multiple shooting with collocation: an efficient approach ... · contents 1. motivation1....

OpenModelica Annual Workshop 2012Linköping , February 6, 2012

Multiple Shooting with Collocation: An Efficient Approach to DynamicEfficient Approach to Dynamic

Optimization, with Possibilities forIntegration in OpenModelicaIntegration in OpenModelica

Ines Mynttinen Jasem Tamimi Pu LiInes Mynttinen, Jasem Tamimi, Pu Li

Simulation and Optimal Processes Group (SOP)Institute of Automation and Systems EngineeringInstitute of Automation and Systems Engineering

Ilmenau University of Technology

Contents

1. Motivation1. Motivation

2. Multiple Shooting

3. Collocation on Finite Elements

4. A Case Study

5 Classification and Implementation5. Classification and Implementation

6. Conclusions

Motivation: NMPC using multiple shootingMotivation: NMPC using multiple shootingwith collocation

NMPC is superior to traditional control in many cases

Critical issues:A• Accuracy

• Problem size - computational effort• Safety specifications – path constraints

Fast algorithm is needed

Nonlinear optimal control problem

ft⎧ ⎫⎪ ⎪

Nonlinear optimal control problem

( ) ( ) ( ) ( )( )0

, , , , , ,f

f f ft

J x y t p L x t y t u t p dt⎧ ⎫⎪ ⎪= Φ +⎨ ⎬⎪ ⎪⎩ ⎭

∫( ) ( ) ( ), , ,min

u t x t y t p

( ) ( ) ( ) ( )( )( ) ( ) ( )( )

, , , ,

0

x t f x t y t u t t p

g x t y t u t t p

= ( )0 0x t x=

t t t⎡ ⎤∈ ⎣ ⎦

s.t.

( ) ( ) ( )( )( ) ( ) ( )( )( ) ( )( )

0 , , , ,

0 , , , ,

g x t y t u t t p

h x t y t u t t p

=

≤

0 , ft t t⎡ ⎤∈ ⎣ ⎦

( ) ( )( )( ) ( )( )

0 , ,

0 , ,e i i

i i i

r x t y t p

r x t y t p

=

≤

Bounds of state variables, controls and parameters

Multiple shootingMultiple shooting

blNMPC blproblemtransformation

NMPCproblem

problemsolution

• discretization

• parametrization

• initial guess

• objective functionp

• initial value problem (IVP)

j

• simulation

• continuity equations • continuity

• path constraints

Multiple shooting – control parametrizationMultiple shooting control parametrization

v v 1N Nv v −=

( )u t( )u t

2v

0v1v2

0t 1t 2t Ntt

Multiple shooting – state parametrization( )2 3 2 2; ,xx t s v

Multiple shooting state parametrization

,upper upperx u

x

( ), 0i ih s v ≥0 0 0xs x− =

v v v v

1xs 2

xs 3xs

,lower lowerx u

0v 1v 2v 3v

0t 1t 2t 3t 4 ft t=

Finite dimensional NLP problemFinite dimensional NLP problem

1N −⎧ ⎫( )1

0, , ( , , )

N

N N i ii

J s p t L s v p=

⎧ ⎫= Φ +⎨ ⎬⎩ ⎭

∑0 10 1

,...,,...,

minNN

v vs s

−−

0 00 s x= −s.t.

( )( )1 10 , , ,

0 , ,i i i i i

i i

s x s v p t

g s v p+ += −

= 0,1,...,i N=

0,1,..., 1i N= −

( )( )

0 , ,

0 ,i i

e N

h s v p

r s p

≤

=

( )0 ,i Nr s p≤

Multiple shooting

( )f t

Problem formulation

( ) [ ] ( )( )

, ,

i i

x f x u t

x t x

=

=( ) [ ]

( )1, ,i i i

i i

u t t t t v

x t s+∈ →

→

[ ]1,i it t t +∈(0) (0)v s

( )0...i N=

,i iv s

Simulator(0) (0),i iv s

( )( ) ( )1,k k

i ix v t +

( ) ( ),k ki iv s

O ti i ( ) ( )( ) /k kij idx t ds* *,i iv s

Optimizer ( ) ( )( ) /k kij idx t dv

Multiple shooting - problem solutionMultiple shooting problem solution

( ) ( )*; 0x xs x t s v− =

,upper upperx u

( ) ( )3 2 3 2 2; , 0s x t s v =

( )*

*0v *

3v( )*2

xs( )*3

xs

( ), 0i ih s v ≥0 0 0xs x− = 0

*1v

*2v

3

( )*1xs

,lower lowerx u1

0t 1t 2t 3t 4 ft t=

Collocation on finite elementsCollocation on finite elements

x( 1)2ix −

( 1)2x

1it −Δ itΔ 1it +Δ

1ix2ix

( 1)1ix +

( 1)2ix +

( 1)1ix −( 1)1ix +

t t

( 1)0ix − ( 1) 0i M ix x− = ( 1)0iM ix x +=

0t ft

Collocation on finite elements

( ) ( )K

∑

Collocation on finite elements

( )K

jτ τ−( ) ( )

0ik k

kx t x l τ

=

= ∑d d d

( )0

jk

k j k j

lτ τ

ττ τ≠ =

= ∏−

ddt

d dd dt

ττ

( )

[ ]1,i i

i i

t t tt t hτ

+∈

= +

( ) ( )0

1=

= ∑K

kik

ki

lx t x

h dττ

( ) ( ),x t f x t=

( ) ( )K

kdlx h f x t

τ=∑ ( )

0,ik i

kx h f x t

dτ=

=∑

Advantages ofcollocation on finite elements

• Single-step method smooth profiles requiredonly within the finite elements, discontinuity ofcontrol profiles at possible

• High-order implicit method provides accurate, 1,...,it i N=

g p pprofiles with relatively few finite elements, i.e. reduces problem sizep

• No stability limitations on element size for stiffsystemssystems

• Path constraints can be held inside the elements

Case study - CSTR

( ) 2 ( ) 2i ( ) ( )f x u

t N Nref ref d∑ ∑∫

Case study CSTR

0

( ) 2 ( ) 2

, 1 1min ( ) ( )ref ref

i i i i i ix u i it

w x x w u u dt= =

− + −∑ ∑∫t d l tis.t. model equations

variable bounds

Using 50 subintervals,g ,IPOPT 3.4.0 optimizer.CPU time is 0.953 sec.

Case study - CSTRCase study CSTR

mol

/l)

el x

1 (m

)

atio

nx2

(m

Tank

leve

r co

ncen

trM

olar

Time (min)

Case study - CSTRCase study CSTR

l/min

)

u2 (K

)

wra

te u

1 (l

mpr

atur

eu

Out

letf

low

oola

ntte

m

O C

Time (min)

Classification of methods – consequencesqfor the optimization framework

Single shooting

Multiple shooting

Simultaneousapproach

Optimizationvariables 0,..., 1

ivi N= −

,0,..., 1

i iv si N= −

,0,...,

i iv xi N K= ×

Problem size small intermediate large

Simulation 1 IVP N IVP`s None

Path constraints

nonefulfilled at fulfilled at

, 0,...,it i N K= × , 0,...,it i N K= ×

Implementation in OpenModelicaWh t i d d?What is needed?

Model Optimization problem

objectiveDiscretizerequations init

objectiveopt. variablesconstraints

Interface: f, gradient,

Simulator ( ) ( ),k ki iv s

f, gradient,g, Jacobian• Collocation

• Sensitivityyequations Optimizer

SummarySummary

Collocation method: fast accurate simulator

Multiple shoothing: reduced problem size, path constraintsMultiple shoothing: reduced problem size, path constraints incorporated

Multiple shooting with collocation: high accuracy and reduced computational effortp

Structure of the algorithm: reveals what is needed forStructure of the algorithm: reveals what is needed for implementation in OpenModelica

Many thanksMany thanksfor your attention!

References[1] J. Tamimi, P. Li, A combined approach to nonlinear model predictive control of fast systems Journal of Process Control 20

References

predictive control of fast systems, Journal of Process Control, 20 (2010), 1092‐1102. [2] B. A. Finlayson, The method of weighted residuals and [ ] y , gvariational principles, Academic Press 5th edition (1972).[3] L. T. Biegler, Solution of dynamic optimization problem by successive quadratic programming and orthogonal collocation, Computers and Chemical Engineering 8 (1984), 243‐248.[4] D B Leineweber H G Bock J P Schlöder An efficient[4] D. B. Leineweber, H. G. Bock, J. P. Schlöder, An efficient multiple shooting based reduced SQP strategy for large‐scale dynamic process optimization, Computers and Chemical y p p , pEngineering 27(2), (2003), 157‐166.

multiple shooting with collocation: an efficient approach ... · contents 1. motivation1....

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