some My ∈ M u = h(y) ⇒ v . = ˙ u = ∂h(y) ∂y ˙ y V = h(y)−h(0)2 ˙ V = 2[h(y)−h(0)]∂h(y) ∂y ˙ y = 2[h(y)−h(0)]v

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-58
SLIDE 58

pollo

Implicit representation

Formula: if ∂h(y) ∂y ∈ M then h(y)−h(0) = Myy, for some My ∈ M u = h(y) ⇒ v . = ˙ u = ∂h(y) ∂y ˙ y V = h(y)−h(0)2 ˙ V = 2[h(y)−h(0)]∂h(y) ∂y ˙ y = 2[h(y)−h(0)]v Claim We can apply the previous result to the ”dual” problem ˙ V = 2M⊤

y v

My ∈ M

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-59
SLIDE 59

pollo

The discrete–time case

Assume that the tuning cannot be continuously performed but only a sequence of experiments is possible.

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-60
SLIDE 60

pollo

The discrete–time case

Assume that the tuning cannot be continuously performed but only a sequence of experiments is possible. Heat exchanger problem q S T1 T 2 T0

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-61
SLIDE 61

pollo

Heat exchanger problem

Use a software to determine the inputs

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-62
SLIDE 62

pollo

Heat exchanger problem

Use a software to determine the inputs fluid flow q, cooler surface S,

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-63
SLIDE 63

pollo

Heat exchanger problem

Use a software to determine the inputs fluid flow q, cooler surface S, in order to produce

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-64
SLIDE 64

pollo

Heat exchanger problem

Use a software to determine the inputs fluid flow q, cooler surface S, in order to produce a desired positive temperature drop ∆T = T1 −T2 > 0, given the inlet temperature T1, a desired exchanged heat h.

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-65
SLIDE 65

pollo

Nonlinear equation solving

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-66
SLIDE 66

pollo

Nonlinear equation solving

.... without the equation...

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-67
SLIDE 67

pollo

Nonlinear equation solving

.... without the equation... y y y

2 3 1

u 1 u 2

3

u software

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-68
SLIDE 68

pollo

Tempra process in siderurgy

  • T SURF

TFIN flow length

Inputs: length and flow Outputs: Thermal variation ∆Tsurf /∆t and final temperature J = ±ε −α −β −γ

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-69
SLIDE 69

pollo

The discrete–time case: technical problem

yk = g(uk) uk+1 = uk +vk

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-70
SLIDE 70

pollo

The discrete–time case: technical problem

yk = g(uk) uk+1 = uk +vk Then yk+1 = g(uk+1) = g(uk +vk) = g(uk)+Guvk = yk +Guvk

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-71
SLIDE 71

pollo

The discrete–time case: technical problem

yk = g(uk) uk+1 = uk +vk Then yk+1 = g(uk+1) = g(uk +vk) = g(uk)+Guvk = yk +Guvk Take the Lyapunov function V (y) = y2.

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-72
SLIDE 72

pollo

The discrete–time case: technical problem

yk = g(uk) uk+1 = uk +vk Then yk+1 = g(uk+1) = g(uk +vk) = g(uk)+Guvk = yk +Guvk Take the Lyapunov function V (y) = y2. min

vk≤ξ

max

Gu∈M yk +Guv2 = max Gu∈M

min

vk≤ξ yk +Guv2

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-73
SLIDE 73

pollo

The discrete–time case: technical problem

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-74
SLIDE 74

pollo

The discrete–time case: technical problem

min

vk≤ξ

max

Gu∈M yk +Guv = min vk≤ξ

max

Gi∈vertM yk +Givk

= min

vk≤ξ φ(vk)

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-75
SLIDE 75

pollo

The discrete–time case: technical problem

min

vk≤ξ

max

Gu∈M yk +Guv = min vk≤ξ

max

Gi∈vertM yk +Givk

= min

vk≤ξ φ(vk)

Theorem If any matrix in the polytope M is right invertible, then the model–free tuning problem can be solved with a control scheme of the form uk+1 = uk +vk, (1) vk = argmin

vk

φ(vk) (2)

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-76
SLIDE 76

pollo

The heater design problem

5 10 15 20 24.5 25 25.5 26 26.5 27 27.5 28 28.5 29 29.5 Outputs step ∆T [K] 5 10 15 20 6.8 6.9 7 7.1 7.2 7.3 7.4 7.5 x 10

5

h [W]

Evolution of the temperature drop [K] (blue) and heat [W] (green).

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-77
SLIDE 77

pollo

The heater design problem

5 10 15 20 26.5 27 27.5 28 28.5 29 29.5 30 30.5 Inputs step S [m2] 5 10 15 20 6 6.1 6.2 6.3 6.4 6.5 6.6 6.7 x 10

−6

q [m3/s]

Evolution of surface [m2] (blue) and flow [l/s] (green).

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-78
SLIDE 78

pollo

A manipulation problem

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-79
SLIDE 79

pollo

A manipulation problem

PoCN Project.

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-80
SLIDE 80

pollo

A manipulation problem

PoCN Project.

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-81
SLIDE 81

pollo

A manipulation problem

PoCN Project.

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-82
SLIDE 82

pollo

A manipulation problem

PoCN Project.

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-83
SLIDE 83

pollo

A manipulation problem

PoCN Project.

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-84
SLIDE 84

pollo

A manipulation problem

PoCN Project.

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-85
SLIDE 85

pollo

A manipulation problem

PoCN Project.

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-86
SLIDE 86

pollo

A manipulation problem

PoCN Project.

θ

A D C B

δ

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-87
SLIDE 87

pollo

Formulation

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-88
SLIDE 88

pollo

Formulation

θ = f (Tr,Tc) δ = g(Tr,Tc) Tc: closing initial time Tr: rotation initial time

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-89
SLIDE 89

pollo

Formulation

θ = f (Tr,Tc) δ = g(Tr,Tc) Tc: closing initial time Tr: rotation initial time J = − + + +

  • ,
  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-90
SLIDE 90

pollo

Convergence of the methods

1 2 3 4 5 6 7 8 9 10 −0.4 −0.2 0.2 0.4 0.6 0.8 1

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-91
SLIDE 91

pollo

Convergence of the methods

−5 −4 −3 −2 −1 1 2 3 4 5 −5 −4 −3 −2 −1 1 2 3 4 5 −5 −4 −3 −2 −1 1 2 3 4 5 −5 −4 −3 −2 −1 1 2 3 4 5 −5 −4 −3 −2 −1 1 2 3 4 5 −5 −4 −3 −2 −1 1 2 3 4 5 −5 −4 −3 −2 −1 1 2 3 4 5 −5 −4 −3 −2 −1 1 2 3 4 5 −5 −4 −3 −2 −1 1 2 3 4 5 −5 −4 −3 −2 −1 1 2 3 4 5 −5 −4 −3 −2 −1 1 2 3 4 5 −5 −4 −3 −2 −1 1 2 3 4 5

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-92
SLIDE 92

pollo

A manipulation problem

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-93
SLIDE 93

pollo

A manipulation problem

Learning stage: profiles and videocamera;

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-94
SLIDE 94

pollo

A manipulation problem

Learning stage: profiles and videocamera; Experiments in different relative positions on the conveyor belt;

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-95
SLIDE 95

pollo

A manipulation problem

Learning stage: profiles and videocamera; Experiments in different relative positions on the conveyor belt; Interpolation.

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-96
SLIDE 96

pollo

Ongoing: a cable robot

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-97
SLIDE 97

pollo

The end

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach

slide-98
SLIDE 98

pollo

The end Grazie!

  • F. Blanchini, G. Fenu, G. Giordano, F.A. Pellegrino

Plant tuning: a robust Lyapunov approach