Regionless Explicit MPC of a Distillation Columndrgona/files/presentation/ECC16.pdf · Regionless...

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

Ján Drgoňa (STU) ECC16 June 30, 2016 1 / 17

Model Predictive Control

Implicit vs Explicit MPC

Implicit MPC Explicit MPC

Implicit MPC

Large systems

Expensive implementation

Matrix inversions

Harder to certify

Explicit MPC

Implicit MPC

Large systems

Matrix inversions

Harder to certify

Explicit MPC

Small systems

Cheap implementation

Division-free

Rigorous analysis

Implicit MPC

Large systems

Matrix inversions

Harder to certify

Explicit MPC

Small systems

Cheap implementation

Division-free

Rigorous analysis

Regionless Explicit MPC

Region-Based Explicit MPC

2UT HU + θT FU

s.t. GU ≤ w + Sθ

Region-Based Explicit MPC

U⋆ = Viθ + vi

λ⋆ = Qiθ + qi

Ri = θ | GU⋆ ≤ w + Sθ, λ⋆ ≥ 0

Active Sets – Geometric Method1

1M. Baotić: Optimal Control of Piecewise Affine Systems – A Multi-parametric Approach, 2005

Pick an arbitrary point in Ω.

Construct an initial critical region. R1

Pick new points in Ω. R1

Construct new critical regions. R1

Repeat recursively until whole Ω iscovered.

R10R11

R17R18

The Idea

Avoid the construction and thestorage of regions!

R10R11

R17R18

The Idea

U⋆ = Viθ + viV (Ai)

λ⋆ = Qiθ + qiQ(Ai)

Ri = θ | GU⋆ ≤ w + Sθ, λ⋆ ≥ 0

R10R11

R17R18

The Idea2

U⋆ = V (Ai)θ + v(Ai)

λ⋆ = Q(Ai)θ + q(Ai)

A = A1, . . . ,AR

A10A11

A17A18

2F. Borrelli and M. Baotić: On the computation of linear model predictive control laws, Automatica 2008

Active Sets – Extensive Enumeration3

. . .. . .

1 2 3 r

1, 2 1, 3 1, r 3, 1 3, 2 3, r

1, 3, 2 1, 3, r 3, 1, 2 3, 1, r

A10A11

A17A18

3A. Gupta et. al.: A novel approach to multiparametric quadratic programming, Automatica 2011

Active Sets – Extensive Enumeration3 – Tree Pruning

. . .. . .

1 2 3 r

1, 2 1, 3 1, r 3, 1 3, 2 3, r

1, 3, 2 1, 3, r 3, 1, 2 3, 1, r

A10A11

A17A18

Explicit MPC Solution Recap

Region-Based Approach

Off-line phase

On-line phase

Off-line phase

1 Active Sets

On-line phase

Off-line phase

1 Active Sets

Geometric method

On-line phase

Off-line phase

1 Active Sets

Geometric methodExtensive enumeration

On-line phase

Off-line phase

1 Active Sets

2 U⋆, λ⋆ from KKT conditions

On-line phase

Off-line phase

1 Active Sets

3 Construction of regions

On-line phase

Off-line phase

1 Active Sets

On-line phase

4 Function evaluation

Regionless Approach

Off-line phase

1 Active Sets

On-line phase

4 Point location

Point Location Problem2

How to find in which region we are,without storing any regions? A1

A10A11

A17A18

U⋆ = V (Ai)θ + v(Ai)

λ⋆ = Q(Ai)θ + q(Ai)

R(Ai) = θ | GU⋆ ≤ w + Sθ, λ⋆ ≥ 0

A10A11

A17A18

U⋆ = V (A1)θ + v(A1)

λ⋆ = Q(A1)θ + q(A1)

R(A1) = θ | GU⋆ ≤ w + Sθ, λ⋆ ≥ 0

A10A11

A17A18

U⋆ = V (A2)θ + v(A2)

λ⋆ = Q(A2)θ + q(A2)

R(A2) = θ | GU⋆ ≤ w + Sθ, λ⋆ ≥ 0

A10A11

A17A18

U⋆ = V (A2)θ + v(A2)

λ⋆ = Q(A2)θ + q(A2)

R(A2) = θ | GU⋆ ≤ w + Sθ, λ⋆ ≥ 0

A10A11

A17A18

Sequential search2

1: procedure Regionless(θ)2: for i ∈ 1, . . . , R do

3: Compute λ = Q(Ai)θ + q(Ai)4: if λ ≥ 0 then

5: Compute U = V (Ai)θ + v(Ai)6: if GU ≤ w + Sθ then

7: U⋆ ← U

8: return U⋆

9: end if

10: end if

11: end for

12: end procedure

Sequential search2

1: procedure Regionless(θ) ⊲ given initial condition θ

2: for i ∈ 1, . . . , R do

3: Compute λ = Q(Ai)θ + q(Ai)4: if λ ≥ 0 then

5: Compute U = V (Ai)θ + v(Ai)6: if GU ≤ w + Sθ then

7: U⋆ ← U

8: return U⋆

9: end if

10: end if

11: end for

12: end procedure

Sequential search2

2: for i ∈ 1, . . . , R do

3: Compute λ = Q(Ai)θ + q(Ai)4: if λ ≥ 0 then ⊲ dual feasibility check5: Compute U = V (Ai)θ + v(Ai)6: if GU ≤ w + Sθ then

7: U⋆ ← U

8: return U⋆

9: end if

10: end if

11: end for

12: end procedure

Sequential search2

2: for i ∈ 1, . . . , R do

3: Compute λ = Q(Ai)θ + q(Ai)4: if λ ≥ 0 then ⊲ dual feasibility check5: Compute U = V (Ai)θ + v(Ai)6: if GU ≤ w + Sθ then ⊲ primal feasibility check7: U⋆ ← U

8: return U⋆

9: end if

10: end if

11: end for

12: end procedure

Sequential search2

2: for i ∈ 1, . . . , R do

8: return U⋆ ⊲ obtain globally optimal U⋆

9: end if

10: end if

11: end for

12: end procedure

Sequential search2

2: for i ∈ 1, . . . , R do

9: end if

10: end if

11: end for

12: end procedure

Q(Ai), q(Ai), V (Ai), v(Ai ), G , w , S are precomputed and stored off-line.2F. Borrelli and M. Baotić: On the computation of linear model predictive control laws, Automatica 2008

Sequential search4

2: for i ∈ 1, . . . , R do

3: Compute λ = Q(Ai)θ + q(Ai)4: if λ ≥ 0 then ⊲ dual feasibility check5: Compute U = −H−1(F T θ + GT

Aiλ) ⊲ U parametrized by λ

6: if GU ≤ w + Sθ then ⊲ primal feasibility check7: U⋆ ← U

9: end if

10: end if

11: end for

12: end procedure

4M. Kvasnica, et. al.: On Region-Free Explicit Model Predictive Control, Conference on Decision and Control 2015

Sequential search4

2: for i ∈ 1, . . . , R do

3: Compute λ = Q(Ai)θ + q(Ai)4: if λ ≥ 0 then ⊲ dual feasibility check5: Compute U = −H−1(F T θ + GT

Aiλ) ⊲ U parametrized by λ

6: if GU ≤ w + Sθ then ⊲ primal feasibility check7: U⋆ ← U

9: end if

10: end if

11: end for

12: end procedure

Q(Ai), q(Ai),A, H−1, F , G , w , S are precomputed and stored off-line.4M. Kvasnica, et. al.: On Region-Free Explicit Model Predictive Control, Conference on Decision and Control 2015

Distillation Column – Process Unit

reboiler

pre-heaterfeed

condenser

distillateaccumulator

reflux valve distillate

waste valve

reboiler

pre-heaterfeed

condenser

waste valve

reboiler

pre-heaterfeed

condenser

waste valve

reboiler

pre-heaterfeed

condenser

waste valve

reboiler

pre-heaterfeed

condenser

waste valve

reboiler

pre-heaterfeed

condenser

waste valve

reboiler

pre-heaterfeed

condenser

waste valve

reboiler

pre-heaterfeed

condenser

waste valve

reboiler

pre-heaterfeed

condenser

waste valve

reboiler

pre-heaterfeed

condenser

waste valve

Distillation Column – Variables

Process variable: y

Temperature at top of the column

Manipulated variable: u

Reflux ratio

reboiler

pre-heaterfeed

condenser

waste valve

Model Identification

xk+1 = Axk + Buk

yk = Cxk

0 1000 2000 3000 4000

Time [s]y

Measurement

Identification

0 1000 2000 3000 4000

Time [s]

Model Identification

xk+1 = Axk + Buk

yk = Cxk

nx = 10

0 1000 2000 3000 4000

Time [s]y

Measurement

Identification

0 1000 2000 3000 4000

Time [s]

Disturbance Modelling

Design model:

xk+1 = Axk + Buk

yk = Cxk

Augmented design model:

xk+1 = Axk + Buk

dk+1 = dk

yk = Cxk + Edk

Augmented design model:

xk+1 = Axk + Buk

dk+1 = dk

yk = Cxk + Edk

Luenberger observer:

A 00 I

uk + L (ym,k − yk)

Model Predictive Control Formulation

minu0,...,uN−1

Nc−1∑

‖∆uk‖2Qu

‖yk − yref‖2Qy

reference tracking

s.t. xk+1 = Axk + Buk

yk = Cxk + Ed0

uk = uNc−1 move blocking

∆uk = uk − uk−1

umin ≤ uk ≤ umax

∆umin ≤ ∆uk ≤ ∆umax

x0 = x(t), d0 = d(t), u−1 = u(t − 1)

13 parameters!

minu0,...,uN−1

Nc−1∑

‖∆uk‖2Qu

reference tracking

yk = Cxk + Ed0

x0 = x(t), d0 = d(t), u−1 = u(t − 1)

13 parameters!

minu0,...,uN−1

Nc−1∑

‖∆uk‖2Qu

reference tracking

yk = Cxk + Ed0

x0 = x(t), d0 = d(t), u−1 = u(t − 1)

17 parameters!

minu0,...,uN−1

Nc−1∑

‖∆uk‖2Qu

reference tracking

yk = Cxk + Ed0

x0 = x(t), d0 = d(t), u−1 = u(t − 1)

17 parameters!

Experimental Results

Ts = 1 s, N = 40, Nc = 5, Qy = 400, Qu = 1

0 5 10 15 20 25 30 35 40

Time [min]

Experimental Results

Ts = 1 s, N = 40, Nc = 5, Qy = 400, Qu = 1

0 5 10 15 20 25 30 35 40

Time [min]

Controller Synthesis and Complexity

on-line

method CPU time suboptimality

on-line

Gurobi 38.74 ms ≈ 0 %

on-line

Gurobi 38.74 ms ≈ 0 %Fiordos 1 13.00 ms ≈ 0 %

on-line

Gurobi 38.74 ms ≈ 0 %Fiordos 1 13.00 ms ≈ 0 %Fiordos 2 13.15 ms ≈ 4 %

on-line

Gurobi 38.74 ms ≈ 0 %Fiordos 1 13.00 ms ≈ 0 %Fiordos 2 13.15 ms ≈ 4 %Regionless 11.95 ms ≈ 0 %

on-line

Execution time improvement by the factor of 6.

on-line

off-line (17 params., 1095 active sets)

Regionless Region-based

on-line

Construction time 1116 s 1360 s

on-line

Construction time 1116 s 1360 sMemory demands 1224 kB 1140 MB

Backup: KKT Conditions

HU⋆ + F T θ + GTA λ⋆ = 0 stationarity

GAU⋆ = wA + SAθ primal equality

GN U⋆ < wN + SN θ primal inequality

λ⋆ ≥ 0 dual inequality

λ⋆T (GAU⋆ − wA − SAθ) = 0 complementarity slackness

Backup: Extensive Enumeration3 – Optimality of A

maxt,U,θ,λ

s.t. HU + F T θ + GTA λ = 0

GAU = wA + SAθ

t ≤ wN + SN θ − GN U

λ ≥ t

t ≥ 0

if LP is feasible with t⋆ > 0 than A is optimal set of active constraints

Backup: Extensive Enumeration3 – Complexity Bound

Rmax =Ncnu∑

k!(nc − k)!

Rmax = maximum number of A

Nc = control horizon

nu = number of control inputs

nc = number of constraints

R ≪ Rmax thanks to pruning

Backup: Regionless Explicit MPC Memory Demands

Regionless

A 13918 integersQ(A), q(A) 54852 floating-point numbersH, F , G , w , S 11584 floating-point numbers

Total memory demands 11224 kB

Region-based

R 10004490 floating-point numbers

Total memory demands 11111140 MB

integer = 2 bytes, single-precision floating-point number = 4 bytes

Backup: Regionally Optimal Map θ → U⋆

Q(A) = −(GAH−1GTA )−1(SA + GAH−1F T )

q(A) = −(GAH−1GTA )−1wA

λ⋆ = Q(A)θ + q(A)

U⋆ = −H−1(F T θ + GTA λ⋆)

Backup: Control Scheme

MPCDistillation

Column

FilterObserver

PWMyref u ym

x , d , u

Backup: Distillation Column – Hardware Solution

Regionless Explicit MPC of a Distillation Columndrgona/files/presentation/ECC16.pdf · Regionless...

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