Structured H∞control of a continuous crystallizer

L. Ravanbod,D. Noll,

P. Apkarian

Institut de Mathématiques de Toulouse

IFAC Workshop on

Control Distributed Parameter Systems

Toulouse, France

July 20-24, 2009

2

Outline

• Industrial Crystallizer : presentation, physical model.

• Why H∞ control?

• H∞ control :

structured controller,

structured + time constraints.

• Simulation results :

application to continuous crystallizer.

3

Continuous Industrial Crystallizer used for mass production of high-purity solids from

liquids.

This crystallizer produces hundreds of tons of amonium sulfate per day.

(a)

(b)

(c)

(d)

(e)

a: body

b: settling region of fine crystals

c: slurry is withdrawn

d: slurry heated and combined with product feed

e: solvent evaporates

4

q, cfq, hf ,n, c

q, hp,n, c

Continuous Crystallizer : Our hypotheses

► q: feed rate

► Cf: solute concentration in feed

► C: solute concentration

► hf: classification function of fines dissolution

► hp: classification function of product removal

► n: number density function

ideal mixing,

isothermal operation,

constant overall volume,

nucleation at negligible size,

size-independence growth rate,

no breakage, no agglomeration

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Continuous Crystallizer:equations

Population balance :

Lf Lp

1

1+R2

R1

0

n L , tt

G cn L , t

Lqv

h f L h p L n L , t

n L,0 n0 L , n 0, tB cG c

G c K g c t csg , B c K b c t cs

b

with initial and boundary conditions:

and the classification functions:

nucleation (birth) rategrowth rate

=(number of crystals/crystal length L)/volume at time t n L , t

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Continuous Crystallizer:equations

mole balance :

with initial condition:

Mdc

dt

q Mc

v

Mc d

dt

q Mcfv

qv

1 K v 0hp L 1 n L , t L3 dL

c 0 c0

Crystal size distribution is represented by mass density function:

and by overall crystal mass

and where: t 1 Kv 0n L,t L3dL

m L , t K v n L , t L3

m t0

K v n L , t L3dL

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Continuous Crystallizer:Why feedback control?

• Nucleation, • crystal growth, • fines dissolution,• classified product removal• …

Undesirable oscillatory behaviour

0 5 10 15 20 25 30 35 40 45 504.0894

4.0896

4.0898

4.09

4.0902

4.0904

4.0906

4.0908

4.091

4.0912

4.0914

t [h]

C [

mol

/l]

solute concentration

As in solute concentration:

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Continuous Crystallizer: Why feedback control?

Or as in mass density function:

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Control strategiesP: plant (crystallizer), K: controller

P

K

y

w

u

z

w z

• guarantying internal stability,

• minimizing impact of on

Find structured controller i.e. K(s) ĸ(s) :

Two families of linear regulators:

• if w white noise:

• if w of finite energy:

C Ref , m Ref , noise

C f , R 1C n , m n

w(t) :

u(t) :

C Err , m Errz(t) :y(t) :

minK

T zw K 22 min

K

12

Trace T zwT j , K T zw j , K d

minK

T zw K minK

max T zw j , K( Supω )

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H∞ control of crystallizer

Population balance finite or infinite dim.model finite dim. H∞ controller (SISO) (Chiu et al 1999, Bosgra et al 1995) (Vollmer and Raisch 2001)

Population balance

Large state linear model

small dim. H∞ controller

We propose:

Advantage:

selection of controller structure,

easily extendable to MIMO,

time constraints conveniently added.

Previous works:

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min K

Tw z K

Constrained structured H∞ control

Minimize:

Subject tozl t z K , t zu t , for all t t t

z K , s Tw z K , s w0 s

K K ΘTime domain constraints,

(w0(t) step, ramps, sinusoid)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8-3

-2

-1

0

1

2

3

4

5

6

7x 10

-3

time

zu t

z l t

Θ decision variable

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Multistage H∞ synthesis

Smooth optimization

(SQP) Non smooth optimization

stabilizing 2nd order H∞ controller

Closed-loop interconnection

Non smooth optimization

stabilizing 2nd order H∞ controller: time constraints are approximately satisfied

stabilizing 2nd order H∞ controller: time constraints satisfied

S ISO : 1 9

M IM O : 1 16

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H∞ control Numerical method

H∞ synthesis is minmax nonsmooth and nonconvex techniques is proposed:

ming x 0

f x

F y , x max f y f x max g x ,0 ; g y max g x ,0

minimized by Cutting-Plane Algorithm.

Smooth optimization (SQP) accelerates creation of good starting points.

Closed-loop stability is guaranteed by constraint:

A B. K. C 0.001

Efficient for large systems due to possibility of structure selection

( Apkarian, Noll, Bompart, Rondepierre,…2006, 2007, 2008)

is handled through a progress function:

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Continuous Crystallizer:modelling

1 Choosing the parameter values: KCl laboratory crystallizer used by : U. Vollmer, J. Raisch, Control Engineering Practice 2001

solute concentration in the feed Cf(t)

solute concentration in the liquid C(t)Cf(t) and disolution rate R1(t)

C(t) and overall crystal mass M(t)

Model input, output choice

• SISO

• MIMO

discretization of n(L,t) w.r.t L

n L 1, t

L

n L 1 L , t n L 1 L , t

2 Ln L 1 L , t n L 1 , t n L 1 L , t

0mm 2mm

N. L

2

3

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Continuous Crystallizer:modelling

Lf Lp

1

1+R2

R1

0

4

Linearization at an equilibrium point

Linear model (for synthesis)

5 Nonlinear model

(for validation)

A , B , C , D

x T t n L , t n 2 L , t n N L , t C t 1 N 1

with state space representation

Equidistant discretization with N=250, et mollifying the classification functions

Equidistant discretization with N=1000

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Simulation results (Input-Output Model precision)

SISO MIMO

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Simulation resultsSISO : solute concentrations

Second order H∞ controller+ time domain constraints

0 5 10 15 20 25 303

4

5

6

7C

f [m

ol/l]

0 5 10 15 20 25 304.09

4.095

4.1

4.105

4.11

C [

mol

/l]

t [min]

constrained H

H

constrained H

H

Solute concentration in the crystallizer

Solute concentration in the feed

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Simulation resultsMIMO: solute concentration, overall mass

Second order H∞ controller+ constraints

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Simulation results Mass density

• evolution in open-loop and in closed-loop • evolution from one

equlibrium point to another one

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New control methodology allows great flexibility of controller structure: small controllers for large systems.

Time constraints can be added and allow to include features of nonlinear systems.

Problem is genuinely nonsmooth and specific algorithm has to be developed.

Conclusion

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Simulation resultsMIMO: solute concentration, overall mass

PID H∞ controller+ time constraints

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Control strategiesP: plant (crystallizer), K: controller

P

K

y

w

u

z

w z

• internal stability,

• minimizing impact of on

Objectif find K(s) ĸ(s) :

Two families of linear regulators can be found:

• if w white noise:

• if w of finite energy: minK

T w z . minK

maxw 0

z t 2

w t 2

minK

T w z . 2 minK

maxw s H

z s 2

w s

C Ref , m Ref , n y , n u

C f , R 1C n , m n

w(t) :

u(t) :

C Err , m Errz(t) :y(t) :