Flatness-Based Model Selection of Benzaldehyde ...

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Flatness-Based Model Selection of Benzaldehyde Lyase Catalysed Biochemical Reaction Network Moritz Schulze, René Schenkendorf, 26. May 2017

Center of Pharmaceutical Engineering (PVZ) TU Braunschweig founded in 2012 19 institutes, ca. 100 scientists

1500 m2 labs & 42 m2 pilot plant area

26. May 2017 Moritz Schulze, René Schenkendorf Page 2 Flatness-Based Model Selection of Benzaldehyde Lyase Catalysed Biochemical Reaction Network

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Center of Pharmaceutical Engineering (PVZ) TU Braunschweig founded in 2012 19 institutes, ca. 100 scientists

1500 m2 labs & 42 m2 pilot plant area

interdisciplinary collaboration low-cost and effective APIs personalised therapy with individualised drug products


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PSE group of InES

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reliable monitoring

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Pharmaceutical Systems Engineering group

First principles models Model selection

ro co bus nt t ro l al

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rod u

q

m

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System understanding

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ua

Uncertainty & fault analysis

ri

lit

al ti m n op sig de

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Parameter identiﬁcation

ct


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Agenda

Motivation Concept of flatness Results and challenges


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Motivation Declining profit margins in pharmaceutical industry Increasing R&D costs and time Strengthened competition (generic drugs)


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High product quality requirements Good system understanding Design depends critically on the used model Set of candidates (reactants?, mechanistics?, kinetics?)


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→ Careful model selection and optimal design of experiments

High product quality requirements Good system understanding Design depends critically on the used model Set of candidates (reactants?, mechanistics?, kinetics?)

OED


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Model discrimination state-of-the-art

OED of dynamic systems requires optimisation of (in general time dependent) control variables → optimal control problem


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Model discrimination state-of-the-art

OED of dynamic systems requires optimisation of (in general time dependent) control variables → optimal control problem Approximation of control inputs by e.g. orthogonal collocation or CVP techniques → Large problems, high computational effort and efficient solvers required


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Flatness-based strategy

New for model selection (widely applied in control problems)


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New for model selection (widely applied in control problems) Experimental conditions are derived analytically


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New for model selection (widely applied in control problems) Experimental conditions are derived analytically Feedforward control


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New for model selection (widely applied in control problems) Experimental conditions are derived analytically Feedforward control Analysis tool


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Concept of flatness x : state variables, u : inputs, x˙ = f(x, u) y = h(x, u)

|

{z

}

model M

yflat

→

y : outputs

x = fx (yflat , y˙ flat , ...) u = fu (yflat , y˙ flat , ...)

|

{z

}

inverse model M −1

yflat

M −1

u

y M

yflat = fflat (x, u, u˙ , ...) and its derivatives fully describe dynamic behaviour of the system.


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Model discrimination ˘flat ) max D(y ˘ yflat

EXP u

M1 M2 M3

˘flat s.t . ||∆u || < → shape functions for y

ydata y ymod 1 ymod 2 ymod 3


t

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Case study 1: Academic example Discrimination criterion ("T-optimal design") D=

m −1 X

m X ui (y flat ) − uj (y flat ) 2

i =1 j =i +1

m model candidates Mi : M1 : x˙ = −0.1x + u M2 : x˙ = −0.2x + u M3 : x˙ = −0.01x 2 + u


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m −1 X


i =1 j =i +1

m model candidates Mi : M1 : x˙ = −0.1x + u M2 : x˙ = −0.2x + u M3 : x˙ = −0.01x 2 + u Flat output y flat = x


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m −1 X


i =1 j =i +1

m model candidates Mi : M1 : x˙ = −0.1x + u M2 : x˙ = −0.2x + u M3 : x˙ = −0.01x 2 + u

max D (y flat , y˙ flat ) y flat

s .t . x < 50 x0 < 10

Flat output y flat = x


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30

50

Controls ui

State x = y flat

Case study 1: Optimisation results

30 10 0

0

M1 M2 M3

20 10 0

100 200 300 Time

0

100 200 300 Time

max D(y flat , y˙ flat ) y flat

y

s.t .

flat

(t ) =

6 X

Ci t i

i =0

x < 50 x0 < 10

D increases as y flat = x incr. → optimisation as expected


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Case study 2: BAL catalysed reaction network

Mechanistics 2 BA

BAL − −−→ step1

1 BZ BAL

1 BZ + 1 ALD − −−→ 1 HPP + 1 BA step3

Dynamic system BA : x˙ 1 = −2νstep1 + νstep3 + u1

ALD : x˙ 2 = −νstep3 + u2 BZ : x˙ 3 = νstep1 − νstep3 HPP : x˙ 4 = νstep3


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Case study 2: Model candidates Candidates (kinetics) M1 : Michaelis-Menten with inhibition M2 : Michaelis-Menten (set KI,2 = ∞) M3 : Power law

   

νstep1   ν step3 νstep1 M3 νstep3

M1 , M2

!2 = [E ]Vmax,1

x1 KM,BA (1+x2 /KI,2 )+x1

= [E ]Vmax,3 KM,BZ (1+xx32 /KI,2 )+x3 = k1 x12 = k3 x2 x3


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Results: System trajectories Model 2:

yflat

=

x2 x4

=

1 30(1 − exp[−t /100])

0.3

u1 u2

0.2 0.1 0

0

100 200 Time [min]

300

30

5 States [mM]

Controls [mM/min]

Flat outputs → controls → states

4 3 2 1 0

x1 x2 x3 x4 0

10 0 300

100 200 Time [min]


20

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Conc. /[mM]

Case study 2: Optimisation 60

20

M1 M2 M3

40 10 0

20 0

100 200 Time /[min]

Controls 1

300

→

0

0

100 200 Time /[min]

300

Controls 2


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Conc. /[mM]

Case study 2: Optimisation 60

20

M1 M2 M3

40 10 0

20 0

100 200 Time /[min]

Controls 1

300

→

0

0

100 200 Time /[min]

300

Controls 2

Is this the optimal discriminating input?


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Case study 2: Results Flatness as a model analysis tool Complex regions Singularities → Constraints on feasible region


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Choice of shape functions (flat outputs) Analytic, non-piecewise functions, e.g. polynomial, rational, exponential Diverse local and global solvers (MATLAB, e.g. fminsearch, fmincon, particlesearch, patternsearch) → No final optimal result in setting up the optimisation problem


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Choice of shape functions (flat outputs) Analytic, non-piecewise functions, e.g. polynomial, rational, exponential Diverse local and global solvers (MATLAB, e.g. fminsearch, fmincon, particlesearch, patternsearch) → No final optimal result in setting up the optimisation problem

Outlook: Splines (increasing degrees of freedom)


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Thanks for your attention!


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