Model Predictive Control Jn Drgoa (STU) ECC16 June 30, 2016 2 / - PowerPoint PPT Presentation

Regionless Explicit MPC of a Distillation Column Ján Drgoňa , Filip Janeček, Martin Klaučo, and Michal Kvasnica Slovak University of Technology in Bratislava, Slovakia June 30, 2016 I AM Ján Drgoňa (STU) ECC16 June 30, 2016 1 / 17

Model Predictive Control Ján Drgoňa (STU) ECC16 June 30, 2016 2 / 17

Implicit vs Explicit MPC Explicit MPC Implicit MPC Ján Drgoňa (STU) ECC16 June 30, 2016 3 / 17

Implicit vs Explicit MPC Explicit MPC Implicit MPC Large systems Expensive implementation Matrix inversions Harder to certify Ján Drgoňa (STU) ECC16 June 30, 2016 3 / 17

Implicit vs Explicit MPC Explicit MPC Implicit MPC Large systems Small systems Expensive implementation Cheap implementation Matrix inversions Division-free Harder to certify Rigorous analysis Ján Drgoňa (STU) ECC16 June 30, 2016 3 / 17

Implicit vs Explicit MPC Explicit MPC Implicit MPC Large systems Small systems Expensive implementation Cheap implementation Matrix inversions Division-free Harder to certify Rigorous analysis Regionless Explicit MPC Ján Drgoňa (STU) ECC16 June 30, 2016 3 / 17

Region-Based Explicit MPC 1 2 U T HU + θ T FU min U s.t. GU ≤ w + S θ Ján Drgoňa (STU) ECC16 June 30, 2016 4 / 17

Region-Based Explicit MPC U ⋆ = V i θ + v i λ ⋆ = Q i θ + q i R i = { θ | GU ⋆ ≤ w + S θ, λ ⋆ ≥ 0 } Ján Drgoňa (STU) ECC16 June 30, 2016 4 / 17

Active Sets – Geometric Method 1 1 M. Baotić: Optimal Control of Piecewise Affine Systems – A Multi-parametric Approach, 2005 Ján Drgoňa (STU) ECC16 June 30, 2016 5 / 17

Active Sets – Geometric Method 1 Pick an arbitrary point in Ω . 1 M. Baotić: Optimal Control of Piecewise Affine Systems – A Multi-parametric Approach, 2005 Ján Drgoňa (STU) ECC16 June 30, 2016 5 / 17

Active Sets – Geometric Method 1 Construct an initial critical region. R 1 1 M. Baotić: Optimal Control of Piecewise Affine Systems – A Multi-parametric Approach, 2005 Ján Drgoňa (STU) ECC16 June 30, 2016 5 / 17

Active Sets – Geometric Method 1 Pick new points in Ω . R 1 1 M. Baotić: Optimal Control of Piecewise Affine Systems – A Multi-parametric Approach, 2005 Ján Drgoňa (STU) ECC16 June 30, 2016 5 / 17

Active Sets – Geometric Method 1 R 3 R 2 R 4 Construct new critical regions. R 5 R 1 R 7 R 6 1 M. Baotić: Optimal Control of Piecewise Affine Systems – A Multi-parametric Approach, 2005 Ján Drgoňa (STU) ECC16 June 30, 2016 5 / 17

Active Sets – Geometric Method 1 R 3 R 2 R 4 Repeat recursively until whole Ω is R 5 R 1 covered. R 7 R 6 1 M. Baotić: Optimal Control of Piecewise Affine Systems – A Multi-parametric Approach, 2005 Ján Drgoňa (STU) ECC16 June 30, 2016 5 / 17

Active Sets – Geometric Method 1 R 11 R 10 R 9 R 12 R 3 R 2 R 4 R 13 Repeat recursively until whole Ω is R 8 R 5 R 1 covered. R 14 R 7 R 19 R 15 R 6 R 17 R 18 R 16 1 M. Baotić: Optimal Control of Piecewise Affine Systems – A Multi-parametric Approach, 2005 Ján Drgoňa (STU) ECC16 June 30, 2016 5 / 17

The Idea R 11 R 10 R 9 R 12 R 3 R 2 R 4 R 13 Avoid the construction and the R 8 R 5 R 1 R 14 storage of regions! R 7 R 19 R 15 R 6 R 17 R 18 R 16 Ján Drgoňa (STU) ECC16 June 30, 2016 6 / 17

The Idea R 11 R 10 R 9 R 12 R 3 R 2 R 4 R 13 U ⋆ = V i θ + v i V ( A i ) R 8 R 5 R 1 λ ⋆ = Q i θ + q i Q ( A i ) R 14 R 7 R 19 R 15 R 6 R i = { θ | GU ⋆ ≤ w + S θ, λ ⋆ ≥ 0 } R 17 R 18 R 16 Ján Drgoňa (STU) ECC16 June 30, 2016 6 / 17

The Idea 2 A 11 A 10 A 9 A 12 A 3 A 2 A 4 A 13 U ⋆ = V ( A i ) θ + v ( A i ) A 8 A 5 A 1 λ ⋆ = Q ( A i ) θ + q ( A i ) A 14 A 7 A 19 A 15 A 6 A = {A 1 , . . . , A R } A 17 A 18 A 16 2 F. Borrelli and M. Baotić: On the computation of linear model predictive control laws, Automatica 2008 Ján Drgoňa (STU) ECC16 June 30, 2016 6 / 17

Active Sets – Extensive Enumeration 3 A 11 A 10 {} A 9 A 12 A 3 A 2 A 4 A 13 { r } { 1 } { 2 } { 3 } . . . A 8 A 5 A 1 A 14 A 7 A 19 A 15 A 6 { 1 , 2 } { 1 , 3 } { 1 , r } { 3 , 1 } { 3 , 2 } { 3 , r } . . . . . . A 17 A 18 A 16 { 1 , 3 , 2 } { 1 , 3 , r } { 3 , 1 , 2 } { 3 , 1 , r } . . . . . . 3 A. Gupta et. al.: A novel approach to multiparametric quadratic programming, Automatica 2011 Ján Drgoňa (STU) ECC16 June 30, 2016 7 / 17

Active Sets – Extensive Enumeration 3 – Tree Pruning A 11 A 10 {} A 9 A 12 A 3 A 2 A 4 A 13 { r } { 1 } { 2 } { 3 } . . . A 8 A 5 A 1 A 14 A 7 A 19 A 15 A 6 { 1 , 2 } { 1 , 3 } { 1 , r } { 3 , 1 } { 3 , 2 } { 3 , r } . . . . . . A 17 A 18 A 16 { 1 , 3 , 2 } { 1 , 3 , r } { 3 , 1 , 2 } { 3 , 1 , r } . . . . . . 3 A. Gupta et. al.: A novel approach to multiparametric quadratic programming, Automatica 2011 Ján Drgoňa (STU) ECC16 June 30, 2016 7 / 17

Explicit MPC Solution Recap Region-Based Approach Off-line phase On-line phase Ján Drgoňa (STU) ECC16 June 30, 2016 8 / 17

Explicit MPC Solution Recap Region-Based Approach Off-line phase Active Sets 1 On-line phase Ján Drgoňa (STU) ECC16 June 30, 2016 8 / 17

Explicit MPC Solution Recap Region-Based Approach Off-line phase Active Sets 1 Geometric method On-line phase Ján Drgoňa (STU) ECC16 June 30, 2016 8 / 17

Explicit MPC Solution Recap Region-Based Approach Off-line phase Active Sets 1 Geometric method Extensive enumeration On-line phase Ján Drgoňa (STU) ECC16 June 30, 2016 8 / 17

Explicit MPC Solution Recap Region-Based Approach Off-line phase Active Sets 1 Geometric method Extensive enumeration U ⋆ , λ ⋆ from KKT conditions 2 On-line phase Ján Drgoňa (STU) ECC16 June 30, 2016 8 / 17

Explicit MPC Solution Recap Region-Based Approach Off-line phase Active Sets 1 Geometric method Extensive enumeration U ⋆ , λ ⋆ from KKT conditions 2 Construction of regions 3 On-line phase Ján Drgoňa (STU) ECC16 June 30, 2016 8 / 17

Explicit MPC Solution Recap Region-Based Approach Off-line phase Active Sets 1 Geometric method Extensive enumeration U ⋆ , λ ⋆ from KKT conditions 2 Construction of regions 3 On-line phase Function evaluation 4 Ján Drgoňa (STU) ECC16 June 30, 2016 8 / 17

Explicit MPC Solution Recap Regionless Approach Off-line phase Active Sets 1 Geometric method Extensive enumeration U ⋆ , λ ⋆ from KKT conditions 2 Construction of regions 3 On-line phase Point location 4 Ján Drgoňa (STU) ECC16 June 30, 2016 8 / 17

Point Location Problem 2 A 11 A 10 A 9 A 12 A 3 A 2 ? A 4 A 13 How to find in which region we are, A 8 A 5 A 1 without storing any regions? A 14 A 7 A 19 A 15 A 6 A 17 A 18 A 16 2 F. Borrelli and M. Baotić: On the computation of linear model predictive control laws, Automatica 2008 Ján Drgoňa (STU) ECC16 June 30, 2016 9 / 17

Point Location Problem 2 A 11 A 10 A 9 A 12 A 3 A 2 ? A 4 A 13 U ⋆ = V ( A i ) θ + v ( A i ) A 8 A 5 A 1 λ ⋆ = Q ( A i ) θ + q ( A i ) A 14 A 7 A 19 A 15 A 6 R ( A i ) = { θ | GU ⋆ ≤ w + S θ, λ ⋆ ≥ 0 } A 17 A 18 A 16 2 F. Borrelli and M. Baotić: On the computation of linear model predictive control laws, Automatica 2008 Ján Drgoňa (STU) ECC16 June 30, 2016 9 / 17

Point Location Problem 2 A 11 A 10 A 9 A 12 A 3 A 2 ? A 4 A 13 U ⋆ = V ( A 1 ) θ + v ( A 1 ) A 8 A 5 A 1 λ ⋆ = Q ( A 1 ) θ + q ( A 1 ) A 14 A 7 A 19 A 15 A 6 R ( A 1 ) = { θ | GU ⋆ ≤ w + S θ, λ ⋆ ≥ 0 } A 17 A 18 A 16 2 F. Borrelli and M. Baotić: On the computation of linear model predictive control laws, Automatica 2008 Ján Drgoňa (STU) ECC16 June 30, 2016 9 / 17

Point Location Problem 2 A 11 A 10 A 9 A 12 A 3 A 2 ? A 4 A 13 U ⋆ = V ( A 2 ) θ + v ( A 2 ) A 8 A 5 A 1 λ ⋆ = Q ( A 2 ) θ + q ( A 2 ) A 14 A 7 A 19 A 15 A 6 R ( A 2 ) = { θ | GU ⋆ ≤ w + S θ, λ ⋆ ≥ 0 } A 17 A 18 A 16 2 F. Borrelli and M. Baotić: On the computation of linear model predictive control laws, Automatica 2008 Ján Drgoňa (STU) ECC16 June 30, 2016 9 / 17

Point Location Problem 2 A 11 A 10 A 9 A 12 A 3 U ⋆ A 2 A 4 A 13 U ⋆ = V ( A 2 ) θ + v ( A 2 ) A 8 A 5 A 1 λ ⋆ = Q ( A 2 ) θ + q ( A 2 ) A 14 A 7 A 19 A 15 A 6 R ( A 2 ) = { θ | GU ⋆ ≤ w + S θ, λ ⋆ ≥ 0 } A 17 A 18 A 16 2 F. Borrelli and M. Baotić: On the computation of linear model predictive control laws, Automatica 2008 Ján Drgoňa (STU) ECC16 June 30, 2016 9 / 17

Sequential search 2 1: procedure Regionless ( θ ) for i ∈ { 1 , . . . , R } do 2: Compute λ = Q ( A i ) θ + q ( A i ) 3: if λ ≥ 0 then 4: Compute U = V ( A i ) θ + v ( A i ) 5: if GU ≤ w + S θ then 6: U ⋆ ← U 7: return U ⋆ 8: end if 9: end if 10: end for 11: 12: end procedure 2 F. Borrelli and M. Baotić: On the computation of linear model predictive control laws, Automatica 2008 Ján Drgoňa (STU) ECC16 June 30, 2016 10 / 17

Model Predictive Control Jn Drgoa (STU) ECC16 June 30, 2016 2 / - PowerPoint PPT Presentation

Regionless Explicit MPC of a Distillation Column Jn Drgoa , Filip Janeek, Martin Klauo, and Michal Kvasnica Slovak University of Technology in Bratislava, Slovakia June 30, 2016 I AM Jn Drgoa (STU) ECC16 June 30, 2016 1 / 17 Model

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