Model-based design of optimal experiments for guaranteed ...€¦ · Model-based design of optimal...

Model-based design of optimal experiments for

guaranteed parameter estimation of nonlinear

dynamic systems

A.R. Gottu Mukkula and R. Paulen

Technische Universitat Dortmund

September 29, 2016

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1 / 16Process Dynamicsand Operations

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Motivation

Classical Parameter Estimation

p1

p2

P0

t

u(t)

Optimal Experiment Design

Guaranteed Parameter Estimation

p1

p2

Pe

P0

t

u(t)

?

Optimal Experiment Design

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model: y = F (p)

{

dynamic system: x = f (x ,u,p), x(0) = g(p)

output equation: y = h(x ,p)

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model: y = F (p)

{



y

t

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model: y = F (p)

{



y , y

t p1

p2

P0

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minp∈P0

‖y − y‖22

s.t. y = F (p)

y , y

t p1

p2

P0

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minp∈P0

‖y − y‖22

s.t. y = F (p)

y , y

t

y(e) ∈ y + [ e, e ]

p1

p2

P0

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e ∈ [e,e] : minp∈P0

‖y+e − y‖22

s.t. y = F (p)

y(e), y

t p1

p2

P0

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Guaranteed Parameter Estimation (GPE)

∀e ∈ [e,e] : minp∈P0

‖y+e − y‖22

s.t. y = F (p)

y , y

t p1

p2

Pe

P0

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findp∈P0

all p

s.t. e ≤ y − y ≤ e

y = F (p)

y , y

t p1

p2

Pe

P0

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findp∈P0

all p

s.t. e ≤ y − y ≤ e

y = F (p)

y , y

tp1

p2

Pe

P0

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Model-based Optimal Experiment Design (OED)

Idea: Improve the reliability of parameter estimates by

optimizing some measure of Fisher Information Matrix (FIM)

minu(t)

φ (FIM) s.t. FIM =

Ne∑

k=1

∂y(k)

∂pTQ∂y(k)

∂p, y = F (p,u)

t

u(t)

p1

p2

P0

A design: φ(FIM) = tr(FIM), D design: φ(FIM) = det(FIM), . . .

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Optimal Experiment Design (OED) for GPE

The classical least-squares and guaranteed parameter

estimation yield different results.

y , y

tp1

p2

Pe

P0

How to perform OED in the context of GPE?

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1 Find an over-approximation of GPE solution set

maxp/p

np∑

j=1

pj − pj

s.t. y(t) = F (π), ∀π ∈ {E(p),p/p}

ei ≤ y(ti ,E(p))− y(ti ,pj/pj) ≤ ei , ∀j ∈ {1, . . . ,np}

t

y , y

p1

p2

Pe

E(p)p

1

p2

p1

p2

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2 Optimize over the over-approximation of GPE solution set

minu

maxp/p

np∑

j=1

pj − pj

(A design)

s.t. y(t) = F (π), ∀π ∈ {E(p),p/p}


t

y , y

p1

p2

Pe

E(p)p

1

p2

p1

p2

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2 Optimize over the over-approximation of GPE solution set

minu

maxp/p

np∏

j=1

pj − pj

(D design)

s.t. y(t) = F (π), ∀π ∈ {E(p),p/p}


t

y , y

p1

p2

Pe

E(p)p

1

p2

p1

p2

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Optimal Experiment Design for GPE

Reformulation into Mathematical Program with Equilibrium

Constraints (MPEC)

minu,λ,λ

np∑

j=1

pUj − pL

j

s.t. y(t) = F (π,u),

0 = ∇πL(x(t),u(t), π, λπλπ)

ei ≤ y(ti ,E(p))− y(ti ,pj/pj) ≤ ei

0 = diagλij(y(ti ,pj/pj)− y(ti ,E(p)) + ei)

0 = diagλij(y(ti ,E(p))− y(ti ,pj/pj)− ei)

0 ≤ λ, λ

u ≤ u(t) ≤ u

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Optimal Experiment Design for GPE

Regularization of the MPEC (Hatz et al., 2013)

minu,λ,λ

w ,w

np∑

j=1

pUj − pL

j + ρ‖w‖1 + ρ‖w‖1

s.t. y(t) = F (π,u),

0 = ∇πL(x(t),u(t), π, λπλπ)

ei ≤ y(ti ,E(p))− y(ti ,pj/pj) ≤ ei

−w ij≥diagλij(y(ti ,pj/pj)− y(ti ,E(p)) + ei)

−w ij≥diagλij(y(ti ,E(p))− y(ti ,pj/pj)− ei)

0 ≤ λ, λ,w ,w

u ≤ u(t) ≤ u

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OED for GPE: Case Study 1

Reaction A+Bk→C run in isothermal semi-batch reaction with the

feed of component B (Srinivasan et al., 2003)

dcA

dt= −kcAcB −

u

VcA

dcB

dt= −kcAcB +

u

V(cB,in − cB)

dV

dt= u, Tj = T +

−∆HkcAcBV

αA

u, cB,in

Tj,in

Tj

Outputs cA, cB ,Tj sampled at ten equidistant time instants,

corrupted with zero-mean noise with standard deviation of

0.005 mol/L and 0.05 K respectively

Estimated parameters (k ,∆H) = (0.05784 L/mol/h, −72 kJ/mol)

Feed u(t) ∈ [0,1]

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Optimal A design for GPE (red) compared to classical A design

(black) confindence ellipsoid

0 0.2 0.4 0.6 0.8 1 1.2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Classical A design

GPE A design

time [h]

u(t)

0.05 0.055 0.06 0.065-8

-7.5

-7

-6.5

k [l/mol h]

∆H

[10

4J/m

ol]

Solved with CasADi + Ipopt + CVODE (sequential approach)

GPE solution sets obtained from CRONOS (Chachuat et al.)

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Lotka-Volterra (predator-prey) model modified to control the prey

population

dx1

dt= −kx (x1 − u(t)), x1(0) = 10,

dx2

dt= (0.05x1 − d)x2, x2(0) = 10.

Output x2 sampled at equidistant time instants, corrupted with

zero-mean noise with standard deviation of 0.1

Estimated parameters (kx ,d) = (0.9,1.5)

Stimulus of growth u(t) ∈ [0,1]

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Optimal A design for GPE (purple) compared to classical A

design (green); “classical” confidence ellipsoids included

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

control input for classical A−design

control input for GPE A−design

t

u(t)

0.82 0.86 0.9 0.94 0.981.42

1.46

1.5

1.54

1.58

99% confidence ellipsoid for classical A−design

99% confidence ellipsoid for GPE A−design

GPE solution set for classical A−design

GPE solution set for GPE A−design

kx

d



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Lotka-Volterra (predator-prey) model modified to control the prey

population

dx1

dt= x1 − p1x1x2 − 0.4x1u, x1(0) = 0.5,

dx2

dt= −x2 + p2x1x2 − 0.2x2u, x2(0) = 0.7.

Outputs x1 and x2 sampled at optimized time instants (τ1, τ2) are

corrupted with zero-mean noise with standard deviation of 0.1

Estimated parameters (p1,p2) = (1,1)

Fishing (yes/no) u(t) ∈ {0,1} (binary)

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Optimal D design for GPE (green) compared to classical D

design (purple); “classical” confidence ellipsoids included

0 2 4 6 8 10 12

0

1

time

u(t)

Classical D designGPE D design

0.85 0.9 0.95 1 1.05 1.1 1.15

0.85

0.9

0.95

1

1.05

1.1

1.15

p1

p2

GPE D designClassical D design



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Interesting observation: Optimal D design for GPE is a local

optimum of classical D design

43

21

06

8

10

0

-400

-1200

-1000

-800

-600

-200

12

τ1τ2

−d

et(

FIM

)

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Conclusions

Problem of optimal experiment design for guaranteed parameter

estimation formulated as a bilevel program

The approximation of the set of guaranteed parameter

estimates realized as a box

Computationally intensive problem but tractable for small-scale

cases

Classical optimal experiment design does not cope well with the

nonlinearity in the model → Results of optimal experiment for

guaranteed parameter estimation differ from the classical

experiment design

Acknowledgement:

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