Control of a solution copolymerization reactor using piecewise linear models
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Control of a Solution Copolymerization Reactor using Piecewise Linear Models. Leyla Özkan APACT-03 York, UK April 30 th , 2003. Presentation Outline. Motivation Multi-Model Predictive Control Formulation Implementation on MMVA Solution Co-polymerization Reactor Stability Analysis

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Control of a Solution Copolymerization Reactor using Piecewise Linear Models

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Control of a solution copolymerization reactor using piecewise linear models

Control of a Solution Copolymerization Reactor using Piecewise Linear Models

Leyla Özkan

APACT-03 York, UK

April 30th , 2003


Presentation outline

Presentation Outline

  • Motivation

  • Multi-Model Predictive Control Formulation

  • Implementation on MMVA Solution Co-polymerization Reactor

  • Stability Analysis

  • Conclusion

APACT03-30 April, 2003


Motivation

Motivation

  • Polymerization reactors

    • Complex nonlinear kinetics and behavior

    • Difficult to specify control objectives

    • Global competition

      • Strict requirements on polymer properties

      • Grade transitions common

    • Control objective

      • Minimize grade transition time

Reduce Off-Specification Product

APACT03-30 April, 2003


Model predictive control

min

J

(

k

)

p

u

(

.

)

Past

Future

Set point

dx

Predicted y

=

f

(

x

,

u

,

t

)

dt

Closed loop y

Open loop u

£

£

u

u

u

min

max

£

£

y

y

y

min

max

Closed loop u

k

k+1

k+Hc

k+Hp

Hc

Hp

Model Predictive Control

  • Model Predictive Control

    • Class of control algorithms that solves optimization problem at every instant

APACT03-30 April, 2003


Multi model predictive control

[

]

¥

T

=

+

+

+

J

x

(

k

m

|

k

)

Q

x

u

Ru

(

k

m

|

k

)

å

¥

I

T

+

+

(

k

)

(

k

m

|

k

)

(

k

m

|

k

)

finite horizon cost

infinite horizon cost

m

=

0

2

å

+

+

x

(

k

m

|

k

)

2

n

+

+

+

+

u

u

(

(

k

k

m

m

|

|

k

k

)

)

x

(

k

m

|

k

)

R

Q

2

2

=

m

n+1

I

å

+

Q

R

I

=

m

0

  • Effective in terminal region

  • Bounded by

  • State feedback controllaw

T

+

+

+

+

x

)

)

(

k

n

1

P x

(

k

n

1

Multi-Model Predictive Control

  • Infinite horizon objective function

  • Free input variables

  • Forces states towards

  • terminal region

APACT03-30 April, 2003


Quasi infinite horizon strategy

x(k+1|k), u(k+1|k)

x(k+2|k), u(k+2|k)

x(k|k), u(k|k)

n+1 m  

(x,u) = (0,0)

u(k+m|k)=Kx(k+m|k)

Terminal region

x(k+n+1|k)

Quasi-infinite horizon strategy

APACT03-30 April, 2003


Illustration of multi model predictive control

  • Define sequence of regions and models

t

J

J

¥

n

OP2

x2

OP1

x1

Illustration of multi-model predictivecontrol

Control Recipe

If x terminal region

quasi-infinite horizon

If x terminal region

infinite horizon

APACT03-30 April, 2003


The resulting lmi problem

g

min

é

ù

1

*

*

*

*

*

L

ê

ú

0

.

5

g

Q

x

(

k

|

k

)

I

0

0

0

0

0

I

ê

ú

0

.

5

ê

ú

g

R

u

(

k

|

k

)

0

I

0

0

0

0

ê

ú

0

.

5

³

+

g

0

Q

x

(

k

1

|

k

)

0

0

I

0

0

0

ê

ú

I

0

.

5

ê

ú

+

g

R

u

(

k

1

|

k

)

0

0

0

I

0

0

ê

ú

M

O

ê

ú

ê

ú

+

+

x

(

k

n

1

|

k

)

0

0

0

0

0

Q

ë

û

é

ù

T

0

.

5

T

0

.

5

T

+

Q

(

A

Q

B

Y

)

QE

QQ

Y

R

t

t

t

t

I

t

ê

ú

T

T

+

+

x

x

(

A

Q

B

Y

)

Q

b

b

b

e

0

0

ê

ú

t

t

t

t

t

t

t

ê

ú

T

T

³

0

x

-

x

-

E

Q

e

b

(

I

e

e

)

0

0

t

t

t

t

t

ê

ú

0

.

5

ê

ú

g

Q

Q

0

0

I

0

I

ê

ú

0

.

5

g

R

Y

0

0

0

I

ë

û

t

The resulting LMI problem

Finite horizon

x(k) terminal region

Infinite horizon

APACT03-30 April, 2003


The resulting lmi problem1

g

min

x

Q

,

,

Y

i

s.t.

é

ù

T

1

x

(

k

|

k

)

³

0

ê

ú

ê

ú

x

(

k

|

k

)

Q

û

ë

and

é

ù

T

0

.

5

T

0

.

5

T

+

Q

(

A

Q

B

Y

)

QE

QQ

Y

R

i

i

i

i

I

i

ê

ú

T

T

+

+

x

x

(

A

Q

B

Y

)

Q

b

b

b

e

0

0

ê

ú

i

i

i

i

i

i

i

ê

ú

T

T

³

0

x

-

x

-

E

Q

e

b

(

I

e

e

)

0

0

i

i

i

i

i

ê

ú

0

.

5

ê

ú

g

Q

Q

0

0

I

0

I

ê

ú

0

.

5

g

R

Y

0

0

0

I

ë

û

i

-

-

-

1

1

1

=

g

x

=

 l

=

P

Q

K

Y

Q

i

i

Özkan, L. et.al. , Control of a solution copolymerization reactor using multi-model predictive control, Chem. E. Science, 58:1207-1221, 2003

The resulting LMI problem

Feasible

ONLY IF

bi=0

(x(k)  terminal region)

APACT03-30 April, 2003


Input constraints

  • Consider

j

=

1

,

2

,

,

n

£

u

(

k

)

u

u

k

0

j

j

,

max

³

Finite horizon cost:

Infinite horizon cost:

Impose directly

(

u

(

k

+

m

|

k

)

k

+

m

|

k

)

=

Kt x

+

£

Exists s.t.

X

u

(

k

m

|

k

)

é

ù

X

Y

j

2

t

³

£

0

X

u

ê

ú

jj

j

,

max

u

Y

Q

ë

û

j

t

,

max

m

=

0

, 1, …, n

j

j

=

=

1

1

,

,

2

2

,

,

,

,

n

n

u

u

m

=

n+1 ,…,

Input Constraints

APACT03-30 April, 2003


Mmva solution copolymerization reactor

Monomer (FA)

Monomer (FB)

Initiator (FI)

Coolant

Solvent (FS)

Coolant

Transfer Agent (FC)

Polymer

Solvent

Unreacted feed

Inhibitor (FZ)

Richards, J. R. et.al. , Feedforward and Feedback Control of a Solution Co-polymerization Reactor, AIChE Journal, 35(6):891-907, 1989

MMVA Solution Copolymerization Reactor

  • Characteristics:

    • Based on free radical mechanism

    • Realistic industrially

      • 12 states, 4 outputs

    • Dynamics depends on monomers’ feed ratio

Challenge: Optimization problem involves 200–300 variables

APACT03-30 April, 2003


Implementation on mmva copolymerization reactor

  • Step input (FA/FB: 0.2 0.25)

4

OP 2

3.8

Mw/E4 (kg/kmol)

3.6

OP 1

3.4

0

10

20

30

40

50

time (h)

Implementation on MMVA copolymerization reactor

  • Manipulated variables

    • FA, FB, FC, Tj

  • Obtaining multiple models

    • Select new desired operating point

    • Assume a trajectory

APACT03-30 April, 2003


Implementation on mmva cont d

(x,u) - (0,0)

  • Desired operating point

    • Driving force

    • Linear models are updated accordingly

  • Norm measure to define sequence of linear models

x3

x

x(k|k)

x2

x1

t

Implementation on MMVA (cont’d)

  • Simplifications

    • Only terminal region approximated as ellipsoid

APACT03-30 April, 2003


Implementation on mmva cont d1

Ap

Y

30

30

0

0

10

10

20

20

time (h)

time (h)

Implementation on MMVA (cont’d)

  • Effect of number of linear models

2 models

4

353.5

3 models

353.4

3.8

T (K)

353.3

4 models

Mw/E4 (kg/kmol)

353.2

3.6

353.1

3.4

Off–spec. (kg)

0.64

27

26

0.62

Open loop302.4

25

0.6

2 models134.8

Gpi (kg/h)

0.58

24

3 models110.1

23

0.56

4 models109.8

0.54

22

APACT03-30 April, 2003


Stability analysis

  • MPC

    • Lyapunov Approach

      • dx(t)/dt=f(x) and x=0 equilibrium

      • V(x)

        • V(x)>0, x 0 and V(x)=0  x=0

        • d V/dt<0, x 0

Asymptotically

stable origin

V(x)

Time

Stability Analysis

  • Conventional control

    • Check eigenvalues of closed-loop system

APACT03-30 April, 2003


Hybrid systems

Discrete

components

Continuous

dynamical systems

Logic commands (switches,automata)

Hybrid systems

Hybrid systems

  • Definition:

    • Dynamical systems with continuous and discrete state variables

APACT03-30 April, 2003


Multiple lyapunov functions

Branicky, M.S., Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Trans. Automatic Control, 43(4):475-482, 1998.

Multiple Lyapunov Functions

  • Consider Vi :i=1,2,…,N

  • Given S={x0: (i0,t0),(i1,t1),…,(iN,tN),..}

  • Denote

    • T: increasing sequence of times t0,t1,…tN

    • (T):even sequence of T:t0,t2,t4,

If Viis monotonically non-increasing on(T) then system is stable in the sense of Lyapunov

APACT03-30 April, 2003


Multiple lyapunov functions1

  • Interpretation for N=2

V1(x)

t0

t1

t2

t3

t4

t5

t6

V2(x)

Multiple Lyapunov Functions

  • Easy to interpret

  • N candidate Lyapunov functions

  • Requires stable subsystems

APACT03-30 April, 2003


Multi model predictive control with stability guarantee

Theorem

Ifx(k)  terminal region, the quasi–infinite horizon optimization problem is solvedincluding the contractive constraint. If x(k)  terminal regionthe infinite horizon optimization problem is solved. The closed loop system is stable if the feasible solutions of the control strategy defined are implemented in a receding horizon fashion

Multi-Model Predictive Control with Stability Guarantee

  • Contractive constraint

    •  (k) <  (k-1)  x  terminal region

  • Stability Analysis

APACT03-30 April, 2003


Multi model predictive control with stability guarantee1

  • Proof

    • Candidate Lyapunov function

terminal region

Ï

g

ì

(

k

)

x

(

k

)

=

V

í

a

(

(

x

))

T

terminal region

Î

x

(

k

)

Px

(

k

)

x

(

k

)

î

Multi-Model Predictive Control with Stability Guarantee

  • Remarks

    • Stability results depend on feasibility

    • States are measurable

    • @ k=ts

      • V(x(ts))-V(x(ts-1))<0; not required

APACT03-30 April, 2003


Mmva solution copolymerization revisited

1

10

0

10

 (k)

xT(k)Px(k)

-1

10

log (V)

-2

10

-3

10

-4

10

0

5

10

15

t(h)

MMVA solution copolymerization revisited

  • Observations

    • V monotonically decreasing

    • ts=7.0 and 7.75 h

APACT03-30 April, 2003


Summary and conclusion

Summary and Conclusion

  • Multi-model control algorithm is developed

    • Hybrid structure

  • Implemented on a high dimensional problem

    • The effect of number of linear models

      • Decrease in transition time

    • Computational difficulty

      • Number of free input variables

    • Large size LMI’s solved on a realistic problem

  • Stability analysis

    • Contractive constraint

    • Multiple Lyapunov Functions

APACT03-30 April, 2003


Control of a solution copolymerization reactor using piecewise linear models

Thank you

for your attention

Leyla Özkan

APACT03-30 April, 2003


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