Polytopic outer-approximation of semialgebraic sets V. Cerone 1 , D. - - PowerPoint PPT Presentation

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Polytopic outer-approximation of semialgebraic sets V. Cerone 1 , D. - - PowerPoint PPT Presentation

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Polytopic outer-approximation of semialgebraic sets V. Cerone 1 , D. Piga 2 , D. Regruto 1 , 1 DAUIN, Politecnico di Torino, Italy 2 Delft Center for Systems and Control, TU Delft, The Netherlands 1 0 x 2 1 2 3 4 2 0 2 4 x

slide-1
SLIDE 1 −4 −2 2 4 −3 −2 −1 1 x1 x2

Polytopic outer-approximation of semialgebraic sets

PIPPP

  • V. Cerone1, D. Piga2, D. Regruto1,

1 DAUIN, Politecnico di Torino, Italy 2 Delft Center for Systems and Control, TU Delft, The Netherlands

  • D. Piga ()

Polytopic outer approximation 1

slide-2
SLIDE 2 −4 −2 2 4 −3 −2 −1 1 x1 x2

Semialgebraic sets

Definition

A semialgebraic set is a subset of Rn defined by a finite sequence of polynomial equality and inequality constraints.

  • D. Piga ()

Polytopic outer approximation 2

slide-3
SLIDE 3 −4 −2 2 4 −3 −2 −1 1 x1 x2

Semialgebraic sets

Definition

A semialgebraic set is a subset of Rn defined by a finite sequence of polynomial equality and inequality constraints.

−2 −1 1 2 −2 −1 1 2 x1 x2

  • D. Piga ()

Polytopic outer approximation 2

slide-4
SLIDE 4 −4 −2 2 4 −3 −2 −1 1 x1 x2

Semialgebraic sets

Definition

A semialgebraic set is a subset of Rn defined by a finite sequence of polynomial equality and inequality constraints.

−2 −1 1 2 −2 −1 1 2 x1 x2 −2 −1 1 2 −2 −1 1 2 x1 x2

x2

1 + x2 2 ≤ 1

x2 ≥ x1

  • D. Piga ()

Polytopic outer approximation 2

slide-5
SLIDE 5 −4 −2 2 4 −3 −2 −1 1 x1 x2

Polytopic outer-approximation

Definition

An Euclidean set P is called a polytopic outer-approximation of S if P is a polytope and S ⊆ P.

  • D. Piga ()

Polytopic outer approximation 3

slide-6
SLIDE 6 −4 −2 2 4 −3 −2 −1 1 x1 x2

Polytopic outer-approximation

Definition

An Euclidean set P is called a polytopic outer-approximation of S if P is a polytope and S ⊆ P.

−2 −1 1 2 −2 −1 1 2 x1 x2

  • D. Piga ()

Polytopic outer approximation 3

slide-7
SLIDE 7 −4 −2 2 4 −3 −2 −1 1 x1 x2

Polytopic outer-approximation

Definition

An Euclidean set P is called a polytopic outer-approximation of S if P is a polytope and S ⊆ P.

−2 −1 1 2 −2 −1 1 2 x1 x2 −2 −1 1 2 −2 −1 1 2 x1 x2

  • D. Piga ()

Polytopic outer approximation 3

slide-8
SLIDE 8 −4 −2 2 4 −3 −2 −1 1 x1 x2

Level of accuracy

Level of accuracy Question

What is a “reasonable” indicator to measure the level of accuracy in approximating a set?

  • D. Piga ()

Polytopic outer approximation 4

slide-9
SLIDE 9 −4 −2 2 4 −3 −2 −1 1 x1 x2

Level of accuracy

Level of accuracy Question

What is a “reasonable” indicator to measure the level of accuracy in approximating a set?

Answer

Volume of the polytopic outer-approximation P, i.e.

  • D. Piga ()

Polytopic outer approximation 4

slide-10
SLIDE 10 −4 −2 2 4 −3 −2 −1 1 x1 x2

Level of accuracy

Level of accuracy Question

What is a “reasonable” indicator to measure the level of accuracy in approximating a set?

Answer

Volume of the polytopic outer-approximation P, i.e.

  • P 1dx.
  • D. Piga ()

Polytopic outer approximation 4

slide-11
SLIDE 11 −4 −2 2 4 −3 −2 −1 1 x1 x2

Problem formulation

Problem

Among all the polytopes P containing S find the one with minimum volume,

  • D. Piga ()

Polytopic outer approximation 5

slide-12
SLIDE 12 −4 −2 2 4 −3 −2 −1 1 x1 x2

Problem formulation

Problem

Among all the polytopes P containing S find the one with minimum volume, i.e. min

  • P

1dx s.t. S ⊆ P

  • D. Piga ()

Polytopic outer approximation 5

slide-13
SLIDE 13 −4 −2 2 4 −3 −2 −1 1 x1 x2

Problem formulation

Problem

Among all the polytopes P containing S find the one with minimum volume, i.e. min

  • P

1dx s.t. S ⊆ P

Problem

Find the liner hull of S.

  • D. Piga ()

Polytopic outer approximation 5

slide-14
SLIDE 14 −4 −2 2 4 −3 −2 −1 1 x1 x2

A motivation example

Design of Robust Controllers for Uncertain LTI Systems

  • D. Piga ()

Polytopic outer approximation 6

slide-15
SLIDE 15 −4 −2 2 4 −3 −2 −1 1 x1 x2

A motivation example

Design of Robust Controllers for Uncertain LTI Systems

r e u y

✲ ❡ ✻ ✲ C(s) ✲ P(s) ✲

  • D. Piga ()

Polytopic outer approximation 6

slide-16
SLIDE 16 −4 −2 2 4 −3 −2 −1 1 x1 x2

A motivation example

Design of Robust Controllers for Uncertain LTI Systems

r e u y

✲ ❡ ✻ ✲ C(s) ✲ P(s) ✲

P(s) = s + 1 s2 + p1s + p2

  • D. Piga ()

Polytopic outer approximation 6

slide-17
SLIDE 17 −4 −2 2 4 −3 −2 −1 1 x1 x2

A motivation example

Design of Robust Controllers for Uncertain LTI Systems

r e u y

✲ ❡ ✻ ✲ C(s) ✲ P(s) ✲

P(s) = s + 1 s2 + p1s + p2 p1, p2 ∈ P

−1 −0.5 0.5 1 −1 −0.5 0.5 1 p1 p2

  • D. Piga ()

Polytopic outer approximation 6

slide-18
SLIDE 18 −4 −2 2 4 −3 −2 −1 1 x1 x2

A motivation example

Design of Robust Controllers for Uncertain LTI Systems

r e u y

✲ ❡ ✻ ✲ C(s) ✲ P(s) ✲

P(s) = s + 1 s2 + p1s + p2 p1, p2 ∈ P

−1 −0.5 0.5 1 −1 −0.5 0.5 1 p1 p2

Well-settled techniques to design robust controllers if p1, p2 ∈ P

  • D. Piga ()

Polytopic outer approximation 6

slide-19
SLIDE 19 −4 −2 2 4 −3 −2 −1 1 x1 x2

A motivation example

Design of Robust Controllers for Uncertain LTI Systems

r e u y

✲ ❡ ✻ ✲ C(s) ✲ P(s) ✲

P(s) = s + 1 s2 + p1s + p2

1

p1 ∈ [0, 2]

  • D. Piga ()

Polytopic outer approximation 7

slide-20
SLIDE 20 −4 −2 2 4 −3 −2 −1 1 x1 x2

A motivation example

Design of Robust Controllers for Uncertain LTI Systems

r e u y

✲ ❡ ✻ ✲ C(s) ✲ P(s) ✲

P(s) = s + 1 s2 + p1s + p2 p1 ∈ [0, 2], p2 = p2

1

  • D. Piga ()

Polytopic outer approximation 7

slide-21
SLIDE 21 −4 −2 2 4 −3 −2 −1 1 x1 x2

A motivation example

Design of Robust Controllers for Uncertain LTI Systems

r e u y

✲ ❡ ✻ ✲ C(s) ✲ P(s) ✲

P(s) = s + 1 s2 + p1s + p2 p1 ∈ [0, 2], p2 = p2

1

p2 ∈ [0, 4]

0.5 1 1.5 2 1 2 3 4 5 p1 p2

  • D. Piga ()

Polytopic outer approximation 7

slide-22
SLIDE 22 −4 −2 2 4 −3 −2 −1 1 x1 x2

A motivation example

Design of Robust Controllers for Uncertain LTI Systems

r e u y

✲ ❡ ✻ ✲ C(s) ✲ P(s) ✲

P(s) = s + 1 s2 + p1s + p2 p1 ∈ [0, 2], p2 = p2

1

p2 ∈ [0, 4], p2 ∈ [0, 4]

0.5 1 1.5 2 1 2 3 4 5 p1 p2

  • D. Piga ()

Polytopic outer approximation 7

slide-23
SLIDE 23 −4 −2 2 4 −3 −2 −1 1 x1 x2

A motivation example

Design of Robust Controllers for Uncertain LTI Systems

r e u y

✲ ❡ ✻ ✲ C(s) ✲ P(s) ✲

P(s) = s + 1 s2 + p1s + p2 p1 ∈ [0, 2], p2 = p2

1

p2 ∈ [0, 4]

0.5 1 1.5 2 1 2 3 4 5 p1 p2

  • D. Piga ()

Polytopic outer approximation 7

slide-24
SLIDE 24 −4 −2 2 4 −3 −2 −1 1 x1 x2

A motivation example

Design of Robust Controllers for Uncertain LTI Systems

r e u y

✲ ❡ ✻ ✲ C(s) ✲ P(s) ✲

P(s) = s + 1 s2 + p1s + p2 p1 ∈ [0, 2], p2 = p2

1

p2 ∈ [0, 4]p1, p2 ∈ P

0.5 1 1.5 2 1 2 3 4 5 p1 p2

  • D. Piga ()

Polytopic outer approximation 7

slide-25
SLIDE 25 −4 −2 2 4 −3 −2 −1 1 x1 x2

A motivation example

Design of Robust Controllers for Uncertain LTI Systems

r e u y

✲ ❡ ✻ ✲ C(s) ✲ P(s) ✲

P(s) = s + 1 s2 + p1s + p2 p1 ∈ [0, 2], p2 = p2

1

p2 ∈ [0, 4]p1, p2 ∈ P

0.5 1 1.5 2 1 2 3 4 5 p1 p2

  • D. Piga ()

Polytopic outer approximation 7

slide-26
SLIDE 26 −4 −2 2 4 −3 −2 −1 1 x1 x2

Volume Computation

min

  • P

1dx s.t. S ⊆ P

  • D. Piga ()

Polytopic outer approximation 8

slide-27
SLIDE 27 −4 −2 2 4 −3 −2 −1 1 x1 x2

Volume Computation

min

  • P

1dx s.t. S ⊆ P

Question

How to compute the volume of P?

  • D. Piga ()

Polytopic outer approximation 8

slide-28
SLIDE 28 −4 −2 2 4 −3 −2 −1 1 x1 x2

Volume Computation

min

  • P

1dx s.t. S ⊆ P

Question

How to compute the volume of P?

Trouble 1

Computing the volume of a polytope is an NP-hard problem.

  • D. Piga ()

Polytopic outer approximation 8

slide-29
SLIDE 29 −4 −2 2 4 −3 −2 −1 1 x1 x2

Volume Computation

min

  • P

1dx s.t. S ⊆ P

Question

How to compute the volume of P?

Trouble 1

Computing the volume of a polytope is an NP-hard problem.

Trouble 2

We don’t know P!

  • D. Piga ()

Polytopic outer approximation 8

slide-30
SLIDE 30 −4 −2 2 4 −3 −2 −1 1 x1 x2

Main Algorithm

Polytopic outer-approximation of S Idea

1 Take an outer-bounding box B of the Euclidean set S

  • D. Piga ()

Polytopic outer approximation 9

slide-31
SLIDE 31 −4 −2 2 4 −3 −2 −1 1 x1 x2

Main Algorithm

Polytopic outer-approximation of S

0.5 1 1.5 2 0.5 1 1.5 x1 x2

  • D. Piga ()

Polytopic outer approximation 10

slide-32
SLIDE 32 −4 −2 2 4 −3 −2 −1 1 x1 x2

Main Algorithm

Polytopic outer-approximation of S Idea

1 Take an outer-bounding box B of the Euclidean set S 2 Generate a sequence L of N random points xi uniformly

distributed in B

  • D. Piga ()

Polytopic outer approximation 11

slide-33
SLIDE 33 −4 −2 2 4 −3 −2 −1 1 x1 x2

Main Algorithm

Polytopic outer-approximation of S

0.5 1 1.5 2 0.5 1 1.5 x1 x2

  • D. Piga ()

Polytopic outer approximation 12

slide-34
SLIDE 34 −4 −2 2 4 −3 −2 −1 1 x1 x2

Main Algorithm

Polytopic outer-approximation of S Idea

1 Take an outer-bounding box B of the Euclidean set S 2 Generate a list L of N random points xi uniformly

distributed in B

3 Compute the half-space H : w⊤x + b ≥ 0 containing the

minimum number of points of the list L and such that S ⊆ H

  • D. Piga ()

Polytopic outer approximation 13

slide-35
SLIDE 35 −4 −2 2 4 −3 −2 −1 1 x1 x2

Main Algorithm

Polytopic outer-approximation of S

0.5 1 1.5 2 0.5 1 1.5 x1 x2

  • D. Piga ()

Polytopic outer approximation 14

slide-36
SLIDE 36 −4 −2 2 4 −3 −2 −1 1 x1 x2

Main Algorithm

Polytopic outer-approximation of S Idea

1 Take an outer-bounding box B of the Euclidean set S 2 Generate a list L of N random points xi uniformly

distributed in B

3 Compute the half-space H : w⊤x + b ≥ 0 containing the

minimum number of points of the list L and such that S ⊆ H

4 Up-to-date the list L by getting rid of all the points that

do not belong to H and go to step 3

  • D. Piga ()

Polytopic outer approximation 15

slide-37
SLIDE 37 −4 −2 2 4 −3 −2 −1 1 x1 x2

Main Algorithm

Polytopic outer-approximation of S

0.5 1 1.5 2 0.5 1 1.5 x1 x2

  • D. Piga ()

Polytopic outer approximation 16

slide-38
SLIDE 38 −4 −2 2 4 −3 −2 −1 1 x1 x2

Main Algorithm

Polytopic outer-approximation of S

0.5 1 1.5 2 0.5 1 1.5 x1 x2

  • D. Piga ()

Polytopic outer approximation 17

slide-39
SLIDE 39 −4 −2 2 4 −3 −2 −1 1 x1 x2

Main Algorithm

Polytopic outer-approximation of S

0.5 1 1.5 2 0.5 1 1.5 x1 x2

  • D. Piga ()

Polytopic outer approximation 18

slide-40
SLIDE 40 −4 −2 2 4 −3 −2 −1 1 x1 x2

Main Algorithm

Polytopic outer-approximation of S

0.5 1 1.5 2 0.5 1 1.5 x1 x2

  • D. Piga ()

Polytopic outer approximation 19

slide-41
SLIDE 41 −4 −2 2 4 −3 −2 −1 1 x1 x2

Main Algorithm

Polytopic outer-approximation of S

0.5 1 1.5 2 0.5 1 1.5 x1 x2

  • D. Piga ()

Polytopic outer approximation 20

slide-42
SLIDE 42 −4 −2 2 4 −3 −2 −1 1 x1 x2

Main Algorithm

Polytopic outer-approximation of S

0.5 1 1.5 2 0.5 1 1.5 x1 x2

  • D. Piga ()

Polytopic outer approximation 21

slide-43
SLIDE 43 −4 −2 2 4 −3 −2 −1 1 x1 x2

Main Algorithm

Polytopic outer-approximation of S

0.5 1 1.5 2 0.5 1 1.5 x1 x2

  • D. Piga ()

Polytopic outer approximation 22

slide-44
SLIDE 44 −4 −2 2 4 −3 −2 −1 1 x1 x2

Main Algorithm

Polytopic outer-approximation of S

0.5 1 1.5 2 0.5 1 1.5 x1 x2

  • D. Piga ()

Polytopic outer approximation 23

slide-45
SLIDE 45 −4 −2 2 4 −3 −2 −1 1 x1 x2

Main Algorithm

Polytopic outer-approximation of S

0.5 1 1.5 2 0.5 1 1.5 x1 x2

  • D. Piga ()

Polytopic outer approximation 24

slide-46
SLIDE 46 −4 −2 2 4 −3 −2 −1 1 x1 x2

Main Algorithm

Polytopic outer-approximation of S

0.5 1 1.5 2 0.5 1 1.5 x1 x2

  • D. Piga ()

Polytopic outer approximation 25

slide-47
SLIDE 47 −4 −2 2 4 −3 −2 −1 1 x1 x2

Main Algorithm

Polytopic outer-approximation of S

0.5 1 1.5 2 0.5 1 1.5 x1 x2

  • D. Piga ()

Polytopic outer approximation 26

slide-48
SLIDE 48 −4 −2 2 4 −3 −2 −1 1 x1 x2

Computation of the half-spaces

Compute the half-space H : w⊤x + b ≥ 0 containing the minimum number of points of the list L and such that S ⊆ H, i.e.

  • D. Piga ()

Polytopic outer approximation 27

slide-49
SLIDE 49 −4 −2 2 4 −3 −2 −1 1 x1 x2

Computation of the half-spaces

Compute the half-space H : w⊤x + b ≥ 0 containing the minimum number of points of the list L and such that S ⊆ H, i.e.

min

w,b N

  • i=1

IH(xi) xi ∈ L s.t. w ⊤x + b ≥ 0 ∀x ∈ S

  • D. Piga ()

Polytopic outer approximation 27

slide-50
SLIDE 50 −4 −2 2 4 −3 −2 −1 1 x1 x2

Computation of the half-spaces

min

w,b N

  • i=1

IH(xi) xi ∈ L s.t. w ⊤x + b ≥ 0 ∀x ∈ S

  • D. Piga ()

Polytopic outer approximation 28

slide-51
SLIDE 51 −4 −2 2 4 −3 −2 −1 1 x1 x2

Computation of the half-spaces

min

w,b N

  • i=1

IH(xi) xi ∈ L s.t. w ⊤x + b ≥ 0 ∀x ∈ S I{H}(xi) = 1 if w ⊤xi + b ≥ 0 if w ⊤xi + b < 0

  • D. Piga ()

Polytopic outer approximation 28

slide-52
SLIDE 52 −4 −2 2 4 −3 −2 −1 1 x1 x2

Computation of the half-spaces

min

w,b N

  • i=1

IH(xi) xi ∈ L s.t. w ⊤x + b ≥ 0 ∀x ∈ S I{H}(xi) = 1 if w ⊤xi + b ≥ 0 if w ⊤xi + b < 0

  • (

)

i

I x

H T i

x b

  • D. Piga ()

Polytopic outer approximation 28

slide-53
SLIDE 53 −4 −2 2 4 −3 −2 −1 1 x1 x2

Computation of the half-spaces

min

w,b N

  • i=1

IH(xi) xi ∈ L s.t. w ⊤x + b ≥ 0 ∀x ∈ S I{H}(xi) = 1 if w ⊤xi + b ≥ 0 if w ⊤xi + b < 0

  • (

)

i

I x

H T i

x b

  • R{H}(xi) =

w ⊤xi + b if w ⊤xi + b ≥ 0 if w ⊤xi + b < 0

  • (

)

i

I x

H T i

x b

  • (

)

i

R x

H

  • D. Piga ()

Polytopic outer approximation 28

slide-54
SLIDE 54 −4 −2 2 4 −3 −2 −1 1 x1 x2

Computation of the half-spaces

min

w,b N

  • i=1

RH(xi) xi ∈ L s.t. w⊤x + b ≥ 0 ∀x ∈ S

slide-55
SLIDE 55 −4 −2 2 4 −3 −2 −1 1 x1 x2

Computation of the half-spaces

min

w,b N

  • i=1

RH(xi) xi ∈ L s.t. w⊤x + b ≥ 0 ∀x ∈ S S = { x :gj(x) ≥ 0, j = 1,. . . ,M}

  • D. Piga ()

Polytopic outer approximation 29

slide-56
SLIDE 56 −4 −2 2 4 −3 −2 −1 1 x1 x2

Computation of the half-spaces

min

w,b N

  • i=1

RH(xi) xi ∈ L s.t. w⊤x + b ≥ 0 ∀x ∈ S S = { x :gj(x) ≥ 0, j = 1,. . . ,M} min w, b

N

  • i=1

RH(xi) xi ∈ L

  • D. Piga ()

Polytopic outer approximation 29

slide-57
SLIDE 57 −4 −2 2 4 −3 −2 −1 1 x1 x2

Computation of the half-spaces

min

w,b N

  • i=1

RH(xi) xi ∈ L s.t. w⊤x + b ≥ 0 ∀x ∈ S S = { x :gj(x) ≥ 0, j = 1,. . . ,M} min w, b Θ0,. . . , ΘM

N

  • i=1

RH(xi) xi ∈ L s.t. w⊤x + b =σ2

0(x,Θ0)+σ2 1(x,Θ1)g1(x) +. . .+ σ2 M(x,ΘM)gM(x)

  • D. Piga ()

Polytopic outer approximation 29

slide-58
SLIDE 58 −4 −2 2 4 −3 −2 −1 1 x1 x2

Computation of the half-spaces

min

w,b N

  • i=1

RH(xi) xi ∈ L s.t. w⊤x + b ≥ 0 ∀x ∈ S S = { x :gj(x) ≥ 0, j = 1,. . . ,M} min w, b Θ0,. . . , ΘM

N

  • i=1

RH(xi) xi ∈ L s.t. w⊤x + b =σ2

0(x,Θ0)+σ2 1(x,Θ1)g1(x) +. . .+ σ2 M(x,ΘM)gM(x)

σ2

0(x,Θ0), σ2 1(x,Θ1), . . . , σ2 M(x,ΘM) SOS

  • D. Piga ()

Polytopic outer approximation 29

slide-59
SLIDE 59 −4 −2 2 4 −3 −2 −1 1 x1 x2

Computation of the half-spaces

min

w,b N

  • i=1

RH(xi) xi ∈ L s.t. w⊤x + b ≥ 0 ∀x ∈ S S = { x :gj(x) ≥ 0, j = 1,. . . ,M} min w, b Θ0,. . . , ΘM

N

  • i=1

RH(xi) xi ∈ L s.t. w⊤x + b =σ2

0(x,Θ0)+σ2 1(x,Θ1)g1(x) +. . .+ σ2 M(x,ΘM)gM(x)

σ2

0(x,Θ0), σ2 1(x,Θ1), . . . , σ2 M(x,ΘM) SOS

Robust Optimization problem

  • D. Piga ()

Polytopic outer approximation 29

slide-60
SLIDE 60 −4 −2 2 4 −3 −2 −1 1 x1 x2

Computation of the half-spaces

min

w,b N

  • i=1

RH(xi) xi ∈ L s.t. w⊤x + b ≥ 0 ∀x ∈ S S = { x :gj(x) ≥ 0, j = 1,. . . ,M} min w, b Θ0,. . . , ΘM

N

  • i=1

RH(xi) xi ∈ L s.t. w⊤x + b =σ2

0(x,Θ0)+σ2 1(x,Θ1)g1(x) +. . .+ σ2 M(x,ΘM)gM(x)

σ2

0(x,Θ0), σ2 1(x,Θ1), . . . , σ2 M(x,ΘM) SOS

Convex Robust Optimization problem

  • D. Piga ()

Polytopic outer approximation 29

slide-61
SLIDE 61 −4 −2 2 4 −3 −2 −1 1 x1 x2

Logo?

−4 −2 2 4 −3 −2 −1 1 x1 x2

  • D. Piga ()

Polytopic outer approximation 30

slide-62
SLIDE 62 −4 −2 2 4 −3 −2 −1 1 x1 x2

Logo?

−4 −2 2 4 −3 −2 −1 1 x1 x2

  • D. Piga ()

Polytopic outer approximation 31