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

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 1 D. Piga () Polytopic outer approximation 1 PIPPP

Semialgebraic sets Definition A semialgebraic set is a subset of R n defined by a finite sequence of polynomial equality and inequality constraints. 1 0 x 2 −1 −2 −3 −4 −2 0 2 4 x 1 D. Piga () Polytopic outer approximation 2

Semialgebraic sets Definition A semialgebraic set is a subset of R n defined by a finite sequence of polynomial equality and inequality constraints. 2 1 x 2 0 −1 −2 −2 −1 0 1 2 x 1 1 0 x 2 −1 −2 −3 −4 −2 0 2 4 x 1 D. Piga () Polytopic outer approximation 2

Semialgebraic sets Definition A semialgebraic set is a subset of R n defined by a finite sequence of polynomial equality and inequality constraints. 2 2 1 1 x 2 x 2 0 0 −1 −1 −2 −2 −2 −1 0 1 2 −2 −1 0 1 2 x 1 x 1 x 2 1 + x 2 2 ≤ 1 x 2 ≥ x 1 1 0 x 2 −1 −2 −3 −4 −2 0 2 4 x 1 D. Piga () Polytopic outer approximation 2

Polytopic outer-approximation Definition An Euclidean set P is called a polytopic outer-approximation of S if P is a polytope and S ⊆ P . 1 0 x 2 −1 −2 −3 −4 −2 0 2 4 x 1 D. Piga () Polytopic outer approximation 3

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 x 2 0 −1 −2 −2 −1 0 1 2 x 1 1 0 x 2 −1 −2 −3 −4 −2 0 2 4 x 1 D. Piga () Polytopic outer approximation 3

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 2 1 1 x 2 0 x 2 0 −1 −1 −2 −2 −2 −1 0 1 2 −2 −1 0 1 2 x 1 x 1 1 0 x 2 −1 −2 −3 −4 −2 0 2 4 x 1 D. Piga () Polytopic outer approximation 3

Level of accuracy Level of accuracy Question What is a “reasonable” indicator to measure the level of accuracy in approximating a set? 1 0 x 2 −1 −2 −3 −4 −2 0 2 4 x 1 D. Piga () Polytopic outer approximation 4

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. 1 0 x 2 −1 −2 −3 −4 −2 0 2 4 x 1 D. Piga () Polytopic outer approximation 4

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 1 dx . 1 0 x 2 −1 −2 −3 −4 −2 0 2 4 x 1 D. Piga () Polytopic outer approximation 4

Problem formulation Problem Among all the polytopes P containing S find the one with minimum volume, 1 0 x 2 −1 −2 −3 −4 −2 0 2 4 x 1 D. Piga () Polytopic outer approximation 5

Problem formulation Problem Among all the polytopes P containing S find the one with minimum volume, i.e. � min 1 dx s . t . S ⊆ P P 1 0 x 2 −1 −2 −3 −4 −2 0 2 4 x 1 D. Piga () Polytopic outer approximation 5

Problem formulation Problem Among all the polytopes P containing S find the one with minimum volume, i.e. � min 1 dx s . t . S ⊆ P P Problem Find the liner hull of S . 1 0 x 2 −1 −2 −3 −4 −2 0 2 4 x 1 D. Piga () Polytopic outer approximation 5

A motivation example Design of Robust Controllers for Uncertain LTI Systems 1 0 x 2 −1 −2 −3 −4 −2 0 2 4 x 1 D. Piga () Polytopic outer approximation 6

A motivation example Design of Robust Controllers for Uncertain LTI Systems ✲ ❡ ✲ C ( s ) ✲ P ( s ) ✲ y r e u ✻ 1 0 x 2 −1 −2 −3 −4 −2 0 2 4 x 1 D. Piga () Polytopic outer approximation 6

A motivation example Design of Robust Controllers for Uncertain LTI Systems ✲ ❡ ✲ C ( s ) ✲ P ( s ) ✲ y r e u ✻ s + 1 P ( s ) = s 2 + p 1 s + p 2 1 0 x 2 −1 −2 −3 −4 −2 0 2 4 x 1 D. Piga () Polytopic outer approximation 6

A motivation example Design of Robust Controllers for Uncertain LTI Systems ✲ ❡ ✲ C ( s ) ✲ P ( s ) ✲ y r e u ✻ 1 s + 1 P ( s ) = s 2 + p 1 s + p 2 0.5 p 2 0 p 1 , p 2 ∈ P −0.5 −1 −1 −0.5 0 0.5 1 p 1 1 0 x 2 −1 −2 −3 −4 −2 0 2 4 x 1 D. Piga () Polytopic outer approximation 6

A motivation example Design of Robust Controllers for Uncertain LTI Systems ✲ ❡ ✲ C ( s ) ✲ P ( s ) ✲ y r e u ✻ 1 s + 1 P ( s ) = s 2 + p 1 s + p 2 0.5 p 2 0 p 1 , p 2 ∈ P −0.5 −1 −1 −0.5 0 0.5 1 p 1 Well-settled techniques to design robust controllers if p 1 , p 2 ∈ P 1 0 x 2 −1 −2 −3 −4 −2 0 2 4 x 1 D. Piga () Polytopic outer approximation 6

A motivation example Design of Robust Controllers for Uncertain LTI Systems ✲ ❡ ✲ C ( s ) ✲ P ( s ) ✲ y r e u ✻ s + 1 P ( s ) = s 2 + p 1 s + p 2 1 p 1 ∈ [0 , 2] 1 0 x 2 −1 −2 −3 −4 −2 0 2 4 x 1 D. Piga () Polytopic outer approximation 7

A motivation example Design of Robust Controllers for Uncertain LTI Systems ✲ ❡ ✲ C ( s ) ✲ P ( s ) ✲ y r e u ✻ s + 1 P ( s ) = s 2 + p 1 s + p 2 p 1 ∈ [0 , 2] , p 2 = p 2 1 1 0 x 2 −1 −2 −3 −4 −2 0 2 4 x 1 D. Piga () Polytopic outer approximation 7

A motivation example Design of Robust Controllers for Uncertain LTI Systems ✲ ❡ ✲ C ( s ) ✲ P ( s ) ✲ y r e u ✻ s + 1 5 P ( s ) = s 2 + p 1 s + p 2 4 3 p 2 2 p 1 ∈ [0 , 2] , p 2 = p 2 1 1 0 p 2 ∈ [0 , 4] 0 0.5 1 1.5 2 p 1 1 0 x 2 −1 −2 −3 −4 −2 0 2 4 x 1 D. Piga () Polytopic outer approximation 7

A motivation example Design of Robust Controllers for Uncertain LTI Systems ✲ ❡ ✲ C ( s ) ✲ P ( s ) ✲ y r e u ✻ s + 1 5 P ( s ) = s 2 + p 1 s + p 2 4 3 p 2 2 p 1 ∈ [0 , 2] , p 2 = p 2 1 1 0 p 2 ∈ [0 , 4] , p 2 ∈ [0 , 4] 0 0.5 1 1.5 2 p 1 1 0 x 2 −1 −2 −3 −4 −2 0 2 4 x 1 D. Piga () Polytopic outer approximation 7

A motivation example Design of Robust Controllers for Uncertain LTI Systems ✲ ❡ ✲ C ( s ) ✲ P ( s ) ✲ y r e u ✻ s + 1 5 P ( s ) = s 2 + p 1 s + p 2 4 3 p 2 2 p 1 ∈ [0 , 2] , p 2 = p 2 1 1 0 p 2 ∈ [0 , 4] 0 0.5 1 1.5 2 p 1 1 0 x 2 −1 −2 −3 −4 −2 0 2 4 x 1 D. Piga () Polytopic outer approximation 7

A motivation example Design of Robust Controllers for Uncertain LTI Systems ✲ ❡ ✲ C ( s ) ✲ P ( s ) ✲ y r e u ✻ s + 1 5 P ( s ) = s 2 + p 1 s + p 2 4 3 p 2 2 p 1 ∈ [0 , 2] , p 2 = p 2 1 1 0 p 2 ∈ [0 , 4] p 1 , p 2 ∈ P 0 0.5 1 1.5 2 p 1 1 0 x 2 −1 −2 −3 −4 −2 0 2 4 x 1 D. Piga () Polytopic outer approximation 7

A motivation example Design of Robust Controllers for Uncertain LTI Systems ✲ ❡ ✲ C ( s ) ✲ P ( s ) ✲ y r e u ✻ s + 1 5 P ( s ) = s 2 + p 1 s + p 2 4 3 p 2 2 p 1 ∈ [0 , 2] , p 2 = p 2 1 1 0 p 2 ∈ [0 , 4] p 1 , p 2 ∈ P 0 0.5 1 1.5 2 p 1 1 0 x 2 −1 −2 −3 −4 −2 0 2 4 x 1 D. Piga () Polytopic outer approximation 7

Volume Computation � min 1 dx s . t . S ⊆ P P 1 0 x 2 −1 −2 −3 −4 −2 0 2 4 x 1 D. Piga () Polytopic outer approximation 8

Volume Computation � min 1 dx s . t . S ⊆ P P Question How to compute the volume of P ? 1 0 x 2 −1 −2 −3 −4 −2 0 2 4 x 1 D. Piga () Polytopic outer approximation 8

Volume Computation � min 1 dx s . t . S ⊆ P P Question How to compute the volume of P ? Trouble 1 Computing the volume of a polytope is an NP-hard problem. 1 0 x 2 −1 −2 −3 −4 −2 0 2 4 x 1 D. Piga () Polytopic outer approximation 8

Volume Computation � min 1 dx s . t . S ⊆ P 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 ! 1 0 x 2 −1 −2 −3 −4 −2 0 2 4 x 1 D. Piga () Polytopic outer approximation 8

Main Algorithm Polytopic outer-approximation of S Idea 1 Take an outer-bounding box B of the Euclidean set S 1 0 x 2 −1 −2 −3 −4 −2 0 2 4 x 1 D. Piga () Polytopic outer approximation 9

Main Algorithm Polytopic outer-approximation of S 1.5 1 x 2 0.5 0 0.5 1 1.5 2 x 1 1 0 x 2 −1 −2 −3 −4 −2 0 2 4 x 1 D. Piga () Polytopic outer approximation 10

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 x i uniformly distributed in B 1 0 x 2 −1 −2 −3 −4 −2 0 2 4 x 1 D. Piga () Polytopic outer approximation 11

Main Algorithm Polytopic outer-approximation of S 1.5 1 x 2 0.5 0 0.5 1 1.5 2 x 1 1 0 x 2 −1 −2 −3 −4 −2 0 2 4 x 1 D. Piga () Polytopic outer approximation 12